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TITLE: How to determine thermal inertia of a resistor?
QUESTION [2 upvotes]: In a recent experiment I've done, I heated up some gas at constant volume using a resistor inside a glass bottle. I turned on the resistor for $\Delta t$ s, but I saw that the pressure rises for $\tau > \Delta t$ s. I attributed this extra heating to the thermal inertia of the resistor.
I've also seen that $t^*:=\tau - \Delta t$ can be neglected for the calculations as $t^* << \Delta t$, but I don't know how to justify this with equations.
The setup consists of a glass container (5.4L) with a resistor inside at 4.69V with a current of 0.432A. The resistor is located on the bottom of the glass bottle and the pressure sensor on top. Once the switch is turned on, the graph of $P$ over $t$ looks like a quadratic growing function until the maximum followed by an exponential decay.
The goal of the experiment is to determine $c_v$ of the air. To do this I have an expression $\Delta P = \frac{1}{c_v}\alpha \Delta t$ $\alpha$ being some constant. This expression is based on an assumption of the process being adiabatic but the right hand of the image being an exponential decay suggests there is a loss of internal energy through heat. This is why, assuming the loss of energy from point one to point three is the same as that from three to five, I just need determine $\Delta Q_{3-5}$ and add that to $\Delta P = P_{max} - P_{base}$. The problem with this is that I only have the time $\Delta t$, so to determine the loss of energy I can only take $\Delta Q_{3-4}$ so I somehow need to justify that $t^*$ is small compared to $\Delta t$ to approximate $\Delta Q_{3-5} \approx \Delta Q_{3-4}$
Can someone tell me whether I'm right in attributing the extra time to thermal inertia and to help me find an expression to justify $t^* << \Delta t$.
REPLY [1 votes]: I don't know the distance between the resistor and your pressure sensor. The pressure rise after the resistor is turned off may be due to the heat exchange between the resistor and the gas, but it can also be due to the finite time required for the gas to achieve thermal equilibrium.
I don't quite understand why you need to justify $t^*<<\Delta t$. You know $\Delta t$, so you know how much heat was generated, this should be enough to calculate the final pressure (assuming the losses are minimal).
EDIT (Oct 31, 2022): Thank you for adding the details of the experiment. So your sensor is relatively far from the resistor. I suspect that resistor inertia is not as important as the time required for equilibration in the container (which is probably defined by convection in air). You may estimate this time, or you can use a resistor providing faster heating and turned off faster, so the heat generated in the resistor is the same. The losses will be smaller over this smaller time.
Your assumption that "the loss of energy from point one to point three es the same as that from three to five" does not seem right, as losses depend on the temperature. | {"set_name": "stack_exchange", "score": 2, "question_id": 734413} |
TITLE: $\triangle ABC$ and $\triangle ADE$ are isosceles. Show that $\angle BAD=\angle EAC$
QUESTION [2 upvotes]: I'm not sure if my answer is correct for the following question:
$\triangle ABC$ and $\triangle ADE$ are isosceles. Show that $\angle BAD=\angle EAC$.
My answer to this question is:
$AB=AD$ (Given, properties of isosceles $\triangle$)
$AC=AE$ (Given, properties of isosceles $\triangle$)
$\angle BAD+\angle BAO=\angle EAC+\angle OAC$ ($AO$ bisects $BC$)
Therefore, $\triangle ABC\cong\triangle ADE$ (Side-Angle-Side) and $\angle BAD=\angle EAC$.
Can someone please check if my answer is correct or is there a better way to solve the question?
REPLY [1 votes]: $AB = AD$ and $AC = AE$ are not correct. Instead,
Since $AD = AE$, we know that $\angle ADE = \angle AED \implies \angle ADB = \angle AEC$.
Also, by the equality $AB =AC$, we have $\angle ABD = \angle ACE$.
Using these two, we can directly conclude that $\angle BAD = \angle AEC$ because we know that $\angle ABD + \angle ADB + \angle BAD = \angle ACE+\angle AEC + \angle EAC\ (= 180^\circ)$. | {"set_name": "stack_exchange", "score": 2, "question_id": 2670052} |
TITLE: Analytic function such that $f(1/n)=(-1)^n/n, n=1,2,\dots?$
QUESTION [4 upvotes]: Question is prove or disprove: There exists an analytic function such that $f(1/n)=(-1)^n/n, n=1,2\dots$, with $0$ in the domain of $f.$
My attempt: If it exists, then clearly $f(z)=z^kg(z)$, where $k\ge0$ is the order of zero [or possible zero,I have included this by $k\gt0$] $z=0$ of $f(z).$Also,$g(0)\ne0$
So, $g(1/n)=f(1/n)n^k\implies g(1/n)=n^k\{\frac{(-1)^n}n\}$
Now the $\lim_{n\to\infty}\{\frac{(-1)^nn^k}n\}$ does not exist[of course,when $k\ge1$]$\implies g(0)$ doesn't exist.
But this is a contradiction to my assumption that $f$ is analytic.So such an $f$ doesn't exist.
And in case of $k=0$,above limit $\lim_{n\to\infty}g(1/n)=0$,which is again a contradiction.
Is there something wrong in above procedure?Please guide me through if there is some.I'll appreciate any help towards this.Thanks in advance!
PS:I stated that $g(0)\implies$ "contradiction to my assumption that $f$ is analytic".Well,I stated so because since by assumption $f$ is analytic and $z^k$ is entire[$k\ge0$],so $g$ must be analytic in the domain of $f$.
REPLY [7 votes]: Another proof: Assume $f$ is entire for convenience and that $f(1/n) = (-1)^n/n, n = 1,2,\dots$ Then $f(z) = -z$ on the set $\{1,1/3,1/5,\dots \},$ which has the limit point $0.$ By the identity principle, $f(z) = -z$ everywhere. But $f(1/2) = 1/2,$ contradiction. Hence there is no such $f.$ | {"set_name": "stack_exchange", "score": 4, "question_id": 1447713} |
\begin{document}
\title
[Integral points close to a space curve]
{Integral points close to a space curve}
\begin{abstract}
We establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion.
\end{abstract}
\maketitle
\section{Integral points on a curve} \label{s1}
A classical theorem of Jarn\'{i}k \cite{J} states that a strictly convex arc in $\R^2$ of length ${q}\ge1$ contains at most $\ll q^{\frac23}$ integral points, and that the exponent $\frac23$ is best possible.
However, the above bound is susceptible to improvement if we start with a fixed strictly convex arc $\Gamma:y=f(x)$ and consider the number of integral points on the dilation $q\Gamma$ by a factor $q\ge1$. Indeed, Swinnerton-Dyer \cite{SD} showed that for a $C^3$ strictly convex arc $\Gamma$ and for any $\varepsilon>0$
$$
\#(q\Gamma\cap\mathbb{Z}^2)\ll_{\Gamma,\varepsilon} q^{\frac35+\varepsilon}.
$$
Later on, W.M. Schmidt \cite{S} obtained a uniform version (with respect to $\Gamma$) of Swinnerton-Dyer's theorem and generalized it to hypersurfaces. In a landmark paper where an ingenious determinant method was developed, Bombieri and Pila \cite{BP} proved the optimal bound
$$
\#(q\Gamma\cap\mathbb{Z}^2)\ll_{\Gamma,\varepsilon} q^{\frac12+\varepsilon}
$$
for strictly convex $f\in C^\infty[0,1]$, which confirms a conjecture of P. Sarnak. In a subsequent paper, Pila \cite{Pi} obtained the same bound under the weaker assumption that $f\in C^{104}[0,1]$ and the determinant
\[f''
\left|
\begin{array}{ccc}
f'''& 3f''& 0\\
f^{(4)}& 4f'''&6f''\\
f^{(5)}& 5f^{(4)}& 20f'''
\end{array}
\right|
\]
is nowhere vanishing.
Better results are available for algebraic curves of degree at least 3. Assuming $\Gamma$ is a subset of an irreducible algebraic curve of degree $d$ inside a square of side $q$, Bombieri and Pila showed, in the same paper cited above, that the number of lattice points on $\Gamma$ is
$$
\ll_{d,\varepsilon} q^{1/d+\varepsilon}.
$$
Similar bounds are also known for the problem of counting rational points on $\Gamma$, or equivalently counting integral points on the corresponding projective curve \cite{EV,HB,W}.
\section{Integral points close to a curve}
Sometimes, we are not only interested in integral/rational points on a curve, but also those lying close to the curve, or equivalently those in a very thin neighborhood of the curve.
The question of estimating the number of lattice points near dilations of the unit circle $x^2+y^2=1$ is of course closely tied with the celebrated circle problem of Gauss, which asks for the best possible error term when approximating the number of lattice points inside a circle centered at the origin with radius $r$ by its area $\pi r^2$. In general, one may investigate the number of lattice points near dilations of a reasonably smooth planar curve. For results on this rather difficult problem and applications to questions such as gaps between squarefull numbers, see \cite{hux2, T} and the references therein. Swinnerton-Dyer's method \cite{SD} still plays an important role in this general setting, but the proofs are much more technical and even the results are too formidable to be reproduced here. However, for the standard parabola $\{(x,x^2), x\in[0,1]\}$, we have established a better bound by incorporating some multiplicative number theory pertaining to the arithmetic nature of the problem \cite{hua4}.
One may as well study this problem for proper submanifolds of the Euclidean space $\R^n$. In general, within a fixed ambient space, the problem becomes more difficult as the codimension of the underlying manifold increases. In fact, Lettington \cite{L1,L2} has proved essentially best possible bounds for convex hypersurfaces (codimension 1). Later, Beresnevich, Vaughan, Velani, Zorin \cite{BVVZ} and Simmons \cite{Si} have made further progress for submanifolds satisfying various rank or curvature conditions, which only hold generically for submanifolds with sufficiently large dimensions in terms of the ambient dimension. Recently, J. Liu and the author \cite{HL1} have obtained optimal results for affine subspaces of $\R^n$ satisfying certain diophantine type conditions.
In this paper, we are primarily concerned with the above counting problem for space curves in $\R^3$, which has been under people's radar for quite a while, but seems to have resisted all the attacks so far. Here we make some progress on this problem.
By the inverse function theorem, any $C^3$ space curve may be parametrized in the Monge form locally. If the curve is also compact, it can be covered by finitely many subcurves, each presented in the Monge form. Hence bounds for the global counting problem can be obtained from local ones, at the expense of losing a constant factor. Therefore, after proper dilation and translation, we may assume that the curve $\mathcal{C}$ is parametrized by
\begin{equation*}
\{(x, f_1(x), f_2(x)):x\in [0,1]\},\quad \textrm{where } f_1, f_2\in C^3([0,1]),
\end{equation*}
and moreover the torsion of $\mathcal{C}$ is nowhere vanishing if and only if
\begin{equation}\label{e1}
\begin{vmatrix}
1 & f_1'(x) & f_2'(x) \\
0 & f_1''(x) & f_2''(x) \\
0 & f_1'''(x) & f_2'''(x)
\end{vmatrix}
=
\begin{vmatrix}
f_1''(x) & f_2''(x) \\
f_1'''(x) & f_2'''(x)
\end{vmatrix}\not=0, \quad\textrm{for all }x\in [0,1].
\end{equation}
Let $$A(q,\delta):=\#\{a\in [0,q]\cap\mathbb{Z}:\|qf_1(a/q)\|<\delta, \|qf_2(a/q)\|<\delta\}.$$ Roughly speaking, $A(q,\delta)$ counts the number of rational points with denominator $q$ lying in the $O(\delta/q)$ neighborhood of $\mathcal{C}$, or equivalently this is the same as counting lattice points within distance $O(\delta)$ to the dilation $q\mathcal{C}$ of the curve $\mathcal{C}$. A simple probabilistic heuristic shows that one expects to have $A(q,\delta)\asymp \delta^2q$, which of course breaks down when $\delta\to0$ and $q$ is fixed. The following theorem confirms this heuristic when $\delta q^{\frac15}(\log q)^\frac25\to\infty$.
\begin{thm}\label{t1}
Let $\mathcal{C}$ be a compact $C^3$ curve in $\mathbb{R}^3$ with nonvanishing torsion. Then
for any $\delta\in(0,1/2)$ and $q\ge1$, we have
$$
A(q,\delta)\ll_{\mathcal{C}}\delta^2q+q^{\frac35}(\log q)^{\frac45}.
$$
Furthermore, there exist positive constants $C$ and $Q_0$ such that when $\delta\ge Cq^{-\frac15}(\log q)^{\frac25}$ and $q\ge Q_0$ we have
$$
A(q,\delta)\gg_{\mathcal{C}} \delta^2q.
$$
\end{thm}
Our Theorem \ref{t1} represents the first nontrivial result of its kind about curves in $\R^3$. The term $\delta^2 q$ is the heuristic main term hence cannot be dispensed with. It remains to be seen whether the other term $q^{\frac35}(\log q)^{\frac45}$ is subject to improvement. It is not unlikely that
$$
A(q,\delta)\ll\delta^2q+q^{\frac12+\varepsilon}
$$
or even
\begin{equation}\label{e3}
A(q,\delta)\ll\delta^2q+q^{\frac13+\varepsilon}.
\end{equation}
We may as well expect that the lower bound $A(q,\delta)\gg\delta^2q$ holds provided that $\delta\gg q^{-\frac14+\varepsilon}$ (or $\delta\gg q^{-\frac13+\varepsilon}$).
In view of the cubic Veronese curve $\mathcal{V}_3=\{(x,x^2,x^3):x\in[0,1]\}$, we must have the lower bound
$$
\#(q\mathcal{V}_3\cap\mathbb{Z}^3)\gg q^{\frac13}.
$$
Therefore the conjectural bound \eqref{e3} is the best that one can hope for. Nevertheless, further improvement should be possible if one averages over $q$. Indeed, it is reasonable to conjecture that
$$
\sum_{q\le Q} A(q,\delta)\ll\delta^2Q^2+Q^{1+\varepsilon}
$$
or even
$$
\sum_{q\le Q} A(q,\delta)\ll\delta^2Q^2+Q^{\frac23+\varepsilon}.
$$
The above conjectures, if true, would have significant consequences in metric diophantine approximations on nondegenerate curves in $\R^3$. We only remark in passing that estimating the number of lattice/rational points close to a manifold is a very active area of research and refer the interested readers to the papers \cite{bere, BDV, BVVZ, BVVZ1, BZ, hua1, hua3, hux, Le, Si, VV} for more background and recent developments.
An alert reader may wonder whether the torsion condition \eqref{e1} is necessary for Theorem \ref{t1} to hold true. Here we provide an example to answer the above question affirmatively. Consider the embedded parabola $\{(x,x^2,0)|x\in[0,1]\}$ in $\R^3$, which clearly fails the torsion condition \eqref{e1}. In this case, the second inequality in the definition of $A(q,\delta)$ is always true, hence the counting problem is equivalent to its analogue in $\R^2$ for the standard parabola. The latter problem has been studied recently by H. Li and the author \cite[Theorem 2]{HL2}\footnote{The result there is more precise than the one quoted here.}, and we have
$$
A(q,\delta)=2\delta q+O(q^{\frac12+\varepsilon}),\quad \text{for any }\varepsilon>0,
$$
which is clearly incompatible with the upper bound in Theorem \ref{t1}. This shows that our Theorem \ref{t1} cannot hold for this embedded parabola, and therefore the nonvanishing torsion condition assumed in Theorem \ref{t1} cannot be completely dispensed with.
The novel feature of our approach is an induction scheme which enables us to reduce the lattice points counting problem for space curves to one that is relevant to counting rational points near planar curves. More precisely, the starting point of our proof of Theorem \ref{t1} is based on an argument of Sprind\v{z}uk \cite[\S 2.9]{Sp}, which is also revisited in \cite{BVVZ, Si}. In a nutshell, the idea is that we approximate the curve by some short line segments in such a way that a thin neighborhood of the curve more or less coincides with a thin neighborhood of the broken line segments. This way, the counting problem is locally linearized, therefore can be better handled by analytic techniques. However, in order to add up the contributions from those linear patches and obtain enough savings to claim victory, one has to assume some rank or curvature conditions in Sprind\v{z}uk's original argument and its later variations, which unfortunately completely exclude curves in $\mathbb{R}^n$ with $n\ge3$. It is at this stage that we deviate from all previous approaches and have to reply on a very delicate analysis which utilizes information from one lower dimension. Finally, we use essentially the best possible result in $\mathbb{R}^2$ c.f. Proposition \ref{l5} as the initial input. In principle, one may use this induction scheme to obtain bounds for lattice points close to a curve in $\mathbb{R}^n$ with $n\ge4$. Unfortunately, the quality of the estimates rendered this way deteriorates fairly fast as $n$ increases. Therefore, we decide not to pursue the full potential of our method herein, and only focus on the pivotal case $n=3$.
\section{Preliminary Lemmata}
The main purpose of this section is to prove the Proposition \ref{l5} below, whose argument draws upon our earlier work \cite{hua2}.
Let $I=[\xi,\eta]$. Suppose that $f\in C^2(I)$ satisfies
\begin{equation}\label{e4}
0<c_1\le|f''(x)|\le c_2.
\end{equation}
Let
$$
\mu(j_1, j_2, \lambda)=\{x\in I: \|j_1x+j_2f(x)\|<\lambda\}
$$
and
\begin{equation*}
\mu(j_1, j_2, p, \lambda)=\{x\in I: |j_1x+j_2f(x)-p|<\lambda\}.
\end{equation*}
\begin{prop}\label{l5} For positive integer $J$ and $\lambda\in(0,\frac12)$,
we have
$$
\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}|\mu(j_1, j_2, \lambda)|\ll \lambda J^2+\lambda^{\frac12}J^{\frac12}\log J.
$$
\end{prop}
\subsection{Lemmata} We state some lemmata first, which will be used in the proof of Proposition \ref{l5}.
\begin{lem}[{\cite[Lemma 9.7]{Ha}}]\label{l2}
Let $h(x)\in C^2(I)$ be such that $\min_{x\in I}|h'(x)|=\delta_1$ and $\min_{x\in I}|h''(x)|=\delta_2$. For $\tau>0$, define
$$E(\tau):=\{x\in I:|h(x)|<\tau\}.$$ Then we have
$$|E(\tau)|\ll\min\left(\frac\tau{\delta_1},\sqrt{\frac{\tau}{\delta_2}}\,\right).$$
\end{lem}
\begin{lem}[{\cite[Lemma 4]{hua2}}]\label{l3}
Suppose that $\phi$ has a continuous second derivative on a bounded interval $K$ which is bounded away from 0, and let $\delta\in(0,\frac14)$. Then for any $\varepsilon>0$ and $U\ge1$,
$$\sum_{U\le u<2U}\sum_{\substack{t/u\in K\\\|u\phi(t/u)\|<\delta}}1\ll_{\varepsilon, K} \delta^{1-\varepsilon}U^2+U\log(2U).$$
\end{lem}
\begin{lem}[{\cite[Lemma 5]{hua2}}]\label{l4}
Under the same conditions with Lemma \ref{l3}, for $\Lambda\in(0,1)$ and $U\ge1$,
$$\sum_{U\le u<2U}\sum_{\substack{t/u\in K\\\|u\phi(t/u)\|\ge\delta}}\left\|u\phi\left(\frac{t}{u}\right)\right\|^{-\Lambda}\ll_K U^2+\delta^{-\Lambda}U\log(2U).$$
\end{lem}
\subsection{Proof of Poposition \ref{l5}}
We will divide the set $[-J,J]\cap\mathbb{Z}^2\backslash(0,0)$ of all possible choices for $(j_1,j_2)$ into a couple of subsets. We also notice that for given $(j_1,j_2)$, there are only finitely many $p\in\mathbb{Z}$ such that $\mu(j_1,j_2,p,\lambda)\not=\emptyset$. Indeed, such $p$ must satisfy
\begin{equation}\label{e3.3}
|p|\le C \max\{|j_1|,|j_2|\}
\end{equation}
where $$C=\max_{x\in I}\{|x|+|f(x)|+1\}.$$
Let
$$M=1+\max_{x\in I}|f'(x)|$$
and
$$
\Theta=[-J,J]^2\cap\mathbb{Z}\times\mathbb{Z}\setminus(0,0),
$$
$$\Theta_1=\{(j_1,j_2)\in \Theta:|j_1|>2M|j_2|\},$$
$$\Theta_2=\Theta\setminus\Theta_1.$$
We consider the case $(j_1,j_2)\in\Theta_1$ first. In this case, we have
$$|j_1+j_2f'(x)|\ge|j_1|-M|j_2|\ge\frac{|j_1|}2.$$
Now by Lemma \ref{l2}, we know
$$
|\mu(j_1,j_2,p,\lambda)|\ll \frac{\lambda}{|j_1|}.
$$
Moreover, for a given $j_1$, there are at most $\ll j_1^2$ possible choices for $j_2$ and $p$.
Therefore
$$
\sum_{\substack{(j_1,j_2)\in\Theta_1\\p\in\mathbb{Z}}}|\mu(j_1,j_2,p,\lambda)|
{\ll}\lambda J^{2}.
$$
Now for $(j_1,j_2)\in\Theta_2$, clearly we have $j_2\not=0$ and $|j_1|\le 2M |j_2|$. For convenience, we may extend the definition of $f(x)$ to $\mathbb{R}$ by taking the second order Taylor expansions at the end points of $I$. Namely
let
$$f(x)=f(\eta)+f'(\eta)(x-\eta)+\frac{f''(\eta)}2(x-\eta)^2$$
when $x>\eta$
and
$$f(x)=f(\xi)+f'(\xi)(x-\xi)+\frac{f''(\xi)}2(x-\xi)^2$$
when $x<\xi$.
Clearly the extended $f$ satisfies $f\in C^2(\mathbb{R})$ and $c_1\le |f''|\le c_2$. Since $f''$ does not change sign throughout $\mathbb{R}$, $f'$ is strictly monotonic on $\mathbb{R}$ and has range $(-\infty,\infty)$. Let $g(y):\mathbb{R}\rightarrow\mathbb{R}$ be the inverse function of $-f'(x)$. To this end,
let $$x_0:=g(j_1/j_2)$$
which is the unique point $x_0\in\mathbb{R}$ such that
\begin{equation}\label{e3.4}
j_1+j_2f'(x_0)=0.
\end{equation}
Let $K=[-2M,2M]$ and $I'=g(K)\supset I$. So $x_0\in I'$.
Note that
$$g'(y)=\frac{-1}{f''(g(y))}$$
and hence that
\begin{equation}\label{e3.12}
c_2^{-1}\le|g'(y)|\le c_1^{-1}
\end{equation}
for all $y\in\mathbb{R}$. Thus by the mean value theorem
\begin{equation}\label{e3.5}
|I'|\le c_1^{-1}|K|= 4c_1^{-1}M.
\end{equation}
Now, we let
$$F(x)=j_1x+j_2f(x)$$ with $j_1/j_2\in K$. Then by \cite[Lemma 3]{hua2} we have the following lemma.
\begin{lem}\label{l1}
$$|F'(x)|\asymp |j_2(F(x)-F(x_0))|^{1/2}$$
where the $\asymp$ constants depend only on $c_1$, $c_2$.
\end{lem}
Now let $p_0$ be the unique integer such that
\begin{equation}\label{e3.8}
-\frac12<F(x_0)-p_0\le\frac12.
\end{equation}
If $p\not=p_0$, then for $x\in\mu(j_1,j_2,p,\lambda)$
\begin{align*}
|F(x)-F(x_0)|&=|p-p_0+F(x)-p-F(x_0)+p_0|\\
&\ge|p-p_0|-|F(x)-p|-|F(x_0)-p_0|\\
&\overset{\eqref{e3.8}}{\ge}|p-p_0|-\lambda-1/2\\
&\ge\frac13|p-p_0|
\end{align*}
provided that
\begin{equation}\label{e3.9}
\lambda\le \frac1{8}.
\end{equation}
Proposition \ref{l5} is trivial when $\lambda>\frac18$, so without loss of generality we may assume \eqref{e3.9} holds.
By Lemma \ref{l1} and then Lemma \ref{l2}, we get, when $p\not=p_0$
\begin{equation}\label{e10}
|\mu(j_1,j_2,p,\lambda)|\ll \lambda(|j_2||p-p_0|)^{-1/2}.
\end{equation}
Therefore for fixed $(j_1,j_2)\in\Theta_2$
\begin{align*}
\sum_{p\not=p_0}|\mu(j_1,j_2,p,\lambda)|&\ll \sum_{p\not=p_0}\lambda(j_2|p-p_0|)^{-1/2}\\
&\overset{\eqref{e3.3}}{\ll}\lambda j_2^{-1/2}j_2^{1-1/2}\\
&\ll \lambda.
\end{align*}
Hence
$$\sum_{\substack{(j_1,j_2)\in\Theta_2\\p\not=p_0}}|\mu(j_1,j_2,p,\lambda)|\ll \lambda J^{2}.$$
Hitherto, we are left with the most difficult case $p=p_0$. For $x\in\mu(j_1,j_2,p_0,\lambda)$, we have
\begin{align*}
|F(x)-F(x_0)|&=|F(x)-p_0+p_0-F(x_0)| \\
&\ge\|F(x_0)\|-\lambda\\
&\ge\frac12\|F(x_0)\|
\end{align*}
provided that
$$\|F(x_0)\|\ge2\lambda.$$
By Lemma \ref{l1} and the inequality above
$$|F'(x)|\gg (j_2\|F(x_0)\|)^{1/2}.$$
Then by Lemma \ref{l2}
\begin{equation}\label{e3.10}
|\mu(j_1,j_2,p_0,\lambda)|\ll\frac{\lambda}{j_2^{1/2}}\|F(x_0)\|^{-1/2},
\end{equation}
when $\|F(x_0)\|\ge2\lambda$.
In the case if $\|F(x_0)\|<2\lambda$, we simply use the upper bound
\begin{equation}\label{e3.11}
|\mu(j_1,j_2,p_0,\lambda)|\overset{\text{Lem.}\ref{l2}}{\ll} \sqrt{\frac{\lambda}{j_2}}.
\end{equation}
We now define the dual curve $f^*(y)$ of $f(x)$, whose derivative is the inverse function of $-f'(x)$. Namely
$$f^*(y):=yg(y)+f(g(y)).$$
It is readily seen that
\begin{equation}\label{e3.13}
(f^*)'(y)=g(y)+yg'(y)+f'(g(y))g'(y)=g(y)
\end{equation}
and that
\begin{equation}\label{e3.14}
j_2f^*(j_1/j_2)=j_1g(j_1/j_2)+j_2f(g(j_1/j_2))=j_1x_0+j_2f(x_0)=F(x_0).
\end{equation}
Finally, we are poised to treat the sum
\begin{equation}\label{e3.15}
\sum_{(j_1,j_2)\in\Theta_2}|\mu(j_1,j_2,p_0,\lambda)|.
\end{equation}
We further divide this into two cases, namely $\|F(x_0)\|<2\lambda$ and $\|F(x_0)\|\ge2\lambda$.
For any nonnegative integer $k$,
\begin{align*}
&\sum_{\substack{(j_1,j_2)\in\Theta_2\\2^{k}\le |j_2|<2^{k+1}\\\|F(x_0)\|<2\lambda}}|\mu(j_1,j_2,p_0,\lambda)|\\
\overset{\eqref{e3.11}\&\eqref{e3.14}}{\ll}& \sum_{2^{k}\le j_2<2^{k+1}}\sum_{\substack{j_1/j_2\in K\\\|j_2f^*(j_1/j_2)\|<2\lambda}}\left(\frac{\lambda}{2^k}\right)^{\frac{1}2}\\
\overset{\text{Lem.}\ref{l3}}{\ll}&\Big(\lambda^{1-\varepsilon}2^{2k}+(k+1)2^k\Big)\left(\frac{\lambda}{2^k}\right)^{\frac{1}2}\\
\ll&\lambda^{\frac{3}2-\varepsilon}2^{3k/2}+(k+1)\lambda^\frac{1}2 2^{k/2}.
\end{align*}
By summing over $0\le k\le \log_2 J$, we have
\begin{equation}\label{e3.17}
\sum_{\substack{(j_1,j_2)\in\Theta_2\\\|F(x_0)\|<2\lambda}}|\mu(j_1,j_2,p_0,\lambda)|\ll \lambda^{\frac32-\varepsilon}J^{\frac32}+\lambda^{\frac12}J^{\frac12}\log J.
\end{equation}
The other case can be treated in a similar fashion.
\begin{align*}
&\sum_{\substack{(j_1,j_2)\in\Theta_2\\2^{k}\le |j_2|<2^{k+1}\\\|F(x_0)\|\ge2\lambda}}|\mu(j_1,j_2,p_0,\lambda)|\\
\overset{\eqref{e3.10}\&\eqref{e3.14}}{\ll}& \sum_{2^{k}\le j_2<2^{k+1}}\sum_{\substack{j_1/j_2\in K\\\|j_2f^*(j_1/j_2)\|\ge2\lambda}}\left(\frac{\lambda}{2^{k/2}}\right)\|j_2f^*(j_1/j_2)\|^{-1/2}\\
\overset{\text{Lem.}\ref{l4}}{\ll}&\left(\lambda^{-1/2}2^{k}(k+1)+2^{2k}\right)\frac{\lambda}{2^{k/2}}\\
\ll&\lambda^{1/2}2^{k/2}(k+1)+\lambda 2^{3k/2}.
\end{align*}
Again by summing over $0\le k\le \log_2 J$, we obtain
\begin{equation}\label{e3.18}
\sum_{\substack{(j_1,j_2)\in\Theta_2\\\|F(x_0)\|\ge2\lambda}}|\mu(j_1,j_2,p_0,\lambda)|\ll \lambda^{\frac12}J^{\frac12}\log J+\lambda J^{\frac32}.
\end{equation}
\section{Proof of Theorem \ref{t1}}
Let
$$
\mathcal{A}(q,\delta)=\{a\in [0,q]\cap\mathbb{Z}:\|qf_1(a/q)\|<\delta, \|qf_2(a/q)\|<\delta\}.
$$
In view of the torsion condition \eqref{e1}, either $f_1''$ or $f_2''$ must be bounded away from 0 in a sufficiently small neighborhood of any $x\in [0,1]$. Therefore, without loss of generality, we may prove Theorem \ref{t1} under the additional assumption that $|f_1''|\ge c_3>0$. The general case would follow immediately by compactness.
Let $c_4=\max(\|f_1\|_{C^3([0,1])},\|f_2\|_{C^3([0,1])},1)$, $q_0=\lfloor(2^{-1}c_4^{-1}\delta q)^{\frac12}\rfloor$ and $r=\lfloor q/q_0\rfloor$. For the time being, we suppose $\delta\ge 2c_4q^{-1}$ so that $q_0\ge1$. For each $a\in [0,q]\cap\mathbb{Z}$ write $a=q_0s+a_0$ with $a_0=a_0(a)\in[0, q_0)\cap\mathbb{Z}$ and $s=s(a)\in[0, r]\cap\mathbb{Z}$.
Let
$$
\mathcal{A}(q,\delta,s)=\{a\in\mathcal{A}(q,\delta): s(a)=s\}
$$
and
$$
{A}(q,\delta,s)=\#\mathcal{A}(q,\delta,s).
$$
By the Taylor theorem, when $a\in\mathcal{A}(q,\delta,s)$, we have for $i\in\{1,2\}$
\begin{equation*}
\left|f_i\left(\frac{a}q\right)-f_i\left(\frac{q_0s}q\right)-\frac{a_0}qf_i'\left(\frac{q_0s}q\right)\right|\le \frac{c_4}2\cdot\frac{a_0^2}{q^2}\le \frac{c_4q_0^2}{2q^2}< \frac{\delta}{2q}.
\end{equation*}
Then it follows by the triangle inequality that for $i\in\{1,2\}$
\begin{equation}\label{e4.1}
\Bigg|\left\|qf_i\left(\frac{q_0s}q\right)+{a_0}f_i'\left(\frac{q_0s}q\right)\right\|-\left\|qf_i\left(\frac{a}q\right)\right\|\Bigg|\le\frac{\delta}2.
\end{equation}
Let $$\mathcal{B}(q,\delta,s):=\left\{a_0\in[0,q_0)\cap\mathbb{Z}:\left\|qf_i\left(\frac{q_0s}q\right)+{a_0}f_i'\left(\frac{q_0s}q\right)\right\|<\delta,\quad 1\le i\le 2\right\},$$
$$
B(q,\delta,s):=\#\mathcal{B}(q,\delta,s),
$$
$$
B_1(q,\delta):=\sum_{0\le s\le r}B(q,\delta,s),
$$
and
$$
B_2(q,\delta):=\sum_{0\le s< r}B(q,\delta,s).
$$
We then observe from \eqref{e4.1} that
$$
a=q_0s+a_0\in\mathcal{A}(q,\delta,s)\Rightarrow a_0\in\mathcal{B}(q,3\delta/2,s),\quad\textrm{when } s\le r
$$
and
$$
a_0\in\mathcal{B}(q,\delta/2,s)\Rightarrow a=q_0s+a_0\in\mathcal{A}(q,\delta,s)\quad\textrm{when } s<r.
$$
Therefore we obtain
$$
{A}(q,\delta,s)\le{B}(q,3\delta/2,s),\quad\textrm{when } s\le r
$$
and
$$
{B}(q,\delta/2,s)\le {A}(q,\delta,s)\quad\textrm{when } s<r,
$$
and hence
\begin{equation}\label{e4.8}
B_2(q,\delta/2)\le A(q,\delta)\le B_1(q,3\delta/2).
\end{equation}
Now to estimate ${B}(q,\delta,s)$, we recall some basic properties of Selberg's magic functions. See \cite[Chapter 1]{mo} for details about the construction of these functions.
Let $\Delta=(\alpha, \beta)$ be an arc of $\R/\Z$ with $\alpha<\beta<\alpha+1$, and $\chi_{_\Delta}(x)$ be its characteristic function. Then there exist finite trigonometric polynomials of degree at most $J$
$$S^{\pm}_J(x)=\sum_{|j|\le J}b_j^{\pm}e(jx)$$ such that
$$
S^{-}_J(x)\le \chi_{_\Delta}(x)\le S^{+}_J(x)
$$
and
$$b_0^\pm=\beta-\alpha\pm\frac1{J+1}
$$
and
$$|b_j^\pm|\le\frac1{J+1}+\min\left(\beta-\alpha,\frac1{\pi|j|}\right)
$$
for $0<|j|\le J$. Here $e(x)=e^{2\pi ix}$.
Here we choose $\alpha=-\delta$, $\beta=\delta$ and $J=\lfloor\frac1{\delta}\rfloor$. With such choices, observe that $|b_j^\pm|\le3\delta$ for all $j\in\mathbb{Z}$ and $b_0^-\ge\delta$.
Let $$
F_i(s,a_0):=qf_i\left(\frac{q_0s}q\right)+{a_0}f_i'\left(\frac{q_0s}q\right),\quad 1\le i\le 2.$$
Then
\begin{align*}
B(q,\delta,s)=& \sum_{0\le a_0<q_0}\chi_{_\Delta}(F_1(s,a_0))\chi_{_\Delta}(F_2(s,a_0))\\
\le&\sum_{0\le a_0<q_0}S^+_J(F_1(s,a_0))S^+_J(F_2(s,a_0))\\
=&\sum_{|j_1|,|j_2|\le J}b_{j_1}^+b_{j_2}^+\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_iF_i(s,a_0)\right)\\
\le&9\delta^2 q_0+\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}b_{j_1}^+b_{j_2}^+\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_iF_i(s,a_0)\right),
\end{align*}
where in the last inequality we single out the zero frequency term $j_1=j_2=0$ which gives the heuristic main term for $B(q,\delta,s)$.
Similarly
\begin{align*}
B(q,\delta,s)=& \sum_{0\le a_0<q_0}\chi_{_\Delta}(F_1(s,a_0))\chi_{_\Delta}(F_2(s,a_0))\\
\ge&\sum_{0\le a_0<q_0}S^-_J(F_1(s,a_0))S^-_J(F_2(s,a_0))\\
=&\sum_{|j_1|,|j_2|\le J}b_{j_1}^-b_{j_2}^-\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_iF_i(s,a_0)\right)\\
\ge&\delta^2 q_0+\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}b_{j_1}^-b_{j_2}^-\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_iF_i(s,a_0)\right).
\end{align*}
Note that
\begin{align*}
\left|\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_iF_i(s,a_0)\right)\right|=&\left|\sum_{0\le a_0<q_0}e\left(\sum_{i=1}^2j_i{a_0}f_i'\left(\frac{q_0s}q\right)\right)\right|\\
\le&\min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0s}q\right)\right\|^{-1}\right),
\end{align*}
where in the last inequality we use the well known linear exponential sum estimate
$$
\left|\sum_{n\le N}e(n\gamma)\right|\le\min(N, \|\gamma\|^{-1}).
$$
Thus
\begin{equation}\label{e4.2}
B_1(q,\delta)\le9\delta^2(q+q_0)+E(q,\delta)
\end{equation}
and
\begin{equation}\label{e4.3}
B_2(q,\delta)\ge\delta^2 (q-q_0)-E(q,\delta),
\end{equation}
where
$$
E(q,\delta)=9\delta^2\sum_{0\le s\le r}\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}\min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0s}q\right)\right\|^{-1}\right).$$
\par
For a given integer $s$, consider the intervals $I_s=[s-1/2,s+1/2]$, unless $s=0$ or $s=r$ in which case we consider $[s, s+1/2]$ or $[s-1/2,s]$ respectively. For $\alpha\in I_s$ we have
$$
\left|f_i'\left(\frac{q_0s}q\right)-f_i'\left(\frac{q_0\alpha}q\right)\right|\le c_4\frac{q_0}{2q},\quad 1\le i\le 2.
$$
Hence
$$
\left|\sum_{i=1}^2j_i\left(f_i'\left(\frac{q_0s}q\right)-f_i'\left(\frac{q_0\alpha}q\right)\right)\right|\le c_4J\frac{q_0^2}{qq_0}\le c_4\frac1{\delta}\frac{\delta q}{2c_4qq_0}\le \frac1{2q_0}.
$$
Thus
\begin{equation}\label{e4.4}
\min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0s}q\right)\right\|^{-1}\right)\le2 \min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0\alpha}q\right)\right\|^{-1}\right).
\end{equation}
Now we integrate over $\alpha\in I_s$ and then sum over $s\in[0, r]$. This way we obtain
\begin{align}
&\nonumber\sum_{0\le s\le r}\min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0s}q\right)\right\|^{-1}\right)\\\nonumber\le&\left(2\int_0^{\frac12}+\int_{\frac12}^{r-\frac12}+2\int_{r-\frac12}^r \right)2\min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0\alpha}q\right)\right\|^{-1}\right)d\alpha\\\nonumber\le&4\int_0^r \min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0\alpha}q\right)\right\|^{-1}\right)d\alpha\\\nonumber
\le&4\int_0^{\frac{q}{q_0}} \min\left(q_0,\left\|\sum_{i=1}^2j_if_i'\left(\frac{q_0\alpha}q\right)\right\|^{-1}\right)d\alpha\\\label{e4.5}
\ll&\frac{q}{q_0}\int_{I} \min\left(q_0,\left\|j_1\beta+j_2f(\beta)\right\|^{-1}\right)d\beta
\end{align}
where
$$
I=[\inf f_1',\sup f_1']
$$
and
$$
f=f_2'\circ(f_1')^{-1}.
$$
In the last inequality, we make the change of variable $\beta=f'_1(q_0\alpha/q)$ and use the assumption that $f''_1$ is bounded away from 0.
\par
Now we verify that the torsion condition \eqref{e1} implies that $f''$ is bounded away from zero. Since $f\circ f_1'=f_2'$, we have
$$
(f'\circ f_1')\cdot f_1''=f_2'',
$$
$$
(f''\circ f_1')\cdot (f_1'')^2+(f'\circ f_1')\cdot f_1'''=f_2'''
$$
and hence
$$
f''\circ f_1'=\frac{f_1''f_2'''-f_2''f_1'''}{(f_1'')^3}
$$
which is bounded away from 0 in view of \eqref{e1}.
We are ready to estimate the resulting integral from \eqref{e4.5}. Our strategy is to decompose $I$ into subsets defined by the inequalities $\left\|j_1\beta+j_2f(\beta)\right\|<\frac{1}{q_0}$ and $\frac{2^{k-1}}{q_0}\le\left\|j_1\beta+j_2f(\beta)\right\|<\frac{2^k}{q_0}$ for $1\le k\le \log_2q_0$. This naturally leads to the set $\mu(j_1,j_2,\lambda)$ defined in the previous section, for which we may invoke Proposition \ref{l5}.
Therefore, we have
\begin{align}
&\nonumber\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}\int_{I} \min\left(q_0,\left\|j_1\beta+j_2f(\beta)\right\|^{-1}\right)d\beta\\
&\nonumber\ll\sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}\left(q_0|\mu(j_1,j_2,1/q_0)|+\sum_{1\le k\le\log_2q_0}\int_{\frac{2^{k-1}}{q_0}\le\left\|j_1\beta+j_2f(\beta)\right\|<\frac{2^k}{q_0}} \left\|j_1\beta+j_2f(\beta)\right\|^{-1}d\beta\right)\\
&\nonumber\ll \sum_{\substack{|j_1|,|j_2|\le J\\(j_1,j_2)\not=(0,0)}}\sum_{0\le k\le \log_2 q_0}\frac{q_0}{2^k}|\mu(j_1, j_2, 2^k/q_0)|\\
&\nonumber\overset{\textrm{Prop. }\ref{l5}}{\ll}\sum_{0\le k\le \log_2 q_0}q_02^{-k}(2^kq_0^{-1}J^2+2^{k/2}q_0^{-1/2}J^{1/2}\log J)\\
&\ll J^2\log q_0+q_0^{1/2}J^{1/2}\log J.\label{e4.6}
\end{align}
Hence we conclude from \eqref{e4.4}, \eqref{e4.5} and \eqref{e4.6}, that
\begin{align*}
E(q,\delta)
\le& C_1\delta^2\frac{q}{q_0}(J^2\log q_0+q_0^{1/2}J^{1/2}\log J)\\
\le &C_2(\delta^{-1/2}q^{1/2}\log q+\delta^{5/4}q^{3/4}\log \frac1\delta),
\end{align*}
where the second inequality follows on noting that $J=\lfloor 1/\delta\rfloor$ and $q_0=\lfloor(2^{-1}c_4^{-1}\delta q)^{\frac12}\rfloor$.
Hence we obtain from \eqref{e4.2} that
$$
B_1(q,\delta)\ll \delta^2 q+\delta^{-1/2}q^{1/2}\log q+\delta^{5/4}q^{3/4}\log \frac1\delta,\quad \text{when } \delta\ge 2c_4q^{-1}.
$$
Now observe that for fixed $q$, $B_1(q,\delta)$ is increasing in $\delta$. Let $\delta_0=q^{-\frac15}(\log q)^{\frac25}$. Therefore when $\delta<2c_4\delta_0$,
$$
B_1(q,\delta)\le B_1(q,2c_4\delta_0)\ll q^{\frac35}(\log q)^{\frac45},
$$
and when $\delta\ge 2c_4\delta_0$,
$$
B_1(q,\delta)\ll \delta^2q.
$$
In any case, we have for all $\delta\in(0,1/2)$ that
\begin{equation}\label{e4.9}
B_1(q,\delta)\ll \delta^2q+q^{\frac35}(\log q)^{\frac45}.
\end{equation}
\par
On the other hand, there exist positive constants $C, Q_0$ such that when $\delta\ge C\delta_0$ and $q\ge Q_0$ we have
$$
\delta^2q_0+C_2(\delta^{-1/2}q^{1/2}\log q+\delta^{5/4}q^{3/4}\log \frac1\delta)
\le \frac1{100}\delta^2q.
$$
Therefore, we obtain the lower bound from \eqref{e4.3},
\begin{equation}\label{e4.10}
B_2(q,\delta)\ge 0.99 \delta^2q, \quad\text{when }\delta\ge C\delta_0, q\ge Q_0.
\end{equation}
Now the proof follows by combining \eqref{e4.8}, \eqref{e4.9} and \eqref{e4.10}.
\proof[Acknowledgments]
The author is grateful to the anonymous referee for carefully reading the manuscript and providing helpful suggestions to improve the presentation. | {"config": "arxiv", "file": "1809.07796.tex"} |
TITLE: Easy example of a herbrand structure
QUESTION [1 upvotes]: Can someone give me an easy example of a Herbrand structure?
I can't really visualise the difference between a Herbrand and a normal structure.
REPLY [2 votes]: Example
Consider the very simple FOL formula : $R(c)$.
The domain of the Herbrand structure is :
the set of all ground terms [i.e. closed terms] of the language.
In the above case, we have only the individual constant $c$ as gorund term. Thus, the domain is $H = \{ c \}$.
With it, we define the Herbrand interpretation :
an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset of the relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows the symbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.
Again, we have a very simple Herbrand inerpretation $H_S$ :
$H_S = (H, R^H)$,
where $H$ is the domain defiend above and $R^H$ is the subset of $H$ interpreting the relation symbol $R$.
Obviously, $R^H = \{ c \}$. | {"set_name": "stack_exchange", "score": 1, "question_id": 3086002} |
TITLE: Solving second order ODE with polynomial coefficients
QUESTION [2 upvotes]: What is the solution
Can some one solve this.....
$T.y''+T'y'+T''y=0$.,
where $T$ is a polynomial. Like $1-2x-x^2$ etc. in $x$........
Is there anyway to assume the answer as polynomial and go for substitution or is there any other way?
If I take the first two terms and write it as a product, may be there is a chance. But is there a general procedures for these kind of questions
REPLY [0 votes]: There is not necessarily an elementary solution. For example, if $T = 1 - 2x - x^2$, the solutions involve some nasty hypergeometric functions. | {"set_name": "stack_exchange", "score": 2, "question_id": 1197915} |
TITLE: Find the transition matrix for a six-sided die
QUESTION [2 upvotes]: For some reason I have been struggling with this problem for the past couple hours.
I believe I have solved part a.
Since there are 6 states (assuming a standard die and the die is fair), then there is an equal chance of landing in any of the 6 states for every dice throw. Therefore I concluded that the transition matrix looks as such:
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
For part b I setup my starting matrix as:
s1 =
[1]
[0]
[0]
[0]
[0]
[0]
I multiplied my transition matrix by s1 5 times and arrived at the same vector s1. Arithmetically, this makes sense. However, logically it does not.
Can somebody help explain where I went wrong?
Edit: Based on the answers I believe the transition matrix would be:
[1/6 0 0 0 0 0]
[1/6 2/6 0 0 0 0]
[1/6 1/6 3/6 0 0 0]
[1/6 1/6 1/6 4/6 0 0]
[1/6 1/6 1/6 1/6 5/6 0]
[1/6 1/6 1/6 1/6 1/6 1]
REPLY [2 votes]: From state $s_1$ you have $1/6$ chance of remaining in that state and a $1/6$ chance of transitioning to any of the other states.
From state $s_2$ you have a $2/6$ chance of remaining there (by getting either a $1$ or a $2$) anad a $1/6$ chance of transitioning to each of $3,4,5,6.$
From state $s_3$ you have a $3/6$ chance of remaining there (by getting $1,$ $2,$ or $3$) and a $1/6$ chance of transitioning to each of $4,5,6.$
From state $s_4$ you have a $4/6$ chance of remaining there (by getting $1,$ $2,$ $3,$ or $4$) and a $1/6$ chance of transitioning to each of $5,6.$
From state $s_5$ you have a $5/6$ chance of remaining there (by getting $1,$ $2,$ $3,$ $4,$ or $5$) and a $1/6$ chance of transitioning to $6.$
From state $s_6$ you can only remain there. | {"set_name": "stack_exchange", "score": 2, "question_id": 2745436} |
TITLE: How to solve recurrence relation $T(n)=T(n-1)+\lceil \log(n) \rceil$
QUESTION [1 upvotes]: Without the ceilings, the solution is reasonable clear (given here).
Is there a way to reach a solution with the ceilings, or the difference between the two?
REPLY [0 votes]: I'll do this solution, although you may not like it, and it might not work.
$$f(n+1)=f(n)+c \cdot \log(n)$$
We'll define c in a little bit. Also we'll start at $n=1$ and use the power and multiplication rules.
$$f(2)=\log(1^c)+\log(2^c)=c \cdot \log(1\cdot2)$$
$$f(3)=c \cdot \log(1\cdot2)+c \cdot \log(3)=c \cdot \log(1\cdot2 \cdot 3)$$
Its fairly obvious this is just a factorial...
$$f(n)=c \cdot \log(n!)$$
For generality, use the gamma function...
$$f(n)=c \cdot \log(\Gamma(n+1))$$
with $c=1$, this is a famous result in and of its self. Here's the part that might upset you. c is actually the operator for the ceiling function!
$$f(n)=ceiling(\log(\Gamma(n+1))$$ | {"set_name": "stack_exchange", "score": 1, "question_id": 1220505} |
\begin{document}
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\FirstPageHeading
\ShortArticleName{Indef\/inite Af\/f\/ine Hyperspheres Admitting a Pointwise Symmetry. Part 2}
\ArticleName{Indef\/inite Af\/f\/ine Hyperspheres\\ Admitting a Pointwise Symmetry. Part 2\footnote{This paper is a
contribution to the Special Issue ``\'Elie Cartan and Dif\/ferential Geometry''. The
full collection is available at
\textit{}\href{http://www.emis.de/journals/SIGMA/Cartan.html}{http://www.emis.de/journals/SIGMA/Cartan.html}}}
\Author{Christine SCHARLACH}
\AuthorNameForHeading{C. Scharlach}
\Address{Technische Universit\"at Berlin,
Fak. II, Inst. f. Mathematik, MA 8-3, 10623 Berlin, Germany}
\Email{\href{mailto:[email protected]}{[email protected]}}
\URLaddress{\url{http://www.math.tu-berlin.de/~schar/}}
\ArticleDates{Received May 08, 2009, in f\/inal form October 06, 2009; Published online October 19, 2009}
\Abstract{An af\/f\/ine hypersurface $M$ is said to admit a pointwise symmetry, if
there exists a subgroup $G$ of $\Aut(T_p M)$ for all $p\in M$, which
preserves (pointwise) the af\/f\/ine metric $h$, the dif\/ference tensor $K$
and the af\/f\/ine shape operator $S$. Here, we consider 3-dimensional
indef\/inite af\/f\/ine hyperspheres, i.e.\ $S= H\Id$ (and thus $S$ is
trivially preserved). In Part 1 we found the possible symmetry groups
$G$ and gave for each $G$ a canonical form of $K$. We started a
classif\/ication by showing that hyperspheres admitting a pointwise
$\Z_2\times \Z_2$ resp.\ $\R$-symmetry are well-known, they have
constant sectional curvature and Pick invariant $J<0$ resp.\
$J=0$. Here, we continue with af\/f\/ine hyperspheres admitting a
pointwise $\Z_3$- or $SO(2)$-symmetry. They turn out to be warped
products of af\/f\/ine spheres ($\Z_3$) or quadrics ($SO(2)$) with a
curve.}
\Keywords{af\/f\/ine hyperspheres; indef\/inite af\/f\/ine metric; pointwise symmetry; af\/f\/ine
dif\/fe\-ren\-tial geometry; af\/f\/ine spheres; warped products}
\Classification{53A15; 53B30}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\newtheorem*{DecProb}{(De)composition Problem}
\section{Introduction}\label{sec:intro}
Let $M^n$ be a connected, oriented manifold. Consider an immersed
hypersurface with relative normalization, i.e., an immersion
$\hs\colon M^n \rightarrow \R^{n+1}$ together with a transverse vector
f\/ield $\xi$ such that $D \xi$ has its image in
$\hs_*T_xM$. Equi-af\/f\/ine geometry studies the properties of such
immersions under equi-af\/f\/ine transformations, i.e.\ volume-preserving
linear transformations ($SL(n+1,\R)$) and translations.
In the theory of nondegenerate equi-af\/f\/ine hypersurfaces there exists
a canonical choice of transverse vector f\/ield $\xi$ (unique up to
sign), called the af\/f\/ine (Blaschke) normal, which induces a connection
$\nabla$, a nondegenerate symmetric bilinear form $h$ and a 1-1 tensor
f\/ield $S$ by
\begin{gather}
D_X Y =\nabla_X Y +h(X,Y)\xi,\label{strGauss}\\
D_X \xi =-SX,\label{strWeingarten}
\end{gather}
for all $X,Y \in {\cal X}(M)$. The connection $\nabla$ is called the
induced af\/f\/ine connection, $h$ is called the af\/f\/ine metric or Blaschke
metric and $S$ is called the af\/f\/ine shape operator. In general
$\nabla$ is not the Levi-Civita connection $\hat\nabla$ of $h$. The
dif\/ference tensor $K$ is def\/ined as
\begin{gather}\label{defK}
K(X,Y)=\nabla_X Y-\hat\nabla_X Y,
\end{gather}
for all $X,Y \in {\cal X}(M)$. Moreover the form $h(K(X,Y),Z)$ is a
symmetric cubic form with the property that for any f\/ixed $X\in {\cal
X}(M)$, $\trace K_X$ vanishes. This last property is called the
apolarity condition. The dif\/ference tensor $K$, together with the
af\/f\/ine metric $h$ and the af\/f\/ine shape operator $S$ are the most
fundamental algebraic invariants for a nondegenerate af\/f\/ine
hypersurface (more details in Section~\ref{sec:basics}). We say that
$M^n$ is indef\/inite, def\/inite, etc.\ if the af\/f\/ine metric $h$ is
indef\/inite, def\/inite, etc.\ (Because the af\/f\/ine metric is a multiple of
the Euclidean second fundamental form, a positive def\/inite
hypersurface is locally strongly convex.) For details of the basic
theory of nondegenerate af\/f\/ine hypersurfaces we refer to \cite{LSZ93}
and~\cite{NS94}.
Here we will restrict ourselves to the case of af\/f\/ine hyperspheres,
i.e.\ the shape operator will be a (constant) multiple of the
identity ($S= H \Id$). Geometrically this means that all af\/f\/ine
normals pass through a f\/ixed point or they are parallel. There are many af\/f\/ine hyperspheres, even in the two-dimensional case only partial classif\/ications are known. This is due to the fact that af\/f\/ine hyperspheres reduce to the study of the Monge-Amp\`ere equations. Our question
is the following: {\em What can we say about a three-dimensional
affine hypersphere which admits a~pointwise $G$-symmetry, i.e.\ there
exists a non-trivial subgroup $G$ of the isometry group such that for
every $p\in M$ and every $L\in G$:}
\[
K(LX_p,L Y_p)= L(K(X_p, Y_p))\qquad \forall\, X_p, Y_p\in T_p M.
\] We
have motivated this question in Part~1~\cite{S07a} (see also
\cite{Bry01,Vr04,LS05}). A classif\/ication of
$3$-dimen\-sional positive def\/inite af\/f\/ine hyperspheres admitting
pointwise symmetries was obtained in \cite{Vr04}. We continue the
classif\/ication in the indef\/inite case. We can assume that the af\/f\/ine
metric has index two, i.e.\ the corresponding isometry group is the
(special) Lorentz group $\SO(1,2)$. In Part 1, it turns out that a
$\SO(1,2)$-stabilizer of a nontrivial traceless cubic form is
isomorphic to either $\SO (2)$, $\SO(1,1)$, $\R$, the group $S_3$ of
order~6, $\Z_2 \times \Z_2$, $\Z_3$, $\Z_2$ or it is trivial. We have shown that hyperspheres admitting a pointwise $\Z_2 \times \Z_2$- resp.\ $\R$-symmetry are well-known, they have constant sectional curvature and Pick invariant $J<0$ resp.\ $J=0$.
In the following we classify the indef\/inite af\/f\/ine hyperspheres which
admit a pointwise $\Z_3$-, $\SO(2)$- or $\SO(1,1)$-symmetry. They turn out to
be warped
products of af\/f\/ine spheres ($\Z_3$) or quadrics ($SO(2)$, $SO(1,1)$) with a
curve. As a result we get a new composition method. Since the methods for the proofs for $SO(1,1)$ are similar to those for $\Z_3$- or $\SO(2)$-symmetry (and as long) we will omit them here. Both methods and results are dif\/ferent in case of $S_3$-symmetry and will be published elsewhere. The paper is organized as follows:
We will state the basic formulas of (equi-)af\/f\/ine hypersurface-theory needed in the further classif\/ication in Section~\ref{sec:basics}. In Section~\ref{sec:type2,3,4}, we show that in case of $\SO(2)$-, $S_3$- or $\Z_3$-symmetry we can extend the canonical form of $K$ (cf.~\cite{S07a}) locally. Thus we can obtain information about the coef\/f\/icients of $K$ and $\nabla$ from the basic equations of Gauss,
Codazzi and Ricci (cf.~Section~\ref{sec:Intcond}). In Section~\ref{sec:type2,4} we show, that in case of $\Z_3$- or $\SO(2)$-symmetry it follows that the hypersurface
admits a warped product structure $\R\times_{e^f}N^2$. Then we
classify such hyperspheres by showing how they can be constructed
starting from $2$-dimensional positive def\/inite af\/f\/ine spheres
resp. quadrics (cf.\ Theorems~\ref{thm:ClassC1}--\ref{thm:ExC3}). We end in Section~\ref{sec:type8} by stating the classif\/ication results in case of $SO(1,1)$-symmetry (cf.\ Theorems~\ref{thm:ClassC1t8}--\ref{thm:ExC3t8}).
The classif\/ication can be seen as a generalization of the well
known Calabi product of hyperbolic af\/f\/ine spheres \cite{Ca72,HLV08}
and of the constructions for af\/f\/ine spheres considered in~\cite{DV94}. The following natural question for a (de)composition theorem,
related to these constructions, gives another motivation for studying $3$-dimensional
hypersurfaces admitting a pointwise symmetry:
\begin{DecProb} Let $M^n$ be a nondegenerate
affine hypersurface in $\R^{n+1}$. Under what conditions do there
exist affine hyperpsheres $M_1^r$ in $\R^{r+1}$ and $M_2^s$ in
$\R^{s+1}$, with $r+s=n-1$, such that $M = I \times_{f_1} M_1
\times_{f_2} M_2$, where $I \subset \R$ and $f_1$ and $f_2$ depend
only on $I$ $($i.e.~$M$ admits a warped product structure$)$? How can the
original immersion be recovered starting from the immersion of the
affine spheres?
\end{DecProb}
Of course the f\/irst dimension in which the above problem can be
considered is three and our results provide an
answer in that case.
\section{Basics of af\/f\/ine hypersphere theory}
\label{sec:basics}
First we recall the def\/inition of the af\/f\/ine normal $\xi$
(cf.~\cite{NS94}). In equi-af\/f\/ine hypersurface theory on the ambient
space $\R^{n+1}$ a f\/ixed volume form $\det$ is given. A transverse
vector f\/ield $\xi$ induces a volume form $\theta$ on $M$ by
$\theta(X_1,\ldots,X_n)=\det(\hs_* X_1,\ldots,\hs_* X_n,\xi)$. Also
the af\/f\/ine metric $h$ def\/ines a volume form $\omega_h$ on $M$, namely
$\omega_h=|\det h|^{1/2}$. Now the af\/f\/ine normal $\xi$ is uniquely
determined (up to sign) by the conditions that $D \xi$ is everywhere
tangential (which is equivalent to $\nabla \theta =0$) and that
\begin{gather}\label{BlaschkeNormal}
\theta = \omega_h.
\end{gather}
Since we only consider 3-dimensional indef\/inite hyperspheres, i.e.
\begin{gather}\label{affHypSphere}
S= H \Id,\qquad H=\text{const},
\end{gather}
we can f\/ix the orientation of the af\/f\/ine normal $\xi$ such that the
af\/f\/ine metric has signature one. Then the sign of $H$ in the
def\/inition of an af\/f\/ine hypersphere is an invariant.
Next we state some of the fundamental equations, which a
nondegenerate hypersurface has to satisfy, see also \cite{NS94} or
\cite{LSZ93}. These equations relate $S$ and $K$ with amongst others
the curvature tensor $R$ of the induced connection $\nabla$ and the
curvature tensor $\hat R$ of the Levi-Civita connection~$\widehat\nabla$ of the af\/f\/ine metric $h$. There are the
Gauss equation for $\nabla$, which states that:
\begin{gather*}
R(X,Y)Z =h(Y,Z)SX -h(X,Z) SY,
\end{gather*}
and the Codazzi equation
\begin{gather*}
(\nabla_X S) Y =(\nabla_Y S) X.
\end{gather*}
Also we have the total symmetry of the af\/f\/ine cubic form
\begin{gather}\label{defC}
C(X,Y,Z)= (\nabla_X h) (Y,Z) = -2 h(K(X,Y),Z).
\end{gather}
The fundamental existence and uniqueness theorem, see~\cite{Dil89} or~\cite{DNV90}, states that given~$h$,~$\nabla$ and~$S$ such that the
dif\/ference tensor is symmetric and traceless with respect to $h$, on a
simply connected manifold $M$ an af\/f\/ine immersion of $M$ exists if and
only if the above Gauss equation and Codazzi equation are satisf\/ied.
From the Gauss equation and Codazzi equation above the Codazzi
equation for $K$ and the Gauss equation for $\widehat\nabla$ follow:
\begin{gather*}
(\widehat{\nabla}_X K)(Y,Z)- (\widehat{\nabla}_Y K)(X,Z)= \half( h(Y,Z)SX-
h(X,Z)SY - h(SY,Z)X + h(SX, Z)Y),\!
\end{gather*}
and
\begin{gather*} \hat R (X,Y)Z =
\tfrac{1}{2} (h(Y,Z) SX - h(X,Z) SY + h(SY,Z)X - h(SX,Z)Y ) - [K_X
,K_Y] Z
\end{gather*}
If we def\/ine the Ricci tensor of the Levi-Civita connection $\widehat
\nabla$ by:
\begin{gather}\label{def:RicLC}
\ricLC(X,Y)=\trace\{ Z \mapsto \hat R(Z,X)Y\}.
\end{gather}
and the Pick invariant by:
\begin{gather}\label{def:Pick}
J= \frac{1}{n (n-1)} h(K,K),
\end{gather}
then from the Gauss equation we get for the scalar
curvature $\hat{\kappa}=\frac{1}{n (n-1)}(\sum_{i,j} h^{ij}
\ricLC_{ij})$:
\begin{gather}\label{TE}
\hat \kappa= H+J.
\end{gather}
For an af\/f\/ine hypersphere the Gauss and Codazzi equations have the form:
\begin{gather}\label{gaussInd}
R(X,Y)Z=H(h(Y,Z)X -h(X,Z) Y),\\
\label{codazziS} (\nabla_X H)Y=(\nabla_Y H)X, \qquad \text{i.e.}
\quad H= {\rm const},\\
\label{CodK} (\widehat{\nabla}_X K)(Y,Z)=(\widehat{\nabla}_Y K)(X,Z),\\
\label{gaussLC} \hat R (X,Y)Z = H(h(Y,Z)X -h(X,Z) Y)- [K_X ,K_Y] Z.
\end{gather}
Since $H$ is constant, we can rescale $\hs$ such that $H\in \{-1,0,1\}$.
\section[A local frame for pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry]{A local frame for pointwise $\boldsymbol{\SO(2)}$-, $\boldsymbol{S_3}$- or $\boldsymbol{\Z_3}$-symmetry}
\label{sec:type2,3,4}
Let $\M$ be a hypersphere admitting a $\SO(2)$-, $S_3$- or
$\Z_3$-symmetry. According to \cite[Theorem~4, 2.--4.]{S07a} there exists
for every $p\in \M$ an ONB $\{ \bt, \bv, \bw\}$ of $T_p \M$ such that
\begin{alignat*}{4}
& K(\bt,\bt) = -2a_4 \bt, \qquad & & K(\bt,\bv) =a_4 \bv, \qquad && K(\bt,\bw) =
a_4 \bw, &\\
& K(\bv,\bv) = -a_4 \bt + a_6 \bv, \qquad && K(\bv,\bw) = -a_6 \bw, \qquad &&
K(\bw,\bw) = -a_4 \bt -a_6 \bv,&
\end{alignat*}
where $a_4 > 0$ and $a_6=0$ in case of $SO(2)$-symmetry, $a_4=0$ and
$a_6>0$ for $S_3$, and $a_4>0$ and $a_6>0$ for $\Z_3$.
We would like to extend the ONB locally. It is well known that
$\ricLC$ (cf.~\eqref{def:RicLC}) is a symmetric operator and we
compute
(some of the computations in this section are done with the
CAS
Mathematica\footnote{See Appendix or \url{http://www.math.tu-berlin.de/~schar/IndefSym_typ234.html}.}):
\begin{lemma} Let $p \in \M$ and $\{\bt,\bv,\bw\}$ the basis constructed
earlier. Then
\begin{alignat*}{3}
&\ricLC(\bt,\bt) =-2(H- 3a_4^2) , \qquad &&\ricLC(\bt,\bv)=0, & \\
&\ricLC(\bt,\bw)=0,\qquad &&
\ricLC(\bv,\bv)=2(H-a_4^2+a_6^2) , & \\
&\ricLC(\bv,\bw)=0,&&\ricLC(\bw,\bw)=2(H-a_4^2+a_6^2) .&
\end{alignat*}
\end{lemma}
\begin{proof}
The proof is a straight-forward computation using the Gauss
equation~\eqref{gaussLC}. It follows e.g.\ that
\begin{gather*}
\hat R(\bt,\bv)\bt = H \bv -K_{\bt}(a_4\bv)+K_{\bv}(-2 a_4 \bt) = H
\bv -a_4^2 \bv -2 a_4^2\bv = (H- 3 a_4^2)\bv,\\
\hat
R(\bt,\bw)\bt = H \bw -K_{\bt}(a_4 \bw) +K_{\bw}(-2 a_4 \bt) =H \bw
- a_4^2 \bw -2 a_4^2 \bw = (H- 3 a_4^2)\bw,\\
\hat
R(\bt,\bv)\bw =-K_{\bt}(-a_6 \bw)+K_{\bv}(a_4\bw)=0.
\end{gather*}
From this it immediately follows that
\[
\ricLC(\bt,\bt) = -2(H-3a_4^2)
\]
and
\[
\ricLC(\bt,\bw)=0.
\]
The other equations follow by similar computations.
\end{proof}
We want to show that the basis we have constructed, at each point $p$,
can be extended dif\/ferentiably to a neighborhood of the point $p$ such
that, at every point, $K$ with respect to the frame $\{T,V,W\}$ has
the previously described form.
\begin{lemma}\label{lem:KfT234}
Let $\M$ be an affine hypersphere in $\mathbb R^4$ which admits a
pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry. Let $p \in M$. Then
there exists an orthonormal frame $\{T,V,W\}$ defined in a~neighborhood of the point $p$ such that $K$ is given by:
\begin{alignat*}{4}
& K(T,T)= -2a_4 T,\qquad && K(T,V) =a_4 V,\qquad && K(T,W) =
a_4 W,& \\
& K(V,V)= -a_4 T + a_6 V,\qquad && K(V,W)= -a_6 W, \qquad &&
K(W,W) = -a_4 T -a_6 V, &
\end{alignat*}
where $a_4 > 0$ and $a_6=0$ in case of $\SO(2)$-symmetry, $a_4=0$ and
$a_6>0$ in case of $S_3$-symmetry, and $a_4>0$ and $a_6>0$ in case of
$\Z_3$-symmetry.
\end{lemma}
\begin{proof}First we want to show that at every point the vector
$\bt$ is uniquely def\/ined (up to sign) and dif\/ferentiable. We
introduce a symmetric operator $\hat A$ by:
\begin{gather*}
\ricLC(Y,Z)= h(\hat A Y,Z).
\end{gather*}
Clearly $\hat A$ is a dif\/ferentiable operator on $M$. Since $2(H-
3a_4^2) \neq 2(H-a_4^2+a_6^2)$, the operator has two distinct
eigenvalues. A standard result then implies that the
eigen distributions are dif\/ferentiable. We take $T$ a local unit
vector f\/ield spanning the 1-dimensional eigen distribution, and local
orthonormal vector f\/ields $\tilde{V}$ and $\tilde{W}$ spanning the
second eigen distribution. If $a_6=0$, we can take $V=\tilde{V}$ and
$W= \tilde{W}$.
As $T$ is (up to sign) uniquely determined, for $a_6\neq 0$ there
exist dif\/ferentiable functions~$a_4$,~$c_6$ and~$c_7$,
$c_6^2+c_7^2\neq 0$, such that
\begin{alignat*}{3}
& K(T,T)= -2a_4 T,\qquad & & K(\tilde{V},\tilde{V})= -a_4 T + c_6 \tilde{V}+
c_7 \tilde{W},& \\
& K(T,\tilde{V}) =a_4 \tilde{V},\qquad & &
K(\tilde{V},\tilde{W})= c_7 \tilde{V} -c_6 \tilde{W},& \\
& K(T,\tilde{W})= a_4 \tilde{W},\qquad && K(\tilde{W},\tilde{W})= -a_4 T -c_6
\tilde{V} -c_7 \tilde{W}.&
\end{alignat*}
As we have shown in \cite{S07a}, in
the proof of Theorem~2 (Case~2), we
can always rotate $\tilde{V}$ and $\tilde{W}$ such that we obtain the
desired frame.
\end{proof}
\begin{remark} It actually follows from the proof of the previous lemma
that the vector f\/ield $T$ is (up to sign) invariantly def\/ined on
$M$, and therefore the function $a_4$, too. Since the Pick invariant~\eqref{def:Pick} $J= \frac{1}{3}(-5 a_4^2 + 2 a_6^2)$, the function
$a_6$ also is invariantly def\/ined on the af\/f\/ine hypersphere~$\M$.
\end{remark}
\section[Gauss and Codazzi for pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry]{Gauss and Codazzi for pointwise $\boldsymbol{\SO(2)}$-, $\boldsymbol{S_3}$- or $\boldsymbol{\Z_3}$-symmetry}
\label{sec:Intcond}
In this section we always will work with the local frame constructed
in the previous lemma. We denote the coef\/f\/icients of the Levi-Civita
connection with respect to this frame by:
\begin{alignat*}{4}
& \widehat{\nabla}_T T = \ac12 V + \ac13 W,\qquad &&
\widehat{\nabla}_T V = \ac12 T - \bc13 W,\qquad &&
\widehat{\nabla}_T W = \ac13 T + \bc13 V,& \\
& \widehat{\nabla}_V T = \ac22 V + \ac23 W, \qquad &&
\widehat{\nabla}_V V = \ac22 T - \bc23 W, \qquad &&
\widehat{\nabla}_V W = \ac23 T + \bc23 V,& \\
& \widehat{\nabla}_W T = \ac32 V + \ac33 W, \qquad &&
\widehat{\nabla}_W V = \ac32 T - \bc33 W, \qquad &&
\widehat{\nabla}_W W = \ac33 T + \bc33 V.&
\end{alignat*}
We will evaluate f\/irst the Codazzi and then the Gauss equations
(\eqref{CodK} and \eqref{gaussLC}) to obtain more information.
\begin{lemma}\label{lem:CodK}
Let $\M$ be an affine hypersphere in $\mathbb R^4$ which admits a
pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry and $\{T,V,W\}$ the
corresponding ONB. If the symmetry group is
\begin{description}\itemsep=0pt
\item[$\mathbf{SO(2)}$,] then $0= \ac12 =\ac13 =\ac23 =\ac32$,
$\ac33=\ac22$ and \\ $T(a_4)=-4\ac22 a_4$, $0=V(a_4) = W(a_4)$,
\item[$\mathbf{S_3}$,] then $0= \ac12 =\ac13$, $ \ac23=-3 \bc13=
-\ac32$, $\ac33=\ac22$ and \\ $T(a_6)=-\ac22 a_6$, $V(a_6) = 3\bc33
a_6$, $W(a_6)=-3 \bc23 a_6$,
\item[$\mathbf{\Z_3}$ and $\mathbf{a_6^2\neq 4 a_4^2}$,] then $0=
\ac12 =\ac13 =\ac23 =\ac32$, $\ac33=\ac22$, $\bc13=0$,\\
$T(a_4)=-4\ac22 a_4$, $0=V(a_4) = W(a_4)$, and \\ $T(a_6)=-\ac22 a_6$,
$V(a_6) = 3\bc33 a_6$, $W(a_6)=-3 \bc23 a_6$,
\item[$\mathbf{\Z_3}$ and $\mathbf{a_6=2 a_4}$,] then $\ac12 =2 \ac22=
-2 \ac33=- \bc33$,\\ $\ac13 =- 2\ac23 = -2 \ac32= \bc23$, $\bc13=0$,
and \\ $T(a_4)=0$, $V(a_4)=-4 \ac22 a_4$, $W(a_4)= 4 \ac23 a_4$.
\end{description}
\end{lemma}
\begin{proof}
An evaluation of the Codazzi equations \eqref{CodK} with the help of
the CAS
Mathematica leads to the following equations (they relate
to eq1--eq6 and eq8--eq9 in the Mathematica notebook):
\begin{gather}
V(a_4)=- 2\ac12 a_4, \qquad T(a_4)=-4 \ac22 a_4 + \ac12 a_6, \qquad
0=4 \ac23 a_4 + \ac13 a_6,\label{CodKeq1} \\ W(a_4)=- 2\ac13
a_4,\qquad 0=4 \ac32 a_4 + \ac13 a_6,\qquad T(a_4)=-4 \ac33 a_4 -
\ac12 a_6, \label{CodKeq2}\\ T(a_6)-V(a_4)=3\ac12 a_4 -\ac22
a_6,\qquad 0=\ac13 a_4 + (\ac23 + 3 \bc13)a_6, \label{CodKeq3}\\
W(a_4)=(\ac23+ \ac32) a_6,\qquad W(a_6)=
(-\ac23+3 \ac32)a_4 - \bc23 a_6,\nonumber \\ V(a_6)=(-\ac22+\ac33) a_4+ 3 \bc33
a_6,
\label{CodKeq4} \\
T(a_6)=-\ac12 a_4 -\ac33 a_6,\qquad
W(a_4)=- 3\ac13 a_4+ (-\ac32+3 \bc13)a_6,\label{CodKeq5}\\
V(a_4)=(- \ac22+\ac33) a_6, \qquad W(a_6)= (3\ac23 - \ac32)a_4 -3
\bc23 a_6,\label{CodKeq6}\\ 0=(\ac23-\ac32) a_4,\label{CodKeq8}\\
W(a_4)=-\ac13 a_4 +(\ac32-3 \bc13) a_6.\label{CodKeq9}
\end{gather}
From the f\/irst equation of \eqref{CodKeq2} (we will use the notation
\eqref{CodKeq2}.1) and \eqref{CodKeq4}.1 resp.\ \eqref{CodKeq1}.3 and~\eqref{CodKeq2}.2 we get:
\begin{gather}\label{a13,a23+a32}
0 =2 \ac13 a_4 +(\ac23+\ac32)a_6,\qquad
0 =2 (\ac23+\ac32)a_4 + \ac13 a_6.
\end{gather}
From \eqref{CodKeq6}.1) and \eqref{CodKeq1}.1
resp.~\eqref{CodKeq1}.2 and \eqref{CodKeq2}.3 we get:
\begin{gather}\label{a12,a22+a33}
0 =-2 \ac12 a_4 +2 (\ac22-\ac33)a_6,\qquad
0 =2 (-\ac22+\ac33)a_4 + \ac12 a_6.
\end{gather}
We consider f\/irst the case, that $\mathbf{a_6^2\neq 4 a_4^2}$. Then
we obtain from the foregoing equations that $\ac13=0$,
$\ac32=-\ac23$, $\ac12=0$ and $\ac33=\ac22$. Furthermore it follows
from \eqref{CodKeq1}.1 that $V(a_4)=0$, from \eqref{CodKeq1}.2 that
$T(a_4)=-4 \ac22 a_4$ and from \eqref{CodKeq1}.3 that $\ac23
a_4=0$. Equation \eqref{CodKeq2}.1 becomes $W(a_4)=0$, equation
\eqref{CodKeq3}.2 $T(a_6)=-\ac22 a_6$ and \eqref{CodKeq3}.3 $(\ac23+3
\bc13)a_6=0$. Finally equation \eqref{CodKeq4}.2 resp. 3 gives
$W(a_6)=-3 \bc23 a_6$ and $V(a_6)= 3 \bc33 a_6$.
In case of $SO(2)$-symmetry ($a_4>0$ and $a_6=0$) it follows that
$\ac23=0$ and thus the statement of the theorem.
In case of $S_3$-symmetry ($a_4=0$ and $a_6>0$) it follows that
$\ac23=-3\bc13$ and thus the statement of the theorem.
In case of $\Z_3$-symmetry ($a_4>0$ and $a_6>0$) it follows that
$\ac23=0$ and $\bc13=0$ and thus the statement of the theorem.
In case that $a_6= \pm 2 a_4$ ($\neq 0$), we can choose $V$, $W$ such
that $\mathbf{a_6 = 2a_4}$. Now equations \eqref{CodKeq8},
\eqref{CodKeq1}.3 and \eqref{CodKeq3}.3 lead to $\ac23=\ac32$,
$\ac13=-2 \ac23$ and $\bc13=0$. A combination of \eqref{CodKeq1}.2
and \eqref{CodKeq2}.3 gives $\ac12=(\ac22-\ac33)$, and then by
equations \eqref{CodKeq3}.2, \eqref{CodKeq1}.1 and \eqref{CodKeq1}.2
that $\ac33=-\ac22$. Thus $T(a_4)=0$ by \eqref{CodKeq1}.2,
$V(a_4)=- 4 \ac22 a_4$ by \eqref{CodKeq1}.1 and $W(a_4)= 4 \ac22
a_4$ by \eqref{CodKeq2}.1. Finally \eqref{CodKeq4}.2 and
\eqref{CodKeq2}.1 resp. \eqref{CodKeq4}.3 and \eqref{CodKeq1}.1 imply
that $\bc23=-\ac23$ resp. $\bc33=-\ac22$.
\end{proof}
An evaluation of the Gauss equations \eqref{gaussLC} with the help of
the CAS Mathematica leads to the following:
\begin{lemma}\label{lem:GaussLC}
Let $\M$ be an affine hypersphere in $\mathbb R^4$ which admits a
pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry and $\{T,V,W\}$ the
corresponding ONB. Then
\begin{gather}
T(\ac22) = -\ac22^2 +\ac23^2 +H-3 a_4^2,\label{Gauss1.1}\\
T(\ac23) =-2\ac22\ac23,\label{Gauss1.2}\\ W(\ac22) + V(\ac23)
=0,\label{Gauss1.3}\\ W(\ac23) - V(\ac22) =0,\label{Gauss1.4}\\
V(\bc13) - T(\bc23) = \ac22\bc23 + (\ac23 +
\bc13)\bc33,\label{Gauss1.5}\\ T(\bc33) - W(\bc13) =
(\ac23+\bc13)\bc23 - \ac22\bc33 ,\label{Gauss1.6}\\ V(\bc33) -
W(\bc23) = -\ac22^2-\ac23^2 +2 \ac23\bc13 +\bc23^2 +\bc33^2 +H +a_4^2
+2 a_6^2. \label{Gauss1.7}
\end{gather}
If the symmetry group is $\Z_3$, then $a_6^2\neq 4 a_4^2$.
\end{lemma}
\begin{proof}
The equations relate to eq11--eq13 and eq16 in the Mathematica
notebook. If $a_6^2= 4 a_4^2 (\neq 0)$, then we obtain by equations
eq11.1 and eq12.3 resp. eq15.3 and eq12.3 that $2 V(\ac22)=-4 \ac22^2
-H+ 3 a_4^2$ resp. $2 W(\ac23)=4 \ac23^2+H-3 a_4^2$, thus $
V(\ac22)-W(\ac23)= -2 \ac22^2 -2 \ac23^2 -H +3 a_4^2$. This gives a
contradiction to eq13.3, namely $ V(\ac22)-W(\ac23)= -2 \ac22^2 -2
\ac23^2 -H -9 a_4^2$.
\end{proof}
\section[Pointwise $\Z_3$- or $\SO(2)$-symmetry]{Pointwise $\boldsymbol{\Z_3}$- or $\boldsymbol{\SO(2)}$-symmetry} \label{sec:type2,4}
The following methods only work in the case of $\Z_3$- or $\SO(2)$-symmetry,
therefore the case of $S_3$-symmetry will be considered elsewhere.
As the vector f\/ield $T$ is globally def\/ined, we can def\/ine the
distributions $L_1=\Span\{T\}$ and $L_2=\Span\{V,W\}$. In the
following we will investigate these distributions. For the terminology
we refer to \cite{Noe96}.
\begin{lemma}\label{L1}
The distribution $L_1$ is autoparallel $($totally geodesic$)$ with respect
to $\widehat\nabla$.
\end{lemma}
\begin{proof} From $\widehat{\nabla}_{T} T = \ac12 V + \ac13 W=0$
(cf.~Lemmas~\ref{lem:CodK} and~\ref{lem:GaussLC}) the claim follows immediately.
\end{proof}
\begin{lemma}\label{L2}
The distribution $L_2$ is spherical with mean curvature normal
$U_2=\ac22 T$.
\end{lemma}
\begin{proof} For $U_2=\ac22 T\in L_1=L_2^{\perp}$ we have
$h(\widehat{\nabla}_{E_a} E_b, T)= h(E_a, E_b) h(U_2,T)$ for $E_a,
E_b\in \{V,W\}$, and $h(\widehat{\nabla}_{E_a} U_2, T)= h(E_a(\ac22) T
+ \ac22 \widehat{\nabla}_{E_a} T, T)=0$ (cf.\ Lemma~\ref{lem:CodK} and
\eqref{Gauss1.3}, \eqref{Gauss1.4}).
\end{proof}
\begin{remark} $\ac22$ is independent of the
choice of ONB $\{V,W\}$. It therefore is a globally def\/ined
function on $\M$.
\end{remark}
We introduce a coordinate function $t$ by $\pt :=T$. Using the
previous lemma, according to \cite{PR93}, we get:
\begin{lemma}\label{warped} $(\M,h)$ admits a warped product structure
$\M=I \times_{e^f}N^2$ with $f: I \to \mathbb R$
satisfying
\begin{gather}\label{deff}
\frac{\partial f}{\partial t}=\ac22.
\end{gather}
\end{lemma}
\begin{proof} Proposition~3 in \cite{PR93} gives the warped product structure with
warping function $\lambda_2:I \to \mathbb R$. If we introduce $f=\ln
\lambda_2$, following the proof we see that $\ac22
T=U_2=-\grad(\ln\lambda_2)=-\grad f$.
\end{proof}
\begin{lemma}\label{lem:curvN2} The curvature of $N^2$ is
${}^NK(N^2)=e^{2f}(H+2 a_6^2 + a_4^2-\ac22^2)$.
\end{lemma}
\begin{proof} From Proposition~2 in \cite{PR93} we get the following relation
between the curvature tensor $\hat R$ of the warped product $\M$ and
the curvature tensor $\tilde R$ of the usual product of
pseudo-Riemannian manifolds ($X,Y,Z\in {\cal X}(M)$ resp. their
appropriate projections):
\begin{gather*} \hat R(X,Y)Z = \tilde{R} (X,Y)Z + h(Y,Z)(\widehat{\nabla}_X U_2 - h(X,U_2)U_2) - h(\widehat{\nabla}_X
U_2- h(X,U_2)U_2,Z)Y \\
\phantom{\hat R(X,Y)Z =}{} - h(X,Z)(\widehat{\nabla}_Y U_2 -
h(Y,U_2)U_2) + h(\widehat{\nabla}_Y U_2- h(Y,U_2)U_2,Z)X \\
\phantom{\hat R(X,Y)Z =}{} +h(U_2,U_2)(h(Y,Z)X-h(X,Z)Y).
\end{gather*}
Now $\tilde{R}(X,Y)Z={}^N\hat R(X,Y)Z$ for all $X,Y,Z\in TN^2$ and
otherwise zero (cf.\ \cite[page~89]{O'N83}, Corollary~58) and
$K(N^2)=K(V,W)= \frac{h(-\hat{R}(V,W)V,W)}{h(V,V)h(W,W)-h(V,W)^2}$
(cf.~\cite[page~77]{O'N83}, the curvature tensor has the opposite
sign). Since $h(X,Y)=e^{2f} {}^Nh(X,Y)$ for $X,Y\in TN^2$, it follows
that
\[
{}^NK(N^2)=e^{2f} h(-{}^N\hat{R}(V,W)V,W).
\] Finally we obtain by
the Gauss equation~\eqref{gaussLC} the last
ingredient for the computation: $\hat R(V,W)V$ $ =-(H+2 a_6^2 + a_4^2)
W$ (cf.~the Mathematica notebook).
\end{proof}
Summarized we have obtained the following structure equations
(cf.\ \eqref{strGauss}, \eqref{strWeingarten} and \eqref{defK}), where
$a_6=0$ in case of $\SO(2)$-symmetry resp. $\bc13=0$ in case of
$\Z_3$-symmetry:
\begin{gather}
D_T T = -2a_4 T- \xi, \label{D11}\\
D_T V = +a_4 V - \bc13 W, \label{D12}\\
D_T W = +\bc13 V + a_4 W, \label{D13}\\
D_V T =+(\ac22 +a_4) V, \label{D21}\\
D_W T =+(\ac22 + a_4)W, \label{D31}\\
D_V V =+ a_6 V -\bc23 W +(\ac22 - a_4)T +\xi, \label{D22}\\
D_V W = +\bc23 V - a_6 W, \label{D23}\\
D_W V =-(\bc33 +a_6) W, \label{D32}\\
D_W W = +(\bc33- a_6) V +(\ac22 - a_4) T +\xi, \label{D33}
\\
\label{Dxi}
D_{X} \xi= -H X.
\end{gather}
The Codazzi and Gauss equations (\eqref{CodK} and \eqref{gaussLC})
have the form (cf.\ Lemmas~\ref{lem:CodK} and \ref{lem:GaussLC}):
\begin{gather}
T(a_4) =-4\ac22 a_4, \qquad 0=V(a_4) = W(a_4),\label{Da4}\\
T(a_6) =-\ac22 a_6,\qquad V(a_6) = 3\bc33 a_6,\qquad W(a_6)=-3 \bc23
a_6,\label{Da6}\\
T(\ac22) = -\ac22^2 +H-3 a_4^2, \qquad
V(\ac22)=0, \qquad W(\ac22) =0,\label{Da22}\\
V(\bc13) - T(\bc23) = \ac22\bc23 + \bc13\bc33,\label{Db1}\\
T(\bc33) - W(\bc13) =\bc13\bc23 - \ac22\bc33 ,\label{Db2}\\
V(\bc33) - W(\bc23) =
-\ac22^2+\bc23^2 +\bc33^2 +H +a_4^2 +2 a_6^2,\label{Db3}
\end{gather}
where $a_6=0$ in case of $\SO(2)$-symmetry resp. $\bc13=0$ in case of
$\Z_3$-symmetry.
Our f\/irst goal is to f\/ind out how $N^2$ is immersed in $\Rf$, i.e.\ to
f\/ind an immersion independent of $t$. A look at the structure
equations \eqref{D11}--\eqref{Dxi} suggests to start with a linear
combination of~$T$ and~$\xi$.
We will solve the problem in two steps. First we look for a vector
f\/ield $X$ with $D_T X=\alpha X$ for some function~$\alpha$: We def\/ine
$X:=A T +\xi$ for some function $A$ on $\M$. Then $D_T X=\alpha X$ if\/f
$\alpha=-A$ and $\pt A= -A^2 +2a_4 A+ H$, and $A:=\ac22- a_4$ solves
the latter dif\/ferential equation. Next we want to multiply $X$ with
some function $\beta$ such that $D_T (\beta X)=0$: We def\/ine a positive
function $\beta$ on $\R$ as the solution of the dif\/ferential equation:
\begin{gather}\label{dtbeta}
\tfrac{\partial}{\partial t} \beta = (\ac22- a_4)\beta
\end{gather}
with initial condition $\beta(t_0)>0$. Then $D_T(\beta X)=0$ and by
\eqref{D21}, \eqref{Dxi} and \eqref{D31} we get (since
$\beta$, $\ac22$ and $a_4$ only depend on $t$):
\begin{gather}
D_{T}(\beta((\ac22- a_4)T +\xi)) =0,\label{eq31}\\
D_{V}(\beta((\ac22- a_4)T +\xi)) =\beta(\ac22^2-a_4^2-H)V ,\label{eq32}\\
D_{W}(\beta((\ac22- a_4)T +\xi)) =\beta(\ac22^2-a_4^2-H)W.\label{eq33}
\end{gather}
To obtain an immersion we need that $\nu:=\ac22^2-a_4^2-H$ vanishes
nowhere, but we only get:
\begin{lemma}\label{nu}
The function $\nu=\ac22^2-a_4^2-H$ is globally defined,
$\pt(e^{2f} \nu)=0$ and $\nu$ vanishes identically or nowhere on $\R$.
\end{lemma}
\begin{proof} Since $0=\pt {}^NK(N^2) = \pt(e^{2f}(2a_6^2-\nu))$
(Lemma~\ref{lem:curvN2}) and $\pt(e^{2f}2 a_6^2)=0$ (cf.~\eqref{Da6} and~\eqref{deff}), we get that $\pt(e^{2f} \nu)=0$. Thus $\pt\nu=-2 (\pt
f)\nu= -2\ac22\nu$.
\end{proof}
\subsection[The first case: $\nu \neq 0$ on $\M$]{The f\/irst case: $\boldsymbol{\nu \neq 0}$ on $\boldsymbol{\M}$}
\label{sec:case1}
We may, by translating $f$, i.e.\ by replacing $N^2$ with a homothetic
copy of itself, assume that $e^{2f} \nu =\eps_1$, where $\eps_1 =\pm 1$.
\begin{lemma}\label{defphi}
$\varPhi:=\beta ((\ac22- a_4)T +\xi)\colon
M^3 \to \R^4$ induces a proper affine sphere structure, say~$\tilde{\phi}$, mapping $N^2$ into a 3-dimensional linear subspace
of $\R^4$. $\tilde{\phi}$ is part of a quadric iff $a_6 =0$.
\end{lemma}
\begin{proof}
By \eqref{eq32} and \eqref{eq33} we have $\varPhi_*(E_a)= \beta
\nu E_a$ for $E_a\in \{V,W\}$. A further dif\/ferentiation, using \eqref{D22}
($\beta$ and $\nu$ only depend on $t$), gives:
\begin{gather*}
D_{V} \varPhi_*(V) = \beta \nu D_{V} V = \beta \nu ((\ac22- a_4)T +a_6 V - \bc23 W +\xi)\\
\phantom{D_{V} \varPhi_*(V)}{} =a_6\varPhi_*(V)-\bc23 \varPhi_*(W) +\nu \varPhi
=a_6\varPhi_*(V)-\bc23 \varPhi_*(W) +\eps_1 e^{-2f}\varPhi.
\end{gather*}
Similarly, we obtain the other derivatives, using \eqref{D23}--\eqref{D33}, thus:
\begin{gather}
D_{V} \varPhi_*(V) = a_6\varPhi_*(V) -\bc23 \varPhi_*(W) +
e^{-2f}\eps_1 \varPhi, \label{Dphivv}\\
D_{V} \varPhi_*(W) = \bc23
\varPhi_*(V) -a_6 \varPhi_*(W), \label{Dphivw}\\
D_{W} \varPhi_*(V) =
-(\bc33+ a_6)\varPhi_*(W), \label{Dphiwv}\\
D_{W} \varPhi_*(W) =
(\bc33-a_6)\varPhi_*(V) +e^{-2f}\eps_1 \varPhi, \label{Dphiww}\\ D_{E_a}
\varPhi = \beta e^{-2f}\eps_1 E_a.\label{Dphiea}
\end{gather}
The foliation at $f=f_0$ gives an immersion of $N^2$ to $M^3$, say
$\pi_{f_0}$. Therefore, we can def\/ine an immersion of $N^2$ to $\R^4$
by $\tilde{\phi}:=\varPhi\circ\pi_{f_0}$, whose structure equations
are exactly the equations above when $f=f_0$. Hence, we know that
$\tilde{\phi}$ maps $N^2$ into
$\Span\{\varPhi_*(V),\varPhi_*(W),\varPhi\}$, an af\/f\/ine hyperplane of
$\R^4$ and $\pt\varPhi=0$ implies $\varPhi(t,v,w)=\tilde{\phi}(v,w)$.
We can read of\/f the coef\/f\/icients of the dif\/ference tensor
$K^{\tilde{\phi}}$ of $\tilde{\phi}$ (cf.~\eqref{strGauss} and
\eqref{defK}): $K^{\tilde{\phi}}(\tilde{V},\tilde{V})=a_6 \tilde{V}$,
$K^{\tilde{\phi}}(\tilde{V},\tilde{W})=-a_6
\tilde{W}$,$K^{\tilde{\phi}}(\tilde{W},\tilde{W})=- a_6 \tilde{V}$, and
see that $\trace (K^{\tilde{\phi}})_X$ vanishes. The af\/f\/ine metric
introduced by this immersion corresponds with the metric on $N^2$.
Thus $\eps_1 \tilde{\phi}$ is the af\/f\/ine normal of $\tilde{\phi}$ and
$\tilde{\phi}$ is a proper af\/f\/ine sphere with mean curvature
$\eps_1$. Finally the vanishing of the dif\/ference tensor
characterizes quadrics.
\end{proof}
Our next goal is to f\/ind another linear combination of $T$ and $\xi$,
this time only depending on $t$. (Then we can express $T$ in terms
of $\phi$ and some function of $t$.)
\begin{lemma}\label{defdelta}
Define $\delta := H T +(\ac22+a_4) \xi$. Then there exist a constant
vector $C \in \R^4$ and a~function $a(t)$ such that
\[
\delta(t)= a(t) C.
\]
\end{lemma}
\begin{proof} Using \eqref{D21} resp. \eqref{D31} and
\eqref{Dxi} we obtain that $D_{V}\delta = 0=D_{W} \delta$. Hence
$\delta$ depends only on the variable $t$. Moreover, we get by
\eqref{D11}, \eqref{Da22}, \eqref{Da4} and \eqref{Dxi} that
\begin{gather*}
\pt\delta =D_{T} (H T+(\ac22+a_4)\xi)\\
\hphantom{\pt\delta}{} =H(-2a_4 T-\xi) + (-\ac22^2
+H -3 a_4^2-4 \ac22 a_4)\xi -(\ac22 + a_4) H T\\
\hphantom{\pt\delta}{} =-(3 a_4+\ac22)(H
T +(\ac22+a_4)\xi) =-(3 a_4+\ac22) \delta.
\end{gather*}
This implies that there exists a constant vector $C$ in $\Rf$ and a
function $a(t)$ such that $\delta(t)=a(t)C$.
\end{proof}
Notice that for an improper af\/f\/ine hypersphere ($H=0$) $\xi$ is
constant and parallel to $C$. Combining $\tilde{\phi}$ and $\delta$ we
obtain for $T$ (cf.\ Lemmas~\ref{defphi} and \ref{defdelta}) that
\begin{gather}\label{T}
T(t,v,w)= -\frac{a}{\nu}C +\frac{1}{\beta\nu}(\ac22+a_4)\tilde{\phi}(v,w).
\end{gather}
In the following we will use for the partial derivatives the
abbreviation $\hs_x:= \frac{\partial}{\partial x}\hs $, $x=t,v,w$.
\begin{lemma}\label{partialF}
\begin{gather*}
\hs_t = -\frac{a}{\nu}C +\pt\left(\frac{1}{\beta \nu}\right)\tilde{\phi},\qquad
\hs_v = \frac{1}{\beta\nu} \tilde{\phi}_v,\qquad
\hs_w = \frac{1}{\beta\nu} \tilde{\phi}_w.
\end{gather*}
\end{lemma}
\begin{proof} As by \eqref{dtbeta} and Lemma~\ref{nu} $\pt
\frac{1}{\beta\nu}= \frac{1}{\beta\nu}(\ac22+a_4)$, we obtain the
equation for $\hs_t =T$ by~\eqref{T}. The other equations follow
from \eqref{eq32} and \eqref{eq33}.
\end{proof}
It follows by the uniqueness theorem of f\/irst order dif\/ferential
equations and applying a~translation that we can write
\[
\hs(t,v,w)= \tilde{a}(t) C +\frac{1}{\beta\nu}(t)
\tilde{\phi}(v,w)
\] for a suitable function $\tilde{a}$ depending only
on the variable $t$. Since $C$ is transversal to the image of
$\tilde{\phi}$ (cf.~Lemmas~\ref{defphi} and \ref{defdelta},
$\nu\not\equiv 0$), we obtain that after applying an equiaf\/f\/ine
transformation we can write: $\hs(t,v,w) =(\gamma_1(t), \gamma_2(t)
\phi(v,w))$, in which $\tilde{\phi}(v,w)=(0,\phi(v,w))$. Thus we have
proven the following:
\begin{theorem}\label{thm:ClassC1} Let $\M$ be an indefinite affine hypersphere
of~$\Rf$ which admits a pointwise $\mathbb
Z_3$- or
$SO(2)$-symmetry. Let $\ac22^2-a_4^2 \neq H$ for some $p\in
\M$. Then
$\M$ is affine equivalent to
\[
\hs:\ I\times N^2\to \Rf: \ (t,v,w)\mapsto (\gamma_1(t), \gamma_2(t)
\phi(v,w)),
\]
where $\phi: N^2 \to \mathbb R^3$ is a $($positive
definite$)$ elliptic or hyperbolic affine sphere and $\gamma:I\to
\mathbb R^2$ is a curve. Moreover, if $\M$
admits a pointwise $SO(2)$-symmetry then $N^2$ is either an
ellipsoid or a two-sheeted hyperboloid.
\end{theorem}
We want to investigate the conditions imposed on the curve $\ga$. For
this we compute the derivatives of $\hs$:
\begin{alignat}{4}
& \hs_t=(\ga_1',\ga_2' \phi ),\qquad & & \hs_v =(0,\ga_2\phi_v ),\qquad && \hs_w
=(0,\ga_2\phi_w ),& \nonumber\\
& \hs_{tt}=(\ga_1'',\ga_2'' \phi),\qquad &&
\hs_{tv}=(0,\ga_2'\phi_{v}),\qquad && \hs_{tw}=(0,\ga_2'\phi_w), & \label{DF} \\
& \hs_{vv}=(0,\ga_2\phi_{vv}),\qquad && \hs_{vw}=(0,\ga_2\phi_{vw}),\qquad &&
\hs_{ww}=(0,\ga_2'\phi_{ww}). &\nonumber
\end{alignat}
Furthermore we have to distinguish if $\M$ is proper ($H=\pm 1$) or
improper ($H=0$).
First we consider the case that $\M$ is proper, i.e. $\xi=-H\hs$. An
easy computation shows that the condition that $\xi$ is a transversal
vector f\/ield, namely $ 0\neq \det(\hs_t ,\hs_v, \hs_w, \xi)=-\ga_2^2
(\ga_1\ga_2' - \ga_1' \ga_2) \det(\phi_v, \phi_w, \phi)$, is
equivalent to $\ga_2\neq 0$ and $\ga_1\ga_2' - \ga_1' \ga_2 \neq
0$. To check the condition that $\xi$ is the Blaschke normal
(cf.~\eqref{BlaschkeNormal}), we need to compute the Blaschke metric
$h$, using \eqref{strGauss}, \eqref{DF},
\eqref{Dphivv}--\eqref{Dphiww} and the notation $r,s\in\{v,w\}$ and
$g$ for the Blaschke metric of $\phi$:
\begin{gather*}
\hs_{tt} = \cdots \hs_t +\frac{\ga_1'\ga_2'' - \ga_1''
\ga_2'}{H(\ga_1\ga_2' - \ga_1' \ga_2)} \xi,\qquad \hs_{tr} = \text{tang}, \\
\hs_{rs}= \text{tang} -\frac{\ga_1' \ga_2}{H(\ga_1\ga_2' - \ga_1'
\ga_2)}\eps_1 g\left(\frac{\p}{\p r}, \frac{\p}{\p s}\right)\xi.
\end{gather*}
We obtain that
\[
\det h = h_{tt}
(h_{vv}h_{ww}-h_{vw}^2)=\frac{\ga_1'\ga_2'' - \ga_1'' \ga_2'}{H^3
(\ga_1\ga_2' - \ga_1' \ga_2)^3}(\ga_1')^2 \ga_2^2 \det g.
\]
Thus
\[
\ga_2^4 (\ga_1\ga_2' - \ga_1'
\ga_2)^2 \det(\phi_v,\phi_w,\phi)^2=\left|\frac{\ga_1'\ga_2'' - \ga_1''
\ga_2'}{(\ga_1\ga_2' - \ga_1' \ga_2)^3} (\ga_1')^2 \ga_2^2 \det
g\right|
\] is equivalent to \eqref{BlaschkeNormal}. Since $\phi$ is a def\/inite proper af\/f\/ine sphere with normal
$-\eps_1\phi$, we can again use \eqref{BlaschkeNormal} to obtain
\[\xi=-H\hs \Longleftrightarrow \ga_2^2|\ga_1\ga_2' - \ga_1' \ga_2|^5=
|\ga_1'\ga_2'' - \ga_1'' \ga_2'| (\ga_1')^2 \neq 0.\] From the
computations above ($g$ is positive def\/inite) also it follows that $\hs$
is indef\/inite if\/f either
\begin{gather*}
H\sign(\ga_1\ga_2' - \ga_1' \ga_2) =
\sign(\ga_1'\ga_2'' - \ga_1'' \ga_2')= \sign(\ga_1'\ga_2
\eps_1)
\qquad \mbox{or}\\
-H\sign(\ga_1\ga_2' - \ga_1' \ga_2) =
\sign(\ga_1'\ga_2'' - \ga_1'' \ga_2')= \sign(\ga_1'\ga_2
\eps_1).
\end{gather*}
Next we consider the case that $\M$ is improper, i.e.~$\xi$ is
constant. By Lemma~\ref{defdelta} $\xi$ is parallel to $C$ and thus
transversal to $\phi$. Hence we can apply an af\/f\/ine transformation to
obtain $\xi=(1,0,0,0)$. An easy computation shows that the condition
that $\xi$ is a transversal vector f\/ield, namely $0\neq \det(\hs_t
,\hs_v, \hs_w, \xi)=-\ga_2^2 \ga_2' \det(\phi_v, \phi_w, \phi)$, is
equivalent to $\ga_2\neq 0$ and $\ga_2' \neq 0$. To check the
condition that $\xi$ is the Blaschke normal
(cf.~\eqref{BlaschkeNormal}) we need to compute the Blaschke metric
$h$, using \eqref{strGauss}, \eqref{DF},
\eqref{Dphivv}--\eqref{Dphiww} and the notation $r,s\in\{v,w\}$ and
$g$ for the Blaschke metric of $\phi$:
\begin{gather*}
\hs_{tt} = \cdots \hs_t -\frac{\ga_1'\ga_2'' - \ga_1'' \ga_2'}{\ga_2'}
\xi,\qquad \hs_{tr}= \text{tang}, \qquad \hs_{rs}= \text{tang} +\frac{\ga_1'
\ga_2}{\ga_2'}\eps_1 g\left(\frac{\p}{\p r}, \frac{\p}{\p s}\right)\xi.
\end{gather*}
We obtain that
\[
\det h = h_{tt}
(h_{vv}h_{ww}-h_{vw}^2)=-\frac{\ga_1'\ga_2'' - \ga_1''
\ga_2'}{(\ga_2')^3}(\ga_1')^2 \ga_2^2 \det g.
\]
Thus
\eqref{BlaschkeNormal} is equivalent to
\[
\ga_2^4 (\ga_2')^2
\det(\phi_v,\phi_w,\phi)^2=\left|\frac{\ga_1'\ga_2'' - \ga_1''
\ga_2'}{(\ga_2')^3}(\ga_1')^2 \ga_2^2 \det g\right|.
\] Since $\phi$ is a
def\/inite proper af\/f\/ine sphere with normal $-\eps_1\phi$, we can again
use \eqref{BlaschkeNormal} to obtain
\[\xi=(1,0,0,0) \Longleftrightarrow \ga_2^2|\ga_2'|^5=
|\ga_1'\ga_2'' - \ga_1'' \ga_2'| (\ga_1')^2 \neq 0 .\] From the
computations above also it follows that $\hs$ is indef\/inite if\/f either
\begin{gather*}
- \sign(\ga_2')= \sign(\ga_1'\ga_2'' - \ga_1'' \ga_2')= \sign(\ga_1'
\ga_2 \eps_1) \qquad \mbox{or}\\
\sign(\ga_2')= \sign(\ga_1'\ga_2'' - \ga_1''
\ga_2')= \sign(\ga_1' \ga_2 \eps_1).
\end{gather*}
So we have seen under which conditions we can construct a 3-dimensional indef\/inite af\/f\/ine hypersphere out of an af\/f\/ine sphere:
\begin{theorem}\label{thm:ConstrC1}
Let $\phi:N^2 \to \mathbb R^3$ be a positive definite elliptic or
hyperbolic affine sphere $($with mean curvature $\eps_1=\pm 1)$, and
let $\gamma=(\ga_1,\ga_2): I \to \mathbb R^2$ be a curve. Define $\hs:I\times N^2\to \Rf$ by $\hs(t,v,w)= (\gamma_1(t), \gamma_2(t)
\phi(v,w))$.
\begin{enumerate}\itemsep=0pt
\item[$(i)$] If $\ga$ satisfies $\ga_2^2|\ga_1\ga_2' - \ga_1'
\ga_2|^5= \sign(\ga_1' \ga_2 \eps_1)(\ga_1'\ga_2'' - \ga_1'' \ga_2')
(\ga_1')^2\neq 0$, then $\hs$ defines \mbox{a~$3$-dimensional} indefinite
proper affine hypersphere.
\item[$(ii)$] If $\ga$ satisfies $\ga_2^2|\ga_2'|^5=
\sign(\ga_1' \ga_2 \eps_1)(\ga_1'\ga_2'' - \ga_1'' \ga_2')
(\ga_1')^2\neq 0$, then $\hs$ defines a $3$-dimensional indefinite
improper affine hypersphere.
\end{enumerate}
\end{theorem}
Now we are ready to check the symmetries.
\begin{theorem}\label{thm:ExC1}
Let $\phi:N^2 \to \mathbb R^3$ be a positive definite elliptic or
hyperbolic affine sphere $($with mean curvature $\eps_1=\pm 1)$, and
let $\gamma=(\ga_1,\ga_2): I \to \mathbb R^2$ be a curve such that
$\hs(t,v,w)=(\gamma_1(t), \gamma_2(t) \phi(v,w ))$ defines a $3$-dimensional indefinite
affine hypersphere. Then $\hs(N^2\times I)$ admits a pointwise $\mathbb Z_3$- or
$SO(2)$-symmetry.
\end{theorem}
\begin{proof}
We already have shown that $\hs$ def\/ines a 3-dimensional indef\/inite
proper resp. improper af\/f\/ine hypersphere. To prove the symmetry we
need to compute~$K$. By assumption, $\phi$ is an af\/f\/ine sphere with
Blaschke normal $\xi^\phi =-\eps_1 \phi$. For the structure equations
\eqref{strGauss} we use the notation $\phi_{rs} = {}^\phi \Ga_{rs}^{u}
\phi_u - g_{rs} \eps_1 \phi$, $r,s,u \in \{v,w\}$. Furthermore we
introduce the notation $\al = \ga_1 \ga_2' - \ga_1' \ga_2$. Note that
$\al'=\ga_1 \ga_2'' - \ga_1'' \ga_2$. If $\hs$ is proper, using \eqref{DF},
we get the structure equations \eqref{strGauss} for $\hs$:
\begin{gather*}
\hs_{tt} = \frac{\al'}{\al} \hs_t +\frac{\ga_1'\ga_2'' - \ga_1''
\ga_2'}{H\al} \xi,\qquad \hs_{tr} = \frac{\ga_2'}{\ga_2} \hs_r, \\ \hs_{rs} =
{}^\phi \Ga_{rs}^{u} \hs_u - g_{rs} \eps_1 \frac{\ga_1 \ga_2}{\al} \hs_t -
g_{rs} \eps_1 \frac{\ga_1' \ga_2}{H\al}\xi.
\end{gather*}
We compute $K$ using \eqref{defC} and obtain:
\begin{gather*}
(\nabla_{\hs_t} h)(\hs_r,\hs_s) = \left(\left(\frac{\ga_1 \ga_2}{\al}\right)'
\frac{\al}{\ga_1 \ga_2} -2 \frac{\ga_2'}{\ga_2}\right) h(\hs_r,\hs_s),\\
(\nabla_{\hs_r} h)(\hs_t,\hs_t) = 0,
\end{gather*}
implying that $K_{\hs_t}$ restricted to the space spanned by $\hs_v$ and
$\hs_w$ is a multiple of the identity. Taking $T$ in direction of $\hs_t$,
we see that $\hs_v$ and $\hs_w$ are orthogonal to $T$. Thus we can
construct an ONB $\{T,V,W\}$ with $V,W$ spanning $\Span\{\hs_v,\hs_w\}$
such that $a_1 = 2 a_4$, $a_2=a_3=a_5=0$. By the considerations in
\cite[Section~4]{S07a} we see that $\hs$ admits a pointwise $\mathbb Z_3$- or
$SO(2)$-symmetry. If~$\hs$ is improper, the proof runs completely analogous.
\end{proof}
\subsection[The second case: $\nu \equiv 0$ and $H\neq 0$ on $\M$]{The second case: $\boldsymbol{\nu \equiv 0}$ and $\boldsymbol{H\neq 0}$ on $\boldsymbol{\M}$}
\label{sec:case2}
Next, we consider the case that $H =\ac22^2 - a_4^2$ and $H\neq 0$ on
$\M$. It follows that $\ac22\neq \pm a_4$ on~$\M$.
We already have seen that $\M$ admits a warped product structure. The
map $\varPhi$ we have constructed in Lemma~\ref{defphi} will not def\/ine
an immersion (cf.~\eqref{eq32} and \eqref{eq33}). Anyhow, for a f\/ixed
point $t_0$, we get from \eqref{D22}--\eqref{D33}, \eqref{eq32} and
\eqref{eq33}, using the notation $\tilde{\xi}=(\ac22-a_4) T + \xi$:
\begin{gather*}
D_V V =a_6 V - \bc23 W +\tilde{\xi},\qquad
D_V W =\bc23 V - a_6 W,\\
D_W V =-(\bc33 + a_6) W,\qquad
D_W W =(\bc33 - a_6) V +\tilde{\xi},\qquad
D_{E_a}\tilde{\xi}=0, \qquad E_a\in \{V,W\}.
\end{gather*}
Thus, if $v$ and $w$ are local coordinates which span the second
distribution $L_2$, then we can interpret $\hs(t_0,v,w)$ as a
positive def\/inite improper af\/f\/ine sphere in a $3$-dimensional
linear subspace.
Moreover, we see that this improper af\/f\/ine sphere is a paraboloid
provided that $a_6(t_0, v,w)$ vanishes identically. From the
dif\/ferential equations \eqref{Da6} determining $a_6$, we see that this
is the case exactly when $a_6$ vanishes identically, i.e.\ when $\M$
admits a pointwise $SO(2)$-symmetry.
After applying a translation and a change of coordinates, we may
assume that
\begin{gather*}
\hs(t_0,v,w)=(v,w,f(v,w),0),
\end{gather*}
with af\/f\/ine normal $\tilde{\xi}(t_0,v,w)=(0,0,1,0)$. To obtain $T$
at $t_0$, we consider \eqref{D21} and \eqref{D31} and get that
\begin{gather*}
D_{E_a}(T-(\ac22+a_4) \hs) = 0,\qquad E_a, E_b\in \{V,W\}.
\end{gather*}
Evaluating at $t=t_0$, this means that there exists a constant vector
$C$, transversal to $\Span\{V,W,\xi\}$, such that
$T(t_0,v,w)=(\ac22+a_4)(t_0) \hs(t_0,v,w) +C$. Since $\ac22+a_4\neq 0$
every\-where, we can write:
\begin{gather}\label{Tt0}
T(t_0,v,w)=\alpha_1 (v,w, f(v,w),\alpha_2),
\end{gather}
where $\alpha_1, \alpha_2\neq 0$ and we applied an equiaf\/f\/ine
transformation so that $C=(0,0,0,\alpha_1\alpha_2)$. To obtain
information about $D_T T$ we have that $D_T T= -2 a_4 T -\xi$
(cf.~\eqref{D11}) and $\xi=\tilde{\xi} - (\ac22- a_4)T$ by the
def\/inition of $\tilde{\xi}$. Also we know that
$\tilde{\xi}(t_0,v,w)=(0,0,1,0)$ and by \eqref{eq31}--\eqref{eq33}
that $D_{X}(\beta\tilde{\xi})=0$, $X\in {\cal X}(M)$. Taking suitable
initial conditions for the function $\beta$ ($\beta(t_0)=1$), we get
that $\beta\tilde{\xi}=(0,0,1,0)$ and f\/inally the following vector
valued dif\/ferential equation:
\begin{gather*}
D_T T= (\ac22 -3 a_4) T -\frac1{\beta} (0,0,1,0).
\end{gather*}
Solving this dif\/ferential equation, taking into account the initial
conditions \eqref{Tt0} at $t=t_0$, we get that there exist functions
$\delta_1$ and $\delta_2$ depending only on $t$ such that
\begin{gather*}
T(t,u,v)= (\delta_1(t) v,\delta_1(t) w, \delta_1(t) (f(v,w)
+\delta_2(t)), \alpha_2 \delta_1(t)),
\end{gather*}
where $\delta_1(t_0)=\alpha_1$, $\delta_2(t_0)=0$, $\delta_1'(t)
=(\ac22 -3 a_4) \delta_1(t)$ and $\delta_2'(t) =\delta_1^{-1}(t)
\beta^{-1}(t)$. As $T(t,v,w) =\tfrac{\partial \hs}{\partial
t}(t,v,w)$ and $\hs(t_0,v,w) =(v,w,f(v,w),0)$ it follows by
integration that
\[
\hs(t,v,w)= (\gamma_1(t) v, \gamma_1(t) w, \gamma_1(t) f(v,w)
+\gamma_2(t) , \alpha_2 (\gamma_1(t)-1)),
\]
where $\gamma_1'(t)
=\delta_1(t)$, $\gamma_1(t_0)=1$, $\gamma_2(t_0)=0$ and $\gamma_2'(t)
=\delta_1(t)\delta_2(t)$. After applying an af\/f\/ine transformation
we have shown:
\begin{theorem} \label{thm:ClassC2} Let $\M$ be an indefinite proper affine
hypersphere of $\,\Rf$ which admits a pointwise $\mathbb Z_3$- or
$SO(2)$-symmetry. Let
$H= \ac22^2 -a_4^2(\neq 0)$ on $M^3$. Then $\M$ is affine equivalent with
\[
\hs: \ I\times N^2\to \Rf: \ (t,v,w)\mapsto (\gamma_1(t) v, \gamma_1(t)
w, \gamma_1(t) f(v,w) +\gamma_2(t),\gamma_1(t)),
\] where $\psi: N^2
\to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ is a positive definite
improper affine sphere with affine normal $(0,0,1)$ and $\gamma:I\to
\mathbb R^2$ is a curve. Moreover, if $\M$ admits a
pointwise $SO(2)$-symmetry then $N^2$ is an elliptic paraboloid.
\end{theorem}
We want to investigate the conditions imposed on the curve $\ga$. For
this we compute the derivatives of $\hs$:
\begin{gather}
\hs_t =(\gamma_1' v,\gamma_1' w,\gamma_1' f(v,w)
+\gamma_2',\gamma_1'),\nonumber\\
\hs_v =(\gamma_1,0,\gamma_1 f_v,0),\qquad
\hs_w =(0,\gamma_1,\gamma_1 f_w,0),\nonumber\\
\hs_{tt} =(\gamma_1'' v, \gamma_1''w ,\gamma_1'' f(v,w) +
\gamma_2'',\gamma_1''), \label{DF2}
\\
\hs_{tv} =\tfrac{\gamma_1'}{\gamma_1}
\hs_v,\qquad \hs_{tw}=\tfrac{\gamma_1'}{\gamma_1} \hs_w,\nonumber\\
\hs_{vv} =(0,0,f_{vv}\gamma_1,0),\qquad
\hs_{vw}=(0,0,\gamma_1 f_{vw},0),\qquad \hs_{ww}=(0,0,\gamma_1 f_{ww},0)
\nonumber.
\end{gather}
$\M$ is a proper hypersphere, i.e.\ $\xi=-H\hs$. An easy computation
shows that the condition that $\xi$ is a transversal vector f\/ield,
namely $0\neq \det(\hs_t ,\hs_v, \hs_w, \xi)=-H \ga_1^2 (\ga_1\ga_2' -
\ga_1'\ga_2)$, is equivalent to $\ga_1\neq 0$ and $\ga_1\ga_2' -
\ga_1'\ga_2\neq 0$. Since $(0,0,1,0)= \frac{\ga_1}{\ga_1\ga_2' -
\ga_1'\ga_2} \hs_t - \frac{\ga_1'}{\ga_1\ga_2' - \ga_1'\ga_2} \hs$, we
have the following structure equations:
\begin{gather}
\hs_{tt} =\left(\frac{\ga_1''}{\ga_1'} + \frac{\ga_1'\ga_2'' -
\ga_1''\ga_2'}{\ga_1'}\frac{\ga_1}{\ga_1\ga_2' - \ga_1'\ga_2}\right) \hs_t +
\frac{\ga_1'\ga_2'' - \ga_1''\ga_2'}{\ga_1\ga_2' -
\ga_1'\ga_2}\frac1H \xi, \nonumber \\
\hs_{tr} =\frac{\gamma_1'}{\gamma_1} \hs_r,\qquad
\hs_{rs} =\frac{\ga_1^2}{\ga_1\ga_2' - \ga_1'\ga_2} f_{rs} \hs_t +
\frac{\ga_1\ga_1'}{\ga_1\ga_2' - \ga_1'\ga_2} f_{rs} \frac1H \xi
.\label{streqC2}
\end{gather}
We obtain:
\[
\det h = h_{tt} (h_{vv}h_{ww}-h_{vw}^2)
=\frac{\ga_1'\ga_2'' - \ga_1'' \ga_2'}{H^3(\ga_1\ga_2' -
\ga_1'\ga_2)^3}\ga_1^2(\ga_1')^2 (f_{vv} f_{ww}-f_{vw}^2).
\] Since
$\psi$ is a positive def\/inite improper af\/f\/ine sphere with af\/f\/ine
normal $(0,0,1)$, we get by \eqref{BlaschkeNormal} that $f_{vv}
f_{ww}-f_{vw}^2 =1$. Now \eqref{BlaschkeNormal} (for $\xi$) is
equivalent to
\[
\ga_1^4 (\ga_1\ga_2' -\ga_1'\ga_2)^2 =
\left|\frac{\ga_1'\ga_2'' - \ga_1'' \ga_2'}{(\ga_1\ga_2'
-\ga_1'\ga_2)^3}\right|\ga_1^2(\ga_1')^2 .
\] It follows that
\[\xi=-H\hs \Longleftrightarrow \ga_1^2|\ga_1\ga_2' -\ga_1'\ga_2|^5=
|\ga_1'\ga_2'' - \ga_1'' \ga_2'| (\ga_1')^2 \neq 0 .\] From the
computations above also it follows that $\hs$ is indef\/inite if\/f either
\begin{gather*}
\sign(\ga_1'\ga_2'' - \ga_1'' \ga_2')= \sign(H(\ga_1\ga_2'
-\ga_1'\ga_2))= -\sign(\ga_1 \ga_1') \qquad \mbox{or}\\
\sign(\ga_1'\ga_2'' -
\ga_1'' \ga_2')= -\sign(H(\ga_1\ga_2' -\ga_1'\ga_2))= -\sign(\ga_1
\ga_1').
\end{gather*}
So we have seen under which conditions we can construct a 3-dimensional indef\/inite af\/f\/ine hypersphere out of an af\/f\/ine sphere:
\begin{theorem}\label{thm:ConstrC2}
Let $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ be a
positive definite improper affine sphere with affine normal
$(0,0,1)$, and let $\gamma: I \to \mathbb R^2$ be a curve. Define $\hs:I\times N^2\to \Rf$ by $\hs(t,v,w)= (\gamma_1(t) v, \gamma_1(t) w, \gamma_1(t) f(v,w)
+\gamma_2(t),\gamma_1(t))$.
If $\ga=(\ga_1,\ga_2)$ satisfies
$\ga_1^2|\ga_1\ga_2' -\ga_1'\ga_2|^5=
-\sign(\ga_1\ga_1')(\ga_1'\ga_2'' - \ga_1'' \ga_2') (\ga_1')^2 \neq
0$, then $\hs$ defines a $3$-dimensional indefinite proper affine
hypersphere.
\end{theorem}
Now we are ready to check the symmetries.
\begin{theorem}\label{thm:ExC2}
Let $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ be a
positive definite improper affine sphere with affine normal
$(0,0,1)$, and let $\gamma: I \to \mathbb R^2$ be a curve such that
$\hs(t,v,w)=(\gamma_1(t) v, \gamma_1(t) w, \gamma_1(t) f(v,w)
+\gamma_2(t),\gamma_1(t))$ defines a $3$-dimensional indefinite proper affine
hypersphere. Then $\hs(N^2\times I)$ admits a pointwise $\mathbb
Z_3$- or $SO(2)$-symmetry.
\end{theorem}
\begin{proof}
We already have shown that $\hs$ def\/ines a 3-dimensional indef\/inite
proper af\/f\/ine hypersphere with af\/f\/ine normal $\xi= -H\hs$. To prove the
symmetry we need to compute $K$. We get the induced connection and the
af\/f\/ine metric from the structure equations \eqref{streqC2}. We
compute~$K$ using \eqref{defC} and obtain:
\begin{gather*}
(\nabla_{\hs_t} h)(\hs_r,\hs_s) = \left(\pt\ln\left(\frac{\ga_1
\ga_1'}{\ga_1\ga_2' -\ga_1'\ga_2}\right)-2 \frac{\ga_1'}{\ga_1}\right)
h(\hs_r,\hs_s),\\ (\nabla_{\hs_r} h)(\hs_t,\hs_t) = 0,
\end{gather*}
implying that $K_{\hs_t}$ restricted to the space spanned by $\hs_v$ and
$\hs_w$ is a multiple of the identity. Taking $T$ in direction of $\hs_t$,
we see that $\hs_v$ and $\hs_w$ are orthogonal to $T$. Thus we can
construct an ONB $\{T,V,W\}$ with $V,W$ spanning $\Span\{\hs_v,\hs_w\}$
such that $a_1 = 2 a_4$, $a_2=a_3=a_5=0$. By the considerations in
\cite[Section~4]{S07a} we see that $\hs$ admits a pointwise
$\mathbb Z_3$- or $SO(2)$-symmetry.
\end{proof}
\subsection[The third case: $\nu \equiv 0$ and $H=0$ on $\M$]{The third case: $\boldsymbol{\nu \equiv 0}$ and $\boldsymbol{H=0}$ on $\boldsymbol{\M}$}
\label{sec:case3}
The f\/inal cases now are that $\nu \equiv 0$ and $H=0$ on the
whole of $M^3$ and hence $\ac22=\pm a_4$.
First we consider the case that $\ac22= a_4 =:a>0$. Again we use that
$M^3$ admits a warped product structure and we f\/ix a parameter
$t_0$. At the point $t_0$, we have by \eqref{D22}--\eqref{Dxi}:
\begin{gather*}
D_V V =+ a_6 V -\bc23 W +\xi, \\
D_V W = +\bc23 V - a_6 W,\\
D_W V =-(\bc33 +a_6) W, \\
D_W W = +(\bc33- a_6) V +\xi,\\
D_{X} \xi= 0.
\end{gather*}
Thus, if $v$ and $w$ are local coordinates which span the second
distribution $L_2$, then we can interpret $\hs(t_0,v,w)$ as a
positive def\/inite improper af\/f\/ine sphere in a $3$-dimensional
linear subspace.
Moreover, we see that this improper af\/f\/ine sphere is a paraboloid
provided that $a_6(t_0, v,w)$ vanishes identically. From the
dif\/ferential equations \eqref{Da6} determining $a_6$, we see that this
is the case exactly when $a_6$ vanishes identically, i.e.\ when $\M$
admits a pointwise $SO(2)$-symmetry.
After applying an af\/f\/ine transformation and a change of coordinates,
we may assume that
\begin{gather}\label{initcond1}
\hs(t_0,v,w)=(v,w,f(v,w),0),
\end{gather}
with af\/f\/ine normal $\xi(t_0,v,w)=(0,0,1,0)$, actually
\[
\xi(t,v,w)=(0,0,1,0)
\] ($\xi$ is constant on $\M$ by
assumption). Furthermore we obtain by \eqref{D21} and \eqref{D31},
that $D_U T= 2 a U$ for all $U\in L_2$. We def\/ine $\delta:= T- 2a
\hs$, which is transversal to $\Span\{V,W, \xi\}$. Since $a$ is
independent of $v$ and $w$ (cf.~\eqref{Da4}), $D_U \delta =0$, and we
can assume that
\begin{gather}\label{initcond2}
T(t_0,v,w) - 2a(t_0) \hs(t_0,v,w)= (0,0,0,1).
\end{gather}
We can integrate \eqref{Da4} ($T(a)= -4 a^2$) and we take $a=
\frac1{4t}$, $t>0$. Thus \eqref{D11} becomes $D_T T=-\frac1{2t} T
-\xi$ and we obtain the following linear second order ordinary
dif\/ferential equation:
\begin{gather}\label{DTT}
\pp{t}\hs + \frac1{2t} \pt \hs = -\xi.
\end{gather}
The general solution is $\hs(t,v,w)= -\frac{t^2}3 \xi + 2\sqrt{t}
A(v,w)+ B(v,w)$. The initial conditions \eqref{initcond1} and
\eqref{initcond2} imply that $A(v,w)=\big(\frac{v}{2\sqrt{t_0}},
\frac{w}{2\sqrt{t_0}}, \frac{f(v,w)}{2\sqrt{t_0}}+\frac23 t_0^{3/2},
\sqrt{t_0}\big)$ and $B(v,w)=(0,0,-t_0^2,-2 t_0)$. Obviously we can
translate $B$ to zero. Furthermore we can translate the af\/f\/ine sphere
and apply an af\/f\/ine transformation to obtain
$A(v,w)=\frac{1}{2\sqrt{t_0}}(v, w, f(v,w), 1)$. After a change of
coordinates we get:
\begin{gather}\label{result3.1}
\hs(t,v,w)= (tv, tw, t f(v,w) - c t^4, t), \qquad c,t>0.
\end{gather}
Next we consider the case that $-\ac22= a_4 =:a>0$. Again we use that
$M^3$ admits a warped product structure and we f\/ix a parameter
$t_0$. A look at \eqref{D22}--\eqref{Dxi} suggests to def\/ine $\tilde{\xi}= -2a T+\xi$, then we get at the point $t_0$:
\begin{gather*}
D_V V =+ a_6 V -\bc23 W +\tilde{\xi}, \\
D_V W = +\bc23 V - a_6 W,\\
D_W V =-(\bc33 +a_6) W, \\
D_W W = +(\bc33- a_6) V +\tilde{\xi},\\
D_{U} \tilde{\xi}= 0.
\end{gather*}
Thus, if $v$ and $w$ are local coordinates which span the second
distribution $L_2$, then we can interpret $\hs(t_0,v,w)$ as a
positive def\/inite improper af\/f\/ine sphere in a $3$-dimensional
linear subspace.
Moreover, we see that this improper af\/f\/ine sphere is a paraboloid
provided that $a_6(t_0, v,w)$ vanishes identically. From the
dif\/ferential equations \eqref{Da6} determining $a_6$, we see that this
is the case exactly when $a_6$ vanishes identically, i.e.\ when $\M$
admits a pointwise $SO(2)$-symmetry.
After applying an af\/f\/ine transformation and a change of coordinates,
we may assume that
\begin{gather}\label{init}
\hs(t_0,v,w)=(v,w,f(v,w),0),
\end{gather}
with af\/f\/ine normal
\begin{gather}\label{initxi}
\tilde{\xi}(t_0,v,w)=(0,0,1,0).
\end{gather}
We have considered $\tilde{\xi}$ before. We can solve
\eqref{dtbeta} ($\pt \beta= -2a \beta$) explicitly by $\beta=c
\frac1{\sqrt{a}}$ (cf.~\eqref{Da4}) and get by
\eqref{eq31}--\eqref{eq33} that $D_X (\frac1{\sqrt{a}} \tilde{\xi})
=0$. Thus $\frac1{\sqrt{a}}(-2a T+\xi)=:C$ for a constant vector $C$,
i.e. $T=-\frac1{2a}(\sqrt{a} C - \xi)$. Notice that by \eqref{Dxi}
$\xi$ is a constant vector, too. We can choose $a=\frac1{4|t|}$, $t<0$
(cf.~\eqref{Da4}), and we obtain the ordinary dif\/ferential equation:
\begin{gather}\label{DT}
\pt \hs= -\sqrt{|t|} C -2t \xi, \qquad t<0.
\end{gather}
The solution (after a
translation) with respect to the initial condition \eqref{init} is
$\hs(t,v,w)=\frac23 |t|^{\frac32} C- t^2 \xi + (v,w,f(v,w),0)$. Notice
that $C$ is a multiple of $\tilde{\xi}$ and hence by \eqref{initxi} a
constant multiple of $(0,0,1,0)$. Furthermore $\xi$ is transversal to
the space spanned by
$\hs(t_0,v,w)$. So we get after an af\/f\/ine transformation and a change
of coordinates:
\begin{gather}\label{result3.2}
\hs(t,v,w)= (v,w, f(v,w) + c t^3, t^4 ),\qquad c,t>0.
\end{gather}
Combining both results \eqref{result3.1} and \eqref{result3.2} we have:
\begin{theorem} \label{thm:ClassC3} Let $\M$ be an indefinite improper affine
hypersphere of $\Rf$ which admits a pointwise $\mathbb
Z_3$- or $SO(2)$-symmetry. Let $\ac22^2 =a_4^2$ on $M^3$. Then $\M$ is affine
equivalent with either
\begin{gather*} \hs: \ I\times N^2\to \Rf: \ (t,v,w)\mapsto (t v, t w,t f(v,w) -c
t^4,t),\qquad (\ac22 =a_4)\qquad \text{or}\\
\hs: \ I\times N^2\to \Rf: \ (t,v,w)\mapsto (v, w, f(v,w) +c
t^3,t^4),\qquad(-\ac22 =a_4),
\end{gather*}
where $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ is a
positive definite improper affine sphere with affine normal
$(0,0,1)$ and $c,t\in \R^+$.
Moreover, if $\M$ admits a pointwise $SO(2)$-symmetry then~$N^2$ is an
elliptic paraboloid.
\end{theorem}
The computations for the converse statement can be done completely
analogous to the previous cases, they even are simpler (the curve is
given parametrized).
\begin{theorem}\label{thm:ExC3}
Let $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ be a
positive definite improper affine sphere with affine normal
$(0,0,1)$. Define $\hs(t,v,w)= (t v, t w,t f(v,w) -c t^4,t)$
or $\hs(t,v,w)= (v, w, f(v,w) +c t^3,t^4)$, where $c,t\in \R^+$.
Then $\hs$ defines a $3$-dimensional indefinite improper affine
hypersphere, which admits a pointwise $\mathbb
Z_3$- or $SO(2)$-symmetry.
\end{theorem}
\section[Pointwise $\SO(1,1)$-symmetry]{Pointwise $\boldsymbol{\SO(1,1)}$-symmetry}
\label{sec:type8}
Let $\M$ be a hypersphere admitting a $\SO(1,1)$-symmetry. We only state the classif\/ication results. The proofs are done quite similar, using a lightvector-frame instead of an orthonormal one, and will appear elsewhere. We denote a lightvector-frame by $\{E,V,F\}$, where $E$ and $F$ are lightvectors and $V$ is spacelike (cf.~\cite{S07a}).
\begin{lemma}\label{lem:KfT8}
Let $\M$ be an affine hypersphere in $\mathbb R^4$ which admits a
pointwise $\SO(1,1)$-symmetry. Let $p \in M$. Then there exists a
lightvector-frame $\{E,V,F\}$ defined in a neighborhood of the point~$p$ and a positive function $b_4$ such that $K$ is given by:
\begin{alignat*}{4}
& K(V,V)= -2b_4 V,\qquad & & K(V,E)=b_4 E, \qquad && K(V,F)=
b_4 F,&\\
& K(E,E)=0,\qquad && K(E,F)= b_4 V, \qquad &&
K(F,F)=0.&
\end{alignat*}
\end{lemma}
In the following we denote the coef\/f\/icients of the Levi-Civita
connection with respect to this frame by:
\begin{alignat*}{4}
& \widehat{\nabla}_E E = \ac11 E + \bc11 V,\qquad &&
\widehat{\nabla}_E V = \ac12 E - \bc11 F,\qquad &&
\widehat{\nabla}_E F =-\ac12 V - \ac11 F, & \\
& \widehat{\nabla}_V E = \ac21 E + \bc21 V, \qquad &&
\widehat{\nabla}_V V = \ac22 E - \bc21 F, \qquad &&
\widehat{\nabla}_V F =-\ac22 V - \ac21 F,& \\
& \widehat{\nabla}_F E = \ac31 E + \bc31 V,\qquad &&
\widehat{\nabla}_F V = \ac32 E - \bc31 F, \qquad &&
\widehat{\nabla}_F F =-\ac32 V - \ac31 F.&
\end{alignat*}
Similar as before, it turns out that the vector f\/ield $V$ is globally def\/ined, and we can def\/ine the
distributions $L_1=\Span\{V\}$ and $L_2=\Span\{E,F\}$. Again, $L_1$ is autoparallel with
respect to~$\widehat\nabla$, and $L_2$ is spherical with mean curvature normal
$-\ac12 V$. We introduce a coordinate function~$v$ by $\ptv:=V$.
\begin{lemma}\label{lem:nut8}
The function $\nu=b_4^2-\ac12^2-H$ is globally defined,
$\ptv(e^{2f} \nu)=0$ and $\nu$ vanishes identically or nowhere on $\R$.
\end{lemma}
Again we have to distinguish three cases.
\subsection[The first case: $\nu \neq 0$ on $\M$]{The f\/irst case: $\boldsymbol{\nu \neq 0}$ on $\boldsymbol{\M}$}
\label{sec:case1t8}
\begin{theorem}\label{thm:ClassC1t8} Let $\M$ be an indefinite affine hypersphere
of $\Rf$ which admits a pointwise $SO(1,1)$-symmetry. Let
$b_4^2-\ac12^2 \neq H$ for some $p\in \M$. Then $\M$ is affine
equivalent to
\[
\hs: \ I\times N^2\to \Rf: \ (v,x,y)\mapsto (\gamma_1(v),
\gamma_2(v) \phi(x,y)),
\]
where
$\phi: N^2 \to \mathbb R^3$ is a one-sheeted hyperboloid
and $\gamma:I\to \mathbb R^2$ is a~curve.
\end{theorem}
\begin{theorem}\label{thm:ConstrC1t8}
Let $\phi:N^2 \to \mathbb R^3$ be a one-sheeted hyperboloid and
let $\gamma: I \to \mathbb R^2$ be a curve. Define $\hs:I\times N^2\to \Rf$ by
$\hs(v,x,y)=(\gamma_1(v), \gamma_2(v) \phi(x,y ))$.
\begin{enumerate}\itemsep=0pt
\item[$(i)$] If $\ga=(\ga_1,\ga_2)$ satisfies $\ga_2^2|\ga_1\ga_2' - \ga_1'
\ga_2|^5= |\ga_1'\ga_2'' - \ga_1'' \ga_2'|
(\ga_1')^2\neq 0$, then $\hs$ defines a~$3$-di\-men\-sional indefinite
proper affine hypersphere.
\item[$(ii)$] If $\ga=(\ga_1,\ga_2)$ satisfies $\ga_2^2|\ga_2'|^5=
|\ga_1'\ga_2'' - \ga_1'' \ga_2'|
(\ga_1')^2\neq 0$, then $\hs$ defines a $3$-dimensional indefinite
improper affine hypersphere.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{thm:ExC1t8}
Let $\phi:N^2 \to \mathbb R^3$ be a one-sheeted hyperboloid and
let $\gamma: I \to \mathbb R^2$ be a curve, such that
$\hs(v,x,y)=(\gamma_1(v), \gamma_2(v) \phi(x,y ))$ defines
a $3$-dimensional indefinite affine hypersphere. Then
$\hs(I\times N^2)$ admits a pointwise $SO(1,1)$-symmetry.
\end{theorem}
\subsection[The second case: $\nu \equiv 0$ and $H\neq 0$ on $\M$]{The second case: $\boldsymbol{\nu \equiv 0}$ and $\boldsymbol{H\neq 0}$ on $\boldsymbol{\M}$}
\label{sec:case2t8}
\begin{theorem} \label{thm:ClassC2t8} Let $\M$ be an indefinite proper affine
hypersphere of $\Rf$ which admits a pointwise
$SO(1,1)$-symmetry. Let
$H= b_4^2-\ac12^2 (\neq 0)$ on $\M$. Then $\M$ is affine equivalent with
\[
\hs: \ I\times N^2\to \Rf: \ (v,x,y)\mapsto (\gamma_1(v) x,\gamma_1(v)
y,\gamma_1(v) f(x,y) + \gamma_2(v), \gamma_1(v)),
\] where $\psi: N^2
\to \mathbb R^3:(x,y) \mapsto (x,y,f(x,y))$ is a hyperbolic
paraboloid with affine normal $(0,0,1)$ and $\gamma:I\to \mathbb
R^2$ is a curve.
\end{theorem}
\begin{theorem}\label{thm:ConstrC2t8}
Let $\psi:N^2 \to \mathbb R^3$ be a hyperbolic paraboloid with
affine normal $(0,0,1)$, and let $\gamma: I \to \mathbb R^2$ be a
curve. Define $\hs:I\times N^2\to \Rf$ by $\hs(v,x,y)= (\gamma_1(v) x,\gamma_1(v)
y,\gamma_1(v) f(x,y) + \gamma_2(v), \gamma_1(v))$. If $\ga=(\ga_1,\ga_2)$ satisfies $\ga_1^2|\ga_1\ga_2' -
\ga_1' \ga_2|^5= |\ga_1'\ga_2'' - \ga_1'' \ga_2'| (\ga_1')^2\neq 0$,
then $\hs$ defines a $3$-dimensional indefinite proper affine
hypersphere.
\end{theorem}
\begin{theorem}\label{thm:ExC2t8}
Let $\psi:N^2 \to \mathbb R^3$ be a hyperbolic paraboloid with
affine normal $(0,0,1)$, and let $\gamma: I \to \mathbb R^2$ be a
curve, such that $\hs(v,x,y)= (\gamma_1(v) x,\gamma_1(v)
y,\gamma_1(v) f(x,y) + \gamma_2(v), \gamma_1(v))$ defines a~$3$-di\-men\-sional indefinite proper affine hypersphere. Then
$\hs(I\times N^2)$ admits a pointwise $SO(1,1)$-symmetry.
\end{theorem}
\subsection[The third case: $\nu \equiv 0$ and $H=0$ on $\M$]{The third case: $\boldsymbol{\nu \equiv 0}$ and $\boldsymbol{H=0}$ on $\boldsymbol{\M}$}
\label{sec:case3t8}
\begin{theorem} \label{thm:ClassC3t8} Let $\M$ be an indefinite improper affine
hypersphere of $\Rf$ which admits a pointwise $SO(1,1)$-symmetry.
Let $\ac12^2 =b_4^2$ on $M^3$. Then $\M$ is affine
equivalent with either
\begin{gather*}
\hs: \ I\times N^2\to \Rf: \ (v,x,y)\mapsto (vx, vy, v f(x,y)
- c v^4, v),\quad(\ac12 =b_4)\qquad \text{or}\\
\hs: \ I\times N^2\to \Rf: \ (v,x,y)\mapsto (x,y, f(x,y) + c v^3, v^4 )
,\qquad(-\ac12 =b_4),
\end{gather*}
where $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ is a
hyperbolic paraboloid with affine normal
$(0,0,1)$ and $c,t\in \R^+$.
\end{theorem}
\begin{theorem}\label{thm:ExC3t8}
Let $\psi: N^2 \to \mathbb R^3:(v,w) \mapsto (v,w,f(v,w))$ be a
hyperbolic paraboloid with affine nor\-mal
$(0,0,1)$. Define $\hs(t,v,w)= (t v, t w,t f(v,w) -c t^4,t)$
or $\hs(t,v,w)= (v, w, f(v,w) +c t^3,t^4)$, where $t\in \R^+$, $c\neq 0$.
Then $\hs$ defines a $3$-dimensional indefinite improper affine
hypersphere, which admits a pointwise $SO(1,1)$-symmetry.
\end{theorem}
\appendix
\section[Computations for pointwise $\SO(2)$-, $S_3$- or $\Z_3$-symmetry]{Computations for pointwise $\boldsymbol{\SO(2)}$-, $\boldsymbol{S_3}$- or $\boldsymbol{\Z_3}$-symmetry}
\noindent\(e[1]\text{:=}\{1,0,0\};e[2]\text{:=}\{0,1,0\};e[3]\text{:=}\{0,0,1\}\);
\subsection*{ONB of SO(1,2): \textit{e}[1]=T, \textit{e}[2]=V, \textit{e}[3]=W}
\subsection*{Af\/f\/ine metric}
\noindent h[\text{y$\_$},\text{z$\_$}]\text{:=}-y[[1]]z[[1]]+\text{Sum}[y[[i]]z[[i]],\{i,2,3\}];
\subsection*{Dif\/ference tensor (r=a1, s=a4, u=a6)}
\noindent K[\text{y$\_$},\text{z$\_$}]\text{:=} \text{Sum}[y[[i]]z[[j]]k[i,j], \{i,1,3\}, \{j,1,3\}];\\
k[1,1]\text{:=}\{-r,0,0\};
k[1,2]\text{:=}\{0,s,0\};
k[1,3]\text{:=}\{0,0,r-s\};\\
k[2,1]\text{:=}\{0,s,0\};
k[2,2]\text{:=}\{-s,u,0\};
k[2,3]\text{:=}\{0,0,-u\};\\
k[3,1]\text{:=}\{0,0,r-s\};
k[3,2]\text{:=}\{0,0,-u\};
k[3,3]\text{:=}\{-(r-s),-u,0\};
\subsection*{Ricci tensor, scalar curvature and Pick invariant}
\subsubsection*{[Kx,Ky]z}
\noindent\(\text{LK}[\text{x$\_$},\text{y$\_$},\text{z$\_$}]\text{:=}K[x,K[y,z]]-K[y,K[x,z]];\)
\noindent\(\text{ListLK}\text{:=}\{\text{LK}[e[1],e[2],e[1]],\text{LK}[e[1],e[2],e[2]],\text{LK}[e[1],e[2],e[3]],\\
\text{LK}[e[1],e[3],e[1]],\text{LK}[e[1],e[3],e[2]],\text{LK}[e[1],e[3],e[3]],\\
\text{LK}[e[2],e[3],e[1]],\text{LK}[e[2],e[3],e[2]],\text{LK}[e[2],e[3],e[3]]\};\\
\text{FullSimplify}[\text{ListLK}]\)
\subsubsection*{Curvature tensor (of the Levi-Civita connection)}
\noindent\(R[\text{x$\_$},\text{y$\_$},\text{z$\_$}]\text{:=}H(h[y,z] x-h[x,z] y)-K[x,K[y,z]]+K[y,K[x,z]];\)
\noindent\(\text{ListR}\text{:=}\{R[e[1],e[2],e[1]],R[e[1],e[2],e[2]],R[e[1],e[2],e[3]],\\
R[e[1],e[3],e[1]],R[e[1],e[3],e[2]],R[e[1],e[3],e[3]],\\
R[e[2],e[3],e[1]],R[e[2],e[3],e[2]],R[e[2],e[3],e[3]]\};\\
\text{Simplify}[\text{ListR}]\)
\subsubsection*{Ricci tensor (of the Levi-Civita connection)}
\noindent\(\text{ric}[\text{x$\_$},\text{y$\_$}]\text{:=}
\text{Simplify}[
(-h[R[e[1],x,y],e[1]]+h[R[e[2],x,y],e[2]]+
h[R[e[3],x,y],e[3]])]\);
\noindent\(\text{Listric}\text{:=}\{\text{ric}[e[1],e[1]],\text{ric}[e[1],e[2]],\text{ric}[e[1],e[3]],
\text{ric}[e[2],e[2]],\text{ric}[e[2],e[3]],\text{ric}[e[3],e[3]]\}; \\
\text{Simplify}[\text{Listric}]\)
\subsubsection*{Scalar curvature (of the Levi-Civita connection)}
\noindent\(\text{sc}\text{:=}1/6(-\text{ric}[e[1],e[1]]+\text{ric}[e[2],e[2]]+\text{ric}[e[3],e[3]]); \\
\text{Simplify}[\text{sc}]\)
\subsubsection*{Pick invariant}
\noindent\(P\text{:=}
1/6(-(k[1,1][[1]]){}^{\wedge}2 + (k[2,2][[2]]){}^{\wedge}2 +(k[3,3][[3]]){}^{\wedge}2 +
3 ((k[1,1][[2]]){}^{\wedge}2+(k[1,1][[3]]){}^{\wedge}2 -(k[2,2][[1]]){}^{\wedge}2 -
(k[3,3][[1]]){}^{\wedge}2 +(k[2,2][[3]]){}^{\wedge}2+(k[3,3][[2]]){}^{\wedge}2)-
6 (k[1,2][[3]]){}^{\wedge}2); \\
\text{Simplify}[P]\)
\subsection*{Lemma 1}
\noindent\(r=2s; \ \text{Simplify}[\text{Listric}]\)
\noindent\(\text{Simplify}[P]\)
\subsection*{Lemma 8}
\noindent\(R[e[2],e[3],e[2]]\)
\subsection*{Lemma 3 }
\subsubsection*{Levi-Civita connection (ONB)}
\noindent\(n[1,1]\text{:=}\{0,\text{a12},\text{a13}\};
n[1,2]\text{:=}\{\text{a12},0,-\text{b13}\};
n[1,3]\text{:=}\{\text{a13},\text{b13},0\};\\
n[2,1]\text{:=}\{0,\text{a22},\text{a23}\};
n[2,2]\text{:=}\{\text{a22},0,-\text{b23}\};
n[2,3]\text{:=}\{\text{a23},\text{b23},0\};\\
n[3,1]\text{:=}\{0,\text{a32},\text{a33}\};
n[3,2]\text{:=}\{\text{a32},0,-\text{b33}\};
n[3,3]\text{:=}\{\text{a33},\text{b33},0\};\)
\noindent\(\text{Na}[\text{y$\_$},\text{z$\_$}]\text{:=}\text{Sum}[y[[i]]z[[j]]n[i,j],\{i,1,3\},\{j,1,3\}]+
\text{Sum}[y[[1]]\text{Dt}[z[[i]],\text{f1}]e[i]\)\\
\(+\, y[[2]]\text{Dt}[z[[i]],\text{f2}]e[i] +
y[[3]]\text{Dt}[z[[i]],\text{f3}]e[i],\{i,1,3\}];\)
\subsubsection*{Codazzi for K (af\/f\/ine hypersphere)}
\noindent
\(\text{codazzi}[\text{x$\_$},\text{y$\_$},\text{z$\_$}]\text{:=}\text{Na}[x,K[y,z]]-K[\text{Na}[x,y],z]-K[y,\text{Na}[x,z]] -
\text{Na}[y,K[x,z]]+K[\text{Na}[y,x],z]+K[x,\text{Na}[y,z]] ;\)
\noindent\(\text{eq1}\text{:=}\text{Simplify}[\text{codazzi}[e[2],e[1],e[1]]];
\text{eq2}\text{:=}\text{Simplify}[\text{codazzi}[e[3],e[1],e[1]]];\\
\text{eq3}\text{:=}\text{Simplify}[\text{codazzi}[e[1],e[2],e[2]]];
\text{eq4}\text{:=}\text{Simplify}[\text{codazzi}[e[3],e[2],e[2]]];\\
\text{eq5}\text{:=}\text{Simplify}[\text{codazzi}[e[1],e[3],e[3]]];
\text{eq6}\text{:=}\text{Simplify}[\text{codazzi}[e[2],e[3],e[3]]];\\
\text{eq7}\text{:=}\text{Simplify}[\text{codazzi}[e[1],e[2],e[3]]];
\text{eq8}\text{:=}\text{Simplify}[\text{codazzi}[e[2],e[3],e[1]]];\\
\text{eq9}\text{:=}\text{Simplify}[\text{codazzi}[e[3],e[1],e[2]]];\)
\noindent\(\text{eq}\text{:=}\{\text{eq1},\text{eq2},\text{eq3},\text{eq4},\text{eq5},\text{eq6},\text{eq7},\text{eq8},\text{eq9}\}; \text{eq}\)
\subsubsection*{1. case: u${}^{\wedge}$2$\neq $4 s${}^{\wedge}$2}
{\bf conclusions from eq1,2,4:}
\noindent\(\text{Simplify}[\text{eq2}[[1]]-2 \text{eq4}[[1]]]\)
\noindent\(\text{Simplify}[\text{eq1}[[3]]+ \text{eq2}[[2]]]\)
\noindent\(\text{a13}=0; \text{a32}=-\text{a23};\)
\noindent\(\text{eq}\)
\noindent {\bf conclusions from eq1,2,6:}
\noindent\(\text{Simplify}[-2 \text{eq6}[[1]]+ \text{eq1}[[1]]]\)
\noindent\(\text{Simplify}[\text{eq1}[[2]]- \text{eq2}[[3]]]\)
\noindent\(\text{a12}=0; \text{a33}=\text{a22};\)
\noindent\(\text{eq}\)
\noindent\(\text{Clear}[\text{a13},\text{a12},\text{a32},\text{a33}]\)
\subsubsection*{2. case: u=2s$\neq $0}
\noindent\(u=2s;\ \text{eq}\)
\noindent {\bf conclusions from eq8, eq1:}
\noindent\(\text{a32}= \text{a23}; \text{a13}=- 2 \text{a23};\text{eq}\)
\noindent {\bf conclusions from eq3:}
\noindent\(\text{b13}=0; \text{eq}\)
\noindent\(\text{Simplify}[ \text{eq1}[[2]]- \text{eq2}[[3]]]\)
\noindent\(\text{a12}=-(\text{a33}- \text{a22}); \text{eq}\)
\noindent\(\text{Simplify}[ \text{eq3}[[2]]-1/2 \text{eq1}[[1]]+2 \text{eq1}[[2]]]\)
\noindent\(\text{a33}=-\text{a22}; \text{eq}\)
It follows that T(a4)=0, V(a4)=-4a22 a4, W(a4)=4a23 a4.
\noindent\(\text{Simplify}[ \text{eq4}[[2]]+\text{eq2}[[1]]]\)
\noindent\(\text{Simplify}[ \text{eq4}[[3]]+ \text{eq1}[[1]]]\)
\noindent\(\text{b23}=- \text{a23}; \text{b33}=- \text{a22};\)
\subsection*{Lemma 4}
\subsubsection*{Gauss for Levi-Civita connection (af\/f\/ine hypersphere)}
\noindent\(\text{gaussLC}[\text{x$\_$},\text{y$\_$},\text{z$\_$}]\text{:=}\text{Na}[x,\text{Na}[y,z]]-\text{Na}[y,\text{Na}[x,z]]-
\text{Na}[\text{Na}[x,y]-\text{Na}[y,x],z]-H h[y,z]x+H h[x,z]y+K[x,K[y,z]]-
K[y,K[x,z]];\)
\noindent\(\text{eq11}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[2],e[2]]];
\text{eq12}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[3],e[2]]];\\
\text{eq13}\text{:=}\text{Simplify}[\text{gaussLC}[e[2],e[3],e[2]]];
\text{eq14}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[2],e[1]]];\\
\text{eq15}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[3],e[1]]];
\text{eq16}\text{:=}\text{Simplify}[\text{gaussLC}[e[2],e[3],e[1]]];\\
\text{eq17}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[2],e[3]]];
\text{eq18}\text{:=}\text{Simplify}[\text{gaussLC}[e[1],e[3],e[3]]];\\
\text{eq19}\text{:=}\text{Simplify}[\text{gaussLC}[e[2],e[3],e[3]]];\)
\noindent\(\text{eqG}\text{:=}\{\text{eq11},\text{eq12},\text{eq13},\text{eq14},\text{eq15},\text{eq16},\text{eq17},\text{eq18},\text{eq19}\};
\)
\subsubsection*{2. case: u=2s $\neq $0}
\noindent\(\text{eqG}\)
\noindent\(\text{Simplify}[\text{eq11}[[1]]-\text{eq12}[[3]]]\)
\noindent\(\text{Simplify}[\text{eq15}[[3]]+\text{eq12}[[3]]]\)
Contradiction to eq13.3
\noindent\(\text{Clear}[\text{b33},\text{b23},\text{a33},\text{a12},\text{b13},\text{a32},\text{a13},u]\)
\subsubsection*{1. case: u${}^{\wedge}$2$\neq $4 s${}^{\wedge}$2}
\noindent\(\text{a13}=0; \text{a32}=-\text{a23};\text{a12}=0; \text{a33}=\text{a22};\text{eqG}\)
\subsection*{Acknowledgements}
Partially supported by the DFG-Project PI 158/4-5
`Geometric Problems and Special PDEs'.
\pdfbookmark[1]{References}{ref} | {"config": "arxiv", "file": "0910.3609/sigma09-097.tex"} |
\begin{document}
\maketitle
\begin{abstract}
Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in the standard EKI.
This paper proposes a strategy to implement $l_p, 0<p\leq 1,$ regularization for EKI to recover sparse structures in the solution. The strategy transforms a $l_p$ problem into a $l_2$ problem, which is then solved by Tikhonov EKI.
The transformation is explicit, and thus the proposed approach has a computational cost comparable to Tikhonov EKI. We validate the proposed approach's effectiveness and robustness through a suite of numerical experiments, including compressive sensing and subsurface flow inverse problems.
\end{abstract}
\section{Introduction}
A wide range of problems in science and engineering are formulated as inverse problems.
Inverse problems aim to estimate a quantity of interest from noisy, imperfect observation or measurement data, such as state variables or a set of parameters that constitute a forward model.
Examples include deblurring and denoising in image processing \cite{CS}, recovery of permeability in subsurface flow using pressure fields \cite{OIL}, and training a neural network in machine learning \cite{NN,MLEKI} to name a few. In this paper, we consider the inverse problem of finding $u\in\mathbb{R}^N$ from measurement data $y\in\mathbb{R}^m$ where $u$ and $y$ are related as follows
\begin{equation}\label{eq:inverse}
y=G(u)+\eta.
\end{equation}
Here $G:\mathbb{R}^N\to\mathbb{R}^m$ is a forward model that can be nonlinear and computationally expensive to solve, for example, solving a PDE problem. The last term $\eta$ is a measurement error. The measurement error is unknown in general, but we assume that it is drawn from a known probability distribution, a Gaussian distribution with mean zero and a known covariance $\Gamma$. By assuming that the forward model $G$ and the observation covariance $\Gamma$ are known, the unknown variable $u$ is estimated by solving an optimization problem
\begin{equation}\label{eq:optimization}
\argmin_{u\in \mathbb{R}^N} \frac{1}{2}\|y-G(u)\|^2_{\Gamma},
\end{equation}
where $\|\cdot\|_{\Gamma}$ is the norm induced from the inner product using the inverse of the covariance matrix $\Gamma$, that is $\|a\|_{\Gamma}^2=\langle a,\Gamma^{-1}a\rangle$ for the standard inner product $\langle,\rangle$ in $\mathbb{R}^m$.
Ensemble Kalman inversion (EKI), pioneered in the oil industry \cite{OIL} and mathematically formulated in an application-neutral setting in \cite{EKI}, is a derivative-free method that lies between the deterministic and the probabilistic approaches for inverse problems.
EKI's key feature is an iterative application of the Kalman update of the ensemble-based Kalman filters \cite{EAKF, EnKF}.
Ensemble-based Kalman filters are well known for their success in numerical weather prediction, stringent inverse problems involving high-dimensional systems.
EKI iterates the ensemble-based Kalman update in which the ensemble mean converges to the solution of the optimization problem \cref{eq:optimization}.
EKI can be thought of as a least-squares method in which the derivatives are approximated from an empirical correlation of an ensemble \cite{statisticalderivative}, not from a variational approach. Thus, EKI is highly parallelizable without calculating the derivatives related to the forward or the adjoint problem used in the gradient-based methods.
Inverse problems are often ill-posed, which suffer from non-uniqueness of the solution and lack stability.
Also, in the context of regression, the solution can show overfitting.
A common strategy to overcome ill-posed problems is regularizing the solution of the optimization problem \cite{regularization}.
That is, a special structure of the solution from prior information, such as sparsity, is imposed to address ill-posedness.
The standard EKI \cite{EKI} implements regularization by restricting the ensemble to the linear span of the initial ensemble reflecting prior information.
The ensemble-based Kalman update is known for that the ensemble remains in the linear span of the initial ensemble \cite{span,EKI}.
Thus, the EKI ensemble always stays in the linear span of the initial ensemble, which regularizes the solution.
Although this approach shows robust results in certain applications, numerical evidence demonstrates that overfitting may still occur \cite{EKI}.
As an effort to address the overfitting of the standard EKI, an iterative regularization method has been proposed in \cite{iterativeregularization}, which approximates the regularizing Levenberg-Marquardt scheme \cite{LM}.
As another regularization approach using a penalty term to the objective function, a recent work called Tikhonov EKI (TEKI) \cite{TEKI} implements the Tikhonov regularization (which imposes a $l_2$ penalty term to the objective function) using an augmented measurement model that adds artificial measurements to the original measurement.
TEKI's implementation is a straightforward modification of the standard EKI method with a marginal increase in the computational cost.
The regularization methods for EKI mentioned above address several issues of ill-posed problems, including overfitting. However, it is still an open problem to implement other types of regularizers, such as $l_1$ or total variation (TV) regularization.
This paper aims to implement $l_p, 0<p\leq 1$, regularization to recover sparse structures in the solution of inverse problems.
In other words, we propose a highly-parallelizable derivative-free method that solves the following $l_p$ regularized optimization problem
\begin{equation}\label{eq:lpinu}
\argmin_{u\in X}\frac{\lambda}{2}\|u\|_p^p+\frac{1}{2}\|y-G(u)\|^2_{\Gamma},
\end{equation}
where $\|u\|_p$ is the $l_p$ norm of u, i.e., $\sum_i^N |u_i|^p$, and $\lambda$ is a regularization coefficient.
The proposed method's key idea is a transformation of variables that converts the $l_p$ regularization problem to the Tikhonov regularization problem. Therefore, a local minimizer of the original $l_p$ problem can be found by a local minimizer of the $l_2$ problem that is solved using the idea of Tikhonov EKI.
As this transformation is explicit and easy to calculate, the proposed method's overall computational complexity remains comparable to the complexity of Tikhonov EKI.
In general, a transformed optimization problem can lead to additional difficulties, such as change of convexity, increased nonlinearity, additional/missing local minima of the original problem, etc. \cite{practicaloptimization}.
We show that the transformation does not add or remove local minimizers in the transformed formulation. A work imposing sparsity in EKI has been reported recently \cite{sparseEKI}. The idea of this work is to use thresholding and a $l_1$ constraint to impose sparsity in the inverse problem solution. The $l_1$ constraint is further relaxed by splitting the solution into positive and negative parts. The split converts the $l_1$ problem to a quadratic problem, while it still has a non-negativity constraint. On the other hand, our method does not require additional constraints by reformulating the optimization problem and works as a solver for the $l_p$ regularized optimization problem \cref{eq:lpinu}.
This paper is structured as follows. \Cref{sec:EKI} reviews the standard EKI and Tikhonov EKI.
In \cref{sec:lpEKI}, we describe a transformation that converts the $l_p$ regularization problem \cref{eq:lpinu}, $0<p\leq 1$, to the Tikhonov (that is, $l_2$) regularization problem, and provide the complete description of the $l_p$ regularized EKI algorithm. We also discuss implementation and computation issues.
\Cref{sec:tests} is devoted to the validation of the effectiveness and robustness of regularized EKI through a suite of numerical tests. The tests include a scalar toy problem with an analytic solution, a compressive sensing problem to benchmark with a convex $l_1$ minimization method, and a PDE-constrained nonlinear inverse problem from subsurface flow. We conclude this paper in \cref{sec:conclusion}, discussing the proposed method's limitations and future work.
\section{Ensemble Kalman inversion}\label{sec:EKI}
The $l_p$ regularized EKI uses a change of variables to transform a $l_p$ problem into a $l_2$ problem, which is then solved by the standard EKI using an augmented measurement model.
This section reviews the standard EKI and the application of the augmented measurement model in Tikhonov EKI to implement $l_2$ regularization. The review is intended to be concise, delivering the minimal ideas for the $l_p$ regularized EKI. Detailed descriptions of the standard EKI and the Tikhonov EKI methods can be found in \cite{EKI} and \cite{TEKI}, respectively.
\subsection{Standard ensemble Kalman inversion}\label{subsec:EKI}
EKI incorporates an artificial dynamics, which corresponds to the application of the forward model to each ensemble member. This application moves each ensemble member to the measurement space, which is then updated using the ensemble Kalman update formula.
The ensemble updated by EKI stays in the linear span of the initial ensemble \cite{EKI, span}. Therefore, by choosing an initial ensemble appropriately for prior information, EKI is regularized as the ensemble is restricted to the linear span of the initial ensemble.
Under a continuous-time limit, when the operator $G$ is linear, it is proved in \cite{EKIanalysis} that EKI estimate converges to the solution of the following optimization problem
\begin{equation}
\argmin_{u\in\mathbb{R}^N}\frac{1}{2}\|y-G(u)\|_{\Gamma}^2.
\end{equation}
In this paper, we consider the discrete-time EKI in \cite{EKI}, which is described below.
\textbf{Algorithm: standard EKI}\\
Assumption: an initial ensemble of size $K$, $\{u_0^{(k)}\}_{k=1}^K$ from prior information, is given.\\
For $n=1,2,...,$
\begin{enumerate}
\item Prediction step using the artificial dynamics:
\begin{enumerate}
\item Apply the forward model $G$ to each ensemble member
\begin{equation}
g_n^{(k)}:=G(u_{n-1}^{(k)})
\end{equation}
\item From the set of the predictions $\{g_n^{(k)}\}_{k=1}^K$, calculate the mean and covariances
\begin{equation}\label{eq:samplemean}
\overline{g}_n=\frac{1}{K}\sum_{k=1}^Kg_n^{(k)},
\end{equation}
\begin{equation}\label{eq:samplecovariance}
\begin{split}
C^{ug}_n&=\frac{1}{K}\sum_{k=1}^K(u_n^{(k)}-\overline{u}_n)\otimes(g_n^{(k)}-\overline{g}_n),\\
C^{gg}_n&=\frac{1}{K}\sum_{k=1}^K(g_n^{(k)}-\overline{g}_n)\otimes(g_n^{(k)}-\overline{g}_n),
\end{split}
\end{equation}
where $\overline{u}_n$ is the mean of $\{u_n^{(k)}\}$, i.e., $\displaystyle\frac{1}{K}\sum_{k=1}^Ku_n^{(k)}$.
\end{enumerate}
\item Analysis step:
\begin{enumerate}
\item Update each ensemble member $u_n^{(k)}$ using the Kalman update
\begin{equation}\label{eq:ensembleupdate}
u_{n+1}^{(k)}=u_{n}^{(k)}+C^{ug}_n(C^{gg}_n+\Gamma)^{-1}(y_{n}^{(k)}-g_n^{(k)}),
\end{equation}
where $y_{n+1}^{(k)}=y+\zeta_{n+1}^{(k)}$ is a perturbed measurement using Gaussian noise $\zeta_{n+1}^{(k)}$ with mean zero and covariance $\Gamma$.
\item Compute the mean of the ensemble as an estimate for the solution
\begin{equation}
\overline{u}_{n+1}=\frac{1}{K}\sum_{k=1}^Ku_n^{(k)}
\end{equation}
\end{enumerate}
\end{enumerate}
\begin{remark}
The term $C^{ug}_n(C^{gg}_n+\Gamma)^{-1}$ in \cref{eq:ensembleupdate} is from the Kalman gain matrix. The standard EKI uses an extended space, $(u,G(u))\in\mathbb{R}^{N+m}$, and then use the Kalman update for the extended space variable. However, as we need to update only $u$ while $G(u)$ is subordinate to $u$, we have the update formula \cref{eq:ensembleupdate}.
\end{remark}
\subsection{Tikhonov ensemble Kalman inversion}\label{subsec:TKEI}
EKI is regularized through the initial ensemble reflecting prior information. However, there are several numerical evidence showing that EKI regularized only through an ensemble may have overfitting \cite{EKI}.
Among other approaches to regularize EKI, Tikhonov EKI \cite{TEKI} uses the idea of an augmented measurement to implement $l_2$ regularization, which is a simple modification of the standard EKI.
For the original measurement $y$, the augmented measurement model extends $y$ by adding the zero vector in $\mathbb{R}^N$, which yields an augmented measurement vector $z\in\mathbb{R}^{m+N}$
\begin{equation}\label{eq:augmentedmeasurement}
\mbox{augmented measurement vector: }z=(y,0).
\end{equation}
The forward model is also augmented to account for the augmented measurement vector, which adds the identity measurement
\begin{equation}\label{eq:augmentedforward}
\mbox{augmented forward model: }F(u)=(G(u), u).
\end{equation}
Using the augmented measurement vector and the model, Tikhonov EKI has the following inverse problem of estimating $u$ from $z$
\begin{equation}
z=F(u)+\zeta.
\end{equation}
Here $\zeta$ is a $m+N$-dimensional measurement error for the augmented measurement model, which is Gaussian with mean zero and covariance
\begin{equation}\label{eq:augmentedcovariance}
\Sigma=\begin{pmatrix}\Gamma&0\\0&\frac{1}{\lambda}I_{N}\end{pmatrix},
\end{equation}
for the $N\times N$ identity matrix $I_N$.
The mechanism enabling the $l_2$ regularization in Tikhonov EKI is the incorporation of the $l_2$ penalty term as a part of the augmented measurement model. From the orthogonality between different components in $\mathbb{R}^{m+N}$, we have
\begin{equation}
\begin{split}
\frac{1}{2}\|z-F(u)\|^2_{\Sigma}&=\frac{1}{2}\|y-G(u)\|^2_{\Gamma}+\frac{1}{2}\|0-u\|^2_{\frac{1}{\lambda}I_N}\\
&=\frac{1}{2}\|y-G(u)\|^2_{\Gamma}+\frac{\lambda}{2}\|u\|^2_2.\\
\end{split}
\end{equation}
Therefore, the standard EKI algorithm applied to the augmented measurement minimizes $\frac{1}{2}\|z-F(u)\|^2_{\Sigma}$, which equivalently minimizes the $l_2$ regularized problem.
\section{$l_p$-regularization for EKI}\label{sec:lpEKI}
This section describes a transformation that converts a $l_p, 0<p\leq 1,$ regularization problem to a $l_2$ regularization problem. $l_p$-regularized EKI ($l_p$EKI), which we completely describe in \cref{subsec:lpEKI}, utilizes this transformation and solves the transformed $l_2$ regularization problem using the idea of Tikhonov EKI \cite{TEKI}, the augmented measurement model.
\subsection{Transformation of $l_p$ regularization into $l_2$ regularization}\label{subsec:transformation}
For $0<p\leq 1$, we define a function $\psi:\mathbb{R}\to\mathbb{R}$ given by
\begin{equation}\label{eq:utovcomponent}
\psi(x)=\sgn(x)|x|^{\frac{p}{2}}, \quad x\in\mathbb{R}.
\end{equation}
Here $\sgn(x)$ is the sign function of $x$, which has 1 for $x>0$, 0 for $x=0$, and -1 for $x<0$. It is straightforward to check that $\psi$ is bijective and has an inverse $\xi:\mathbb{R}\to\mathbb{R}$ defined as
\begin{equation}\label{eq:vtoucomponent}
\xi(x)=\sgn(x)|x|^{\frac{2}{p}}, \quad x\in\mathbb{R}.
\end{equation}
For $u$ in $\mathbb{R}^N$, we define a nonlinear map $\Psi:\mathbb{R}^N\to\mathbb{R}^N$, which applies $\psi$ to each component of $u=(u_1,u_2,...,u_N)$,\begin{equation}\label{eq:utov}
\Psi(u)=(\psi(u_1),\psi(u_2),...,\psi(u_N)).
\end{equation}
As $\psi$ has an inverse, the map $\Psi$ also has an inverse, say $\Xi$
\begin{equation}\label{eq:vtou}
\Xi(u)=\Psi^{-1}(u)=(\xi(u_1),\xi(u_2),...,\xi(u_N)).
\end{equation}
For $v=\Psi(u)$, it can be checked that for each $i=1,2,...,N$,
\[|v_i|^2=|\psi(u_i)|^2=|u_i|^{p},\]
and thus we have the following norm relation
\begin{equation}\label{eq:normequivalence}
\|v\|_2^2=\|u\|_p^p.
\end{equation}
This relation shows that the map $v=\Psi(u)$ converts the $l_p$-regularized optimization problem in $u$ \cref{eq:lpinu} to a $l_2$ regularized problem in $v$,
\begin{equation}\label{eq:l2inv}
\argmin_{v\in \mathbb{R}^N}\frac{\lambda}{2}\|v\|_2^2+\frac{1}{2}\|y-\tilde{G}(v)\|^2_{\Gamma},
\end{equation}
where $\tilde{G}$ is the pullback of $G$ by $\Xi$
\begin{equation}
\tilde{G}=G\circ \Xi.
\end{equation}
A transformation between $l_1$ and $l_2$ regularization terms has already been used to solve an inverse problem in the Bayesian framework \cite{l1RTO}. In the context of the randomize-then-optimize framework \cite{RTO}, the method in \cite{l1RTO} draws a sample from a Gaussian distribution, which is then transformed to a Laplace distribution. As this method needs to match the corresponding densities of the variables (the original and the transformed variables) as random variables, the transformation involves calculations related to cumulative distribution functions. For the scalar case, $v\in\mathbb{R}$, the transformation from $l_2$ to $l_1$, denoted as $gl$, is given by
\begin{equation}\label{eq:l1rto}
gl(v)=-\sgn(v)\log\left(1-2\left|\phi(v)-\frac{1}{2}\right|\right).
\end{equation}
where $\phi(u)$ is the cumulative distribution function of the standard Gaussian distribution.
\cref{fig:transformations} shows the two transformations $\xi$ \cref{eq:vtoucomponent} and $gl$ \cref{eq:l1rto}; the former is based on the norm relation \cref{eq:normequivalence} and the latter is based on matching densities as random variables.
\begin{figure}[!ht]
\centering
\includegraphics[width=.5\textwidth]{./fig_transformations.png}
\caption{$\xi$: transformation matching the norm relation \cref{eq:normequivalence}, $gl$: transformation from Gaussian to Laplace distributions.}\label{fig:transformations}
\end{figure}
We note that the transformation $\xi$ has a region around 0 flatter than the transformation $gl$, but $\xi$ diverts quickly as $v$ moves further away from $0$. From this comparison, we expect that the flattened region of $\xi$ plays another role in imposing sparsity by trapping the ensemble to the flattened area.
In general, a reformulation of an optimization problem using a transformation has the following potential issues \cite{practicaloptimization}: i) the degree of nonlinearity may be significantly increased, ii) the desired minimum may be inadvertently excluded, or iii) an additional local minimum can be included.
In \cite{EKIconvergence}, for a non-convex problem, it is shown that TEKI converges to an approximate local minimum if the gradient and Hessian of the objective function are bounded. It is straightforward to check that the transformed objective function has bounded gradient and Hessian if $0<p\leq 1$ regardless of the convexity of the problem. Therefore, if we can show that the original and the transformed problems have the same number of local minima, then it is guaranteed to find a local minimum of the original problem by finding a local minimum of the transformed problem using TEKI. We want to note the importance of the sign function in defining $\psi$ and $\xi$. The sign function is not necessary to satisfy the norm relation \cref{eq:normequivalence}, but it is essential to make the transformation $\Psi$ and its inverse $\Xi$ bijective. Without being bijective, the transformed $l_2$ problem can have more or less local minima than the original problem.
The following theorem shows that the transformation does not add or remove local minima.
\begin{theorem} For an objective function $J(u):\mathbb{R}^N\to\mathbb{R}$, if $u^*$ is a local minimizer of $J(u)$, $\Psi(u^*)$ is also a local minimizer of $\tilde{J}(v)=J\circ\Xi(v)$. Similarly, if $v^*$ is a local minimizer of $\tilde{J}(v)$, then $\Xi(v^*)$ is also a local minimizer of $J(u)=\tilde{J}\circ \Psi(u)$.
\end{theorem}
\begin{proof}
From the definition \cref{eq:utov} and \cref{eq:vtou}, $\Psi$ and $\Xi$ are continuous and bijective. Thus for $u\in\mathbb{R}^N$, both $\Psi$ and $\Xi$ map a neighborhood of $u\in\mathbb{R}^N$ to neighborhoods of $\Psi(u)$ and $\Xi(u)$, respectively. As $u^*$ is a local minimizer, there exists a neighborhood $\mathcal{N}$ of $u^*$ such that
\begin{equation}
J(u^*)\leq J(w)\quad \mbox{for all }w\in\mathcal{N}.
\end{equation}
Let $v=\Psi(u^*)$ and $\mathcal{M}:=\Psi(\mathcal{N})$ that is a neighborhood of $v$. For any $w\in\mathcal{M}$, $\Xi(w)\in\mathcal{N}$ and thus we have
\begin{equation}
\tilde{J}(v)=J(\Xi(v))=J(u)\leq J(\Xi(w))=\tilde{J}(w),
\end{equation}
which shows that $v$ is a local minimizer of $\tilde{J}$. The other direction is proved similarly by changing the roles of $\Psi$ and $\Xi$ and of $J$ and $\tilde{J}$.
\end{proof}
We note that an insolated local minimizer can replace the local minimizer in the theorem.
If there is a unique global minimizer of the $l_p$ regularization problem \cref{eq:lpinu}, the theorem guarantees that we can find it by finding the global minimizer of the $l_2$ regularized problem \cref{eq:l2inv}.
\begin{corollary}
For $0<p\leq 1$, if the $l_p$ regularized optimization \cref{eq:lpinu} has a unique global minimizer, say $u^{\dag}$, the $l_2$ regularized optimization \cref{eq:l2inv} also has a unique global minimizer. By finding the minimizer $u^{\dag}$ of \cref{eq:l2inv}, say $v^{\dag}$, $u^{\dag}$ is given by
\begin{equation}u^{\dag}=\Xi(v^{\dag}).
\end{equation}
\end{corollary}
\subsection{Algorithm}\label{subsec:lpEKI}
$l_p$-regularized EKI ($l_p$EKI) solves the transformed $l_2$ regularization problem using the standard EKI with the augmented measurement model. For the current study's completeness to implement $l_p$EKI, this subsection describes the complete $l_p$EKI algorithm and discuss issues related to implementation. Note that the Tikhonov EKI (TEKI) part in $l_p$EKI is slightly modified to reflect the setting assumed in this paper. The general TEKI algorithm and its variants can be found in \cite{TEKI}.
We assume that the forward model $G$ and the measurement error covariance $\Gamma$ are known, and measurement $y\in\mathbb{R}^m$ is given (and thus $z=(y,0)$ is also given). We also fix the regularization coefficient $\lambda$ and $p$. Under this assumption, $l_p$EKI uses the following iterative procedure to update the ensemble until the ensemble mean $\overline{v}=\displaystyle\frac{1}{K}\sum_{k=1}^Kv^{(k)}$ converges.
\vspace{0.02\textwidth}
\textbf{Algorithm: $l_p$-regularized EKI}\\
Assumption: an initial ensemble of size $K$, $\{v_0^{(k)}\}_{k=1}^K$, is given.\\
For $n=1,2,...,$
\begin{enumerate}
\item Prediction step using the forward model:
\begin{enumerate}
\item Apply the augmented forward model $F$ to each ensemble member
\begin{equation}
f_n^{(k)}:=F(v_n^{(k)})=(\tilde{G}(v_n^{(k)}), v_n^{(k)})
\end{equation}
\item From the set of the predictions $\{f_n^{(k)}\}_{k=1}^K$, calculate the mean and covariances
\begin{equation}\label{eq:rEKI:samplemean}
\overline{f}_n=\frac{1}{K}\sum_{k=1}^Kf_n^{(k)},
\end{equation}
\begin{equation}\label{eq:rEKI:samplecovariance}
\begin{split}
C^{vf}_n&=\frac{1}{K}\sum_{k=1}^K(v_n^{(k)}-\overline{v}_n)\otimes(f_n^{(k)}-\overline{f}_n),\\
C^{ff}_n&=\frac{1}{K}\sum_{k=1}^K(f_n^{(k)}-\overline{f}_n)\otimes(f_n^{(k)}-\overline{f}_n)
\end{split}
\end{equation}
where $\overline{v}_n$ is the ensemble mean of $\{v_n^{(k)}\}$, i.e., $\displaystyle\frac{1}{K}\sum_{k=1}^Kv_n^{(k)}$.
\end{enumerate}
\item Analysis step:
\begin{enumerate}
\item Update each ensemble member $v_n^{(k)}$ using the Kalman update
\begin{equation}\label{eq:rEKI:ensembleupdate}
v_{n+1}^{(k)}=v_{n}^{(k)}+C^{vf}_n(C^{ff}_n+\Sigma)^{-1}(z_{n+1}^{(k)}-f_n^{(k)}),
\end{equation}
where $z_{n+1}^{(k)}=z+\zeta_{n+1}^{(k)}$ is a perturbed measurement using Gaussian noise $\zeta_{n+1}^{(k)}$ with mean zero and covariance $\Sigma$.
\item For the ensemble mean $\overline{v}_n$, the $l_p$EKI estimate, $u_n$, for the minimizer of the $l_p$ regularization is given by
\begin{equation}\label{eq:rEKI:finalestimate}
u = \Xi(\overline{v}_n).
\end{equation}
\end{enumerate}
\end{enumerate}
\begin{remark} In EKI and TEKI, the covariance of $\zeta_{n+1}^{(k)}$ can be set to zero so that all ensemble member uses the same measurement $z$ without perturbations. In our study, we focus on the perturbed measurement using the covariance matrix $\Gamma$.
\end{remark}
\begin{remark} The above algorithm is equivalent to TEKI, except that the forward model $G$ is replaced with the pullback of $G$ by the transformation $\Xi$. In comparison with TEKI, the additional computational cost for $l_p$EKI is to calculate the Transformation $\Xi(v)$. In comparison with the standard EKI, the additional cost of $l_p$EKI, in addition to the cost related to the transformation, is the matrix inversion $(C^{gg}_n+\Sigma)^{-1}$ in the augmented measurement space $\mathbb{R}^{m+N}$ instead of a matrix inversion in the original measurement space $\mathbb{R}^m$. As the covariance matrices are symmetric positive definite, the matrix inversion can be done efficiently.
\end{remark}
\begin{remark} In $l_p$EKI, it is also possible to consider estimating $u$ by transforming each ensemble member and take average of the transformed members, that is,
\begin{equation}\label{eq:rEKI:finalestimate2}
u=\frac{1}{K}\sum_{k=1}^K \Xi(v_n^{(k)})
\end{equation}
instead of \cref{eq:rEKI:finalestimate}. If the ensemble spread is large, these two approaches will make a difference. In our numerical tests in the next section, we do not incorporate covariance inflation. Thus the ensemble spread becomes relatively small when the estimate converges, and thus \cref{eq:rEKI:finalestimate} and \cref{eq:rEKI:finalestimate2} are not significantly different. In this study, we use \cref{eq:rEKI:finalestimate} to measure the performance of $l_p$EKI.
\end{remark}
In recovering sparsity using the $l_p$ penalty term, if the penalty term's convexity is not necessary, it is preferred to use a small $p<1$ as a smaller $p$ imposes stronger sparsity. The transformation in $l_p$EKI works for any positive $p$, but the transformation can lead to an overflow for a small $p$; the function $\xi$ depends on an exponent $\frac{2}{p}$ that becomes large for a small $p$. Therefore, there is a limit for the smallest $p$. In our numerical experiments in the next section, the smallest $p$ is 0.7 in the compressive sensing test.
There is a variant of $l_p$EKI worth further consideration. In \cite{EKIanalysis}, a continuous-time limit of EKI has been proposed, which rescales $\Gamma\to h^{-1}\Gamma$ using $h>0$ so that the matrix inversion $(C^{gg}_n+h^{-1}\Gamma)^{-1}$ is approximated by $h\Gamma^{-1}$ as a limit of $h\to 0$. In many applications, the measurement error covariance is assumed to be diagonal. That is, the measurement error corresponding to different components are uncorrelated. Thus the inversion $\Gamma^{-1}$ becomes a cheap calculation in the continuous-time limit. The continuous-time limit is then discretized in time using an explicit time integration method with a finite time step. The latter is called `learning rate' in the machine learning community, and it is known that an adaptive time-stepping to solve an optimization often shows improved results \cite{adaptivelearningrate,adaptivestepping}. The current study focuses on the discrete-time update described in \cref{eq:ensembleupdate} and we leave adaptive time-stepping for future work.
\section{Numerical tests}\label{sec:tests}
We apply $l_p$-regularized EKI ($l_p$EKI) to a suite of inverse problems to check its performance in regularizing EKI and recovering sparse structures of solutions.
The tests include: i) a scalar toy model where an analytic solution is available, ii) a compressive sensing problem to recover a sparse signal from random measurements of the signal, iii) an inverse problem in subsurface flow; estimation of permeability from measurements of hydraulic pressure field whose forward model is described by a 2D elliptic partial differential equation \cite{Darcy, OIL}. In all tests, we run $l_p$EKI for various values of $p\leq 1$, and compare with the result of Tikhonov EKI. We analyze the results to check how effectively $l_p$EKI implements $l_p$ regularization and recover sparse solutions. When available, we also compare $l_p$EKI with a $l_1$ convex minimization method. As quantitative measures for the estimation performance, we calculate the $l_1$ error of the $l_p$EKI estimates and the data misfit $\|y-G(u)\|_{2}$.
Several parameters are to be determined in $l_p$EKI to achieve robust estimation results, the regularization coefficient $\lambda$, ensemble size, and its initialization. The regularization coefficient can be selected, for example, using cross-validation. As the coefficient can significantly affect the performance, we find the coefficient by hand-tuning so that $l_p$EKI achieves the best result for a given $p$.
Ensemble initialization plays a role in regularizing EKI, restricting the estimate to the linear span of the initial ensemble.
In our experiments, instead of tuning the initial ensemble for improved results,
we initialize the ensemble using a Gaussian distribution with mean zero and a constant diagonal covariance matrix (the variance will be specified later for each test). As this initialization does not utilize any prior information, a sparse structure in the solution, we regularize the solution mainly through the $l_p$ penalty term.
For each test, we run 100 trials of $l_p$EKI through 100 realizations of the initial ensemble distribution and use the estimate averaged over the trials along with its standard deviation to measure the performance difference.
Regarding the ensemble size, for the scalar toy and the compressive sensing problems, we test ensemble sizes larger than the dimension of $u$, the unknown variable of interest. The purpose of a large ensemble size is to minimize the sampling error while we focus on the regularization effect of $l_p$EKI. To show the applicability of $l_p$EKI for high-dimensional problems, we also test a small ensemble size using the idea of multiple batches used in \cite{batch}. The multiple batch approach runs several batches where small magnitude components are removed after each batch. After removing small magnitude components from the previous batch, the ensemble is used for the next batch. The multiple batch approach enables a small ensemble size, 50 ensemble members, for the compressive sensing and the 2D elliptic inversion problems where the dimensions of $u$ are 200 and 400, respectively.
In ensemble-based Kalman filters, covariance inflation is an essential tool to stabilize and improve the performance of the filters. In a connection with the inflation, an adaptive time-stepping has been investigated to improve the performance of EKI. Although the adaptive time-stepping can be incorporated in $l_p$EKI for performance improvements, we use the discrete version $l_p$EKI described in \cref{subsec:lpEKI} focusing on the effect of different types of regularization on inversion. We will report a thorough investigation along the line of adaptive time-stepping in another place.
\subsection{A scalar toy problem}\label{subsec:scalar}
The first numerical test is a scalar problem for $u\in\mathbb{R}$ with an analytic solution.
As this is a scalar problem, there is no effect of regularization from ensemble initialization, and we can see the effect from the $l_p$ penalty term. The scalar optimization problem we consider here is the minimization of an objective function $J(u)=\frac{1}{4}|u|^p+\frac{1}{2}(1-u)^2$
\begin{equation}\label{eq:scalar}
\argmin_{u\in\mathbb{R}}J(u)=\argmin_{u\in\mathbb{R}}\frac{1}{4}|u|^p+\frac{1}{2}(1-u)^2.
\end{equation}
This setup is equivalent to solving the optimization problem \cref{eq:lpinu} using $l_p$ regularization with $\lambda=1/2$, where $y=1$, $G(u)=u$, and $\eta$ is Gaussian with mean zero and variance 1. Using the transformation $v=\Psi(u)=\psi(u)=\sgn(u)|u|^{\frac{p}{2}}$ defined in \cref{eq:utovcomponent}, $l_p$EKI minimizes a transformed objective function $\tilde{J}(v)=\frac{1}{4}|v|^2+\frac{1}{2}(1-\sgn(v)|v|^{2/p})^2$
\begin{equation}\label{eq:scalartransformed}
\argmin_{v\in\mathbb{R}}\tilde{J}(v)=\argmin_{v\in\mathbb{R}}\frac{1}{4}|v|^2+\frac{1}{2}(1-\sgn(v)|v|^{2/p})^2,
\end{equation}
which is an $l_2$ regularization of $\frac{1}{2}(1-\sgn(v)|v|^{\frac{2}{p}})^2$.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\textwidth]{./fig_scalar_objective.png}
\caption{Objective functions of \cref{eq:scalar} and \cref{eq:scalartransformed} for $p=1$ (first row) and $p=0.5$ (second row).}\label{fig:scalarobjective}
\end{figure}
For $p=1$, the first row of \cref{fig:scalarobjective} shows the objective functions of $l_p$ \cref{eq:scalar} and the transformed $l_2$ \cref{eq:scalartransformed} formulations. Each objective function has a unique global minimum without other local minima. The minimizers are $\frac{3}{4}$ and $\frac{\sqrt{3}}{2}$ for $l_1$ and $l_2$, respectively. We can check that the transformation does not add/remove local minimizers, but the convexity of the objective function changes. The transformed objective function $\tilde{J}$ has an inflection point at $u=0$, which is also a stationary point. Note that the original function has no other stationary points than the global minimizer.
When $p=0.5$, a potential issue of the transformation can be seen explicitly. The original objective and the transformed objective functions are shown in the second row of \cref{fig:scalarobjective}. Due to the regularization term with $p=0.5$, the objective functions are non-convex and have a local minimizer at $u=v=0$ in addition to the global minimizers. In the transformed formulation (bottom right of \cref{fig:scalarobjective}), the objective function flattens around $v=0$, which shows a potential issue of trapping ensemble members around $v=0$. Numerical experiments show that if the ensemble is initialized with a small variance, the ensemble is trapped around $v=0$.
On the other hand, if the ensemble is initialized with a sufficiently large variance (so that some of the ensemble members are initialized out of the well around $v=0$), $l_p$EKI shows convergence to the true minimizer, $v=0.9304$ (or $u=0.8656$) even when it is initialized around $0$.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{./fig_scalar_timeseries.png}
\caption{Time series of $l_p$EKI estimate, $\xi(\overline{v}_n)$, which is averaged over 100 different trials. }
\label{fig:scalartimeseries}
\end{figure}
We use 100 different realizations for the ensemble initialization and each trial uses 50 ensemble members. The estimates at each iteration, which is averaged over different trials, are shown in \cref{fig:scalartimeseries}.
For $p=1$ (first row) and $p=0.5$ (second row), the left and right columns show the results when the ensemble is initialized with mean 1 and 0, respectively. When $p=1$ and initialized around $1$, the ensemble estimate quickly converges to the true value $0.75$ as the objective function is convex, and the initial guess is close to the true value. When $p=0.5$, as the objective function is non-convex due to the regularization term, the convergence is slower than the $p=1$ case. When the ensemble is initialized around 0 for $p=0.5$, a local minimizer, the ensemble needs to be initialized with a large variance. Using variance 1, which is 10 times larger than 0.1, the variance for the ensemble initialization around 1, $l_p$EKI converges to the true value. The performance difference between different trials is marginal. The standard deviations of the estimate after 50 iterations are $6.62\times 10^{-3}$ ($p=1$ initialized with 1), $7.95\times 10^{-3}$ ($p=1$ initialized with 0), $8.79\times 10^{-3}$ ($p=0.5$ initialized with 1), and $1.14\times 10^{-2}$ ($p=0.5$ initialized with 0). As a reference, the estimate using the transformation \cref{eq:l1rto} based on matching the densities of random variables converges to 0.71.
\subsection{Compressive sensing}
The second test is a compressive sensing problem. The true signal $u$ is a vector in $\mathbb{R}^{200}$, which is sparse with only four randomly selected non-zero components (their magnitudes are also randomly chosen from the standard normal distribution.)
The forward model $G:\mathbb{R}^{200}\to\mathbb{R}^{20}$ is a random Gaussian matrix of size $20\times 200$, which yields a measurement vector in $\mathbb{R}^{20}$. The measurement $y$ is obtained by applying the forward model to the true signal $u$ polluted by Gaussian noise with mean zero and variance $0.01$. As the forward model is linear, several robust methods can solve the sparse recovery problem, including the $l_1$ convex minimization method \cite{cvx}. This test aims to compare the performance of $l_p$EKI for various $p$ values, rather than to advocate the use of $l_p$EKI over other standard methods. As the forward model is linear and cheap to calculate, the standard methods are preferred over $l_p$EKI for this test.
We first use a large ensemble size, 2000 ensemble members, to run $l_p$EKI. The ensemble is initialized by drawing samples from a Gaussian distribution with mean zero and a diagonal covariance (which yields variance 0.1 for each component).
For $p=1$ and $0.7$, the tuned regularization coefficients, $\lambda$, are 100 and 300. When $p=2$, which corresponds to TEKI, the best result can be obtained using $\lambda$ ranging from 10 to 200; we use the result of $\lambda=50$ to compare with the other cases. For $p=1$, we also compare the result of the convex $l_1$ minimization method using the KKT solver in the Python library CVXOPT \cite{cvxopt}.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\textwidth]{./fig_cs_estimates.png}
\caption{Compressive sensing. Reconstruction of sparse signal using $l_p$EKI for $p$=2, 1, and 0.7. Ensemble size is 2000. The bottom right plot is the reconstruction using the convex $l_1$ minimization method. For the true signal, only the nonzero components are marked}.\label{fig:csestimate}
\end{figure}
\Cref{fig:csestimate} shows the $l_p$EKI estimates after 20 iterations averaged over 100 trials for $p=2$ (top left), $p=1$ (top right), and $p=0.7$ (bottom left), along with the estimate by the convex optimization (bottom right).
As it is well known in compressive sensing, $l_2$ regularization fails to capture the true signal's sparse structure. As $p$ decreases to 1, $l_p$EKI develops sparsity in the estimate, comparable to the estimate of the convex $l_1$ minimization method. The slightly weak magnitudes of the three most significant components by $l_p$EKI improve as $p$ decreases to $0.7$. When $p=0.7$, $l_p$EKI captures the correct magnitudes at the cost of losing the smallest magnitude component. We note that the smallest magnitude component is difficult to capture; the magnitude is comparable to the measurement error $0.1=\sqrt{0.01}$.
\begin{table}[t!]
\centering
\begin{tabular}{|c|c|c|}
\hline
Method&$l_1$ error&data misfit\\
\hline
$p=2$, ens size 2000&14.0802&0.0515\\
$p=1$, ens size 2000&0.7848&0.8018\\
$p=0.7$, ens size 2000&0.2773&1.2737\\
$p=1$, ens size 50&1.6408&1.4095\\
$p=0.7$, ens size 50&0.6027&1.8958\\
$l_1$ convex minimization&0.5623&0.9030\\
\hline
\end{tabular}
\caption{Compressive sensing. $l_p$EKI estimate $l_1$ error and data misfit for $p=2,1$ and $0.7$. }\label{tb:csmisfitl1err}
\end{table}
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\textwidth]{./fig_cs_misfitl1err.png}
\caption{Compressive sensing. $l_1$ error of the $l_p$EKI estimate and data misfit.}\label{fig:csmisfitl1err}
\end{figure}
Another cost of using $p<1$ to impose stronger sparsity than $p=1$ is a slow convergence rate of$l_p$EKI. The time series of the $l_1$ estimation error and the data misfit of $l_p$EKI averaged over 100 trials are shown in \cref{fig:csmisfitl1err} alongside those of the convex optimization method. The results show that $p=0.7$ converges slower than $p=1$ (see Table \cref{tb:csmisfitl1err} for the numerical values of the error and the misfit). Although there is a slowdown in convergence, it is worth noting that $l_p$EKI with $p=0.7$ converges in a reasonably short time, 15 iterations, to achieve the best result. $l_p$EKI with $p=2$ converges fast with the smallest data misfit. However, the $l_2$ regularization is not strong enough to impose sparsity in the estimate and yields the largest estimation error, which is 20 times larger than the case of $p=1$. Note that the convex optimization method has the fastest convergence rate; it converges within three iterations and captures the four nonzero components with slightly smaller magnitudes than $p=0.7$ for the three most significant components.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\textwidth]{./fig_cs_estimates_nsamp50.png}
\caption{Compressive sensing. Reconstruction of sparse signal using $l_p$EKI for p=1 and 0.7. Ensemble size is 50. For the true signal, only the nonzero components are marked.}\label{fig:csestimate_nsamp50}
\end{figure}
The ensemble size 2000 is larger than the dimension of the unknown vector $u$, 200. A large ensemble size can be impractical for a high-dimensional unknown vector. To see the applicability of $l_p$EKI using a small ensemble size, we use 50 ensemble members and two batches following the multiple batch approach \cite{batch}. The first batch runs 10 iterations, and all components whose magnitudes are less than 0.1 (the square root of the observation variance) are removed. The problem's size the second batch solves ranges from 30-45 (depending on a realization of the initial ensemble), which is then solved for another 10 iterations. The estimates using 50 ensemble members for $p=1$ and $p=0.7$ after two batch runs (i.e., 20 iterations) are shown in \cref{fig:csestimate_nsamp50}. Compared with the large ensemble size case, the small ensemble size run also captures the most significant components at the cost of fluctuating components larger than the large ensemble size test. We note that the estimates are averaged over 100 trials, and thus there are components whose magnitudes are less than the threshold value 0.1 used in the multiple batch run.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\textwidth]{./fig_cs_std.png}
\caption{Compressive sensing. Standard deviation of the estimates using 100 trials.}\label{fig:csstdrest}
\end{figure}
As a measure to check the performance difference for different trials, \cref{fig:csstdrest} shows the standard deviations of $l_p$EKI estimates for $p=1$ and $0.7$ after 20 iterations. The first row shows the results using 2000 ensemble members, while the second row shows the ones using 50 ensemble members.
The standard deviations of the large ensemble size are smaller than those of the small ensemble size case as the large ensemble size has a smaller sampling error. In all cases, the standard deviations are smaller than 6\% of the magnitude of the most significant components. In terms of $p$, the standard deviations of $p=0.7$ are smaller than those of $p=1$.
\subsection{2D elliptic problem}
Next, we consider an inverse problem where the forward model is given by an elliptic partial differential equation.
The model is related to subsurface flow described by Darcy flow in the two-dimensional unit square $(0,1)^2\subset\mathbb{R}^2$
\begin{equation}\label{eq:2d}
-\nabla \cdot (k(x)\nabla p(x)) = f(x), \quad x=(x_1,x_2)\in(0,1)^2.
\end{equation}
The scalar field $k(x)>\alpha>0$ is the permeability, and another field $p(x)$ is the piezometric head or the pressure field of the flow. For a known source term $f(x)$, the inverse problem estimates the permeability from measurements of the pressure field $p$. This model is a standard model for an inverse problem in oil reservoir simulations and has been actively used to measure EKI's performance and its variants, including TEKI \cite{EKI, TEKI}.
We follow the same setting used in TEKI \cite{TEKI} for the boundary conditions and the source term.
The boundary conditions consist of Dirichlet and Neumann boundary conditions
\[p(x_1,0)=100, \frac{\partial p}{\partial x_1}(1,x_2)=0, -k\frac{\partial p}{\partial x_1}(0,x_2)=500, \frac{\partial p}{\partial x_2}(x_1,1)=0,\]
and the source term is piecewise constant
\[\displaystyle f(x_1,x_2)=\left\{
\begin{array}{ll}
0&\mbox{if } 0\leq x_2\leq \frac{4}{6},\\
137&\mbox{if }\frac{4}{6}< x_2\leq \frac{5}{6},\\
274&\mbox{if }\frac{5}{6}<x_2\leq 1.
\end{array}\right.\]
A physical motivation of the above configuration can be found in \cite{Darcy}.
We use $15\times 15$ regularly spaced points in $(0,1)^2$ to measure the pressure field with a small measurement error variance $10^{{-6}}$. For a given $k$, the forward model is solved by a FEM method using the second-order polynomial basis on a $60\times 60$ uniform mesh.
In addition to the standard setup, we impose a sparse structure in the permeability. We assume that the
log permeability, $\log k$, can be represented by 400 components in the cosine basis $\phi_{ij}=\cos(i\pi x_1)\cos(j\pi x_2), i,j=0,1,...,19,$
\begin{equation}\log k(x)=\sum_{i,j=0}^{19}u_{ij}\phi_{ij}(x),\end{equation}
where only six of $\{u_{ij}\}$ are nonzero. That is, we assume that the discrete cosine transform of $\log k$ is sparse with only 6 nonzero components out of 400 components.
Thus, the problem we consider here can be formulated as an inverse problem to recover $u=\{u_{ij}\}\in\mathbb{R}^{400}$ (which has only six nonzero components) from a measurement $y\in\mathbb{R}^{225}$, the measurement of $p$ at $15\times 15$ regularly spaced points.
In terms of sparsity reconstruction, the current setup is similar to the previous compressive sensing problem, but the main difference lies in the forward model.
In this test, the forward model is nonlinear and computationally expensive to solve, where the forward model in the compressive sensing test was linear using a random measurement matrix.
For this test, we run $l_p$EKI using only a small ensemble size due to the high computational cost of running the forward model. As in the previous test, we use the multiple batch approach. First, the $l_p$EKI ensemble of size 50 is initialized around zero with Gaussian perturbations of variance 0.1. After the first five iterations, all components whose magnitudes less than $5\times 10^{-3}$ are removed at each iteration. The threshold value is slightly smaller than the smallest magnitude component of the true signal. Over 100 different trials, the average number of nonzero components after 30 iterations is 18 that is smaller than the ensemble size.
\begin{figure}[!htp]
\centering
\subfloat[true]{\includegraphics[width=.98\textwidth]{./fig_2d_utrue.png}}\\
\subfloat[$p=2$]{\includegraphics[width=.98\textwidth]{./fig_2d_up20.png}}\\
\subfloat[$p=1$]{\includegraphics[width=.98\textwidth]{./fig_2d_up10.png}}\\
\subfloat[$p=0.8$]{\includegraphics[width=.98\textwidth]{./fig_2d_up08.png}}
\caption{2D elliptic problem. Left column: the true $u$ and $l_p$EKI estimates for $p=2, 1,$ and $0.8$. Right column: $\log k$ of the true and $l_p$EKI estimates. All plots have the same grey scale. $p=1$ and $0.8$ use the results after 20 iterations while $p=2$ uses the result after 50 iterations. For the true signal, only the nonzero components are marked.}\label{fig:2destimates}
\end{figure}
The true value of $u$ used in this test and its corresponding log permeability, $\log k$, are shown in the first row of \cref{fig:2destimates} ($u$ is represented as a one-dimensional vector by concatenating the row vectors of $\{u_{ij}\}$).
The $l_p$EKI estimates for $p=2,1$, and $0.8$ are shown in the second to the fourth rows of \cref{fig:2destimates}. Here $p=0.8$ was the smallest value we can use for $l_p$EKI due to the numerical overflow in the exponentiation of $\log k$. A smaller $p$ can be used with a smaller variance for ensemble initialization, but the gain is marginal.
The results of $l_p$EKI are similar to the compressive sensing case.
$p=0.8$ has the best performance recovering the four most significant components of $u$.
$p=1$ has slightly weak magnitudes missing the correct magnitudes of the two most significant components (corresponding to one-dimensional indices 141 and 364).
Both cases converge within 20 iterations to yield the best result (see \cref{fig:2dmisfitl1err} and Table \cref{tb:2dmisfitl1err} for the time series and numerical values of the $l_1$ error and data misfit). When $p=2$, $l_p$EKI performs the worst; it has the largest $l_1$ error and data misfit. We note that $p=2$ uses the result after running 50 iterations at which the estimate converges.
\begin{figure}[t!]
\centering
\includegraphics[width=.98\textwidth]{./fig_2d_misfitl1err.png}
\caption{2D elliptic problem. $l_1$ error of the $l_p$EKI estimates and data misfit.}\label{fig:2dmisfitl1err}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=.98\textwidth]{./fig_2d_std.png}
\caption{2D elliptic problem. Standard deviation of the estimates using 100 trials.}\label{fig:2dstd}
\end{figure}
The performance difference between different trials is not significant. The standard deviations of the $l_p$EKI estimates using 100 trials are shown in \cref{fig:2dstd}. The standard deviations for nonzero components are larger than the other components, but the largest standard deviation is less than 3\% of the magnitude of the true signal. As in the compressive sensing test, the deviations are slightly smaller for $p<1$ than $p=1$.
\begin{table}[t!]
\centering
\begin{tabular}{|c|c|c|}
\hline
$p$&$l_1$ error&data misfit\\
\hline
2&21.3389&4.1227\\
1&0.1553&0.5707\\
0.8&0.0719&0.5682\\
\hline
\end{tabular}
\caption{2D elliptic problem. $l_p$EKI estimate $l_1$ error and data misfit for $p=2,1$ and $0.8$. }\label{tb:2dmisfitl1err}
\end{table}
\section{Discussions and conclusions}\label{sec:conclusion}
We have proposed a strategy to implement $l_p, 0<p\leq 1$, regularization in ensemble Kalman inversion (EKI) to recover sparse structures in the solution of an inverse problem.
The $l_p$-regularized ensemble Kalman inversion ($l_p$EKI) proposed here uses a transformation to convert the $l_p$ regularization to the $l_2$ regularization, which is then solved by the standard EKI with an augmented measurement model used in Tikhonov EKI. We showed a one-to-one correspondence between the local minima of the original and the transformed formulations. Thus a local minimum of the original problem can be obtained by finding a local minimum of the transformed problem.
As other iterative methods for non-convex problems, initialization plays a vital role in the proposed method's performance.
The effectiveness and robustness of regularized EKI are validated through a suite of numerical tests, showing robust results in recovering sparse solutions using $p\leq1$.
In implementing $l_p$ regularization for EKI, there is a limit for $p<1$ due to an overflow. One possible workaround is to use a nonlinear augmented measurement model related to the transformation $\Psi$, not the transformation $\Xi$. The nonlinear measurement model is general to incorporate the $l_p$ regularization term directly instead of using the transformed $l_2$ problem. However, this approach lacks a mathematical framework to prevent the inadvertent addition of local minima. This approach is under investigation and will be reported in another place.
For successful applications of $l_p$EKI for high-dimensional inverse problems, it is essential to maintain a small ensemble size for efficiency. In the current study, we considered the multiple batch approach. The approach removes non-significant components after each batch, and thus the problem size (i.e., the dimension of the unknown signal) decreases over different batch runs. This approach enabled $l_p$EKI to use only 50 ensemble members to solve 200 and 400-dimensional inverse problems. Other techniques, such as variance inflation and localization, improve the performance of the standard EKI using a small ensemble size \cite{EKIanalysis}. It would be natural to investigate if these techniques can be extended to $l_p$EKI to decrease the sampling error of $l_p$EKI.
In the current study, we have left several variants of $l_p$EKI for future work. Weighted $l_1$ has been shown to recover sparse solutions using fewer measurements than the standard $l_1$ \cite{weightedl1}. It is straightforward to implement weighted $l_1$ (and further weighted $l_p$ for $p<1$) in $l_p$EKI by replacing the identity matrix in \cref{eq:augmentedcovariance} with another type of covariance matrix corresponding to the desired weights. We plan to study several weighting strategies to improve the performance of$l_p$EKI. As another variant of$l_p$EKI, we plan to investigate the adaptive time-stepping under the continuous limit. The time step for solving the continuous limit equation, which is called `learning rate' in the machine learning community, is known to affect an optimization solver \cite{adaptivelearningrate}. The standard ensemble Kaman inversion has been applied to machine learning tasks, such as discovering the vector fields defining a differential equation, using time series data \cite{MLEKI} and sparse learning using thresholding \cite{sparseEKI}. We plan to investigate the effect of an adaptive time-stepping for performance improvements and compare with the sparsity EKI method using thresholding in dimension reduction in machine learning.
\bibliographystyle{siamplain}
\bibliography{regularizedEKI}
\end{document} | {"config": "arxiv", "file": "2009.03470/Arxiv_regularizedEKI.tex"} |
\begin{document}
\maketitle
\begin{abstract}
For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing a subpath $S_{xy}$ of $S$ with an internally disjoint subpath $T_{xy}$ of $T$. We prove that $\mathcal{P}(G_{uv})$ is always connected and give a necessary and a sufficient condition for connectedness in cases where the cycles formed by the replacing subpaths are restricted to a specific family of cycles of $G$.
\end{abstract}
\section{Introduction}
For any vertices $x,y$ of a path $L$, we denote by $L_{xy}$ the subpath of $L$ that joins $x$ and $y$. Let $G$ be a $2$-connected graph and $u$ and $v$ be vertices of $G$. The \emph{$uv$ path graph of $G$} is the graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where two paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing a subpath $S_{xy}$ of $S$ with an internally disjoint subpath $T_{xy}$ of $T$. The $uv$ path graph $\mathcal{P}(G_{uv})$ is closely related to the graph $G(P,f)$ of $f$-monotone paths on a polytope $P$ (see C. A. Athanasiadis \emph{et al} \cite{ALZ,AER}), whose vertices are the $f$-monotone paths on $P$ and where two paths $S$ and $T$ are adjacent if there is a $2$-dimensional face $F$ of $P$ such that $T$ is obtained from $S$ by replacing an $f$-monotone subpath of $S$ contained in $F$ with the complementary $f$-monotone subpath of $T$ contained in $F$.
In Section \ref{Prem} we show that the graphs $\mathcal{P}(G_{uv})$ are always connected as is the case for the graphs $G(P,f)$.
If $S$ and $T$ are adjacent paths in a $uv$-path graph $\mathcal{P}(G_{uv})$, then $S\cup T$ is a subgraph of $G$ consisting of a unique cycle $\sigma$ joined to $u$ and $v$ by disjoint paths $P_{u}$ and $P_{v}$. See Figure \ref{figmonocle}.
\begin{figure}[h!]
\centering
\includegraphics[width= 4in]{figmonocle}
\caption{$S \cup T$}
\label{figmonocle}
\end{figure}
Let $\mathcal{C}$ be a set of of cycles of $G$; the \emph{$uv$-path graph of $G$ defined by $\mathcal{C}$} is the spanning subgraph $\mathcal{P}_\mathcal{C}(G_{uv})$ of $\mathcal{P}(G_{uv})$ where two paths $S$ and $T$ are adjacent if and only if the unique cycle $\sigma$ which is contained in $S\cup T$ lies in $\mathcal{C}$.
A graph $\mathcal{P}_\mathcal{C}(G_{uv})$ may be disconnected.
The $uv$ path graph $\mathcal{P}(G_{uv})$ is also related to the well-known \emph{tree graph} $\mathcal{T}(G)$ of a connected graph $G$, studied by R. L. Cummins \cite{C66}, in which the vertices are the spanning trees of $G$ and the edges correspond to pairs of trees $S$ and $R$ which are obtained from each other by a single edge exchange. As in the $uv$ path graph, if two trees $S$ and $R$ are adjacent in $\mathcal{T}(G)$, then $S \cup R$ is a subgraph of $G$ containing a unique cycle. X. Li \textit{et al} \cite{LNR} define, in an analogous way, a subgraph $\mathcal{T}_\mathcal{C}(G)$ of $\mathcal{T}(G)$ for a set of cycles $\mathcal{C}$ of $G$ and give a necessary condition and a sufficient condition for $\mathcal{T}_\mathcal{C}(G)$ to be connected. In sections \ref{sectionnecessary} and \ref{sectionsufficient} we show that the same conditions apply to $uv$ path graphs $\mathcal{P}_\mathcal{C}(G_{uv})$.
Similar results are obtained by A. P. Figueroa \textit{et al} \cite{FFR} with respect to the \emph{perfect matching graph} $\mathcal{M}(G)$ of a graph $G$ where the vertices are the perfect matchings of $G$ and in which two matchings $L$ and $N$ are ajacent if their symmetric difference is a cycle of $G$. Again, if $L$ and $N$ are adjacent matchings in $\mathcal{M}(G)$, then $L\cup M$ contains a unique cycle of $G$.
For any subgraphs $F$ and $H$ of a graph $G$, we denote by $F \Delta H$ the subgraph of $G$ induced by the set of edges $(E(F) \setminus E(H)) \cup (E(H) \setminus E(F))$.
\section{Preliminary results}
\label{Prem}
In this section we prove that the $uv$ path graph is connected for any $2$-connected graph $G$ and give an upper bound for the diameter of a graph $\mathcal{P}(G_{uv})$.
\begin{theorem}
\label{connected}
Let $G$ be a $2$-connected graph. The $uv$-path graph $\mathcal{P}(G_{uv})$ is connected for every pair of vertices $u,v$ of $G$.
\end{theorem}
\begin{proof}
For any different $uv$ paths $Q$ and $R$ in $G$ denote by $n(Q,R)$ the number of consecutive initial edges $Q$ and $R$ have in common. Assume the result is false and choose two $uv$ paths $S: u = x_0, x_1, \ldots, x_s = v$ and $T: u = y_0, y_1, \ldots, y_t = v$ in different components of $\mathcal{P}(G_{uv})$ for which $n^{*} = n(S,T)$ is maximum.
Since edges $x_{n^{*}}x_{n^{*}+1}$ and $y_{n^{*}}y_{n^{*}+1}$ are not equal, $x_{n^{*}+1} \neq y_{n^{*}+1}$. Let $j = min \{i: x_{n^{*}+i} \in V(T)\}$ and $k = min \{i: y_{n^{*}+i} \in V(S)\}$ and let $l$ and $m$ be integers such that $y_{l} = x_{n^{*}+j} $, $x_m = y_{n^{*}+k}$. Consider the path:
$$S': u = x_0, x_1, \ldots, x_{n^*}, y_{n^* +1}, y_{n^* +2}, \ldots, y_{n^*+k}, x_{m+1},x_{m+2}, \ldots , x_s = v$$
Paths $S$ and $S'$ are adjacent in $\mathcal{P}(G_{uv})$ since $S'$ is obtained from $S$ by replacing the subpath $x_{n^{*}}, x_{n^{*}+1}, \ldots, x_{m}$ of $S$ with the subpath $y_{n^{*}}, y_{n^{*}+1}, \ldots, \allowbreak y_{n^{*}+k}$ of $S'$. Notice that $n(S', T) \geq n(S, T) + 1$ since $x_0x_1, x_1x_2, \ldots, x_{n^{*}-1}x_{n^{*}}, \allowbreak x_{n^{*}}y_{n^{*}+1} \in E(S') \cap E(T)$. By the choice of $S$, and $T$, paths $S'$ and $T$ are connected in $P(G_{uv})$. This implies that $S$ and $T$ are also connected in $\mathcal{P}(G_{uv})$ which is a contradiction.
\end{proof}
For any two vertices $u$ and $v$ of a connected graph $G$ we denote by $d_G(u,v)$ the distance between $u$ and $v$ in $G$, that is the length of a shortest $uv$ path in $G$. The diameter of a connected graph $G$ is the maximum distance among pairs of vertices of $G$. For a path $P$, we denote by $l(P)$ the length of $P$.
\begin{theorem}
\label{diameter}
Let $u$ and $v$ be vertices of a $2$-connected graph $G$. The diameter of the graph $\mathcal{P}(G_{uv})$ is at most $2d_G(u,v)$.
\end{theorem}
\begin{proof}
Let $S$ and $T$ be $uv$ paths in $G$ and let $P$ be a shortest $uv$ path in $G$. From the proof of Theorem \ref{connected} one can see that there are two paths $Q_S$ and $Q_T$ in $\mathcal{P}(G_{uv})$, each with length at most $l(P)$, joining $S$ to $P$ and $T$ to $P$, respectively. Clearly $Q_S \cup Q_T$ contains a path joining $S$ and $T$ in $\mathcal{P}(G_{uv})$ with length at most $2l(P)=2d_G(u,v)$.
\end{proof}
In Figure \ref{figtight} we show a $2$-connected graph $G^2$ and paths $S$ and $T$ joining vertices $u$ and $v$ of $G^2$ such that $d_{G^2}(u,v) = 2$ and $d_{\mathcal{P}(G^2_{uv})}(S,T) = 4$. For any positive integer $k > 2$ the graph $G^2$ can be extended to a graph $G^k$ such that $d_{G^k}(u,v) = k$ and that the diameter of $\mathcal{P}(G^k_{uv}))$ is $2k$. This shows that Theorem \ref{diameter} is tight.
\begin{figure}[h!]
\centering
\includegraphics[width= 4in]{figtight}
\caption{Graph $G^2$ and paths $S$ and $T$.}
\label{figtight}
\end{figure}
\section{Necessary condition}
\label{sectionnecessary}
Let $u$ and $v$ be vertices of a $2$-connected graph $G$ and $S$ and $T$ be two $uv$ paths adjacent in $\mathcal{P}(G_{uv})$. Since $T$ is obtained from $S$ by replacing a subpath $S_{xy}$ of $S$ with an internally disjoint subpath $T_{xy}$ of $T$, the graph $S \Delta T$ is the cycle $S_{xy} \cup T_{xy}$.
\begin{theorem}
\label{necessary}
Let $G$ be a $2$-connected graph, $u$ and $v$ be vertices of $G$ and $\mathcal{C}$ be a set of cycles of $G$. If the graph $\mathcal{P}_\mathcal{C}(G_{uv})$ is connected, then $\mathcal{C}$ spans the cycle space of $G$.
\end{theorem}
\begin{proof}
Let $\sigma$ be a cycle of $G$. Since $G$ is $2$-connected, there are two disjoint paths $P_u$ and $P_v$ joining, respectively, $u$ and $v$ to $\sigma$. Denote by $u'$ and $v'$ the unique vertices of $P_u$ and $P_v$, respectively, that lie in $\sigma$. Vertices $u'$ and $v'$ partition cycle $\sigma$ into two internally disjoint paths $Q$ and $R$. Let $S = P_u \cup Q\cup P_v$ and $T = P_u \cup R \cup P_v$. Clearly $S$ and $T$ are two different $uv$ paths in $G$ such that $S \Delta T = \sigma$.
Since $\mathcal{P}_\mathcal{C}(G_{uv})$ is connected, there are $uv$ paths $S = W_0, W_1, \ldots, W_k = T$ such that for $i=1, 2, \ldots k$, paths $W_{i-1}$ and $W_i$ are adjacent in $\mathcal{P}_\mathcal{C}(G_{uv})$. For $i=1, 2, \ldots k$ let $\alpha_i = W_{i-1} \Delta W_i$. Then $\alpha_1, \alpha_2, \ldots, \alpha_k$ are cycles in $\mathcal{C}$ such that:
$$\alpha_1 \Delta \alpha_2 \Delta \cdots \Delta \alpha_k = (W_0 \Delta W_1) \Delta (W_1 \Delta W_2) \Delta \cdots \Delta (W_{k-1} \Delta W_k ) = W_0 \Delta W_k = \sigma$$
Therefore $\mathcal{C}$ spans $\sigma$.
\end{proof}
Let $G$ be a complete graph with four vertices $u, x, y ,v$ and let $\mathcal{C} = \{\alpha, \beta, \delta \}$, where $\alpha = uxv$, $\beta = uyv$ and $\delta = uxyv$. Set $\mathcal{C}$ spans the cycle space of $G$ but the graph $\mathcal{P}_\mathcal{C}(G_{uv})$ is not connected since the $uv$ path $uyxv$ is an isolated vertex of $\mathcal{P}_\mathcal{C}(G_{uv})$, see Fig \ref{figK4}. This shows that the condition in Theorem \ref{necessary} is not sufficient for $P_\mathcal{C}(G_{uv})$ to be connected.
\begin{figure}[h!]
\centering
\includegraphics[width= 4in]{figK4}
\caption{Graph $G$, set $\mathcal{C} = \{\alpha, \beta, \delta\}$ and graph $\mathcal{P}_C(G_{uv})$.}
\label{figK4}
\end{figure}
\section{Sufficient condition}
\label{sectionsufficient}
A \emph{unicycle} of a connected graph $G$ is a spanning subgraph $\mathcal{U}$ of $G$ that contains a unique cycle. Let $u$ and $v$ be vertices of a $2$-connected graph $G$. A $uv$-\emph{monocle} of $G$ is a subgraph of $G$ that consists of a cycle $\sigma$ and two disjoint paths $P_u$ and $P_v$ that join, respectively $u$ and $v$ to $\sigma$, see Fig. \ref{figmonocle}. Clearly for each $uv$-monocle $\mathcal{M}$ of a $2$-connected graph $G$, there is a unicycle $\mathcal{U}$ of $G$ that contains $\mathcal{M}$.
Let $\mathcal{C}$ be a set of cycles of $G$. A cycle $\sigma$ of $G$ has \emph{Property $\Delta^*$} with respect to $\mathcal{C}$ if for every unicycle $\mathcal{U}$ containing $\sigma$ there is an edge $e$ of $G$, not in $\mathcal{U}$ and two cycles $\alpha, \beta \in \mathcal{C}$, contained in $\mathcal{U} + e$, such that $\sigma = \alpha \Delta \beta$.
\begin{lemma}
\label{principal}
Let $G$ be a $2$-connected graph and $u$ and $v$ be vertices of $G$. Also let $\mathcal{C}$ be a set of cycles of $G$ and $\sigma$ be a cycle having Property $\Delta^*$ with respect to $\mathcal{C}$. The graph $\mathcal{P}_{\mathcal{C \cup \{\sigma\}}}(G_{uv})$ is connected if and only if $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{lemma}
\begin{proof}
If $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected, then $\mathcal{P}_{\mathcal{C \cup \{\sigma\}}}(G_{uv})$ is connected since the former is a subgraph of the latter.
Assume now $\mathcal{P}_{\mathcal{C \cup \{\sigma\}}}(G_{uv})$ is connected and let $S$ and $T$ be $uv$ paths in $G$ which are adjacent in $\mathcal{P}_{\mathcal{C \cup \{\sigma\}}}(G_{uv})$. We show next that $S$ and $T$ are connected in $\mathcal{P}_\mathcal{C}(G_{uv})$ by a path of length at most 2.
If $\omega = S \Delta T \in \mathcal{C}$, then $S$ and $T$ are adjacent in $\mathcal{P}_\mathcal{C}(G_{uv})$. For the case $\omega = \sigma$ denote by $\mathcal{M}$ the $uv$-monocle given by $S \cup T$.
Let $\mathcal{U}$ be a unicycle of $G$ containing $\mathcal{M}$. Since $\sigma$ has Property $\Delta^*$ with respect to $\mathcal{C}$, there exists an edge $e=xy$ of $G$, not in $\mathcal{U}$, and two cycles $\alpha, \beta \in \mathcal{C}$ contained in $\mathcal{U} + e $ such that $\sigma = \alpha \Delta \beta$.
Let $x'$ and $y'$ denote the vertices in $\mathcal{M}$ which are closest in $\mathcal{U}$ to $x$ and $y$, respectively. Then there exists a path $R_{x'y'}$ in $G$, with edges in $E(\mathcal{U}+e) \setminus E(\mathcal{M})$ joining $x'$ and $y'$ and such that cycles $\alpha$ and $\beta$ are contained in $\mathcal{M} \cup R_{x'y'}$. We analyze several cases according to the location of $x'$ and $y'$ in $\mathcal{M}$.
Denote by $P_u$ and $P_v$ the unique paths, contained in $\mathcal{M}$, that join $u$ and $v$ to $\sigma$ and by $u'$ and $v'$ the vertices where $P_u$ and $P_v$, respectively, meet $\sigma$.\\
\noindent \emph{Case} 1.- $x' \in V(P_u)$, $y' \in V(P_v)$. Without loss of generality we assume $\alpha = S_{x'y'} \cup R_{y'x'}$ and $\beta = T_{x'y'} \cup R_{y'x'}$, see Fig. \ref{figcase1}.
\begin{figure}[h!]
\centering
\includegraphics[width= 4.3in]{figcase1}
\caption{Left: $\mathcal{M} \cup R_{y'x'}$. Right: Cycles $\alpha$ and $\beta$.}
\label{figcase1}
\end{figure}
Let $Q$ be the $uv$-path obtained from $S$ by replacing $S_{x'y'}$ with $R_{x'y'}$. Notice that $Q$ can also be obtained from $T$ by replacing $T_{x'y'}$ with $R_{x'y'}$.\\
\noindent \emph{Case} 2.- $x' \in V(P_u)$, $y' \in S \cap \sigma$. Without loss of generality we assume $\alpha = S_{x'y'} \cup R_{y'x'}$ and $\beta = T_{x'v'} \cup S_{v'y'} \cup R_{y'x'}$, see Fig. \ref{figcase2}.
\begin{figure}[h!]
\centering
\includegraphics[width= 4.3in]{figcase2}
\caption{Left: $\mathcal{M} \cup R_{y'x'}$. Right: Cycles $\alpha$ and $\beta$.}
\label{figcase2}
\end{figure}
Again let $Q$ be the $uv$-path obtained from $S$ by replacing $S_{x'y'}$ with $R_{x'y'}$. In this case, $Q$ can also be obtained from $T$ by replacing $T_{x'v'}$ with $R_{x'y'} \cup S_{y'v'}$.\\
\noindent \emph{Case} 3.- $x', y' \in S \cap \sigma$. Without loss of generality we assume $\alpha = S_{x'y'} \cup R_{y'x'}$ and $\beta = S_{u'x'} \cup R_{x'y'} \cup S_{y'v'} \cup T_{v'u'}$ , see Fig. \ref{figcase3}.
\begin{figure}[h!]
\centering
\includegraphics[width= 4.3in]{figcase3}
\caption{Left: $\mathcal{M} \cup R_{y'x'}$. Right: Cycles $\alpha$ and $\beta$.}
\label{figcase3}
\end{figure}
Let $Q$ be the $uv$-path obtained from $S$ by replacing $S_{x'y'}$ with $R_{x'y'}$. Path $Q$ is also obtained from $T$ by replacing $T_{u'v'}$ with $S_{u'x'} \cup R_{x'y'} \cup S_{y'v'}$.\\
\noindent \emph{Case} 4.- $x' \in S \cap \sigma$ and $y' \in T \cap \sigma$. Without loss of generality we assume $\alpha = S_{u'x'} \cup R_{x'y'} \cup T_{y'u'}$ and $\beta = T_{y'v'} \cup S_{v'x'} \cup R_{x'y'} $. , see Fig. \ref{figcase4}.
\begin{figure}[h!]
\centering
\includegraphics[width= 4.3in]{figcase4}
\caption{Left: $\mathcal{M} \cup R_{y'x'}$. Right: Cycles $\alpha$ and $\beta$.}
\label{figcase4}
\end{figure}
Let $Q$ be the $uv$-path obtained from $S$ by replacing $S_{u'x'}$ with $T_{u'y'} \cup R_{y'x'}$. Now $Q$ can also be obtained from $T$ by replacing $T_{y'v'}$ with $R_{y',x'} \cup S_{x'v'}$\\
In each case $S \Delta Q = \alpha$ and $Q \Delta T = \beta$. Since $\alpha, \beta \in \mathcal{C}$, path $S$ is adjacent to $Q$ and path $Q$ is adjacent to $T$ in $P_\mathcal{C}(G_{uv})$. Therefore $S$ and $T$ are connected in $\mathcal{P}_{\mathcal{C}}(G_{uv})$ by a path with length at most $2$.
All remaining cases are analogous to either Case 2 or to Case 3.
\end{proof}
Consider a $2$-connected graph $G$ with two specified vertices $u$ and $v$ and let $\mathcal{C}$ be a set of cycles of $G$. Construct a sequence of sets of cycles $\mathcal{C} = \mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_k$ as follows: If there is a cycle $\sigma_1$ not in $\mathcal{C}_0$ that has Property $\Delta^*$ with respect to $\mathcal{C}_0$ add $\sigma_1$ to $\mathcal{C}_0$ to obtain $\mathcal{C}_1$. At step $t$ add to $\mathcal{C}_t$ a new cycle $\sigma_{t+1}$ (if it exists) that has Property $\Delta
^*$ with respect to $\mathcal{C}_t$ to obtain $\mathcal{C}_{t+1}$. Stop at a step $k$ where there are no cycles, not in $\mathcal{C}_k$, having Property $\Delta^*$ with respect to $\mathcal{C}_k$. We denote by $Cl(\mathcal{C})$ the final set obtained with this process. Li \emph{et al} \cite{LNR} proved that the final set of cycles obtained is independent of which cycle $\sigma_t$ is added at each step in the case of multiple possibilities.
A set of cycles of $G$ is \emph{$\Delta^*$-dense} if $Cl(\mathcal{C})$ is the whole set of cycles of $G$.
\begin{theorem}
\label{sufficient}
If $\mathcal{C}$ is $\Delta^*$-dense, then $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{theorem}
\begin{proof}
Since $\mathcal{C}$ is $\Delta^*$-dense, $Cl(\mathcal{C})$ is the set of cycles of $G$ and therefore $\mathcal{P}_{Cl(\mathcal{C})}(G_{uv}) = \mathcal{P}(G_{uv})$ which is connected by Theorem \ref{connected}.
Let $\mathcal{C}= \mathcal{C}_0 , \mathcal{C}_1, \ldots, \mathcal{C}_k = Cl(\mathcal{C})$ be a sequence of sets of cycles obtained from $\mathcal{C}$ as above. By Lemma \ref{principal}, all graphs $\mathcal{P}_{Cl(\mathcal{C})}(G_{uv}) = \mathcal{P}_{\mathcal{C}_{k}}(G_{uv}), \mathcal{P}_{\mathcal{C}_{k-1}}(G_{uv}), \allowbreak \ldots, \mathcal{P}_{\mathcal{C}_{0}}(G_{uv}) = \mathcal{P}_{\mathcal{C}}(G_{uv})$ are connected.
\end{proof}
Li \emph{et al} \cite{LNR} proved the following:
\begin{theorem}
\label{carasinternas}
If $G$ is a plane $2$-connected graph and $\mathcal{C}$ is the set of internal faces of $G$, then $\mathcal{C}$ is $\Delta^*$-dense.
\end{theorem}
\begin{theorem}
\label{porunaarista}
If $G$ is a $2$-connected graph and $\mathcal{C}$ is the set of cycles that contain a given edge $e$ of $G$, then $\mathcal{C}$ is $\Delta^*$-dense.
\end{theorem}
We end this section with the following immediate corollaries.
\begin{corollary}
\label{corcarasinternas}
Let $u$ and $v$ be vertices of a $2$-connected plane graph $G$. If $\mathcal{C}$ is the set of internal faces of $G$, then $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{corollary}
\begin{proof}
By Theorem \ref{carasinternas}, $\mathcal{C}$ is $\Delta^*$-dense and by Theorem \ref{sufficient}, $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{proof}
\begin{corollary}
\label{corporunaarista}
Let $u$ and $v$ be vertices of a $2$-connected graph $G$. If $\mathcal{C}$ is the set of cycles of $G$ that contain a given edge $e$, then $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{corollary}
\begin{proof}
By Theorem \ref{porunaarista}, $\mathcal{C}$ is $\Delta^*$-dense and by Theorem \ref{sufficient}, $\mathcal{P}_{\mathcal{C}}(G_{uv})$ is connected.
\end{proof}
\begin{corollary}
\label{corporunvertice}
Let $u$ and $v$ be vertices of a $2$-connected graph $G$. If $\mathcal{C}_u$ is the set of cycles of $G$ that contain vertex $u$, then $\mathcal{P}_{\mathcal{C}_u}(G_{uv})$ is connected.
\end{corollary}
\begin{proof}
Let $e$ be an edge of $G$ incident with vertex $u$. Clearly the set $\mathcal{C}(e)$ of cycles that contain edge $e$ is a subset of the set $\mathcal{C}_u$. Therefore $\mathcal{P}_{\mathcal{C}(e)}(G_{uv})$ is a subgraph of $\mathcal{P}_{\mathcal{C}_u}(G_{uv})$. By Corollary \ref{corporunaarista}, the graph $\mathcal{P}_{\mathcal{C}(e)}(G_{uv})$ is connected.
\end{proof} | {"config": "arxiv", "file": "2104.00481/main.tex"} |
TITLE: If $P_n=\alpha^n+\beta^n\;, \alpha+\beta=1, \;\alpha \cdot \beta=-1,\;P_{n-1}=11,\; P_{n+1}=29$ Find $(P_n)^2,\;$ where $n\in \mathbb N$
QUESTION [2 upvotes]: If $P_n=\alpha^n+\beta^n\;, \alpha+\beta=1, \;\alpha \cdot \beta=-1,\;P_{n-1}=11,\; P_{n+1}=29$, $\alpha$ and $\beta$ are real numbers. Find $(P_n)^2,\;$ where $n\in \mathbb N$
My Approach:
Method $1$ :
$P_{n-1}\cdot P_{n+1}=11\cdot 29 \implies\; (\alpha^{n-1}+\beta^{n-1})\cdot (\alpha^{n+1}+\beta^{n+1})=\alpha^{2n}+\beta^{2n}+(\alpha \beta)^{n-1}\cdot(\alpha^2+\beta^2)$
$\implies \; 319=P_{2n}+3 (-1)^{n-1}$
$\implies\;P_{2n}=319-3(-1)^{n-1} \implies P_{2n}=319+3(-1)^n$
Now
$(P_{n})^2=(\alpha^n+\beta^n)^2=\alpha^{2n}+\beta^{2n}+2(\alpha \beta)^n\;\implies\;(P_{n})^2=P_{2n}+2(-1)^n$
$\implies \; (P_{n})^2=319+5(-1)^n$
$\implies (P_n)^2=324\;$ If $n$ is even and $(P_n)^2=314\;$ if $n$ is odd.
But given answer is $324$ only.
Method $2$:
Form a quadratic equation with sum of roots and product of roots is given
$\alpha^2-\alpha-1=0\quad$ and $\quad\beta^2-\beta-1=0$
$\implies \;\alpha^{n+1}=\alpha^n+\alpha^{n-1}\cdots(1)\quad$ and $\quad \beta^{n+1}=\beta^{n}+\beta^{n-1}\cdots (2)$
Now add both equation $(1)+(2)$
$\implies$ $P_{n+1}=P_{n}+P_{n-1} \; \implies \; P_{n}=18$
$\implies \; (P_n)^2=324$.
My doubt: What is wrong with my method $1$ ?
REPLY [1 votes]: Since $P_n = \alpha^n + \beta^n $, then $P_n$ satisfies the difference equation
$ (E - \alpha)(E - \beta) P_n = 0 $
where $E$ is the advance operator, i.e. $E\left(P_n\right) = P_{n+1}$
Hence, multiplying the operators, we get the difference equation
$P_{n+2} - (\alpha + \beta) P_{n+1} + \alpha \beta P_{n} = 0 $
Substituting the given values of $(\alpha + \beta)$ and $\alpha \beta$,
$P_{n+2} - P_{n+1} - P_{n} = 0 $
That is,
$P_{n+2} = P_{n+1} + P_n$
Now
$P_1 = \alpha + \beta = 1 $
$P_2 = \alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2 \alpha \beta = 3 $
Therefore, the sequence $\{P_n\}$ is as follows:
$1, 3, 4, 7, 11, 18, 29, 47, ... $
We already see our terms $11$ and $29$ , hence the one between them is $ 18 $, whose square is $324$ | {"set_name": "stack_exchange", "score": 2, "question_id": 4426543} |
\begin{document}
\title{The Bramble-Hilbert Lemma\thanks{I place this document in the public domain
and you can do with it anything you want.}}
\author{Jan Mandel\\University of Colorado Denver}
\date{\relax}
\maketitle
\begin{abstract}
This is an introductory document surveying several results in polynomial
approximation, known as the Bramble-Hilbert lemma.
\end{abstract}
\section{Introduction}
In numerical analysis, the Bramble-Hilbert lemma bounds the error of an
approximation of a function $u$ by a polynomial of order at most $m-1$ in
terms of derivatives of $u$ of order $m$. Both the error of the approximation
and the derivatives of $u$ are measured by $L^{p}$ norms on a bounded domain
in $\mathbb{R}^{n}$. This is similar to classical numerical analysis, where,
for example, the error of interpolation $u$ on an interval by a linear
function (that is, approximation by a polynomial of order one) can be bounded
using the second derivative of $u$. The difference is that the Bramble-Hilbert
lemma applies in any number of dimensions, not just one dimension, and the
approximation error and the derivatives of $u$ are measured by more general
norms involving averages, not just the maximum norm.
Additional assumptions on the domain are needed for the Bramble-Hilbert lemma
to hold. Essentially, the boundary of the domain must be \textquotedblleft
reasonable\textquotedblright. For example, domains that have a spike or a slit
with zero angle at the tip are excluded. Domains that are reasonable enough
include all convex domains and Lipschitz domains, which includes all domains
with a continuously differentiable boundary.
The main use of the Bramble-Hilbert lemma is to prove bounds on the error of
interpolation of function $u$ by an operator that preserves polynomials of
order up to $m-1$, in terms of the derivatives of $u$ of order $m$. This is an
essential step in error estimates for the finite element method. The
Bramble-Hilbert lemma is applied there on the domain consisting of one element.
\section{The one dimensional case}
Before stating the lemma in full generality, it is useful to look at some
simple special cases. In one dimension and for a function $u$ that has $m$
derivatives on interval $\left( a,b\right) $, the lemma reduces to
\[
\inf_{v\in P_{m-1}}\bigl\Vert u^{\left( k\right) }-v^{\left( k\right)
}\bigr\Vert_{L^{p}\left( a,b\right) }\leq C\left( m\right) \left(
b-a\right) ^{m-k}\bigl\Vert u^{\left( m\right) }\bigr\Vert_{L^{p}\left(
a,b\right) },
\]
where $P_{m-1}$ is the space of all polynomials of order at most $m-1$.
In the case when $p=\infty$, $m=2$, $k=1$, and $u$ is twice differentiable,
this means that there exists a polynomial $v$ of degree one such that for all
$x\in\left( a,b\right) $,
\[
\left\vert u\left( x\right) -v\left( x\right) \right\vert \leq C\left(
b-a\right) ^{2}\sup_{\left( a,b\right) }\left\vert u^{\prime\prime
}\right\vert ,\text{\quad for all }x\in\left( a,b\right) .
\]
This inequality follows from the well-known error estimate for linear
interpolation by choosing $v$ as the linear interpolant of $u$.
\section{Statement of the lemma}
Suppose $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$, with
boundary $\partial\Omega$ and diameter $d$. $W_{p}^{k}(\Omega)$ is the Sobolev
space of all function $u$ on $\Omega$ with weak derivatives $D^{\alpha}u$ of
order $\left\vert \alpha\right\vert $ up to $k$ in $L^{p}(\Omega)$. Here,
$\alpha=\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{n}\right) $ is a
multiindex, $\left\vert \alpha\right\vert =$ $\alpha_{1}+\alpha_{2}
+\cdots+\alpha_{n}$, and $D^{\alpha}$ denotes the derivative $\alpha_{1}$
times with respect to $x_{1}$, $\alpha_{2}$ times with respect to $\alpha_{2}
$, and so on. The Sobolev seminorm on $W_{p}^{m}(\Omega)$ consists of the
$L^{p}$ norms of the highest order derivatives,
\[
\left\vert u\right\vert _{W_{p}^{m}(\Omega)}=\left( \sum_{\left\vert
\alpha\right\vert =m}\left\Vert D^{\alpha}u\right\Vert _{L^{p}(\Omega)}
^{p}\right) ^{1/p}\text{ if }1\leq p<\infty
\]
and
\[
\left\vert u\right\vert _{W_{\infty}^{m}(\Omega)}=\max_{\left\vert
\alpha\right\vert =m}\left\Vert D^{\alpha}u\right\Vert _{L^{\infty}(\Omega)}
\]
$P_{k}$ is the space of all polynomials of order up to $k$ on $\mathbb{R}^{n}
$. Note that $D^{\alpha}v=0$ for all $v\in P_{m-1}$. and $\left\vert
\alpha\right\vert =m$, so $\left\vert u+v\right\vert _{W_{p}^{m}(\Omega)}$ has
the same value for any $v\in P_{k-1}$.
\textbf{Lemma (Bramble and Hilbert)} Under additional assumptions on the
domain $\Omega$, specified below, there exists a constant $C=C\left(
m,\Omega\right) $ independent of $p$ and $u$ such that for any $u\in
W_{p}^{k}(\Omega)$ there exists a polynomial $v\in P_{m-1}$ such that for all
$k=0,\ldots,m$,
\[
\left\vert u-v\right\vert _{W_{p}^{k}(\Omega)}\leq Cd^{m-k}\left\vert
u\right\vert _{W_{p}^{m}(\Omega)}.
\]
\section{The original result}
The lemma was proved by Bramble and Hilbert \cite{Bramble-1970-ELF} under the
assumption that $\Omega$ satisfies the \emph{strong cone property;} that is,
there exists a finite open covering $\left\{ O_{i}\right\} $ of
$\partial\Omega$ and corresponding cones $\{C_{i}\}$ with vertices at the
origin such that $x+C_{i}$ is contained in $\Omega$ for any $x$ $\in\Omega\cap
O_{i}$.
The statement of the lemma here is a simple rewriting of the right-hand
inequality stated in Theorem 1 in \cite{Bramble-1970-ELF}. The actual
statement in \cite{Bramble-1970-ELF} is that the norm of the factorspace
$W_{p}^{m}(\Omega)/P_{m-1}$ is equivalent to the $W_{p}^{m}(\Omega)$ seminorm.
The $W_{p}^{m}(\Omega)$ is not the usual one but the terms are scaled with $d$
so that the right-hand inequality in the equivalence of the seminorms comes
out exactly as in the statement here.
In the original result, the choice of the polynomial is not specified, and the
dependence of the constant on the domain $\Omega$ is not given either.
\section{A constructive form}
An alternative result was given by Dupont and Scott \cite{Dupont-1980-PAF}
under the assumption that the domain $\Omega$ is \emph{star-shaped;} that is,
there exists a ball $B$ such that for any $x\in\Omega$, the closed convex hull
of $\left\{ x\right\} \cup B$ is a subset of $\Omega$. Suppose that
$\rho_{\max}$ is the supremum of the diameters of such balls. The ratio
$\gamma=d/\rho_{\max}$ is called the chunkiness of $\Omega$.
Given a fixed ball $B$ as above, and a function $u$, the averaged Taylor
polynomial $Q^{m}u$ is defined as
\[
Q^{m}u=\int\limits_{B}T_{y}^{m}u\left( x\right) \psi\left( y\right) dx,
\]
where
\[
T_{y}^{m}u\left( x\right) =\sum\limits_{k=0}^{m-1}\sum\limits_{\left\vert
\alpha\right\vert =k}\frac{1}{\alpha!}D^{\alpha}u\left( y\right) \left(
x-y\right) ^{\alpha}
\]
is the Taylor polynomial of degree at most $m-1$ of $u$ centered at $y$
evaluated at $x$, and $\psi\geq0$ is a function that has derivatives of all
orders, equals to zero outside of $B$, and such that
\[
\int\limits_{B}\psi dx=1.
\]
Such function $\psi$ always exists.
Then the lemma holds with the constant $C=C\left( m,n,\gamma\right) $, that
is, the constant depends on the domain $\Omega$ only through its chunkiness
$\gamma$ and the dimension of the space $n$. For more details and a tutorial
treatment, see the monograph by Brenner and Scott \cite{Brenner-2002-MTF}. The
result can be extended to the case when the domain $\Omega$ is the union of a
finite number of star-shaped domains and more general polynomial spaces than
the space of all polynomials up to a given degree \cite{Dupont-1980-PAF}.
\section{Bound on linear functionals}
This result follows immediately from the above lemma, and it is also called
sometimes the Bramble-Hilbert lemma, for example by Ciarlet
\cite{Ciarlet-2002-FEM}. It is essentially Theorem 2 from
\cite{Bramble-1970-ELF}.
\textbf{Lemma} Suppose that $\ell$ is a continuous linear functional on
$W_{p}^{m}(\Omega)$ and $\left\Vert \ell\right\Vert _{W_{p}^{m}(\Omega
)^{^{\prime}}}$ its dual norm. Suppose that $\ell\left( v\right) =0$ for all
$v\in P_{m-1}$. Then there exists a constant $C=C\left( \Omega\right) $ such
that
\[
\left\vert \ell\left( u\right) \right\vert \leq C\left\Vert \ell\right\Vert
_{W_{p}^{m}(\Omega)^{^{\prime}}}\left\vert u\right\vert _{W_{p}^{m}(\Omega)}.
\]
\bibliographystyle{plain}
\bibliography{../../bddc/bibliography/bddc}
\end{document} | {"config": "arxiv", "file": "0710.5148/Bramble-Hilbert_lemma.tex"} |
TITLE: Maximize $l_1$ norm with unitary matrix
QUESTION [3 upvotes]: Given an invertible matrix $A \in \mathbb{C}^{n\times n}$. How to find
$$
U^* = \max_{\text{$U$ with $U^H U = I$}} \lVert U A\rVert_1,
$$
where $\lVert\cdot\rVert_1$ is the entrywise 1-norm, i.e., $\lVert A\rVert_1 = \sum_{i,j} \lvert A_{ij}\rvert$ and $\cdot^H$ denotes the complex conjugate transpose?
For $A = I$, the solution is any complex Hadamard matrix, e.g., a scaled discrete Fourier matrix.
REPLY [0 votes]: Here an alternative solution, which outperformed @DSM solution (4) in all tested cases.
Without loss of generality, we assume that $\lVert A \rVert_F = 1$. The optimal value for the $\ell_1$ norm is attained by the unitary Hadamard matrix $H$, e.g., DFT matrix.
As a proxy cost function, we use therefore
$$
\hat{U} = \min_U \| |U A| - |H| \|_F,
$$
where $|\cdot|$ is the element-wise absolute values. This is equivalent for an optimal set of phases $\hat{P}$ with $|\hat{P}_{ij}| = 1$ such that
$$
\hat{U} = \min_U \| U A - |H| \circ \hat{P} \|_F,
$$
where $\circ$ denotes the element-wise (Hadamard) product. This can be solved iteratively by unitary Procrustes solution such that
$$
U^{(i+1)} = \min_U \| U A - |H| \circ P^{(i)} \|_F \\
P^{(i+1)} = U^{(i+1)}A \oslash |U^{(i+1)}A|,
$$
where $\oslash$ is the element-wise (Hadamard) division. These iterations are guaranteed to converge:
Because of the Procrustes solution is the global minimum for the Frobenius norm, we have
$$
\| U^{(i+1)} A - |H| \circ P^{(i)} \|_F \leq \| U^{(i)} A - |H| \circ P^{(i)} \|_F
$$
Then, updating the phase also reduces the error
$$
\| U^{(i+1)} A - |H| \circ P^{(i+1)} \|_F \leq \| U^{(i+1)} A - |H| \circ P^{(i)} \|_F,
$$
which is essentially a element-wise version of
$$
\phi = \min_\theta \left(a e^{\imath \phi} - b e^{\imath \theta} \right)^2,
$$
where $a$, $b$, $\phi$, $\theta$ are real valued. | {"set_name": "stack_exchange", "score": 3, "question_id": 359023} |
TITLE: Use Maximum Likelihood Estimation to guess which dice got selected
QUESTION [0 upvotes]: We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the eyes is 7. Estimate which two they selected by using the maximum likelihood principle.
This is a problem on an old probability theory exam. Unfortunately we only ever calculated pretty straightforward examples in class and I have no idea how to tackle this one. However, since the expected value for the sum of the eyes of two six-sided dice is $7$ I guess they likely selected both of these! How can we approach this problem?
REPLY [0 votes]: Hint:
What is the probability of a sum of $7$ from two six-sided dice?
What is the probability of a sum of $7$ from two four-sided dice? | {"set_name": "stack_exchange", "score": 0, "question_id": 1647866} |
\begin{definition}[Definition:Homotopy/Path]
Let $X$ be a [[Definition:Topological Space|topological space]].
Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be [[Definition:Path (Topology)|paths]].
We say that $f$ and $g$ are '''path-homotopic''' if they are [[Definition:Relative Homotopy|homotopic relative]] to $\left\{ {0, 1}\right\}$.
\end{definition} | {"config": "wiki", "file": "def_27735.txt"} |
TITLE: What is the temperature of a pure quantum state?
QUESTION [5 upvotes]: I was wondering about temperatures and pure quantum states. I'm currently working on thermalization of isolated quantum systems, which can be described by pure quantum states (kets). How do we define a temperature for these?
Normally you work within the density matrix formalism so you can define an ensemble, which is a mixed state, for a certain temperature. In my scenario we prepare a system in a lab in a pure state, that is isolated, and let it thermalize to its equilibrium state. This must be (in very good approximation) still a pure state if the system is strongly isolated, because of the unitarity of evolution operator.
We can then probably assign a certain temperature to this equilibrium system, as we could bring this system in contact with a heath bath with a certain temperature and monitor if energy flows out of the system or into the system (aka it has a higher or lower temperature). But from a theoretical viewpoint I have no idea how to assign a temperature to a pure quantum state, so I'm quite dazzled by this.
I know the ground state is supposed to have $T = 0 K$, but that's all I could think of. The Internet is not giving me much more information.
Edit: Ofcourse I am talking about a certain many-body system with a corresponding Hamiltonian, which should be non-integrable and chaotic. This way the system will thermalize (in most cases) following the Eigenstate Thermalization Hypothesis. You could take the 1D spin chain non-integrable Ising model as example.
REPLY [2 votes]: The same question could be asked in classical physics: What is the temperature of a specific microstate of an ideal gas? Conceptually, the answer to the quantum version is essentially the same as the answer to the classical version.
If the question is how to actually calculate the temperature of a given pure state (at least in principle), then one way is to take a partial trace over half of the system. If the system is large enough so that the energy of interaction between the two halves is negligible, then the result should have the form $\exp(-\beta H)$ where $H$ is the self-Hamiltonian of the remaining half. This assumes that the whole-system pure state has attained "equilibrium", in the sense that it has evolved long enough to be in a "typical" state for the given macro-conditions.
This paper may be of interest:
"Canonical Typicality of Energy Eigenstates of an Isolated Quantum System," https://arxiv.org/abs/1511.06680
From the abstract:
Currently there are two main approaches to describe how quantum statistical physics emerges from an isolated quantum many-body system in a pure state...
And here's an older review paper:
"Pure State Quantum Statistical Mechanics," http://arxiv.org/abs/1003.5058 | {"set_name": "stack_exchange", "score": 5, "question_id": 540850} |
TITLE: Transition matrix of polynomial.
QUESTION [1 upvotes]: Good night, i need help with this.
Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\} $
I found a basis for W, $\left\{ 2,x,x^{2}+1,x^{3}\right\} $ and i work in a linear combination of basis W to V and i make a system, but then I had problems with the system
$\begin{cases}
2=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\
x=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\
x^{2}+1=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\
x^{3}=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)
\end{cases}$
REPLY [2 votes]: All you need to do now is to organize the four sets of the four $\alpha$'s that you will find from your four systems of equations (do you see why each identity you wrote is a system of equations?) into a $4 \times 4$ matrix $A$.
Finally, whether it's $A$, $A^T$, $A^{-1}$, or $(A^T)^{-1}$ that is the answer you're looking for depends on your definition of transition matrix. | {"set_name": "stack_exchange", "score": 1, "question_id": 1756143} |
TITLE: Generalize Implicit Differentiation to find Tangent Plane
QUESTION [1 upvotes]: For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ show that at the point $(a,b,c)$,
$\frac{\partial z}{\partial x}=g_x(a,b)=-\frac{F_x(a,b,c)}{F_z (a,b,c)}$
I start by using the chain rule to differentiate w.r.t $x$
$F_x=\frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx} + \frac{\partial F}{\partial z} \frac{dz}{dx}$
In the teacher's hint, he stated that $\frac{dy}{dx}$ is equal to zero, which using this I can solve the question, I don't however understand why $\frac{dy}{dx}=0$. I was wondering if someone could explain this.
REPLY [1 votes]: Get the total differential of F(x,y,z)=k
$$dF(x,y,z)=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz=0$$
Since z is implicitly defined by x and y
$$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$
Now replacing dz to dF
$$dF(x,y,z)=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}(\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy)=0$$
Rearranging the terms
$$dF(x,y,z)=\bigg(\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}\bigg)dx+\bigg(\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial y}\bigg)dy=0$$
To make this equation hold we need to satisfy below conditions
$$\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0 \Rightarrow \frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$$
$$\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial y}=0 \Rightarrow \frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$$
The first condition is what you are looking for. | {"set_name": "stack_exchange", "score": 1, "question_id": 1639876} |
TITLE: Let $f(x)=\frac{1-2x^2}{1+x^2} (x\in \mathbb{R})$. Find its Chebyshev norm $||f||_\infty$
QUESTION [0 upvotes]: Let $f(x)=\frac{1-2x^2}{1+x^2} (x\in \mathbb{R})$. Find its Chebyshev norm $||f||_\infty$
I'm not to sure how to go about answering this question, I think the way to go by answering it is finding the supremum of $|f(x)|$?
REPLY [0 votes]: We have $|f(x)| \to 2$ for $x \to \pm \infty$ and $|f(x)| \le 2$ for all $x \in \mathbb R$.
It is your turn to verify this inequality !
Conclusion: $||f||_\infty=2$. | {"set_name": "stack_exchange", "score": 0, "question_id": 2755944} |
TITLE: How to compute the determinant of this block matrix?
QUESTION [2 upvotes]: $$M = \left[\begin{matrix} -C & -A \\ A^\top & 0
\end{matrix} \right]$$
I found a paper using $\det(M) = \det(A^\top C^{-1}A)$ but don't know how to prove this.
REPLY [2 votes]: Performing Gaussian elimination,
$$\begin{bmatrix} \mathrm I & \mathrm O\\ \mathrm A^\top \mathrm C^{-1} & \mathrm I \end{bmatrix} \begin{bmatrix} - \mathrm C & -\mathrm A \\ \mathrm A^\top & \mathrm O \end{bmatrix} = \begin{bmatrix} - \mathrm C & -\mathrm A \\ \mathrm O & -\mathrm A^\top \mathrm C^{-1} \mathrm A \end{bmatrix}$$
Note that the determinant of a block triangular matrix is the product of the determinants of the diagonal blocks. | {"set_name": "stack_exchange", "score": 2, "question_id": 3418744} |
TITLE: How to prove that the given set is not uncountable?
QUESTION [2 upvotes]: I was trying to solve the question given in my assignment on metric spaces.
Let $S$ be a subset of $R$. Let $C$ be the set of points $x$ in $R$ with the property that $S\cap (x-\delta,x+\delta )$ is uncountable for every $\delta > 0.$ Prove that $S - C$ is finite or countable.
I started like this:
Let $x$ $\in$ $S-C$ $=>$ $S\cap (x-\delta,x+\delta )$ is countable for some $\delta>0$.
Also, $(S-C)\cap(x-\delta,x+\delta )\subset S\cap (x-\delta,x+\delta )$.
So, $(S-C)\cap(x-\delta,x+\delta )$ is countable.
Hence, for every $x$ $ \in$ $S-C$ , $\exists$ a $\delta>0$ such that $(S-C)\cap(x-\delta,x+\delta )$ is countable.
But after that I couldn't advance. Any help would be appreciated.
Thanks.
REPLY [1 votes]: For any $x\in S\setminus C$, let $a_x,b_x$ be defined so that:
$$a_x=\inf\{a\mid S\cap (a,x) \text{ is countable or finite}\}\\
b_x=\sup\{b\mid S\cap (x,b)\text{ is countable or finite}\}$$
There is always such $a_x,b_x$ because $x\notin C$ so $(x-\delta,x+\delta)$ is finite or countable for some $\delta$, and we can show that $(a_x,b_x)\cap S$ is countable or finite, since it is the countable union of $S\cap\left(a_x+\frac{1}{n},b_x-\frac{1}{n}\right)$ each of which must be countable.
The values $a_x,b_x$ can be $-\infty$ or $+\infty$, respectively.
Now, if $x,y\in S\setminus C$ and $y\in (a_x,b_x)$ then $(a_y,b_y)=(a_x,b_x)$.
So we have an equivalence relation on $S\setminus C$ defined as $x\sim y$ if and only if $(a_x,b_x)=(a_y,b_y)$. Modulo this equivalence relation, there is only countably many classes, because $x\sim y$, and the interals $(a_x,b_x)$ and $(a_y,b_y)$ are disjoint of the are not equal.
But then:
$$S\setminus C \subseteq S\cap \bigcup_{i=1}^{\infty}(a_{x_i},b_{x_i})=\bigcup_{i=1}^{\infty} S\cap (a_{x_i},b_{x_i})$$ is countable or finite.
A more direct way to write this same proof is to use that $\mathbb R$ has a countable basis. This also lets you generalize the theorem.
For each $x\in S\setminus C$, let $\delta_x>0$ be chosen so that $I_x=(x-\delta_x,x+\delta_x)$ has the property that $I_x\cap S$ is finite or countable.
Then let $U=\bigcup_{x\in S\setminus C} I_x$.
We see immediately that $S\setminus C\subseteq U$, and that $U$ is open, since it is a union of open sets.
Let $U_1,U_2,U_3,\dots$ be a countable basis for $\mathbb R$. Then pick the countable subset $V_1,V_2,\dots$ such that $V_i\subseteq I_x$ for some $x\in S\setminus C$. Show $U=\bigcup V_i$.
Now, since $V_i\subseteq I_x$ for some $x$, $V_i\cap S\subseteq I_x\cap S$ which is finite or countable.
So $S\cap U=S\cap\bigcup V_i= \bigcup S\cap V_i$ is finite or countable.
But $S\setminus C\subseteq S\cap U$. So $S\setminus C$ is also countable or finite.
Generalization: Let $X$ be a topological space with a countable basis. Let $S\subseteq X$ and let $C$ be the set of elements of $x\in X$ such that $U\cap S$ is uncountable for every open set $U$ of $X$ containing $x$. Then $S\setminus C$ is countable or finite. | {"set_name": "stack_exchange", "score": 2, "question_id": 1710599} |
\begin{document}
\begin{abstract}
A \textit{meander} is a topological configuration of a line and a simple closed curve in the plane (or
a pair of simple closed curves on the 2-sphere) intersecting transversally. In physics,
meanders provide a model of polymer \textit{folding}, and their enumeration is directly
related to the entropy of the associated dynamical systems.
We combine recent results on Masur--Veech volumes of the moduli
spaces of meromorphic quadratic differentials in genus zero and our
previous result that horizontal and
vertical separatrix diagrams of integer quadratic differentials
are asymptotically uncorrelated
to derive two applications to asymptotic enumeration of meanders.
First, we get simple asymptotic formulae for the number of pairs of
transverse simple closed curves on a sphere and for the number of
closed meanders of fixed combinatorial type when the number of
crossings $2N$ goes to infinity.
Second, we compute the asymptotic probability of getting a simple
closed curve on a sphere by identifying the endpoints of two arc
systems (one on each of the two hemispheres) along the common
equator. Here the total number of minimal arcs of the two arc
systems is considered as a fixed parameter while the number of all
arcs (same for each of the two hemispheres) grows.
The number of all meanders with $2N$
crossings grows exponentially when $N$ grows.
However, the additional combinatorial constraints we impose
in this article yield polynomial asymptotics.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction and statements of main results}
\label{s:introduction:and:main:results}
In the seminal paper~\cite{Mirzakhani:grouth:of:simple:geodesics}
M.~Mirzakhani has computed the asymptotics for the number of simple
closed hyperbolic geodesics on a hyperbolic surface
of constant negative curvature and frequencies of simple closed
hyperbolic geodesics of fixed combinatorial type.
We count the asymptotics for the number of \textit{pairs} of
transverse simple closed curves of a fixed combinatorial type
on a sphere
when the number of
intersections tends to infinity. The similar enumeration problems in higher genera
will be considered in the sequel.
M.~Mirzakhani establishes a relations between counting of simple
closed curves and Weil--Petersson volumes of the moduli
spaces of bordered hyperbolic surfaces. Counting pairs of transverse
simple closed curves leads naturally to Masur--Veech volumes of the
moduli spaces of meromorphic quadratic differentials with at most
simple poles.
In this section we state our results on meander
enumeration. The link with quadratic differentials and Masur-Veech
volumes will be explained in Section~\ref{s:strategy}.
\subsection{Counting meanders with given number of minimal arcs}
\label{ss:Asymptotic:number:of:meanders:with:given:number:of:minimal:arcs}
A closed \textit{plane meander} is a smooth
closed curve in the plane transversally intersecting the horizontal
line as in Figure~\ref{fig:meander:types}. According to the
paper~\cite{Lando:Zvonkin} of S.~Lando and A.~Zvonkine (serving as a
reference paper in the literature on meanders) the notion ``meander''
was suggested by V.~I.~Arnold in~\cite{Arnold} though meanders were
studied already by H.~Poincar\'e~\cite{Poincare}. Meanders appear in
various contexts, in particular in physics,
see~\cite{DiFrancesco:Golinelli:Guitter}. Counting meanders has a
reputation of a notoriously difficult problem. The number of meanders
with $2n$ crossings is conjecturally asymptotic to $R^{2n}
n^{-\alpha}$ where $R$ and $\alpha$ are some constants. A conjectural
value of the critical exponent $\alpha$ is given
in~\cite{DiFrancesco:Golinelli:Guitter:2}.
\begin{figure}[hbt]
\special{
psfile=meander_II.eps
hscale=22
vscale=22
voffset=-100
hoffset=10
}
\special{
psfile=meander_I.eps
hscale=22
vscale=22
voffset= -87.5
hoffset=160
}
\begin{picture}(260,10)(0,0)
\put(-20,-105){Contributes to $\cM^+_5(N)$}
\put(160,-105){Contributes to $\cM^-_5(N)$}
\end{picture}
\vspace{100bp}
\caption{
\label{fig:meander:types}
Meander with a maximal arc (``rainbow'')
on the left and without one on the right.
Both meanders have $5$ minimal arcs (``pimples'').
}
\end{figure}
We say that a closed meander has a \textit{maximal arc}
(``\textit{rainbow}'' in terminology of~\cite{Andersen:Chekhov:Penner:Reidys:Sulkowski})
if it has an
arc joining the leftmost and the rightmost crossings with the
horizontal line. Otherwise meander \textit{does not a have maximal
arc}. Meander on the left of Figure~\ref{fig:meander:types} has
maximal arc, while the one on the right -- does not.
By \textit{minimal arc} (``\textit{pimple}'' in terminology
of~\cite{Andersen:Chekhov:Penner:Reidys:Sulkowski}, or
``\textit{internal arch}'' in terminology
of~\cite{DiFrancesco:Golinelli:Guitter}) we call an arc which does
not have any crossings inside. The areas between the horizontal line
and the minimal arcs of meanders are colored in black in
Figure~\ref{fig:meander:types}; each of the two meanders has $p=5$
minimal arcs.
By convention, in this paper we do not consider the trivial closed
meander represented by a circle. All other closed meanders satisfy
$p\ge 3$ when they have a maximal arc and $p\ge 4$ when they do not.
Let $\cM^+_p(N)$ and $\cM^-_p(N)$ be the numbers of closed
meanders respectively with and without maximal arc (``rainbow'') and
having at most $2N$ crossings with the horizontal line and exactly
$p$ minimal arcs (``pimples''). We consider $p$ as a parameter and we
study the leading terms of the asymptotics of $\cM^+_p(N)$ and
$\cM^-_p(N)$ as $N\to+\infty$.
\begin{Theorem}
\label{th:meander:counting}
For any fixed $p$ the numbers $\cM^+_p(N)$ and $\cM^-_p(N)$ of closed
meanders with $p$ minimal arcs (pimples) and with at most $2N$ crossings have
the following asymptotics as $N\to+\infty$:
\begin{align}
\label{eq:asymptotics:with}
\cM^+_p(N) &=2(p+1)\cdot
\frac{\cyl_{1,1}\big(\cQ(1^{p-3},-1^{p+1})\big)}
{(p+1)!\, (p-3)!}
\cdot \frac{N^{2p-4}}{4p-8}\ +\ o(N^{2p-4})=
\\ \notag
&=
\frac{2}{ p!\, (p-3)!}
\left(\frac{2}{\pi^2}\right)^{p-2}\cdot
\binom{2p-2}{p-1}^2\cdot \frac{N^{2p-4}}{4p-8}\ +\ o(N^{2p-4})\,.
\\
\notag
\\
\label{eq:asymptotics:without}
\cM^-_p(N) &=
\frac{2\,\cyl_{1,1}\big(\cQ(1^{p-4},0,-1^p)\big)}
{p!\,(p-4)!}
\cdot \frac{N^{2p-5}}{4p-10}\ +\ o(N^{2p-5})=
\\ \notag
&=
\frac{4}{ p!\, (p-4)!}
\left(\frac{2}{\pi^2}\right)^{p-3}\cdot
\binom{2p-4}{p-2}^2
\cdot \frac{N^{2p-5}}{4p-10}\ +\ o(N^{2p-5})\,.
\end{align}
\end{Theorem}
The quantities $\cyl_{1,1}\big(\cQ(1^{p-3},-1^{p+1})\big)$
and $\cyl_{1,1}\big(\cQ(1^{p-4},0,-1^p)\big)$ in the above formulae
are related to Masur--Veech volumes of the moduli space of meromorphic
quadratic differentials. Their definition and role would be discussed
in section~\ref{s:strategy}. Theorem~\ref{th:meander:counting} is proved in
section~\ref{ss:Proofs:number:of:meanders} with exception for the
explicit expressions for these two quantities evaluated in
Corollary~\ref{cor:principal} in
section~\ref{s:Computations:for:pillowcase:covers}.
Note that the number $\cM^+_p(N)$ grows as $N^{2p-4}$ while
$\cM^-_p(N)$ grows as $N^{2p-5}$. This means that for large $N$ all
but negligible fraction of meanders having any given number $p$ of
minimal arcs (pimples) do have a maximal arc (rainbow) as the left
one in Figure~\ref{fig:meander:types}.
As the reader could observe in the statement of Theorem~\ref{th:meander:counting},
our approach to counting meanders differs from the traditional one:
we fix the combinatorics of the meander and then count the asymptotic
number of meanders of chosen combinatorial type as the number of
intersections $N$ tends to infinity.
Our settings can be seen as a zero temperature
limit in the thermodynamical sense, where the complexity of a
meander is measured in terms of the number of minimal arcs.
Applying Stirling's formula we get the following asymptotics for the
coefficients in formulae~\eqref{eq:asymptotics:with}
and~\eqref{eq:asymptotics:without} for large values of parameter $p$:
\begin{align*}
\frac{2}{ p!\, (p-3)!}
\left(\frac{2}{\pi^2}\right)^{p-2}\cdot
\binom{2p-2}{p-1}^2\cdot\frac{1}{4p-8}
&\ \sim\ \frac{\pi^2}{256}\cdot \left(\frac{32 e^2}{\pi^2 p^2}\right)^p
\hspace*{15pt} \text{ for }p\gg1\,.
\\
\frac{4}{ p!\, (p-4)!}
\left(\frac{2}{\pi^2}\right)^{p-3}\cdot
\binom{2p-4}{p-2}^2\cdot\frac{1}{4p-10}
&\ \sim\ \frac{\pi^2 e^2}{128 p}\cdot \left(\frac{32 e^2}{\pi^2 p^2}\right)^{p-1}
\text{ for }p\gg1\,.
\end{align*}
(we again recall that in our setting we always assume that $N\gg p$).
In section~\ref{ss:Proofs:number:of:meanders} we provide
an analogous statement, Theorem~\ref{gth:meanders:fixed:stratum},
which counts meanders in the setting where the combinatorial type is
specified in a more detailed way.
\subsection{Counting meanders with given reduced arc systems}
\label{ss:Meanders:and:arc:systems}
Extending the horizontal segment of a plane meander
to the infinite line and passing to a one-point compactification of
the plane we get a meander on the 2-sphere. A meander on the
sphere is a pair of transversally intersecting
labeled simple closed curves. It will be always clear from the context whether we
consider meanders in the plane or on the sphere. Essentially, we
follow the following dichotomy: enumerating meanders, as in the
previous section, we work with meanders in the plane, while
considering frequencies of pairs of simple closed curves among more
complicated pairs of multicurves, as in the current section, we work
with meanders on the sphere.
Each meander defines a pair of arc systems in discs as in
Figure~\ref{fig:meander}. An arc system on the disc (also known as a
``chord diagram'') can be encoded by the dual tree, see the trees in
dashed lines on the right pictures in Figure~\ref{fig:meander}.
Namely, the vertices of the tree correspond to the faces in which the
arc system cuts the disc; two vertices are joined by an edge if and
only if the corresponding faces have common arc. It is convenient to
simplify the dual tree by forgetting all vertices of valence two. We call the resulting tree the
\textit{reduced} dual tree.
\begin{figure}[hbt]
\special{
psfile=meander0.eps
hscale=40
vscale=40
voffset=-100
hoffset=15
}
\special{
psfile=meander1.eps
hscale=40
vscale=40
voffset=-111
hoffset=140
}
\special{
psfile=meander_North.eps
hscale=40
vscale=40
voffset=-60
hoffset=270
angle=0
}
\special{
psfile=meander_South.eps
hscale=40
vscale=40
voffset=-135
hoffset=270
angle=0
}
\vspace{135bp}
\caption{
\label{fig:meander}
A closed meander on the left. The associated pair of arc systems
in the middle. The same arc systems on the discs and the associated
dual trees on the right.
}
\end{figure}
It is much easier to count arc systems (for example, arc systems
sharing the same reduced dual tree). However, this does not simplify
counting meanders since identifying a pair of arc systems with the
same number of arcs by the common equator, we sometimes get a meander
and sometimes --- a curve with several connected components, see
Figure~\ref{fig:pairs:of:arc:systems}.
\begin{figure}[htb]
\special{
psfile=meander_tennis_ball_1.eps
hscale=20
vscale=20
voffset=-83
hoffset=56
}
\special{
psfile=meander_tennis_ball_2.eps
hscale=20
vscale=20
voffset=-83
hoffset=210.5
}
\vspace{80bp}
\caption{
\label{fig:pairs:of:arc:systems}
Gluing two hemispheres with arc systems along the common
equator we may get either a single simple closed curve (as on the left
picture) or a multicurve with several connected components (as on the
right picture).
}
\end{figure}
We now consider the more specialized setting where we fix a pair of
plane trees and count meanders whose corresponding pair of arc systems
have these given dual trees. Let us mention that everywhere in this
paper we consider only \textit{plane trees}, that is trees
embedded into the plane.
Let $(\cT_{top}, \cT_{bottom})$ be a pair of plane trees
with no vertices of valence 2. We consider arc
system with the same number of arcs $n\le N$ on a labeled pair of oriented
discs having $\cT_{top}$ and $\cT_{bottom}$ as reduced dual trees. We
draw the arc system corresponding to $\cT_{top}$ on the northern
hemisphere, and the arc system corresponding to $\cT_{bottom}$ on
the southern hemisphere. To simplify gluing of the two hemispheres,
we assume that all segments of the boundary
circle between adjacent endpoints of the arcs have equal length
and that the arcs are orthogonal to the boundary circle.
There are $2n$ ways to isometrically identify the boundaries of two hemispheres
into the sphere in such way that the endpoints of the arcs match.
We consider all possible triples
$$
(\text{$n$-arc system of type $\cT_{top}$;
$n$-arc system of type $\cT_{bottom}$;
identification})
$$
as described above for all $n\le N$. Define
\begin{equation}
\label{eq:p:connected:iota:kappa}
\prob_{\mathit{connected}}(\cT_{top}, \cT_{bottom}; N):=
\frac{\text{
number of triples giving rise to meanders
}}
{\text{
total number of different triples
}}\,.
\end{equation}
\begin{Theorem}
\label{th:trivalent:trees:connected:proportion}
For any pair of \textit{trivalent} plane trees
$\cT_{bottom},\cT_{top}$,
having the total number $p$ of leaves
(vertices of valence one)
the following limit exists:
\begin{multline}
\label{eq:p1:fixed:number:of:poles}
\lim_{N\to+\infty} \prob_{\mathit{connected}}(\cT_{bottom},\cT_{top}; N)
\ =\
\prob_1(\cQ(1^{p-4},-1^p))
\ =\\=\
\frac{\cyl_1(\cQ(1^{p-4},-1^p))}
{\Vol\cQ(1^{p-4},-1^p)}
\ =\
\frac{1}{2}\left(\frac{2}{\pi^2}\right)^{p-3}\cdot
\binom{2p-4}{p-2}\,.
\end{multline}
\end{Theorem}
The quantity $\cyl_{1}\big(\cQ(1^{p-4},-1^p)\big)$ in the above
formula is related to Masur--Veech volume of the moduli space of
meromorphic quadratic differentials. Its definition and role would be
discussed in section~\ref{s:strategy}.
The quantity in this Theorem should be interpreted as the asymptotic probability
$\prob_{\mathit{connected}}(\cT_{bottom},\cT_{top})$ with which a random choice of twist
identifying a pair of random arc systems of fixed combinatorial types
of the same cardinality defines a meander. To be more accurate,
one should rather speak of asymptotic \textit{frequency} of meanders
among resulting multicurves.
Theorem~\ref{th:trivalent:trees:connected:proportion} is proved at the end of
section~\ref{ss:Proofs:fractions}.We will actually
state and prove a more general statement,
Theorem~\ref{th:any:trees:connected:proportion}, where not only trivalent trees
are considered.
The fact that this asymptotic frequency is nonzero is already somehow
unexpected. For example, for the pair of trees as on the right side
of Figure~\ref{fig:meander} the corresponding asymptotic
frequency is equal to
$$
\prob_{\mathit{connected}}(\hspace*{14pt},\hspace*{14pt})=\frac{280}{\pi^6}\approx 0.291245\,,
$$
\mbox{
\special{
psfile=meandre3_up.eps
hscale=40
vscale=40
voffset=20
hoffset=145
angle=60
}
\special{
psfile=meandre3_up.eps
hscale=40
vscale=40
voffset=20
hoffset=153
angle=0
}
}
which is not even close to 0.
Stirling's formula gives the following asymptotics for
$\prob_1(\cQ(1^{p-4},-1^p))$ for large values of parameter $p$:
$$
\prob_1(\cQ(1^{p-4},-1^p))
=
\frac{1}{2}\left(\frac{2}{\pi^2}\right)^{p-3}\cdot
\binom{2p-4}{p-2}
\sim
\frac{2}{\sqrt{\pi p}}\cdot\left(\frac{8}{\pi^2}\right)^{p-3}
\ \text{for}\ p\gg 1
$$
(we recall that in our setting we always assume that $N\gg p$).
Another unexpected fact that follows from
Theorem~\ref{th:trivalent:trees:connected:proportion} is that the way the leaves
(univalent vertices) are distributed between the two trees is
irrelevant: the answer depends only on the total number $p$ of
leaves. This observation suggests an alternative (and much less
restrictive) way to fix combinatorics of the meanders. Namely, we can
fix only the total number $p$ of leaves (vertices of valence one) of
the two trees together, where $p\ge 4$.
\begin{Theorem}
\label{th:p:fixed:number:of:leaves:general}
Let $p \geq 4$.
The frequency $\prob_{\mathit{connected}}(p; N)$ of meanders
obtained by all possible identifications of all arc systems with at most $N$ arcs represented by
all possible pairs of (not necessarily trivalent) plane trees having the total number $p$ of leaves
(vertices of valence one) has the same limit $\prob_1(\cQ(1^{p-4},-1^p))$
as the frequency $\prob_{\mathit{connected}}(\cT_{bottom},\cT_{top}; N)$
of meanders represented by any individual pair of trivalent
plane trees with the total number $p$ of leaves:
\begin{multline}
\label{eq:p1:fixed:number:of:poles:general}
\lim_{N\to+\infty}
\prob_{\mathit{connected}}(p; N)
\ =\
\prob_1(\cQ(1^{p-4},-1^p))
\ =\\=\
\frac{\cyl_1(\cQ(1^{p-4},-1^p))}
{\Vol\cQ(1^{p-4},-1^p)}
\ =\
\frac{1}{2}\left(\frac{2}{\pi^2}\right)^{p-3}\cdot
\binom{2p-4}{p-2}\,.
\end{multline}
\end{Theorem}
Theorem~\ref{th:p:fixed:number:of:leaves:general}
is proved at the end of
section~\ref{s:From:arc:systems:and:meanders:to:pillowcase:covers}.
The proof is based on the fact that the contribution of any pair of
trees where at least one of the trees has a vertex of valency $4$ or
higher is negligible in comparison with the contribution of any pair
of trivalent trees.
\medskip
\noindent\textbf{Acknowledgements.}
The vague idea that counting results concerning
linear involutions (see Appendix~\ref{s:linear:involutions})
might have applications to meanders was discussed
by the authors in independent conversations with M.~Kontsevich and
with \mbox{M.~Mirzakhani} on the early stage of the
project~\cite{DGZZ}. We are grateful to
\mbox{M.~Kontsevich} and to M.~Mirzakhani for
these discussions and for their insights in enumerative geometry
which were very inspiring for us.
We thank MPIM in Bonn, where part of this work was
performed, for providing us with friendly and stimulating environment.
\section{Idea of proof}
\label{s:strategy}
\subsection{Pairs of transverse multicurves on the
sphere as pillowcase covers}
\label{ss:pairs:of:multicurves:as:pillowcase:covers}
A \textit{multicurve} on the sphere, is a collection of pairwise
nonintersecting smooth simple closed curves in the sphere.
\begin{Definition}
\label{def:pair:of:multicurves}
We say that two multicurves on the sphere form a
\textit{transverse connected pair} if any intersection between any connected component of the
first curve and any connected component of the second curve is
transverse and if in addition the union of two multicurves is
connected.
\end{Definition}
Having a transverse connected pair of multicurves we always assume
that the pair is ordered. By convention, the first multicurve is
called ``horizontal'' and the second one --- ``vertical''. We
consider natural equivalence classes of
transverse connected pairs of multicurves
up to diffeomorphisms preserving the orientation of the
sphere and respecting horizontal and vertical labelling.
Let $\cG$ be the graph defined by a transverse connected
pair of multicurves. The vertices of $\cG$ are intersections of the
multicurves, so all vertices of $\cG$ have valence $4$. Hence, all
faces of the dual graph $\cG^\ast$ are $4$-gons. The edges of
$\cG^\ast$ dual to horizontal edges of $\cG$ will be called vertical,
and those dual to the vertical edges of $\cG$ will be called
horizontal. By construction, any two non-adjacent edges of any face
of $\cG^\ast$ are either both horizontal or both vertical.
Let us choose metric squares with side $\frac{1}{2}$ as faces of
$\cG^\ast$. By the Two Color Theorem every plane graph whose faces
all have even numbers of sides is bipartite (see e.~g. \cite{Soifer},
pp.~136--137), so $\cG$ is bipartite. This means that the squares of
$\cG^\ast$ can be colored in the chessboard manner. A square-tiled
surface which admits a chessboard coloring is called \textit{pillowcase
cover}. It is a ramified cover over the standard ``pillow''
(obtained by isometric identification of the boundaries of two
squares with side $\frac{1}{2}$) ramified only over the corners of
the pillow, see Figure~\ref{fig:square:pillow}. By convention we
memorize which sides of the pillow are horizontal, and which ones are
vertical. Conversly, every pillowcase cover defines a transverse
connected pair of multicurves: collections of closed horizontal and
vertical curves passing through the centers of the squares of the
tiling.
\begin{figure}[htb]
\special{
psfile=multicurves_as_pillowcase_cover.eps
hscale=18
vscale=18
voffset=-75
hoffset=30
angle=0
}
\special{
psfile=pillowcase_cover_and_pillow.eps
hscale=50
vscale=50
voffset=-75
hoffset=180
angle=0
}
\vspace{70pt}
\caption{
\label{fig:square:pillow}
Graph dual to a transverse connected pair of multicurves on
a sphere defines a pillowcase cover.}
\end{figure}
We have proved the following statement.
\begin{Proposition}
\label{prop:pairs:multicurves:pillowcase:covers}
There is a natural one-to-one correspondence between
transverse connected pairs of multicurves on the sphere
and pillowcase covers of genus 0, where the square tiling is
given by the graph dual to the graph formed by the union of two
multicurves.
\end{Proposition}
The ``pillow'' as above defines a unique quadratic differential $q_0$
on $\CP$ having simple poles at the four corners of the pillow and no
other singularities. A pillowcase cover is thus endowed with the
meromorphic quadratic differential $q=\pi^\ast q_0$ that is a pullback of the
pillow differential $q_0$ by the covering map $\pi$. Simple poles of
$q$ correspond to bigons of $\cG$; zeroes of order $\degofz\in\N$
correspond to $(2\degofz+4)$-gons.
\begin{NNRemark}
``Pillowcase covers'' are defined differently by different authors. In
the original paper~\cite{Eskin:Okounkov:pillowcase}, A.~Eskin and
A.~Okounkov define pillowcase covers as ramified covers $\pi$ of
degree $2d$ over the sphere having the following ramification
type. They fix a partition $\mu$ with entries $\mu_i\leq 2d$ and a partition $\nu$ of an even
number bounded from above by $2d$ into odd parts. The cover $\pi$ has the profile $[\nu,
2^{d-|\nu|/2}]$ over one corner of the pillow and the profile $[2^d]$
over three other corners. Additionally, $\pi$ has profile $[\mu_i,
1^{2d-\mu_i}]$ over $\ell(\mu)$ distinct non-corner points of the
pillow.
Lemma~B.2 in~\cite{AEZ:genus:0} relates the asymptotic numbers of
pillowcase covers of degree at most $2N$ defined in these two
different ways in the strata of genus 0 moduli spaces as $N\to+\infty$.
\end{NNRemark}
\subsection{ Couting pillowcase covers}
\label{ss:General:strategy}
The moduli space of meromorphic quadratic
differentials on $\CP$ with exactly $p$ simple poles is naturally
stratified by the strata $\cQ(\nu, -1^{|\nu|+4})$ of quadratic
differentials with prescribed orders of zeroes $\nu$ ($\nu_i$ zeroes
of order $i$) and with $p=|\nu|+4$ simple poles (see e.g.
\cite{Zorich:flat:surfaces} for references). Here
\begin{equation}
\label{eq:def:abs:nu}
|\nu|=1\cdot \nu_1 + 2\cdot \nu_2 + 3\cdot \nu_3 + \dots\,.
\end{equation}
Under the above interpretation,
transverse connected pairs of multicurves having fixed number of bigonal faces correspond to
pillowcase covers with fixed number of simple poles.
The transverse connected pairs of multicurves having fixed number $\nu_1$ of
hexagonal faces, fixed number $\nu_2$ of octagonal faces, fixed
number $\nu_\degofz$ of $2(\degofz+2)$-gonal faces for $\degofz\in\N$ correspond to
pillowcase covers in the fixed stratum $\cQ(\nu,-1^{|\nu|+4})$.
In particular, the number of bigonal faces equals $|\nu|+4$.
The number of squares is the total number of crossings between
the two multicurves. Generally speaking, pillowcase covers play the role of integer points
in strata of moduli spaces of quadratic differentials.
Gluing two hemispheres with arc systems along
the common equator, we get a transverse connected pair of multicurves.
The horizontal multicurve has a single connected
component, it is just a simple closed curve represented by the
equator, whereas the vertical multicurve may have several connected components.
Such transverse connected pairs of multicurves correspond to
pillowcase covers having a single horizontal cylinder of height
$\frac{1}{2}$.
Labeled connected pairs of transverse simple closed curves correspond to
pillowcase covers having a single horizontal cylinder of height
$\frac{1}{2}$ and a single vertical cylinder of height $\frac{1}{2}$.
Closed meanders in the plane correspond to pillowcase covers as above
with a marked vertical side of one of the squares of the tiling.
Having translated our counting problems into the language of pillowcase
covers, we are ready to present our approach in detail.
\medskip
\noindent\textbf{Pillowcase covers of fixed combinatorial type and Masur--Vech volumes.}
For any (generalized) partition $\nu=[0^{\nu_0} 1^{\nu_1} 2^{\nu_2}\dots]$
denote by $\Vol\cQ(\nu,-1^{|\nu|+4})$ the Masur--Veech volume of the stratum
$\cQ(\nu,-1^{|\nu|+4})$ of genus 0 meromorphic quadratic differentials with
at most simple poles (for the precise definition of the Masur--Veech volume see e.~g.~\cite{AEZ:genus:0}).
Then the following formula holds:
\begin{equation}
\label{eq:volume}
\Vol\cQ(\nu,-1^{|\nu|+4}):=2\pi^2\cdot
(f(0)\big)^{\nu_0}(f(1)\big)^{\nu_1}(f(2)\big)^{\nu_2}\cdots\,,
\end{equation}
where $|\nu|=1\cdot\nu_1+2\cdot\nu_2+\dots$ and
$$
f(\degofz)=
\frac{\degofz\,!!}{(\degofz+1)!!}\cdot\pi^{\degofz}\cdot
\begin{cases}
\pi&\text{if $\degofz$ is odd}\\
2 &\text{if $\degofz$ is even}\,.
\end{cases}
$$
(here we use the notation
$$
\degofz\,!!:=
\begin{cases}
1\cdot 3\cdot 5 \dots \cdot \degofz, &\text{when $\degofz$ is odd,}
\\
2\cdot 4\cdot 6 \dots \cdot \degofz, &\text{when $\degofz$ is even}\,.
\end{cases}
$$
and the common convention $0!!:=1$).
This formula was originally
conjectured by M. Kontsevich and recently proved in~\cite{AEZ:genus:0}.
In this setting zeroes and poles of quadratic differentials
are \textit{labeled}.
As it follows from the definition of the Masur--Veech volume,
the number of pillowcase covers in the stratum
$\cQ(\nu,-1^{|\nu|+4})$ in the moduli space of meromorphic quadratic
differentials with labeled zeroes and poles tiled with at
most $2N$ squares has asymptotics
\begin{equation}
\label{eq:VolQ:N:d}
\Vol\cQ(\nu,-1^{|\nu|+4}) \cdot \frac{N^d}{2d} + o(N^d)
\ \text{ as }\ N\to+\infty\,,
\end{equation}
where
\begin{equation}
\label{eq:dim}
d=\dim_{\mathbb{C}}\cQ(\nu,-1^{|\nu|+4})=\ell(\nu)+|\nu|+2
\end{equation}
and
\begin{equation}
\label{eq:def:ell:nu}
\ell(\nu):=\nu_0+\nu_1+\dots\,.
\end{equation}
As ``combinatorial type'' of a pillowcase cover one can use the
number $p$ of bigons, as in Theorems~\ref{th:meander:counting}
and~\ref{th:p:fixed:number:of:leaves:general}.
In this setting formulae~\eqref{eq:VolQ:N:d}
and~\eqref{eq:dim} imply that all but negligible part of
transverse connected pairs of multicurves having large
number $N$ of intersections would have only
bigons, squares, and hexagons as faces and would correspond to
pillowcase covers in the principal stratum $\cQ(1^{p-4},-1^p)$.
As an alternative choice of ``combinatorial type'' of a pillowcase
cover one can specify the number of hexagons, octagons, etc,
separately, thus fixing the stratum $\cQ(\nu,-1^{|\nu|+4})$. This
corresponds to the setting of
Theorems~\ref{th:any:trees:connected:proportion}
and~\ref{gth:meanders:fixed:stratum} below.
Under either choice we have a simple asymptotic formula for the number of
transverse connected pairs of multicurves of fixed combinatorial type
with at most $2N$ intersections.
\begin{Remark}[Labeled versus non-labeled zeroes and poles]
\label{rm:labeled:zeroes:and:poles}
When we introduced pillowcase covers in
section~\ref{ss:pairs:of:multicurves:as:pillowcase:covers} and
identified them with transverse connected pairs of multicurves in
Proposition~\ref{prop:pairs:multicurves:pillowcase:covers}, we did
not label zeroes and poles of the corresponding quadratic differential,
which was quite natural in this setting.
Traditionally, one labels zeroes and poles of a pillowcase cover in
the contex of Masur--Veech volumes. So we do label zeroes and poles
in the current section.
We remind the setting every time when there may be any ambiguity.
\end{Remark}
\medskip
\noindent\textbf{Pillowcase covers with a single horizontal cylinder.}
Enumeration of pillowcase covers with a single
horizontal cylinder was performed in~\cite{DGZZ}. In
Section~\ref{s:Computations:for:pillowcase:covers} we reproduce the
relevant computations in the case of the sphere. The number of
pillowcase covers tiled with at most $2N$ squares lying in the stratum $\cQ(\nu,-1^{|\nu|+4})$
and having a single horizontal cylinder of minimal possible height
$\frac{1}{2}$ has asymptotics
\begin{equation}
\label{eq:c1:N:d}
\cyl_1(\cQ(\nu,-1^{|\nu|+4})) \cdot \frac{N^d}{2d} + o(N^d)
\ \text{ as }\ N\to+\infty\,,
\end{equation}
where the coefficient $\cyl_1(\cQ(\nu,-1^{|\nu|+4}))$
is positive and is given by the explicit formula (see~\eqref{eq:c1:answer}) that is particularly simple
for the principal stratum, see \eqref{eq:c1:principal:answer}. (Here we assume that zeroes and poles of the corresponding quadratic differentials are labeled.)
\medskip
\noindent\textbf{Pillowcase covers with a single horizontal and a single
vertical cylinder.}
We are particularly interested in counting
pillowcase covers having
a single horizontal cylinder of height $\frac{1}{2}$ and a single
vertical cylinder of width $\frac{1}{2}$.
The number $\cP^{\mathit{labeled}}_\nu(N)$ of genus 0 pillowcase covers in the stratum $\cQ(\nu, -1^{|\nu|+4})$
with at most $2N$ squares having a single horizontal cylinder
of height $1/2$ and a single vertical cylinder of width $1/2$ has
asymptotics
\begin{equation}
\label{eq:c11:Q:nu}
\cP^{\mathit{labeled}}_\nu(N)=
\cyl_{1,1}\left(\cQ(\nu,-1^{|\nu|+4})\right) \cdot \frac{N^d}{2d} + o(N^d)
\text{ as } N\to+\infty\,,
\end{equation}
where the constant $\cyl_{1,1} \left(\cQ(\nu,-1^{|\nu|+4}\right)$
satisfies the relation
\begin{equation}
\label{eq:c11:as:c1:squared:over:Vol}
\cyl_{1,1}(\cQ(\nu,-1^{|\nu|+4}))=
\frac{\big(\cyl_1(\cQ(\nu,-1^{|\nu|+4}))\big)^2}{\Vol\cQ(\nu,-1^{|\nu|+4})}\,.
\end{equation}
The both statements are formulated more precisely in Theorem~\ref{th:c1:in:genus:0}
in Section~\ref{s:Computations:for:pillowcase:covers} and follow from the results
of~\cite{DGZZ}.
The relation~\ref{eq:c11:as:c1:squared:over:Vol} can be viewed as a
statement about independence of horizontal and vertical decompositions
of pillowcase covers: the asymptotic fraction
of pillowcase covers having a single horizontal cylinder of height
$\frac{1}{2}$ among all pillowcase covers in $\cQ(\nu,-1^{|\nu|+4})$
tiled with at most $2N$ squares is the same as the asymptotic fraction
of pillowcase covers having a single horizontal cylinder of height
$\frac{1}{2}$ and a single vertical cylinder of width $\frac{1}{2}$
among all pillowcase covers having a single vertical cylinder of width
$\frac{1}{2}$ and tiled with at most $2N$ squares.
Forgetting the labeling of zeroes and poles we get the asymptotics of
the number of connected pairs of transverse simple closed curves of fixed
combinatorial type with at most $2N$ crossings.
\medskip
\noindent\textbf{Further remarks.}
It is worth mentioning that all the above quantities have combinatorial nature,
but were computed by alternative methods. The Masur--Veech volumes in
genus zero are closely related to Hurwitz numbers counting
covers of the sphere of some very special
ramification type. However, all attempts to compute these volumes by purely
combinatorial methods have (up to now) failed even for covers
of the simplest ramification type, see e.~g.~\cite{AEZ:Dedicata}.
The proof in~\cite{AEZ:genus:0} of the
formula for the Masur--Veech volumes implicitly uses the analytic
Riemann--Roch theorem in addition to combinatorics.
The result about pillowcase covers with a single horizontal and a single
vertical cylinder is proved in~\cite{DGZZ} using ergodicity of the
Teichm\"uller geodesic flow with respect to the Masur--Veech measure
and Moore's ergodicity theorem. The proof was inspired by the approach of
M.~Mirzakhani to counting simple closed geodesics on hyperbolic surfaces.
\section{From arc systems and meanders to pillowcase covers}
\label{s:From:arc:systems:and:meanders:to:pillowcase:covers}
In this section we give precise bijections between meanders and
pillowcase covers with
a single maximal cylinder in both horizontal and
vertical directions. We consider meanders in the plane
in sections~\ref{ss:orientation}--\ref{ss:Meanders:and:pilowcase:covers:in:a:given:stratum}
and meanders on the sphere in section~\ref{ss:Proofs:fractions}.
\subsection{Orientation, marking and weight}
\label{ss:orientation}
We have seen in
Proposition~\ref{prop:pairs:multicurves:pillowcase:covers} from
Section~\ref{ss:pairs:of:multicurves:as:pillowcase:covers} that
transverse connected pairs of multicurves on the sphere are in bijection with pillowcase covers of genus 0.
A pillowcase cover arising from a pair of arc systems has
a single horizontal cylinder of height $\frac{1}{2}$. In particular, a
pillowcase cover arising from a meander has a single horizontal and
a single vertical cylinder of height
(respectively width) $\frac{1}{2}$.
However, pairs of arc systems and meanders (both
in the plane and on the sphere) carry an extra marking.
Namely, a pair of arc systems comes with a given choice of a top and
bottom sides. Furthermore, the pillowcase cover
corresponding to a \textit{plane} meander has a special square corresponding
to the leftmost intersection. Summarizing, we get the following
result:
\begin{Lemma}
\label{lem:bijection:meander:pillowcase:cover}
There is a natural bijection between
meanders in the plane and
pillowcase covers with a marked
oriented vertical side of one of the squares that
have a single horizontal and a single vertical cylinder of height (width)
$\frac{1}{2}$.
\end{Lemma}
In order to provide exact counting of meanders
we present the conventions for counting pillowcase covers and see
how these quantities are related to arc systems and meander counting. The pillowcase
covers considered in Lemma~\ref{lem:bijection:meander:pillowcase:cover}
are not well suited for counting. We will consider pillowcase covers
with a marked \textit{vertex} of the square tiling.
\begin{Convention}
\label{conv:marked:on:top}
By convention, the marked vertex is located at the
end of the marked oriented vertical edge on the top boundary
component of the single horizontal cylinder.
\end{Convention}
Note that the two boundary components of the single
horizontal cylinder do not intersect. Thus, the marked vertex
uniquely defines the top boundary component and, hence,
provides us with the canonical orientation of the
waist curve of the single horizontal cylinder.
Let us reconstruct the labeled pair of arc systems in the plane from
a pillowcase cover of genus zero tiled with a single horizontal band
of squares and having a marked vertex. If the marked vertex of the square tiling is a simple pole of the
quadratic differential, there is a single vertical side of the
square tiling adjacent to it, and the choice of the vertical side is
canonical. If the marked vertex of the square tiling is a regular point of the
quadratic differential, there are two adjacent vertical sides, so
there are two ways to chose a distinguished vertical side which,
generally, lead to two different arc systems. We say ``generally''
because it might happen that the pillowcase cover is particularly
symmetric (like pillowcase covers associated to arc systems from
Figure~\ref{fig:pairs:of:arc:systems}) and the resulting two arc
systems are isomorphic.
As soon as we are interested only in the asymptotic counting we can
simply neglect this issue: the pillowcase covers with extra
symmetries occur too rarely to affect the asymptotics. To perform
exact count we establish the following standard Convention.
\begin{Convention}
\label{conv:symmetry}
We always count a marked or non-marked pillowcase cover with a weight
reciprocal to the order $|\Aut|$ of the automorphism group of the
cover. In the current context we keep track of which sides
of the pillowcase cover are horizontal and which ones are vertical,
but we do not label either the sides or the vertices of the pillowcase
cover. By definition, the automorphism group $\Aut$ acts by flat
isometries sending horizontal (respectively vertical) sides of the
tiling to horizontal (respectively vertical) sides and keeping the
marked point (if any) fixed.
\end{Convention}
In particular, if we have a marked point at a regular vertex of a
pillowcase cover, the automorphism group is either trivial or $\Z/2\Z$.
If we have a marked point at a zero of degree $\degofz$ of a pillowcase
cover, the automorphism group is a (usually trivial) subgroup of the
cyclic group $\Z/(\degofz+2)\Z$.
\subsection{Meanders with a given number of minimal arcs and pillowcase covers}
In this section and in the next one we continue to work
with \textit{plane} meanders.
Under Conventions~\ref{conv:marked:on:top} and~\ref{conv:symmetry}, any
collection of weighted pillowcase covers on the sphere with a single
band of horizontal squares and with a marked regular point defines
twice as much arc systems; the weighted collection of pillowcase covers
as above with a marked zero of degree $\degofz$ defines
$(\degofz+2)$ times more arc systems for any $\degofz\in\N$.
\begin{Lemma}
Let the initial closed meander in the plane have $p$ minimal arcs,
where $p\ge 3$. The associated pillowcase cover has $p+1$ simple
poles if the initial meander has a maximal arc and $p$ simple poles
if it does not.
\end{Lemma}
\begin{proof}
A maximal arc becomes indistinguishable from a minimal arc after
passing to a labeled pair of transverse simple closed curves on the
sphere. Minimal and maximal arcs are in bijective correspondence with
bigons in the partition of the sphere by the union of these transverse
simple closed curves. Bigons, in turn, are in bijective
correspondence with simple poles of the associated pillowcase cover.
\end{proof}
Recall that $\cM_p^+(N)$ and $\cM_p^-(N)$ denote the number of
meanders with $p$ minimal arcs and respectively with and without
a maximal arc. Denote $\cP_p(N)$ the number of pillowcase covers of genus zero
tiled with at most $2N$ squares, having exactly $p$ simple poles,
a single horizontal cylinder of height $\frac{1}{2}$ and a single
vertical cylinder of width $\frac{1}{2}$. Denote by
$\cP_{p,\degofz}(N)$, where $\degofz=0,1,2,\dots$, the number of
pillowcase covers as above having in addition a marked point at a
regular vertex when $\degofz=0$ and at a zero of order $\degofz$ when
$\degofz>0$.
Note that a pillowcase cover of genus $0$ with $p$
simple poles cannot have zeroes of order greater than $p-4$.
\begin{Lemma}
\label{lm:M:through:P}
Under Convention~\ref{conv:symmetry} on the weighted count of
pillowcase covers the following equalities hold:
\begin{align}
\label{eq:Mplus:P}
\cM^+_p(N)&= 2(p+1)\cdot \cP_{p+1}(N)
\\
\label{eq:Mminus:P}
\cM^-_p(N) &= \sum_{\degofz=0}^{p-4} (\degofz+2)\cdot \cP_{p,\degofz}(N)
\,-\,\frac{1}{2}\,\cM_{p-1}^+(N)
\,.
\end{align}
\end{Lemma}
\begin{proof}
If the meander has $2n$ intersections, then the associated pillowcase
cover is tiled with $2n$ squares with side $\frac{1}{2}$.
To every closed meander with a maximal arc and with $p$ minimal arcs
we associated a canonical pillowcase cover of genus zero with $p+1$
simple poles, a single horizontal cylinder of height $\frac{1}{2}$ and
a single vertical cylinder of width $\frac{1}{2}$, see
Proposition~\ref{prop:pairs:multicurves:pillowcase:covers}.
Conversely, to every such pillowcase cover we can associate $2(p+1)$
meanders with one maximal arc and $p$ minimal arcs. Indeed, choose
any of the $(p+1)$ simple poles and choose independently one of the
two possible orientations of the waist curve of the horizontal
cylinder. Cutting this waist curve at the intersection with the
single vertical edge of the square tiling adjacent to the selected
pole we get a closed meander in the plane with a maximal arc.
It might happen that some of the resulting $2(p+1)$ meanders are
pairwise isomorphic. However, this implies that the automorphism group of
the pillowcase cover is nontrivial, and
Convention~\ref{conv:symmetry} provides the exact count. This
completes the proof of equality~\eqref{eq:Mplus:P}.
Similarly, to every closed meander without a maximal arc and with $p$
minimal arcs we assigned a canonical pillowcase cover of genus zero
having $p$ simple poles, a single horizontal cylinder of height
$\frac{1}{2}$, a single vertical cylinder of width $\frac{1}{2}$, and a
marked vertex following Convention~\ref{conv:marked:on:top}. The
assumption that the initial meander does not have any maximal arc
excludes coincidence of the marked point with a simple pole on
the top side. In order to exclude a maximal arc on the bottom side,
one needs to subtract a half of $\cM_{p-1}^+(N)$ (that is, $p\cdot\cP_{p,-1}(N)$). At
the end of section~\ref{ss:orientation} we have seen that under
Convention~\ref{conv:symmetry} on the weight with which we count
pillowcase covers with a marked vertex, any collection of weighted
pillowcase covers on the sphere with a single horizontal cylinder of
height $\frac{1}{2}$ and with a marked regular point defines twice as
much closed meanders in the plane; the weighted collection of
pillowcase covers as above with a marked zero of degree $\degofz$
defines $(\degofz+2)$ times more closed meanders in the plane for any
$\degofz\in\N$. As before, if some of the resulting
meanders are isomorphic we do not count them several times since by
definition of the automorphism group $\Aut$ of the corresponding
pillowcase cover, the resulting multiplicity coincides with the order
$|\Aut|$ of the automorphism group. This completes the proof of
equality~\eqref{eq:Mminus:P}.
\end{proof}
\subsection{Meanders and pilowcase covers in a given stratum}
\label{ss:Meanders:and:pilowcase:covers:in:a:given:stratum}
We now introduce finer counting with respect to a fixed stratum.
For a partition $\nu=[1^{\nu_1} 2^{\nu_2} \dots]$ denote by
$\cM^+_\nu(N)$ and $\cM^-_\nu(N)$ the number of meanders
leading to pillowcase covers in the
stratum $\cQ(\nu, -1^{|\nu|+4})$ of meromorphic quadratic
differentials
respectively with a maximal arc and
without maximal arcs.
We say that such meanders are \textit{of type} $\nu$.
Similarly, let $\cP_\nu(N)$ be
the number of pillowcase covers in the stratum $\cQ(\nu,-1^{|\nu|+4})$ of
genus zero tiled with at most $2N$ squares, with a single
horizontal cylinder of height $\frac{1}{2}$ and a single vertical
cylinder of width $\frac{1}{2}$. Denote by $\cP_{\nu,\degofz}(N)$,
$\degofz=0,1,2,\dots$, the number of pillowcase covers as above having in
addition a marked point at a regular vertex when $\degofz=0$ and at a zero
of order $\degofz$ when $\degofz>0$. By definition, we let $\cP_{\nu,\degofz}(N)=0$ for
any $N$ when $\nu_\degofz=0$. Recall that by Convention~\ref{conv:symmetry}
we count pillowcase covers with weights reciprocal to the orders of
their automorphism groups.
\begin{Lemma}
\label{lm:M:through:P:nu}
Under Convention~\ref{conv:symmetry} on weights with which we count
pillowcase covers the following equalities hold
\begin{align}
\label{eq:Mplus:P:nu}
\cM^+_\nu(N)&= 2(|\nu|+4)\cdot \cP_\nu(N)\,
\\
\label{eq:Mminus:P:nu}
\cM^-_\nu(N) &= \sum_{\degofz=0}^{|\nu|} (\degofz+2)\cdot \cP_{\nu,\degofz}(N)
\,-\,\frac{1}{2}\,\cM_{\nu}^+(N)
\,.
\end{align}
\end{Lemma}
\begin{proof}
The proof is completely analogous to the proof of
Lemma~\ref{lm:M:through:P}.
\end{proof}
\subsection{Asymptotic frequency of meanders: general setting}
\label{ss:Proofs:fractions}
In this section we return to meanders on the sphere.
Let $\cT$ be a plane tree. We associate to $\cT$
a generalized integer partition
$\nu = \nu(\cT) = [0^{\nu_0} 1^{\nu_1} 2^{\nu_2} \ldots]$ where $\nu_\degofz$ denotes the number
of internal vertices of valence $\degofz+2$ for $\degofz\in\N$. The number of leaves, or
equivalently of vertices of valence $1$, is then expressed in terms of the (generalized)
partition $\nu$ as $2+|\nu|$ where $|\nu|$ is the sum of the partition (see~\eqref{eq:def:abs:nu}).
Given two generalized partitions $\iota = [0^{\iota_0} 1^{\iota_1} 2^{\iota_2} \dots]$ and
$\kappa= [0^{\kappa_0} 1^{\kappa_1} 2^{\kappa_2} \dots]$ we define their sum as
$\nu=\iota+\kappa= [0^{\iota_0+\kappa_0} 1^{\iota_1+\kappa_1} 2^{\iota_2+\kappa_2}
\dots]$. We say that $\iota$ is a \textit{subpartition} of $\nu$ and denote it
as $\iota \subset \nu$. For a subpartition $\iota\subset\nu$ we define
the difference $\kappa=\nu-\iota$.
The following Lemma recalls what graphs
of horizontal saddle connections have horizontally one-cylinder
pillowcase covers in a given stratum of meromorphic quadratic
differentials in genus zero.
\begin{Lemma}
A ribbon graph $\cD$ represents the graph of horizontal saddle
connections of some pillowcase cover in a stratum
$\cQ(\nu,-1^{|\nu|+4})$ having a single horizontal cylinder if and only
if it is represented by a pair of plane trees with associated
partitions $\nu_{top}$ and $\nu_{bottom}$
such that the sum $\nu_{top}+\nu_{bottom} = \nu$.
\end{Lemma}
\begin{proof}
The Lemma was proved in section~\ref{ss:General:strategy}.
\end{proof}
We formulate and prove the following generalization of
Theorem~\ref{th:trivalent:trees:connected:proportion} giving a formula for the
limit of the fraction~\eqref{eq:p:connected:iota:kappa} of meanders
which we get identifying arc systems of types $\cT_{top}$ and
$\cT_{bottom}$ with the same number of arcs, see
Figure~\ref{fig:pairs:of:arc:systems}.
Though we agreed in section~\ref{ss:Meanders:and:arc:systems}
to consider reduced trees, suppressing
the vertices of valence $2$, it is often convenient to keep several
marked points, so we state the Theorem below in this slightly more
general setting. Note that since $f(0)=2$, the number $\nu_0$ of zeroes in the
partition $\nu$ affects the value of the
function $\Vol\cQ(\nu,-1^{|\nu|+4})$. Adding an extra marked point
we double the Masur--Veech volume of the corresponding stratum.
\begin{Theorem}
\label{th:any:trees:connected:proportion}
For any pair of plane trees $\cT_{top}, \cT_{bottom}$ with
associated generalized partitions $\nu_{top}$ and $\nu_{bottom}$
the following limit exists and is positive:
$$
\lim_{N\to+\infty} \prob_{\mathit{connected}}(\cT_{top}, \cT_{bottom}; N)
=
\prob_1(\cQ(\nu,-1^{|\nu|+4}))>0\,
$$
where $\nu = \nu_{top} + \nu_{bottom}$ and $\prob_1(\cQ(\nu,-1^{|\nu|+4}))$
is defined by
\begin{equation}
\label{eq:probability}
\prob_1\left(\cQ(\nu,-1^{|\nu|+4})\right)
=
\frac{\cyl_1(\cQ(\nu,-1^{|\nu|+4}))}
{\Vol\cQ(\nu,-1^{|\nu|+4})}\,.
\end{equation}
Here $\Vol\cQ(\nu,-1^{|\nu|+4})$ is given by formula~\eqref{eq:volume},
and $\cyl_1(\cQ(\nu,-1^{|\nu|+4}))$ takes the value
\begin{equation}
\label{eq:c1:nu}
\cyl_1(\cQ(\nu,-1^{|\nu|+4}))
= 2\sum_{\mu\subset\nu}
\binom{|\nu|+4}{|\mu|+2}
\binom{\nu_0}{\mu_0}
\binom{\nu_1}{\mu_1}
\binom{\nu_2}{\mu_2}
\cdots
\,.
\end{equation}
\end{Theorem}
\begin{proof}
The trees $\cT_{top}$ and $\cT_{bottom}$ represent the trees formed
by the horizontal saddle connections of the pillowcase cover.
Vertices of valence one are in bijective correspondence with simple
poles. Vertices of valence two represent marked points (if any).
Vertices of valence $\degofz+2$ are in bijective correspondence with
zeroes of degree $\degofz$ for $\degofz\in\N$. Recall that the type
$\nu=[1^{\nu_1} 2^{\nu_2} 3^{\nu_3} \dots]$ of the graph
$\cT_{bottom}\sqcup\cT_{top}$ encodes the total number $\nu_\degofz$
of vertices of valence $\degofz+2$ in $\cT_{bottom}\sqcup\cT_{top}$
for $\degofz\in\N$. We conclude that a pair of arc systems having
$\cT_{bottom}$ and $\cT_{top}$ as dual trees defines a pillowcase
cover in the stratum $\cQ(\nu,-1^{|\nu|+4})$ of meromorphic
quadratic differentials.
We are ready to express the numerator and the denominator
of~\eqref{eq:p:connected:iota:kappa} in terms of pillowcase covers.
First, note that arc systems
are defined on a pair of labeled
oriented discs (called top and bottom, or northern and southern
hemispheres).
When $\cT_{bottom}$ and $\cT_{top}$ are not isomorphic as ribbon
graphs, the ``total number of different triples'' in the denominator
of~\eqref{eq:p:connected:iota:kappa} is equal to the weighted number
of pillowcase covers tiled with at most $2N$ squares that form a single
horizontal band and having the non-labeled ribbon graph
$\cD:=\cT_{bottom}\sqcup\cT_{top}$ as the diagram of horizontal
saddle connections. Here we identify triples leading to isomorphic
pairs of labeled multicurves. We do not label either of components,
vertices, or edges of the graph $\cD$, but we consider the plane
trees $\cT_{bottom}$ and $\cT_{top}$ as ribbon graphs, so the
corresponding topological discs are oriented.
When $\cT_{bottom}$ and $\cT_{top}$ are
isomorphic as ribbon graphs, the ``total number of different
triples'' in the denominator of~\eqref{eq:p:connected:iota:kappa} is
equal to twice the weighted number of pillowcase covers as above.
When $\cT_{bottom}$ and $\cT_{top}$ are not isomorphic as ribbon
graphs, the ``number of triples leading to meander'' in the numerator
of~\eqref{eq:p:connected:iota:kappa} is equal to the weighted number
of pillowcase covers as above which in addition have a single vertical
cylinder. When $\cT_{bottom}$ and $\cT_{top}$ are isomorphic, the
``number of triples leading to a meander'' is twice the weighted number
of pillowcase covers as above that have a single vertical cylinder.
Thus, the limit in
Theorem~\ref{th:any:trees:connected:proportion} is the asymptotic fraction of
pillowcase covers having a single horizontal cylinder of height
$\frac{1}{2}$ corresponding to the separatrix diagram $\cD$ of
horizontal saddle connections, and a single vertical cylinder of
width $\frac{1}{2}$ among all pillowcase covers with a single
horizontal cylinder of height $\frac{1}{2}$ corresponding to the
separatrix diagram $\cD$ of horizontal saddle connections.
Theorem~1.19 in~\cite{DGZZ} asserts that such limit exists and that the
``horizontal and vertical cylinder decompositions are asymptotically
uncorrelated'', so the above limit coincides with the asymptotic
fraction of pillowcase covers having a single vertical cylinder of
width $\frac{1}{2}$ among all pillowcase covers -- we can omit the
conditions on the horizontal foliation. This proves existence of the
limit in Theorem~\ref{th:any:trees:connected:proportion}. By definition, the
latter asymptotic fraction is precisely the quantity
$\prob_1(\cQ(\nu,-1^{|\nu|+4}))$ which proves the second statement
in Theorem~\ref{th:any:trees:connected:proportion} together with
formula~\eqref{eq:probability}. The remaining
formula~\eqref{eq:c1:nu} for the quantity
$\cyl_1(\cQ(\nu,-1^{|\nu|+4}))$ introduced in~\eqref{eq:c1:N:d} will be
derived in Theorem~\ref{th:c1:in:genus:0} of
section~\ref{s:Computations:for:pillowcase:covers}. This completes
the proof of Theorem~\ref{th:any:trees:connected:proportion}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:trivalent:trees:connected:proportion}]
Theorem~\ref{th:trivalent:trees:connected:proportion} is a particular case
of the Theorem~\ref{th:any:trees:connected:proportion} when the
plane trees $\cT_{bottom},\cT_{top}$
are trivalent and
have the total number $p$ of leaves
(vertices of valence one). In this situation $\nu=[1^{p-4}\ -1^p]$.
By Theorem~\ref{th:any:trees:connected:proportion} we have
$$
\lim_{N\to+\infty} \prob_{\mathit{connected}}(\cT_{bottom},\cT_{top}; N)
=
\prob_1(\cQ(1^{p-4},-1^p))\,.
$$
We apply now formula~\eqref{eq:p1:with:and:without:0}
proved in Corollary~\ref{cor:leading:0} in section~\ref{s:Computations:for:pillowcase:covers}
which states that
$$
\prob_1(\cQ(1^{p-4},-1^p))=
\frac{\cyl_1(\cQ(1^{p-4},-1^p))}
{\Vol\cQ(1^{p-4},-1^p)}\,.
$$
It remains to apply
formula~\eqref{eq:c1:principal:answer} for the numerator and
equation~\eqref{eq:volume} for the denominator of the latter quantity
to complete the proof. We have proved
Theorem~\ref{th:trivalent:trees:connected:proportion} conditional to
Corollaries~\ref{cor:principal} and~\ref{cor:leading:0} left to
section~\ref{s:Computations:for:pillowcase:covers}.
\end{proof}
\subsection{Counting meanders of special combinatorial types}
\label{ss:Proofs:number:of:meanders}
In this section we return to plane meanders
with exception for the
Proof of Theorem~\ref{th:p:fixed:number:of:leaves:general} at the
very end of the section, where we work with meanders on the sphere.
We state now an analog of Theorem~\ref{th:meander:counting}, where
instead of the number of minimal arcs (pimples) we use the partition
$\nu$ as a combinatorial passport of the meander.
\begin{Theorem}
\label{gth:meanders:fixed:stratum}
For any partition $\nu=[1^{\nu_1} 2^{\nu_2} 3^{\nu_3} \dots]$,
the number
$\cM^+_\nu(N)$ (respectively $\cM^-_\nu(N)$) of closed
plane meanders of type $\nu$, with (respectively without)
a maximal arc and with at most $2N$ crossings has
the following asymptotics as $N\to+\infty$:
\begin{align}
\label{eq:asymptotics:nu:plus}
\cM^+_\nu(N)&=
2(|\nu|+4)\cdot
\frac{\cyl_{1,1}\big(\cQ(\nu,-1^{|\nu|+4})\big)}
{(|\nu|+4)!\cdot\prod_j \nu_j!}
\cdot
\frac{N^{\ell(\nu)+|\nu|+2}}{2\ell(\nu)+2|\nu|+4}
\ +\\
\notag
&\hspace{2.7in} +
o\big(N^{\ell(\nu)+|\nu|+2}\big)\,,
\\
\label{eq:asymptotics:nu:minus}
\cM^-_\nu(N)&=
\frac{2\,\cyl_{1,1}\big(\cQ(\nu,0,-1^{|\nu|+4})\big)}
{(|\nu|+4)!\cdot\prod_j \nu_j!}
\cdot
\frac{N^{\ell(\nu)+|\nu|+3}}{2\ell(\nu)+2|\nu|+6}
\ +\\
\notag
&\hspace{2.7in} +
o\big(N^{\ell(\nu)+|\nu|+3}\big)\,,
\end{align}
where
\begin{equation}
\label{eq:c11:with:0:through:c11:without:0}
\cyl_{1,1}\big(\cQ(\nu,0,-1^{|\nu|+4})\big)
=
2\cdot \cyl_{1,1}\big(\cQ(\nu,-1^{|\nu|+4})\big)
\end{equation}
and
\begin{multline}
\label{eq:c11:nu:in:Th4}
\cyl_{1,1}\big(\cQ(\nu,-1^{|\nu|+4})\big)
=\\=
\frac{4}{\Vol\cQ(\nu,-1^{|\nu|+4})}\cdot
\left(
\sum_{\iota_1=0}^{\nu_1}
\sum_{\iota_2=0}^{\nu_2}
\sum_{\dots}^{\dots}
\binom{\nu_1}{\iota_1}
\binom{\nu_2}{\iota_2}
\cdots
\binom{|\nu|+4}{|\iota|+2}
\right)^2
\end{multline}
\end{Theorem}
Note that contrary to the original Theorem~\ref{th:meander:counting},
where the setting is somewhat misleading, in the setting of
Theorem~\ref{gth:meanders:fixed:stratum} we get more natural
formula $\cM^+_\nu(N)=o\big(\cM^-_\nu(N)\big)$ as $N\to+\infty$.
Up to now we performed the exact count. The Lemma below gives the
term with dominating contribution to the asymptotic count when
the bound $2N$ for
the number of squares in the pillowcase cover tends to
infinity.
\begin{Lemma}
\label{lm:P:Pprincipal}
The following limits hold
\begin{align}
\label{eq:P:pN:to:principal}
\lim_{N\to+\infty} \frac{1}{\cP_{1^{p-4}}(N)}\cdot
\cP_p(N)
&= 1\,,
\\
\label{eq:Pd:pN:to:principal:0}
\lim_{N\to+\infty} \frac{1}{2\,\cP_{1^{p-4},0}(N)}\cdot
\left(\sum_{\degofz=0}^{p-4} (\degofz+2)\cdot\cP_{p,\degofz}(N)\right)
&= 1\,,
\\
\label{eq:P:nu:d:to:P:nu:0}
\lim_{N\to+\infty} \frac{1}{2\,\cP_{\nu,0}(N)}\cdot
\left(\sum_{\degofz=0}^{|\nu|} (\degofz+2)\cdot \cP_{\nu,\degofz}(N)\right) &=1\,.
\end{align}
\end{Lemma}
\begin{proof}
Let $\nu=[1^{\nu_1} 2^{\nu_2}\dots]$, where $|\nu|=p-4$ be a
partition of the number $p-4$ into the sum of positive
integers $\degofz_1,\dots,\degofz_\noz$, where
$\degofz_1+\dots+\degofz_\noz=p-4$. By Theorem~1.19 in~\cite{DGZZ},
\begin{equation}
\label{eq:P:nu:as:c11}
\cP_\nu(N) = \frac{\cyl_{1,1}\big(\cQ(\nu,-1^p)\big)}
{p!\cdot \prod_j \nu_j!}
\cdot
\frac{N^d}{2d} + o(N^d)\,,
\text{ when }N\to+\infty\,,
\end{equation}
where the constant $\cyl_{1,1}\left(\cQ(\nu,-1^p)\right)$ defined
in~\eqref{eq:c11:Q:nu} is positive, and
\begin{equation}
\label{eq:d}
d=\dim_{\mathbb{C}} \cQ(\nu,-1^{|\nu|+4})=
\dim_{\mathbb{C}} \cQ(\degofz_1,\dots,\degofz_\noz,-1^p)=\noz+p-2\,.
\end{equation}
For a given number $p\ge 4$ of simple poles, the only stratum of the
maximal dimension is the principal stratum
$\cQ(1^{p-4},-1^p)$, where all zeroes are simple. This is the only
stratum which contributes a term of order $N^{2p-6}$ to $\cP_p(N)$,
where $2p-6=\dim_{\mathbb{C}}\cQ(1^{p-4},-1^p)$.
This proves equation~\eqref{eq:P:pN:to:principal}.
For $\degofz\ge 1$ the quantity $\cP_{\nu,\degofz}(N)$ counts pillowcase covers
with a marked zero of order $\degofz$ in the stratum
$\cQ(\nu,-1^{|\nu|+4})$. Hence, it has the asymptotic growth rate of the
same order as the quantity $\cP_{\nu}(N)$ counting pillowcase covers
in the same stratum without any marking, i.~e. it grows like $N^d$, where $d=\dim_{\mathbb{C}}\cQ(\nu,-1^{|\nu|+4})$.
The dimensional count as above implies that
the contribution of any term $\cP_{\nu,\degofz}(N)$ with $\degofz\ge 1$
to the sum in the right-hand side of~\eqref{eq:Pd:pN:to:principal:0}
has the order at most $N^{2p-6}$.
Let us now analyse the contribution of various strata to $\cP_{p,0}(N)$.
It follows from Theorem~1.19 in~\cite{DGZZ} that
\begin{multline}
\label{eq:P:nu:0:as:c11}
\cP_{\nu,0}(N) =
\frac{\cyl_{1,1}\big(\cQ(\nu,0,-1^{|\nu|+4})\big)}
{(|\nu|+4)!\cdot\prod_j \nu_j!}
\cdot
\frac{N^{\ell(\nu)+|\nu|+3}}{2\ell(\nu)+2|\nu|+6}
\ +\\+\
o\big(N^{\ell(\nu)+|\nu|+3}\big)\,,
\text{ when }N\to+\infty\,,
\end{multline}
where the constant $\cyl_{1,1}\big(\cQ(\nu,0,-1^{|\nu|+4})\big)$ is
positive. Here we used the same notation as in formula~\eqref{eq:dim}
for the dimension of the stratum $\cQ(\nu,0,-1^{|\nu|+4})$.
This implies that for any partition $\nu$ of $p-4$
different from $1^{p-4}$, its contribution
$\cP_{\nu,0}$ also has order at most $N^{2p-6}$, which means that
$\cP_{p,0}(N)$ behaves like $N^{2p-5}$ for $N$ large, and
that the only stratum which gives a contribution of this order is
the principal stratum with a marked point $\cQ(1^{p-4},0,-1^p)$. This
proves equality~\eqref{eq:Pd:pN:to:principal:0}.
By the same reason the summand $2\,\cP_{\nu,0}(N)$ dominates in the sum in the right-hand side
of~\eqref{eq:P:nu:d:to:P:nu:0}. It is the only term whose
contribution is of order $N^{d+1}$, where
$d=\dim_{\mathbb{C}}\cQ(\nu,-1^{|\nu|+4})$. The asymptotics of other
terms in the sum have lower orders in $N$ as $N\to+\infty$. This proves
equality~\eqref{eq:P:nu:d:to:P:nu:0}.
\end{proof}
Now we have everything for the proofs of
Theorem~\ref{th:meander:counting} and of Theorem~\ref{gth:meanders:fixed:stratum}.
\begin{proof}[Proof of Theorem~\ref{th:meander:counting}]
The chain of relations including \eqref{eq:Mplus:P} from
Lemma~\ref{lm:M:through:P}, \eqref{eq:P:pN:to:principal}
from Lemma~\ref{eq:P:pN:to:principal} and
\eqref{eq:P:nu:as:c11} yields
\begin{multline*}
\cM^+_p(N)= 2(p+1)\cdot \cP_{p+1}(N)
=
2(p+1)\cdot \cP_{1^{p-3}}(N) + o(N^{2p-4})\
=\\=
2(p+1)\cdot
\frac{\cyl_{1,1}\big(\cQ(1^{p-3},-1^{p+1})\big)}
{(p+1)!\,(p-3)!}
\cdot
\frac{N^{2p-4}}{4p-8} + o(N^{2p-4})\,
\text{ when }N\to+\infty\,.
\end{multline*}
This proves the first equality in~\eqref{eq:asymptotics:with}. The
constant $\cyl_{1,1}\big(\cQ(1^{p-3},-1^{p+1})\big)$ is expressed by our
main formula~\eqref{eq:c11:as:c1:squared:over:Vol} in terms of
$\cyl_1\big(\cQ(1^{p-3},-1^{p+1})\big)$ computed in
Corollary~\ref{cor:principal} in
section~\ref{s:Computations:for:pillowcase:covers} and in terms of
the Masur--Veech volume of the stratum $\cQ(1^{p-3},-1^{p+1})$ given
by formula~\eqref{eq:volume}.
Similarly, the chain of relations including \eqref{eq:Mminus:P} from
Lemma~\ref{lm:M:through:P}, \eqref{eq:Pd:pN:to:principal:0}
from Lemma~\ref{eq:P:pN:to:principal} and
\eqref{eq:P:nu:0:as:c11} implies
\begin{multline*}
\cM^-_p(N) = \sum_{\degofz=0}^{p-4} (\degofz+2)\cdot \cP_{p,\degofz}(N)
\,-\,\frac{1}{2}\,\cM_{p-1}^+(N)
\ =\
2\,\cP_{1^{p-4},0}(N)+ o(N^{2p-5})\
=\\=
\frac{2\,\cyl_{1,1}\big(\cQ(1^{p-4},0,-1^p)\big)}
{p!\,(p-4)!}\cdot
\frac{N^{2p-5}}{4p-10} + o(N^{2p-5})\,,
\text{ when }N\to+\infty\,.
\end{multline*}
This proves the first equality in~\eqref{eq:asymptotics:without}. The
constant $\cyl_{1,1}\big(\cQ(1^{p-4},0,-1^p)\big)$ is expressed by our
main formula~\eqref{eq:c11:as:c1:squared:over:Vol} in terms of
$\cyl_1\big(\cQ(1^{p-4},0,-1^p)\big)$ computed in
Corollary~\ref{cor:principal} of
Section~\ref{s:Computations:for:pillowcase:covers} and in terms of
the Masur--Veech volume of the stratum $\cQ(1^{p-4},0,-1^p)$ given by
formula~\eqref{eq:volume}.
Thus, the proof of Theorem~\ref{th:meander:counting} is conditional subject to the
explicit count of $\cyl_1\big(\cQ(1^{p-3},-1^{p+1})\big)$ and of
$\cyl_1\big(\cQ(1^{p-4},0,-1^p)\big)$ performed in
Corollary~\ref{cor:principal} below.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{gth:meanders:fixed:stratum}]
The proof of Theorem~\ref{gth:meanders:fixed:stratum} is completely
analogous to the proof of Theorem~\ref{th:meander:counting}.
Composing relation~\eqref{eq:Mplus:P:nu} from
Lemma~\ref{lm:M:through:P:nu} with
relation~\eqref{eq:P:nu:as:c11}, we get
\begin{multline*}
\cM^+_\nu(N)= 2(|\nu|+4)\cdot \cP_\nu(N)
\ =\\=\
2(|\nu|+4)\cdot
\frac{\cyl_{1,1}\big(\cQ(\nu,-1^{|\nu|+4})\big)}
{(|\nu|+4)!\cdot\prod_j \nu_j!}
\cdot
\frac{N^{\ell(\nu)+|\nu|+2}}{2\ell(\nu)+2|\nu|+4}
\ +\\+\ o\big(N^{\ell(\nu)+|\nu|+2}\big)\,,
\text{ when }N\to+\infty\,,
\end{multline*}
where we translated formula~\eqref{eq:d} for the dimension $d$ of the
stratum $\cQ(\nu,-1^{|\nu|+4})$ to notations~\eqref{eq:dim}.
This proves formula~\eqref{eq:asymptotics:nu:plus}
in Theorem~\ref{gth:meanders:fixed:stratum}.
Composing relation~\eqref{eq:Mminus:P:nu} from
Lemma~\ref{lm:M:through:P:nu} with
relation~\eqref{eq:P:nu:d:to:P:nu:0}, followed
by~\eqref{eq:P:nu:0:as:c11} we get
\begin{multline*}
\cM^-_\nu(N)
=
\sum_{\degofz=0}^{|\nu|} (\degofz+2)\cdot \cP_{\nu,\degofz}(N)
\,-\,\frac{1}{2}\,\cM_{\nu}^+(N)
=
2\,\cP_{\nu,0}(N)+o\big(N^{2\ell(\nu)+2|\nu|+3}\big)
=\\=
\frac{2\,\cyl_{1,1}\big(\cQ(\nu,0,-1^{|\nu|+4})\big)}
{(|\nu|+4)!\cdot\prod_j \nu_j!}
\cdot
\frac{N^{\ell(\nu)+|\nu|+3}}{2\ell(\nu)+2|\nu|+6}
\ +\\+\
o\big(N^{\ell(\nu)+|\nu|+3}\big)\,,
\text{ when }N\to+\infty\,.
\end{multline*}
This proves formula~\eqref{eq:asymptotics:nu:minus}
in Theorem~\ref{gth:meanders:fixed:stratum}.
We have proved Theorem~\ref{gth:meanders:fixed:stratum} conditional to
expressions~\eqref{eq:c11:with:0:through:c11:without:0}
and~\eqref{eq:c11:nu:in:Th4} for the quantities
$\cyl_{1,1}(\cQ(\nu,0,-1^{|\nu|+4}))$ and
$\cyl_{1,1}(\cQ(\nu,-1^{|\nu|+4}))$ left to
Theorem~\ref{th:c1:in:genus:0} in
section~\ref{s:Computations:for:pillowcase:covers}.
\end{proof}
We conclude this section with the proof of
Theorem~\ref{th:p:fixed:number:of:leaves:general}.
\begin{proof}[Proof of Theorem~\ref{th:p:fixed:number:of:leaves:general}]
The number of reduced plane trees with fixed number $p$ of leaves is
finite. By definition of the ratio
$\prob_{\mathit{connected}}(p; N)$ its numerator
is the sum of the numerators of~\eqref{eq:p:connected:iota:kappa}
over all such pairs of
trees, and the
denominator of the ratio $\prob_{\mathit{connected}}(p; N)$ is the
sum of the denominators
of~\eqref{eq:p:connected:iota:kappa}
over all such pairs of trees. Applying the dimensional
argument as in the proof of Theorem~\ref{th:meander:counting} we
conclude that contributions of pairs of trees, where at least one of
the trees is not trivalent, to the numerator or to the denominator of
the above ratio is of the order $o(N^{2p-6})$ for such contributions
correspond to strata $\cQ(\nu,-1^p)$ of meromorphic quadratic differentials in
genus zero different from the principal one. Thus, for large $N$
these contributions are negligible compared to the contribution of
any pair of trivalent trees. The contribution
of any pair of trivalent trees is of the order $N^{2p-6}$, where
$2p-6=\dim_{\mathbb{C}}\cQ(1^{p-4},-1^p)$ is the dimension
of the principal
stratum. Thus, studying the asymptotics of the ratio
$\prob_{\mathit{connected}}(p; N)$ we can ignore the pairs of trees
where at least one of the trees is not trivalent.
To complete the proof of
Theorem~\ref{th:p:fixed:number:of:leaves:general} it remains to notice that by
Theorem~\ref{th:trivalent:trees:connected:proportion} for any pair of
\textit{trivalent} plane trees $\cT_{bottom},\cT_{top}$, the
ratio of positive quantities~\eqref{eq:p:connected:iota:kappa}
has the same limit
$\prob_1(\cQ(1^{p-4},-1^p))$.
\end{proof}
\section{Computations for pillowcase covers}
\label{s:Computations:for:pillowcase:covers}
Let $\nu=[0^{\nu_0} 1^{\nu_1} 2^{\nu_2} \dots]$ be a
(generalized) partition of a
natural number $|\nu|$ into the sum of nonnegative integer numbers
(in this section we allow entries $0$):
$$
|\nu|:=
\underbrace{0+\dots+0}_{\nu_0}+
\underbrace{1+\dots+1}_{\nu_1}+
\underbrace{2+\dots+2}_{\nu_2}+
\dots
$$
The common convention on Masur--Veech volumes of the strata of
meromorphic quadratic differentials with at most simple poles
suggests to label (give names) to all zeroes and poles. Denote by
$\cP^{\mathit{labeled}}_\nu(N)$ the number of pillowcase covers with
labeled zeroes and poles in the stratum $\cQ(\nu,-1^{|\nu|+4})$ in
genus zero tiled with at most $2N$ squares with the side
$\frac{1}{2}$ and having a single horizontal cylinder of height
$\frac{1}{2}$ and a single vertical cylinder of width $\frac{1}{2}$. It
is easy to see that a pillowcase cover as above cannot have any
symmetries. Convention~\ref{conv:symmetry} on weights with which we
count pillowcase covers with non-labeled zeroes and poles is designed
to assure the following relation between the two counts valid for any
$N\in\N$:
\begin{equation}
\label{eq:P:labeled:through:non}
\cP^{\mathit{labeled}}_\nu(N)=
\left(\prod_{\degofz=0}^\infty \nu_\degofz !\right)
\cdot(|\nu|+4)!\,\cdot\,
\cP_\nu(N)\,,
\end{equation}
where the product above contains, actually, only finite number of factors.
\begin{Theorem}
\label{th:c1:in:genus:0}
The number $\cP^{\mathit{labeled}}_\nu(N)$ of pillowcase covers with
labeled zeroes and poles in the stratum $\cQ(\nu,-1^{|\nu|+4})$ tiled
with at most $2N$ squares $(\frac{1}{2})\times(\frac{1}{2})$ and
having a single horizontal cylinder of height $\frac{1}{2}$ and a single
vertical cylinder of width $\frac{1}{2}$ has the following
asymptotics as $N\to+\infty$:
$$
\cP^{\mathit{labeled}}_\nu(N)=
\cyl_{1,1}\left(\cQ(\nu,-1^{|\nu|+4})\right)\cdot\frac{N^d}{2d} +
o\left(N^{d}\right)\text{ as } N\to+\infty\,,
$$
were
$$
\cyl_{1,1}\left(\cQ(\nu,-1^{|\nu|+4})\right)=
\frac{\left(\cyl_1\left(\cQ(\nu,-1^{|\nu|+4})\right)\right)^2}
{\Vol\cQ(\nu,-1^{|\nu|+4})}
$$
and
\begin{equation}
\label{eq:c1:answer}
\cyl_1\left(\cQ(\nu,-1^{|\nu|+4})\right)=
2\cdot
\sum_{\iota_0=0}^{\nu_0}
\sum_{\iota_1=0}^{\nu_1}
\sum_{\iota_2=0}^{\nu_2}
\sum_{\dots}^{\dots}
\binom{\nu_0}{\iota_0}
\binom{\nu_1}{\iota_1}
\binom{\nu_2}{\iota_2}
\cdots
\binom{|\nu|+4}{|\iota|+2}\,.
\end{equation}
Here $\iota=[0^{\iota_0} 1^{\iota_1} 2^{\iota_2} \dots]$
and
$
d=\dim_{\mathbb{C}}\cQ(\nu,-1^{|\nu|+4})=
\ell(\nu)+|\nu|+2
$\,.
\end{Theorem}
Before proving Theorem~\ref{th:c1:in:genus:0} we prove the following
Corollary~\ref{cor:principal}.
\begin{Corollary}
\label{cor:principal}
The number $\cP^{\mathit{labeled}}_{1^k}(N)$ of pillowcase covers
with labeled zeroes and poles in the stratum $\cQ(1^k,-1^{k+4})$
tiled with at most $2N$ squares and having a single horizontal cylinder
of height $\frac{1}{2}$ and a single vertical cylinder of width
$\frac{1}{2}$ has the following asymptotics as $N\to+\infty$:
$$
\cP^{\mathit{labeled}}_{1^k}(N)=
\cyl_{1,1}\left(\cQ(1^k,-1^{k+4})\right)\cdot\frac{N^{2k+2}}{4k+4} +
o\left(N^{2k+2}\right)\text{ as } N\to+\infty\,,
$$
were
$$
\cyl_{1,1}\left(\cQ(1^k,-1^{k+4})\right)=
\frac{\left(\cyl_1\left(\cQ(1^k,-1^{k+4})\right)\right)^2}
{4\left(\cfrac{\pi^2}{2}\right)^{k+1}}
$$
and
\begin{equation}
\label{eq:c1:principal:answer}
\cyl_1\left(\cQ(1^k,-1^{k+4})\right)=
2\cdot\binom{2k+4}{k+2}
\end{equation}
The number $\cP^{\mathit{labeled}}_{1^k,0}(N)$ of pillowcase covers
as above with a marked regular vertex of the tiling
has the following asymptotics as
$N\to+\infty$:
$$
\cP^{\mathit{labeled}}_{1^k,0}(N)=
2\cdot \cyl_{1,1}\left(\cQ(1^k,-1^{k+4})\right)\cdot\frac{N^{2k+3}}{4k+6} +
o\left(N^{2k+3}\right)\text{ as } N\to+\infty\,,
$$
\end{Corollary}
\begin{proof}
By~\eqref{eq:volume} we have
$$
\Vol\cQ(1^k,-1^{k+4})=2\pi^2\cdot\left(\frac{\pi^2}{2}\right)^k
=4\cdot\left(\frac{\pi^2}{2}\right)^{k+1}
$$
To prove~\eqref{eq:c1:principal:answer} we apply the following
combinatorial identity to simplify formula~\eqref{eq:c1:answer} in
the particular case when $\nu=[1^k]$:
$$
\sum_{\iota_1=0}^k \binom{k}{\iota_1} \binom{k+4}{\iota_1+2}
= \binom{2k+4}{k+2}\,,
$$
see (3.20) in~\cite{Gould}.
It remains to prove that
\begin{equation}
\label{eq:tmp}
\cyl_{1,1}\left(\cQ(1^k,0,-1^{k+4})\right)=
2\cdot \cyl_{1,1}\left(\cQ(1^k,-1^{k+4})\right)\,.
\end{equation}
By~\eqref{eq:c11:as:c1:squared:over:Vol} we have
$$
\cyl_{1,1}\left(\cQ(1^k,0,-1^{k+4})\right)=
\frac{\Big(\cyl_1\left(\cQ(1^k,0,-1^{k+4})\right)\Big)^2}
{\Vol\cQ(1^k,0,-1^{k+4})}\,.
$$
Equation~\ref{eq:c1:answer} implies that
$$
\cyl_1\left(\cQ(1^k,0,-1^{k+4})\right)=2\cdot
\cyl_1\left(\cQ(1^k,-1^{k+4})\right)
$$
Finally, by~\eqref{eq:volume} we have
$$
\Vol\cQ(1^k,0,-1^{k+4})=2\Vol\cQ(1^k,-1^{k+4})\,.
$$
and~\eqref{eq:tmp} follows.
\end{proof}
We also prove the following elementary technical Corollary
of Theorem~\ref{th:c1:in:genus:0}.
\begin{Corollary}
\label{cor:leading:0}
Consider a
(generalized)
partition $\nu=[0^{\nu_0} 1^{\nu_1} 2^{\nu_2}\dots]$ and its
subpartition $\nu'=[1^{\nu_1} 2^{\nu_2} \dots]$ obtained by suppressing
all zero entries. The following formulae are valid:
\begin{align}
\label{eq:c1:with:and:without:0}
\cyl_1\big(\nu,-1^{|\nu|+4}\big)
&=
2^{\nu_0}\cdot \cyl_1\big(\nu',-1^{|\nu'|+4}\big)
\\
\label{eq:p1:with:and:without:0}
\prob_1\big(\nu,-1^{|\nu|+4}\big)
&=
\prob_1\big(\nu',-1^{|\nu'|+4}\big)
\,.
\end{align}
\end{Corollary}
\begin{proof}
Note that $|\nu'|=|\nu|$. Similarly, having any subpartion
$\iota'\subset\iota$ obtained from a partition $\iota$ by suppressing
all zero entries we have $|\iota'|=|\iota|$. Thus we can rewrite
formula~\eqref{eq:c1:answer} as
\begin{multline*}
\cyl_1\left(\cQ(\nu,-1^{|\nu|+4})\right)=
2\cdot
\sum_{\iota_0=0}^{\nu_0}
\sum_{\iota_1=0}^{\nu_1}
\sum_{\iota_2=0}^{\nu_2}
\sum_{\dots}^{\dots}
\binom{\nu_0}{\iota_0}
\binom{\nu_1}{\iota_1}
\binom{\nu_2}{\iota_2}
\cdots
\binom{|\nu|+4}{|\iota|+2}
=\\=
\left(\sum_{\iota_0=0}^{\nu_0}
\binom{\nu_0}{\iota_0}
\right)\cdot
\left(
2\sum_{\iota_1=0}^{\nu_1}
\sum_{\iota_2=0}^{\nu_2}
\sum_{\dots}^{\dots}
\binom{\nu_1}{\iota_1}
\cdots
\binom{|\nu'|+4}{|\iota'|+2}
\right)=
2^{\nu_0} \cyl_1\left(\cQ(\nu',-1^{|\nu'|+4})\right),
\end{multline*}
which proves~\eqref{eq:c1:with:and:without:0}.
To prove~\eqref{eq:p1:with:and:without:0} it suffices
to note that by formula~\eqref{eq:volume}, we have
$$
\Vol\cQ(\nu,-1^{|\nu|+4})=(f(0))^{\nu_0}
\Vol\cQ(\nu',-1^{|\nu'|+4})= 2^{\nu_0}\Vol\cQ(\nu',-1^{|\nu'|+4})\,.
$$
Passing to the ratios
\begin{multline*}
\prob_1\left(\cQ(\nu,-1^{|\nu|+4})\right):=\
\frac{\cyl_1\left(\cQ(\nu,-1^{|\nu|+4})\right)}
{\Vol\left(\cQ(\nu,-1^{|\nu|+4})\right)}
\ =\\=\
\frac{\cyl_1\left(\cQ(\nu',-1^{|\nu'|+4})\right)}
{\Vol\left(\cQ(\nu',-1^{|\nu'|+4})\right)}
\ =:
\prob_1\left(\cQ(\nu',-1^{|\nu'|+4})\right)
\end{multline*}
we get the desired equation~\eqref{eq:c1:with:and:without:0}.
\end{proof}
Recall that a type $\iota=[0^{\iota_0} 1^{\iota_1} 2^{\iota_2}
\dots]$ of a plane tree $\cT$ records the number $\iota_\degofz$ of
vertices of valence $\degofz+2$ for $\degofz=0,1,2,\dots$. Note that
in section~\ref{s:Computations:for:pillowcase:covers} we allow to the
tree have several vertices of valence $2$. Recall also that $|\nu|$
denotes the sum of the entries of the partition $\nu=[0^{\nu_0}
1^{\nu_1} 2^{\nu_2}\dots]$; by $\ell(\nu)$ we denote the length of
$\nu$, where this time we count the entries $0$ if any:
\begin{align*}
|\nu|&:=1\cdot\nu_1+2\cdot\nu_2+3\cdot\nu_3+\dots
\\
\ell(\nu)&:=\nu_0+\nu_1+\nu_2+\nu_3+\dots
\end{align*}
In the Lemma below we reproduce
formula~(2.2) from Proposition (2.2) in~\cite{DGZZ}
adapting it to the language of the current paper.
\begin{Lemma}
\label{lm:contribution:quadratic}
Consider a separatrix diagram $\cD=\cT(\iota)\sqcup\cT(\nu-\iota)$
represented by a non-labeled pair of plane trees $\cT(\iota)$ and
$\cT(\nu-\iota)$ with profiles $\iota\subset\nu$ and $\nu-\iota$
respectively.
The number of pillowcase covers with labeled zeroes and poles, tiled
with at most $2N$ squares and having a single horizontal cylinder of
height $\frac{1}{2}$ representing a given separatrix diagram
$\cD$
has the following asymptotics when $N\to+\infty$
\begin{equation}
\label{eq:general:contribution:of:D:with:N}
\cyl_1(\cD)\cdot \frac{N^d}{2d} + o(N^d)\,,
\end{equation}
where the dimension $d$ of the ambient stratum $\cQ(\nu,-1^{|\nu|+4})$
is defined by equation~\eqref{eq:dim} and
\begin{equation}
\label{eq:c1:D:iota}
\cyl_1(\cD) =
\cfrac{4}{|\Aut(\cD)|}\cdot
\frac{(|\nu|+4)!\cdot\mult_0!\cdot\mult_1!\cdot \mult_2! \cdots}
{\big(|\iota|+\ell(\iota)\big)!
\cdot
\big(|\nu-\iota|+\ell(\nu-\iota)\big)!
}\,.
\end{equation}
\end{Lemma}
\begin{Remark}
\label{rm:zeta:d}
In this paper we denote by $\cyl_1(\cD)$ the coefficient of the leading term in the
asymptotics of the number of pillowcase covers tiled with at most
$2N$ squares and having a single horizontal cylinder of \textit{minimal
possible} height $\frac{1}{2}$. In the companion paper~\cite{DGZZ} we
used a similar notation $c_1(\cD)$ for the coefficient
in asymptotics where we made no restriction on the height of the cylinder.
It is easy to see that the two coefficients
differ by the factor $\zeta(d)$, namely,
$$
c_1(\cD) =
\zeta(d) \cdot \cyl_1(\cD)\,,
$$
where $d=\dim_{\mathbb{C}}\cQ(\nu,-1^{|\nu|+4})$ is given by
formula~\eqref{eq:dim}.
\end{Remark}
\begin{proof}
The number of edges $m$ of $\cT(\iota)$, and $n$ of
$\cT(\nu-\iota)$ are expressed as
\begin{align*}
m&=|\iota|+\ell(\iota)+1
\\
n&=|\nu-\iota|+\ell(\nu-\iota)+1
\end{align*}
and the dimension $d$ of the stratum satisfies relation $d=m+n$.
Consider any pillowcase cover having the diagram $\cD$ as the diagram
of horizontal saddle connections. Cut it open along all horizontal
saddle connections. By definition of $\cD$ it has $m$ pairs of saddle
connections on one side of the cylinder; $n$ pairs of saddle
connections on the other side of the cylinder, all saddle connection
has its twin on the same side.
The proof now follows line by line the second part of the proof of
the more general Proposition~2.2 in~\cite{DGZZ}. Note that the
parameter $l$ used in Proposition~2.2 to denote the number of
saddle connections which after the surgery as above appear on both
sides of the cylinder is equal to zero in genus zero. One extra
simplification comes from the fact that in the proof of
Proposition~(2.2) in~\cite{DGZZ} we sum over various possible heights
of the horizontal cylinder, while in our context it equals to
$\frac{1}{2}$, see Remark~\eqref{rm:zeta:d}. As a result we do not
get the extra factor $\zeta(d)$ present in the original
expression~(2.2) in Proposition~(2.2) in~\cite{DGZZ}.
\end{proof}
Consider a separatrix diagram $\cD=\cT(\iota)\sqcup\cT(\nu-\iota)$ as
above. Defining the automorphism group $\Aut(\cD)$ we assume that none
of the vertices, edges, or boundary components of the ribbon graph
$\cD$ is labeled; however, we assume that the orientation of the
ribbons is fixed. Thus
\begin{equation}
\label{eq:order:Gamma:D}
|\Aut(\cD)|=|\Aut(\cT(\iota))|\cdot|\Aut(\cT(\nu-\iota))|\cdot
\begin{cases}
2&\text{if } \cT(\iota)\simeq\cT(\nu-\iota)\\
1&\textit{otherwise}
\end{cases}
\end{equation}
Here $\simeq$ stands for an isomorphism of plane (``ribbon'')
trees.
The following counting Theorem for plane trees is well known; see,
for example, \cite[2, p.6]{Moon}. It is the last element needed for
proof of Theorem~\ref{th:c1:in:genus:0}.
\begin{NNTheorem}
For any partition $\iota=[0^{\iota_0} 1^{\iota_1} 2^{\iota_2}\dots]$
the following expression holds
$$
\sum_{\cT(\iota)} \frac{1}{|\Aut(\cT(\iota))|}=
\frac
{\big(|\iota|+\ell(\iota)\big)!}
{\big(|\iota|+2\big)!\cdot \iota_0!\cdot \iota_1!\cdot\iota_2!\cdots}\,,
$$
where we sum over all plane trees corresponding to a partition
$\iota$ and $|\Aut(\cT(\iota))|$ is the order of the automorphism group
of the tree $\cT(\iota)$.
\end{NNTheorem}
\begin{proof}[Proof of Theorem~\ref{th:c1:in:genus:0}]
The first two statements of Theorem~\ref{th:c1:in:genus:0} are,
a particular case of Theorem 1.19 in~\cite{DGZZ} which, morally,
claims that ``horizontal and vertical decompositions of pillowcase
covers are asymptotically uncorrelated''.
It only remains to prove expression~\eqref{eq:c1:answer}.
Combining equation~\eqref{eq:c1:D:iota} with the above Theorem we
conclude that the sum of $\cyl_1(\cD)$ over all realizable one-cylinder
separatrix diagrams $\cD$ in any given stratum
$\cQ(\nu,-1^{|\nu|+4})$ in genus zero can be expressed as follows
\begin{multline*}
\cyl_1\left(\cQ(\nu,-1^{|\nu|+4})\right)=
\sum_{\cD} \cyl_1(\cD)=
\frac{1}{2}\sum_{\iota\subset\nu}
\left(
\frac{4\cdot(|\nu|+4)!\cdot\mult_0!\cdot\mult_1!\cdot \mult_2! \cdots}
{\big(|\iota|+\ell(\iota)\big)!
\cdot
\big(|\nu-\iota|+\ell(\nu-\iota)\big)!
}
\right)
\cdot\\ \cdot
\left(
\frac
{\big(|\iota|+\ell(\iota)\big)!}
{\big(|\iota|+2\big)!\cdot \iota_0!\cdot \iota_1!\cdots}
\right)
\cdot
\left(
\frac
{\big(|\nu-\iota|+\ell(\nu-\iota)\big)!}
{\big(|\nu-\iota|+2\big)!\cdot (\nu_0-\iota_0)!\cdot (\nu_1-\iota_1)!\cdots}
\right)
=\\=
2\sum_{\iota\subset\nu}
\binom{|\nu|+4}{|\iota|+2}
\binom{\nu_0}{\iota_0}
\binom{\nu_1}{\iota_1}
\binom{\nu_2}{\iota_2}
\cdots
\end{multline*}
\end{proof}
\appendix
\section{Meanders and pairs of arc systems satisfying additional
combinatorial constraints}
\label{s:number:of:meanders:of:given:combinatorial:type}
In this appendix we count the frequency of
meanders among all pairs of arc systems imposing additional
constraints on combinatorics of the pair of arc systems.
In section~\ref{ss:Meanders:and:arc:systems} we have assigned to any
closed meander two arc systems on discs (considered as hemispheres).
Passing to the one-point compactification of the plane we place our
closed meander curve on the resulting sphere. By construction, this
meander curve is the simple closed curve on the sphere obtained from
the two arc systems by identifying the two hemispheres along the
common equator. By convention we call the equator the
\textit{horizontal} curve and the simple closed meander curve --- the
\textit{vertical curve} on the resulting sphere.
\begin{figure}[hbt]
\special{
psfile=meander_and_ddual_trees.eps
hscale=40
vscale=40
voffset=-103
hoffset=100
}
\vspace{100bp}
\caption{
\label{fig:dual:trees:for:horizontal}
We can change the roles of the horizontal and vertical curves and
construct the reduced dual trees for the horizontal curve of a closed
meander. The tree in the bounded region (which is shaded in the
picture) is denoted by $\cT^\ast_0$. The tree in the complementary
unbounded region is denoted by $\cT^\ast_\infty$.
}
\end{figure}
We have constructed two reduced dual graphs to the two arc systems,
see the right picture in Figure~\ref{fig:meander}. We can change the
roles of the horizontal and vertical curves and consider the vertical
curve as the new ``equator'' of the sphere. Then, the horizontal
curve (former equator) takes the role of the meander curve and
defines a pair of arc systems and reduced dual trees, see
Figure~\ref{fig:dual:trees:for:horizontal}. The meander cuts the
plane in two regions: one bounded and one unbounded. We denote the
reduced dual tree as above staying in the bounded domain of the plane
by $\cT^\ast_0$ and the one in the complementary unbounded domain --- by
$\cT^\ast_\infty$.
In the setting where a pair of transverse labeled simple closed
curves on a sphere comes from arc systems on two hemispheres, we do
not have the distinction between ``bounded'' and ``unbounded''
domains. In this case the graph
$\cD^\ast=\cT^\ast_0\sqcup\cT^\ast_\infty$ obtained as a disjoint
union of the trees $\cT^\ast_0$ and $\cT^\ast_\infty$ does not have
any canonical labeling of connected components.
As before, consider a pair of arc systems on two hemispheres
containing the same number of arcs and identify them along the
equator matching the endpoints of arcs. We get a simple closed curve
on the sphere and a multicurve transverse to it. Recall that we
associate to any transverse connected pair of multicurves its
type $\nu=[1^{\nu_1} 2^{\nu_2} 3^{\nu_3} \dots]$, where
the entry $\nu_\degofz$ records the number of $(2\degofz+4)$-gons,
for $\degofz\in\N$, among the faces in which the pair of multicurves
cuts the sphere.
When the two multicurves are simple
closed curves, the entry $\nu_\degofz$ also records the number of vertices of
valence $\degofz+2$ in the graph $\cT^\ast_0\sqcup\cT^\ast_\infty$, and in
the graph $\cT_{bottom}\sqcup\cT_{top}$. We will call $\nu$ the
\textit{type} of the graphs $\cT_{bottom}\sqcup\cT_{top}$ and
$\cT^\ast_0\sqcup\cT^\ast_\infty$.
By duality, if we specify the trees $\cT^\ast_0$ and
$\cT^\ast_\infty$ instead of the trees $\cT_{bottom}$ and
$\cT_{top}$ in the setting of Theorem~\ref{th:any:trees:connected:proportion}
we get the completely parallel statement.
We can state the following more detailed version of
Theorem~\ref{th:any:trees:connected:proportion}. This time we chose two pairs of
trees: $\cT_{bottom},\cT_{top}$ and $\cT^\ast_0, \cT^\ast_\infty$,
such that the graphs $\cD^\ast:=\cT^\ast_0\sqcup\cT^\ast_\infty$ and
$\cT_{bottom}\sqcup\cT_{top}$ share any given type $\nu$.
The two connected components of the graph $\cT_{bottom}\sqcup\cT_{top}$
are labeled; the two components of the graph $\cD^\ast$ --- not.
As before,
we consider all possible triples
$$
(\text{$n$-arcs system of type $\cT_{top}$; $n$-arcs system of type $\cT_{bottom}$;
identification})
$$
as described above for all $n\le N$. In analogy
with~\eqref{eq:p:connected:iota:kappa} we define the fraction
$p(\cT_{bottom},\cT_{top}; \cT^\ast_0,\cD^\ast; N)$ of triples
as above which lead to a meander with the graph
$\cD^\ast$ dual to the equator, among all triples as above.
\begin{Proposition}
\label{prop:fix:two:pairs:of:trees}
For any two pairs of plane trees $\cT_{bottom},\cT_{top}$ and
$\cT^\ast_0, \cT^\ast_\infty$, such that the graphs
$\cD^\ast:=\cT^\ast_0\sqcup\cT^\ast_\infty$ and
$\cT_{bottom}\sqcup\cT_{top}$ share any given type $\nu$, the
following limit exists and has the following strictly positive value:
$$
\lim_{N\to+\infty} p(\cT_{bottom},\cT_{top}; \cD^\ast; N)
=
\prob_1(\cD^\ast)\,.
$$
The limit $\prob_1(\cD^\ast)$ is
expressed by the following formula
\begin{equation}
\label{eq:probability:trees}
\prob_{1}(\cD^\ast)=\frac{\cyl_1(\cD^\ast)}{\Vol\cQ(\nu,-1^{|\nu|+4})}
\end{equation}
Denote by $\iota$ and $\nu-\iota$ profiles of the plane trees
$\cT^\ast_0$ and $\cT^\ast_\infty$. The coefficient $\cyl_1(\cD^\ast)$
in the above formula is given by equation
\begin{equation}
\label{eq:c1:D:theorem}
\cyl_1(\cD(\iota))=
\cfrac{4}{|\Aut(\cD^\ast)|}\cdot
\frac{(|\nu|+4)!\cdot\mult_0!\cdot\mult_1!\cdot \mult_2! \cdots}
{\big(|\iota|+\ell(\iota)\big)!
\cdot
\big(|\nu-\iota|+\ell(\nu-\iota)\big)!
}
\end{equation}
Here $|\Aut(\cD^\ast)|$ denotes the order of the automorphism group of
the ribbon graph $\cD^\ast$, where neither of connected components,
edges or vertices of $\cD^\ast$ are labeled, but the orientation of
the ribbon graph is fixed.
\end{Proposition}
\begin{proof}
As in Theorem~\ref{th:any:trees:connected:proportion} we consider
labled pairs of arc systems on oriented discs assuming that the
number of arcs in two systems is the same. ``All possible triples''
in the denominator of the ratio $p(\cT_{bottom},\cT_{top}; \cD^\ast;
N)$ is exactly the same as above: the graph $\cD^\ast$ does not carry
any information in the definition of the ``possible triples''.
Computing the numerator we impose now a given separatrix diagram
$\cD^\ast$ as the graph of vertical saddle connections of the single
vertical cylinder.
Thus, the limit in Proposition~\ref{prop:fix:two:pairs:of:trees} is the
asymptotic fraction of pillowcase covers having a single horizontal
cylinder of height $\frac{1}{2}$ corresponding to the separatrix
diagram $\cD=\cT_{bottom}\sqcup\cT_{top}$ of horizontal saddle
connections, and a single vertical cylinder of width $\frac{1}{2}$
corresponding to the separatrix diagram $\cD^\ast$ of vertical saddle
connections among all pillowcase covers having a single horizontal
cylinder of height $\frac{1}{2}$ corresponding to the separatrix
diagram $\cD$ of horizontal saddle connections.
Theorem~1.19 in~\cite{DGZZ} asserts that such limit exists and that
``horizontal and vertical cylinder decompositions are asymptotically
uncorrelated'', so the above limit coincides with the asymptotic
fraction of pillowcase covers having a single vertical cylinder of
width $\frac{1}{2}$ corresponding to the separatrix diagram
$\cD^\ast$ among all pillowcase covers. As before we can omit
conditions on the horizontal foliation. This proves existence of the
limit.
By definition, the latter asymptotic fraction is the quantity
$\prob_1(\cD^\ast)$.
This proves the first
equality in formula~\eqref{eq:probability:trees}. The value of
$\cyl_1(\cD^\ast)$ is computed in equation~\eqref{eq:c1:D:iota} in
Lemma~\ref{eq:general:contribution:of:D:with:N}, which completes the
proof of Proposition~\ref{prop:fix:two:pairs:of:trees}.
\end{proof}
\section{Arc systems as linear involutions}
\label{s:linear:involutions}
We have seen that every closed meander in the plane defines a pair of
arc systems as in Figure~\ref{fig:meander}. A pair of arc systems can
be encoded by a \textit{linear involution}
(see~\cite{Danthony:Nogueira} and~\cite{Boissy:Lanneau}) generalizing
an interval exchange transformation. In the example of
Figure~\ref{fig:meander} we get the linear involution
$$
\begin{pmatrix}
A,B,B,C,C,A\\
D,E,E,F,F,D,G,G
\end{pmatrix}\,,
$$
see Figure~\ref{fig:arc:systems:as:linear:involution}.
\begin{figure}[htb]
\special{
psfile=meander1shadowed.eps
hscale=40
vscale=40
hoffset=120
voffset=-106
}
\begin{picture}(0,0)(59,-117)
\put(12,-168){\small A}
\put(35,-168){\small B}
\put(58,-168){\small B}
\put(74,-168){\small C}
\put(86,-168){\small C}
\put(103,-168){\small A}
\end{picture}
\begin{picture}(0,0)(60,-104)
\put(4,-168){\small D}
\put(21,-168){\small E}
\put(44,-168){\small E}
\put(62,-168){\small F}
\put(74,-168){\small F}
\put(85,-168){\small D}
\put(95,-168){\small G}
\put(106,-168){\small G}
\end{picture}
\vspace{100bp}
\caption{
\label{fig:arc:systems:as:linear:involution}
Pair of arc systems as a linear involution.
}
\end{figure}
We define the distance between two consecutive intersections of the
horizontal segment with the meander to be $\frac{1}{2}$. Thus, in our
example we assign the following lengths to the intervals under
exchange:
$$
|A|=1,\
|B|=1,\
|C|=\frac{1}{2},\
|D|=\frac{1}{2},\
|E|=1,\
|F|=\frac{1}{2},\
|G|=\frac{1}{2}\,.
$$
Consider the trees $\cT^\ast_0$ and $\cT^\ast_\infty$ dual to the
horizontal line as in Figure~\ref{fig:dual:trees:for:horizontal}.
Follow each half-edge of these trees from the vertex till the first
intersection with the horizontal line. The resulting intersection
points are exactly the extremities of the intervals under exchange.
\begin{figure}[htb]
\special{
psfile=meandre2.eps
hscale=60
vscale=60
voffset=-70
hoffset=80
}
\special{
psfile=meandre3_up.eps
hscale=100
vscale=100
voffset=-5
hoffset=282
angle=180
}
\special{
psfile=meandre3_up.eps
hscale=100
vscale=100
voffset=-75
hoffset=250
angle=0
}
\begin{picture}(0,0)(100,-130)
\put(31,-145){\small A}
\put(9,-159){\small B}
\put(53,-164){\small B}
\put(87,-161){\small C}
\put(110,-150){\small C}
\put(80,-145){\small A}
\end{picture}
\begin{picture}(0,0)(100,-110)
\put(40,-165){\small D}
\put(12,-168){\small E}
\put(27,-184){\small E}
\put(62,-185){\small F}
\put(85,-183){\small F}
\put(106,-172){\small D}
\put(83,-167){\small G}
\put(60,-165){\small G}
\end{picture}
\vspace{70bp}
\caption{
\label{fig:meander:as:cylinder}
Jenkins--Strebel differential with a single horizontal cylinder on
$\CP$ associated to a linear involution and the ribbon graph of its
horizontal saddle connections.}
\end{figure}
Reciprocally, every linear involution of intervals of lengths in
$\frac{1}{2}\N$ and such that twins of the top interval are located
on top and twins of the bottom intervals are located on the bottom
naturally defines a pair of arc systems in the plane and a pillowcase
cover of genus zero having a single horizontal cylinder. Namely,
consider a rectangle of height $\frac{1}{2}$ and with a horizontal
side having the same length as the base segment of the linear
involution (see the shadowed rectangle in the middle
of Figure~\ref{fig:arc:systems:as:linear:involution}). Identify the
two vertical sides of the rectangle by a parallel translation and
identify the subsegments of the horizontal sides as prescribes the
linear involution, see Figure~\ref{fig:meander:as:cylinder}. We get a
Jenkins--Strebel differential with a single horizontal cylinder.
\section{Meanders of low combinatorial complexity}
In this section we present an explicit formula for meanders whose underlying
pillowcase cover belongs to $\cQ(-1^4)$. We also present some numerical
experiments for meanders leading to pillowcase covers
with $5$ poles (stratum $\cQ(1, -1^5)$) and with
$6$ poles (strata $\cQ(1, -1^6)$ and $\cQ(2, -1^6)$).
The reader can see that these experiments
provide strong numerical evidence for
our asymptotic results.
Let us recall from Theorem~\ref{th:meander:counting} that
$\cM_p^+(N)$ (respectively $\cM_p^-(N)$) counts the number of
plane meanders with at most $N$ arcs (or,
equivalently, with at most $2N$ crossings) and with (resp.
without) a maximal arc and $p$ minimal arcs. In the language of
flat surfaces, $\cM_{p-1}^+(N)$
and $\cM_p^-(N)$ correspond to pillowcases covers
in one of the strata $\cQ(\nu, -1^p)$
where $\nu = [1^{\nu_1} 2^{\nu_2} \ldots]$ is a partition of $p-4$
(see Lemma~\ref{lm:M:through:P}).
We consider here a refined counting. Namely, let $M_{n,p}$ be the
number of meanders with exactly $n$ arcs (or,
equivalently, with exactly $2n$ crossings) and
either with $p$ minimal arcs and no maximal arcs, or with $p-1$ minimal
arcs and a maximal arc.
In other words, $M_{n,p}$ denotes the number of meanders
that correspond to pillowcase covers of degree $n$ whose associated
quadratic differential has exactly $p$ poles. We have the following relation
\[
\cM_{p-1}^+(N) + \cM_p^-(N) = \sum_{n=1}^N M_{n,p}.
\]
Note that by equation~\eqref{eq:asymptotics:with} we have
$\cM_{p-1}^+(N)=O(N^{2p-6})$ while
by equation~\eqref{eq:asymptotics:without}
$\cM_p^-(N)=O(N^{2p-5})$. Thus,
the contribution of $\cM_{p-1}^+(N)$ becomes negligible for large $N$.
The following array presents the values $M_{n,p}$ where the number of arcs
$n$ ranges from 1 to 9 and $p$ ranges from 4 to 8. These values were obtained by
listing all meanders and filtering them by the total number $p$ of minimal and maximal arcs.
\vspace{1em}
\begin{center}
\begin{tabular}{rr||c|c|c|c|c|c|c|c|c}
&$n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
[-\halfbls]
$p$&
\begin{picture}(0,0)(0,0)
\put(-22,9){\line(2,-1){25}}
\end{picture}
\hspace*{-5pt}
&&&&&&&&&
\\ \hline \hline
4 \hspace*{-5pt}&& 1 & 2 & 6 & 8 & 20 & 12 & 4 & 32 & 54 \\ \hline
5 \hspace*{-5pt}&& 0 & 0 & 0 & 16 & 40 & 168 & 280 & 544 & 1152 \\ \hline
6 \hspace*{-5pt}&& 0 & 0 & 2 & 16 & 110 & 416 & 1470 & 4128 & 9102 \\ \hline
7 \hspace*{-5pt}&& 0 & 0 & 0 & 0 & 60 & 576 & 3276 & 13632 & 45468 \\ \hline
8 \hspace*{-5pt}&& 0 & 0 & 0 & 2 & 30 & 462 & 4228 & 26424 & 130410 \\
\end{tabular}\end{center} \vspace{1em}
The sum of entries $\sum_{p=4}^{+\infty} M_{n,p}$ in each column of this
biinfinite array is the mysterious number of meanders with $n$ arcs: 1, 2,
8, 42, 262, 1828, 13820, 110954, 933458, \ldots (sequence
A005315 from \cite{oeis}). Here we study
this array by lines.
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}{0.45\textwidth}
\special{
psfile=num_meanders_p4.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{Number of meanders with exactly $n$ arcs divided by $n^2$, i.~e. the function $n \mapsto \phi(n)/n$.}
\end{subfigure}
\hspace{.08\textwidth}
\begin{subfigure}{0.45\textwidth}
\special{
psfile=num_meanders_p4_cumulative.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{Number of meanders with at most $N$
arcs divided by $N^3$ (represented by blue points) and the asymptotic value $\frac{2}{\pi^2}$ (red line).}
\end{subfigure}
\end{center}
\caption{The number $M_{n,4}$ of meanders with $4$ minimal arcs or,
equivalently, the number of meanders
whose associated pillowcase covers belongs to $\cQ(-1^4)$.}
\label{fig:Q1p4}
\end{figure}
For $p=4$, there is only one stratum $\cQ(-1^4)$ and the corresponding
generalized interval exchange transformations
(see Appendix~\ref{s:linear:involutions}) are reduced to rotations. It is
then easy to deduce the following
\begin{Lemma}
We have $M_{n,4} = n \phi(n)$ where $\phi$ is the Euler totient function.
In particular
\[
\sum_{n=1}^N M_{n,4} = \cM_3^+(N) + \cM_4^-(N) \sim \frac{2 N^3}{\pi^2}.
\]
\end{Lemma}
This result is coherent with the one suggested
by formula~\eqref{eq:asymptotics:without} for $\cM_4^-(N)$;
see also Figure~\ref{fig:Q1p4} for graphics related to $M_{n,4}$.
We do not hope to get a closed formula for $M_{n,5}$. However, it is not hard to
compute these numbers using generalized interval exchanges and Rauzy induction.
We were able to compute 400 of these numbers represented in the second line
of the above table and the list starts with
\vspace*{10pt}\begin{center}\begin{minipage}{0.9\textwidth}
0, 0, 0, 16, 40, 168, 280, 544, 1152, 1560, 2640, 3504, 5824, 6552, 12000,
11456, 19176, 18648, 31312, 30640, 50064, 43736, 71392, 62304, 104800,
87672, 141048, 121968, 191632, 154200, 255192, 209536, \ldots
\end{minipage}\end{center}
\bigskip
Figure~\ref{fig:Q1p5}
represents first $400$ terms of this sequence. It agrees with the value
$$
\prob_1\left(\cQ(1,-1^5)\right)\ =\
\frac{16}{3 \pi^4}
$$
predicted
by formula~\eqref{eq:asymptotics:without} for $\cM_5^-(N)$.
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}{0.4\textwidth}
\special{
psfile=num_meanders_p5.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{
Normalized number of meanders $\cfrac{M_{5,n}}{n^4}$ with exactly $n$ arcs}
\end{subfigure}
\hspace{.08\textwidth}
\begin{subfigure}{0.4\textwidth}
\vspace*{22pt}
\special{
psfile=num_meanders_p5_cumulative.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{
Normalized number of meanders $\cfrac{\sum_{n=6}^N M_{5,n}}{N^5}$ with
at most $N$ arcs
(blue points) and the asymptotic value $\frac{16}{3 \pi^4}$ (red line).}
\end{subfigure}
\end{center}
\caption{The number $M_{n,5}$ of meanders
whose associated pillowcase covers belongs to $\cQ(1, -1^5)$.}
\label{fig:Q1p5}
\end{figure}
We now present some
numerical evidence for the theoretical prediction of
Theorem~\ref{th:any:trees:connected:proportion}.
There are three pairs of trees with
$6$ univalent vertices in total. For these three pairs of trees
Theorem~\ref{th:any:trees:connected:proportion} gives
\begin{align*}
\prob_{\mathit{connected}}(\hspace*{16pt},\hspace*{4pt}) &=
\prob_1\left(\cQ(2,-1^6)\right)=
\frac{45}{2\pi^4} \sim 0.231
\\
\prob_{\mathit{connected}}(\hspace*{14pt},\hspace*{14pt}) =
\prob_{\mathit{connected}}(\hspace*{14pt},\hspace*{4pt}) &=
\prob_1\left(\cQ(1^2,-1^6)\right)=
\frac{280}{\pi^6} \sim 0.291\,.
\end{align*}
\special{
psfile=meander4.eps
hscale=40
vscale=40
voffset=36
hoffset=159
angle=0
}
\special{
psfile=meander2.eps
hscale=40
vscale=40
voffset=36
hoffset=179
angle=0
}
\special{
psfile=meandre3_up.eps
hscale=40
vscale=40
voffset=12
hoffset=72
angle=60
}
\special{
psfile=meandre3_up.eps
hscale=40
vscale=40
voffset=12
hoffset=82
angle=0
}
\special{
psfile=meander5.eps
hscale=40
vscale=40
voffset=11
hoffset=161
angle=0
}
\special{
psfile=meander2.eps
hscale=40
vscale=40
voffset=11
hoffset=179
angle=0
}
These limit ratios
are represented by horizontal lines
dominating the plots of exact values of
$\prob_{connected}(\cT_{top}, \cT_{bottom}; N)$
in Figure~\ref{fig:Qp6:ratio}. We denote
by $\cT_{[]}$, $\cT_{[1]}$, $\cT_{[2]}$
and $\cT_{[1,1]}$ respectively the unique trees with no internal vertices, with
a single internal vertex of valence 3, with a single internal
vertex of valence 4 and with two internal vertices of valence 3.
\begin{figure}[!ht]
\begin{subfigure}{0.45\textwidth}
\special{
psfile=num_meanders_p6_cumulative_conf3.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{
Trees $\cT_{[1,1]},\ \cT_{[\ ]}$.
}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\special{
psfile=num_meanders_p6_cumulative_conf2.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{
Trees
$\cT_{[1]},\ \cT_{[1]}$.
}
\end{subfigure}
\\
\begin{subfigure}{0.5\textwidth}
\special{
psfile=num_meanders_p6_cumulative_conf1.eps
hscale=100
vscale=100
voffset=-460
hoffset=-225
}
\vspace*{125pt}
\caption{
Trees $\cT_{[2]}, \cT_{[\ ]}$.
}
\end{subfigure}
\caption{
Proportion $\prob_{connected}(\cT_{top}, \cT_{bottom}; N)$
of pairs of arcs systems which lead to meanders
among all pairs of arc systems with at most $N$ arcs.
We consider all pairs of trees
$(\cT_{top}, \cT_{bottom})$ with $6$ leaves.}
\label{fig:Qp6:ratio}
\end{figure} | {"config": "arxiv", "file": "1705.05190/meanders_may_12.tex"} |
TITLE: Ordinary Differential equation question help
QUESTION [0 upvotes]: So i saw a question on ODE online and it stated that:
For the given Differential equation $y'=-2x+3y-5$ , a solution comes in the form $y(x)=mx+b$.
Then the instructor proceeded to state that if y(x) is a solution it needs to be true for all x's, im really confused in that last line "true for all x's" what does that mean.
REPLY [1 votes]: When you substitute back into the differential equation you get $m=(3m-2)x+3b-5$. You may have an instinct to solve the resulting equation for $x$, getting $x=\frac{m+5-3b}{3m-2}$. However that is how you solve an algebraic equation for a number, not a function equation (such as a differential equation) for a function. Stating that two functions are equal means they take the same output for every input. A functional (or differential) equation must be satisfied not just at a single $x$ value, but rather for all $x$ values. A polynomial equation can be solved for isolated $x$ values which satisfy it (for example by using the quadratic formula). But two polynomials can only be equal for all $x$ if all their coefficients match.
For the equation in question this can only happen if the linear terms and constant terms have the same coefficient. In other words $3m-2=0$ and $3b-5=m$, giving $y=2x/3+17/9$ as your answer, a function of $x$ which satisfies the functional equation at every $x$ value. | {"set_name": "stack_exchange", "score": 0, "question_id": 1967702} |
TITLE: Prove that for each $p \in Y$ the quotient field of $O_p$ is isomorphic to the field $K(Y)$.
QUESTION [4 upvotes]: I'm reading Algebraic Geometry of Hartshorne. I have a question about a proof. First of all I'll write the important definitions that Hartshorne uses.
Let $Y\subset \mathbb{A}^n$ be a quasi-affine variety.
A function $f:Y\to \mathbb{k}$ is said to be regular at $P\in Y$ if is an open neighborhood $U$ of $P$ such that $U\subset Y$ and polynomials $g,h\in A=\mathbb{k}[x_1,...,x_n]$ such that $h$ is nowhere zero on $U$ and $f=g/h$ on $U$. We say that $f$ is regular at $Y$ if is regular at each $P\in Y$.
We denote by $O(Y)$ the ring of all regular functions on $Y$. If $P\in Y$, we define the local ring of $P$ on $Y$, $O_P$ to be the ring of germs of regular functions on $Y$ near $P$. In other words, an element of $O_P$ is a pair $\left<U,f\right>$ where $U$ is an open neighborhood of $P$ and $f$ is regular on $U$. And where we identify two pairs $\left<U,f \right>$ and $\left<V,g \right>$ if $f=g$ on $U\cap V$.
Finally we define the function field $K(Y)$ as the set of equivalence relations, given by elements of the form $\left<U,f \right>$ for some $U$ open subset of $Y$ (not necessary containing $P$).
The operations that turns $O_p$ and $K(Y)$ into a rings an actually $K(Y)$ into a field are the usual for germs.
I'm reading the proof of Theorem $3.2$, I don't understand the proof of part $d)$. It uses the following observation:
My question: Prove that for each $p \in Y$ the quotient field of $O_p$ is isomorphic to the field $K(Y)$.
An element $z \in Frac(O_P)$ is writen in the form $z=f/g$ where $f,g\in O_P$ and $g$ is not the zero on $O_P$. Recall that $f,g$ are equivalence classes of the form $\left<U,f \right>$ and $\left<V,g \right>$, without lost of generality we can assume that $U=V$ otherwise we intersect the open sets and restrict the functions. I think that the most natural map is given by $\phi: Frac(O_p) \to K(Y)$ defined by:
$$ \left<U,f \right> / \left<U,g \right> \mapsto \left<U,f/g \right>$$
I proved that $\phi$ is an injective ring homomorphism. I think that this is the map that gives the isomorphism, but I'm not sure how to prove that it's surjective (or it's not but I don't think so).
Please help me
REPLY [2 votes]: Fix $[(U,f)] \in K(Y)$ an equivalence class of regular functions. Let $V \subseteq U$ be any open subset such that there exist polynomials $g$ and $h$ with $Z(h) \cap V = \emptyset$ and $(U,f) \sim (V,g/h)$. We then have both $[(Y,g)]$ and $[(Y,h)]$ elements of $\mathscr{O}_P$. Since $h$ is not the zero polynomial, it is not the zero element of $\mathscr{O}_P$, and also $Z(h) \neq Y$. So $\phi$ sends the element $[(Y,g)]/[(Y,h)]$ in the field of fractions of $\mathscr{O}_P$ to $[(Y\setminus Z(h),g/h)] \in K(Y)$, and this is precisely $[(U,f)]$.
By the way, you have made a slight error in your definition of $\phi$. You will need to set
$$
\phi\left(\frac{[(U,f)]}{[(V,g)]}\right) = [(W,f/g)],
$$
where $W = (U \cap V) \setminus Z(g)$. | {"set_name": "stack_exchange", "score": 4, "question_id": 1186323} |
TITLE: Time complexity of recursive functions
QUESTION [1 upvotes]: I am trying to derive the number of times * is performed in the following function.
int f(n) {
if (n == 1)
return 2*3;
else
return f(n-2)*f(n-2)*f(n-2);
}
Can someone help me in deriving this?
REPLY [1 votes]: Let $T(n)$ be the number of times $*$ is performed in evaluating $T(n)$. The base case is easy enough; for any other case, we must first evaluate $f(n-2)$ 3 times (incurring $3T(n-2)$ multiplications) and then two multiplications to put everything together. Hence,
$$T(1) = 1$$
$$T(2k+1) = 3T(2k-1) + 2$$
We may solve this recurrence to get $$T(2k+1) = \frac{2}{3} 3^{k+1} - 1 $$ | {"set_name": "stack_exchange", "score": 1, "question_id": 2458323} |
\begin{document}
\maketitle
\begin{abstract}
\noindent We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation. We consider discrete as well as continuous distributions, proving convergence rates of the proposed algorithm in both settings. Key elements of our analysis are a new result showing that the Sinkhorn divergence on compact domains has Lipschitz continuous gradient with respect to the Total Variation and a characterization of the sample complexity of Sinkhorn potentials. Experiments validate the effectiveness of our method in practice.
\end{abstract}
\section{Introduction}
\footnotetext[1]{Computer Science Department, University College London, WC1E 6BT London, United Kingdom}\footnotetext[2]{Computational Statistics and Machine Learning - Istituto Italiano di Tecnologia, 16100 Genova, Italy}\footnotetext[3]{Electrial and Electronics Engineering Department, Imperial College London, SW7 2BT, United Kingdom.}
Aggregating and summarizing collections of probability measures is a key task in several machine learning scenarios. Depending on the metric adopted, the properties of the resulting average (or {\em barycenter}) of a family of probability measures vary significantly. By design, optimal transport metrics are better suited at capturing the geometry of the distribution than Euclidean distance or other $f$-divergence \cite{cuturi14}. In particular, Wasserstein barycenters have been successfully used in settings such as texture mixing \cite{rabin2011wasserstein}, Bayesian inference \cite{srivastava2018scalable}, imaging \cite{gramfort2015fast}, or model ensemble \cite{dognin2019wasserstein}.
The notion of barycenter in Wasserstein space was first introduced by \cite{AguehC11} and then investigated from the computational perspective for the original Wasserstein distance \cite{staib2017parallel,stochwassbary} as well as its entropic regularizations (e.g. Sinkhorn) \cite{cuturi14, BenamouCCNP15, decentralized2018}. Two main challenges in this regard are: $i$) how to efficiently identify the support of the candidate barycenter and $ii$) how to deal with continuous (or infinitely supported) probability measures. The first problem is typically addressed by either fixing the support of the barycenter a-priori \cite{staib2017parallel,decentralized2018} or by adopting an alternating minimization procedure to iteratively optimize the support point locations and their weights \cite{cuturi14,stochwassbary}. While fixed-support methods enjoy better theoretical guarantees, free-support algorithms are more memory efficient and practicable in high dimensional settings. The problem of dealing with continuous distributions has been mainly approached by adopting stochastic optimization methods to minimize the barycenter functional \cite{stochwassbary,staib2017parallel,decentralized2018}
In this work we propose a novel method to compute the barycenter of a set of probability distributions with respect to the Sinkhorn divergence \cite{genevay2018learning} that does not require to fix the support beforehand. We address both the cases of discrete and continuous probability measures. In contrast to previous free-support methods, our algorithm does not perform an alternate minimization between support and weights. Instead, we adopt a Frank-Wolfe (FW) procedure to populate the support by incrementally adding new points and updating their weights at each iteration, similarly to kernel herding strategies \cite{bach2012equivalence} and conditional gradient for sparse inverse problem \cite{bredies2013inverse,boyd2017alternating}. Upon completion of this paper, we found that an idea with similar flavor, concerning the application a Frank-Wolfe scheme in conjunction with Sinkhorn functionals has been very recently considered in distributional regression settings for the case of Sinkhorn negentropies [35]. However, the analysis in this paper focuses on the theoretical properties of the proposed algorithm, specifically for the case of an {\em inexact} Frank-Wolfe procedure, which becomes critical in the case of continuous measures. In particular, we prove the convergence and rates of the proposed optimization method for both finitely and infinitely supported distribution settings. A central result in our analysis is the characterization of regularity properties of Sinkhorn potentials (i.e. the dual solutions of the Sinkhorn divergence problem), which extends recent work in \cite{feydy2018interpolating, genevay2018sample} and which we consider of independent interest. We empirically evaluate the performance of the proposed algorithm.
\paragraph{Contributions} The analysis of the proposed algorithm hinges on the following contributions: $i$) we show that the gradient of the Sinkhorn divergence is Lipschitz continuous on the space of probability measures with respect to the Total Variation. This grants us convergence of the barycenter algorithm in finite settings. $ii$) We characterize the sample complexity of Sinkhorn potentials of two empirical distributions sampled from arbitrary probability measures. This latter result allows us to $iii$) provide a concrete optimization scheme to approximately solve the barycenter problem for arbitrary probability measures with convergence guarantees. $iv$) A byproduct of our analysis is the generalization of the \fw{} algorithm to settings where the objective functional is defined only on a set with empty interior, which is the case for Sinkhorn divergence barycenter problem.
The rest of the paper is organized as follows: \cref{sec:background} reviews standard notions of optimal transport theory. \cref{sec:algorithm-theory} introduces the barycenter functional, and proves the Lipschitz continuity of its gradient.
\cref{sec:algorithm-practice} describes the implementation of our algorithm and \cref{sec:algorithm-convergence} studies its convergence rates. Finally, \cref{sec:experiments} evaluates the proposed methods empirically and \cref{sec:conclusion} provides concluding remarks.
\section{Background}\label{sec:background}
The aim of this section is to recall definitions and properties of Optimal Transport theory with entropic regularization. Throughout the work, we consider a compact set $\dom\subset\R^d$ and a symmetric cost function $\cost\colon\dom\times\dom\to\R$. We set $\diam := \sup_{x,y\in\X}~\dist(x,y)$ and denote by $\prob(\dom)$ the space of probability measures on $\dom$ (positive Radon measures with mass $1$). For any $\alpha,\beta \in \prob(\dom)$, the Optimal Transport problem with entropic regularization is defined as follow \cite{peyre2017computational,cuturi2013sinkhorn,genevay2016}
\begin{equation}\label{eq:primal_pb}
\oteps(\alpha,\beta) = \min_{\pi\in\Pi(\alpha,\beta)}~\int_{\dom^2}\cost(x,y)\,d\pi(x,y) + \eps\kl(\pi|\alpha\otimes\beta),\qquad \eps\geq0
\end{equation}
where $\kl(\pi|\alpha\otimes\beta)$ is the \emph{Kullback-Leibler divergence} between the candidate transport plan $\pi$ and the product distribution $\alpha \otimes \beta$, and $\Pi(\alpha,\beta)=\{\pi\in\mathcal{M}_{+}^1(\dom^2)\colon \mathsf{P}_{1\#}\pi=\alpha,\,\,\mathsf{P}_{2\#}\pi=\beta\}$, with $\mathsf{P}_{i}\colon\dom\times \dom\rightarrow\dom$ the projector onto the $i$-th component and $\#$ the push-forward operator. The case $\eps = 0$ corresponds to the classic Optimal Transport problem introduced by Kantorovich \cite{kantorovich1942transfer}. In particular, if $\cost = \nor{\cdot}^p$ for $p\in [1,\infty)$, then $\OT_0$ is the well-known $p$-Wasserstein distance \cite{villani2008optimal}.
Let $\eps>0$. Then, the dual problem
of \cref{eq:primal_pb},
in the sense of Fenchel-Rockafellar,
is \cite{chizat2018scaling,feydy2018interpolating}
\begin{equation}\label{eq:dual_pb}
\oteps(\alpha,\beta) = \max_{u,v\in \cont(\dom)} \int u(x)\,d\alpha(x) + \int v(y)\,d\beta(y) -\eps \int e^{\frac{u(x) + v(y) - \cost(x,y)}{\eps}}\,d\alpha(x)d\beta(y),
\end{equation}
where $\cont(\dom)$ denotes the space of real-valued continuous functions on $\dom$, endowed with $\norm{\cdot}_{\infty}$. Let $\mu\in\prob(\dom)$. We denote by $\rmap_\mu\colon\cont(\X)\to\cont(\X)$ the map such that, for any $w\in\cont(\X)$,
\begin{equation}\label{eq:rmap}
\rmap_\mu(w)\colon x\mapsto -\eps\log \int e^{\frac{w(y) - \cost(x,y)}{\eps}}\,d\mu(y).
\end{equation}
The first order optimality conditions for \cref{eq:dual_pb} are (see \cite{feydy2018interpolating} or \cref{subsec:sinkiter})
\begin{equation}\label{eq:fixed-point-sinkhorn}
u = \rmap_\beta(v) \quad \alpha \text{- a.e.} \qquad\text{and}\qquad v = \rmap_\alpha(u) \quad \beta \text{- a.e}.
\end{equation}
Pairs $(u,v)$ satisfying \cref{eq:fixed-point-sinkhorn} exist \cite{knopp1968note} and are referred to as {\em Sinkhorn potentials}. They are unique $(\alpha,\beta)$ - a.e. up to additive constant, i.e. $(u+t,v-t)$ is also a solution for any $t\in\R$. In line with \cite{genevay2018sample,feydy2018interpolating} it will be useful in the following to assume $(u,v)$ to be the Sinkhorn potentials such that: $i)$ $u(\anchor)=0$ for an arbitrary anchor point $\anchor\in\dom$ and $ii)$ \cref{eq:fixed-point-sinkhorn} is satisfied pointwise on the entire domain $\dom$. Then, $u$ is a fixed point of the map $\rmap_{\beta\alpha} = \rmap_\beta\circ\rmap_\alpha$ (analogously for $v$). This suggests a fixed point iteration approach to minimize \cref{eq:dual_pb}, yielding the well-known Sinkhorn-Knopp algorithm which has been shown to converge linearly in $\cont(\dom)$ \cite{sinkhorn1967,knopp1968note}
. We recall a key result characterizing the differentiability of $\oteps$ in terms of the Sinkhorn potentials that will be useful in the following.
\begin{proposition}[Prop $2$ in \cite{feydy2018interpolating}]\label{prop:derivaties}
Let $\nabla\oteps\colon\prob(\dom)^2\to\cont(\dom)^2$ be such that,
$\forall\alpha,\beta \in \prm(\X)$
\begin{equation}
\nabla\oteps(\alpha,\beta) =(u,v), \qquad \text{with} \qquad u = \rmap_\beta(v),~~ v = \rmap_\alpha(u)~~\text{on } \dom, \quad u(\anchor) = 0.
\end{equation}
Then, $\forall\alpha,\alpha^\prime,\beta,\beta^\prime\in\prob(\dom)$, the directional derivative of $\oteps$ along $(\mu,\nu) = (\alpha^\prime-\alpha,\beta^\prime-\beta)$ is
\begin{equation}\label{eq:directional-derivative-oteps-intro}
\oteps^\prime(\alpha,\beta; \mu,\nu) = \scal{\nabla \oteps(\alpha,\beta)}{(\mu,\nu)} = \scal{u}{\mu} + \scal{v}{\nu},
\end{equation}
where $\scal{w}{\rho} = \int w(x)\,d\rho(x)$ denotes the canonical pairing between the spaces $\cont(\dom)$ and $\meas(\dom)$.
\end{proposition}
Note that $\nabla\oteps$ is not a gradient in the standard sense. In particular note that the directional derivative in \cref{eq:directional-derivative-oteps-intro} is not defined for any pair of signed measures, but only along {\em feasible directions} $(\alpha^\prime-\alpha,\beta^\prime-\beta)$.
\paragraph{Sinkhorn Divergence} The fast convergence of Sinkhorn-Knopp algorithm makes $\oteps$ (with $\eps>0$) preferable to $\OT_0$ from a computational perspective \cite{cuturi2013sinkhorn}. However, when $\eps>0$ the entropic regularization introduces a bias in the optimal transport problem, since in general $\oteps(\mu,\mu)\neq 0$. To compensate for this bias, \cite{genevay2018learning} introduced the Sinkhorn {\em divergence}
\begin{equation}\label{eq:sink_divergence}
\sink\colon\prob(\dom)\times\prob(\dom)\to\R, \qquad
(\alpha,\beta) \mapsto \oteps(\alpha,\beta) - \frac{1}{2}\oteps(\alpha,\alpha) -\frac{1}{2}\oteps(\beta,\beta),
\end{equation}
which was shown in \cite{feydy2018interpolating} to be nonnegative, bi-convex and to metrize the convergence in law under mild assumptions. We characterize the gradient of $\sink(\cdot,\beta)$ for a fixed $\beta\in\prob(\X)$, which will be key to derive our optimization algorithm for computing Sinkhorn barycenters.
\begin{remark}\label{rem:gradient-sinkhorn-divergence}
Let $\nabla_1\oteps:\prob(\dom)^2\to\cont(\dom)$ be the first component of $\nabla\oteps$ (informally, the $u$ of the Sinkhorn potentials). As in \cref{prop:derivaties}, for any $\beta\in\prob(\dom)$ the gradient of $S_\eps(\cdot,\beta)$ is
\begin{equation}\label{eq:grad_sink}
\nabla [S_\eps(\cdot, \beta)]\colon\prob(\X)\to\cont(\X) \qquad \alpha \mapsto \nabla_1\oteps(\alpha,\beta) - \frac{1}{2}\nabla_1\oteps(\alpha,\alpha) = u-p,
\end{equation}
with $u=\rmap_{\beta\alpha}(u)$ and $p = \rmap_{\alpha}(p)$ the Sinkhorn potentials of $\oteps(\alpha,\beta)$ and $\oteps(\alpha,\alpha)$ respectively.
\end{remark}
\section{Sinkhorn barycenters with Frank-Wolfe}\label{sec:algorithm-theory}
Given $\beta_1,\dots\beta_m\in\prob(\dom)$ and $\omega_1,\dots,\omega_m\geq0$ a set of weights such that $\sum_{j=1}^m \omega_j = 1$, the main goal of this paper is to solve the following {\em Sinkhorn barycenter} problem
\begin{equation}\label{eq:sinkhorn-barycenter}
\min _{\alpha \in \prob(\dom)} \bary(\alpha), \qquad\textnormal{with}\qquad \bary(\alpha) = \sum_{j=1}^m ~\omega_j~\sink(\alpha, \beta_j).
\end{equation}
Although the objective functional $\bary$ is convex, its domain $\prm(\dom)$ has \textit{empty} interior in the space of finite signed measure $\meas(\X)$. Hence standard notions of Fr\'echet or G\^ateaux differentiability do not apply.
This, in principle causes some difficulties in devising optimization methods.
To circumvent this issue, in this work we adopt
the Frank-Wolfe (\fw{}) algorithm. Indeed,
one key advantage of this method is that it is formulated in terms of
directional derivatives along feasible directions
(i.e., directions that locally remain inside the constraint set). Building upon \cite{dem1967,dem1968,dunn1978conditional},
which study the algorithm in Banach spaces, we show that
the ``weak'' notion of directional differentiability of $\sink$ (and hence of $\bary$) in \cref{rem:gradient-sinkhorn-divergence} is sufficient
to carry out the convergence analysis.
While full details are provided in \cref{sec:frank-wolfe}, below we give an overview of the main result.
\paragraph{Frank-Wolfe in dual Banach spaces} Let $\VV$ be a real Banach space with topological dual $\BB$ and let $\CC\subset\BB$ be a nonempty, convex, closed and bounded set. For any $\xx\in\BB$ denote by $\feas_\CC(\xx)=\R_+(\CC-\xx)$ the set of feasible direction of $\CC$ at $\xx$ (namely $s=t(\xx^\prime - \xx)$ with $\xx^\prime\in\CC$ and $t>0$). Let $\func\colon\CC\to\R$ be a convex function and assume that there exists a map $\nabla\func\colon\CC\to\VV$ (not necessarily unique) such that $\scal{\nabla \func(\xx)}{s} = \func^\prime(\xx;s)$ for every $s\in\feas_\CC(\xx)$.
In \cref{alg:abstract-FW-intro} we present a method to minimize $\func$. The algorithm is structurally equivalent to the standard \fw{} \cite{dunn1978conditional,jaggi2013revisiting}
and accounts for possible inaccuracies in solving the minimization
in step $(i)$. This will be key in \cref{sec:algorithm-convergence}
when studying the barycenter problem for $\beta_j$ with infinite
support. The following result (see proof in \cref{sec:frank-wolfe}) shows that under the additional
assumption that $\nabla\func$ is Lipschitz-continuous and with sufficiently fast decay of the errors, the above
procedure converges in value to the minimum of $\func$
with rate $O(1/k)$. Here $\Diam(\CC)$ denotes the diameter of $\CC$
with respect to the dual norm.
\begin{theorem}\label{thm:fw-informal}
Under the assumptions above, suppose in addition that $\nabla \func$ is $L$-Lipschitz continuous with $L>0$. Let $(\xx_k)_{k \in \N}$ be obtained according to \cref{alg:abstract-FW-intro}. Then, for every integer $k\geq1$,
\begin{equation}
\label{eq:20190418a}
\func(\xx_k) - \min \func \leq \frac{2}{k+2} L\,\Diam(\CC)^2 + \precision_k.
\end{equation}
\end{theorem}
\begin{algorithm}[t]
\caption{{\sc Frank-Wolfe in Dual Banach Spaces}}
\label{alg:abstract-FW-intro}
\begin{algorithmic}
\vspace{0.25em}
\STATE {\bfseries Input:} initial $\xx_0\in\CC$, precision
$(\precision_k)_{k \in \N} \in \R_{++}^\N$, such that $\precision_k(k+2)$ is nondecreasing.
\vspace{0.45em}
\STATE {\bfseries For} $k=0,1,\dots$
\STATE \qquad Take $\zz_{k+1}$ such that ${\func^\prime(\xx_k, \zz_{k+1} - \xx_k) \leq \min_{\zz \in \CC} \func^\prime(\xx_k, \zz - \xx_k) + \frac{\precision_k}{2}}$
\STATE \qquad ${\xx_{k+1} = \xx_k + \frac{2}{k+2}(\zz_{k+1} - \xx_k)}$
\end{algorithmic}
\end{algorithm}
\paragraph{Frank-Wolfe Sinkhorn barycenters} We show that the barycenter problem \cref{eq:sinkhorn-barycenter} satisfies the setting and hypotheses of \cref{thm:fw-informal} and can be thus approached via \cref{alg:abstract-FW-intro}.
\emparagraph{Optimization domain} Let $\VV = \cont(\dom)$, with dual $\BB=\meas(\X)$. The constraint set $\CC = \prob(\dom)$ is convex, closed, and bounded.
\emparagraph{Objective functional} The objective functional $\func = \bary\colon\prob(\dom)\to\R$, defined in \cref{eq:sinkhorn-barycenter}, is convex since it is a convex combination of $\sink(\cdot,\beta_j)$, with
$j= 1 \dots m$. The gradient $\nabla\bary\colon\prob(\dom)\to\cont(\dom)$ is $\nabla\bary = \sum_{j=1}^m~\omega_j~ \nabla \sink(\cdot,\beta_j)$, where $\nabla \sink(\cdot,\beta_j)$ is given in \cref{rem:gradient-sinkhorn-divergence}.
\emparagraph{Lipschitz continuity of the gradient.} This is the most critical condition and is addressed in the following theorem.
\begin{restatable}{theorem}{TLipschitzContinuityTV}\label{thm:lip-continuity-total-variation-informal}
The gradient $\nabla\oteps$
defined in \cref{prop:derivaties}
is Lipschitz continuous. In particular, the first component $\nabla_1\oteps$ is $2\eps e^{3\diameps}$-Lipschitz continuous, i.e., for every $\alpha,\alpha^\prime,\beta,\beta^\prime\in\prob(\dom)$,
\begin{equation}
\supnor{u - u^\prime} = \supnor{\nabla_1\oteps(\alpha,\beta)-\nabla_1\oteps(\alpha^\prime,\beta^\prime)} \leq 2\eps e^{3\diameps}~(\nor{\alpha - \alpha^\prime}_{TV} + \nor{\beta-\beta^\prime}_{TV}),
\end{equation}
where $\diam = \sup_{x,y\in\X}~\dist(x,y)$, $u = \rmap_{\beta\alpha}(u), u^\prime = \rmap_{\beta^\prime,\alpha^\prime}(u^\prime)$, and $u(\anchor)=u^\prime(\anchor)=0$. Moreover, it follows from \cref{eq:grad_sink} that $\nabla \sink(\cdot,\beta)$ is $6\eps e^{3\diameps}$-Lipschitz continuous. The same holds for $\nabla \bary$.
\end{restatable}
\cref{thm:lip-continuity-total-variation-informal} is one of the main contributions of this paper. It can be rephrased by saying that the operator that maps
a pair of distributions to their Sinkhorn potentials is Lipschitz continuous. This result is significantly deeper than the one given in \cite[Lemma 1]{decentralized2018}, which establishes the Lipschitz continuity of the gradient in the \textit{semidiscrete} case. The proof (given in \cref{sec:app-frank-wolfe-algorithm}) relies on non-trivial tools from Perron-Frobenius theory for Hilbert's metric \cite{lemmens2012nonlinear}, which is a well-established framework to study Sinkhorn potentials \cite{peyre2017computational}. We believe this result is interesting not only for the application of \fw{} to the Sinkhorn barycenter problem,
but also for further understanding regularity properties of entropic optimal transport.
\section{Algorithm: practical Sinkhorn barycenters}\label{sec:algorithm-practice}
According to \cref{sec:algorithm-theory}, \fw{} is a valid approach to tackle the barycenter problem \cref{eq:sinkhorn-barycenter}. Here we describe how to implement in practice the abstract procedure of \cref{alg:abstract-FW-intro} to obtain a sequence of distributions $(\alpha_k)_{k\in\N}$ minimizing $\bary$.
A main challenge in this sense resides in finding a minimizing feasible direction for $\bary^\prime(\alpha_k;\mu-\alpha_k) = \scal{\nabla\bary(\alpha_k)}{\mu-\alpha_k}$. According to \cref{rem:gradient-sinkhorn-divergence}, this amounts to solve
\begin{equation}\label{eq:inner-fw}
\mu_{k+1} \in \argmin_{\mu\in\prob(\dom)} ~\sum_{j=1}^m ~\omega_j~ \scal{u_{jk}-p_{k}}{\mu} \qquad\text{where}\qquad u_{jk}-p_{k} = \nabla\sink[(\cdot,\beta_j)](\alpha_k),
\end{equation}
with $p_k = \nabla_1\oteps(\alpha_k,\alpha_k)$ not depending on $j$. In general \cref{eq:inner-fw} would entail a minimization over the set of all probability distributions on $\dom$. However, since the objective functional is linear in $\mu$ and $\prob(\X)$ is a weakly-$*$ compact convex set, we can apply Bauer maximum principle (see e.g., \cite[Thm. 7.69]{aliprantis2006}). Hence, solutions are achieved at the extreme points of the optimization domain, namely Dirac's deltas for the case of $\prob(\X)$ \cite[p. 108]{choquet1969}. Now, denote by $\delta_x\in\prob(\dom)$ the Dirac's delta centered at $x\in\dom$. We have $\scal{w}{\delta_x} = w(x)$ for every $w\in\cont(\dom)$. Hence \cref{eq:inner-fw} is equivalent to
\begin{equation}\label{eq:inner-fw-pointwise}
\mu_{k+1} = \delta_{x_{k+1}} \qquad \text{with} \qquad x_{k+1} \in \argmin_{x\in\dom}~ \sum_{j=1}^m ~\omega_j~ \big(u_{jk}(x)-p_{k}(x)\big).
\end{equation}
Once the new support point $x_{k+1}$ has been obtained, the update in \cref{alg:abstract-FW-intro} corresponds to
\begin{equation}\label{eq:fw-bary-update}
\alpha_{k+1} = \alpha_k + \frac{2}{k+2} (\delta_{x_{k+1}} -\alpha_k) = \frac{k}{k+2} \alpha_k + \frac{2}{k+2} \delta_{x_{k+1}}.
\end{equation}
In particular, if \fw{} is initialized with a distribition with finite support, say $\alpha_0 = \delta_{x_0}$ for some $x_0\in\dom$, then also every further iterate $\alpha_k$ will have at most $k+1$ support points.
According to \cref{eq:inner-fw-pointwise}, the inner optimization for \fw{} consists in minimizing the functional $x\mapsto\sum_{j=1}^m ~\omega_j~ \big(u_{jk}(x)-p_{k}(x)\big)$ over $\dom$. In practice, having access to such functional poses already a challenge, since it requires computing the Sinkhorn potentials $u_{jk}$ and $p_{k}$, which are infinite dimensional objects. Below we discuss how to estimate these potentials when the $\beta_j$ have finite support. We then address the general setting.
\paragraph{Computing $\nabla_1\oteps$ for probability distributions with finite support}
Let $\alpha,\beta\in\prob(\dom)$, with $\beta = \sum_{i=1}^{n} b_i \delta_{y_i}$ a probability measure with finite support, with $\msf{b} = (b_i)_{i=1}^{n}$ nonnegative weights summing up to $1$. It is useful to identify $\beta$ with the pair $(\mbf{Y},\msf{b})$, where $\mbf{Y}\in\R^{d \times n}$ is the matrix with $i$-th column equal to $y_i$. Let now $(u,v)\in\cont(\dom)^2$ be the pair of Sinkhorn potentials associated to $\alpha$ and $\beta$ in \cref{prop:derivaties}, recall that $u = \rmap_\beta(v)$. Denote by $\msf v\in\R^n$ the {\em evaluation vector} of the Sinkhorn potential $v$, with $i$-th entry $\msf{v}_i = v(y_i)$. According to the definition of $\rmap_\beta$ in \cref{eq:rmap}, for any $x\in\dom$
\begin{equation}\label{eq:sinkhorn-gradient-routine}
[\nabla_1\oteps(\alpha,\beta)](x) = u(x) = [\rmap_{\beta}(v)](x) = -\eps\log \sum_{i=1}^n ~e^{(\msf{v}_i - \cost(x,y_i))/\eps}~b_i,
\end{equation}
since the integral $\rmap_\beta(v)$ reduces to a sum
over the support of $\beta$. Hence, the gradient of $\oteps$ (i.e. the potential $u$), {\em is uniquely characterized in terms of the finite dimensional vector $\msf{v}$ collecting the values
of the potential $v$ on the support of $\beta$
}. We refer as {\sc SinkhornGradient} to the routine which associates to each triplet $(\mbf{Y},\mbf{b},\mbf{v})$ the map
$x\mapsto-\eps~\log \sum_{i=1}^n ~e^{(\msf{v}_i - \cost(x,y_i))/\eps}~b_i$.
\begin{algorithm}[t]
\caption{{\sc Sinkhorn Barycenter}}\label{alg:practical-FW}
\begin{algorithmic}
\vspace{0.25em}
\STATE {\bfseries Input:} $\beta_j = (\mbf{Y}_j,\msf{b}_j)$ with $\mbf{Y}_j\in\R^{d \times n_j}, \msf{b}_j\in\R^{n_j},\omega_j>0$ for $j=1,\dots,m$, $x_0 \in\R^d$, $\eps>0$, $K\in\N$.
\vspace{0.45em}
\STATE {\bfseries Initialize:} $\alpha_0 = (\mbf{X}_0,\msf{a}_0)$ with $\mbf{X}_0=x_0$, $\msf{a}_0 = 1$.
\vspace{0.45em}
\STATE {\bfseries For} $k=0,1,\dots,K-1$
\vspace{0.2em}
\STATE \qquad $\msf{p} =$ {\sc SinkhornKnopp}$(\alpha_k,\alpha_k,\eps)$
\STATE \qquad $p(\cdot) =$ {\sc SinkhornGradient}$(\mbf{X}_k,\msf{a}_k,\msf{p})$
\vspace{0.25em}
\STATE \qquad {\bfseries For} $j=1,\dots m$
\STATE \qquad\qquad $\msf{v}_j = $ {\sc SinkhornKnopp}$(\alpha_k,\beta_j,\eps)$
\STATE \qquad\qquad $u_j(\cdot) =$ {\sc SinkhornGradient}$(\mbf{Y}_j,\msf{b}_j,\msf{v}_j)$
\STATE \qquad {\bfseries Let} $\innerfunc\colon x
\mapsto \sum_{j=1}^m\omega_j ~u_j(x) - p(x)$
\STATE \qquad $x_{k+1} = $ {\sc Minimize}$(\innerfunc)$
\vspace{0.25em}
\STATE \qquad $\mbf{X}_{k+1} = [\mbf{X}_k,x_{k+1}]$ ~and~ $\msf{a}_{k+1} = \frac{1}{k+2}\left[k~\msf{a}_k,2\right]$
\STATE \qquad $\alpha_{k+1} = (\mbf{X}_{k+1},\msf{a}_{k+1})$
\vspace{0.45em}
\STATE {\bfseries Return:} $\alpha_K$
\end{algorithmic}
\end{algorithm}
\paragraph{Sinkhorn barycenters: finite case}
\cref{alg:practical-FW} summarizes \fw{} applied to the barycenter problem \cref{eq:sinkhorn-barycenter} when the $\beta_j$'s have finite support. Starting from a Dirac's delta $\alpha_0 = \delta_{x_0}$, at each iteration $k\in\N$ the algorithm proceeds by: $i)$ finding the corresponding evaluation vectors $\msf{v}_j$'s and $\msf{p}$ of the Sinkhorn potentials for $\oteps(\alpha_k,\beta_j)$ and $\oteps(\alpha_k,\alpha_k)$ respectively, via the routine {\sc SinkhornKnopp} (see \cite{cuturi2013sinkhorn,feydy2018interpolating} or \cref{algo:sinkalgo_disc}). This is possible since both $\beta_j$ and $\alpha_k$ have finite support and therefore the problem of approximating the evaluation vectors $\msf{v}_j$ and $\msf{p}$ reduces to an optimization problem over finite vector spaces that can be efficiently solved \cite{cuturi2013sinkhorn}; $ii)$ obtain the gradients $u_{j} = \nabla_1\oteps(\alpha_k,\beta_j)$ and $p = \nabla_1\oteps(\alpha_k,\alpha_k)$ via {\sc SinkhornGradient}; $iii)$ minimize $\innerfunc:x\mapsto\sum_{j=1}^n \omega_j ~ u_j(x) - p(x)$ over $\dom$ to find a new point $x_{k+1}$ (we comment on this meta-routine {\sc Minimize} below); $iv)$ finally update the support and weights of $\alpha_k$ according to \cref{eq:fw-bary-update} to obtain the new iterate $\alpha_{k+1}$.
A key feature of \cref{alg:practical-FW} is that the support of the candidate barycenter is updated {\em incrementally} by adding at most one point at each iteration, a procedure similar in flavor to the kernel herding strategy in \cite{bach2012equivalence,lacoste2015sequential}. This contrasts with previous methods for barycenter estimation \cite{cuturi14,BenamouCCNP15,staib2017parallel,decentralized2018}, which require the support set, or at least its cardinality, to be fixed beforehand. However, indentifying the new support point requires solving the nonconvex problem \cref{eq:inner-fw-pointwise}, a task addressed by the meta-routine {\sc Minimize}. This problem is typically smooth (e.g., a linear combination of Gaussians when $\cost(x,y) = \nor{x-y}^2$) and first or second order nonlinear optimization methods can be adopted to find stationary points. We note that all free-support methods in the literature for barycenter estimation are also affected by nonconvexity since they typically require solving a bi-convex problem (alternating minimization between support points and weights) which is not jointly convex \cite{cuturi14,stochwassbary}. We conclude by observing that if we restrict to the setting of \cite{staib2017parallel,decentralized2018} with fixed support set, then {\sc Minimize} can be solved exactly by evaluating the functional in \cref{eq:inner-fw-pointwise} on each candidate support point.
\paragraph{Sinkhorn barycenters: general case} When the $\beta_j$'s have infinite support, it is not possible to apply Sinkhorn-Knopp in practice. In line with \cite{genevay2018sample, staib2017parallel}, we can randomly sample empirical distributions $\hat\beta_j = \frac{1}{n}\sum_{i=1}^n \delta_{x_{ij}} $ from each $\beta_j$ and apply Sinkhorn-Knopp to $(\alpha_k,\hat\beta_j)$ in \cref{alg:abstract-FW-intro} rather than to the ideal pair $(\alpha_k,\beta_j)$. This strategy is motivated by \cite[Prop 13]{feydy2018interpolating}, where it was shown that Sinkhorn potentials vary continuously with the input measures. However, it opens two questions: $i)$ whether this approach is theoretically justified (consistency) and $ii)$ how many points should we sample from each $\beta_j$ to ensure convergence (rates). We answer these questions in \cref{thm:sinkhorn-barycenters-infinite-case} in the next section.
\section{Convergence analysis}\label{sec:algorithm-convergence}
We finally address the convergence of \fw{} applied to both the finite and infinite settings discussed in \cref{sec:algorithm-practice}. We begin by considering the finite setting.
\begin{restatable}{theorem}{TLSinkhornBarycenterFiniteCase}
\label{thm:sinkhorn-barycenters-finite-case}
Suppose that $\beta_1,\dots\beta_m\in\prob(\dom)$ have finite support and let $\alpha_k$ be the $k$-th iterate of \cref{alg:practical-FW} applied to \cref{eq:sinkhorn-barycenter}. Then,
\begin{equation}\label{eq:convergence_finite_case}
\bary(\alpha_k) - \min_{\alpha\in\prob(\dom)}\bary(\alpha) \leq \frac{48\,\eps\, e^{3\diameps}}{k+2}.
\end{equation}
\end{restatable}
The result follows by the convergence result of \fw{} in \cref{thm:fw-informal} applied with the Lipschitz constant computed in \cref{thm:lip-continuity-total-variation-informal}, and recalling that $\Diam(\prob(\X))=2$ with respect to the Total Variation. We note that \cref{thm:sinkhorn-barycenters-finite-case} assumes {\sc SinkhornKnopp} and {\sc Minimize} in \cref{alg:practical-FW} to yield exact solutions. In \cref{sec:app-frank-wolfe-algorithm} we comment how approximation errors in this context affect the bound in \cref{eq:convergence_finite_case}.
\paragraph{General setting} As mentioned in \cref{sec:algorithm-practice}, when the $\beta_j$'s are not finitely supported we adopt a sampling approach. More precisely we propose to {\em replace} in \cref{alg:practical-FW} the ideal Sinkhorn potentials of the pairs $(\alpha,\beta_j)$ with those of $(\alpha,\hat\beta_j)$, where each $\hat\beta_j$ is an empirical measure randomly sampled from $\beta_j$. In other words we are performing the \fw{} algorithm with a (possibly rough) approximation of the correct gradient of $\bary$. According to \cref{thm:fw-informal}, \fw{} allows errors in the gradient estimation (which are captured into the precision $\Delta_k$ in the statement). To this end, the following result \textit{quantifies} the approximation error between $\nabla_1\oteps(\cdot,\beta)$ and $\nabla_1\oteps(\cdot,\hat\beta)$ in terms of the sample size of $\hat\beta$.
\begin{restatable}[Sample Complexity of Sinkhorn Potentials]{theorem}{TLSampleComplexitySinkhornPotentials}
\label{thm:sample-complexity-sinkhorn-gradients}
Suppose that $\cost \in \cont^{s+1}(\X\times\X)$ with $s>d/2$.
Then,
there exists a constant $\overline\sinkconst=\overline\sinkconst(\dom,\cost,d)$ such that for any $\alpha,\beta\in\prob(\dom)$ and any empirical measure $\hat\beta$ of a set of $n$ points independently sampled from $\beta$, we have, for every $\tau\in(0,1]$
\begin{equation}\label{eq:uniform-approximation-empirical-continuous-sinkhorn-potentials}
\supnor{u - u_n} = \lVert \nabla_1\oteps(\alpha,\beta)-\nabla_1\oteps(\alpha,\hat\beta) \rVert_{\infty}\leq \frac{8\varepsilon~\overline\sinkconst e^{3\diameps}\log\frac{3}{\tau}}{\sqrt{n}}
\end{equation}
with probability at least $1-\tau$, where $u= \rmap_{\beta\alpha}(u), u_n = \rmap_{\hat\beta\alpha}(u_n)$ and $u(\anchor) = u_n(\anchor) = 0$.
\end{restatable}
\noindent\cref{thm:sample-complexity-sinkhorn-gradients} is one of the main results of this work. We point out that it {\em cannot} be obtained by means of the Lipschitz continuity of $\nabla_1\oteps$ in \cref{thm:lip-continuity-total-variation-informal}, since empirical measures do not converge in $\nor{\cdot}_{TV}$ to their target distribution \cite{devroye1990no}. Instead, the proof consists in considering the weaker {\em Maximum Mean Discrepancy (MMD)} metric associated to a universal kernel \cite{song2008learning}, which metrizes the topology of the convergence in law of $\prob(\dom)$ \cite{sriperumbudur2011universality}. Empirical measures converge in MMD metric to their target distribution \cite{song2008learning}. Therefore, by proving the Lipschitz continuity of $\nabla_1\oteps$ with \textit{respect to \mmd{}} in \cref{prop:lipschitz-continuity-mmd} we are able to conclude that \cref{eq:uniform-approximation-empirical-continuous-sinkhorn-potentials} holds. This latter result relies on higher regularity properties of Sinkhorn potentials, which have been recently shown \cite[Thm.2]{genevay2018sample} to be uniformly bounded in Sobolev spaces under the additional assumption $c\in\cont^{s+1}(\X\times\X)$. For sufficiently large $s$, the Sobolev norm is in duality with the MMD \cite{muandet2017kernel} and allows us to derive the required Lipschitz continuity. We conclude noting that while \cite{genevay2018sample} studied the sample complexity of the Sinkhorn {\em divergence}, \cref{thm:sample-complexity-sinkhorn-gradients} is a sample complexity result for Sinkhorn {\em potentials}. In this sense, we observe that the constants appearing in the bound are tightly related to those in \cite[Thm.3]{genevay2018sample} and have similar behavior with respect to $\eps$. We can now study the convergence of \fw{} in continuous settings.
\begin{restatable}{theorem}{TLSInkhornBarycenterInfiniteDim}
\label{thm:sinkhorn-barycenters-infinite-case}
Suppose that $\cost \in \cont^{s+1}(\X\times\X)$ with $s>d/2$.
Let $n \in \N$ and
$\hat\beta_1,\dots,\hat\beta_m$ be empirical distributions with $n$ support points, each independently sampled from $\beta_1,\dots,\beta_m$.
Let $\alpha_k$ be the $k$-th iterate of \cref{alg:practical-FW} applied to $\hat\beta_1,\dots,\hat\beta_m$. Then for any $\tau\in(0,1]$, the following holds with probability larger than $1-\tau$
\begin{equation}
\bary(\alpha_k) - \min_{\alpha\in\prob(\dom)} \bary(\alpha) \leq
\frac{64 \bar\sinkconst \varepsilon e^{3\diameps} \log\frac{3}{\tau} }{\min(k,\sqrt{n})}.
\end{equation}
\end{restatable}
The proof is shown in \cref{sec:sample-complexity-sinkhorn-potentials}.
A consequence of \cref{thm:sinkhorn-barycenters-infinite-case} is that the accuracy of \fw{} depends simultaneously on the number of iterations and the sample size used in the approximation of the gradients: by choosing $n = k^2$ we recover the $O(1/k)$ rate of the finite setting, while for $n=k$ we have a rate of $O(k^{-1/2})$, which is reminiscent of typical sample complexity results, highlighting the statistical nature of the problem.
\begin{remark}[Incremental Sampling]
The above strategy requires sampling the empirical distributions for $\beta_1,\dots,\beta_m$ beforehand. A natural question is whether it is be possible to do this {\em incrementally}, sampling new points and updating $\hat\beta_j$ accordingly, as the number of \fw{} iterations increase. To this end, one can perform an intersection bound and see that this strategy is still consistent, but the bound in \cref{thm:sinkhorn-barycenters-infinite-case}
worsens the logarithmic term, which becomes $\log (3m k/\tau)$.
\end{remark}
\section{Experiments}
\label{sec:experiments}
In this section we show the performance of our method in a range of experiments \footnote{\url{https://github.com/GiulsLu/Sinkhorn-Barycenters}}.
\paragraph{Discrete measures: barycenter of nested ellipses} We compute the barycenter of $30$ randomly generated nested ellipses on a $50\times50$ grid similarly to \cite{cuturi14}. We interpret each image as a probability distribution in $2$D. The cost matrix is given by the squared Euclidean distances between pixels. \cref{fig:ellipses} reports $8$ samples of the input ellipses and the barycenter obtained with \cref{alg:practical-FW}. It shows qualitatively that our approach captures key geometric properties of the input measures.
\begin{figure}[t]
\begin{minipage}[t]{0.3\textwidth}
\centering
\includegraphics[height=4.05cm,trim={0 -0.05cm 0 0},clip]{images/ellipses.png} \caption{Ellipses}
\label{fig:ellipses}
\end{minipage}
\begin{minipage}[t]{0.7\textwidth}
\centering
\includegraphics[height=4cm]{images/gauss1.png}
\includegraphics[height=4cm]{images/gauss2.png}
\caption{Barycenters of Gaussians}
\label{fig:gauss}
\end{minipage}
\end{figure}
\paragraph{Continuous measures: barycenter of Gaussians}
We compute the barycenter of $5$ Gaussian distributions $\mathcal{N}(m_i,C_i)$ $i=1,\dots,5$ in $\R^2$, with mean $m_i\in\R^2$ and covariance $C_i$ randomly generated. We apply \cref{alg:practical-FW} to empirical measures obtained by sampling $n=500$ points from each $\mathcal{N}(m_i,C_i)$, $i=1,\dots,5$. Since the (Wasserstein) barycenter of Gaussian distributions can be estimated accurately (see \cite{AguehC11}), in \cref{fig:gauss}
we report both the output of our method (as a scatter plot) and the true Wasserstein barycenter (as level sets of its density). We observe that our estimator recovers both the mean and covariance of the target barycenter. See the supplementary material for additional experiments also in the case of mixtures of Gaussians.
\paragraph{Image ``compression'' via distribution matching}
Similarly to \cite{stochwassbary}, we test \cref{alg:practical-FW} in the special case of computing the ``barycenter'' of a single measure $\beta\in\prm(\dom)$. While the solution of this problem is the distribution $\beta$ itself, we can interpret the intermediate iterates $\alpha_k$ of \cref{alg:practical-FW} as compressed version of the original measure. In this sense $k$ would represent the level of compression since $\alpha_k$ is supported on {\em at most} $k$ points. \cref{fig:cheeta} (Right) reports iteration $k=5000$ of \cref{alg:practical-FW} applied to the $140\times140$ image in \cref{fig:cheeta} (Left) interpreted as a probability measure $\beta$ in $2$D. We note that the number of points in the support is $\sim 3900$: indeed, \cref{alg:practical-FW} selects the most relevant support points points multiple times to accumulate the right amount of mass on each of them (darker color = higher weight). This shows that \fw{} tends to greedily search for the most relevant support points, prioritizing those with higher weight
\begin{figure}[t]
\begin{minipage}{0.6\textwidth}
\centering
\includegraphics[height = 4.7cm]{images/cheetaorig_copy.png}\quad
\includegraphics[height = 4.7cm]{images/cheeta4k_copy_better_crop.png}
\caption{(left) original image 140x140 pixels, sample (right) }
\label{fig:cheeta}
\end{minipage}
\begin{minipage}{0.39\textwidth}
\centering
\includegraphics[height = 4.7cm,trim={0 0.5cm 0 0.5cm},clip]{images/centroids2.png}
\caption{$k$-means}
\label{fig:centroids}
\end{minipage}
\end{figure}
\paragraph{k-means on MNIST digits} We tested our algorithm on a $k$-means clustering experiment. We consider a subset of $500$ random images from the MNIST dataset. Each image is suitably normalized to be interpreted as a probability distribution on the grid of $28\times28$ pixels with values scaled between $0$ and $1$. We initialize $20$ centroids according to the $k$-means++ strategy \cite{kmeans++}. \cref{fig:centroids} deipcts the $20$ centroids obtained by performing $k$-means with \cref{alg:practical-FW}. We see that the structure of the digits is successfully detected, recovering also minor details (e.g. note the difference between the $2$ centroids).
\paragraph{Real data: Sinkhorn propagation of weather data}
We consider the problem of Sinkhorn {\em propagation} similar to the one in \cite{Solomon:2014:WPS}. The goal is to predict the distribution of missing measurements for weather stations in the state of Texas, US by ``propagating'' measurements from neighboring stations in the network. The problem can be formulated as minimizing the functional $\sum_{(v,u)\in\mathcal{V}} \omega_{uv}\sink(\rho_v,\rho_u)$ over the set $\{\rho_v\in\prob(\R^2) | v\in\mathcal{V}_0\}$ with: $\mathcal{V}_0\subset\mathcal{V}$ the subset of stations with missing measurements, $G = (\mathcal{V},\mathcal{E})$ the whole graph of the stations network, $\omega_{uv}$ a weight inversely proportional to the geographical distance between two vertices/stations $u,v\in\mathcal{V}$. The variable $\rho_v\in\prob(\R^2)$ denotes the distribution of measurements at station $v$ of daily {\em temperature} and {\em atmospheric pressure} over one year. This is a generalization of the barycenter problem \cref{eq:sinkhorn-barycenter} (see also \cite{peyre2017computational}).
From the total $|\mathcal{V}|=115$, we randomly select $10\%,20\%$ or $30\%$ to be {\em available} stations, and use \cref{alg:practical-FW} to propagate their measurements to the remaining ``missing'' ones.
We compare our approach (\fw{}) with the Dirichlet (DR) baseline in \cite{Solomon:2014:WPS} in terms of the error $d(C_T,\hat C)$ between the covariance matrix $C_T$ of the groundtruth distribution and that of the predicted one. Here $d(A,B) = \norm{\log(A^{-1/2} B A^{-1/2})}$ is the geodesic distance on the cone of positive definite matrices. The average prediction errors are: $2.07$ (\fw{}), $2.24$ (DR) for $10\%$, $1.47$ (\fw{}), $1.89$(DR) for $20\%$ and $1.3$ (\fw{}), $1.6$ (DR) for $30\%$. \cref{fig:propagation} qualitatively reports the improvement $\Delta = d(C_T,C_{DR}) - d(C_T,C_{FW})$ of our method on individual stations: a higher color intensity corresponds to a wider gap in our favor between prediction errors, from light green $(\Delta\sim 0)$ to red $(\Delta\sim 2)$. Our approach tends to propagate the distributions to missing locations with higher accuracy.
\begin{figure}[t]
\centering
\includegraphics[height = 4cm]{images/fig_10_from_pdf.png}\qquad
\includegraphics[height = 4cm]{images/fig_meteo6.png}\qquad
\includegraphics[height = 4cm]{images/fig_35_from_pdf.png}
\caption{From Left to Right: propagation of weather data with $10\%,20\%$ and $30\%$ stations with available measurements (see text). \label{fig:propagation}}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We proposed a Frank-Wolfe-based algorithm to find the Sinkhorn barycenter of probability distributions with either finitely or infinitely many support points. Our algorithm belongs to the family of barycenter methods with free support since it adaptively identifies support points rather than fixing them a-priori. In the finite settings, we were able to guarantee convergence of the proposed algorithm by proving the Lipschitz continuity of gradient of the barycenter functional in the Total Variation sense. Then, by studying the sample complexity of Sinkhorn potential estimation, we proved the convergence of our algorithm also in the infinite case. We empirically assessed our method on a number of synthetic and real datasets, showing that it exhibits good qualitative and quantitative performance. While in this work we have considered \fw{} iterates that are a convex combination of Dirac's delta, models with higher regularity (e.g. mixture of Gaussians) might be more suited to approximate the barycenter of distributions with smooth density. Hence, future work will investigate how the perspective adopted in this work could be extended also to other barycenter estimators.
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\newpage
\appendix
\crefname{assumption}{Assumption}{Assumptions}
\crefname{equation}{}{}
\Crefname{equation}{Eq.}{Eqs.}
\crefname{figure}{Figure}{Figures}
\crefname{table}{Table}{Tables}
\crefname{section}{Section}{Sections}
\crefname{theorem}{Theorem}{Theorems}
\crefname{proposition}{Proposition}{Propositions}
\crefname{fact}{Fact}{Facts}
\crefname{lemma}{Lemma}{Lemmas}
\crefname{corollary}{Corollary}{Corollaries}
\crefname{example}{Example}{Examples}
\crefname{remark}{Remark}{Remarks}
\crefname{algorithm}{Algorithm}{Algorithms}
\crefname{enumi}{}{}
\crefname{appendix}{Appendix}{Appendices}
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\section*{\Huge\textbf{Supplementary Material}}
Below we give an overview of the structure of the supplementary material and highlight the main novel results of this work.
\paragraph{ \cref{sec:frank-wolfe}: abstract Frank-Wolfe algorithm in dual Banach spaces} This section contains full details on Frank-Wolfe algorithm. The novelty stands in the relaxation of the differentiability assumptions.\\
\paragraph{\cref{sec:PFtheory}: DAD problems and convergence of Sinkhorn-Knopp algorithm} This section is a brief review of basic concepts from the nonlinear Perrom-Frobeius theory, DAD problems, and applications to the study of Sinkorn algorithm.\\
\paragraph{ \cref{subsec:lipschitz-total-variation}: Lipschitz continuitity of the gradient of the Sinkhorn divergence with respect to Total Variation} This section contains one of the main contributions of our work, \cref{prop:lipschitz-continuity-total-variation2}, from which we derive \cref{thm:lip-continuity-total-variation-informal} in the main text. \\
\paragraph{\cref{sec:app-frank-wolfe-algorithm}: Frank-Wolfe algorithm for Sinkhorn barycenters} This section contains the complete analysis of FW algorithm for Sinkhorn barycenters, which takes into account the error in the computation of Sinkhorn potentials and the error in their minimization. The main result is the convergence of the Frank-Wolfe scheme for finitely supported distributions in \cref{thm:full_convergence_FW_with_error}.\\
\paragraph{\cref{sec:sample-complexity-sinkhorn-potentials}: Sample complexity of Sinkhorn potential and convergence of \cref{alg:practical-FW} in case of continuous measures} This section contains the discussion and the proofs of two of main results of the work \cref{thm:sample-complexity-sinkhorn-gradients}, \cref{thm:sinkhorn-barycenters-infinite-case}.\\
\paragraph{ \cref{sec:additional_exp}: additional experiments} This section contains additional experiment on barycenters of mixture of Gaussian, barycenter of a mesh in 3D (dinosau) and additional figures on the experiment on Sinkhorn propagation described in \cref{sec:experiments}.
\section{The Frank-Wolfe algorithm in dual Banach spaces}\label{sec:frank-wolfe}
In this section we detail the convergence analysis of the Frank-Wolfe algorithm in abstract dual Banach spaces and under mild directional differentiablility assumptions so to cover the setting of Sinkhorn barycenters described in \cref{sec:algorithm-theory} of the paper.
Let $\VV$ be a real Banach space and let
be $\BB$ its topological dual.
Let $\CC\subset \BB$ be a nonempty, closed,
convex, and bounded set and let
$\func\colon\CC \to \R$ be a convex function. We address the following optimization problem
\begin{equation}
\label{eq:minprob}
\min_{\xx \in \CC} \func(\xx),
\end{equation}
assuming that the set of solutions is nonemtpy.
We recall the concept of the tangent cone of feasible directions.
\begin{definition}\label{def:cone-of-feasible-directions}
Let $\xx\in \CC$. Then \emph{the cone of feasible directions of $\CC$ at $\xx$} is
$\feas_{\CC}(\xx) = \R_+ (\CC - \xx)$ and the \emph{tangent cone of $\CC$ at $\xx$}
is
\begin{equation*}
\mathcal{T}_{\CC}(\xx) = \overline{\feas_{\CC}(\xx)} = \big\{ \uu \in \BB \,\vert\, (\exists (t_{k})_{k \in \N} \in \R_{++}^{\N})(t_k \to 0) (\exists (\xx_{k})_{k \in \N} \in \CC^\N)\ t_k^{-1}(\xx_k - \xx) \to \uu \big\}.
\end{equation*}
\end{definition}
\begin{remark}
\normalfont
$\feas_{\CC}(\xx)$ is the cone generated by $\CC-\xx$, and it is a convex cone.
Indeed, if $t>0$ and $\uu \in \feas_{\CC}(\xx)$, then $t \uu \in \feas_{\CC}(\xx)$. Moreover, if $\uu_1, \uu_2 \in \feas_{\CC}(\xx)$, then there exists $t_1,t_2>0$ and $\xx_1,\xx_2 \in \CC$ such that $\uu_i = t_i(\xx_i - \xx)$,
$i=1,2$. Thus,
\begin{equation*}
\uu_1 + \uu_2 = (t_1 + t_2)
\Big( \frac{t_1}{t_1 + t_2} \xx_1
+ \frac{t_2}{t_1 + t_2} \xx_2 - \xx \Big) \in \R_+ (\CC - \xx).
\end{equation*}
So, $\mathcal{T}_{\CC}(\xx)$ is a closed convex cone too.
\end{remark}
\begin{definition}\label{def:directional-derivative}
Let $\xx \in \CC$ and $\uu \in \feas_{\CC}(\xx)$. Then,
\emph{the directional
derivative of $\func$ at $\xx$ in the direction $\uu$} is
\begin{equation*}
\func^\prime(\xx;\uu) = \lim_{t \to 0^+} \frac{\func(\xx + t \uu) - \func(\xx)}{t} \in \left[-\infty, +\infty\right[.
\end{equation*}
\end{definition}
\begin{remark}\label{rem:properties-of-directional-derivative}
\normalfont
The above definition is well-posed. Indeed,
since $v$ is a feasible direction of $\CC$ at $x$, there exists $t_1>0$
and $\xx_1 \in \CC$ such that $\uu = t_1 (\xx_1-\xx)$; hence
\begin{equation*}
(\forall\, t\in \left]0,1/t_1\right])\quad
x + t \uu = x + t\,t_1(\xx_1-\xx) = (1 - t\,t_1) \xx+ t\,t_1 \xx_1 \in \CC.
\end{equation*}
Moreover, since $\func$ is convex, the function $t\in \left]0,1/t_1\right] \mapsto (\func(\xx + t \uu) - \func(\xx))/t$ is increasing,
hence
\begin{equation}
\label{eq:20190410a}
\lim_{t \to 0^+} \frac{\func(\xx + t \uu) - \func(\xx)}{t}=
\inf_{t \in \left]0,1/t_1\right]} \frac{\func(\xx + t \uu) - \func(\xx)}{t}.
\end{equation}
\end{remark}
It is easy to prove that the function
\begin{equation*}
\uu \in \feas_{\CC}(\xx) \mapsto \func^\prime(\xx;\uu) \in \left[-\infty,+\infty\right[
\end{equation*}
is positively homogeneous and sublinear (hence convex), that is,
\begin{enumerate}[{\rm (i)}]
\item $(\forall\, \uu \in \feas_{\CC}(\xx))(\forall\,t \in \R_+)$\
$\func^\prime(\xx;t \uu) = t \func^\prime(\xx;\uu)$;
\item $(\forall\,\uu_1,\uu_2 \in \feas_{\CC}(\xx))$\
$\func^\prime(\xx;\uu_1 + \uu_2) \leq \func^\prime(\xx;\uu_1) + \func^\prime(\xx;\uu_2)$.
\end{enumerate}
We make the following assumptions about $\func$:
\begin{enumerate}[{\rm H$1$}]
\item\label{H1} $(\forall\, \xx \in \CC)$\ the function
$\uu\mapsto \func^\prime(\xx;\uu)$ is finite, that is, $\func^\prime(\xx;\uu) \in \R$.
\item\label{H2} The \emph{curvature of $\func$} is finite, that is,
\begin{equation}\label{eq:curvature}
C_{\func} = \sup_{\substack{\xx,\zz \in \CC\\ \gamma \in [0,1]}}
\frac{2}{\gamma^2}\big( \func(\xx + \gamma(\zz-\xx)) - \func(\xx) - \gamma \func^\prime(\xx, \zz - \xx) \big)<+\infty.
\end{equation}
\end{enumerate}
\begin{remark}
\normalfont
For every $\xx,\zz \in \CC$, we have
\begin{equation}
\label{eq:sort-of-convexity-directional-derivative}
\func(\zz) - \func(\xx) \geq \func^\prime(\xx,\zz-\xx).
\end{equation}
This follows from \eqref{eq:20190410a} with $\xx_1=\zz$ and $t = 1$ ($t_1 =1$).
\end{remark}
The (inexact) Frank-Wolfe algorithm is detailed in \cref{alg:abstract-FW}.
\begin{algorithm}
\caption{Frank-Wolfe in Dual Banach Spaces}
\label{alg:abstract-FW}
Let $(\gamma_k)_{k \in \N} \in \R_{++}^\N$ be such that $\gamma_0= 1$ and, for every $k \in \N$, $1/\gamma_k \leq 1/\gamma_{k+1} \leq 1/2 + 1/\gamma_{k}$ (i.e., $\gamma_k = 2/(k+2)$).
Let $\xx_0 \in \CC$ and $(\Delta_k)_{k \in \N} \in \R_{+}^\N$ be such that $(\Delta_k/\gamma_k)_{k \in \N}$ is nondecreasing.
Then
\begin{equation*}
\begin{array}{l}
\text{for}\;k=0,1,\dots\\[0.7ex]
\left\lfloor
\begin{array}{l}
\text{find }\zz_{k+1}\in \CC \text{ is such that } \func^\prime(\xx_k; \zz_{k+1} - \xx_k)
\leq \inf_{\zz \in \CC} \func^\prime(\xx_k; \zz - \xx_k) + \frac 1 2 \Delta_k\\[1ex]
\xx_{k+1} = \xx_k + \gamma_k(\zz_{k+1} - \xx_k)
\end{array}
\right.
\end{array}
\end{equation*}
\end{algorithm}
\begin{remark}\
\normalfont
\begin{enumerate}[{\rm (i)}]
\item
\cref{alg:abstract-FW} does not
require the sub-problem $\min_{\zz \in \CC} \func^\prime(\xx_k, \zz - \xx_k)$ to have solutions. Indeed it only requires
computing a $\precision_k$-minimizer of $\func^\prime(\xx_k;\cdot - \xx_k)$ on $\CC$, which always exists.
\item
Since $\CC$ is weakly-$*$ compact (by Banach-Alaoglu theorem),
if $\func^\prime(\xx_k,\cdot - \xx_k)$ is weakly-$*$ continuous on $\CC$, then the sub-problem $\min_{\zz \in \CC} \func^\prime(\xx_k, \zz - \xx_k)$ admits solutions.
Note that this occurs when the directional derivative
$\func^\prime(\xx;\cdot)$ is linear and can be represented in $\VV$.
This case is addressed in the subsequent \cref{p:inexactgrad}.
\end{enumerate}
\end{remark}
\vspace{0.5ex}
\begin{theorem}
\label{thm:FWA}
Let $(\xx_k)_{k \in \N}$ be defined according to \cref{alg:abstract-FW}.
Then, for every integer $k\geq 1$,
\begin{equation}
\label{eq:20190418a}
\func(\xx_k) - \min \func \leq C_{\func} \gamma_k + \Delta_k.
\end{equation}
\end{theorem}
\begin{proof}
Let $\xx_* \in \CC$ be a solution of problem \eqref{eq:minprob}.
It follows from \cref{H2} and the definition of $\xx_{k+1}$ in \cref{alg:abstract-FW},
that
\begin{equation*}
\func(\xx_{k+1}) \leq \func(\xx_k) + \gamma_k \func^\prime(\xx_k;\zz_{k+1} - \xx_k) + \frac{\gamma_k^2}{2} C_{\func}.
\end{equation*}
Moreover, it follows from the definition of $\zz_{k+1}$ in \cref{alg:abstract-FW}
and \eqref{eq:sort-of-convexity-directional-derivative} that
\begin{align*}
\func^\prime(\xx_k; \zz_{k+1} - \xx_k)
&\leq \inf_{\zz \in \CC} \func^\prime(\xx_k; \zz - \xx_k)
+ \frac 1 2 \Delta_k\\
&\leq \func^\prime(\xx_k; \xx_* - \xx_k) + \frac1 2 \Delta_k\\
&\leq -( \func(\xx_k) - \func(\xx_*))+ \frac1 2 \Delta_k.
\end{align*}
Then,
\begin{equation}
\label{eq:20190418b}
\func(\xx_{k+1}) - \func(\xx_*) \leq (1 - \gamma_k) (\func(\xx_k) - \func(\xx_*))
+\frac{\gamma_k^2}{2} \Big(C_{\func}+ \frac{\Delta_k}{\gamma_k}\Big).
\end{equation}
Now, similarly to \cite[Theorem~2]{Jaggi2013}, we can prove \eqref{eq:20190418a} by induction.
Since $\gamma_0 = 1$, $1/\gamma_1 \leq 1/2 + 1/\gamma_0$,
and $\Delta_0/\gamma_0 \leq \Delta_1/ \gamma_1$,
it follows from \eqref{eq:20190418b} that
\begin{equation}
\func(\xx_{1}) - \func(\xx_*) \leq \frac{1}{2}
\Big(C_{\func} +\frac{\Delta_0}{\gamma_0}\Big)
\leq \gamma_1 \Big(C_{\func} +\frac{\Delta_1}{\gamma_1}\Big),
\end{equation}
hence \eqref{eq:20190418a} is true for $k=1$.
Set, for the sake of brevity, $C_k = C_{\func} +\Delta_k/\gamma_k$ and
suppose that \eqref{eq:20190418a} holds for $k \in \N$, $k\geq 1$. Then, it follows from \eqref{eq:20190418b} and the properties of $(\gamma_k)_{k \in \N}$ that
\begin{align*}
\func(\xx_{k+1}) - \func(\xx_*) &\leq (1 - \gamma_k) \gamma_k C_k
+\frac{\gamma_k^2}{2}C_k\\
& = C_k\gamma_k \Big( 1 - \frac{\gamma_k}{2}\Big)\\
& \leq C_k\gamma_k \Big( 1 - \frac{\gamma_{k+1}}{2}\Big) \\
&\leq C_k \dfrac{1}{1/\gamma_{k+1} - 1/2}\Big( 1 - \frac{\gamma_{k+1}}{2}\Big)\\[0.8ex]
&= C_k \gamma_{k+1}\\[0.8ex]
&\leq C_{k+1} \gamma_{k+1}.
\qedhere
\end{align*}
\end{proof}
\begin{corollary}
Under the assumptions of \cref{thm:FWA}, suppose in addition that $\precision_k = \precision \gamma_k^{\zeta}$, for some $\zeta \in [0,1]$ and $\precision \geq 0$. Then we have
\begin{equation}
\func(\xx_k) - \min \func \leq C_{\func} \gamma_k + \precision \gamma_k^{\zeta}.
\end{equation}
\end{corollary}
\begin{proof}
It follows from \cref{thm:FWA} by noting that the sequence $\precision_k/\gamma_k = 1/\gamma_k^{1 - \zeta}$ is nondecreasing.
\end{proof}
\begin{proposition}
\label{p:inexactgrad}
Suppose that there exists a mapping $\nabla \func\colon \CC \to \VV$
such that\footnote{This mapping does not need to be unique.},
\begin{equation}
\label{eq:20190410e}
(\forall\, \xx \in \CC)(\forall\,\zz \in \CC)\quad \scal{\nabla \func(\xx)}{\zz-\xx} = \func^\prime(\xx;\zz-\xx).
\end{equation}
Then the following holds.
\begin{enumerate}[{\rm (i)}]
\item\label{p:inexactgrad_i} Let $k \in \N$
and suppose that there exists $u_k \in \VV$ such that
$\nor{u_k - \nabla \func(\xx_k)} \leq \Delta_{1,k}/4$
and that $\zz_{k+1} \in \CC$ satisfies
\begin{equation*}
\scal{u_k}{\zz_{k+1}} \leq \min_{\zz \in \CC} \scal{u_k}{\zz}
+ \frac{\Delta_{2,k}}{2},
\end{equation*}
for some $\Delta_{1,k},\Delta_{2,k}>0$. Then
\begin{equation}
\label{eq:inexactgrad}
\func^\prime(\xx_k; \zz_{k+1} - \xx_k)
\leq \min_{\zz \in \CC} \func^\prime(\xx_k; \zz - \xx_k) +
\frac1 2( \Delta_{1,k} \Diam(\CC)+ \Delta_{2,k}).
\end{equation}
\item\label{p:inexactgrad_ii}
Suppose that $\nabla \func\colon \CC \to \VV$ is $L$-Lipschitz continuous for some $L>0$. Then, for every $\xx,\zz \in \CC$
and $\gamma \in [0,1]$,
\begin{equation*}
\func(\xx + \gamma(\zz-\xx)) - \func(\xx) - \gamma\scal{\zz - \xx}
{\nabla \func(\xx)} \leq \frac{L}{2}
\gamma^2\nor{\zz-\xx}^2
\end{equation*}
and hence $C_{\func} \leq L \mathrm{diam}(\CC)^2$.
\end{enumerate}
\end{proposition}
\begin{proof}
\cref{p:inexactgrad_i}:
We have
\begin{align}
\label{eq:20190410c}
\nonumber\scal{\nabla \func(\xx_k)}{\zz_{k+1} - \xx_k}
&= \scal{u_k}{\zz_{k+1} - \xx_k} + \scal{\nabla \func(\xx_k) - u_k}{\zz_{k+1} - \xx_k}\\[1ex]
& \leq \min_{\zz \in \CC} \scal{u_k}{\zz - \xx_k}
+ \frac{\Delta_{2,k}}{2} +
\frac{\Delta_{1,k}}{4}\mathrm{diam}(\CC).
\end{align}
Moreover,
\begin{align*}
(\forall\, \zz \in \CC)\quad \scal{u_k}{\zz - \xx_k}
&= \scal{\nabla \func(\xx_k)}{\zz - \xx_k}
+ \scal{u_k - \nabla \func(\xx_k)}{\zz - \xx_k}\\[1ex]
&\leq \scal{\nabla \func(\xx_k)}{\zz - \xx_k}
+ \frac{\Delta_{1,k}}{4} \Diam(\CC),
\end{align*}
hence
\begin{equation}
\label{eq:20190410d}
\min_{\zz \in \CC} \scal{u_k}{\zz - \xx_k} \leq \min_{\zz \in \CC} \scal{\nabla \func(\xx_k)}{\zz - \xx_k}
+ \frac{\Delta_{1,k}}{2} \Diam(\CC).
\end{equation}
Thus, \eqref{eq:inexactgrad} follows from \eqref{eq:20190410c}, \eqref{eq:20190410d}, and \eqref{eq:20190410e}.
\cref{p:inexactgrad_ii}:
Let $\xx,\zz \in \CC$, and define
$\psi\colon[0,1]\to \VV^*$ such that,
$\forall\, \gamma \in [0,1]$,
$\psi(\gamma) = \func(\xx + \gamma(\zz-\xx))$.
Then, it is easy to see that for every $\gamma \in \left]0,1\right[$, $\psi$ is differentiable at $\gamma$
and $\psi^\prime(\gamma) = \func^\prime(\xx + \gamma(\zz-\xx);\zz-\xx) = \scal{\nabla \func(\xx+\gamma(\zz-\xx))}{\zz-\xx}$. Moreover,
$\psi$ is continuous on $[0,1]$. Therefore,
the fundamental theorem of calculus yields
\begin{equation*}
\psi(\gamma) - \psi(0) = \int_0^\gamma \psi\prime(t) d t
\end{equation*}
and hence
\begin{align*}
\func(\xx+\gamma(\zz-\xx)) - \func(\xx) - \scal{\nabla \func(\xx)}{\zz-\xx}
&= \int_0^\gamma \scal{\nabla \func(\xx + t(\zz-\xx)) - \nabla \func(\xx)}{\zz-\xx} dt\\[1ex]
&\leq \int_0^\gamma\nor{\nabla \func(\xx + t(\zz-\xx)) - \nabla \func(\xx)} \nor{\zz-\xx} dt \\[1ex]
&\leq \int_0^\gamma L t\nor{\zz-\xx}^2 dt
\\[1ex]
&= L \frac{\gamma^2}{2}\nor{\zz-\xx}^2.
\qedhere
\end{align*}
\end{proof}
The following result is an extension of a classical result on the directional differentiability of a max function \cite[Theorem~4.13]{bonnans2013perturbation} which relaxes the inf-compactness condition and allows the parameter space to be a convex set, instead of the entire Banach space. This result provide a prototype of functions (of which the entropic regularization of the Wasserstein distance is an instance) which are directionally differentiable only along the feasible directions of their domain and satisfies the hypotheses of \cref{p:inexactgrad}.
\begin{proposition}
\label{p:diff_of_max}
Let $Z$ and $\VV$ be real Banach spaces and
let $\BB$ be the topological dual of $\VV$. Let $\CC\subset \BB$
be a nonempty closed convex set, and let
$g\colon Z\times \BB\to \R$ be such that
\begin{enumerate}[$1)$]
\item for every $z \in Z$, $g(z,\cdot)\colon \BB \to \R$
is G\^ateaux differentiable with derivative in $\VV$, and the partial derivative with respect to the second variable $D_2 g\colon Z \times \BB \to \VV$ is continuous.
\item for every $\xx \in \CC$, $S(\xx):=\argmax_{Z} g(\cdot, \xx) \neq \varnothing$.
\item there exists a continuous mapping $\varphi\colon \CC \to Z$ such that, for every $\xx \in \CC$, $\varphi(\xx) \in S(\xx)$.
\end{enumerate}
Let $\func\colon \CC \to \R$ be defined
as
\begin{equation}
\func(\xx) = \max_{z \in Z} g(z,\xx).
\end{equation}
Then, $\func$ is continuous, directionally differentiable, and, for every $\xx \in \CC$ and $\uu \in \feas_{\CC}(\xx)$
\begin{equation}
\label{eq:20190513b}
\func^\prime(\xx;\uu) = \max_{z \in S(\xx)} \scal{D_2 g(z,\xx)}{\uu} = \scal{D_2 g(\varphi(\xx),\xx)}{\uu}.
\end{equation}
\end{proposition}
\begin{proof}
The function $\func$ is well defined,
since by assumption $2)$, for every $\xx \in \CC$,
$\argmax_{Z}g(\cdot, \xx) \neq \varnothing$.
Let $\xx,u \in \CC$ with $\xx \neq u$.
Then, since $\varphi(\xx) \in S(\xx)$, we have
$\func(\xx) = g(\varphi(\xx),\xx)$ and hence
\begin{multline}
\label{eq:20190511f}
\frac{\func(u) - \func(\xx) - \scal{D_2 g(\varphi(\xx), \xx)}{u - \xx}}{\nor{u - \xx}} \\
\geq \frac{g(\varphi(\xx), u) - g(\varphi(\xx), \xx)-\scal{D_2 g(\varphi(\xx), \xx)}{u - \xx}}{\nor{u-\xx}} \to 0,
\end{multline}
since $g(\varphi(\xx),\cdot)$ is Fr\'echet differentiable\footnote{continuously G\^ateaux differentiable function are Fr\'echet differentiable \cite[pp.34-35]{bonnans2013perturbation}.} at $\xx$ with gradient $D_2 g(\varphi(\xx), \xx)$.
Now,
$\varphi(u) \in S(u)$, and hence
$\func(u) = g(\varphi(u), u)$.
Moreover, $g(\varphi(u), \xx) \leq \func(\xx)$. Therefore,
\begin{multline}
\label{eq:20190503b}
\frac{\func(u) - \func(\xx) - \scal{D_2 g(\varphi(\xx), \xx)}{u - \xx}}{\nor{u - \xx}} \\
\leq \frac{g(\varphi(u), u) - g(\varphi(u), \xx)-\scal{D_2 g(\varphi(\xx), \xx)}{u - \xx}}{\nor{u-\xx}}.
\end{multline}
Let $\varepsilon>0$. Since $D_2 g$ is continuous, there exists $\delta>0$ such that, for every $z^\prime\in Z$ and $\xx^\prime \in \BB$
\begin{equation}
\label{eq:20190503a}
\nor{z^\prime - \varphi(\xx)}\leq \delta\ \text{and}\
\nor{\xx^\prime - \xx} \leq \delta\ \implies\
\nor{D_2 g(z^\prime,\xx^\prime) - D_2 g(\varphi(\xx), \xx)} \leq \varepsilon.
\end{equation}
Moreover, since $\varphi\colon \CC \to Z$ is continuous,
there exists $\eta>0$ such that,
\begin{equation}
\label{eq:20190503c}
\nor{u - \xx} \leq \eta\ \implies\ \nor{\varphi(u) - \varphi(\xx)} \leq \delta.
\end{equation}
Let $z^\prime \in Z$ and suppose that $\nor{z^\prime - \varphi(\xx)} \leq \delta$
and $\nor{u - \xx} \leq \delta$.
Define $\psi\colon [0,1] \to \R$ such that,
for every $s \in [0,1]$, $\psi(s) = g(z^\prime,\xx+s(u - \xx))$.
Then, $\psi$ is continuously differentiable on $[0,1]$ and $\psi^\prime(s) = \scal{D_2 g(z^\prime,\xx+s (u - \xx))}{u - \xx}$. Therefore,
\begin{equation}
\psi(1) - \psi(0) = \int_0^1 \psi^\prime(s) d s
\end{equation}
and hence, it follows from \eqref{eq:20190503a} that
\begin{align*}
\lvert g(z^\prime,u) &- g(z^\prime, \xx) -
\scal{D_2 g(\varphi(\xx),\xx)}{u -\xx} \rvert \\
&= \Big\lvert \int_0^1 \scal{D_2 g(z^\prime,\xx + s (u - \xx))
- D_2 g(\varphi(\xx), \xx)}{u - \xx} ds \Big\rvert\\
&\leq \int_0^1 \nor{D_2 g(z^\prime,\xx + s (u - \xx)) - D_2 g(\varphi(\xx),\xx)} \nor{u - \xx} ds\\
&\leq \varepsilon \nor{u - \xx}.
\end{align*}
Therefore, we derive from \eqref{eq:20190503c}, that for every $u \in \CC$ such that
$\nor{u - \xx} \leq \min\{\eta, \delta\}$,
we have
\begin{equation*}
\bigg\lvert \frac{g(\varphi(u),u) - g(\varphi(u), \xx) -
\scal{D_2 g(\varphi(\xx),\xx)}{u -\xx}}{\nor{u - \xx}} \bigg\rvert \leq \varepsilon.
\end{equation*}
This shows that
\begin{equation}
\label{eq:20190511g}
\lim_{\substack{u \in \CC\\u \to \xx}}\frac{g(\varphi(u),u) - g(\varphi(u), \xx) -
\scal{D_2 g(\varphi(\xx),\xx)}{u -\xx}}{\nor{u - \xx}} = 0.
\end{equation}
Then, we derive from \cref{eq:20190511f}, \eqref{eq:20190503b}, and \cref{eq:20190511g} that
\begin{equation}
\label{eq:20190513a}
\lim_{\substack{u \in \CC\\u \to \xx}}\frac{\func(u) - \func(\xx) -
\scal{D_2 g(\varphi(\xx),\xx)}{u -\xx}}{\nor{u - \xx}} = 0.
\end{equation}
This implies that $\lim_{u \in \CC,u \to \xx} \func(u)=\func(\xx)$.
Moreover, if $\uu \in \feas_{\CC}(\xx)$, there exists $\lambda>0$
and $u \in \CC$ such that $\uu = \lambda(u - \xx)$ and, for every $t \in \left]0,1/\lambda\right]$,
\begin{multline}
\frac{\func(\xx + t \uu) - \func(\xx)}{t} - \scal{D_2 g(\varphi(\xx),\xx)}{\uu}\\
= \nor{\lambda(u - \xx)} \frac{\func(\xx + t \lambda(u - \xx)) - \func(\xx) - \scal{D_2 g(\varphi(\xx),\xx)}{t \lambda(u - \xx)}}{\nor{t \lambda(u - \xx)}}
\end{multline}
and the right hand side goes to zero as $t \to 0^+$, because of \cref{eq:20190513a}.
Therefore, for every $z \in S(\xx)$, since $\func(\xx) = g(z,\xx)$ and $\func(\xx + t\uu) \geq g(z, \xx + t\uu)$, we have
\begin{equation*}
\scal{D_2 g(\varphi(\xx),\xx)}{\uu} = \lim_{t\to 0^+}\frac{\func(\xx + t \uu) - \func(\xx)}{t}
\geq \lim_{t\to 0^+}\frac{g(z, \xx + t \uu) - g(z, \xx)}{t} = \scal{D_2 g(z,\xx)}{\uu}
\end{equation*}
and \cref{eq:20190513b} follows.
\end{proof}
\section{DAD problems and convergence of Sinkhorn-Knopp algorithm}
\label{sec:PFtheory}
In this section we review the basic concepts of the nonlinear Perron-Frobenius theory \cite{lemmens2012nonlinear} which provides tools for dealing with DAD problems and ultimately to study the key properties of the Sinkhorn potentials. This analysis will allow us to provide
in \cref{subsec:lipschitz-total-variation} an upper bound estimate for the Lipschitz constant of the gradient of $\bary$, which is needed in the Frank-Wolfe algorithm.
\subsection{Hilbert's metric and the Birkhoff-Hopf theorem}
In the rest of the appendix we will assume $\X\subset\R^d$ to be a compact set. We denote by $\cont(\X)$ the space of continuous functions on $\X$ endowed with the sup norm, namely $\supnor{f} = \sup_{x\in\X} \abs{f(x)}$.
Let $\nneg(\X)$ be the cone of non-negative continuous functions, that is, $f\in\cont(\X)$ such that $f(x)\geq0$ for every $x\in\X$. Also, we denote by $\posi(\X)$ the set of continuous and (strictly) positive functions on $\X$, which turns out to be the interior of $\nneg(\X)$.
Let $\dist:\X\times\X\to\R_+$ be a positive, symmetric, and continuous function and define $\kerfun:\X\times\X\to\R_{++}$ as
\begin{equation}
\label{eq:kerfunc}
(\forall x,y\in\X) \qquad \kerfun(x,y) = e^{ -\frac{\cost(x,y)}{\varepsilon} }.
\end{equation}
Set $\diam = \sup_{x,y\in\X}~\dist(x,y)$. Then, we have $\kerfun(x,y)\in[e^{-\diameps},1]$ for all $x,y\in\X$.
Let $\alpha\in\prob(\X)$. The operator $\lmap_\alpha\colon \cont(\X)\to\cont(\X)$ is defined as
\begin{equation}
\label{def:lmap}
(\forall f\in\cont(\X))\qquad \lmap_\alpha f\colon x\mapsto \int \kerfun(x,z) f(z)~d\alpha(z).
\end{equation}
Note that $\lmap_\alpha$ is linear and continuous. In particular, since $k(x,y)\in[0,1]$ for all $x,y\in\X$, we have
\begin{equation}
\label{eq:L-maps-nneg-to-nneg}
(\forall\,f\in\nneg(\X))\qquad\lmap_\alpha f \geq 0
\end{equation}
and
\begin{equation}
\label{eq:L_norm}
(\forall\,f\in\cont(\X))\qquad \nor{\lmap_\alpha f}_\infty \leq \nor{f}_\infty.
\end{equation}
\paragraph{Hilbert's Metric} The cone $\nneg(\X)$ induces a partial ordering $\leq$ on $\cont(\X)$, such that
\begin{equation}
(\forall\, f,f^\prime \in \cont(\X))
\qquad f \leq f^\prime \Leftrightarrow\ f^\prime - f \in \nneg(\X).
\end{equation}
According to \cite{lemmens2012nonlinear}, we say that a function $f^\prime\in\nneg(\X)$ {\em dominates} $f\in\cont(\X)$ if there exist $t,s\in\R$ such that
\begin{equation}
t f^\prime \leq f \leq s f^\prime.
\end{equation}
This notion induces an equivalence relation on $\nneg(\X)$, denoted $f\sim f^\prime$, meaning that $f$ dominates $f^\prime$ and $f^\prime$ dominates $f$. The corresponding equivalence classes are called {\em parts} of $\nneg(\X)$.
Let $f,f^\prime \in \nneg(\X)$ be such that $f \sim f^\prime$.
We define
\begin{equation}
M(f/f^\prime) = \inf\{s \in \R \,\vert\, f \leq s f^\prime \}
\qquad \text{and} \qquad m(f/f^\prime)
= \sup\{t\in \R \,\vert\, t f^\prime\leq f\}.
\end{equation}
Note that $m(f/f^\prime)\leq M(f/f^\prime)$. Moreover,
for every $f,f^\prime\in\nneg(\X)$ such that $f \sim f^\prime$, we have that $\supp(f) = \supp(f^\prime)$
and if $f^\prime\neq 0$ (hence $f\neq 0$), then
\begin{equation}
M(f/f^\prime) =
\max_{x \in \supp(f^\prime)} \frac{f(x)}{f^\prime(x)}>0
\quad\text{and}\quad
m(f/f^\prime) = \min_{x \in \supp(f^\prime)} \frac{f(x)}{f^\prime(x)}>0.
\end{equation}
The {\em Hilbert's metric} is defined as
\begin{equation}
d_H(f,f^\prime) = \log\frac{M(f/f^\prime)}{m(f/f^\prime)},
\end{equation}
for all $f\sim f^\prime$ with $f\neq 0$ and $f^\prime\neq 0$, $d_H(0,0) = 0$ and $d_H(f,f^\prime) = +\infty$ otherwise. Direct calculation shows that \cite[Proposition~2.1.1]{lemmens2013birkhoff}
\begin{enumerate}[{\rm (i)}]
\item $d_H(f,f^\prime) \geq 0$ and $d_H(f,f^\prime) = d_H(f^\prime,f)$, for every $f,f^\prime\in\nneg(\X)$;
\item $d_H(f,f^{\prime\prime}) \leq d_H(f,f^\prime) + d_H(f^\prime,f^{\prime\prime})$, for every $f,f^\prime, f^{\prime\prime}\in\nneg(\X)$ with $f \sim f^\prime$ and
$f^\prime \sim f^{\prime\prime}$;
\item $d_H(s f,t f^\prime) = d_H(f,f^\prime)$, for every $f,f^\prime\in\nneg(\X)$ and $s,t>0$.
\end{enumerate}
Note that $d_H$ is not a metric on the parts of $\nneg(\X)$. However the set $\posi(\X)\cap \partial B_1(0) = \{f\in\posi(\X) ~|~ \supnor{f} = 1\}$ equipped with $d_H$ is a complete metric space \cite{nussbaum1988hilbert}. Also, $d_H$ induces a metric on the rays of the parts of $\nneg(\X)$ \cite[Lemma 2.1]{lemmens2013birkhoff}.
We now focus on $\posi(\X)$.
A direct consequence of Hilbert's metric properties is the following.
\begin{lemma}[Hilbert's Metric on $\posi(\X)$]\label{lem:hilbert-metric-on-the-interior}
The interior of $\nneg(\X)$ corresponds to the set of (strictly) positive functions $\posi(\X)$ and is a part of $\nneg(\X)$ with respect to the equivalence relation induced by dominance.
For every $f,f^\prime\in\posi(\X)$,
\begin{equation}
M(f/f^\prime) = \max_{x\in\X} ~\frac{f(x)}{f^\prime(x)} \qquad m(f/f^\prime) = \min_{x\in\X}~\frac{f(x)}{f^\prime(x)},
\end{equation}
and $M(f/f^\prime) \geq m(f/f^\prime) >0$. Therefore
\begin{equation}
\label{eq:20190509b}
d_H(f,f^\prime) = \log \max_{x,y\in\X}~\frac{f(x)~f^\prime(y)}{f(y)~f^\prime(x)}.
\end{equation}
\end{lemma}
\begin{proof}
Since $\X$ is compact it is straightfoward to see that $\posi(\X)$ is the interior of $\nneg(\X)$. By applying \cite[Lemma 1.2.2]{lemmens2012nonlinear} we have that $\posi(\X)$ is a part of $\nneg(\X)$. The characterization of $M(f/f^\prime)$ and $m(f/f^\prime)$ follow by direct calculation from the definition using the fact that $\inf_\X h = \min_\X h >0$ for any $h
\in\posi(\X)$ since $\X$ is compact. Finally, the characterization of Hilbert's metric on $\posi(\X)$ is obtained by recalling that $(\min_{x\in\X}h(x))^{-1} = \max_{x\in\X}h(x)^{-1}$ for every $h\in\posi(\X)$.
\end{proof}
\begin{lemma}[Ordering properties of $\lmap_\alpha$]\label{lem:order-preserving}
Let $\alpha\in\prob(\X)$. Then the following holds:
\begin{enumerate}[{\rm (i)}]
\item\label{lem:order-preserving_i}
the operator $\lmap_\alpha$ is {\em order-preserving} (with respect to the cone $\nneg(\X)$),
that is,
\begin{equation}
(\forall\, f, f^\prime \in \cont(\X))
\qquad f\leq f^\prime\ \Rightarrow\
\lmap_\alpha f \leq \lmap_\alpha f^\prime;
\end{equation}
\item\label{lem:order-preserving_ii}
$\lmap_\alpha$ maps parts of $\nneg(\X)$ to parts of $\nneg(\X)$, that is,
\begin{equation}
(\forall\, f, f^\prime \in \cont(\X))
\qquad f\sim f^\prime\ \Rightarrow\ \lmap_\alpha f \sim \lmap_\alpha f^\prime;
\end{equation}
\item\label{lem:order-preserving_iii}
$\lmap_\alpha(\nneg(\X)) \subset \posi(\X)\cup\{0\}$ and $\lmap_\alpha(\posi(\X))\subset\posi(\X)$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{lem:order-preserving_i}:
Let $f,f^\prime \in \cont(\X)$ with
$f\leq f^\prime$. Then $f^\prime-f\in\nneg(\X)$ and by linearity of $\lmap_\alpha$ combined with \cref{eq:L-maps-nneg-to-nneg}, we have
$\lmap_\alpha f^\prime - \lmap_\alpha f = \lmap_\alpha(f-f^\prime) \geq 0$.
\ref{lem:order-preserving_ii}:
Let $f,f^\prime \in \nneg(\X)$ with
$f\sim f^\prime$. Then there exist
$t,s\in\R$ and $s^\prime,t^\prime \in \R$ such that
$t f^\prime \leq f \leq s f^\prime$
and $t^\prime f \leq f^\prime \leq s^\prime f$.
Since $L_\alpha$ is linear and order-preserving, we have
$\lmap_\alpha f \sim \lmap_\alpha f^\prime$.
\ref{lem:order-preserving_iii}:
Let $f\in\nneg(\X)$. By \cref{eq:L-maps-nneg-to-nneg} and
\cref{eq:L_norm}, for any $x\in\X$
\begin{equation}\label{eq:max-upper-bound-L-alpha}
0 \leq (\lmap_\alpha f)(x) \leq \supnor{\lmap_\alpha f} \leq \int f(x)~ d\alpha(x) = \nor{f}_{L^1(\X,\alpha)}.
\end{equation}
Moreover,
\begin{equation}\label{eq:min-lower-bound-L-alpha}
\lmap_\alpha f(x) = \int k(y,x) f(y)~d\alpha(y) \geq e^{-\diameps}~\nor{f}_{L^1(\X,\alpha)}.
\end{equation}
Therefore, if $\nor{f}_{L^1(\X,\alpha)} = 0$ then by \cref{eq:max-upper-bound-L-alpha} $\lmap_\alpha f = 0$ while, if $\nor{f}_{L^1(\X,\alpha)}>0$ then by \cref{eq:min-lower-bound-L-alpha} $\lmap_\alpha f\in\posi(\X)$. We conclude that the operator $\lmap_\alpha$ maps $\nneg(\X)$ in $\posi(\X)\cup\{0\}$. Moreover, $\lmap_\alpha(\posi(\X))\subset\posi(\X)$, since for every $f\in\posi(\X)$ we have $\nor{f}_{L^1(\X,\alpha)}\geq \min_{\X} f>0$.
\end{proof}
Following \cite[Section~A.4]{lemmens2012nonlinear} we now introduce a quantity which plays a central role in our analysis.
\begin{definition}[Projective Diameter of $\lmap_\alpha$]
Let $\alpha\in\prob(\X)$. The {\em projective diameter of $\lmap_\alpha$} is
\begin{equation}\label{eq:projective-diameter}
\Delta(\lmap_\alpha) = \sup\{d_H(\lmap_\alpha f, \lmap_\alpha f^\prime) ~|~ f,f^\prime\in\nneg(\X),~ \lmap_\alpha f\sim \lmap_\alpha f^\prime\}.
\end{equation}
\end{definition}
The following result shows that it is possible to find a finite upper bound on $\Delta(\lmap_\alpha)$ that is independent on $\alpha$.
\begin{proposition}[Upper bound on the Projective Diameter of $\lmap_\alpha$]\label{rem:projective-diameter-upper-bound}
Let $\alpha\in\prob(\X)$. Then
\begin{equation}
\Delta(\lmap_\alpha) \leq 2\diameps.
\end{equation}
\end{proposition}
\begin{proof}
Let $f,f^\prime \in \nneg(\X)$.
Recall that $\lmap_\alpha$ maps $\nneg(\X)$ into $\posi(\X)\cup\{0\}$ (see \cref{lem:order-preserving}\cref{lem:order-preserving_iii}) and that $\{0\}$ and $\posi(\X)$ are two parts of $\nneg(\X)$ with respect to the relation $\sim$ (see \cite[Lemma 1.2.2]{lemmens2012nonlinear}). Now, if $\lmap_\alpha f = \lmap_\alpha f^\prime = 0$, then we have $d_H(\lmap_\alpha f, \lmap_\alpha f^\prime) = d_H(0,0) = 0$. Therefore it is sufficient to study the case that $\lmap_\alpha f,\lmap_\alpha f^\prime\in\posi(\X)$.
Following the characterization of Hilbert's metric on $\posi(\X)$ given in \cref{lem:hilbert-metric-on-the-interior},
we have
\begin{align*}
d_H(\lmap_\alpha f,\lmap_\alpha f^\prime) & = \log~ \max_{x,y\in\X}~ \frac{(\lmap_\alpha f)(x)~(\lmap_\alpha f^\prime)(y)}{(\lmap_\alpha f)(y)~(\lmap_\alpha f^\prime)(x)} \\
& = \log~ \max_{x,y\in\X}~\frac{\int \kerfun(x,z)f(z)~d\alpha(z)~\int \kerfun(y,w)f^\prime(w)~d\alpha(w)}{\int \kerfun(y,z)f(z)~d\alpha(z)~\int \kerfun(x,w)f^\prime(w)~d\alpha(w)} \\
& = \log~ \max_{x,y\in\X}~\frac{\int \kerfun(x,z)\kerfun(y,w)~f(z)f^\prime(w)~d\alpha(z)d\alpha(w)}{\int \kerfun(y,z)\kerfun(x,w)~f(z)f^\prime(w)~d\alpha(z)d\alpha(w)} \\
& = \log~ \max_{x,y\in\X}~\frac{\int \frac{\kerfun(x,z)\kerfun(y,w)}{\kerfun(y,z)\kerfun(x,w)}~\kerfun(y,z)\kerfun(x,w)~f(z)f^\prime(w)~d\alpha(z)d\alpha(w)}{\int \kerfun(y,z)\kerfun(x,w)~f(z)f^\prime(w)~d\alpha(z)d\alpha(w)} \\
& \leq \log~ \max_{x,y,z,w\in\X}~\frac{\kerfun(x,z)\kerfun(y,w)}{\kerfun(y,z)\kerfun(x,w)}.
\end{align*}
Since, for every $x,y\in\X$, $\dist(x,y)\in[0,\diam]$, we have $k(x,y)\in[e^{-\diameps},1]$ and hence
\begin{equation*}
d_H(\lmap_\alpha f, \lmap_\alpha f^\prime) \leq 2\diameps.
\qedhere
\end{equation*}
\end{proof}
A consequence of \cref{rem:projective-diameter-upper-bound} is a special case of Birkhoff-Hopf theorem.
\begin{theorem}[Birkhoff-Hopf Theorem]\label{thm:birkhoff-hopf}
Let $\lambda = \frac{e^{\diameps}-1}{e^{\diameps}+1}$ and $\alpha\in\prob(\X)$. Then, for every $f,f^\prime\in\nneg(\X)$ such that $f\sim f^\prime$, we have
\begin{equation}
\label{eq:contraction1}
d_H(\lmap_\alpha f,\lmap_\alpha f^\prime) \leq \lambda~d_H(f,f^\prime).
\end{equation}
\end{theorem}
\begin{proof}
The statement is a direct application of the Birkhoff-Hopf theory \cite[Sections A.4 and A.7]{lemmens2012nonlinear}
The {\em Birkhoff contraction ratio} of $\lmap_\alpha$ is defined as
\begin{equation*}
\kappa(\lmap_\alpha) =
\inf\big\{ \hat{\lambda} \in \R_+ ~ \big\vert ~ d_H(\lmap_\alpha f, \lmap_\alpha f^\prime)\leq \hat{\lambda} d_H(f,f^\prime)~~ \forall f,f^\prime\in\nneg(\X),~~ f\sim f^\prime\big\}.
\end{equation*}
Then it follows from
Birkhoff-Hopf theorem
\cite[Theorem~A.4.1]{lemmens2012nonlinear} that
\begin{equation}
\kappa(\lmap_\alpha) = \tanh\left(\frac{1}{4}\Delta(\lmap_\alpha)\right).
\end{equation}
Recalling
the upper bound on the projective diameter f $\lmap_\alpha$ given in \cref{rem:projective-diameter-upper-bound}, we have
\begin{equation*}
\kappa(\lmap_\alpha) \leq \tanh\left(\frac{\diam}{2\varepsilon}\right) = \frac{e^{\diameps}-1}{e^{\diameps}+1} = \lambda,
\end{equation*}
and \eqref{eq:contraction1}
follows.
\end{proof}
\subsection{DAD problems}
\label{subsec:sinkiter}
\paragraph{The map $\tmap_\alpha$}
Let $\alpha \in \prm(\X)$.
We define
the map $\tmap_\alpha\colon\posi(\X)\to\posi(\X)$, such that
\begin{equation}
\label{eq:Ta}
(\forall\,f\in\posi(\X))\qquad
\tmap_\alpha(f) = \inv \circ \lmap_\alpha (f) = 1/(\lmap_\alpha f),
\end{equation}
where
$\inv\colon\posi(\X)\to\posi(\X)$ is defined by $\inv (f) = 1/f$ with
\begin{equation}
(1/f)\colon x\mapsto \frac{1}{f(x)}.
\end{equation}
Note that $\tmap_\alpha$ is well defined since, by \cref{lem:order-preserving}\cref{lem:order-preserving_iii}, $\lmap_\alpha(\posi(\X))\subset\posi(\X)$ and,
for every $f \in \posi(\X)$, $\min_\X f>0$, being $\X$ compact.
Moreover, it follows from \cref{eq:20190509b} in \cref{lem:hilbert-metric-on-the-interior}, that, for any two
$f,f^\prime\in\posi(\X)$
\begin{equation}\label{eq:hilbert-metric-for-inversion}
d_H(1/f,1/f^\prime) = \log~\max_{x,y\in\X} \frac{f(y)f^\prime(x)}{f(x)f^\prime(y)} = d_H(f,f^\prime).
\end{equation}
We highlight here the connection between $\rmap_\alpha$ introduced in the main text in \cref{eq:rmap} and $\tmap_\alpha$, namely for any $\alpha\in\prob(\X)$ and $u\in\cont(\X)$
\begin{equation}
\rmap_\alpha(u) = \eps\log(\tmap_\alpha(e^{u/\eps})).
\end{equation}
\paragraph{Dual $\oteps$ Problem}
focus on the dual problem \cref{eq:dual_pb} of the optimal transport problem with entropic regularization. Let $\alpha,\beta\in\prob(\X)$ and $\varepsilon>0$, we consider
\begin{equation}\label{eq:ot-dual-problem}
\max_{u,v\in\cont(\X)} ~ \int u(x)~d\alpha + \int v(y)~d\beta(y) - \varepsilon\int e^{\frac{u(x) + v(y) - \dist(x,y)}{\varepsilon}}~d\alpha(x)d\beta(y).
\end{equation}
The optimality conditions for problem \eqref{eq:ot-dual-problem}
are
\begin{equation}
\label{eq:Dopt}
\begin{cases}
e^{-\frac{u(x)}{\varepsilon}} = {\displaystyle\int_{\X}}
e^{\frac{v(y) - \cost(x,y)}{\varepsilon}}\,d\beta(y)\quad(\forall\, x \in \supp(\alpha))\\[3ex]
e^{-\frac{v(y)}{\varepsilon}} = {\displaystyle\int_{\X}} e^{\frac{u(x) - \cost(x,y)}{\varepsilon}}\,d\alpha(x)\quad(\forall\, y \in \supp(\beta)),
\end{cases}
\end{equation}
which are equivalent to
\begin{equation}
\label{eq:Dopt2}
\begin{cases}
g(y)^{-1} = {\displaystyle\int_{\X}} e^{\frac{- \cost(x,y)}{\varepsilon}} f(x)\,d\alpha(x)\quad(\forall\, y \in \supp(\beta))\\[3ex]
f(x)^{-1} = {\displaystyle\int_{\X}} e^{\frac{-\cost(x,y)}{\varepsilon}} g(y)\,d\beta(y)\quad(\forall\, x \in \supp(\alpha)),
\end{cases}
\end{equation}
where $f = e^{u/\varepsilon} \in \posi(\X)$ and $g = e^{v/\varepsilon} \in \posi(\X)$.
In the rest of the section we will consider
the following \emph{DAD problem} \cite{lemmens2012nonlinear,nussbaum1993entropy}\begin{equation}
\label{eq:DAD}
(\forall\,y \in \X)\ \
\int_{\X} f(x) \kerfun(x,y) g(y)\,d\alpha (x)=1
\ \ \text{and}\ \
(\forall\,x \in \X)\ \
\int_{\X} f(x) \kerfun(x,y) g(y)\,d\beta (y) = 1.
\end{equation}
It is clear that a solution of \cref{eq:DAD}
is also a solution of \cref{eq:Dopt2}.
However, the vice versa is in general not true, even though
there is a canonical way to build solutions of \cref{eq:DAD}
starting from solutions of \cref{eq:Dopt2}: indeed
if $(f,g)$ is a solution of \cref{eq:Dopt2},
then the functions $\bar f,\bar g\colon \X \to \R$ defined through $\bar{f}(x)^{-1} = \int_{\X} \kerfun(x,y) g(y)\,d\beta(y)$ and $\bar{g}(y)^{-1} = \int_{\X} \kerfun(x,y) g(y)\,d\beta(y)$ provide a solution of \cref{eq:DAD}.
So, the dual $\oteps$ problem \cref{eq:ot-dual-problem} admits a solution if and only if the corresponding DAD problem \cref{eq:DAD} admits a solution.
Recalling the definition of $\tmap_\alpha$ in \eqref{eq:Ta},
problem \eqref{eq:DAD} can be more compactly written as
\begin{equation}\label{eq:fixed-point-f-g}
f = \tmap_\beta(g) \qquad \text{and} \qquad g = \tmap_\alpha(f),
\end{equation}
or equivalently, by setting $\tmap_{\beta\alpha} = \tmap_\beta\circ\tmap_\alpha$ and
$\tmap_{\alpha\beta} = \tmap_\alpha\circ\tmap_\beta$,
\begin{equation}\label{eq:fixed-point-f-f}
f =\tmap_{\beta\alpha}(f) \qquad \text{and} \qquad g = \tmap_{\alpha\beta}(g).
\end{equation}
This shows that the solutions of the DAD problem
\cref{eq:DAD}
are the fixed points of $\tmap_{\alpha\beta}$ and $\tmap_{\beta\alpha}$ respectively.
Note that the operators $\tmap_{\beta \alpha}$
and $\tmap_{\alpha\beta}$
are positively homogeneous,
that is, for every $t \in \R_{++}$ and $f \in \posi(\X)$,
$\tmap_{\beta\alpha}(t f) = t\tmap_{\beta\alpha}(f)$
and
$\tmap_{\alpha\beta}(t f) = t\tmap_{\alpha\beta}(f)$.
Thus, if $f$ is a fixed point of $\tmap_{\beta\alpha}$,
then $t f$ is also a fixed point of $\tmap_{\beta\alpha}$,
for every $t>0$.
If $(f,g)$ is a solution of the DAD problem \cref{eq:DAD}, then the pair $(u,v)$, with $u = \varepsilon\log f$ and $v = \varepsilon\log g$ is a solution of \cref{eq:ot-dual-problem}. We refer
to these solutions as {\em Sinkhorn potentials} of the pair $(\alpha,\beta)$.
Finally, note that, it follows from \cref{eq:Dopt}
that solutions of \cref{eq:ot-dual-problem}
are determined $(\alpha,\beta)$-a.e. on $\X$ and
up to a translation of the form $(u+t,v-t)$, for some $t\in\R$.
The following result is essentially the specialization of \cite[Thm. 7.1.4]{lemmens2012nonlinear} to the case of the map $\tmap_{\beta\alpha}$. We report the proof here for convenience and completeness.
\begin{theorem}[Hilbert's metric contraction for $\tmap_{\beta\alpha}$]\label{thm:fixed-point-sinkhorn-iteration}
The map $\tmap_{\beta\alpha}:\posi(\X)\to\posi(\X)$ has a unique fixed point up to positive scalar multiples. Moreover, let $\lambda = \frac{e^{\diameps}-1}{e^{\diameps}+1}$. Then, for every $f,f^\prime\in\posi(\X)$,
\begin{equation}\label{eq:birkhoff-contration-varphi-a-b}
d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta\alpha}(f^\prime)) \leq \lambda^2 ~d_H(f,f^\prime).
\end{equation}
\end{theorem}
\begin{proof}
By combining \cref{eq:hilbert-metric-for-inversion} with \cref{thm:birkhoff-hopf} we obtain that, for any $f,f^\prime\in\posi(\X)$
\begin{equation}
d_H(\tmap_\alpha(f),\tmap_\alpha(f^\prime)) ~=~ d_H(1/(\lmap_\alpha f),1/(\lmap_\alpha f^\prime)) ~=~ d_H(\lmap_\alpha f,\lmap_\alpha f^\prime) ~\leq~ \lambda~ d_H(f,f^\prime).
\end{equation}
Since the same holds for $\tmap_\beta$ then \cref{eq:birkhoff-contration-varphi-a-b} is satisfied.
Now, let $C = \posi(\X) \cap \partial B_1(0)$. Let $\overline\tmap_{\beta\alpha}\colon C\to C$ be the map
such that
\begin{equation}
(\forall f\in C) \qquad \overline\tmap_{\beta\alpha}(f) = \frac{\tmap_{\beta\alpha} (f)}{\supnor{\tmap_{\beta\alpha} (f)}}.
\end{equation}
Then, since $d_H(s f,t f^\prime) = d_H(f,f^\prime)$ for any $s,t>0$ and $f,f^\prime\in C$, we have
\begin{equation}
d_H(\overline\tmap_{\beta\alpha}(f),\overline\tmap_{\beta\alpha}(f^\prime)) = d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta\alpha}(f^\prime)) \leq \lambda^2 ~ d_H(f,f^\prime).
\end{equation}
Since $(C,d_H)$ is a complete metric space \cite[Theorem~1.2]{nussbaum1988hilbert} and $\overline\tmap_{\beta\alpha}$ is a contraction, we can apply Banach's contraction theorem and conclude that there exists a unique fixed point of $\overline\tmap_{\beta\alpha}$, namely a function $\bar f\in C$ such that
\begin{equation}
\bar f = \overline\tmap_{\beta\alpha}(\bar f) = \frac{\tmap_{\beta\alpha}(\bar f)}{\supnor{\tmap_{\beta\alpha}(\bar f)}}.
\end{equation}
Hence $\bar f$ is an eigenvector for $\tmap_{\beta\alpha}$ with eigenvalue
$t=\lVert \tmap_{\beta\alpha}(\bar{f}) \rVert_{\infty}>0$. Now, we note that
\begin{equation}
\label{eq:20190517a}
(\forall\,f,g \in \posi(\X))\quad
\scal {g \lmap_\alpha f}{\beta}
= \scal{f \lmap_\beta g}{\alpha}
= \int_{\X\times\X} f(x) k(x,y) g(y) d (\alpha\otimes\beta)(x,y).
\end{equation}
Set $\bar{g} = \tmap_{\alpha} (\bar{f})$,
so that $\tmap_\beta (\bar{g}) = t \bar{f}$.
Then, recalling the definitions of
$\tmap_\alpha$ and $\tmap_\beta$,
we have
$\bar{g} \lmap_\alpha \bar{f} \equiv 1$ and
$t^{-1} \equiv \bar{f} \lmap_\beta \bar{g}$. Hence
$t^{-1} = \scal{\bar{f} \lmap_\beta \bar{g}}{\alpha} =
\scal{\bar{g} \lmap_\alpha \bar{f}}{\beta} = 1$. Therefore $\bar{f}$ is a fixed point
of $\tmap_{\beta\alpha}$.
Finally,
if $\bar f^\prime\in\posi(\X)$ is a fixed point of $\tmap_{\beta\alpha}$, then,
since $\tmap_{\beta\alpha}$
is positively homogeneous,
we have
\begin{equation}
\overline\tmap_{\beta\alpha}(\bar{f}^\prime/\lVert \bar{f}^\prime \rVert_{\infty}) =
\dfrac{\tmap_{\beta\alpha}(\bar{f}^\prime/\lVert \bar{f}^\prime \rVert_{\infty})}
{\lVert \tmap_{\beta\alpha}(f^\prime/\lVert \bar{f}^\prime \rVert_{\infty}) \rVert_{\infty}} =
\dfrac{\tmap_{\beta\alpha}(\bar{f}^\prime)}
{\lVert \tmap_{\beta\alpha}(\bar{f}^\prime)\rVert_{\infty}} = \dfrac{\bar{f}^\prime}{\lVert \bar{f}^\prime \rVert_{\infty}},
\end{equation}
that is,
$\bar f^\prime/\supnor{\bar f^\prime}$ is a fixed point of $\overline\tmap_{\beta\alpha}$.
Thus, $\bar f^\prime/\supnor{\bar f^\prime} = \bar f$ and hence $\bar f^\prime$ is a multiple of $\bar f$.
\end{proof}
\begin{corollary}[Existence and uniqueness of Sinkhorn potentials]
Let $\alpha, \beta \in \prm(\X)$. Then,
the DAD problem \cref{eq:DAD} admits a solution $(f,g)$ and every other solution is of type $(t f, t^{-1} g)$, for some $t>0$.
Moreover, there exists a pair $(u,v) \in \cont(\X)^2$ of Sinkhorn potentials and every other pair of Sinkhorn potentials
is of type $(u + s, v- s)$,
for some $s \in \R$.
In particular, for every $x_o \in \X$,
there exist a unique pair $(u,v)$
of Sinkhorn potentials such that $u(x_0) = 0$.
\end{corollary}
\begin{proof}
It follows from \cref{thm:fixed-point-sinkhorn-iteration} and the discussion after \cref{eq:fixed-point-f-f}.
\end{proof}
\paragraph{Bounding $(f,g)$ point-wise} We conclude this section by providing additional properties of the solutions $(f,g)$ of the DAD problem \cref{eq:fixed-point-f-g}. In particular, we show that there exists one such solution for which it is possible to provide a point-wise upper and lower bound independent on $\alpha$ and $\beta$.
\begin{remark}
Let $f \in \posi(\X)$ and set $g=\tmap_\alpha (f)$. Then,
recalling \cref{eq:Ta} and \eqref{eq:L_norm}, we have that, for every $x \in \X$,
\begin{equation*}
1= g(x) (\lmap_\alpha~ f)(x)
\leq g(x) \supnor{\lmap_\alpha f} \leq g(x) \supnor{f}
\end{equation*}
and
\begin{equation*}
1= g(x) (\lmap_\alpha~ f)(x) \geq g(x)(\min_{\X} f) \int \kerfun(x,z)~d\alpha(z) \geq g(x)(\min_{\X} f)e^{-\diameps}.
\end{equation*}
Therefore,
\begin{equation}
\label{eq:20190430a}
\min_{\X} g \geq \frac{1}{\supnor{f}}
\quad\text{and}\quad
\supnor{g} \leq \frac{e^{\diameps}}{\min_{\X} f}.
\end{equation}
\end{remark}
\begin{lemma}(Auxiliary Cone)
\label{lem:auxiliary-cone}
Consider the set
\begin{equation}
\label{eq:defconeK}
K = \{f \in \nneg(\X) ~ | ~ f(x) \leq f(y)~ e^{\diameps} ~~ \forall x,y\in\X \}.
\end{equation}
Let $\alpha \in \prm(\X)$. Then the following holds.
\begin{enumerate}[{\rm (i)}]
\item\label{lem:auxiliary-cone_i}
$K$ is a closed convex cone and $K\subset\posi(\X)\cup\{0\}$;
\item\label{lem:auxiliary-cone_ii} $\lmap_\alpha(\cont_+(\X)) \subset K$;
\item\label{lem:auxiliary-cone_iii} $\mathsf{R}(K) \subset K$;
\item\label{lem:auxiliary-cone_iv}
$\ran(\tmap_{\alpha}) \subset K$;
\item\label{lem:auxiliary-cone_v} If $f \in K$ and $g=\tmap_\alpha f$,
then $g \in K$ and $1\leq (\min_{\X}g)\supnor{f} \leq \supnor{g} \supnor{f} \leq e^{2 \diameps}$.
\item\label{lem:auxiliary-cone_vi}
If $f \in K$ is such that $f(x_o)=1$ for some $x_o \in \X$, then $\nor{\varepsilon \log f}_{\infty} \leq \diam$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{lem:auxiliary-cone_i}:
We see that for any $f\in K$,
\begin{equation}
\label{eq:20190430b}
\max_{\X} f \leq (\min_{\X} f)~ e^{\diameps},
\end{equation}
so, if $f(x) = 0$ for some $x\in\X$, then $f(x) = 0$ on all $\X$. Hence $K\subseteq\posi(\X)\cup\{0\}$.
It is straightforward to verify that $K$ is a convex cone. Moreover $K$ is also closed. Indeed if $(f_n)_{n \in \N}$ is a sequence in $K$ which converges uniformly to $f \in \cont(\X)$, then, for every
$x,y \in \X$ and every $n \in \N$, $f_n(x)\leq f_n(y) e^{\diameps}$ and hence,
letting $n\to +\infty$,
we have $f(x) \leq f(y) e^{\diameps}$.
\ref{lem:auxiliary-cone_ii}:
For every $f\in \cont_+(\X)$ and $x,y\in\X$, we have
\begin{align*}
(\lmap_\alpha f)(x) & = \int \kerfun(x,z) f(z) ~d\alpha(z) \\
& = \int \frac{\kerfun(x,z)}{\kerfun(y,z)} ~ \kerfun(y,z)f(z)~d\alpha(z) \\
& \leq e^{\diameps} \int \kerfun(y,z) f(z) ~d\alpha(z) \\
& = e^{\diameps} (\lmap_\alpha f)(y).
\end{align*}
\ref{lem:auxiliary-cone_iii}:
For every $f\in K$,
\begin{equation*}
(\forall\, x,y \in \X)\qquad
f(x) \leq f(y)~ e^{\diameps} ~\Leftrightarrow~ \frac{1}{f(y)} \leq \frac{1}{f(x)}~ e^{\diameps}.
\end{equation*}
\ref{lem:auxiliary-cone_iv}
It follows from \cref{lem:auxiliary-cone_ii} and \cref{lem:auxiliary-cone_iii}
and the definitions of $\tmap_\alpha$.
\ref{lem:auxiliary-cone_v}:
It follows from \ref{lem:auxiliary-cone_iv}, \eqref{eq:20190430a},
and \eqref{eq:20190430b}.
\ref{lem:auxiliary-cone_vi}:
Let $f \in K$ be such that $f(x_o)=1$. Then $\min_{\X} f \leq 1 \leq \max_{\X} f$.
Thus, it follows from \eqref{eq:20190430b} that
\begin{equation}
\label{eq:20190516b}
\max_{\X} f \leq e^{\diameps}\quad \text{and} \quad
\min_{\X} f \geq e^{-\diameps}
\end{equation}
and hence, for every $x \in \X$, $-\diam \leq \varepsilon \log f(x) \leq \diam$.
\end{proof}
As a direct consequence of \cref{lem:auxiliary-cone} we can establish a uniform point-wise upper and lower bound for the value of DAD solutions.
\begin{corollary}
\label{cor:dad-solutions-bounded}
Let $\alpha,\beta\in\prob(\X)$.
Let $x_o \in \X$ and let $(f,g)$ be
the solution of \cref{eq:fixed-point-f-g} such that $f(x_o) = 1$.
Then $\nor{f}_\infty \leq e^{\diameps}$ and $\nor{g}_{\infty} \leq e^{2\diameps}$. Moreover,
the corrisponding pair $(u,v)$
of Sinkhorn potentials satifies
$\supnor{u}\leq \diam$ and $\supnor{v} \leq 2\diam$.
\end{corollary}
\begin{proof}
Since $f$ and $g$ are fixed points of $\tmap_{\beta\alpha}$ and $\tmap_{\alpha\beta}$
respectively, it follows from \cref{lem:auxiliary-cone}\cref{lem:auxiliary-cone_iv} that $f,g \in K$.
Then, \cref{lem:auxiliary-cone}\cref{lem:auxiliary-cone_vi}
yields $\norm{f}_{\infty} \leq e^{\diameps}$, whereas by the second of \cref{eq:20190430a} and \cref{eq:20190516b} we derive that $\nor{g}_{\infty} \leq e^{2\diameps}$.
\end{proof}
\subsection{Sinkhorn-Knopp algorithm in infinite dimension}
In the context of optimal transport, Sinkhorn-Knopp algorithm is often presented and studied in finite dimension \cite{cuturi2013sinkhorn,peyre2017computational}.
The algorithm
originates from so called
\emph{matrix scaling problems},
also called \emph{DAD problems}, which consists in finding, for a given matrix $A$ with nonnegative entries, two diagonal matrices $D_1$, $D_2$
such that $D_1 A D_2$ is doubly stochastic \cite{sinkhorn1967}.
In our setting it is crucial to analyze the algorithm in infinite dimension.
\cref{thm:fixed-point-sinkhorn-iteration} shows that $\tmap_{\beta\alpha}$ is a contraction with respect to the Hilbert's metric. This suggests a direct approach to find the solutions of the DAD problem by adopting a fixed-point strategy, which turns out to applying the operators $\tmap_\alpha$ and $\tmap_{\beta}$ alternatively, starting from some $f^{(0)}\in\posi(\X)$. This is exactly the approach to the Sinkhorn algorithm pioneered by \cite{menon1967,Frank1989}
and further developed in an infinite dimensional setting in \cite{nussbaum1993entropy}.
In this section we review the algorithm and give the convergence properties
for the special kernel $\kerfun$ in \cref{eq:kerfunc}. In particular we provide rate of convergence in the sup norm $\supnor{\cdot}$.
\begin{algorithm}
\caption{Sinkhorn-Knopp algorithm (infinite dimensional case)}
\label{alg:Sinkhorn_cont}
Let $\alpha,\beta \in \prm(\X)$. Let $f^{(0)} \in \posi(\X)$ and define,
\begin{equation*}
\begin{array}{l}
\text{for}\;\Siter=0,1,\dots\\[0.7ex]
\left\lfloor
\begin{array}{l}
g^{(\Siter+1)} = \tmap_{\alpha}(f^{(\Siter)})\\[1ex]
f^{(\Siter+1)} = \tmap_{\beta}(g^{(\Siter+1)})
\end{array}
\right.
\end{array}
\end{equation*}
\end{algorithm}
\begin{theorem}[Convergence of Sinkhorn-Knopp algorithm]
\label{thm:Sinkhornalgo}
Let $(f^{(\Siter)})_{\Siter \in \N}$ be defined according to \cref{alg:Sinkhorn_cont}.
Let $x_o \in \X$ and let $(f, g)$ be the solution of the
DAD problem \eqref{eq:Dopt2}
such that $f(x_o) = 1$. Then,
defining, for every $\Siter \in \N$,
$\tilde{f}^{(\Siter)} = f^{(\Siter)}/f^{(\Siter)}(x_o)$ and
$\tilde{g}^{(\Siter+1)} = g^{(\Siter+1)} f^{(\Siter)}(x_o)$, we have
\begin{equation}
\label{eq:sinkalgo2}
\begin{cases}
\lVert \log \tilde{f}^{(\Siter)} - \log f\rVert_{\infty} \leq \lambda^{2\Siter}
\bigg(\dfrac{\diam}{\varepsilon}+ \log \dfrac{\nor{f^{(0)}}_{\infty}}{\min_{\X} f^{(0)}} \bigg) \\[2.5ex]
\lVert \log \tilde{g}^{(\Siter+1)} - \log g\rVert_{\infty} \leq e^{3\diameps} \lVert \log \tilde{f}^{(\Siter)} - \log f\rVert_{\infty}.
\end{cases}
\end{equation}
Moreover, let the potentials $(u,v) = (\varepsilon\log f, \varepsilon\log g)$ and, for every $\Siter \in \N$, $(\tilde{u}^{(\Siter)}, \tilde{v}^{(\Siter)})
= (\varepsilon\log \tilde{f}^{(\Siter)}, \varepsilon\log \tilde{g}^{(\Siter)})$.
Then we have
\begin{equation}
\label{eq:sinkalgo3}
\lVert \tilde{u}^{(\Siter)} - u\rVert_{\infty} \leq \lambda^{2\Siter}
\bigg(\frac{\diam + \max\nolimits_{\X} u^{(0)} - \min\nolimits_{\X} u^{(0)}}{\varepsilon}\bigg).
\end{equation}
\end{theorem}
\begin{proof}
Let $\mathcal{A}$ be the set in \cref{lem:relation-supnor-hilbert}.
Clearly, for every $\Siter \in \N$, we have
$f^{(\Siter+1)} = \tmap_{\beta \alpha} (f^{(\Siter)})$
and $\bar{f}, \tilde{f}^{\Siter} \in \mathcal{A}$.
Thus, it follows from \cref{thm:fixed-point-sinkhorn-iteration}
and \cref{eq:relation-supnor-hilbert2} in \cref{lem:relation-supnor-hilbert} that,
for every $\Siter \in \N$,
\begin{equation*}
\lVert \log \tilde{f}^{(\Siter)} - \log f\rVert_{\infty} \leq d_H(\tilde{f}^{\Siter}, f)
= d_H(\tmap_{\beta \alpha}^{(\Siter)}(f^{(0)}), f) \leq \lambda^{2\Siter} d_H(f^{(0)},f).
\end{equation*}
Moreover, recalling \cref{eq:20190509b}, we have
\begin{equation*}
d_H(f^{(0)},f) = d_H (1/f^{(0)}, \lmap_{\beta} g)
= \log \max_{x,y \in \X} \frac{f^{(0)}(y) \lmap_\beta g (y)}{f^{(0)}(x) \lmap_\beta g (x)}
\leq \log \bigg[e^{\diameps} \max_{x,y \in \X} \frac{f^{(0)}(y) }{f^{(0)}(x) } \bigg]
\end{equation*}
where we used the fact that $\lmap_\beta(\cont_{++}(\X)) \subset K$
and the definition \cref{eq:defconeK}. Thus, the first inequality in \cref{eq:sinkalgo2} follows.
The second inequality in \cref{eq:sinkalgo2} and \cref{eq:sinkalgo3} follow
directly from \cref{lem:Lipschitz2} and the fact that $u^{(0)} = \varepsilon \log f^{(0)}$.
\end{proof}
\begin{algorithm}
\caption{Sinkhorn-Knopp algorithm (finite dimensional case)}
\label{algo:sinkalgo_disc}
Let $\mathsf{M} \in \R_{++}^{n_1\times n_2}$,
$\mathsf{a} \in \R^{n_1}_+$,
with $\mathsf{a}^\top \mathsf{1}_{n_1} = 1$,
and $\mathsf{b} \in \R^{n_2}_+$, with
$\mathsf{b}^\top \mathsf{1}_{n_2}=1$.
Let $\mathsf{f}^{(0)} \in \R^{n_1}_{++}$ and define
\begin{equation*}
\begin{array}{l}
\text{for}\;\Siter=0,1,\dots\\[0.7ex]
\left\lfloor
\begin{array}{l}
\mathsf{g}^{(\Siter+1)}= \dfrac{\mathsf{b}}{\mathsf{M}^\top \mathsf{f}^{(\Siter)}}\\[2ex]
\mathsf{f}^{(\Siter+1)}= \dfrac{\mathsf{a}}{\mathsf{M} \mathsf{g}^{(\Siter+1)}}.
\end{array}
\right.
\end{array}
\end{equation*}
\end{algorithm}
\begin{proposition}
\label{rmk:discrete-sinkhorn}
Suppose that $\alpha$ and $\beta$
are probability measures with finite support. Then \cref{alg:Sinkhorn_cont} can be reduced to
the finite dimensional \cref{algo:sinkalgo_disc}. More specifically,
suppose that $\alpha = \sum_{i=1}^{n_1} a_{i} \delta_{x_{i}}$,
and $\beta = \sum_{i=1}^{n_2} b_{i} \delta_{y_{i}}$, where
$\mathsf{a} = (a_i)_{1 \leq i \leq n_1} \in \R^{n_1}_+$, $\sum_{i=1}^n a_i = 1$ and
$\mathsf{b} = (b_i)_{1 \leq i \leq n_2} \in \R^{n_2}_+$, $\sum_{i=1}^n b_{i} = 1$.
Let $\mathsf{K} \in \R^{n_1\times n_2}$ be such that $\mathsf{K}_{i_1,i_2} = \kerfun(x_{i_1},y_{i_2})$
and let $\mathsf{M} = \mathrm{diag}(\mathsf{a})\mathsf{K}\mathrm{diag}(\mathsf{b}) \in \R^{n_1\times n_2}$.
Let $(\mathsf{f}^{(\Siter)})_{\Siter \in \N}$ and $(f^{(\Siter)})_{\Siter \in \N}$ be defined according to \cref{algo:sinkalgo_disc} and \cref{alg:Sinkhorn_cont} respectively, with
$\mathsf{f}^{(0)} = (f^{(0)}(x_i))_{1 \leq i \leq n_1}$.
Then, for every $\Siter \in \N$,
\begin{equation*}
(\forall\, x \in \X)(\forall\, y \in \X)\
g^{(\Siter +1)}(y)^{-1} = \sum_{i_1=1}^{n_1} k(x_{i_1},y)a_{i_1} \mathsf{f}^{(\Siter)}_{i_1}
\ \text{and}\ f^{(\Siter +1)}(x)^{-1} = \sum_{i_2=1}^{n_2} k(x,y_{i_2})
b_{i_2} \mathsf{g}_{i_2}^{(\Siter+1)}.
\end{equation*}
Moreover, setting $u^{(\Siter)} = \varepsilon \log f^{(\Siter)}$, $v^{(\Siter)} = \varepsilon \log g^{(\Siter)}$, $\mathsf{u}^{(\Siter)} = \varepsilon \log \mathsf{f}^{(\Siter)}$,
and $\mathsf{v}^{(\Siter)} = \varepsilon \log \mathsf{g}^{(\Siter)}$, we have
\begin{equation}
\label{eq:20190523c}
\begin{cases}
\displaystyle
(\forall\, y \in \X)\quad
v^{(\Siter +1)}(y) = - \varepsilon \log \sum_{i_1=1}^{n_1} \exp( \mathsf{u}_{i_1}^{(\Siter)} - \cost(x_{i_1},y) ) a_{i_1}\\[1ex]
\displaystyle
(\forall\, x \in \X)\quad u^{(\Siter +1)}(x) = - \varepsilon \log \sum_{i_2=1}^{n_2}
\exp(\mathsf{v}_{i_2}^{(\Siter + 1)} - \cost(x,y_{i_2}) ) b_{i_2}.
\end{cases}
\end{equation}
\end{proposition}
\begin{proof}
Since $\alpha$ and $\beta$ have finite support,
we derive from the definitions of $f^{(\Siter+1)}$ and $g^{(\Siter+1)}$
in \cref{alg:Sinkhorn_cont} and that of $\tmap_\alpha$ and $\tmap_\beta$ that
\begin{equation*}
\begin{cases}
\displaystyle(\forall\, x \in \X)\quad
g^{(\Siter+1)}(y)^{-1} = (\lmap_\alpha f^{(\Siter)}))(y)
= \sum_{i_1=1}^{n_1} a_{i_1} \kerfun(x_{i_1}, y) f^{(\Siter)}(x_{i_1})\\[2ex]
\displaystyle (\forall\, y \in \X)\quad
f^{(\Siter+1)}(x)^{-1} = (\lmap_\beta g^{(\Siter+1)}))(x)
= \sum_{i_2=1}^{n_2} \kerfun(x, y_{i_2})b_{i_2} g^{(\Siter+1)}(y_{i_2}).
\end{cases}
\end{equation*}
Now, multiplying the above equations by $b_{i_2}$ and $a_{i_1}$ respectively, and recalling that $\mathsf{M}_{i_1,i_2} = a_{i_1} \kerfun(x_{i_1},y_{i_2}) b_{i_2}$, we have
\begin{equation*}
\begin{bmatrix}
b_{1} g^{(\Siter+1)}(y_1)^{-1}\\
\vdots\\[0.8ex]
b_{n_2} g^{(\Siter+1)}(y_{n_2})^{-1}
\end{bmatrix}
= \mathsf{M}^\top
\begin{bmatrix}
f^{(\Siter)}(x_{1})\\
\vdots\\
f^{(\Siter)}(x_{n_1})
\end{bmatrix},
\ \
\begin{bmatrix}
a_{1} f^{(\Siter+1)}(x_1)^{-1}\\
\vdots\\[0.8ex]
a_{n_1} f^{(\Siter+1)}(x_{n_1})^{-1}
\end{bmatrix}
= \mathsf{M}
\begin{bmatrix}
g^{(\Siter+1)}(y_{1})\\
\vdots\\[0.8ex]
g^{(\Siter+1)}(y_{n_2})
\end{bmatrix},
\end{equation*}
and hence
\begin{equation*}
\begin{bmatrix}
g^{(\Siter+1)}(y_1)\\
\vdots\\
g^{(\Siter+1)}(y_{n_2})
\end{bmatrix}
= \mathsf{b} \bigg/ \mathsf{M}^\top
\begin{bmatrix}
f^{(\Siter)}(x_{1})\\
\vdots\\
f^{(\Siter)}(x_{n_1})
\end{bmatrix},
\ \
\begin{bmatrix}
f^{(\Siter+1)}(x_1)\\
\vdots\\
f^{(\Siter+1)}(x_{n_1})
\end{bmatrix}
= \mathsf{a} \bigg/ \mathsf{M}
\begin{bmatrix}
g^{(\Siter+1)}(y_{1})\\
\vdots\\
g^{(\Siter+1)}(y_{n_2})
\end{bmatrix}.
\end{equation*}
Therefore, since
$\mathsf{f}^{(0)} = (f^{(0)}(x_i))_{1 \leq i \leq n_1}$, recalling \cref{algo:sinkalgo_disc}, it follows by induction that, for every $\Siter \in \N$, $\mathsf{f}^{(\Siter)} = (f^{(\Siter)}(x_i))_{1 \leq i \leq n_1}$
and $\mathsf{g}^{(\Siter)} = (g^{(\Siter)}(x_i))_{1 \leq i \leq n_1}$. Thus, the first part of the statement follows. The second part follows directly from the definitions of $u^{(\Siter)}$, $v^{(\Siter)}$,
$\mathsf{u}^{(\Siter)}$, and $\mathsf{v}^{(\Siter)}$.
\end{proof}
\begin{remark}\
\normalfont
\begin{enumerate}[{\rm (i)}]
\item
Algorithm \cref{algo:sinkalgo_disc} is the classical (discrete) Sinkhorn algorithm
which was recently studied in several papers \cite{cuturi2013sinkhorn}.
It follows from \cref{thm:Sinkhornalgo} that
considering the solution $(f,g)$ of the DAD problem
such that $f(x_1) = 1$ and
defining
$\tilde{\mathsf{f}}^{(\Siter)} = \mathsf{f}^{(\Siter)}/\mathsf{f}^{(\Siter)}_0$
and $\tilde{\mathsf{g}}^{(\Siter)} = \mathsf{g}^{(\Siter)}\mathsf{f}^{(\Siter)}_0$,
and $\mathsf{f}_i = f(x_i)$ and $\mathsf{g}_j = g(y_j)$,
we have
\begin{equation*}
\lVert \log \tilde{\mathsf{f}}^{(\Siter)} - \log \mathsf{f}\rVert_{\infty} \leq \lambda^{2\Siter} \bigg(\dfrac{\diam}{\varepsilon}+ \log \dfrac{\max_i \mathsf{f}_i^{(0)}}{\min_i \mathsf{f}_i^{(0)}} \bigg).
\end{equation*}
\item The procedure {\sc SinkhornKnopp} discussed in the paper and called in
\cref{alg:practical-FW}, actually output the vector $\mathsf{v}=\varepsilon \log \mathsf{g}^{(\Siter)}$ for sufficiently large $\Siter$.
\item Referring to \cref{sec:algorithm-practice}
in the paper, we recognize that
the expressions on the right hand side of \cref{eq:20190523c} are precisely $\rmap_\alpha(u^{(\Siter)})(x)$ and
$\rmap_\beta(v^{(\Siter+1)})(x)$ respectively.
\end{enumerate}
\end{remark}
\section{Lipschitz continuity of the gradient of Sinkhorn divergence with respect to the Total Variation}\label{subsec:lipschitz-total-variation}
In this section we show that the gradient of the Sinkhorn divergence is Lipschitz continuous with respect to the Total Variation on $\prob(\X)$.
We start by characterizing the relation between Hilbert's metric between functions of the form $f = e^{u/\varepsilon}$ and the $\supnor{\cdot}$ norm between functions of the form $u = \varepsilon \log f$.
\begin{lemma}
\label{lem:relation-supnor-hilbert}
Let $f,f^\prime \in \posi(\X)$ and set $u = \varepsilon\log f$ and $u^\prime = \varepsilon\log f^\prime$. Then
\begin{equation}\label{eq:relation-supnor-hilbert}
d_H(f,f^\prime) \leq 2\supnor{\log f - \log f^\prime} \quad \text{or, equivalently} \quad d_H(e^{u/\varepsilon},e^{u^\prime/\varepsilon}) \leq \frac{2}{\varepsilon} \supnor{u-u^\prime}.
\end{equation}
Moreover, let $ x_o\in\X$, consider the sets $\mathcal{A} = \{h\in\posi(\X) ~|~ h(x_o) = 1\}$
and $\mathcal{B} = \{w \in \cont(\X) ~|~ w(x_o) = 0\}$. Suppose that
$f,f^\prime\in \mathcal{A}$ (or equivalently that $u,u^\prime\in \mathcal{B}$). Then
\begin{equation}
\label{eq:relation-supnor-hilbert2}
\frac{1}{2} d_H(f,f^\prime) \leq \supnor{\log f - \log f^\prime} \leq d_H(f,f^\prime).
\end{equation}
and
\begin{equation}
\label{eq:relation-supnor-hilbert3}
\frac{\varepsilon}{2} d_H(e^{u/\varepsilon},e^{u^\prime/\varepsilon}) \leq \supnor{u - u^\prime} \leq \varepsilon~ d_H(e^{u/\varepsilon},e^{u^\prime/\varepsilon}).
\end{equation}
\end{lemma}
\begin{proof}
We have
\begin{align*}
d_H(f,f^\prime) & = \log~\max_{x,y\in\X}~\frac{f(x)f^\prime(y)}{f(y)f^\prime(x)} \\
& = \log~\max_{x\in\X}~\frac{f(x)}{f^\prime(x)} + \log~\max_{y\in\X}~\frac{f^\prime(y)}{f(y)} \\
& = \max_{x\in\X} \log\frac{f(x)}{f^\prime(x)} + \max_{y\in\X}~\log\frac{f^\prime(y)}{f(y)} \\
& \leq 2\max_{x\in\X}~\left|\log\frac{f(x)}{f^\prime(x)}\right| \\
& = 2\supnor{\log(f/f^\prime)}\\[1ex]
& = 2\supnor{\log f - \log f^\prime}
\end{align*}
and \eqref{eq:relation-supnor-hilbert} follows. Suppose that
$f,f^\prime\in \mathcal{A}$. Then
\begin{align*}
\supnor{\log f - \log f^\prime} & = \max\left\{\log \max_{x\in\X} \frac{f(x)}{f^\prime(x)}, \log \max_{x\in\X} \frac{f^\prime(x)}{f(x)}\right\} \\
& = \max\left\{\log \max_{x\in\X} \frac{f(x)f^\prime(\bar x)}{f(\bar x)f^\prime(x)}, \log \max_{x\in\X} \frac{f(\bar x)f^\prime(x)}{f(x)f^\prime(\bar x)}\right\} \\
& \leq \max\left\{\log \max_{x,y\in\X} \frac{f(x)f^\prime(y)}{f(y)f^\prime(x)}, \log \max_{x,y\in\X} \frac{f(y)f^\prime(x)}{f(x)f^\prime(y)}\right\} \\
& = d_H(f,f^\prime),
\end{align*}
since $f(x_o)/f^\prime(_ox) = f^\prime(x_o)/f(x_o) = 1$. Therefore,
\eqref{eq:relation-supnor-hilbert2} follows.
\end{proof}
\begin{lemma}
\label{lem:20190510a}
For every $x,y \in \R_{++}$ we have
\begin{equation}
\lvert \log x- \log y \rvert \leq
\max\big\{x^{-1}, y^{-1}\big\}
\lvert x - y \rvert.
\end{equation}
\end{lemma}
The following result allows to extend the previous observations on a pair $f,f^\prime$ to the corresponding $g = \tmap_\alpha f$ and $g^\prime = \tmap_\alpha f^\prime$.
\begin{lemma}
\label{lem:Lipschitz2}
Let $x_o \in \X$ and $K\subset\nneg(\X)$ the cone from \cref{lem:auxiliary-cone}. Let $f,f^\prime \in K$ be such that
$f(x_o)=f^\prime(x_o)=1$, and set $g = \tmap_\alpha f$ and $g^\prime = \tmap_\alpha f^\prime$. Then,
\begin{equation}
\supnor{\log g - \log g^\prime} \leq e^{3 \diameps} \supnor{\log f - \log f^\prime}.
\end{equation}
\end{lemma}
\begin{proof}
It follows from \cref{eq:Ta} and \cref{lem:20190510a} that
\begin{equation*}
\lvert \log g - \log g^\prime \rvert
= \Big\lvert \log \frac{g}{g^\prime} \Big\rvert
= \Big\lvert \log \frac{\lmap_\alpha f^\prime}{\lmap_\alpha f} \Big\rvert
\leq \max\big\{g^\prime,
g\big\} \lvert \lmap_\alpha f - \lmap_\alpha f^\prime \rvert.
\end{equation*}
Therefore, since $1 \leq \supnor{f}, \supnor{f^\prime}$, and recalling \cref{lem:auxiliary-cone}\cref{lem:auxiliary-cone_v} and \cref{eq:L_norm},
we have
\begin{align*}
\supnor{\log g - \log g^\prime}
&\leq \max\{\supnor{g},\supnor{g^\prime}\}
\supnor{\lmap_\alpha f-\lmap_\alpha f^\prime}\\
&\leq \max\{\supnor{f}\supnor{g},\supnor{f^\prime}\supnor{g^\prime}\}
\supnor{\lmap_\alpha f-\lmap_\alpha f^\prime}\\
&\leq e^{2\diameps} \supnor{f - f^\prime}\\
&= e^{2\diameps} \lVert e^{\log f} - e^{\log f^\prime} \rVert_{\infty}.
\end{align*}
Now, since $f,f^\prime\leq e^{\diameps}$,
we have $\log f, \log f^\prime \leq \diameps$. Thus,
the statement follows by noting that the exponential function
is Lipschitz continuous on $\left]-\infty, \diameps\right]$
with constant $e^{\diameps}$.
\end{proof}
We are ready to prove the main result of the section.
\begin{theorem}[Lipschitz continuity of the Sinkhorn potentials with respect to the total variation]\label{prop:lipschitz-continuity-total-variation2}
Let $\alpha,\beta,\alpha',\beta'\in\prob(\X)$ and let $x_o \in \X$. Let $(u,v),(u^\prime,v')\in\cont(\X)^2$ be the two pairs of Sinkhorn potentials corresponding to the solution of the regularized OT problem in \cref{eq:ot-dual-problem} for $(\alpha,\beta)$ and $(\alpha',\beta')$ respectively such that $u(x_o) = u^\prime(x_o) = 0$.
Then
\begin{equation}
\label{eq:20190503d}
\supnor{u - u^\prime} \leq 2\varepsilon e^{3\diameps}\nor{(\alpha-\alpha',\beta - \beta')}_{TV}.
\end{equation}
Hence, the map which, for each pair of probability distributions $(\alpha,\beta)\in\prob(\X)^2$ associates the component $u$ of the corresponding Sinkhorn potentials is $2\varepsilon e^{3\diameps}$-Lipschitz continuous {\em with respect to the total variation}.
\end{theorem}
\begin{proof}
The functions $f = e^{u/\varepsilon}$ and $f^\prime = e^{u^\prime/\varepsilon}$ are fixed points of the maps $\tmap_{\beta\alpha}$ and $\tmap_{\beta'\alpha'}$ respectively. Then,
it follows from \cref{thm:fixed-point-sinkhorn-iteration} that
\begin{align*}
d_H(f,f^\prime) & = d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha'}(f^\prime)) \\
& \leq d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha'}(f)) + d_H(\tmap_{\beta'\alpha'}(f),\tmap_{\beta'\alpha'}(f^\prime)) \\
& \leq d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha'}(f)) + \lambda^2 d_H(f,f^\prime),
\end{align*}
hence,
\begin{equation}
\label{eq:20190511b}
d_H(f,f^\prime) \leq \frac{1}{1 - \lambda^2}~d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha'}(f)).
\end{equation}
Moreover, using \cref{eq:relation-supnor-hilbert}, we have
\begin{align}
\nonumber d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha'}(f)) & \leq d_H(\tmap_{\beta\alpha}(f),\tmap_{\beta'\alpha}(f)) + d_H(\tmap_{\beta'\alpha}(f),\tmap_{\beta'\alpha'}(f)) \\
\nonumber & \leq d_H(\tmap_{\beta}(g),\tmap_{\beta'}(g)) + \lambda d_H(\tmap_\alpha(f),\tmap_{\alpha'}(f)) \\
\label{eq:20190511a} & \leq 2\supnor{\log~\frac{\tmap_\beta(g)}{\tmap_{\beta'}(g)}} +2\lambda \supnor{\log~\frac{\tmap_\alpha(f)}{\tmap_{\alpha'}(f)} }.
\end{align}
Now, note that by \cref{lem:20190510a}
\begin{equation}
\bigg\lvert \log~\frac{\tmap_\beta(g)}{\tmap_{\beta^\prime}(g)}\bigg\rvert = \bigg\lvert \log~\frac{\lmap_{\beta^\prime} g}{\lmap_\beta g}\bigg\rvert \leq \max\{1/\lmap_\beta g, 1/\lmap_{\beta^\prime} g\} \lvert (\lmap_{\beta'}-\lmap_\beta) g \rvert
\end{equation}
and that, for every $x \in \X$,
\begin{equation}
\begin{aligned}\label{eq:pairing-for-TV-lipschitz}
[(\lmap_{\beta^\prime}-\lmap_\beta) g](x) & = \int \kerfun(x,z) g(z)~d(\beta-\beta^\prime)(z) \\
& = \scal{\kerfun(x,\cdot) g}{\beta-\beta^\prime}
\leq \supnor{g}\nor{\beta - \beta^\prime}_{TV},
\end{aligned}
\end{equation}
and, similarly, $[(\lmap_{\beta}-\lmap_{\beta^\prime}) g](x) \leq \supnor{g}\nor{\beta - \beta'}_{TV}$.
Therefore, since $1/(\lmap_\beta g) = \tmap_\beta(g) = f$ and $\lmap_{\beta^\prime}g \geq e^{-\diameps} \min g$, it follows from \cref{lem:auxiliary-cone}\cref{lem:auxiliary-cone_v}
and \cref{eq:20190430b} (applied to $g$) that
\begin{equation}
\label{eq:20190511c}
\supnor{\log~\frac{\tmap_\beta(g)}{\tmap_{\beta'}(g)} } \leq
\max\left\{\supnor{f},\frac{e^{\diameps}}{\min g}\right\} \supnor{g}\nor{\beta-\beta'}_{TV} \leq
e^{2\diameps}~\nor{\beta-\beta'}_{TV}.
\end{equation}
Analogously, we have
\begin{equation}
\label{eq:20190511d}
\supnor{\log~\frac{\tmap_\alpha(f)}{\tmap_{\alpha'}(f)}} \leq e^{2\diameps}~\nor{\alpha-\alpha'}_{TV}.
\end{equation}
Putting \cref{eq:20190511b}, \cref{eq:20190511a}, \cref{eq:20190511c}, and \cref{eq:20190511d} together, we have
\begin{equation}
d_H(f,f^\prime) \leq \frac{2e^{2\diameps}}{1-\lambda^2}\left(\lambda \nor{\alpha-\alpha'}_{TV} + \nor{\beta - \beta'}_{TV}\right).
\end{equation}
Now, note that since $e^{\diameps}\geq1$
\begin{equation}
\frac{1}{1-\lambda^2} = \frac{(e^{\diameps} + 1)^2}{4e^\diameps} \leq e^{\diameps}.
\end{equation}
Finally, recalling \cref{eq:relation-supnor-hilbert3}, we have
\begin{equation}
\supnor{u - u^\prime} \leq 2\varepsilon e^{3\diameps}\nor{(\alpha-\alpha',\beta - \beta')}_{TV},
\end{equation}
where $\nor{(\alpha-\alpha',\beta - \beta')}_{TV} = \nor{\alpha-\alpha'}_{TV} + \nor{\beta - \beta'}_{TV}$ is the total variation norm on $\meas(\X)^2$.
\end{proof}
\begin{corollary}
\label{cor:Lipschitz2}
Under the assumption of \cref{prop:lipschitz-continuity-total-variation2}, we have
\begin{equation}
\supnor{u - u^\prime}+\supnor{v - v^\prime} \leq 2 \varepsilon e^{3\diameps}(1+ \varepsilon e^{3\diameps})
\nor{(\alpha-\alpha',\beta - \beta')}_{TV}.
\end{equation}
\end{corollary}
\begin{proof}
It follows from \cref{prop:lipschitz-continuity-total-variation2} and \cref{lem:Lipschitz2}.
\end{proof}
We finally address the issue of the differentiability of the Sinkhorn divergence.
We first recall a few facts about the directional differentiability of $\oteps$ briefly recalled in \cref{sec:background} of the main text. For a more in-depth analysis on this topic we refer the reader to \cite{feydy2018interpolating} (in particular Proposition $2$).
\begin{fact}
\label{rem:directional-derivatives-oteps}
Let $x_{o}\in\X$, $\alpha,\beta\in\prob(\X)$ and $(u,v)\in\cont(\X)^2$ be the pair of corresponding Sinkhorn potentials with $u(x_o) = 0$. The function $\oteps$
is directionally differentiable and
the directional derivative of $\oteps$ in $(\alpha,\beta)$ along a feasible direction $(\mu,\nu)\in\feas_{\prob(\X)^2}\big((\alpha,\beta)\big)$ (see \cref{def:directional-derivative}) is
\begin{equation}\label{eq:directional-derivative-oteps}
\oteps^\prime(\alpha,\beta;\mu,\nu) = \int u(x)~d\mu(x) + \int v(y)~d\nu(y) = \scal{(u,v)}{(\mu,\nu)}.
\end{equation}
Let $\nabla\oteps\colon\prob(\X)^2\to\cont(\X)^2$
be the operator that maps every pair of probability distributions $(\alpha,\beta)\in\prob(\X)^2$ to the corresponding pair of Sinkhorn potentials $(u,v)\in\cont(\X)^2$ with $u(x_o)=0$. Then \cref{eq:directional-derivative-oteps} can be written as
\begin{equation}
\oteps^\prime(\alpha,\beta;\mu,\nu) = \scal{\nabla \oteps(\alpha,\beta)}{(\mu,\nu)}.
\end{equation}
\end{fact}
\begin{remark}
\normalfont
In \cref{rem:directional-derivatives-oteps}, the requirement $u(x_o) = 0$ is only a convention to remove ambiguities. Indeed, for every $t\in\R$, replacing the Sinkhorn potential $(u+t,u-t)$ in \cref{def:cone-of-feasible-directions} does not affect \cref{eq:directional-derivative-oteps}.
\end{remark}
\begin{fact}
\label{f:Sinkhorn}
Let $\beta \in \prm(\X)$ and let $\nabla_1 \oteps$ be the first component of the gradient operator defined in \cref{rem:directional-derivatives-oteps}. Then the
Sinkhorn divergence function
$S_\varepsilon(\cdot, \beta)\colon \prm(\X) \to \R$ in \cref{eq:sink_divergence} is directionally differentiable
and, for every $\alpha \in \prm(\X)$ and every $\mu \in \feas_{\prm(\X)}(\alpha)$,
\begin{equation*}
[S_\varepsilon(\cdot, \beta)]^\prime(\alpha; \mu)
= \scal{\nabla_1 \oteps(\alpha,\beta)- \nabla_1 \oteps(\alpha,\alpha)}{\mu}.
\end{equation*}
So, one can define
$\nabla S_\varepsilon(\cdot, \beta)\colon \prob(\X) \to \cont(\X)$ such that, for every $\alpha \in \prm(\X)$, $\nabla [S_\varepsilon(\cdot, \beta)](\alpha) = \nabla_1 \oteps(\alpha,\beta) - \nabla_1 \oteps(\alpha,\alpha)$ and we have
\begin{equation}
[S_\varepsilon(\cdot, \beta)]^\prime(\alpha; \mu)
=\scal{\nabla S_\varepsilon(\cdot, \beta)}{\mu}.
\end{equation}
Finally, if $\kerfun$
in \cref{eq:kerfunc} is a positive definite kernel,
then the Sinkhorn divergence $S_\varepsilon(\cdot, \beta)$ is convex.
\end{fact}
We are now ready to prove \cref{thm:lip-continuity-total-variation-informal} in the paper.
We recall also the statement for reader's convenience.
\TLipschitzContinuityTV*
\begin{proof}
The first part is just a consequence of \cref{prop:lipschitz-continuity-total-variation2}
and \cref{rem:directional-derivatives-oteps}. The second part, follows from the first part and \cref{f:Sinkhorn}.
\end{proof}
\begin{remark}
\normalfont
It follows from the optimality conditions
\cref{eq:Dopt} that, for every $x \in \supp(\alpha)$
and $y \in \supp(\beta)$,
\begin{equation*}
1 = \int_{\X} e^{\frac{u(x) + v(y) - \cost(x,y)}{\varepsilon}} d\beta(y)
\quad\text{and}\quad
1 = \int_{\X} e^{\frac{u(x) + v(y) - \cost(x,y)}{\varepsilon}} d\alpha(x),
\end{equation*}
hence,
\begin{equation}
\int_{\X} e^{\frac{u\oplus v - \cost}{\varepsilon}} d \alpha\otimes\beta = 1.
\end{equation}
Then, recalling the definition of $\oteps$ in \cref{eq:dual_pb}
and that of its gradient, given above,
we have
\begin{equation}
\oteps(\alpha,\beta) =
\scal{\nabla \oteps(\alpha,\beta)}{(\alpha,\beta)} - \varepsilon.
\end{equation}
Since, $\nabla \oteps$ is bounded and Lipschitz continuous, it follows that $\oteps$ is Lipschitz continuous with respect to the total variation.
\end{remark}
We end the section by providing an independent proof of \cref{rem:directional-derivatives-oteps}, which is based on \cref{p:diff_of_max} and \cref{cor:Lipschitz2}.
\begin{proposition}
The function
$\oteps\colon \prm(\dom)^2 \to \R$, defined in \cref{eq:dual_pb}, is continuous with respect to the total variation,
directionally differentiable, and,
for every $(\alpha,\beta) \in \prm(\dom)^2$
and every feasible direction
$(\mu,\nu) \in \feas_{\prm(\dom)^2}(\alpha,\beta)$, we have
\begin{equation}
\oteps^\prime(\alpha,\beta;\mu,\nu)
= \scal{(u,v)}{(\mu,\nu)},
\end{equation}
where $(u,v) \in \cont(\X)^2$ is any
solution of problem \cref{eq:dual_pb}.
\end{proposition}
\begin{proof}
Let $g\colon \cont(\X)^2\times \finmeas(\X)^2 \to \R$
be such that,
\begin{equation}
g((u,v),(\alpha,\beta)) = \scal{u}{\alpha} + \scal{v}{\beta}-\varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\alpha\otimes \beta}.
\end{equation}
Then, for every $(\alpha,\beta) \in \prm(\X)^2$,
\begin{equation}
\label{eq:20190512a}
\oteps(\alpha,\beta) = \max_{(u,v) \in \cont(\X)^2} g((u,v),(\alpha,\beta)).
\end{equation}
Thus, $\oteps$ is of the type considered in \cref{p:diff_of_max}.
Let $(u,v) \in \cont(\X)$. Then the function $g((u,v), \cdot)$
admits directional derivatives and, for every $(\alpha,\beta), (\mu,\nu) \in \finmeas(\X)^2$, we have
\begin{multline}
\label{eq:20190511e}
[g((u,v),\cdot)]^\prime((\alpha,\beta);(\mu,\nu)) \\[1ex]=
\Big\langle u - \varepsilon e^{\frac{u}{\varepsilon}} \int_{\X} e^{\frac{v - c(\cdot,y)}{\varepsilon}} d\beta(y), \mu \Big\rangle
+ \Big\langle v - \varepsilon e^{\frac{v}{\varepsilon}} \int_{\X} e^{\frac{u - c(x,\cdot)}{\varepsilon}} d\alpha(x), \nu \Big\rangle.
\end{multline}
Indeed, for every $t >0$,
\begin{align*}
\frac{1}{t} \big[g((u,v),& (\alpha,\beta) + t(\mu,\nu)) - g((u,v),(\alpha,\beta)) \big] \\
&= \frac{1}{t}
\big[ \scal{u}{\alpha+ t \mu} + \scal{v}{\beta + t \nu}-\varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{(\alpha + t \mu)\otimes (\beta + t \nu)}\\
&\qquad- \scal{u}{\alpha} -\scal{v}{\beta}
+\varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\alpha\otimes \beta} \big]\\
&=
\scal{u}{\mu} + \scal{v}{\nu}
- \varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\alpha\otimes \nu}
- \varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\mu\otimes \beta}\\[0.8ex]
&\qquad - t\varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\mu\otimes \nu},
\end{align*}
hence
\begin{multline*}
[g((u,v),\cdot)]^\prime((\alpha,\beta);(\mu,\nu)) \\[0.8ex]
= \scal{u}{\mu} + \scal{v}{\nu}
- \varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\alpha\otimes \nu}
- \varepsilon\scal{\exp( (u \oplus v - \cost)/\varepsilon)}{\mu\otimes \beta}
\end{multline*}
and \cref{eq:20190511e} follows. Thus,
the function $g$ is G\^ateaux differentiable
with respect to the second variable, with derivative
\begin{align*}
D_{2}g ((u,v),(\alpha,\beta))
&= \Big(u - \varepsilon e^{\frac{u}{\varepsilon}} \int_{\X} e^{\frac{v - c(\cdot,y)}{\varepsilon}} d\beta(y),
v - \varepsilon e^{\frac{v}{\varepsilon}} \int_{\X} e^{\frac{u - c(x,\cdot)}{\varepsilon}} d\alpha(x)\Big)\\
&= (u,v) - \varepsilon (e^{\frac{u}{\varepsilon}} \lmap_{\beta} e^{\frac{v}{\varepsilon}}, e^{\frac{v}{\varepsilon}} \lmap_{\alpha} e^{\frac{u}{\varepsilon}}) \in \cont(\X)^2,
\end{align*}
which is jointly continuous, since
the maps $(u,\alpha)\mapsto \lmap_{\alpha} e^{u/\varepsilon}$ and
$(v,\beta)\mapsto \lmap_{\beta} e^{v/\varepsilon}$
are continuous.
Moreover, it follows from \cref{cor:Lipschitz2} that there exists a continuous selection of Sinkhorn potentials.
Therefore, it follows from \cref{p:diff_of_max} that
$\oteps$ is directionally differentiable and
\begin{equation}
\oteps^\prime((\alpha,\beta);(\mu,\nu))
= \max_{(u,v)\text{ solution of } \cref{eq:20190512a}}
\scal{D_{2}g ((u,v),(\alpha,\beta))}{(\mu,\nu)}.
\end{equation}
However, if $(u,v)$ is a solution of \cref{eq:20190512a}, it follows from the optimality conditions \cref{eq:Dopt} that
\begin{equation}
e^{\frac{u}{\varepsilon}} \int_{\X} e^{\frac{v - c(\cdot,y)}{\varepsilon}} d\beta(y) = 1
\quad\text{and}\quad
e^{\frac{v}{\varepsilon}} \int_{\X} e^{\frac{u - c(x,\cdot)}{\varepsilon}} d\alpha(x) = 1,
\end{equation}
hence
\begin{equation}
\scal{D_{2}g ((u,v),(\alpha,\beta))}{(\mu,\nu)}
= \scal{(u - \varepsilon, v - \varepsilon)}{(\mu, \nu)} = \scal{(u , v )}{(\mu, \nu)},
\end{equation}
where we used the fact that, since $(\mu,\nu) = t(\mu_1 - \mu_2, \nu_1- \nu_2)$
for some $t>0$ and $\mu_1,\mu_2,\nu_1,\nu_2 \in \prm(\X)$, we have $\scal{1}{\mu} = t\scal{1}{\mu_1 - \mu_2} = 0$ and $\scal{1}{\nu}=t \scal{1}{\nu_1 - \nu_2} =0$.
\end{proof}
\section{The Frank-Wolfe algorithm for Sinkhorn barycenters }\label{sec:app-frank-wolfe-algorithm}
In this section we finally analyze the Frank-Wolfe algorithm for the Sinkhorn barycenters and give convergence results.
The following result is a direct consequence of \cref{thm:Sinkhornalgo} and \cref{rem:directional-derivatives-oteps}.
\begin{theorem}
\label{thm:Sinkhorn2}
Let $(\tilde{u}^{(\Siter)})_{\Siter \in \N}$ be generated according to
\cref{alg:Sinkhorn_cont}.
Then,
\begin{equation}
\label{eq:Sinkhorn2}
(\forall\, \Siter \in \N)\quad\lVert \tilde{u}^{(\Siter)} - \nabla_1 \oteps(\alpha,\beta)\rVert_{\infty} \leq
\lambda^{2\Siter}
\bigg(\frac{\diam + \max\nolimits_{\X} u^{(0)} - \min\nolimits_{\X} u^{(0)}}{\varepsilon}\bigg),
\end{equation}
where $u^{(\Siter)} = \varepsilon \log f^{(\Siter)}$ and
$\tilde{u}^{(\Siter)} = u^{(\Siter)} - u^{(\Siter)}(x_o)$.
\end{theorem}
Therefore, in view of \cref{f:Sinkhorn}, \cref{thm:Sinkhorn2}, and \cref{p:inexactgrad},
we can address the problem of the Sinkhorn barycenter \cref{eq:sinkhorn-barycenter} via the Frank-Wolfe \cref{alg:abstract-FW}. Note that,
according to \cref{p:inexactgrad}\ref{p:inexactgrad_ii}, since the diameter of $\prob(\X)$ with respect to $\nor{\cdot}_{TV}$ is $2$, have that the curvature of $\bary$ is upper bounded by
\begin{equation}
\label{eq:OTcurvature}
C_{\bary} \leq 24\varepsilon e^{3\diameps}.
\end{equation}
Let $k \in \N$
and $\alpha_k$ be the current iteration.
For every $j \in \{1,\dots, m\}$,
we can compute $\nabla_1 \oteps(\alpha_k,\beta_j)$ and
$\nabla_1 \oteps(\alpha_k,\alpha_k)$ by the Sinkhorn-Knopp algorithm.
Thus, by \cref{eq:Sinkhorn2}, we find $\Siter \in \N$ large enough so that $\lVert \tilde{u}_j^{(\Siter)} - \nabla_1 \oteps(\alpha_k,\beta_q) \rVert_\infty \leq \precision_{1,k}/8$ and $\lVert \tilde{p}^{(\Siter)} - \nabla_1 \oteps(\alpha_k,\alpha_k) \rVert_\infty \leq \precision_{1,k}/8$ and we set
\begin{equation}
\tilde{u}^{(\Siter)}:= \sum_{j=1}^m \omega_j
\tilde{u}_j^{(\Siter)} - \tilde{p}^{(\Siter)}.
\end{equation}
Then,
\begin{equation}
\lVert \tilde{u}^{(\Siter)}
- \nabla \bary(\alpha_k)\rVert_\infty \leq \frac{\precision_{1,k}}{4}.
\end{equation}
Now, Frank-Wolf \cref{alg:abstract-FW} (in the version
considered in \cref{p:inexactgrad}\ref{p:inexactgrad_i}) requires finding
\begin{equation}
\label{eq:20190508a}
\eta_{k+1} \in \argmin_{\eta \in \prob(\X)} \langle \tilde{u}^{(\Siter)}, \eta - \alpha_k \rangle
\end{equation}
and make the update
\begin{equation}
\alpha_{k+1} = (1 - \gamma_k) \alpha_k + \gamma_k \eta_{k+1}.
\end{equation}
Since the solution of \cref{eq:20190508a} is a Dirac measure (see \cref{sec:algorithm-practice} in the paper),
the algorithm reduces to
\begin{equation}
\begin{cases}
\text{find } x_{k+1} \in \X \text{ such that }
\tilde{u}^{(\Siter)}(x_{k+1}) \leq
\min_{x \in \X} \tilde{u}^{(\Siter)}(x) + \dfrac{\precision_{2,k}}{2} \\[1ex]
\alpha_{k+1} = (1 - \gamma_k) \alpha_k + \gamma_k \delta_{x_{k+1}}.
\end{cases}
\end{equation}
So, if we initialize the algorithm with $\alpha_0 = \delta_{x_0}$, then any $\alpha_k$ will be a discrete probability measure with support contained in $\{x_0, \dots, x_k\}$. This implies that
if all the $\beta_j$'s are probability measures with finite support, the computation of
$\nabla_1 \oteps(\alpha_k,\beta_j)$ by the Sinkhorn algorithm can be reduced to a fully discrete algorithm, as showed in \cref{rmk:discrete-sinkhorn}. More precisely,
assume that
\begin{equation}
(\forall\, j=1,\dots, m)\quad
\beta_j = \sum_{i_2=0}^n b_{j,i_2} \delta_{y_{j,i_2}}.
\end{equation}
and that at iteration $k$ we have
\begin{equation}
\alpha_k = \sum_{i_1=0}^k a_{k,i_1} \delta_{x_{i_1}}.
\end{equation}
Set
\begin{equation}
\mathsf{a}_k =
\begin{bmatrix}
a_{k,0} \\
\vdots\\
a_{k,k}
\end{bmatrix} \in \R^{k+1},
\ \ \mathsf{M}_0 =
\begin{bmatrix}
a_{k,0}\kerfun(x_{0},x_{0}) a_{k,0} & \dots & a_{k,0}\kerfun(x_{0},x_{k}) a_{k,k}\\
\vdots & \ddots & \vdots\\
a_{k,k}\kerfun(x_{k},x_{0}) a_{k,0} & \dots & a_{k,k}\kerfun(x_{k},x_{k}) a_{k,k}
\end{bmatrix} \in \R^{(k+1)\times(k+1)}
\end{equation}
and, for every $j=1\,\dots, m$,
\begin{equation}
\mathsf{b}_j =
\begin{bmatrix}
b_{j,0} \\
\vdots\\
b_{j,n}
\end{bmatrix} \in \R^{n+1},
\ \
\mathsf{M}_j =
\begin{bmatrix}
a_{k,0}\kerfun(x_{0},y_{j,0}) b_{j,0} & \dots & a_{k,0}\kerfun(x_{0},y_{j,n}) b_{j,n}\\
\vdots & \ddots & \vdots\\
a_{k,k}\kerfun(x_{k},y_{j,0}) b_{j,0} & \dots & a_{k,n}\kerfun(x_{k},y_{j,n}) b_{j,n}
\end{bmatrix} \in \R^{(k+1)\times(n+1)}.
\end{equation}
Then, run \cref{algo:sinkalgo_disc},
with input $\mathsf{a}_k$, $\mathsf{a}_k$,
and $\mathsf{M}_0$
to get $(\mathsf{e}^{(\Siter)},\mathsf{h}^{(\Siter)})$,
and, for every $j=1,\dots, m$,
with input $\mathsf{a}_k$, $\mathsf{b}_j$,
and $\mathsf{M}_j$ to get $(\mathsf{f}^{(\Siter)}_j,\mathsf{g}^{(\Siter)}_j)$.
So, we have,
\begin{equation}
(\forall\, \Siter \in \N) \quad
\begin{cases}
\phantom{(\forall\,j=1,\dots,m)\ \ }
\mathsf{h}^{(\Siter+1)}= \dfrac{\mathsf{a}_k}{\mathsf{M}_0^\top \mathsf{e}^{(\Siter)}},
\quad
\mathsf{e}^{(\Siter+1)}= \dfrac{\mathsf{a}_k}{\mathsf{M}_0 \mathsf{h}^{(\Siter+1)}}\\[2ex]
(\forall\,j=1,\dots,m)\ \ \mathsf{g}_j^{(\Siter+1)}= \dfrac{\mathsf{b}_j}{\mathsf{M}_j^\top \mathsf{f}_j^{(\Siter)}},
\quad
\mathsf{f}_j^{(\Siter+1)}= \dfrac{\mathsf{a}_k}{\mathsf{M}_j \mathsf{g}_j^{(\Siter+1)}}.
\end{cases}
\end{equation}
Then, according to \cref{rmk:discrete-sinkhorn}, for every $\Siter \in \N$, we have
\begin{equation}
\label{eq:20190523b}
(\forall\, x \in \X)\
\begin{cases}
\displaystyle e^{(\Siter)}(x)^{-1} = \sum_{i_2=0}^k \kerfun(x,x_{i_2}) \mathsf{h}^{(\Siter-1)}_{i_2} a_{k,i_2},\\[1ex]
\displaystyle p^{(\Siter)}(x) = \varepsilon \log e^{(\Siter)}(x) = - \varepsilon \log \sum_{i_2=0}^k \kerfun(x,x_{i_2}) \mathsf{h}^{(\Siter-1)}_{i_2} a_{k,i_2}\\[1ex]
\tilde{p}^{(\Siter)}(x) = p^{(\Siter)}(x) - p^{(\Siter)}(x_o).
\end{cases}
\end{equation}
and, for every $j=1,\dots, m$,
\begin{equation}
\label{eq:20190523a}
(\forall\, x \in \X)\
\begin{cases}
\displaystyle f_j^{(\Siter)}(x)^{-1} = \sum_{i_2=0}^n \kerfun(x,y_{i_2}) \mathsf{g}^{(\Siter-1)}_{j,i_2} b_{j,i_2},\\[1ex]
\displaystyle u_j^{(\Siter)}(x) = \varepsilon \log f_j^{(\Siter)}(x) = - \varepsilon \log \sum_{i_2=0}^n \kerfun(x,y_{i_2}) \mathsf{g}^{(\Siter-1)}_{j,i_2} b_{j,i_2}\\[1ex]
\tilde{u}_j^{(\Siter)}(x) = u_j^{(\Siter)}(x) - u_j^{(\Siter)}(x_o).
\end{cases}
\end{equation}
Since the
$\tilde{u}_j^{(\Siter)}$'s and $u_j^{(\Siter)}$'s,
and $\tilde{p}^{(\Siter)}$ and $p^{(\Siter)}$,
differ for a constant only, the final algorithm can be written as in \cref{algo:FW-Baricenters}.
We stress that this algorithm is even more general than
\cref{alg:practical-FW} since, in the computation of the Sinkhorn potentials and in their minimization, errors have been taken into account.
\begin{algorithm}
\caption{Frank-Wolfe algorithm for Sinkhorn barycenter}
\label{algo:FW-Baricenters}
Let $\alpha_0 = \delta_{x_0}$ for some $x_0 \in \X$.
Let $(\precision_k)_{k \in \N} \in \R_{+}^\N$ be such that
$\Delta_k/\gamma_k$ is nondecreasing. Define
\begin{equation*}
\begin{array}{l}
\text{for}\;k=0,1,\dots\\[0.7ex]
\left\lfloor
\begin{array}{l}
\text{run \cref{algo:sinkalgo_disc}
with input $\mathsf{a}_k, \mathsf{a}_k, \mathsf{M}_0$ till } \lambda^{2\Siter} \diameps \leq \frac{\precision_{1,k}}{8} \rightarrow \mathsf{h} \in \R^{k+1} \\[0.7ex]
\text{compute } p \text{ via \cref{eq:20190523b}} \text{ with } \mathsf{h}\\[0.7ex]
\text{for}\;j=1,\dots m\\[0.7ex]
\left\lfloor
\begin{array}{l}
\text{run \cref{algo:sinkalgo_disc}
with input $\mathsf{a}_k, \mathsf{b}_j, \mathsf{M}_j$ till } \lambda^{2\Siter} \diameps \leq \frac{\precision_{1,k}}{8} \rightarrow \mathsf{g}_j \in \R^{n+1} \\[0.7ex]
\text{compute } u_j \text{ via \cref{eq:20190523a}} \text{ with } \mathsf{g}_j \\[0.7ex]
\end{array}
\right.\\[1ex]
\text{set } u
= \sum_{j=1}^m \omega_j u_j - p\\[1ex]
\text{find } x_{k+1} \in \X \text{ such that }
u(x_{k+1}) \leq
\min_{x \in \X} u(x) + \dfrac{\precision_{2,k}}{2}\\[1ex]
\alpha_{k+1} = (1 - \gamma_k) \alpha_k + \gamma_k \delta_{x_{k+1}}.
\end{array}
\right.
\end{array}
\end{equation*}
\end{algorithm}
We now give a final converge theorem, of which
\cref{thm:sinkhorn-barycenters-finite-case} in the paper is a special case.
\begin{theorem}\label{thm:full_convergence_FW_with_error}
Suppose that $\beta_1, \dots, \beta_m \in \prm(\X)$ are probability measures with finite support, each of cardinality $n \in \N$. Let $(\alpha_k)_{k \in \N}$ be generated by \cref{algo:FW-Baricenters}. Then, for every $k \in \N$,
\begin{equation}
\bary(\alpha_k) - \min_{\alpha \in \prm(\X)} \bary(\alpha) \leq \gamma_k
24\varepsilon e^{3\diameps} + 2 \Delta_{1,k} + \Delta_{2,k}
\end{equation}
\end{theorem}
\begin{proof}
It follows from \cref{thm:FWA}, \cref{p:inexactgrad}, and \cref{eq:OTcurvature},
recalling that $\Diam(\prm(\X))=2$.
\end{proof}
\section{Sample complexity of Sinkhorn potential}\label{sec:sample-complexity-sinkhorn-potentials}
In the following we will denote by $\cont^s(\X)$ the space of $s$-differentiable functions with continuous derivatives and by $W^{s,p}(\X)$ the Sobolev space of functions $f\colon \X \to \R$ with $p$-summable weak derivatives up to order $s$ \cite{adams2003sobolev}. We denote by $\nor{\cdot}_{s,p}$ the corresponding norm.
The following result shows that under suitable smoothness assumptions on the cost function $\dist$, the Sinkhorn potentials are uniformly bounded as functions in a suitable Sobolev space of corresponding smoothness. This fact will play a key role in approximating the Sinkhorn potentials of general distributions in practice.
\begin{theorem}[Proposition 2 in \cite{genevay2018sample}]\label{thm:sinkhorn-potentials-uniformly-bounded}
Let $\X$ be a closed bounded domain with Lipschitz boundary
in $\R^d$ (\cite[Definition 4.9]{adams2003sobolev}) and let $\dist \in \cont^{s+1}(\X\times\X)$. Then for every $(\alpha,\beta)\in\prob(\X)^2$, the associated Sinkhorn potentials $(u,v)\in\cont(\X)^2$ are functions in $W^{s,\infty}(\X)$. Moreover, let $x_o\in\X$. Then there exists a constant $\sinkconst>0$, depending only on $\varepsilon,s$ and $\X$, such that for every $(\alpha,\beta)\in\prob(\X)^2$ the associated Sinkhorn potentials $(u,v)\in\cont(\X)^2$ with $u(x_o) = 0$ satisfies $\nor{u}_{s,\infty},\nor{v}_{s,\infty}\leq\sinkconst$.
\end{theorem}
In the original statement of \cite[Proposition 2]{genevay2018sample} the above result is formulated for $\dist\in\cont^{\infty}(\X)$ for simplicity. However, as clarified by the authors, it holds also for the more general case $\dist\in\cont^{s+1}(\X)$.
\begin{lemma}\label{lem:s-infty-norm-of-products-and-exponentials}
Let $\X\subset\R^d$ be a closed bounded domain with Lipschitz boundary
and let $u,u'\in W^{s,\infty}(\X)$. Then the following holds
\begin{enumerate}[{\rm(i)}]
\item \label{item:timesconst} $\nor{uu'}_{s,\infty} \leq \timesconst \nor{u}_{s,\infty}\nor{u'}_{s,\infty}$,
\item \label{item:expconst} $\nor{e^{u}}_{s,\infty}
\leq \nor{e^{u}}_{\infty} (1 + \expconst \nor{u}_{s,\infty})$,
\end{enumerate}
where $\timesconst = \timesconst(s,d)$ and $\expconst = \expconst(s,d)>0$ depend only
on the dimension $d$ and the order of differentiability $s$
but not on $u$ and $u'$.
\end{lemma}
\begin{proof}
\cref{item:timesconst} follows directly from Leibniz formula. To see \cref{item:expconst}, let $\mathbf{i} = (i_1,\dots,i_d)\in\N^d$ be a multi-index with $|\mathbf{i}| = \sum_{\ell=1}^d i_\ell \leq s$ and note that by chain rule the derivatives of $e^{u}$
\eqals{
D^\mathbf{i} ~e^{u} = e^{u}~ P_\mathbf{i}\Big((D^\mathbf{j} u)_{\mathbf{j} \leq \mathbf{i}}\Big),
}
where $P_\mathbf{i}$ is a polynomial of degree $|\mathbf{i}|$ and $\mathbf{j} \leq \mathbf{i}$ is the ordering associated to the cone of non-negative vectors in $\R^d$. Note that $P_0 = 1$, while for $|\mathbf{i}|>0$, the associated polyomial $P_\mathbf{i}$ has a root in zero (i.e. it does not have constant term). Hence
\eqals{
\nor{e^{u}}_{s,\infty} & \leq \nor{e^{u}}_\infty \left(1 + |P|\Big((\nor{D^\mathbf{i} u}_\infty)_{|\mathbf{i}|\leq s}\Big)~\right),
}
where we have denoted by $P = \sum_{0<|\mathbf{i}|\leq s} P_\mathbf{i}$ and by $|P|$ the polynomial with coefficients corresponding to the absolute value of the coefficients of $P$. Therefore, since $\nor{D^\mathbf{i} u}_{\infty}\leq \nor{u}_{s,\infty}$ for any $|\mathbf{i}|\leq s$, by taking
\eqals{
\expconst = |P|\Big((1)_{|\mathbf{i}|\leq s}\Big),
}
namely the sum of all the coefficients of $|P|$, we obtain the desired result. Indeed note that the coefficients of $P$ do not depend on $u$ but only on the smoothness $s$ and dimension $d$.
\end{proof}
\begin{lemma}\label{lem:uniform-bound-products-exponential-sinkhorn-potentials}
Let $\X\subset\R^d$ be a closed bounded domain with Lipschitz boundary and let $ x_o\in\X$. Let $\cost\in\cont^{s+1}(\X\times\X)$,
for some $s\in\N$. Then for any $\alpha,\beta\in\prob(\X)$ and corresponding pair of Sinkhorn potentials $(u,v)\in\cont(\X)^2$ with $u(x_o) = 0$, the functions $\kerfun(x,\cdot)e^{u/\varepsilon}$ and $\kerfun(x,\cdot)e^{v/\varepsilon}$ belong to $W^{s,2}(\X)$ for every $x\in\X$. Moreover, they admit an extension to $\hh = W^{s,2}(\R^d)$ and there exists a constant $\bar\sinkconst$ independent on $\alpha$ and $\beta$, such that for every $x\in\X$
\begin{equation}\label{eq:uniform-bound-products-exponential-sinkhorn-potentials}
\big\lVert \kerfun(x,\cdot) e^{u/\varepsilon}\big\rVert_{\hh},~ \big\lVert \kerfun(x,\cdot) e^{v/\varepsilon} \big\rVert_{\hh} \leq \bar\sinkconst
\end{equation}
(with some abuse of notation, we have identified $\kerfun(x,\cdot) e^{u/\varepsilon}$ and $\kerfun(x,\cdot)e^{v/\varepsilon}$ with their extensions to $\R^d$).
\end{lemma}
\begin{proof}
In the following we denote by $\nor{\cdot}_{s,2} = \nor{\cdot}_{s,2,\X}$ the norm of $W^{s,2}(\X)$ and by $\nor{\cdot}_\hh=\nor{\cdot}_{s,2,\R^d}$ the norm of $\hh = W^{s,2}(\R)$. Let $x\in \X$.
Then, since $u - \cost(x,\cdot) \in W^{s,\infty}(\X)$ and $\nor{u}_{s,\infty} \leq \sinkconst$,
it follows from \cref{lem:s-infty-norm-of-products-and-exponentials} that
\begin{align*}
\big\lVert \kerfun(x,\cdot) e^{u/\varepsilon} \big\rVert_{s,\infty} & =
\big\lVert e^{(u - \cost(x,\dot))/\varepsilon} \big\rVert_{s,\infty}\\
& \leq \big\lVert e^{(u - \cost(x,\dot))/\varepsilon} \big\rVert_{\infty}
(1 + \expconst \nor{u - \cost(x,\cdot)}_{s,\infty})\\
& = \big\lVert \kerfun(x,\cdot) e^{u/\varepsilon} \big\rVert_{\infty}
(1 + \expconst \nor{u - \cost(x,\cdot)}_{s,\infty})\\
& \leq \big\lVert e^{u/\varepsilon}\big\rVert_{\infty}
(1 + \expconst (\sinkconst+ \nor{\cost}_{s,\infty}))\\
& \leq e^{\diameps}(1 + \expconst (\sinkconst+ \nor{\cost}_{s,\infty})),
\end{align*}
where we used the fact that $D^\mathbf{i} [\cost(x,\cdot)] = (D^\mathbf{i} \cost )(x,\cdot)$.
This implies
\begin{equation*}
\big\lVert\kerfun(x,\cdot) e^{u/\varepsilon}\big\rVert_{s,2} \leq
|\X|^{1/2}e^{\diameps}(1 + \expconst (\sinkconst+ \nor{\cost}_{s,\infty}))
\end{equation*}
where $|\X|$ is the Lebesgue measure of $\X$.
Now, we can proceed analogously to \cite[Proposition 2]{genevay2018sample}, and use Stein's Extension Theorem \cite[Theorem 5.24]{adams2003sobolev},\cite[Chapter 6]{stein2016singular}, to guarantee the existence of a {\em total extension operator} \cite[Definition 5.17]{adams2003sobolev}. In particular, there exists a constant $\extensionconst = \extensionconst(s,2,\X)$ such that for any $\varphi\in W^{s,2}(\X)$ there exists $\tilde \varphi \in W^{s,2}(\R^d)$ such that
\begin{equation}
\nor{\tilde \varphi}_{\hh} = \nor{\tilde \varphi}_{s,2,\R^d} \leq \extensionconst\nor{\varphi}_{s,2,\X} = \extensionconst\nor{\varphi}_{s,2}.
\end{equation}
Therefore, we conclude
\begin{equation}
\big\lVert\kerfun(x,\cdot) e^{u/\varepsilon}\big\rVert_{\hh} \leq \extensionconst |\X|^{1/2}e^{\diameps}(1 + \expconst (\sinkconst+ \nor{\cost}_{s,\infty})) =: \bar\sinkconst.
\end{equation}
The same argument applies to $\kerfun(x,\cdot) e^{v/\varepsilon}$ with the only exception that now, in virtue of \cref{cor:dad-solutions-bounded},
we have $\lVert e^{v/\varepsilon}\rVert_{\infty} \leq e^{2\diameps}$. Note that $\bar\sinkconst$ is a constant depending only on $\X$, $\cost$, $s$ and $d$ but it is independent on the probability distributions $\alpha$ and $\beta$.
\end{proof}
\paragraph{Sobolev spaces and reproducing kernel Hilbert spaces}
Recall that for $s>d/2$ the space $\hh = W^{s,2}(\R^d)$, is a reproducing kernel Hilbert space (RKHS) \cite[Chapter 10]{wendland2004scattered}. In this setting we denote by $\kersob:\X\times\X\to\R$ the associated reproducing kernel,
which is continuous and bounded and satisfies the reproducing property
\begin{equation}
(\forall\,x\in\X)(\forall\,f\in\hh)\qquad \scal{f}{\kersob(x,\cdot)}_{\hh} = f(x).
\end{equation}
We can also assume that $\kersob$ is {\em normalized}, namely, $\nor{\kersob(x,\cdot)}_\hh=1$ for all $x\in\X$ \cite[Chapter 10]{wendland2004scattered}.
\paragraph{Kernel mean embeddings}
For every $\beta\in\prob(\X)$, we denote by $\kersob_\beta\in\hh$ the {\em Kernel Mean Embedding} of $\beta$ in $\hh$ \cite{smola2007hilbert,muandet2017kernel}, that is, the vector
\begin{equation}
\kersob_\beta = \int \kersob(x,\cdot) ~d\beta(x).
\end{equation}
In other words, the kernel mean embedding of a distribution $\beta$ corresponds to the expectation of $\kersob(x,\cdot)$ with respect to $\beta$. By the linearity of the inner product and the integral, for every $f\in\hh$, the inner product
\begin{equation}
\scal{f}{\kersob_\beta}_\hh = \int \scal{f}{\kersob(x,\cdot)}~d\beta(x) = \int f(x)~d\beta(x),
\end{equation}
corresponds to the expectation of $f(x)$ with respect to $\beta$.
The {\em Maximum Mean Discrepancy (MMD)} \cite{song2008learning,sriperumbudur2011universality,muandet2017kernel} between two probability distributions $\beta,\beta^\prime\in\prob(\X)$ is defined as
\begin{equation}
\mmd(\beta,\beta^\prime) = \nor{\kersob_\beta - \kersob_{\beta^\prime}}_\hh.
\end{equation}
In the case of the Sobolev space $\hh = W^{s,2}(\R^d)$, the MMD metrizes the weak-$*$ topology of $\prob(\X)$ \cite{sriperumbudur2010hilbert,sriperumbudur2011universality}.
A well-established approach to approximate a distribution $\beta\in\prob(\X)$ is to independently sample a set of points $x_1,\dots,x_n\in\X$ from $\beta$ and consider the empirical distribution $\beta_n = \frac{1}{n}\sum_{i=1}^n \delta_{x_i}$. The following result shows that $\beta_n$ converges to $\beta$ in MMD with high probability. The original version of this result can be found in \cite{song2008learning}, we report an independent proof for completeness.
\begin{lemma}\label{lem:mmd-concentration-inequality}
Let $\beta\in\prob(\X)$. Let $x_1,\dots,x_n\in\X$ be indepedently sampled according to $\beta$ and denote by $\beta_n = \frac{1}{n} \sum_{i=1}^n \delta_{x_i}$. Then, for any $\tau\in(0,1]$, we have
\begin{equation}
\mmd(\beta_n,\beta) \leq \frac{4\log \frac{3}{\tau}}{\sqrt{n}}
\end{equation}
with probability at least $1-\tau$.
\end{lemma}
\begin{proof}
The proof follows by applying Pinelis' inequality \cite{yurinskiui1976exponential,pinelis1994optimum,smale2007learning} for random vectors in Hilbert spaces. More precisely, for $i=1,\dots,n$, denote by $\zeta_i = \kersob(x_i,\cdot)\in\hh$ and recall that $\nor{\zeta_i} = \nor{\kersob(x,\cdot)}=1$ for all $x\in\X$. We can therefore apply \cite[Lemma 2]{smale2007learning} with constants $\widetilde{M}=1$ and $\sigma^2 =\sup_i \mathbb{E} \|\zeta_i\|^2 \leq 1$, which guarantees that, for every $\tau\in(0,1]$
\begin{equation}\label{eq:pinelis-general-form}
\nor{\frac{1}{n}\sum_{i=1}^n \Big[\zeta_i - \mathbb{E}~\zeta_i\Big]}_\hh \leq \frac{2\log\frac{2}{\tau}}{n} + \sqrt{\frac{2\log\frac{2}{\tau}}{n}} \leq \frac{4\log \frac{3}{\tau}}{\sqrt{n}},
\end{equation}
holds with probability at least $1-\tau$. Here, for the second inequality we have used the fact that $\log\frac{2}{\tau}\leq\log\frac{3}{\tau}$ and $\log\frac{3}{\tau}\geq1$ for every $\tau\in(0,1]$. The desired result follows by observing that
\begin{equation}
\kersob_\beta = \int \kersob(x,\cdot)~d\beta(x) = \mathbb{E}~\zeta_i
\end{equation}
for all $i=1,\dots,n$, and
\begin{equation}
\kersob_{\beta} = \frac{1}{m}\sum_{i=1}^m \kersob(x_i,\cdot) = \frac{1}{m}\sum_{i=1}^m \zeta_i.
\end{equation}
Therefore,
\begin{equation}
\mmd(\beta_k,\beta) = \nor{\kersob_{\beta_k} - \kersob_\beta}_\hh
= \nor{\frac{1}{n}\sum_{i=1}^n \Big[\zeta_i - \mathbb{E}~\zeta_i\Big]}_\hh,
\end{equation}
which combined with \cref{eq:pinelis-general-form} leads to the desired result.
\end{proof}
\begin{proposition}[Lipschitz continuity of the Sinkhorn Potentials with respect to the MMD]\label{prop:lipschitz-continuity-mmd}
Let $\X\subset\R^d$ be a compact Lipschitz domain and $\dist\in\cont^{s+1}(\X\times\X)$, with $s>d/2$.
Let $\alpha,\beta,\alpha',\beta'\in\prob(\X)$.
Let $x_o\in\X$ and let $(u,v),(u^\prime,v')\in\cont(\X)^2$ be the two Sinkhorn potentials corresponding to the solution of the regularized OT problem in \cref{eq:ot-dual-problem} for $(\alpha,\beta)$ and $(\alpha',\beta')$ respectively such that $u(x_o) = u^\prime(x_o) = 0$. Then
\begin{equation}
\label{eq:lipschitz-continuity-mmd}
\supnor{u - u^\prime} \leq 2\varepsilon\bar\sinkconst e^{3\diameps}\left(\mmd(\alpha,\alpha') + \mmd(\beta,\beta')\right),
\end{equation}
with $\bar\sinkconst$ from \cref{lem:uniform-bound-products-exponential-sinkhorn-potentials}.
In other words, the operator $\nabla_1\oteps\colon\prob(\X)^2\to\cont(\X)$, defined in \cref{rem:directional-derivatives-oteps},
is $2\varepsilon\bar\sinkconst e^{3\diameps}$-Lipschitz continuous with respect to the \textnormal{MMD}.
\end{proposition}
\begin{proof}
Let $f= e^{u/\varepsilon}$ and $g = e^{v/\varepsilon}$. By relying on \cref{lem:uniform-bound-products-exponential-sinkhorn-potentials} we can now refine the analysis in \cref{prop:lipschitz-continuity-total-variation2}. More precisely, we observe that in \cref{eq:pairing-for-TV-lipschitz} we have
\begin{equation*}
\begin{aligned}{}
[(\lmap_{\beta^\prime}-\lmap_\beta) g](x) & = \int \kerfun(x,z) g(z)~d(\beta-\beta^\prime)(z) \\
& = \int \scal{\kerfun(x,\cdot)g}{\kersob(z,\cdot)}_\hh~d(\beta-\beta')(z) \\
& = \scal{\kerfun(x,\cdot) g}{\kersob_\beta-\kersob_{\beta'}}_{\hh}\\
& \leq \nor{\kerfun(x,\cdot) g}_\hh~ \nor{\kersob_\beta-\kersob_{\beta'}}_{\hh} \\
& \leq \bar\sinkconst~ \mmd(\beta,\beta'),
\end{aligned}
\end{equation*}
where in the first equality, with some abuse of notation, we have implicitly considered the extension of $\kerfun(x,\cdot) g$ to $\hh = W^{s,2}(\R^d)$ as discussed in \cref{lem:uniform-bound-products-exponential-sinkhorn-potentials}. The rest of the analysis in \cref{prop:lipschitz-continuity-total-variation2} remains invaried, eventually leading to \cref{eq:lipschitz-continuity-mmd}.
\end{proof}
It is now clear that \cref{thm:sample-complexity-sinkhorn-gradients} in the paper is just a consequence of
\cref{lem:mmd-concentration-inequality} and
\cref{prop:lipschitz-continuity-mmd}.
We give the statement of the theorem for reader's convenience.
\TLSampleComplexitySinkhornPotentials*
We finally provide the proof of \cref{thm:sinkhorn-barycenters-infinite-case} in the paper.
\TLSInkhornBarycenterInfiniteDim*
\begin{proof}
Let $\widehat\bary(\alpha) = \sum_{j=1}^m \omega_j \sink(\alpha,\hat\beta_j)$. Then,
it follows from the definition of $\bary$ and \cref{thm:sample-complexity-sinkhorn-gradients}
that, for every $k \in \N$, and with probability larger than $1-\tau$, we have
\begin{align*}
\lVert \nabla \widehat\bary(\alpha_k) - \nabla \bary(\alpha_k) \rVert_{\infty}
&\leq \sum_{j=1}^m \omega_j
\lVert \nabla
[\sink(\cdot, \hat{\beta}_j)](\alpha_k) -
\sink(\cdot, \beta_j)](\alpha_k)
\rVert_{\infty}\\
& = \sum_{j=1}^m \omega_j
\lVert \nabla_1 \oteps(\alpha_k, \hat\beta_j)- \nabla_1\oteps(\alpha_k,\beta_j)
\rVert_{\infty}\\
&\leq
\frac{8\varepsilon~\overline\sinkconst e^{3\diameps}\log\frac{3}{\tau}}{\sqrt{n}}\\
&= \frac{\Delta_{1}}{4},
\end{align*}
where
\begin{equation*}
\Delta_1 := \frac{32\varepsilon~\overline\sinkconst e^{3\diameps}\log\frac{3}{\tau}}{\sqrt{n}}.
\end{equation*}
Now, let $\gamma_k = 2/(k+2)$. Since
\cref{alg:practical-FW} is applied to
$\hat{\beta}_1,\dots \hat{\beta}_m$, we have
\begin{equation*}
\delta_{x_{k+1}} \in \argmin_{\prm(\X)} \langle \nabla \widehat\bary(\alpha_k),\cdot \rangle\quad\text{and}\quad
\alpha_{k+1} = (1- \gamma_k) \alpha_k + \gamma_k \delta_{x_{k+1}}.
\end{equation*}
Therefore, it follows from \cref{thm:FWA}, \cref{p:inexactgrad},
and \cref{thm:lip-continuity-total-variation-informal} that,
with probability larger than $1-\tau$, we have
\begin{equation*}
\bary(\alpha_k) - \min_{\prm(\X)} \bary \leq 6 \varepsilon \bar{\sinkconst} e^{3\diameps}\Diam(\prm(\X))^2 \gamma_k + \Delta_1 \Diam(\prm(\X)).
\end{equation*}
The statement follows by noting that $\Diam(\prm(\X))=2$.
\end{proof}
\newpage
\section{Additional experiments}\label{sec:additional_exp}
\paragraph{Sampling of continuous measures: mixture of Gaussians}
We perform the barycenter of 5 mixtures of two Gaussians $\mu_j$, centered at $(j/2, 1/2)$ and $(j/2, 3/2)$ for $j-0,\dots,4$ respectively. Samples are provided in \cref{fig:input_mixture_gauss}. We use different relative weights pairs in the mixture of Gaussians, namely $(1/10,9/10), (1/4,3/4), (1/2,1/2)$. At each iteration, a sample of $n=500$ points is drawn from $\mu_j$, $j=0\dots,4$. Results are reported in \cref{fig:bary_mixture_gauss}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.28]{images/input_1su2_1su2.png}
\includegraphics[scale=0.28]{images/input_1su4_3su4.png}
\includegraphics[scale=0.28]{images/input_1su10_9su10.png}
\caption{Samples of input measures}
\label{fig:input_mixture_gauss}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.25]{images/scatter1su2_1su2.png}
\includegraphics[scale=0.25]{images/scatter1su4_3su4.png}
\includegraphics[scale=0.25]{images/scatter1su10_9su10.png}
\caption{Barycenters of Mixture of Gaussians}
\label{fig:bary_mixture_gauss}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[scale=0.3]{images/bary_input_crop.jpg}\qquad\qquad
\includegraphics[scale=0.3]{images/bary_dino_grid_noaxis.jpg}
\caption{3D dinosaur mesh (left), barycenter of 3D meshes (right)\label{fig:dinosaur}}
\end{figure}
\paragraph{Propagation} We extend the description on the experiment about propagation in \cref{sec:experiments}. Edges $\mathcal{E}$ are selected as follows: we created a matrix $D$ such that $D_{ij}$ contains the distance between station at vertex $i$ and station at vertex $j$, computed using the geographical coordinates of the stations. Each node $v$ in $\mathcal{V}$, is connected to those nodes $u\in\mathcal{V}$ such that $D_{vu} \leq 3$. If the number of nodes $u$ that meet this condition is \textit{less} than $5$, we connect $v$ with its $5$ nearest nodes. If the number of nodes $u$ that meet this condition is \textit{more} than $10$, we connect $v$ with its $10$ nearest nodes. Each edge $e_{uv}$ is weighted with $\omega_{uv}:=D_{uv}$. Since intuitively we may expect that nearer nodes should have more influence in the construction of the histograms of unknown nodes, in the propagation functional we weight $\sink(\rho_v,\rho_u)$ with use $\exp(-\omega_{uv}/\sigma)$ or $1/\omega_{vu}$ suitably normalized.
\paragraph{Large scale discrete measures: meshes} We perform the barycenter of two discrete measures with support in $\R^3$. Meshes of the dinosaur are taken from \cite{solomon2015convolutional}
and rescaled by a 0.5 factor. The internal problem in Frank-Wolfe algorithm is solved using L-BFGS-B SciPy optimizer. Formula of the Jacobian is passed to the method. The barycenter is displayed in \cref{fig:dinosaur} together with an example of the input.
\end{document} | {"config": "arxiv", "file": "1905.13194/main_arxiv.tex"} |
TITLE: logistic differential equation, carrying capacity.
QUESTION [0 upvotes]: Assume that a population grows according to the below logistic differential equation $$\frac{\mathrm{dP} }{\mathrm{d} t}=0.01P-0.0002P^2$$
Then what is the maximum population that this model holds?
I think the answer is 50000(I can be wrong!!). Can anyone show me the steps of how to do this? using direction fields? or solve the differential equation directly? and then how do you get the maximum population? Thank you.
REPLY [1 votes]: This equation is separable. It is also a Ricatti equation (thus linearisable) if you are interested. We have
$$\frac{dP}{dt} = aP(1-bP) \implies \int\frac{dP}{P(1-bP)} = \int a\,dt$$
where $a=\frac{1}{100}$ and $b=\frac{1}{50}$. By partial fractions
$$\frac{1}{P(1-bP)} = \frac{1}{P} + \frac{b}{1-bP}$$
and substituting into the integral
$$\ln|P|-\ln|1-bP| = at+c \implies \frac{P}{1-bP} = Ce^{at}$$
where $c$ is a constant of integration and $C=\pm e^c$. Rearranging we get
$$P=\frac{Ce^{at}}{1+bCe^{at}} = \frac{Ce^{\frac{1}{100}t}}{1+\frac{1}{50}Ce^{\frac{1}{100}t}} = 50\frac{Ce^{\frac{1}{100}t}}{50+Ce^{\frac{1}{100}t}} = \frac{50}{Ae^{-\frac{1}{100}t}+1}$$
for some arbitrary $A=\frac{50}{C}$. Thus the long run population is $P\to 50$.
Note also that $P\equiv 0$ is a solution to the equation. Assuming that $0<P_0<50$, then the solution monotone increases from $P_0$ to $50$. | {"set_name": "stack_exchange", "score": 0, "question_id": 3890741} |
TITLE: Valuation over the algebraically closed field of rational number
QUESTION [2 upvotes]: How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
REPLY [2 votes]: For any finite Galois extension $K/\mathbb{Q}_p$, there is a unique extension of the norm that respects the $p$-adic norm on $\mathbb{Q}_p$, and this is Galois-invariant. Therefore, it must be given by $|x|_K = |Norm(x)|_p^{1/[K:\mathbb{Q}_p]}$. By uniqueness, if we have a tower of field extensions $L/K/\mathbb{Q}_p$, then restricting the norm on $|\cdot |_L$ to $K$ gives $|\cdot |_K$. Since any element of $\overline{Q}$ lives in a finite Galois extension of $\mathbb{Q}_p$, this gives a way to extend the norm to all of $\overline{Q}$. | {"set_name": "stack_exchange", "score": 2, "question_id": 271336} |
TITLE: Is it true that $\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c$?
QUESTION [5 upvotes]: Let $a,b,c\in\mathbb{R}_{>0}$. Is it true that:
$$
\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c
$$
I remarked that the inequality is (a bit weirdly) homogeneous, but couldn't use it. Also directly taking the logarithm doesn't seem to help; how to decide wether it's true?
REPLY [4 votes]: Since $a,b,c>0$,
\begin{align}
\frac{a \log a + b \log b + c \log c}{a+b+c} \le \log \left(\sum_{cyc}a\times \frac{a}{a+ b+ c}\right)
\end{align}
by Jensen's inequality on the $\log$. Taking exponent gives the required result. | {"set_name": "stack_exchange", "score": 5, "question_id": 1841545} |
TITLE: How many numbers between $1$ and $9999$ have sum of their digits equal to $8$? $16$?
QUESTION [3 upvotes]: How many numbers between $1$ and $9999$ have sum of their digits equal to $8$? $16$?
Can someone tell me if I got the right answers? I solved both cases and I've got $148$ for $8$ and $633$ for $16$. I solved this problem using $x_1+x_2+x_3+x_4=8$ then $16$. After that I decided to take the case when $1$ number is bigger than $10$, but I'm not sure if that's the way I should do it for $8$. And then I substract the $4$ cases when $x_1=0$ and $7$ cases when $x_1=9$. Thank you.
REPLY [3 votes]: We treat a number with fewer than four digits as a number with leading zeros. For instance, we regard the number $235$ as $0235$ and the number $8$ as $0008$. Then the number of positive integers between $1$ and $9999$ with digit sum $8$ is equal to the number of solutions of the equation
$$x_1 + x_2 + x_3 + x_4 = 8 \tag{1}$$
in the non-negative integers. A particular solution corresponds to the placement of three addition signs in a row of eight ones. For instance,
$$+ + + 1 1 1 1 1 1 1 1$$
corresponds to the choice $x_1 = x_2 = x_3 = 0$ and $x_4 = 8$. Thus, the number of positive integers with up to four digits that have digit sum $8$ is
$$\binom{8 + 3}{3} = \binom{11}{3}$$
since we must choose which three of the eleven symbols (eight ones and three addition signs) will be addition signs.
The number of positive integers between $1$ and $9999$ which have digit sum $16$ is the number of solutions of the equation
$$x_1 + x_2 + x_3 + x_4 = 16 \tag{2}$$
in the non-negative integers subject to the restrictions that $x_k \leq 9$ for $1 \leq k \leq 4$. The number of solutions of equation 2 in the non-negative integers is
$$\binom{16 + 3}{3}$$
From these, we must exclude those solutions in which at least one of the variables exceeds $9$. Notice that since $2 \cdot 10 = 20 > 16$, at most one of the variables may exceed $9$.
Suppose $x_1 > 9$. Let $y_1 = x_1 - 10$. Then $y_1$ is a non-negative integer. Substituting $y_1 + 10$ for $x_1$ in equation 2 yields
\begin{align*}
y_1 + 10 + x_2 + x_3 + x_4 & = 16\\
y_1 + x_2 + x_3 + x_4 & = 6 \tag{3}
\end{align*}
Equation $3$ is an equation in the non-negative integers with
$$\binom{6 + 3}{3} = \binom{9}{3}$$
solutions. By symmetry, there are $\binom{9}{3}$ solutions of equation 2 in which $x_k$ exceeds $9$ for $1 \leq k \leq 4$. Hence, the number of solutions of equation 2 in which one of the variables exceeds $9$ is
$$\binom{4}{1}\binom{9}{3}$$
Hence, the number of positive integers between $1$ and $9999$ which have digit sum $16$ is
$$\binom{19}{3} - \binom{4}{1}\binom{9}{3}$$ | {"set_name": "stack_exchange", "score": 3, "question_id": 1654809} |
TITLE: Probability of standard brownian motion
QUESTION [4 upvotes]: Let $\{W_t, t\geq 0\}$ be a standard Brownian motion process. An assignment asks me to calculate the following:
$$\begin{align} &\mathbb{P}(W_4<0),\\
&\mathbb{P}(W_{100} < W_{80}),\\
&\mathbb{P}(W_{100} < W_{80}+2), \\
&\mathbb{P}(W_{3} < W_2 + 2 \text{ and } W_1<0).
\end{align}$$
Now I don't need the answers, I would just like to confirm that my reasoning is correct (since no answers are provided by the course).
For $\mathbb{P}(W_4<0)$, I think that $W_4 \sim N(0,4)?$ For a normal random variable $N(0, \sigma^2)$, we know that the probability distribution function equals
$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-x^2}{2\sigma^2}}.$$
Then $$\mathbb{P}(W_4 <0) = \int_{-\infty}^0\frac{1}{\sqrt{2\pi\cdot16}}e^{\frac{-x^2}{2\cdot 16}}\quad ?$$
For $\mathbb{P}(W_{100} < W_{80})$, I thought that since $W_{t+s}-W_s \sim N(0,t)$, that
$$\mathbb{P}(W_{100} < W_{80}) = \mathbb{P}(W_{20} <0)$$
which would then be solved in similar fashion to the first probability?
For $\mathbb{P}(W_{100} < W_{80}+2)$, we use the same method as above, only we integrate from $-\infty$ to $2$ instead of $0$?
And for the last, we know that $\mathbb{P}(W_3 < W_2 + 2 \text{ and } W_1<0) = \mathbb{P}(W_3 - W_2 <2 \text{ and } W_1<0) = \mathbb{P}(W_3-W_2 < 2)\mathbb{P}(W_1<0)?$
Is my reasoning correct? Any advise on easier methods or mistakes I've made is very welcome.
REPLY [3 votes]: Your thoughts are correct… For probabilities of the form $P(X < 0)$ for a centered normal random variable $X$ you can directly conclude $$P(X < 0) = P(X\le 0) = \frac{1}{2}$$
No calculation needed.
For the probabilities of the form $P(X < 2)$ you have to do it your way or use tables of a normal distribution | {"set_name": "stack_exchange", "score": 4, "question_id": 2825947} |
TITLE: Find the maximum flux path between two nodes
QUESTION [0 upvotes]: Given a graph $G$ and two vertices $s$ and $t$, I want the maximum flux path from $s$ to $t$.That is, imagine $G$ to be a flow network with capacities on the edges. I want to find a single path that can carry the maximum flow between $s$ and $t$.
Note that this is not the same as the maximum flow problem where you can compute flows from multiple paths.
Can someone point me to fast algorithms for computing the max flux path?
REPLY [6 votes]: The problem is known as "Maximum capacity path problem" or "Widest path problem"
See the original paper: T. C. Hu, "The Maximum Capacity Route Problem", Operations Research Vol. 9, No. 6 (Nov. - Dec., 1961), pp. 898-900
or the linear time algorithm described in: A. P. Punnen, "A linear time algorithm for the maximum capacity path problem" (but I didn't download/read it) | {"set_name": "stack_exchange", "score": 0, "question_id": 16175} |
\begin{document}
\title{\bf Well-posedness of the linearized problem \\ for MHD contact discontinuities}
\author{{\bf Alessandro Morando}\\
DICATAM, Sezione di Matematica, Universit\`a di Brescia \\ Via Valotti, 9, 25133 Brescia, Italy\\
E-mail: [email protected]
\and
{\bf Yuri Trakhinin}\footnote{Supported in part by the
Landau Network--Centro Volta--Cariplo Foundation and the Ministry of Education and Science of the Russian Federation (contract 14.B37.21.0355).}\\
Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk, Russia\\
E-mail: [email protected]
\and
{\bf Paola Trebeschi}\\
DICATAM, Sezione di Matematica, Universit\`a di Brescia \\ Via Valotti, 9, 25133 Brescia, Italy\\
E-mail: [email protected]
}
\date{
}
\maketitle
\begin{abstract}
We study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows.
\end{abstract}
\section{Introduction}
\label{intro}
We consider the equations of ideal compressible MHD:
\begin{equation}\label{1}
\left\{
\begin{array}{l}
\partial_t\rho +{\rm div}\, (\rho {v} )=0,\\[6pt]
\partial_t(\rho {v} ) +{\rm div}\,(\rho{v}\otimes{v} -{H}\otimes{H} ) +
{\nabla}q=0, \\[6pt]
\partial_t{H} -{\nabla}\times ({v} {\times}{H})=0,\\[6pt]
\partial_t\bigl( \rho e +{\textstyle \frac{1}{2}}|{H}|^2\bigr)+
{\rm div}\, \bigl((\rho e +p){v} +{H}{\times}({v}{\times}{H})\bigr)=0,
\end{array}
\right.
\end{equation}
where $\rho$ denotes density, $v\in\mathbb{R}^3$ plasma velocity, $H \in\mathbb{R}^3$ magnetic field, $p=p(\rho,S )$ pressure, $q =p+\frac{1}{2}|{H} |^2$ total pressure, $S$ entropy, $e=E+\frac{1}{2}|{v}|^2$ total energy, and $E=E(\rho,S )$ internal energy. With a state equation of gas, $\rho=\rho(p ,S)$, and the first principle of thermodynamics, \eqref{1} is a closed system for the unknown $ U =U (t, x )=(p, v,H, S)$.
System (\ref{1}) is supplemented by the divergence constraint
\begin{equation}
{\rm div}\, {H} =0
\label{2}
\end{equation}
on the initial data ${U} (0,{x} )={U}_0({x})$. As is known, taking into account \eqref{2}, we can easily symmetrize system \eqref{1} by rewriting it in the nonconservative form
\begin{equation}
\left\{
\begin{array}{l}
{\displaystyle\frac{1}{\rho c^2}\,\frac{{\rm d} p}{{\rm d}t} +{\rm div}\,{v} =0,\qquad
\rho\, \frac{{\rm d}v}{{\rm d}t}-({H},\nabla ){H}+{\nabla}q =0 ,}\\[9pt]
{\displaystyle\frac{{\rm d}{H}}{{\rm d}t} - ({H} ,\nabla ){v} +
{H}\,{\rm div}\,{v}=0},\qquad
{\displaystyle\frac{{\rm d} S}{{\rm d} t} =0},
\end{array}\right. \label{3}
\end{equation}
where $c^2 =1/\rho_p$ is the square of the sound speed, $\rho_p=\partial\rho /\partial p$, ${\rm d} /{\rm d} t =\partial_t+({v} ,{\nabla} )$ and by $(\ ,\ )$ we denote the scalar product. Equations (\ref{3}) form the symmetric system
\begin{equation}
A_0(U )\partial_tU+\sum_{j=1}^3A_j(U )\partial_jU=0
\label{3Dsys}
\end{equation}
which is hyperbolic if the matrix $A_0= {\rm diag} (\rho_p/\rho ,\rho ,\rho ,\rho , 1,1,1,1)$ is positive definite, i.e.,
\begin{equation}
\rho >0,\quad \rho_p >0. \label{5}
\end{equation}
The symmetric matrices $A_j$ can be easily written down.
Within this paper we assume that the plasma obeys the state equation of a polytropic gas
\begin{equation}
\rho(p,S)= A p^{\frac{1}{\gamma}} e^{-\frac{S}{\gamma}}, \qquad A>0,\quad \gamma>1.
\label{polgas}
\end{equation}
In this case $c^2 =\gamma p/\rho$ and \eqref{5} becomes equivalent to\footnote{In fact, for us it is only important that for a polytropic gas the value $\rho c^2=\gamma p$ is continuous if the pressure is continuous, i.e., our results hold true for state equations for which the value $\rho c^2$ has the same property.}
\begin{equation}\label{Hiper2}
\rho>0, \quad p>0.
\end{equation}
Moreover, we manage to carry out the well-posedness analysis for contact discontinuities (see their definition just below) only for 2D planar MHD flows, i.e., when the space variables $x=(x_1,x_2)\in \mathbb{R}^2$ and the velocity and the magnetic field have only two components: $v=(v_1,v_2)\in \mathbb{R}^2$,
$H=(H_1,H_2)\in \mathbb{R}^2$. In the 2D planar case and for a polytropic gas, the MHD system \eqref{3} reads
\begin{equation}
\label{4}
A_0(U )\partial_tU+A_1(U )\partial_1U+A_2(U )\partial_2U=0
\end{equation}
with $A_0= {\rm diag} (1/(\gamma p) ,\rho ,\rho ,1,1,1)$ and
\[
A_1=\left( \begin{array}{cccccc} \frac{v_1}{\gamma p}& 1 & 0 & 0 & 0 & 0\\[6pt]
1 & \rho v_1 & 0 & 0& H_2& 0 \\
0& 0& \rho v_1 & 0& -H_1& 0 \\
0& 0& 0& v_1 & 0& 0\\
0& H_2& - H_1& 0& v_1 & 0\\
0& 0& 0& 0& 0& v_1
\end{array} \right),\quad
A_2=\left( \begin{array}{cccccc} \frac{v_2}{\gamma p}& 0 & 1& 0 & 0 & 0\\[6pt]
0 & \rho v_2 & 0 & - H_2& 0& 0 \\
1& 0& \rho v_2 & H_1& 0& 0 \\
0& -H_2& H_1& v_2 & 0& 0\\
0& 0& 0& 0& v_2 & 0\\
0& 0& 0& 0& 0& v_2
\end{array} \right).
\]
We consider the MHD equations \eqref{1} for $t\in [0,T]$ in the unbounded space domain $\mathbb{R}^3$ and suppose that $\Gamma (t)=\{ x_1-\varphi (t,x')=0\}$ is a smooth hypersurface in $[0,T]\times\mathbb{R}^3$, where
$x'=(x_2,x_3)$ are tangential coordinates. We assume that $\Gamma (t)$ is a surface of strong discontinuity for the conservation laws (\ref{1}), i.e., we are interested in solutions of (\ref{1}) that are smooth on either side of $\Gamma (t)$. To be weak solutions of (\ref{1}) such piecewise smooth solutions
should satisfy the MHD Rankine-Hugoniot conditions (see, e.g., \cite{LL})
\begin{equation}
\left\{
\begin{array}{l}
[\mathfrak{j}]=0,\quad [H_{\rm N}]=0,\quad \mathfrak{j}\left[v_{\rm N}\right] +
[q]=0,\quad \mathfrak{j}\left[{v}_{\tau}\right]=H_{\rm N}[{H}_{\tau}],\\[6pt]
H_{\rm N}[{v}_{\tau}]=\mathfrak{j}\left[{H}_{\tau}/\rho\right],\quad \mathfrak{j}\left[
e+\frac{1}{2}(|{H} |^2/\rho )\right] + \left[qv_{\rm N} -H_{\rm N}({H} ,{v} )\right] =0
\end{array}
\right.
\label{6}
\end{equation}
at each point of $\Gamma$, where $[g]=g^+|_{\Gamma}-g^-|_{\Gamma}$ denotes the jump of $g$, with $g^{\pm}:=g$ in the domains
\[
\Omega^{\pm}(t)=\{\pm (x_1- \varphi (t,x'))>0\},
\]
and
\[
\mathfrak{j}^{\pm}=\rho (v_{\rm N}^{\pm}-\partial_t\varphi),\quad v_{\rm N}^{\pm}=({v}^{\pm} ,{N}),\quad H_{\rm N}=({H}^{\pm} ,{N}),\quad {N}=(1,-\partial_2\varphi,-\partial_3\varphi ),
\]
\[
{v}^{\pm}_{\tau}=(v^{\pm}_{\tau _1},v^{\pm}_{\tau _2}),\quad {H}^{\pm}_{\tau}= (H^{\pm}_{\tau _1}, H^{\pm}_{\tau _2}),\quad v^{\pm}_{\tau _i}=({v}^{\pm} ,{\tau}_i),
\]
\[
H^{\pm}_{\tau
_i}=({H}^{\pm} ,{\tau}_i),\quad
{\tau}_1=(\partial_2\varphi,1,0),\quad
{\tau}_2=(\partial_3\varphi,0,1),\quad H_{\rm N}|_{\Gamma}:=H_{\rm N}^{\pm}|_{\Gamma};
\]
$\mathfrak{j}:=\mathfrak{j}^{\pm}|_{\Gamma}$ is the mass transfer flux across the discontinuity surface.
From the mathematical point of view, there are two types of strong discontinuities: shock waves and characteristic discontinuities. Following Lax \cite{Lax57}, characteristic discontinuities, which are characteristic free boundaries, are called contact discontinuities. For the Euler equations of gas dynamics contact discontinuities are indeed contact from the physical point of view, i.e., there is no flow across the discontinuity ($\mathfrak{j}=0$).
In MHD the situation with characteristic discontinuities is richer than in gas dynamics. Namely, besides shock waves ($\mathfrak{j}\neq 0$, $[\rho ]\neq 0$) there are three types of characteristic discontinuities \cite{BThand,LL}: tangential discontinuities or current-vortex
sheets ($\mathfrak{j} =0$, $H_{\rm N}|_{\Gamma}= 0$), Alfv\'{e}n or rotational discontinuities ($\mathfrak{j}\neq 0$, $[\rho ] = 0$), and contact discontinuities ($\mathfrak{j}=0$, $H_{\rm N}|_{\Gamma} \neq 0$).
Current-vortex sheets and MHD contact discontinuities are contact from the physical point of view, but Alfv\'{e}n discontinuities are not.
Strong discontinuities formally introduced for the MHD equations do not necessarily exist (at least, locally in time) as piecewise smooth solutions for the full range of admissible initial flow parameters. As is known (see \cite{BS,BThand,Maj,Met}), the fulfilment of the Kreiss-Lopatinski condition \cite{Kreiss} for the linearized constant coefficients problem for a planar discontinuity is necessary (but not sufficient in general) for the local-in-time existence of corresponding nonplanar discontinuities. The violation of the Kreiss-Lopatinski condition is equivalent to the ill-posedness of the linearized constant coefficient problem. This is the same as {\it Kelvin-Helmholtz instability}, and the corresponding planar discontinuity is called unstable or violently unstable.
The more restrictive condition is the uniform Kreiss-Lopatinski condition \cite{Kreiss} and it requires the nonexistence of not only unstable but also neutral modes. The corresponding planar discontinuity is called {\it uniformly stable}. The {uniform stability} condition satisfied at each point of the initial strong discontinuity usually implies its local-in-time existence. At least, this is so for gas dynamical shock waves \cite{BThand,Maj}, and more recently the same was proved for MHD shock waves \cite{MZ,Kwon}. However, {\it neutral (or weak) stability} sometimes also implies local-in-time existence. For example, for the isentropic Euler equations, Coulombel and Secchi \cite{CS2} have proved the local-in-time-existence of neutrally stable vortex sheets (in 2D) and neutrally stable shock waves.
In MHD there are two types of Lax shocks: fast and slow shock waves (see, e.g., \cite{LL}). A complete 2D stability analysis of fast MHD shock waves was carried out in \cite{T} for a polytropic gas equation of state. Taking into account the results of \cite{MZ,Kwon} extending the Kreiss--Majda theory \cite{Kreiss,Maj} to a class of hyperbolic symmetrizable systems with characteristics of variable multiplicities (this class contains the MHD system), uniformly stable fast shock waves found in \cite{T} exist locally in time. Regarding slow shock waves, some results about their stability can be found in \cite{BThand,Fil}.
Current-vortex sheets can be only neutrally stable and a sufficient condition for the neutral stability of planar compressible current-vortex sheets\footnote{For the well-posedness and stability analysis for {\it incompressible} current-vortex sheets we refer the reader to \cite{CMST,MoTraTre,Ticvs} and references therein.} was found in \cite{T05}. The local-in-time existence of solutions with a surface of a current-vortex sheet of the MHD equations was proved in \cite{T09}, provided that the sufficient stability from \cite{T05} holds at each point of the initial discontinuity.
By numerical testing of the Kreiss-Lopatinski condition the parameter domains of stability and violent instability of planar Alfv\'{e}n discontinuities were found in \cite{IT}. Alfv\'{e}n discontinuities are violently unstable for a wide range of flow parameters \cite{IT} but, as current-vortex sheets, they can be only neutrally stable. As was shown in \cite{IT}, the violation of uniform stability for stable Alfv\'{e}n discontinuities is the direct consequence of the fact that the symbol associated to the free boundary is {\it not elliptic}. It means that the boundary conditions cannot be resolved for the time-space gradient $\nabla_{t,x}\varphi=(\partial_t\varphi ,\partial_2\varphi ,\partial_3\varphi )$ of the front function.
In fact, for a general class of free boundary problems one can show that the non-ellipticity of the front symbol implies the existence of neutral modes associated with zeroing of this symbol. At the same time, neutral stability in the case of non-elliptic front symbol is, in some sense, weaker than usual neutral stability. As was noted in \cite{Tcpam}, this kind of neutral stability is, in general, not enough for the well-posedness of the linearized problem with variable coefficients (and, of course, of the original nonlinear problem). That is, this problem can be ill-posed even if the constant coefficients problem satisfies the (weak) Kreiss-Lopatinski condition.
The classical example of a free boundary problem with non-elliptic front symbol is the problem for the incompressible Euler equations with the vacuum boundary condition $p|_{\Gamma}=0$ on a free boundary $\Gamma (t)$ moving with the velocity of fluid particles (see \cite{Lind_incomp} and references therein). The front symbol is also not elliptic for the counterpart of this problem for the compressible Euler equations describing the motion of a compressible perfect liquid (with $\rho|_{\Gamma}>0$) in vacuum \cite{Lind,Tcpam}.
One can easily check that the linearization of the vacuum problem (for compressible or incompressible liquid) associated with a planar boundary always satisfies the Kreiss-Lopatinski condition, i.e., Kelvin-Helmholtz instability does not occur. Nevertheless, the local-in-time existence in Sobolev spaces was managed to be proved in \cite{Lind_incomp,Lind,Tcpam} for the nonlinear free boundary problem only under the physical assumption
\begin{equation}
\frac{\partial p}{\partial N}\leq -\epsilon <0\quad \mbox{on}\ \Gamma (0)
\label{vacc}
\end{equation}
on the normal derivative of the pressure on the initial free boundary, where $N$ is the outward normal to $\Gamma$ and $\epsilon$ is a fixed constant. As is known, the violation of \eqref{vacc} is associated with {\it Rayleigh-Taylor instability} occurring on the level of variable coefficients of the linearized problem. Moreover, for the case of incompressible liquid Ebin \cite{Ebin}
showed the ill-posedness in Sobolev spaces of the nonlinear problem when the physical condition \eqref{vacc} is not satisfied.
In this paper, we are interested in MHD contact discontinuities and for them the front symbol is also always not elliptic. For contact discontinuities, in view of the requirements $\mathfrak{j}=0$ and $H_{\rm N}|_{\Gamma} \neq 0$, the Rankine-Hugoniot conditions \eqref{6} give the boundary conditions
\begin{equation}
[p]=0,\quad [v]=0,\quad [H]=0,\quad \partial_t\varphi =v^+_N\quad \mbox{on}\ \Gamma (t)
\label{bcond}
\end{equation}
which indeed cannot be resolved for $\nabla_{t,x}\varphi$.
At the same time, the density and the entropy may undergo any jump: $[\rho]\neq 0$, $[S]\neq 0$.
\begin{remark}
{\rm
Observe that the continuity of the magnetic field in \eqref{bcond} is equivalent to
\begin{equation}
[H_N]=0,\quad [H_{\tau}]=0\quad \mbox{on}\ \Gamma (t).
\label{bcondH}
\end{equation}
One can show that the first condition in \eqref{bcondH} coming from the constraint equation \eqref{2} is not a real boundary condition and must be regarded as a restriction (boundary constraint) on the initial data (see Proposition \ref{p1} for more details).
}
\label{r1}
\end{remark}
Our final goal is to find conditions on the initial data
\begin{equation}
{U}^{\pm} (0,{x})={U}_0^{\pm}({x}),\quad {x}\in \Omega^{\pm} (0),\quad \varphi (0,{x}')=\varphi _0({x}'),\quad {x}'\in\mathbb{R}^2,\label{indat}
\end{equation}
providing the existence and uniqueness in Sobolev spaces on some time interval $[0,T]$ of a solution $(U^{\pm},\varphi )$ to the free boundary problem \eqref{1}, \eqref{bcond}, \eqref{indat}, where $U^{\pm}:=U$ in $\Omega^{\pm}(t)$, and $U^{\pm}$ is smooth in $\Omega^{\pm}(t)$. Because of the general properties of hyperbolic conservation laws it is natural to expect only the local-in-time existence of solutions with a surface of contact discontinuity. Therefore, from the mathematical point of view, the question on its nonlinear Lyapunov's stability has no sense.
At the same time, the study of the linearized stability of contact discontinuities is not only a necessary step towards the proof of their local-in-time existence but is also of independent interest in connection with various astrophysical applications. As is noted in \cite{Goed}, the boundary conditions \eqref{bcond} are most typical for astrophysical plasmas. Contact discontinuities are usually observed in the solar wind, behind astrophysical shock waves bounding supernova remnants or due to the interaction of multiple shock waves driven by fast coronal mass ejections.
The absence of Kelvin-Helmholtz instability for contact discontinuities follows from a conserved energy integral \cite{BThand} which can be trivially obtained for the constant coefficients problem resulting from the linearization of problem \eqref{1}, \eqref{bcond}, \eqref{indat} about its piecewise constant solution associated with a planar discontinuity. That is, planar MHD contact discontinuities are always neutrally stable (see Section \ref{sec:3} for more details).\footnote{The surprising thing is that neutral stability trivially shown by the energy method cannot be proved by the usual spectral analysis due to the impossibility to solve analytically (in the general case) the dispersion relation for the magnetoacoustics system (see Section \ref{sec:3}).} But, the non-ellipticity of the front symbol causes principal difficulties if we want to extend the a priori $L^2$ estimate \cite{BThand} easily deduced for the constant coefficients problem to the linearized problem with variable coefficients.
In this paper, under a suitable stability condition for the unperturbed flow (see just below) we manage to prove the well-posedness in Sobolev spaces of the linearized variable coefficients problem for contact discontinuities in 2D planar MHD (see \eqref{4}). It is amazing that the classical {\it Rayleigh-Taylor sign condition}
\begin{equation}
\left[\frac{\partial p}{\partial N}\right]\leq -\epsilon <0
\label{RT}
\end{equation}
on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed nonplanar contact discontinuity naturally appears in our energy method as the condition sufficient for the well-posedness of the linearized problem. It is interesting to note that, unlike the condition $[\partial q/\partial N] <0$ considered in \cite{Tjde} for the plasma-vacuum interface problem, the magnetic field does not enter \eqref{RT}. That is, for MHD contact discontinuities condition \eqref{RT} appears in its classical (purely hydrodynamical) form as a condition for the pressure $p$ but not for the total pressure $q=p+\frac{1}{2}|{H} |^2$.
The well-posedness result of our paper is a necessary step to prove the local-in-time existence of MHD contact discontinuities provided that the Rayleigh-Taylor sign condition is satisfied at each point of the initial discontinuity. Since in the basic a priori estimate obtained in this paper for the variable coefficients linearized problem we have a {\it loss of derivatives} from the source terms to the solution, we plan to prove the existence of solutions to the original nonlinear problem by a suitable Nash-Moser-type iteration scheme. We suppose that the scheme of the proof will be really similar to that in \cite{CS2,T09,Tcpam}. We do not see any principal difficulties in this direction but postpone nonlinear analysis to a forthcoming paper.
The extension of our result to the general 3D case is still an open difficult problem. It is worth noting that the Rayleigh-Taylor instability of contact discontinuities was earlier detected in numerical MHD simulations of astrophysical plasmas as fingers near the contact discontinuity in the contour maps of density (see \cite{FangZhang} and references therein). Our hypothesis is that Rayleigh-Taylor instability is associated with the violation of the classical condition \eqref{RT}. The proof of this hypothesis is also an interesting open problem for future research.
The rest of the paper is organized as follows. In Section \ref{sec:2}, we reduce the free boundary problem \eqref{1}, \eqref{bcond}, \eqref{indat} to an initial-boundary value problem in a fixed domain and discuss properties of the reduced problem. In Section \ref{sec:3}, we obtain the linearized problem and formulate the main result for it which is Theorem \ref{t1} about its well-posedness in Sobolev spaces under the Rayleigh-Taylor sign condition for the 2D planar unperturbed flow. Moreover, in Section \ref{sec:3} we briefly discuss properties of the constant coefficients linearized problem for a planar contact discontinuity. In Section \ref{sec:4} we derive the energy a priori estimate for the variable coefficients linearized problem and in Section \ref{sec:exist} prove the existence of solutions of this problem.
\section{Reduced nonlinear problem in a fixed domain}
\label{sec:2}
The function $\varphi (t,x')$ determining the contact discontinuity surface $\Gamma$ is one of the unknowns of
the free boundary problem \eqref{1}/\eqref{3Dsys}, \eqref{bcond}, \eqref{indat}. To reduce this problem to that in a fixed domain we straighten the interface $\Gamma$ by using the same simplest change of independent variables as in \cite{T09,Tcpam}. That is, the unknowns $U^+$ and $U^-$ being smooth in $\Omega^{\pm}(t)$ are replaced by the vector-functions
\begin{equation}
\widetilde{U}^{\pm}(t,x ):= {U}^{\pm}(t,\Phi^{\pm} (t,x),x')
\label{change}
\end{equation}
which are smooth in the half-space $\mathbb{R}^3_+=\{x_1>0,\ x'\in \mathbb{R}^2\}$,
where
\begin{equation}
\Phi^{\pm}(t,x ):= \pm x_1+\Psi^{\pm}(t,x ),\quad \Psi^{\pm}(t,x ):= \chi (\pm x_1)\varphi (t,x'),
\label{change2}
\end{equation}
and $\chi\in C^{\infty}_0(\mathbb{R})$ equals to 1 on $[-1,1]$, and $\|\chi'\|_{L_{\infty}(\mathbb{R})}<1/2$. Here, as in \cite{Met}, we use the cut-off function $\chi$ to avoid assumptions about compact support of the initial data in our (future) nonlinear existence theorem. Alternatively, we could use the same change of variables as in \cite{CMST,MoTraTre-vacuum,SecTra,ST} inspired by Lannes \cite{Lannes} (see Remark \ref{r3} below).
\begin{remark}
{\rm
The above change of variable is admissible if $\partial_1\Phi^{\pm}\neq 0$. The latter is guaranteed, namely, the inequalities $\partial_1\Phi^+> 0$ and $\partial_1\Phi^-< 0$ are fulfilled, if we consider solutions for which $\|\varphi\|_{L_{\infty}([0,T]\times\mathbb{R}^2)}\leq 1$. This holds if,
without loss of generality, we consider the initial data satisfying $\|\varphi_0\|_{L_{\infty}(\mathbb{R}^2)}\leq 1/2$, and the time $T$ in our existence theorem is sufficiently small.}
\label{r2}
\end{remark}
\begin{remark}
{\rm
In the change of variables used in \cite{CMST,MoTraTre-vacuum,SecTra,ST} the function $\Psi^{\pm}$ in \eqref{change2} is defined in a different way. An important point of this change of variables is the regularization of one half of derivative of the lifting function $\Psi^{\pm}$ with respect to $\varphi$. This property was crucial for deriving a priori estimates for the nonlinear problem for incompressible current-vortex sheets in \cite{CMST}. At the same time, this change of variables gave no advantages for the linear analysis for the plasma-vacuum interface problem in \cite{MoTraTre-vacuum,SecTra}, and it just gave a gain of one half of derivative in the subsequent nonlinear existence theorem in \cite{ST} proved by Nash-Moser iterations. This gain of one half of derivative in \cite{ST} is only a nonprincipal technical improvement, and we expect the same for MHD contact discontinuities. At least, for the linearized problem for contact discontinuities, it does not matter whether we define the function $\Psi^{\pm}$ as in \eqref{change2} or as in \cite{CMST,MoTraTre-vacuum,SecTra,ST}. Therefore, choosing now the change of variables \eqref{change}, \eqref{change2}, we postpone the final choice to the forthcoming nonlinear analysis.}
\label{r3}
\end{remark}
Dropping for convenience tildes in $\widetilde{U}^{\pm}$, we reduce \eqref{3Dsys}, \eqref{bcond}, \eqref{indat} to the initial-boundary value problem
\begin{equation}
\mathbb{L}(U^+,\Psi^+)=0,\quad \mathbb{L}(U^-,\Psi^-)=0\quad\mbox{in}\ [0,T]\times \mathbb{R}^3_+,\label{11}
\end{equation}
\begin{equation}
\mathbb{B}(U^+,U^-,\varphi )=0\quad\mbox{on}\ [0,T]\times\{x_1=0\}\times\mathbb{R}^{2},\label{12}
\end{equation}
\begin{equation}
U^+|_{t=0}=U^+_0,\quad U^-|_{t=0}=U^-_0\quad\mbox{in}\ \mathbb{R}^3_+,
\qquad \varphi |_{t=0}=\varphi_0\quad \mbox{in}\ \mathbb{R}^{2},\label{13}
\end{equation}
where $\mathbb{L}(U,\Psi)=L(U,\Psi)U$,
\[
L(U,\Psi)=A_0(U)\partial_t +\widetilde{A}_1(U,\Psi)\partial_1+A_2(U )\partial_2+A_3(U )\partial_3,
\]
\[
\widetilde{A}_1(U^{\pm},\Psi^{\pm})=\frac{1}{\partial_1\Phi^{\pm}}\Bigl(
A_1(U^{\pm})-A_0(U^{\pm})\partial_t\Psi^{\pm}-\sum_{k=2}^3A_k(U^{\pm})\partial_k\Psi^{\pm}\Bigr)
\]
($\partial_1\Phi^{\pm}=\pm 1 +\partial_1\Psi^{\pm}$), and (\ref{12}) is the compact form of the boundary conditions
\begin{equation}
[p]=0,\quad [v]=0,\quad [H_{\tau}]=0,\quad \partial_t\varphi-v_{N}^+|_{x_1=0}=0,
\label{12'}
\end{equation}
with $[g]:=g^+|_{x_1=0}-g^-|_{x_1=0}$ for any pair of values $g^+$ and $g^-$. Moreover, recall that, according to the definition of contact discontinuity, $H^{\pm}_N|_{x_1=0}\neq 0$. Here
\[
v_{N}^{\pm}=v_1^{\pm}- v_2^{\pm}\partial_2\Psi^{\pm}- v_3^{\pm}\partial_3\Psi^{\pm},\quad H_{N}^{\pm}=H_1^{\pm}-H_2^{\pm}\partial_2\Psi^{\pm}-H_3^{\pm}\partial_3\Psi^{\pm},
\]
\[
{H}^{\pm}_{\tau}= (H^{\pm}_{\tau _1}, H^{\pm}_{\tau _2}),\quad H^{\pm}_{\tau
_i}=H_1^\pm \partial_{i+1}\Psi^{\pm}+H_{i+1}^{\pm}, \quad i=1,2.
\]
There appear two natural questions. The first one: Why have the boundary condition $[H_N]=0$ not been included in \eqref{12} (see Remark \ref{r1})? And the second question: Why are systems (\ref{1}) and
\eqref{3Dsys} equivalent on solutions with a surface of contact discontinuity, i.e., why is system (\ref{1}) in the straightened variables equivalent to (\ref{11})? The answer to these questions is given by the following proposition.
\begin{proposition}
Let the initial data \eqref{13} satisfy
\begin{equation}
{\rm div}\, h^+=0,\quad {\rm div}\, h^-=0
\label{14}
\end{equation}
and the boundary condition
\begin{equation}
[H_{N}]=0,
\label{15}
\end{equation}
where $h^{\pm}=(H_{N}^{\pm},H_2^{\pm}\partial_1\Phi^{\pm},H_3^{\pm}\partial_1\Phi^{\pm})$. If problem \eqref{11}--\eqref{13} has a sufficiently smooth solution, then this solution satisfies \eqref{14} and \eqref{15} for all $t\in [0,T]$. The same is true for solutions with a surface of contact discontinuity of system \eqref{1}.
\label{p1}
\end{proposition}
\begin{proof}
The proof that equations \eqref{14} are satisfied for all $t\in [0,T]$ if they are true at $t=0$ is absolutely the same as in \cite{T09}. Regarding the boundary condition \eqref{15}, again, exactly as in Appendix A in \cite{T09}, we consider the equations for $H^{\pm}$ contained in \eqref{11} on the boundary $x_1=0$, we use the last condition in \eqref{12'} and its counterpart for $v^-$ to obtain
\[
\partial_t H_{N}^{\pm} +v_2^{\pm}\partial_2H_{N}^{\pm}+v_3^{\pm}\partial_3H_{N}^{\pm}+
\left(\partial_2v_2^{\pm}+\partial_3v_3^{\pm}\right) H_{N}^{\pm}=0\quad \mbox{on}\ x_1=0.
\]
In view of $[v]=0$, the last equations imply
\[
\partial_t [H_{N}] +v_2^+\partial_2[H_{N}]+v_3^+\partial_3[H_{N}]+ \left(\partial_2v_2^++\partial_3v_3^+\right) [H_{N}]=0\quad \mbox{on}\ x_1=0.
\]
Using the standard method of characteristic curves, we conclude that (\ref{15}) is fulfilled for all $t\in [0,T]$ if it is satisfied for $t=0$.
\end{proof}
Equations (\ref{14}) are just the divergence constraint (\ref{2}) on either side of the straightened front. Using (\ref{14}), we can prove that system (\ref{1}) in the straightened variables is equivalent to (\ref{11}).
Concerning the boundary condition \eqref{15}, we must regard it as the restriction on the initial data (\ref{13}). Otherwise, the hyperbolic problem (\ref{11}), (\ref{12}), (\ref{15}) does not have a correct number of boundary conditions.
Indeed, the boundary matrix reads
\[
A_{\nu}={\rm diag}\, \bigl(\widetilde{A}_1({\bf U}^+,\Psi^+),\widetilde{A}_1({\bf U}^-,\Psi^-)\bigr),
\]
where
\[
\widetilde{A}_1(U^{\pm},\Psi^{\pm}) = \frac{1}{\partial_1 \Phi^{\pm}}\begin{pmatrix} \frac{w_1^{\pm}}{\rho^{\pm}(c^{\pm})^2} & N^{\pm} & 0& 0\\[6pt]
(N^{\pm})^T & \rho w_1^{\pm} I_3 & N^{\pm} \otimes H^{\pm} - h_1^{\pm} I_3 & 0^T\\[6pt]
0^T & H^{\pm} \otimes N^{\pm} - h_1^{\pm} I_3 & w_1^{\pm} I_3 & 0^T\\[3pt]
0& 0& 0 & w_1^{\pm}
\end{pmatrix},
\]
\[
w^{\pm}=u^{\pm}-(\partial_t\Psi^{\pm},0,0),\quad u^{\pm}=(v_N^{\pm},v_2^{\pm}\partial_1\Phi^{\pm},v_3^{\pm}\partial_1\Phi^{\pm}),
\]
$I_3$ is the unit matrix of order 3, $N^{\pm}=(1, -\partial_2\Psi^{\pm}, -\partial_3\Psi^{\pm})$, and $w_1^{\pm}$ and $h_1^{\pm}$ are the first components of the vectors $w^{\pm}$ and $h^{\pm}$ respectively, and $\otimes$ denotes the tensor product. In view of the last condition in \eqref{12'} and its counterpart for $v^-$, we have $w_1^{\pm}|_{x_1=0}=0$ and
\begin{equation}
\widetilde{A}_1(U^{\pm},\Psi^{\pm})|_{x_1=0} = \pm \begin{pmatrix} 0 & N & 0& 0\\[6pt]
N^T & O_3 & N \otimes H^{\pm} - h_1^{\pm} I_3 & 0^T\\[6pt]
0^T & H^{\pm} \otimes N - h_1^{\pm} I_3 & O_3 & 0^T\\[3pt]
0& 0& 0 & 0
\end{pmatrix}_{|x_1=0},
\label{A1tilde}
\end{equation}
where $O_3$ is the zero matrix of order 3. For the matrix
\[
\widehat{\mathcal{A}}^{\pm}=\widetilde{A}_1(\widehat{U}^{\pm},\hat{\Psi}^{\pm})|_{x_1=0}
\]
calculated on a certain background $(\widehat{U}^{\pm},\hat{\varphi} )$ we have
\begin{equation}
(\widehat{\mathcal{A}}^{\pm}U^{\pm},U^{\pm})=((J^{\pm})^T\widehat{\mathcal{A}}^{\pm}J^{\pm}W^{\pm},W^{\pm})=(\widehat{\mathcal{B}}^{\pm}W^{\pm},W^{\pm}),
\label{boundmatr}
\end{equation}
where $\widehat{\mathcal{B}}^{\pm}=\mathcal{B}^{\pm}(\widehat{U}^{\pm}_{|x_1=0},\hat{\varphi} )$,
\[
W^{\pm}= (\check{q}^{\pm},\check{v}^{\pm}_N,v_2^{\pm},v_3^{\pm},\check{H}^{\pm}_N,H^{\pm}_2,H^{\pm}_3,S^{\pm}),\quad \check{q}^{\pm}=p^{\pm}+(\widehat{H}^{\pm},H^{\pm}),
\]
\[
\check{v}^{\pm}_N=v_1^{\pm}-v_2^{\pm}\partial_2\hat{\varphi}-v_3^{\pm}\partial_3\hat{\varphi},\quad
\check{H}^{\pm}_N=H_1^{\pm}-H_2^{\pm}\partial_2\hat{\varphi}-H_3^{\pm}\partial_3\hat{\varphi},
\]
and $U^{\pm}=J^{\pm}W^{\pm}$, with $\det J^{\pm}\neq 0$.
After calculations analogous to those in \cite{SecTra} we find
\begin{equation}
\mathcal{B}^{\pm}=\pm\begin{pmatrix}
0 & e_1 & 0 & 0 \\[3pt]
e_1^T &O_3& -h_1^{\pm}a_0 & 0^T\\[3pt]
0^T & -h_1^{\pm}a_0 & O_3& 0^T\\[3pt]
0 & 0 &0 & 0
\end{pmatrix}_{|x_1=0},
\label{boundmatr'}
\end{equation}
where $e_1= (1,0,0)$ and $a_0=a_0(\varphi )$ is the symmetric positive definite matrix\footnote{The matrix $a_0$ is the matrix $\hat{a}_0$ from \cite{SecTra} calculated on the boundary $x_1=0$.}
\[
a_0=\begin{pmatrix}
1 & \partial_2\varphi & \partial_3\varphi \\[3pt]
\partial_2\varphi & 1+(\partial_2\varphi)^2 &\partial_2\varphi\partial_3\varphi\\[3pt]
\partial_3\varphi &\partial_2\varphi\partial_3\varphi & 1+(\partial_3\varphi)^2
\end{pmatrix}.
\]
Since $h_1^{\pm}|_{x_1=0}\neq 0$, the matrix $\mathcal{B}^{\pm}$ has tree positive, three negative and two zero eigenvalues. Therefore, the boundary matrix $A_{\nu}$ on the boundary $x_1=0$ has six positive, six negative and four zero eigenvalues. That is, the boundary $x_1=0$ is {\it characteristic}, and since one of the boundary conditions is needed for determining the function $\varphi $, the correct number of boundary conditions is seven (that is the case in \eqref{12'}).
\begin{remark}
{\rm
In fact, the signature of the symmetric matrix $A_{\nu}|_{x_1=0}$ associated with a nonplanar front $x_1=\varphi (t,x')$ is determined from the well-known formulae for the eigenvalues of the matrix $A_1$ associated with the planar front $x_1=0$ and calculated through the Alfv\'{e}n and (fast and slow) magnetosonic velocities (see, e.g., \cite{LL}). In this sense, the calculations above were not really necessary. At the same time, for the linearized problem we will need formulae \eqref{boundmatr} and \eqref{boundmatr'} for deriving a priori estimates by the energy method.}
\label{r4}
\end{remark}
In the next section, we linearize problem \eqref{11}--\eqref{13} around a given basic state (``unperturbed flow") and we will have to make reasonable assumptions about it. Since in our future nonlinear analysis by Nash-Moser iterations the basic state playing the role of an intermediate state $(U^{\pm}_{n+1/2},\varphi_{n+1/2})$ (see \cite{CS2,ST,T09,Tcpam}) should finally converge to a solution of the nonlinear problem, we need to know a certain a priori information about solutions of problem \eqref{11}--\eqref{13}. This information is contained in the following proposition.
\begin{proposition}
Assume that problem \eqref{11}--\eqref{13} (with the initial data satisfying \eqref{14} and \eqref{15}) has a sufficiently smooth solution $(U^{\pm},\varphi)$ on a time interval $[0,T]$. Assume also that the plasma obeys the state equation \eqref{polgas} of a
polytropic gas. Then the normal derivatives $\partial_1U^{\pm}$ satisfy the jump conditions
\begin{equation}
[\partial_1H_{N}]=0
\label{jc1}
\end{equation}
and
\begin{equation}
[\partial_1v]=0
\label{jc2}
\end{equation}
\label{p2}
for all $t\in [0,T]$, where due to the fact that we have transformed the domains $\Omega^\pm (t)$ into the same half-space $\mathbb{R}^3_+$ (but not into the different half-spaces $\mathbb{R}^3_+$ and $\mathbb{R}^3_-$) the jump of a normal derivative is defined as follows
\begin{equation}
[\partial_1a]:=\partial_1a^+_{|x_1=0}+\partial_1a^-_{|x_1=0}.
\label{norm_jump}
\end{equation}
\end{proposition}
\begin{proof}
It follows from \eqref{14} that ${\rm div}\, h^+_{|x_1=0}+{\rm div}\, h^-_{|x_1=0}=0$. Then, since $[H]=0$, we get \eqref{jc1}. For a polytropic gas, restricting equations \eqref{11} from $\mathbb{R}^3_+$ to the boundary $x_1=0$ and using the last condition in \eqref{12'} and its counterpart for $v^-$, we obtain
\[
\frac{1}{\gamma p^\pm }\partial^{\pm}_0p^\pm \pm {\rm div}\,u^\pm=0\quad\mbox{on}\ x_1=0,
\]
where $\partial^{\pm}_0:=\partial_t+v_2^\pm\partial_2 +v_3^\pm\partial_3$.
Passing to the jump and using the continuity of the pressure and the velocity, we get ${\rm div}\, u^+_{|x_1=0}+{\rm div}\, u^-_{|x_1=0}=0$, and then
\begin{equation}
[\partial_1v_N]=0.
\label{vn}
\end{equation}
Again considering equations \eqref{11} on the boundary $x_1=0$ and passing to the jump, we obtain
\[
[\partial_0 H]-H^+_N[\partial_1 v] -H^+_2[\partial_2 v]-H^+_3[\partial_3 v]+H^+({\rm div}\, u^+ +{\rm div}\, u^-)=0\quad\mbox{on}\ x_1=0,
\]
i.e.,
\[
H^+_N[\partial_1 v]=H^+({\rm div}\, u^+ +{\rm div}\, u^-) =H^+[\partial_1v_N]\quad\mbox{on}\ x_1=0.
\]
In view of \eqref{vn} and the condition $H_N^+|_{x_1=0}\neq 0$, this gives \eqref{jc2}.
\end{proof}
\section{Linearized problem and main result}
\label{sec:3}
\subsection{Basic state}
We first describe a basic state upon which we perform linearization. Let
\begin{equation}
(\widehat{U}^+(t,x ),\widehat{U}^-(t,x ),\hat{\varphi}(t,{x}'))
\label{a21}
\end{equation}
be a given sufficiently smooth vector-function with $\widehat{U}^{\pm}=(\hat{p}^{\pm},\hat{v}^{\pm},\widehat{H}^{\pm},\widehat{S}^{\pm})$ and
\begin{equation}
\|\widehat{U}^+\|_{W^2_{\infty}(\Omega_T)}+
\|\widehat{U}^-\|_{W^2_{\infty}(\Omega_T)}
+\|\hat{\varphi}\|_{W^3_{\infty}(\partial\Omega_T)} \leq K,
\label{a22}
\end{equation}
where $K>0$ is a constant and
\[
\Omega_T:= (-\infty, T]\times\mathbb{R}^3_+,\quad \partial\Omega_T:=(-\infty ,T]\times\{x_1=0\}\times\mathbb{R}^{2}.
\]
If the basic state \eqref{a21} upon which we shall linearize problem \eqref{11}--\eqref{13} is a solution of this problem (its existence should be proved), then it is natural to call it unperturbed flow. The trivial example of the unperturbed flow is the constant solution $(\overline{U}^+,\overline{U}^-,0)$ associated with the planar contact discontinuity $x_1=0$, where $\overline{U}^{\pm}\in \mathbb{R}^{8}$ are constant vectors.
Considering from now on the case of a polytropic gas, we assume that the basic state defined in ${\Omega_T}$ satisfies there the hyperbolicity condition \eqref{5},
\begin{equation}
\rho (\hat{p}^{\pm},\widehat{S}^{\pm})\geq \bar{\rho}_0 >0,\quad \hat{p}^{\pm} \geq \bar{p}_0 >0 \label{a5}
\end{equation}
(with some fixed constants $\bar{\rho}_0$ and $\bar{p}_0$), the boundary conditions \eqref{12'},
\begin{equation}
[\hat{p}]=0,\quad [\hat{v}]=0,\quad [\widehat{H}_{\tau}]=0, \quad \partial_t\hat{\varphi}-\hat{v}_{N}^+|_{x_1=0}=0,
\label{a12'}
\end{equation}
the condition
\begin{equation}
|\widehat{H}_N^{\pm}|_{x_1=0}|\geq \bar{\kappa}_0 >0
\label{cdass}
\end{equation}
(with some fixed constant $\bar{\kappa}_0$), the equations for $H^{\pm}$ contained in \eqref{11},
\begin{equation}
\partial_t\widehat{H}^{\pm}+\frac{1}{\partial_1\widehat{\Phi}^{\pm}}\left\{ (\hat{w}^{\pm} ,\nabla )
\widehat{H}^{\pm} - (\hat{h}^{\pm} ,\nabla ) \hat{v}^{\pm} + \widehat{H}^{\pm}{\rm div}\,\hat{u}^{\pm}\right\} =0 ,
\label{b21}
\end{equation}
the divergence constraints \eqref{14} and the boundary constraint \eqref{15} for $t\leq 0$,
\begin{equation}
{\rm div}\, \hat{h}^{\pm}|_{t\leq 0}=0,\quad [\widehat{H}_N]|_{t\leq 0}=0,
\label{b14}
\end{equation}
and the jump condition \eqref{vn},
\begin{equation}
[\partial_1\hat{v}_N]=0,
\label{avn}
\end{equation}
where, from now on, the jump of the normal derivatives has to be intended as in \eqref{norm_jump} and
where all of the ``hat'' values are determined like corresponding values for $(U^\pm, \varphi)$, e.g.,
\[
\widehat{\Phi}^{\pm}(t,x )=\pm x_1 +\widehat{\Psi}^{\pm}(t,x ),\quad
\widehat{\Psi}^{\pm}(t,x )=\chi(\pm x_1)\hat{\varphi}(t,x'),
\]
\[
\hat{v}_{N}^{\pm}=\hat{v}_1^{\pm}- \hat{v}_2^{\pm}\partial_2\widehat{\Psi}^{\pm}- \hat{v}_3^{\pm}\partial_3\widehat{\Psi}^{\pm},\quad
\widehat{H}_{N}^{\pm}=\widehat{H}_1^{\pm}- \widehat{H}_2^{\pm}\partial_2\widehat{\Psi}^{\pm}- \widehat{H}_3^{\pm}\partial_3\widehat{\Psi}^{\pm},
\]
\[
\widehat{H}^{\pm}_{\tau}= (\widehat H^{\pm}_{\tau _1}, \widehat H^{\pm}_{\tau _2}),\quad \widehat H^{\pm}_{\tau
_i}=\widehat H_1^\pm \partial_{i+1}\Psi^{\pm}+\widehat H^{\pm}_{i+1}, \quad i=1,2.
\]
Moreover, without loss of generality we assume that $\|\hat{\varphi}\|_{L_{\infty}(\partial\Omega_T)}<1$ (see Remark \ref{r2}). This implies
\[
\partial_1\widehat{\Phi}^+\geq 1/2,\quad \partial_1\widehat{\Phi}^-\leq - 1/2.
\]
Note that \eqref{a22} yields
\[
\|\widehat{W}\|_{W^2_{\infty}(\Omega_T)} \leq C(K),
\]
where $\widehat{W}:=(\widehat{U}^+,\widehat{U}^-,\nabla_{t,x}\widehat{\Psi}^+,
\nabla_{t,x}\widehat{\Psi}^-)$, $\nabla_{t,x}=(\partial_t, \nabla )$, and $C=C(K)>0$ is a constant depending on $K$.
As in Proposition \ref{p1}, equations \eqref{b21} and constraints \eqref{b14} imply
\begin{equation}
{\rm div}\, \hat{h}^+=0, \quad {\rm div}\, \hat{h}^-=0
\label{b14'}
\end{equation}
and
\begin{equation}
[\widehat{H}_N]=0
\label{b15}
\end{equation}
for all $t\in (-\infty ,T]$. Then, as in the proof of Proposition \ref{p2}, using \eqref{a12'}, \eqref{avn} and \eqref{b15}, from \eqref{b21} and \eqref{b14'} we deduce for the basic state the jump conditions \eqref{jc1} and \eqref{jc2},
\begin{equation}
[\partial_1\widehat{H}_{N}]=0,\quad [\partial_1\hat{v}]=0.
\label{jc1'}
\end{equation}
\begin{remark}
{\rm
For the linearized problem we will need equations associated to the divergence constraints \eqref{14}. However, exactly as in \cite{T09}, to deduce them it is not enough that these constraints are satisfied by the basic state (\ref{a21}) (see \eqref{b14'}) and we need actually that the equations for $H^\pm$ themselves contained in \eqref{11} are fulfilled for (\ref{a21}), i.e., assumption \eqref{b21} holds.}
\label{r5}
\end{remark}
\begin{remark}
{\rm
Assumptions \eqref{a5}--\eqref{avn} are nonlinear constraints on the basic state which are automatically satisfied if the basic state is an exact solution of problem \eqref{11}--\eqref{13} (unperturbed flow). We will really need them while deriving a priori estimates for the linearized problem. In the forthcoming nonlinear analysis we plan to use the Nash-Moser method. As in \cite{CS2,T09,Tcpam}, the Nash-Moser procedure will be not completely standard. Namely, at each $n$th Nash-Moser iteration step we will have to construct an intermediate state $(U^{\pm}_{n+1/2},\varphi_{n+1/2})$ satisfying constraints \eqref{a5}--\eqref{avn}. Without assumption \eqref{avn} such an intermediate state can be constructed in exactly the same manner as in \cite{T09}. Assumption \eqref{avn} does not however cause additional difficulties and we postpone corresponding arguments to the nonlinear analysis.}
\label{r6}
\end{remark}
\subsection{Linearized problem}
The linearized equations for (\ref{11}), (\ref{12}) read:
\[
\mathbb{L}'(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})(\delta U^{\pm},\delta\Psi^{\pm}):=
\frac{d}{d\varepsilon}\mathbb{L}(U_{\varepsilon}^{\pm},\Psi_{\varepsilon}^{\pm})|_{\varepsilon =0}={f}^{\pm}
\quad \mbox{in}\ \Omega_T,
\]
\[
\mathbb{B}'(\widehat{U}^+,\widehat{U}^-,\hat{\varphi})(\delta U^+,\delta U^-,\delta \varphi):=
\frac{d}{d\varepsilon}\mathbb{B}(U_{\varepsilon}^+,U_{\varepsilon}^-,\varphi_{\varepsilon})|_{\varepsilon =0}={g}
\quad \mbox{on}\ \partial\Omega_T
\]
where $U_{\varepsilon}^{\pm}=\widehat{U}^{\pm}+ \varepsilon\,\delta U^{\pm}$,
$\varphi_{\varepsilon}=\hat{\varphi}+ \varepsilon\,\delta \varphi$, and
\[
\Psi_{\varepsilon}^{\pm}(t,{x} ):=\chi (\pm x_1)\varphi _{\varepsilon}(t,{x}'),\quad
\Phi_{\varepsilon}^{\pm}(t,{x} ):=\pm x_1+\Psi_{\varepsilon}^{\pm}(t,{x} ),
\]
\[
\delta\Psi^{\pm}(t,{x} ):=\chi (\pm x_1)\delta \varphi (t,{x} ).
\]
Here, as usual, we introduce the source terms ${f}^{\pm}(t,{x} )=(f_1^{\pm}(t,{x} ),\ldots ,f_8^{\pm}(t,{x} ))$ and ${g}(t,{x}' )=(g_1(t,{x}' ),\ldots ,g_7(t,{x}' ))$ to make the interior equations and the boundary conditions inhomogeneous.
We easily compute the exact form of the linearized equations (below we drop $\delta$):
\[
\mathbb{L}'(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm})({U}^{\pm},\Psi^{\pm})\\
=
L(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm}){U}^{\pm} +{\cal C}(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm})
{U}^{\pm} - \bigl\{L(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm})\Psi^{\pm}\bigr\}\frac{\partial_1\widehat{U}{\pm}}{\partial_1\widehat{\Phi}^{\pm}},
\]
\[
\mathbb{B}'(\widehat{{U}}^+,\widehat{{U}}^-,\hat{\varphi})({U}^+,{U}^-,f)=
\left(
\begin{array}{c}
p^+-p^-\\[3pt]
v^+-v^-\\[3pt]
H_{\tau}^+-H_{\tau}^-\\[3pt]
\partial_t\varphi +\hat{v}_2^+\partial_2\varphi +\hat{v}_3^+\partial_3\varphi -v_{N}^+
\end{array}
\right),
\]
where
\[
v_{N}^{\pm}=
v_1^{\pm}-v_2^{\pm}\partial_2\widehat{\Psi}^\pm -v_3^{\pm}\partial_3\widehat{\Psi}^\pm , \quad
{H}^{\pm}_{\tau}= (H^{\pm}_{\tau _1}, H^{\pm}_{\tau _2}),
\]
\[
H^{\pm}_{\tau
_i}=H_1^\pm \partial_{i+1}\widehat{\Psi}^{\pm}+H^{\pm}_{i+1}, \quad i=1,2,
\]
and the matrix
${\cal C}(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm})$ is determined as follows:
\begin{multline*}
{\cal C}(\widehat{{U}}^{\pm},\widehat{\Psi}^{\pm}){Y}
= ({Y} ,\nabla_yA_0(\widehat{{U}}^{\pm} ))\partial_t\widehat{{U}}^{\pm}
+({Y} ,\nabla_y\widetilde{A}_1(\widehat{\bf U}^{\pm},\widehat{\Psi}^{\pm}))\partial_1\widehat{{U}}^{\pm}\\[6pt]
+ ({Y} ,\nabla_yA_2(\widehat{{U}}^{\pm} ))\partial_2\widehat{{U}}^{\pm}
+ ({Y} ,\nabla_yA_3(\widehat{{U}}^{\pm} ))\partial_3\widehat{{U}}^{\pm},
\end{multline*}
\[
({Y} ,\nabla_y A(\widehat{{U}}^{\pm})):=\sum_{i=1}^8y_i\left.\left(\frac{\partial A ({Y} )}{
\partial y_i}\right|_{{Y} =\widehat{{U}}^{\pm}}\right),\quad {Y} =(y_1,\ldots ,y_8).
\]
The differential operator $\mathbb{L}'(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})$ is a first order operator in
$\Psi^{\pm}$. This fact can give some trouble in obtaining a priori estimates for the linearized problem by the energy method. Following
\cite{Al}, we overcome this difficulty by introducing the ``good unknowns'':
\begin{equation}
\dot{U}^+:=U^+ -\frac{\Psi^+}{\partial_1\widehat{\Phi}^+}\,\partial_1\widehat{U}^+,\quad
\dot{U}^-:=U^- -\frac{\Psi^-}{\partial_1\widehat{\Phi}^-}\,\partial_1\widehat{U}^- .
\label{b23}
\end{equation}
Omitting detailed calculations, we rewrite the linearized interior equations in terms of the new unknowns \eqref{b23}:
\begin{equation}
L(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\dot{U}^{\pm} +{\cal C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})
\dot{U}^{\pm} - \frac{\Psi^{\pm}}{\partial_1\widehat{\Phi}^{\pm}}\,\partial_1\bigl\{\mathbb{L}
(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\bigr\}={f}^{\pm}.
\label{b24}
\end{equation}
Dropping as in \cite{Al,CS2,MoTraTre-vacuum,SecTra,ST,T09,Tcpam,Tjde} the zero-order terms in $\Psi^+$ and $\Psi^-$ in \eqref{b24},\footnote{In the future nonlinear analysis the dropped terms in \eqref{b24} should be considered as error terms at each Nash-Moser iteration step.} we write down the final form of our linearized problem for $(\dot{U}^+,\dot{U}^-,\varphi )$:
\begin{equation}
L(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\dot{U}^{\pm} +\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})
\dot{U}^{\pm} =f^{\pm}\qquad \mbox{in}\ \Omega_T,\label{b48}
\end{equation}
\begin{equation}
\left(
\begin{array}{c}
\dot{p}^+-\dot{p}^- + \varphi [\partial_1\hat{p}]\\[3pt]
\dot{v}^+-\dot{v}^-\\
\dot{H}_{\tau}^+-\dot{H}_{\tau}^-+ \varphi [\partial_1\widehat{H}_{\tau}]\\[3pt]
\partial_t\varphi +\hat{v}_2^+\partial_2\varphi +\hat{v}_3^+\partial_3\varphi -\dot{v}_{N}^+ - \varphi \partial_1\hat{v}_N^+
\end{array}
\right)=g \qquad \mbox{on}\ \partial\Omega_T,
\label{b50}
\end{equation}
\begin{equation}
(\dot{U}^+,\dot{U}^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{b51}
\end{equation}
where
\[
\dot{v}_{N}^{\pm}=
\dot{v}_1^{\pm}-\dot{v}_2^{\pm}\partial_2\widehat{\Psi}^\pm -\dot{v}_3^{\pm}\partial_3\widehat{\Psi}^\pm , \quad
\dot{H}^{\pm}_{\tau}= (\dot{H}^{\pm}_{\tau_1}, \dot{H}^{\pm}_{\tau_2}),
\]
\[
\dot{H}^{\pm}_{\tau_i}=\dot{H}_1^\pm \partial_{i+1}\widehat{\Psi}^{\pm}+\dot{H}^{\pm}_{i+1}, \quad i=1,2.
\]
We used the important condition $[\partial_1\hat{v}]=0$ for the basic state, cf. \eqref{jc1'}, while writing down the second line in the left-hand side of the boundary conditions \eqref{b50}. We assume that $f^{\pm}$ and $g$ vanish in the past and consider the case of zero initial data, which is the usual assumption.\footnote{The case of nonzero initial data is postponed to the nonlinear analysis (construction of a so-called approximate solution; see, e.g., \cite{CS2,T09}).}
From problem \eqref{b48}--\eqref{b51} we can deduce nonhomogeneous equations associated with the divergence constraints \eqref{14} and the ``redundant'' boundary condition \eqref{15} for the nonlinear problem. More precisely, we have the following.
\begin{proposition}
Let the basic state \eqref{a21} satisfies assumptions \eqref{a5}--\eqref{avn}.
Then solutions of problem \eqref{b48}--\eqref{b51} satisfy
\begin{equation}
{\rm div}\,\dot{h}^+=f_9^+,\quad {\rm div}\,\dot{h}^-=f_9^-\quad\mbox{in}\ \Omega_T,
\label{b43}
\end{equation}
\begin{equation}
\dot{H}_{N}^+-\dot{H}_N^-=g_8\quad\mbox{on}\ \partial\Omega_T.
\label{b44}
\end{equation}
Here
\[
\dot{h}^{\pm}=(\dot{H}_{N}^{\pm},\dot{H}_2^{\pm}\partial_1\widehat{\Phi}^{\pm},\dot{H}_3^{\pm}\partial_1\widehat{\Phi}^{\pm}),\quad
\dot{H}_{N}^{\pm}=\dot{H}_1^{\pm}-\dot{H}_2^{\pm}\partial_2\widehat{\Psi}^{\pm}-\dot{H}_3^{\pm}
\partial_3\widehat{\Psi}^{\pm}
\]
and the functions $f_9^{\pm}=f_9^{\pm}(t,x )$ and $g_8= g_8(t,x')$, which vanish in the past, are determined by the source terms and the basic state as solutions to the linear inhomogeneous equations
\begin{equation}
\partial_t a^{\pm}+ \frac{1}{\partial_1\widehat{\Phi}^{\pm}}\left\{ (\hat{w}^{\pm} ,\nabla a^{\pm}) + a^{\pm}\,{\rm div}\,\hat{u}^{\pm}\right\}={\mathcal F}^{\pm}\quad \mbox{in}\ \Omega_T,
\label{aa1}
\end{equation}
\begin{equation}
\partial_t g_8 +\partial_2(\hat{v}_2^+g_8)+\partial_3(\hat{v}_3^+g_8)
={\mathcal G}\quad \mbox{on}\ \partial\Omega_T,
\label{aa2}
\end{equation}
where $a^{\pm}=f_9^{\pm}/\partial_1\widehat{\Phi}^{\pm},\quad {\mathcal F}^{\pm}=({\rm div}\,{f}_{h}^{\pm})/\partial_1\widehat{\Phi}^{\pm}$,
\[
{f}_{h}^{\pm}=(f_{N}^{\pm} ,\partial_1\widehat{\Phi}^{\pm}f_6^{\pm},\partial_1\widehat{\Phi}^{\pm}f_7^{\pm}),\quad f_{N}^{\pm}=f_5^{\pm}-f_6^{\pm}\partial_2\widehat{\Psi}^{\pm}-
f_7^{\pm}\partial_3\widehat{\Psi}^{\pm},
\]
\[
{\mathcal G}=\left\{\left.[f_{N}]+\partial_2\bigl(\widehat{H}^+_2g_N\bigr)+\partial_3\bigl(\widehat{H}^+_3g_N\bigr)-\partial_2\bigl(\widehat{H}^+_Ng_3\bigr)
-\partial_3\bigl(\widehat{H}^+_Ng_4\bigr)\right\}\right|_{x_1=0},
\]
\[
[f_{N}]=f_{N}^+|_{x_1=0}-f_{N}^-|_{x_1=0},\quad g_N=g_2-g_3\partial_2\hat{\varphi}-g_4\partial_3\hat{\varphi}.
\]
\label{p3.1}
\end{proposition}
\begin{proof}
The proof of equations \eqref{b43} is absolutely the same as the corresponding proof in \cite{T09}. That is, we write down the equation for $\dot{H}^{\pm}$ contained in \eqref{b48}:
\begin{multline}
{\displaystyle
\partial_t\dot{H}^{\pm}+\frac{1}{\partial_1\widehat{\Phi}^{\pm}}\Bigl\{ (\hat{w}^{\pm} ,\nabla )
\dot{H}^{\pm} - (\hat{h}^{\pm} ,\nabla ) \dot{v}^{\pm} + \widehat{H}^{\pm}{\rm div}\,\dot{u}^{\pm}
}\\
+(\dot{u}^{\pm} ,\nabla )\widehat{H}^{\pm} - (\dot{h}^{\pm} ,\nabla ) \hat{v}^{\pm} + \dot{H}^{\pm}{\rm div}\,\hat{u}^{\pm} \Bigr\} ={f}^{\pm}_{H},
\label{aA3}
\end{multline}
where $\dot{u}^{\pm} =(\dot{v}_{N}^{\pm},\dot{v}_2^{\pm}\partial_1\widehat{\Phi}^{\pm}, \dot{v}_3^{\pm}\partial_1\widehat{\Phi}^{\pm})$ and ${f}^{\pm}_{H}=(f_5^{\pm},f_6^{\pm},f_7^{\pm})$. Then, using not only the divergence constraints \eqref{b14'} for the basic state but also equations \eqref{b21} for $\widehat{H}^{\pm}$ themselves, after long calculations, which are omitted, from (\ref{aA3}) we obtain that $f_9^{\pm}={\rm div}\,\dot{h}^{\pm}$ satisfy (\ref{aa1}) (with $a^{\pm}=f_9^{\pm}/\partial_1\widehat{\Phi}^{\pm}$).
Regarding the additional boundary condition \eqref{b44}, for its derivation we again use (\ref{aA3}) but now being considered at $x_1=0$. Namely, using the boundary conditions (\ref{b50}), system (\ref{b21}) and constraints (\ref{b14'}) at $x_1=0$ as well as \eqref{b15}, from (\ref{aA3}) being considered at $x_1=0$ we get equation (\ref{aa2}) for $g_8=[\dot{H}_N]$. That is, the proof of Proposition \ref{p3.1} is complete.
\end{proof}
\subsection{Constant coefficients linearized problem for a planar discontinuity}
\label{constcoeff}
If we ``freeze'' coefficients of problem \eqref{b48}--\eqref{b51}, drop the zero-order terms in \eqref{b48} and the zero-order terms in $\varphi$ in \eqref{b50}, assume that $\partial_t\hat{\varphi}=\partial_2\hat{\varphi}=\partial_3\hat{\varphi}=0$, and in the change of variables take $\chi\equiv 1$, then we obtain a constant coefficients linear problem which is the result of the linearization of the original nonlinear free boundary value problem \eqref{3Dsys}, \eqref{bcond}, \eqref{indat} about its {\it exact} piecewise constant solution
\[
U^{\pm}=\widehat{U}^{\pm}=(\hat{p},0,\hat{v}_2,\hat{v}_3,\widehat{H}_1,\widehat{H}_2,\widehat{H}_3,\widehat{S}^{\pm})={\rm const}\qquad \mbox{for}\ x_1\gtrless0
\]
for the planar contact discontinuity $x_1=0$. This exact piecewise constant solution satisfies \eqref{5} and \eqref{cdass}:\footnote{For the constant coefficients problem we do not restrict ourselves to the case of a polytropic gas and consider a general equation of state $\rho =\rho (p,S)$.}
\[
\hat{\rho}^{\pm}=\rho (\hat{p},\widehat{S}^{\pm})>0,\quad (\hat{c}^{\pm})^2= 1/\rho_p (\hat{p},\widehat{S}^{\pm})>0, \quad \widehat{H}_1\neq 0.
\]
Since we can perform the Galilean transformation
\[
\tilde{x}_2=x_2 -\hat{v}_2t,\quad \tilde{x}_3=x_3 -\hat{v}_3t,
\]
without loss of generality we may assume that $\hat{v}_2=\hat{v}_3=0$.
Taking into account the above, we have the following constant coefficients problem:
\begin{equation}
\widehat{A}_0^{\pm}\partial_t{U}^{\pm}\pm \widehat{A}_1\partial_1{U}^{\pm}+\sum_{j=2}^{3}\widehat{A}_j\partial_j{U}^{\pm}=f^{\pm} \qquad \mbox{in}\ \Omega_T,\label{99}
\end{equation}
\begin{equation}
\left(
\begin{array}{c}
{p}^+-{p}^- \\[3pt]
{v}^+-{v}^-\\
{H}_2^+-{H}_2^-\\[3pt]
{H}_3^+-{H}_3^-\\[3pt]
\partial_t\varphi -{v}_1^+
\end{array}
\right)=g \qquad \mbox{on}\ \partial\Omega_T,
\label{100}
\end{equation}
\begin{equation}
({U}^+,{U}^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{101}
\end{equation}
where $\widehat{A}_0^\pm= {\rm diag} (1/(\hat{\rho}^{\pm}(\hat{c}^{\pm})^2) ,\hat{\rho}^{\pm} ,\hat{\rho}^{\pm} ,\hat{\rho}^{\pm} ,1 ,1,1,1)$,
\[
\widehat{A}_1=\left( \begin{array}{cccccccc} 0&1&0&0&0&0&0&0\\[6pt]
1& 0&0&0&0&\widehat{H}_2&\widehat{H}_3&0\\
0&0& 0&0&0&-\widehat{H}_1&0&0\\ 0&0&0& 0&0&0&-\widehat{H}_1&0\\
0&0&0&0&0&0&0&0\\
0&\widehat{H}_2&-\widehat{H}_1&0&0&0&0&0\\
0&\widehat{H}_3&0&-\widehat{H}_1&0&0&0&0\\ 0&0&0&0&0&0&0&0\\
\end{array} \right) ,
\]
and the matrices $\widehat{A}_2$ and $\widehat{A}_3$ can be also easily written down.
We can reduce \eqref{99}--\eqref{101} to a problem with homogeneous boundary conditions, i.e., with $g=0$. To this end we use the classical argument suggesting to subtract from the solution a more regular function $\widetilde{U}^{\pm}\in H^1(\Omega_T)$ satisfying the boundary conditions. Then, the new unknown $U^{\natural\pm}={U}^\pm-\widetilde{U}^\pm$ satisfies problem \eqref{99}--\eqref{101} with $g=0$ and $f^\pm =F^\pm$, where
\[
F^\pm =f^\pm-\widehat{A}_0^{\pm}\partial_t\widetilde{U}^\pm \mp \widehat{A}_1\partial_1\widetilde{U}^\pm-\sum_{j=2}^{3}\widehat{A}_j\widetilde{U}^\pm
\]
and
\begin{multline}
\sum_{\pm}\|F^\pm\|_{L^2(\Omega_T)}\leq C \sum_{\pm}\left\{\|f^\pm \|_{L^{2}(\Omega_T)}+ \|\widetilde{U}^\pm \|_{H^{1}(\Omega_T)}\right\}\\
\leq C\biggl\{\sum_{\pm}\|f^\pm \|_{L^{2}(\Omega_T)}+ \|g\|_{H^{1/2}(\partial\Omega_T)} \biggr\}.
\label{nonhom_F}
\end{multline}
Here and later on $C$ is a constant that can change from line to line, and it may depend from another constants.
Since, by virtue of the homogenous boundary conditions \eqref{100} for $U^{\natural\pm}$, we have
\begin{multline*}
[(\widehat{A}_1U^{\natural},U^{\natural})] = (\widehat{A}_1U^{\natural +},U^{\natural +})|_{x_1=0} -(\widehat{A}_1U^{\natural -},U^{\natural -})|_{x_1=0} \\
=2[v_1^{\natural}(p^{\natural} +\widehat{H}_2H_2^{\natural}+\widehat{H}_3H_3^{\natural})] -2\widehat{H}_1[v_2^{\natural}H_2^{\natural}+v_3^{\natural}H_3^{\natural}]=0,
\end{multline*}
standard simple arguments of the energy method applied to systems \eqref{99} give the conserved integral \cite{BThand}
\[
\frac{d}{dt} \biggl(\sum_{\pm}\int_{\mathbb{R}^3_+}(\widehat{A}_0^\pm U^{\natural \pm},U^{\natural \pm}) dx \biggr) =0
\]
if $F^{\pm}=0$. For the general case when $F^{\pm}\neq 0$, we easily get the priori estimate
\[
\sum_{\pm}\|U^{\natural\pm} \|_{L^{2}(\Omega_T)}\leq C \sum_{\pm}\|F^\pm\|_{L^2(\Omega_T)},
\]
for $U^{\natural\pm}$ with no loss of derivatives from $F^{\pm}$. Taking into account \eqref{nonhom_F}, this estimate gives the a priori estimate
\begin{equation}
\sum_{\pm}\|U^{\pm} \|_{L^{2}(\Omega_T)}\leq C \biggl\{\sum_{\pm}\|f^\pm \|_{L^2(\Omega_T)}+ \|g\|_{H^{1/2}(\partial\Omega_T)}\biggr\}
\label{orig_U}
\end{equation}
for the original unknown $U^\pm$ with the loss of ``1/2 derivative'' from $g$ (regarding an a priori estimate for the front perturbation $\varphi$, see Remark \ref{r7} below).
It follows from estimate \eqref{orig_U} that Kelvin-Helmholtz instability never happens for planar contact MHD discontinuities, i.e., they are always (at least, neutrally) stable. It is interesting to note that it seems technically impossible to show this by the spectral analysis because already at the first stage we come to the dispersion relations
\[
\det (s\widehat{A}_0^{\pm}+\lambda \widehat{A}_1 + i\omega_2\widehat{A}_2+i\omega_3\widehat{A}_3)=0
\]
which cannot be analytically solved for the eigenvalues $\lambda$ (even in the 2D planar case \cite{T}). Here $s=\eta +i\xi$, with $\eta >0$, $\xi\in \mathbb{R}$, and $\omega =(\omega_2, \omega_3)\in\mathbb{R}^2$ are the Laplace and Fourier variables respectively.
The fact that the uniform Kreiss-Lopatinski condition \cite{Kreiss} is never satisfied, i.e., planar contact MHD discontinuities are only neutrally stable follows from the non-ellipticity of the front symbol discussed in Section \ref{intro}. Indeed, as was noticed in \cite{IT}, the non-ellipticity of the front symbol always implies the existence of neutral modes. For contact MHD discontinuities, this neutral mode is $s=0$.\footnote{In the original reference frame where $\hat{v}'=(\hat{v}_2,\hat{v}_3)$ is not necessarily zero this neutral mode is $s=-i(\omega,\hat{v}')$.} It corresponds to the following non-trivial normal mode solution of problem \eqref{99}, \eqref{100} with $f^\pm=0$ and $g=0$:
\[
\varphi =\bar{\varphi}e^{i(\omega ,x')},\quad (p^\pm ,v^\pm ,H^\pm )=0,\quad S^{\pm}=\bar{S}^{\pm}e^{i(\omega ,x')},
\]
where $\bar{S}^{\pm}$ and $\bar{\varphi}$ are arbitrary constants. It is natural that in our case when the Kreiss-Lopatinski condition holds only in a weak sense we have a loss of derivatives in the a priori estimate \eqref{orig_U}.
\begin{remark}
{\rm
For estimating the front perturbation $\varphi$ we have to prolong problem \eqref{99}, \eqref{100} up to first-order tangential derivatives (with respect to $t$, $x_2$ and $x_3$). Then, we can estimate the $L^2$ norms of the tangential derivatives of $U^{\pm}$. Taking into account the structure of the boundary matrix $\widehat{A}_1$, we can express $\partial_1v_1^{\pm}$ through these tangential derivatives. After this, using the last boundary condition in \eqref{100} and the trace theorem for $v_1^+$, we get an a priori estimate for $\varphi$ in $L^2$ (see also \cite{BThand}).}
\label{r7}
\end{remark}
\subsection{Linearized problem for the 2D planar case and main result}
In the rest of the paper, we restrict ourselves to 2D planar MHD flows, i.e., when
\[
x=(x_1,x_2)\in \mathbb{R}^2,\quad v=(v_1,v_2)\in \mathbb{R}^2,\quad H=(H_1,H_2)\in \mathbb{R}^2.
\]
Then, for a polytropic gas the symmetric form of the MHD equations is system \eqref{4}. For the 2D planar case the counterpart of the linearized problem \eqref{b48}--\eqref{b51} reads:
\begin{equation}
A_0(\widehat{U}^{\pm})\partial_t\dot{U}^{\pm} +\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\partial_1\dot{U}^{\pm}+A_2(\widehat{U}^{\pm} )\partial_2\dot{U}^{\pm} \\ +\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\dot{U}^{\pm} =f^{\pm}\qquad \mbox{in}\ \Omega_T,\label{b48^}
\end{equation}
\begin{equation}
\left(
\begin{array}{c}
\dot{p}^+-\dot{p}^- + \varphi [\partial_1\hat{p}]\\[3pt]
\dot{v}_1^+-\dot{v}_1^-\\[3pt]
\dot{v}_2^+-\dot{v}_2^-\\[3pt]
\dot{H}_{\tau}^+-\dot{H}_{\tau}^-+ \varphi [\partial_1\widehat{H}_{\tau}]\\[3pt]
\partial_t\varphi +\hat{v}_2^+\partial_2\varphi -\dot{v}_{N}^+ - \varphi \partial_1\hat{v}_N^+
\end{array}
\right)=g \qquad \mbox{on}\ \partial\Omega_T,
\label{b50^}
\end{equation}
\begin{equation}
(\dot{U}^+,\dot{U}^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{b51^}
\end{equation}
where
\[
\Omega_T:= (-\infty, T]\times\mathbb{R}^2_+,\quad \partial\Omega_T:=(-\infty ,T]\times\{x_1=0\}\times\mathbb{R},
\]
\[
\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})=\frac{1}{\partial_1\widehat{\Phi}^{\pm}}\left(
A_1(\widehat{U}^{\pm})-A_0(\widehat{U}^{\pm})\partial_t\widehat{\Psi}^{\pm}-A_2(\widehat{U}^{\pm})\partial_2\widehat{\Psi}^{\pm}\right),
\]
the matrices $A_0$, $A_1$ and $A_2$ are described just after system \eqref{4},
\[
\dot{U}^{\pm}=(\dot{p}^{\pm},\dot{v}_1^\pm ,\dot{v}_2^\pm ,\dot{H}_1^\pm ,\dot{H}_2^\pm ,\dot{S}^\pm ) ,\quad
f^{\pm} =(f^{\pm}_1,\ldots ,f^{\pm}_6),\quad g=(g_1,\ldots , g_5),
\]
\[
\dot{v}_{N}^{\pm}=
\dot{v}_1^{\pm}-\dot{v}_2^{\pm}\partial_2\widehat{\Psi}^{\pm}, \quad \hat{v}_{N}^{\pm}=\hat{v}_1^{\pm}-\hat{v}_2^{\pm}\partial_2\widehat{\Psi}^{\pm},\quad
\dot{H}_{\tau}^{\pm}=\dot{H}_1^{\pm}\partial_2\widehat{\Psi}^{\pm} +\dot{H}_2^{\pm},\quad \mbox{etc.}
\]
In view of the counterparts of \eqref{boundmatr} and \eqref{boundmatr'} for the 2D planar case, the matrices $\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})$ have the following structure:
\begin{equation}
\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})=\widehat{\mathcal{A}}_1^\pm +\widehat{\mathcal{A}}_0^\pm ,
\label{bm_2d}
\end{equation}
where $\widehat{\mathcal{A}}_0^\pm |_{x_1=0}=0$ (the exact form of the matrices $\widehat{\mathcal{A}}_0^\pm$ is of no interest) and
\begin{equation}
(\widehat{\mathcal{A}}_1^\pm \dot{U}^{\pm},\dot{U}^{\pm})=((\widehat{J}^\pm )^T \widehat{\mathcal{A}}_1^\pm\widehat{J}^\pm\dot{W}^\pm,\dot{W}^\pm )=
(\widehat{\mathcal{B}}_1^\pm \dot{W}^\pm,\dot{W}^\pm ),
\label{bm_2d'}
\end{equation}
\begin{equation}
\widehat{\mathcal{B}}_1^{\pm}=\frac{1}{\partial_1\widehat{\Phi}^{\pm}}\begin{pmatrix}
0 & e_1 & 0 & 0 \\[3pt]
e_1^T &O_2& -\widehat{H}_N^{\pm}a_0^\pm & 0^T\\[3pt]
0^T & -\widehat{H}_N^{\pm}a_0^\pm & O_2& 0^T\\[3pt]
0 & 0 &0 & 0
\end{pmatrix},
\label{bm_2d"}
\end{equation}
with $e_1= (1,0)$, $\dot{U}^\pm =\widehat{J}^\pm \dot{W}^\pm$, $\det \widehat{J}^\pm \neq 0$,
\begin{equation}
\dot{W}^{\pm}= (\dot{q}^{\pm},\dot{v}^{\pm}_N,\dot{v}_2^{\pm},\dot{H}^{\pm}_N,\dot{H}^{\pm}_2,\dot{S}^{\pm}),\quad \dot{q}^{\pm}=\dot{p}^{\pm}+\widehat{H}_1^{\pm}\dot{H}_1^{\pm}+\widehat{H}_2^{\pm}\dot{H}_2^{\pm},
\label{bm_2d^}
\end{equation}
and
\[
a_0^{\pm}=\begin{pmatrix}
1 & \partial_2\widehat{\Psi}^{\pm} \\[3pt]
\partial_2\widehat{\Psi}^{\pm} & 1+(\partial_2\widehat{\Psi}^{\pm})^2
\end{pmatrix} >0.
\]
Moreover, for writing down the quadratic forms with the matrices $\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})$ on the boundary we will use their exact form (cf. \eqref{A1tilde})
\begin{equation}
\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})|_{x_1=0} = \pm \begin{pmatrix} 0 & 1 & -\partial_2\hat{\varphi} & 0& 0 & 0\\[6pt]
1 &0 &0 &\widehat{H}_2^\pm\partial_2\hat{\varphi} & \widehat{H}_2^\pm & 0 \\[6pt]
-\partial_2\hat{\varphi} & 0 & 0 & -\widehat{H}_1^\pm\partial_2\hat{\varphi} & -\widehat{H}_1^\pm & 0 \\[6pt]
0 & \widehat{H}_2^\pm\partial_2\hat{\varphi} & -\widehat{H}_1^\pm\partial_2\hat{\varphi} & 0& 0& 0 \\[6pt]
0& \widehat{H}_2^\pm & -\widehat{H}_1^\pm & 0 & 0 & 0 \\[6pt]
0 & 0 &0 &0 &0 &0
\end{pmatrix}_{|x_1=0}.
\label{A1tilde2D}
\end{equation}
As for the case of constant coefficients in \eqref{99}--\eqref{101}, for problem \eqref{b48^}--\eqref{b51^} we will not use the boundary constraint associated with \eqref{15}. Therefore, we do not include it in the 2D counterpart of Proposition \ref{p3.1} below.
\begin{proposition}
Let the basic state \eqref{a21} satisfies assumptions \eqref{a5}--\eqref{avn} (to be exact, their 2D planar analogs).
Then solutions of problem \eqref{b48^}--\eqref{b51^} satisfy
\begin{equation}
{\rm div}\,\dot{h}^+=f_7^+,\quad {\rm div}\,\dot{h}^-=f_7^-\quad\mbox{in}\ \Omega_T.
\label{b43^}
\end{equation}
Here $\dot{h}^{\pm}=(\dot{H}_{N}^{\pm},\dot{H}_2^{\pm}\partial_1\widehat{\Phi}^{\pm})$ and the functions $f_7^{\pm}=f_7^{\pm}(t,x )$, which vanish in the past, are determined by the source terms and the basic state as solutions to the linear inhomogeneous equations
\begin{equation}
\partial_t a^{\pm}+ \frac{1}{\partial_1\widehat{\Phi}^{\pm}}\left\{ (\hat{w}^{\pm} ,\nabla a^{\pm}) + a^{\pm}\,{\rm div}\,\hat{u}^{\pm}\right\}={\mathcal F}^{\pm}\quad \mbox{in}\ \Omega_T,
\label{aa1^}
\end{equation}
where $a^{\pm}=f_7^{\pm}/\partial_1\widehat{\Phi}^{\pm},\quad {\mathcal F}^{\pm}=({\rm div}\,{f}_{h}^{\pm})/\partial_1\widehat{\Phi}^{\pm}$,
${f}_{h}^{\pm}=(f_{N}^{\pm} ,\partial_1\widehat{\Phi}^{\pm}f_5^{\pm})$, $f_{N}^{\pm}=f_4^{\pm}-f_5^{\pm}\partial_2\widehat{\Psi}^{\pm}$, and the vectors $\hat{w}^\pm$ and $\hat{u}^\pm$ are the 2D analogs of the corresponding vectors introduced above.
\label{p3.1'}
\end{proposition}
\begin{remark}
{\rm
In view of \eqref{a12'}, $\hat{w}_1^\pm |_{x_1=0}=0$. Then equations \eqref{aa1^} do not need any boundary conditions and from them we easily get the estimates
\begin{equation}
\|f_7^\pm (t)\|_{L^2(\mathbb{R}^2_+)}\leq C\|{\rm div}\,{f}_{h}^{\pm}\|_{L^2(\Omega_T)}\leq C\|f^\pm \|_{H^1(\Omega_T)}
\label{f7'}
\end{equation}
and
\begin{equation}
\|f_7^\pm \|_{L^2(\Omega_T)}\leq C\|f^\pm \|_{H^1(\Omega_T)}.
\label{f7}
\end{equation}
}
\label{rf7}
\end{remark}
We are now in a position to state the main result of this paper.
\begin{theorem}
Let the basic state \eqref{a21} satisfies assumptions \eqref{a5}--\eqref{avn} (in the 2D planar case). Let also
\begin{equation}
[\partial_1\hat{p} ]\geq \epsilon >0
\label{RTL}
\end{equation}
where $[\partial_1\hat{p} ]=\partial_1\hat{p}^+_{|x_1=0} +\partial_1\hat{p}^-_{|x_1=0}$ (see \eqref{norm_jump}) and $\epsilon$ is a fixed constant. Then, for all $f^{\pm} \in H^1(\Omega_T)$ and $g\in H^{3/2}(\partial\Omega_T)$ which vanish in the past, problem \eqref{b48^}--\eqref{b51^} has a unique solution $((\dot{U}^+, \dot{U}^-) ,\varphi )\in H^1(\Omega_T)\times H^1(\partial\Omega_T)$. Moreover, this solution obeys the a priori estimate
\begin{equation}
\sum_{\pm}\|\dot{U}^\pm \|_{H^{1}(\Omega_T)}+\|\varphi\|_{H^1(\partial\Omega_T)} \leq C\biggl\{\sum_{\pm}\|f^\pm \|_{H^{1}(\Omega_T)}+ \|g\|_{H^{3/2}(\partial\Omega_T)}\biggr\},
\label{main_est}
\end{equation}
where $C=C(K,\bar{\rho}_0,\bar{p}_0,\bar{\kappa}_0,\epsilon,T)>0$ is a constant independent of the data $f^\pm$ and $g$.
\label{t1}
\end{theorem}
Note that inequality \eqref{RTL} is just the Rayleigh-Taylor sign condition \eqref{RT} written for the straightened unperturbed discontinuity (with the equation $x_1=0$).
\section{Energy a priori estimate for the 2D planar case}
\label{sec:4}
\subsection{Reduction to homogeneous boundary conditions}
As for the case of constant coefficients in subsection \ref{constcoeff}, we reduce \eqref{b48^}--\eqref{b51^} to a problem with homogeneous boundary conditions, i.e., with and $g=0$. Using again the classical argument, we subtract from the solution a more regular function $\widetilde{U}^{\pm}\in H^2(\Omega_T)$ satisfying the boundary conditions \eqref{b50^}. Then, the new unknown
\begin{equation}
U^{\pm\natural}=\dot{U}^\pm-\widetilde{U}^\pm ,\label{a87'}
\end{equation}
with
\begin{equation}
\|\widetilde{U}^\pm \|_{H^1(\Omega_T)}\leq C\|g \|_{H^{1/2}(\partial\Omega_T)},
\label{tildU}
\end{equation}
satisfies problem \eqref{b48^}--\eqref{b51^} with $f^\pm =F^\pm =(F^\pm_1,\ldots , F^\pm_6)$, where
\begin{equation}
F^\pm =f^\pm-A_0(\widehat{U}^{\pm})\partial_t\widetilde{U}^{\pm} -\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\partial_1\widetilde{U}^{\pm} -A_2(\widehat{U}^{\pm} )\partial_2\widetilde{U}^{\pm}-\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\widetilde{U}^\pm
\label{a87''}
\end{equation}
and $F^\pm$ satisfy estimates (cf. \eqref{nonhom_F}):
\begin{equation}
\sum_{\pm}\|F^\pm\|_{H^1(\Omega_T)}\leq C\biggl\{\sum_{\pm}\|f^\pm \|_{H^1(\Omega_T)}+ \|g\|_{H^{3/2}(\partial\Omega_T)} \biggr\}.
\label{nonhom_F'}
\end{equation}
Dropping for convenience the indices $^{\natural}$ in \eqref{a87'}, we get our reduced linearized problem:
\begin{equation}
A_0(\widehat{U}^{\pm})\partial_t{U}^{\pm} +\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\partial_1{U}^{\pm}+A_2(\widehat{U}^{\pm} )\partial_2{U}^{\pm} +\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm}){U}^{\pm} =F^{\pm}\qquad \mbox{in}\ \Omega_T,\label{b48b}
\end{equation}
\begin{eqnarray}
{[}p{]}=- \varphi {[}\partial_1\hat{p}{]}, & \label{b50b.1}\\[3pt]
{[}v{]}=0, & \label{b50b.2}\\[3pt]
{[}H_{\tau}{]}=-\varphi {[}\partial_1\widehat{H}_{\tau}{]}, & \label{b50b.3} \\[3pt]
{v}_{N}^+ = \partial_0\varphi - \varphi \partial_1\hat{v}_N^+ & \qquad \mbox{on}\ \partial\Omega_T
\label{b50b.4}
\end{eqnarray}
\begin{equation}
({U}^+,{U}^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{b51b}
\end{equation}
where
\begin{equation}
\partial_0:= \partial_t +\hat{v}_2^+\partial_2 \qquad \mbox{in}\ \Omega_T.
\label{d0}
\end{equation}
It is not a big mistake to call $\partial_0$ the material derivative because on the boundary $\partial_0$ coincides with the material derivative $\partial_t + (\hat{w}^\pm ,\nabla )$ (in the reference frame related to the discontinuity). In view of \eqref{b43^} and \eqref{f7}, solutions to problem \eqref{b48b}--\eqref{b51b} satisfy
\begin{equation}
{\rm div}\,{h}^\pm=F_7^\pm\quad\mbox{in}\ \Omega_T
\label{b43b}
\end{equation}
where
\begin{equation}
\|F_7^\pm \|_{L^2(\Omega_T)}\leq C\|F^\pm \|_{H^1(\Omega_T)}
\label{f7^}
\end{equation}
and we will also need the layerwise counterpart of \eqref{f7^} (cf. \eqref{f7'})
\begin{equation}
\|F_7^\pm (t)\|_{L^2(\mathbb{R}^2_+)}\leq C\|F^\pm \|_{H^1(\Omega_T)}.
\label{f7"}
\end{equation}
Taking into account \eqref{tildU} and \eqref{nonhom_F'}, the following lemma for the reduced problem \eqref{b48b}--\eqref{b51b} yields estimate \eqref{main_est} for problem \eqref{b48^}--\eqref{b51^}.
\begin{lemma}
Let the basic state \eqref{a21} satisfies assumptions \eqref{a5}--\eqref{avn} (in the 2D planar case) together with condition \eqref{RTL}.
Then for all $F^{\pm} \in H^1(\Omega_T)$ which vanish in the past the a priori estimate
\begin{equation}
\sum_{\pm}\|{U}^\pm \|_{H^{1}(\Omega_T)}+\|\varphi\|_{H^1(\partial\Omega_T)} \leq C\sum_{\pm}\|F^\pm \|_{H^{1}(\Omega_T)}
\label{main_est'}
\end{equation}
holds for problem \eqref{b48b}--\eqref{b51b}, where $C=C(K,\bar{\rho}_0,\bar{p}_0,\bar{\kappa}_0,\epsilon,T)>0$ is a constant independent of the data $F^\pm$.
\label{lem1}
\end{lemma}
That is, it remains to prove Lemma \ref{lem1} and the rest of this section will be devoted to this proof. In what follows we assume by default that the assumptions of Lemma \ref{lem1} are satisfied.
\subsection{Estimate of normal derivatives through tangential ones} In spite of the fact that for problem \eqref{b48b}--\eqref{b51b} the boundary $x_1=0$ is characteristic, using the structure of the boundary matrix (see \eqref{bm_2d}--\eqref{bm_2d^}) and the divergence constraints \eqref{b43b}, we can estimate the $L^2-$norms of the normal derivatives $\partial_1U^{\pm}$ through the $L^2-$norms of $U^\pm$ and the tangential derivatives $\partial_tU^{\pm}$ and $\partial_2U^\pm $ (and the $H^1-$ norms of the source terms $F^\pm$).
\begin{proposition}
The solutions to problem \eqref{b48b}--\eqref{b51b} obey the estimate
\begin{equation}
\|\partial_1{U}^\pm\|^2_{L^2(\Omega_t)} \leq C\left\{ \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI^\pm (s)ds \right\}
\label{d1U}
\end{equation}
for all $t\leq T$ and $C=C(K,\bar{\rho}_0,\bar{p}_0,\bar{\kappa}_0,T)>0$ being a constant independent of the source terms $F^\pm$, where $\Omega_t= (-\infty ,t]\times\mathbb{R}^2_+$ and
\begin{equation}
I^\pm (t) =\|U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}+
\|\partial_t U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}+ \|\partial_2U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}
\label{Ipm}
\end{equation}
(clearly, $
\int_0^tI^\pm (s)ds =
\|U^\pm\|^2_{L^2(\Omega_t)}+
\|\partial_t U^\pm\|^2_{L^2(\Omega_t)}+ \|\partial_2U^\pm\|^2_{L^2(\Omega_t)}$).
\label{p5}
\end{proposition}
\begin{proof}
First of all, for any linear symmetric hyperbolic system in the half-plane $\mathbb{R}^2_+$ we can easily get an estimate for the weighted normal derivative $\sigma\partial_1$ of the unknown, where the weight $\sigma (x_1)\in C^{\infty}(\mathbb{R}_+)$ is a monotone increasing function such that $\sigma (x_1)=x_1$ in a neighborhood of the origin and $\sigma (x_1)=1$ for $x_1$ large enough. Indeed, to estimate the weighted normal derivatives of $U^\pm$ we do not need boundary conditions because the weight $\sigma |_{x_1=0}=0$. By applying to systems \eqref{b48b} the operator $\sigma\partial_1$ and using standard arguments of the energy method, we obtain the inequality
\begin{equation}
\|\sigma\partial_1U^{\pm} (t)\|^2_{L^2(\mathbb{R}^2_+)}\leq C\biggl\{ \|F^\pm \|^2_{H^1(\Omega_T)} + \int_0^{\tau}I^\pm (s)ds +\int_0^{t}\|\sigma\partial_1U^{\pm} (s)\|^2_{L^2(\mathbb{R}^2_+)}ds\biggr\}
\label{4.92}
\end{equation}
for all $t\leq \tau \leq T$. Applying then Gronwall's lemma to the function of $t$ staying in the left-hand side of \eqref{4.92}, we get
\[
\|\sigma\partial_1U^{\pm} (t)\|^2_{L^2(\mathbb{R}^2_+)}\leq C\biggl\{ \|F^\pm \|^2_{H^1(\Omega_T)} + \int_0^{\tau}I^\pm (s)ds\biggr\}.
\]
Integrating the last inequality over the time interval $(0,t )$ and setting $\tau =t$ in the result, we come to the estimate
\begin{equation}
\|\sigma\partial_1{U}^\pm\|^2_{L^2(\Omega_t)} \leq C\left\{ \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI^\pm (s)ds \right\}
\label{sigma_d1U}
\end{equation}
for all $t\leq T$ and with a constant $C=C(K,\bar{\rho}_0,\bar{p}_0,T)>0$.
Taking into account the structure of the boundary matrices $\widetilde{A}_1(\widehat{U}^\pm , \widehat{\Psi}^\pm )$ detailed in \eqref{bm_2d}--\eqref{bm_2d^}, we can estimate $\partial_1U^{\pm}$ in a neighborhood of the boundary. Indeed, in view of assumption \eqref{cdass} and the continuity of the basic state \eqref{a21}, there exist such a small but fixed constant $\delta >0$ depending on $\bar{\kappa}_0$ that
\begin{equation}
|\widehat{H}_N^{\pm}|\geq \frac{\bar{\kappa}_0}{2}>0 \qquad\mbox{in}\ \Omega_t^{\delta},
\label{cdass"}
\end{equation}
where
\[
\Omega_t^{\delta}=(-\infty ,t]\times \mathbb{R}^2_{\delta},\quad \mathbb{R}^2_{\delta} =(0,\delta )\times\mathbb{R}.
\]
Rewriting systems \eqref{b48b} in terms of the vectors
\[
{W}^{\pm}= ({q}^{\pm},{v}^{\pm}_N,{v}_2^{\pm},{H}^{\pm}_N,{H}^{\pm}_2,{S}^{\pm})
\]
(with ${q}^{\pm}={p}^{\pm}+\widehat{H}_1^{\pm}{H}_1^{\pm}+\widehat{H}_2^{\pm}{H}_2^{\pm}$, see \eqref{bm_2d^}), using the divergence constraints \eqref{b43b} and taking into account the structure of the matrices $\widehat{\mathcal{B}}_1^\pm$ in \eqref{bm_2d"}, by virtue of \eqref{cdass"}, we can resolve the rewritten systems for the normal derivatives of $(q^\pm ,v_N^\pm ,v_2^\pm ,H_N^\pm,H_2^\pm )$ in $\Omega_t^{\delta}$. By returning to the original unknowns and applying \eqref{f7"} this gives the estimates
\begin{multline}
\| \partial_1V^\pm (t)\|^2_{L^2(\mathbb{R}^2_{\delta})}
\leq C \left\{ \|F^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} + \|F_7^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}+ I^\pm (t)\right\}
\\
\leq C \left\{ \|F^\pm \|^2_{H^1(\Omega_T)} + I^\pm (t)\right\}
\label{1Vt}
\end{multline}
and
\begin{equation}
\| \partial_1V^\pm \|^2_{L^2(\Omega_t^{\delta})} \leq C \left\{ \|F^\pm \|^2_{H^1(\Omega_T)} + \int_0^t I^\pm (s)ds\right\}
\label{1V}
\end{equation}
with a constant $C=C(K,\bar{\kappa}_0,T)>0$, where
\[
V^\pm = (p^\pm ,v^\pm ,H^\pm).
\]
While deriving \eqref{1Vt} we exploited the elementary inequality
\begin{equation}
\| F^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} \leq \| F^\pm \|^2_{L^2(\Omega_t)} +\| \partial_tF^\pm \|^2_{L^2(\Omega_t)}
\label{elem}
\end{equation}
following from the trivial relation
\[
\frac{d}{dt}\,\|F^\pm (t)\|^2_{L_2(\mathbb{R}^2_+)}=2\int_{\mathbb{R}^2_+}(F^\pm,\partial_tF^\pm)dx
\]
(clearly, \eqref{elem} is roughened as $\| F^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} \leq \| F^\pm \|^2_{H^1(\Omega_T)}$).
Since the weight $\sigma$ in \eqref{sigma_d1U} is not zero outside the boundary, it follows from estimate \eqref{sigma_d1U} that
\begin{equation}
\|\partial_1{U}^\pm\|^2_{L^2(\Omega_t \setminus \Omega_t^{\delta})} \leq C\left\{ \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI^\pm (s)ds \right\},
\label{d1U_out}
\end{equation}
where the constant $C$ depends, in particular, on $\delta$ and so on $\bar{\kappa}_0$. Combining \eqref{1V} and \eqref{d1U_out}, we get
\begin{equation}
\|\partial_1{V}^\pm\|^2_{L^2(\Omega_t )} \leq C\left\{ \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI^\pm (s)ds \right\}.
\label{d1V}
\end{equation}
For obtaining the desired estimate \eqref{d1U} it remains to include the normal derivative of $S^\pm$ into estimate \eqref{d1V}. To this end we consider the last equations in systems \eqref{b48b}:
\begin{equation}
\partial_tS^{\pm}+\frac{1}{\partial_1\widehat{\Phi}^{\pm}} (\hat{w}^{\pm} ,\nabla S^{\pm} ) + \mbox{l.o.t.} =F_6^\pm \quad\mbox{in}\ \Omega_T,
\label{entropy}
\end{equation}
where l.o.t are unimportant lower-order terms. Since $\hat{w}_1^\pm |_{x_1=0}=0$, the linear equations \eqref{entropy} considered as equations for $S^\pm$ do not need any boundary conditions. Writing equations similar to \eqref{entropy} for $\partial_1 S^{\pm}$ and using \eqref{d1V} to estimate the normal derivatives of $V^{\pm}$ that appear in the right-hand side of the equations for $\partial_1 S^{\pm}$, we easily derive the estimate
\begin{equation}
\|\partial_1S^\pm\|^2_{L^2(\Omega_t )} \leq C\left\{ \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI^\pm (s)ds \right\}.
\label{ent_est}
\end{equation}
Estimates \eqref{d1V} and \eqref{ent_est} give \eqref{d1U}.
\end{proof}
Note that below we will need the ``local'' layerwise estimate \eqref{1Vt} together with \eqref{d1U} for obtaining the a priori estimate \eqref{main_est'}.
\subsection{Preparatory estimation of material derivatives}
We apply the differential operator $\partial_0$ (see \eqref{d0}) to systems \eqref{b48b}. Then, by standard arguments of the energy method applied to the resulting symmetric hyperbolic systems (considered as systems for $\partial_0U^\pm $ with lower-order terms depending on $U^\pm$, $\partial_tU^\pm$, $\partial_1U^\pm$ and $\partial_2U^\pm$), in view of \eqref{d1U}, we obtain
\begin{multline}
\sum_{\pm}\int_{\mathbb{R}^2_+}\bigl(A_0(\widehat{U}^\pm )\partial_0U^\pm , \partial_0U^\pm \bigr)(t)\,dx +2 \int_{\partial\Omega_t}Q_0dx_2ds \\
\leq C\biggl\{ \sum_{\pm}\|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds \biggr\},
\label{q0}
\end{multline}
where $I(t)=I^+(t)+I^-(t)$, with $I^\pm (t)$ defined in \eqref{Ipm}, and
\[
Q_0 =-\frac{1}{2}\sum_{\pm}\bigl(\widetilde{A}_1(\widehat{U}^\pm, \widehat\Psi^{\pm} )\partial_0U^\pm , \partial_0U^\pm \bigr)\bigr|_{x_1=0}\,.
\]
Using \eqref{A1tilde2D} and the boundary conditions \eqref{b50b.2}, we first calculate
\begin{multline}
-\frac{1}{2}\sum_{\pm}\bigl(\widetilde{A}_1(\widehat{U}^\pm, \widehat\Psi^{\pm} )U^\pm , U^\pm \bigr)\bigr|_{x_1=0}\\=\left.\left(-v_N^+[p]+
(\widehat{H}_1^+v_2^+-\widehat{H}_2^+v_1^+)[H_{\tau}]\right)\right|_{x_1=0}\\
=\left.\left(-v_N^+[p]+
(\widehat{H}_N^+v_2^+-\widehat{H}_2^+v_N^+)[H_{\tau}]\right)\right|_{x_1=0}.
\label{quadr}
\end{multline}
Then
\begin{multline*}
Q_0=
\left.\left(-\partial_0v_N^+[\partial_0p]+
(\widehat{H}_N^+\partial_0v_2^+-\widehat{H}_2^+\partial_0v_N^+)[\partial_0H_{\tau}]\right)\right|_{x_1=0} +
\big({\rm coeff}\,v_2^+[\partial_0p] \\ +{\rm coeff}\,v_2^+ [\partial_0H_{\tau}] + {\rm coeff}\,[H_1] \partial_0v_2^+ + {\rm coeff}\,[H_1] \partial_0v_N^+ + {\rm coeff}\,v_2^+[H_1]\big)\big|_{x_1=0}.
\end{multline*}
Here and later on \,coeff\, denotes a coefficient that can change from line to line, is determined by the basic state, and its concrete form is of no interest.
Taking into account the boundary conditions \eqref{b50b.1}, \eqref{b50b.3}, \eqref{b50b.4}, we obtain
\begin{equation}
\partial_0[p] = {\rm coeff}\,\partial_0\varphi + {\rm coeff}\,\varphi \\= {\rm coeff}\,v_N^+|_{x_1=0} + {\rm coeff}\,\varphi ,
\label{d0q}
\end{equation}
\begin{equation}
\partial_0[H_{\tau}] = {\rm coeff}\,v_N^+|_{x_1=0} + {\rm coeff}\,\varphi .
\label{d0H2}
\end{equation}
By substituting \eqref{d0q} and \eqref{d0H2} into the quadratic form $Q_0$, one gets
\begin{multline}
Q_0=\bigl({\rm coeff}\,v_N^+\partial_0v_N^+ +{\rm coeff}\,v_N^+\partial_0v_2^+ +{\rm coeff}\,\varphi \partial_0v_N^+ +{\rm coeff}\,\varphi \partial_0v_2^+ \\
+{\rm coeff}\,v_2^+v_N^+ + {\rm coeff}\,v_2^+\varphi + {\rm coeff}\,[H_1] \partial_0v_2^+ \\+ {\rm coeff}\,[H_1] \partial_0v_N^+ +{\rm coeff}\,v_2^+[H_1]
\bigr)\bigr|_{x_1=0}\,.
\label{Q_0}
\end{multline}
To treat the integrals of the ``lower-order'' terms like $\,{\rm coeff} v_N^+\partial_{\alpha}v_2^+|_{x_1=0}$ contained in the right-hand side of \eqref{Q_0}, where $\alpha =2$ or we have the time derivative $\partial_t$, we use the same standard arguments as in \cite{T05,T09,Tcpam}. That is, we pass to the volume integral and integrate by parts:
\begin{multline}
\int_{\partial\Omega_t}\hat{c}\,{v}_N^+\partial_{\alpha}{v}_2^+|_{x_1=0}\,dx_2ds
=-\int_{\Omega_t}\partial_1\bigl(\tilde{c}{v}_N^+\partial_{\alpha}{v}_2^+\bigr)dxds \\
=\int_{\Omega_t}\Bigl\{\tilde{c}\partial_{\alpha}{v}_N^+\partial_1{v}_2^+ +(\partial_{\alpha}\tilde{c}){v}_N^+\partial_1{v}_2^+ -\tilde{c}\partial_1{v}_N^+\partial_{\alpha}{v}_2^+\\-(\partial_1\tilde{c}){v}_N^+\partial_\alpha{v}_2^+\Bigr\}dxds
-\int_{\Omega_t}\partial_{\alpha}\bigl(\tilde{c}{v}_N^+\partial_1{v}_2^+\bigr)dxds,
\label{intbypart}
\end{multline}
where $\hat{c}$ is a coefficient and $\tilde{c}|_{x_1=0}=\hat{c}$.
If $\alpha =2$ the last integral in \eqref{intbypart} is equal to zero. But if $\partial_{\alpha}$ denotes the time derivative, the last integral does not disappear. In this case we use a cut-off in the passage to the volume integral. That is, we may assume that the coefficient $\tilde{c}$ appearing in the volume integrals in \eqref{intbypart} is zero for $x_1>\delta $, where $\delta$ is the same as in \eqref{cdass"}. For example, if $\hat{c}=\widehat{H}_N^+|_{x_1=0}$, then we take $\tilde{c} = \eta\widehat{H}_N^+$, where $\eta (x_1)\in C^{\infty}(\mathbb{R}_+)$ is such a rapidly decreasing function that $\eta (0)=1$ and $\eta (x_1)=0$ for $x_1>\delta$.\footnote{If $\hat{c}=1$, we just take $\tilde{c} = \eta$.} Then, all the integrals over $\Omega_t$ in \eqref{intbypart} are replaced by the same integrals over $\Omega^{\delta}_t$ and the last integral reads:
\begin{equation}
-\int_{\Omega_t^{\delta}}\partial_s\bigl(\tilde{c}{v}_N^+\partial_1{v}_2^+\bigr)dxds=
-\int_{\mathbb{R}^2_{\delta}}\left(\tilde{c}{v}_N^+\partial_1{v}_2^+\right)(t)\,dx.
\label{cut-off}
\end{equation}
Using the Young inequality, the elementary inequality (see \eqref{elem})
\begin{equation}
\| U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} \leq \int_0^tI (s)ds
\label{elem'}
\end{equation}
and \eqref{1Vt}, we estimate the last integral as follows:
\begin{equation}
-\int_{\mathbb{R}^2_{\delta}}\tilde{c}{v}_N^+\partial_{1}{v}_2^+\,dx\leq C\biggl\{
\sum_\pm\|F^\pm\|^2_{H^1(\Omega_T)}+\tilde{\varepsilon}\,I(t) +\frac{1}{\tilde{\varepsilon}}\,
\int_0^tI (s)ds \biggr\},
\label{intbypart1}
\end{equation}
where $\tilde{\varepsilon} $ is a small positive constant. The last but one integral in \eqref{intbypart} is estimated by using \eqref{d1U}:
\begin{multline}
\int_{\Omega_t}\Bigl\{\tilde{c}\partial_{\alpha}{v}_N^+\partial_1{v}_2^+ +(\partial_{\alpha}\tilde{c}){v}_N^+\partial_1{v}_2^+ -\tilde{c}\partial_1{v}_N^+\partial_{\alpha}{v}_2^+ -(\partial_1\tilde{c}){v}_N^+\partial_\alpha{v}_2^+\Bigr\}dxds \\
\leq C\biggl\{ \sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds \biggr\}.
\label{intbypart2}
\end{multline}
The integrals of the terms like $\,{\rm coeff} \varphi\partial_0 v_N^+|_{x_1=0}$ contained in \eqref{Q_0} are treated by the integration by parts, using the Young inequality, the cut-off argument as above (when we pass to the volume integral), estimates \eqref{d1U} and \eqref{1Vt} and the estimate
\begin{equation}
\|\varphi (t)\|^2_{L^2(\mathbb{R})}\leq C\left\{ \sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds+ \|\varphi \|^2_{L^2(\partial\Omega_t)} \right\}
\label{frontL2}
\end{equation}
following from the boundary condition \eqref{b50b.4} and estimate \eqref{d1U}:
\begin{multline}
\int_{\partial\Omega_t}{\rm coeff}\, \varphi\partial_0v_N^+|_{x_1=0}\, dx_2ds=\int_{\mathbb{R}}{\rm coeff}\, \varphi v_N^+|_{x_1=0}\, dx_2 \\ +
\int_{\partial\Omega_t}\left.\left({\rm coeff}\, \varphi v_N^+ + {\rm coeff}\, v_N^+\partial_0\varphi \right)\right|_{x_1=0} dx_2ds =
\int_{\mathbb{R}}{\rm coeff}\, \varphi v_N^+|_{x_1=0}\, dx_2 \\ +
\int_{\partial\Omega_t}\left.\left({\rm coeff}\, \varphi v_N^+ + {\rm coeff}\, (v_N^+)^2 \right)\right|_{x_1=0} dx_2ds \\ \leq C\biggl\{ \frac{1}{\tilde{\varepsilon}}\,\|\varphi (t)\|^2_{L^2(\mathbb{R})} + \tilde{\varepsilon}\,\| v_N^+(t) \|^2_{L^2(\mathbb{R}^2_+)} + \tilde{\varepsilon}\,\| \partial_1v_N^+(t) \|^2_{L^2(\mathbb{R}^2_{\delta})}\\ +\|\varphi \|^2_{L^2(\partial\Omega_t)} +\| v_N^+ \|^2_{L^2(\Omega_t)} + \| \partial_1v_N^+ \|^2_{L^2(\Omega_t)}\biggr\} \\ \leq \tilde{\varepsilon}C I(t)+\widetilde{C}(\tilde{\varepsilon})\biggl\{\sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds+ \|\varphi \|^2_{L^2(\partial\Omega_t)} \biggr\}.
\label{intbypart3}
\end{multline}
Here and below $\widetilde{C}=\widetilde{C}(\tilde{\varepsilon})$ is a positive constant depending on $\tilde{\varepsilon} $.
In \eqref{intbypart3} we also used the boundary condition \eqref{b50b.4} for expressing $\partial_0\varphi$ through $v_N^+|_{x_1=0}$ and $\varphi$.
Thus, taking into account the above, we estimate the quadratic form $Q_0$ as follows:
\begin{equation}
Q_0\leq \tilde{\varepsilon}C I(t)+\widetilde{C}(\tilde{\varepsilon})\biggl\{\sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds+ \|\varphi \|^2_{L^2(\partial\Omega_t)} \biggr\}.
\label{estQ0}
\end{equation}
Then, taking into account the positive definiteness of the symmetric matrices $A_0(\widehat{U}^\pm )$, from \eqref{q0}, \eqref{elem'}, \eqref{frontL2} and \eqref{estQ0} we obtain
\begin{multline}
\sum_\pm \Bigl\{ \| U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} +\| \partial_0U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} \Bigr\} + \|\varphi (t)\|^2_{L^2(\mathbb{R})}\\
\leq \tilde{\varepsilon}C\sum_\pm\Bigl(\| \partial_tU^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} +\|\partial_2 U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}\Bigr)\\
+\widetilde{C}(\tilde{\varepsilon})\left\{\sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds+ \|\varphi \|^2_{L^2(\partial\Omega_t)} \right\}.
\label{estd0}
\end{multline}
\subsection{Estimation of $x_2$-derivatives: deduction of the main energy inequality}
We now differentiate systems \eqref{b48b} with respect to $x_2$. Then, similarly to \eqref{q0} by the energy method we get
\begin{multline}
\sum_{\pm}\int_{\mathbb{R}^2_+}\bigl(A_0(\widehat{U}^\pm )\partial_2U^\pm , \partial_2U^\pm \bigr)dx +2 \int_{\partial\Omega_t}Q_2dx_2ds \\
\leq C\biggl\{ \sum_{\pm}\|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds \biggr\},
\label{q2}
\end{multline}
where, cf. \eqref{quadr},
\[
Q_2 =-\frac{1}{2}\sum_{\pm}\bigl(\widetilde{A}_1(\widehat{U}^\pm, \widehat\Psi^{\pm} )\partial_2U^\pm , \partial_2U^\pm \bigr)\bigr|_{x_1=0} \\
=\left.\left(-\partial_2v_N^+[\partial_2p]+
R[\partial_2H_{\tau}]\right)\right|_{x_1=0} +\mathcal{Q}_2
\]
and
\[
R=
(\widehat{H}_N^+\partial_2v_2^+-\widehat{H}_2^+\partial_2v_N^+)|_{x_1=0},
\]
\[
\mathcal{Q}_2 = \big({\rm coeff}\,v_2^+[\partial_2p] +{\rm coeff}\,v_2^+ [\partial_2H_{\tau}] + {\rm coeff}\,[H_1] \partial_2v_2^+ + {\rm coeff}\,[H_1] \partial_2v_N^+ + {\rm coeff}\,v_2^+ [H_1]\big)\big|_{x_1=0}.
\]
To treat the quadratic form $Q_2$ we use not only the boundary conditions but also the interior equations considered on the boundary. Namely, by multiplying the equations for $H^+$ contained in \eqref{b48b} (see their 3D counterpart for $\dot{H} ^\pm$ in \eqref{aA3}) by the vector $(1,-\partial_2\widehat{\Psi}^+)$, considering the result at $x_1=0$ and taking \eqref{b50b.4} into account, we obtain
\begin{equation}
R = -\partial_0H_N^+ +{\rm coeff}\,v_N^+ +{\rm coeff}\,v_2^+ +{\rm coeff}\,H_N^+ +F_N^+\qquad \mbox{on}\ \partial\Omega_T,
\label{eqH}
\end{equation}
with $F_N^+=F_4^+-F_5^+\partial_2\widehat{\Psi}^+$.
Using \eqref{eqH} (as well as the definition of $R$ itself) and the boundary conditions \eqref{b50b.3} and \eqref{b50b.4}, we calculate:
\begin{multline}
Q_2 = [\partial_1\hat{p}]\partial_2\varphi \,\partial_2v_N^+-[\partial_1\widehat{H}_{\tau}]R\partial_2\varphi -[\partial_2\partial_1\widehat{H}_{\tau}]\varphi\,\partial_2v_2^+ \\
+ \big([\partial_2\partial_1\hat{p}]+\widehat{H}_2^+[\partial_2\partial_1\widehat{H}_{\tau}]\big)\varphi\,\partial_2v_N^+ +\mathcal{Q}_2 \\[3pt]
=\underbrace{[\partial_1\hat{p}]\,\partial_t\partial_2\varphi\, \partial_2\varphi} + \underline{{\rm coeff}\,\partial_0H_N^+\,\partial_2\varphi|_{x_1=0} } \\ + \Bigl( {\rm coeff}\,\partial_2^2\varphi\, \partial_2\varphi +
{\rm coeff}\,( \partial_2\varphi )^2+{\rm coeff}\,\varphi\, \partial_2\varphi +{\rm coeff}\,v_N^+ \partial_2\varphi \\+{\rm coeff}\,v_2^+\partial_2\varphi +{\rm coeff}\,H_N^+\partial_2\varphi +{\rm coeff}\,\varphi\, \partial_2v_N^+ + {\rm coeff}\,\varphi \,\partial_2v_2^+ \\
+{\rm coeff}\,v_2^+\varphi + {\rm coeff}\,[H_1] \partial_2v_2^+ + {\rm coeff}\,[H_1] \partial_2v_N^+
\\+ {\rm coeff}\,v_2^+ [H_1]+ {\rm coeff}\,F_N^+\partial_2\varphi +{\rm coeff}\, F_N^+ \varphi\Bigr)\Bigr|_{x_1=0}.
\label{Q2"}
\end{multline}
The underbraced term in \eqref{Q2"} is {\it most important} because under the Rayleigh-Taylor sign condition \eqref{RTL} it gives us the control on the $L^2$ norm of $\partial_2\varphi$ (see below). Having in hand this control, only the underlined term in \eqref{Q2"} needs an additional care whereas the rest terms in \eqref{Q2"} can be easily treated by the integration by parts, etc. Indeed,
\begin{multline}
2 \int_{\partial\Omega_t}Q_2dx_2ds=\int_{\mathbb{R}}[\partial_1\hat{p}](t)(\partial_2\varphi )^2(t)\,dx_2 -K(t) -M(t) \\
\geq \epsilon \, \|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})}-K(t) -M(t),
\label{IntQ2}
\end{multline}
where
\[
K(t)=\int_{\partial\Omega_t}{\rm coeff}\,\partial_0H_N^+\,\partial_2\varphi|_{x_1=0}dx_2ds
\]
and
\begin{equation}
M(t)\leq C\biggl\{ \sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds \\+ \|\varphi \|^2_{L^2(\partial\Omega_t)} +\|\partial_2 \varphi\|^2_{L^2(\partial\Omega_t)}\biggr\}.
\label{Mt}
\end{equation}
Here we used the integration by parts, the elementary inequality
\begin{equation}
\|U^\pm _{|x_1=0} (t)\|^2_{L^2(\partial\Omega_t)}\leq \|U^\pm (t)\|^2_{L^2(\Omega_t)} +\|\partial_1 U^\pm (t)\|^2_{L^2(\Omega_t)},
\label{tr}
\end{equation}
estimate \eqref{d1U} and the trace theorem for $F_N^+$ (or even the inequality like \eqref{tr}).
It remains to estimate the boundary integral $K(t)$ of the underlined term in \eqref{Q2"}. To this end we first integrate by parts and use the boundary condition \eqref{b50b.4}:
\begin{multline*}
K(t)=L(t) +\int_{\partial\Omega_t}\bigl({\rm coeff}\,H_N^+\,\partial_2\varphi +{\rm coeff}\,H_N^+\,\partial_2(\partial_0\varphi)\bigr)\bigr|_{x_1=0}dx_2ds \\ = L(t)+
\int_{\partial\Omega_t}\bigl({\rm coeff}\,H_N^+\,\partial_2\varphi +{\rm coeff}\,H_N^+\,\varphi +{\rm coeff}\,H_N^+\,\partial_2v_N^+\bigr)\bigr|_{x_1=0}dx_2ds ,
\end{multline*}
where
\[
L(t)=\int_{\mathbb{R}}{\rm coeff}\,H_N^+\,\partial_2\varphi|_{x_1=0}dx_2.
\]
Then we apply \eqref{tr}, estimate \eqref{d1U} and arguments as in \eqref{intbypart} with $\alpha =2$:
\begin{equation}
K(t) \leq L(t) + C\biggl\{ \sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tI (s)ds \\ + \|\varphi \|^2_{L^2(\partial\Omega_t)} +\|\partial_2 \varphi\|^2_{L^2(\partial\Omega_t)}\biggr\}.
\label{Kt}
\end{equation}
We now estimate the integral $L(t)$ by using the Young inequality, the passage to the volume integral, the relation $\partial_1H_N^+={\rm coeff}\,H_2^++{\rm coeff}\,\partial_2H_2^++F_7^+$ following from \eqref{b43b}, estimate \eqref{f7"}, and the elementary inequality \eqref{elem'}:
\begin{multline}
L(t) \leq C \left\{ \tilde{\varepsilon}\|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})} +\frac{1}{\tilde{\varepsilon}}\int_{\mathbb{R}}(H_N^+)^2\bigr|_{x_1=0}dx_2\right\}\\
= C \biggl\{ \tilde{\varepsilon}\|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})} -\frac{2}{\tilde{\varepsilon}}\int_{\mathbb{R}^2_+}H_N^+\partial_1H_N^+ dx\biggr\}\\
\leq C \biggl\{ \tilde{\varepsilon}\|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})} +\frac{1}{\tilde{\varepsilon}} \biggl(\frac{1}{\tilde{\varepsilon}^2} \|H_N^+(t)\|^2_{L^2(\mathbb{R}^2_+)}\\ +\tilde{\varepsilon}^2\bigl( \| H_2^+(t)\|^2_{L^2(\mathbb{R}^2_+)} +\|\partial_2H_2^+(t)\|^2_{L^2(\mathbb{R}^2_+)} + \|F_7^+(t)\|^2_{L^2(\mathbb{R}^2_+)}\bigr)\biggr)\biggr\} \\
\leq \tilde{\varepsilon}C \left\{ \|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})} +\|\partial_2U^+(t)\|^2_{L^2(\mathbb{R}^2_+)}\right\} + \widetilde{C}(\tilde{\varepsilon})\left\{ \|F^+ \|^2_{H^1(\Omega_T)}+ \int_0^tI (s)ds\right\},
\label{Lt}
\end{multline}
where we may consider the same $\tilde{\varepsilon}$ as in \eqref{estd0}. Then,
\eqref{q2}, \eqref{IntQ2}, \eqref{Mt}, \eqref{Kt} and \eqref{Lt} imply
\begin{equation} \begin{split}
&\sum_\pm \| \partial_2U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} + \|\partial_2\varphi (t)\|^2_{L^2(\mathbb{R})}\\
\leq &\tilde{\varepsilon}C\left\{ \|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})} +\|\partial_2U^+(t)\|^2_{L^2(\mathbb{R}^2_+)}\right\} + \widetilde{C}(\tilde{\varepsilon})\biggl\{\sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)}\\
& +\int_0^tI (s)ds+ \|\varphi \|^2_{L^2(\partial\Omega_t)} +\|\partial_2 \varphi\|^2_{L^2(\partial\Omega_t)}\biggr\}.
\label{estd2}
\end{split} \end{equation}
At last, choosing $\tilde{\varepsilon} $ to be small enough and taking into account that
\[
\|\partial_0U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} +c_1\|\partial_2U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} \geq c_2\left\{ \|\partial_tU^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)} +\|\partial_2U^\pm (t)\|^2_{L^2(\mathbb{R}^2_+)}\right\},
\]
with suitable positive constants $c_1$ and $c_2$ depending on the basic state, the combination of \eqref{estd0} and \eqref{estd2} yields the energy inequality
\begin{equation}
J(t) \leq C \left\{ \sum_\pm \|F^\pm \|^2_{H^1(\Omega_T)} +\int_0^tJ (s)ds\right\},
\label{main_energy_in}
\end{equation}
where
\[
J(t)=I(t) + \|\varphi (t) \|^2_{L^2(\mathbb{R})} +\|\partial_2 \varphi (t)\|^2_{L^2(\mathbb{R})}.
\]
Then, applying Gronwall's lemma and integrating in time over $(-\infty,T)$, from \eqref{main_energy_in} we derive the energy a priori estimate
\begin{multline}
\sum_{\pm}\left(
\|U^\pm\|_{L^2(\Omega_T)}+
\|\partial_t U^\pm\|_{L^2(\Omega_T)}+ \|\partial_2U^\pm\|_{L^2(\Omega_T)}\right) \\
+\|\varphi\|_{L^2(\partial\Omega_T)} + \|\partial_2\varphi\|_{L^2(\partial\Omega_T)}\leq C\sum_{\pm}\|F^\pm \|_{H^{1}(\Omega_T)}.
\label{main_estim}
\end{multline}
In view of \eqref{tr}, from the boundary condition \eqref{b50b.4} we obtain the estimate
\begin{equation}
\|\partial_t\varphi\|_{L^2(\partial\Omega_T)} \leq C \bigl\{\|U^+\|_{L^2(\Omega_T)}+
\|\partial_1 U^+\|_{L^2(\Omega_T)} +\|\varphi\|_{L^2(\partial\Omega_T)} + \|\partial_2\varphi\|_{L^2(\partial\Omega_T)}\bigr\}.
\label{front_t}
\end{equation}
Combining \eqref{d1U} (with $t=T$), \eqref{main_estim} and \eqref{front_t}, we deduce the desired a priori estimate \eqref{main_est'}. Thus, the proof of Lemma \ref{lem1} is complete. Recall that Lemma \ref{lem1} implies estimate \eqref{main_est} in Theorem \ref{t1}.
\section{Well-posedness of the linearized problem}
\label{sec:exist}
\subsection{Existence of solutions for a fully noncharacteristic ``strictly dissipative'' re\-gularization} We prove the existence of a unique solution $((\dot{U}^+,\dot{U}^- ),\varphi )\in H^1(\Omega_T)\times H^1(\partial\Omega_T)$ to problem \eqref{b48^}--\eqref{b51^} by using its fully noncharacteristic ``strictly dissipative'' regularization containing a small parameter of regularization $\varepsilon$:
\begin{multline}
A_0(\widehat{U}^{\pm})\partial_t\dot{U}^{\pm \varepsilon} -\varepsilon\partial_1\dot{U}^{\pm \varepsilon} +\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\partial_1\dot{U}^{\pm \varepsilon}\\+A_2(\widehat{U}^{\pm} )\partial_2\dot{U}^{\pm \varepsilon} +\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\dot{U}^{\pm \varepsilon} =f^{\pm}\qquad \mbox{in}\ \Omega_T,\label{r48^}
\end{multline}
\begin{equation}
\left(
\begin{array}{c}
\dot{p}^{+ \varepsilon}-\dot{p}^{- \varepsilon} + \varphi^{\varepsilon} [\partial_1\hat{p}]\\[3pt]
\dot{v}_1^{+ \varepsilon}-\dot{v}_1^{- \varepsilon}\\[3pt]
\dot{v}_2^{+ \varepsilon}-\dot{v}_2^{- \varepsilon}\\[3pt]
\dot{H}_{\tau}^{+ \varepsilon}-\dot{H}_{\tau}^{- \varepsilon}+ \varphi^{\varepsilon} [\partial_1\widehat{H}_{\tau}]\\[3pt]
\partial_t\varphi^{\varepsilon} +\hat{v}_2^{+}\partial_2\varphi^{\varepsilon} -\dot{v}_{N}^{+ \varepsilon} - \varphi^{\varepsilon} \partial_1\hat{v}_N^+
\end{array}
\right)=g \qquad \mbox{on}\ \partial\Omega_T,
\label{r50^}
\end{equation}
\begin{equation}
(\dot{U}^{+ \varepsilon},\dot{U}^{- \varepsilon},\varphi^{\varepsilon} )=0\qquad \mbox{for}\ t<0.\label{r51^}
\end{equation}
The boundary conditions \eqref{r50^} for $(\dot{U}^{+ \varepsilon},\dot{U}^{- \varepsilon},\varphi^{\varepsilon} )$ coincide with \eqref{b50^} whereas the interior equations \eqref{r48^} contain the additional terms $-\varepsilon\partial_1\dot{U}^{\pm \varepsilon} $ compared to the original (not regularized) equations \eqref{b48^}.
\begin{lemma}
Let assumptions \eqref{a5}--\eqref{cdass} be satisfied and the constant $\varepsilon >0$ be small enough compared to the constant $\bar{\kappa}_0$ in \eqref{cdass}. Then, for any fixed constant $\varepsilon$ and for all $f^{\pm} \in H^1(\Omega_T)$ and $g\in H^{1}(\partial\Omega_T)$ which vanish in the past, problem \eqref{r48^}--\eqref{r51^} has a unique solution $((\dot{U}^{+ \varepsilon}, \dot{U}^{- \varepsilon}) ,\varphi^{\varepsilon} )\in H^1(\Omega_T)\times H^1(\partial\Omega_T)$.
\label{lem2}
\end{lemma}
\begin{proof}
Since the constant $\varepsilon >0$ is small enough compared to the constant $\bar{\kappa}_0$ in \eqref{cdass}, the number of incoming characteristics for systems \eqref{r48^} coincides with that for systems \eqref{b48^}. Considering for a moment ${\varphi}^{\,\varepsilon}$ as a given function and taking into account \eqref{quadr}, we easily see that the boundary conditions \eqref{r50^} (without the last one) are {\it strictly dissipative}:
\[
\sum_{\pm}\bigl(\bigl\{\varepsilon I_6- \widetilde{A}_1(\widehat{U}^\pm,\widehat\Psi^{\pm}\ddot{})\bigr\}\dot{U}^{\pm \varepsilon} , \dot{U}^{\pm \varepsilon} \bigr)\bigr|_{x_1=0}\geq
\sum_{\pm}\frac{\varepsilon}{2}|\dot{U}^{\pm \varepsilon}_{|x_1=0}|^2- \frac{C}{\varepsilon}(|{g}|^2+|{\varphi}^{\,\varepsilon}|^2),
\]
where $I_6$ is the unit matrix of order 6. Then, by standard arguments of the energy method we obtain
\begin{multline}
\sum_{\pm}\bigl(\|\dot{U}^{\pm \varepsilon} (t)\|^2_{L^{2}(\mathbb{R}^2_+)} +\|\dot{U}^{\pm \varepsilon}_{|x_1=0} \|^2_{L^{2}(\partial\Omega_t)} \bigr) \\ \leq \frac{C}{\varepsilon^2} \biggl\{\sum_{\pm}\big(\|f^\pm \|^2_{L^{2}(\Omega_T)} + \|\dot{U}^{\pm \varepsilon} \|^2_{L^{2}(\Omega_t)}\big) + \|{g}\|^2_{L^{2}(\partial\Omega_T)}
+\|{\varphi}^{\,\varepsilon} \|^2_{L^2(\partial\Omega_t)}\biggr\}.
\label{est_reg}
\end{multline}
Using the Young inequality, from the last boundary condition in \eqref{r50^} we easily deduce
\begin{equation}
\|{\varphi}^{\,\varepsilon} (t)\|^2_{L^{2}(\mathbb{R}^)} \leq C\Big( \delta\|\dot{U}^{\pm \varepsilon}_{|x_1=0} \|^2_{L^{2}(\partial\Omega_t)} +\frac{1}{\delta}\|{\varphi}^{\,\varepsilon} \|^2_{L^2(\partial\Omega_T)}\Big),
\label{est_reg^}
\end{equation}
with a positive constant $\delta$. Combining inequalities \eqref{est_reg} and \eqref{est_reg^}, for a sufficiently small $\delta$ we obtain
\begin{multline*}
\sum_{\pm}\|\dot{U}^{\pm \varepsilon} (t)\|^2_{L^{2}(\mathbb{R}^2_+)} +\|{\varphi}^{\,\varepsilon} (t)\|^2_{L^{2}(\mathbb{R}^)} \\ \leq \frac{C}{\varepsilon^2} \biggl\{\sum_{\pm}\big(\|f^\pm \|^2_{L^{2}(\Omega_T)} + \|\dot{U}^{\pm \varepsilon} \|^2_{L^{2}(\Omega_t)}\big) + \|{g}\|^2_{L^{2}(\partial\Omega_T)}
+\|{\varphi}^{\,\varepsilon} \|^2_{L^2(\partial\Omega_t)}\biggr\}.
\end{multline*}
Applying then Gronwall's lemma, we get the $L^2$ estimate
\begin{equation}
\sum_{\pm}\|\dot{U}^{\pm \varepsilon} \|_{L^{2}(\Omega_T)} +\|{\varphi}^{\,\varepsilon} \|_{L^2(\partial\Omega_T)} \leq {C}({\varepsilon} ) \biggl\{\sum_{\pm}\|f^\pm \|_{L^{2}(\Omega_T)} + \|{g}\|_{L^{2}(\partial\Omega_T)}\biggr\},
\label{est_reg_a}
\end{equation}
where $C(\varepsilon )\rightarrow +\infty$ as $\varepsilon\rightarrow 0$, but now the constant $\varepsilon$ is {\it fixed}. Note that we could also include the $L^2$ norm of the trace of solution in this estimate.
Using tangential differentiation and the fact that the boundary $x_1=0$ is not characteristic for $\varepsilon \neq 0$, we also easily obtain the $H^1$ estimate
\begin{equation}
\sum_{\pm}\|\dot{U}^{\pm \varepsilon} \|_{H^{1}(\Omega_T)} +\|{\varphi}^{\,\varepsilon} \|_{H^1(\partial\Omega_T)} \\ \leq {C}({\varepsilon}) \biggl\{\sum_{\pm}\|f^\pm \|_{H^{1}(\Omega_T)} + \|{g}\|_{H^{1}(\partial\Omega_T)}\biggr\}.
\label{est_reg'}
\end{equation}
Having in hand the $L^2$ estimate \eqref{est_reg_a}, the existence of a weak $L^2$ solution to problem \eqref{r48^}--\eqref{r51^} can be obtained by the classical duality argument. To this end we should obtain an $L^2$ a priori estimate for the dual problem. The boundary conditions \eqref{r50^} are not strictly dissipative in the classical sense but in some sense they are similar to them and it is natural to expect the same from the dual problem. This will enable us to derive an a priori estimate for the dual problem.
To avoid overloading the paper and hiding simple ideas in the shadow of unimportant technical calculations, for the dual problem we restrict ourselves to ``frozen'' coefficients and the case $\hat{\varphi}=0$. More precisely, we consider the following constant coefficients version of problem \eqref{r48^}--\eqref{r51^} (we also drop the superscript $\varepsilon$ by the unknowns):
\begin{equation}
L^{\pm}U^{\pm}:=A_0\partial_t{U}^{\pm} -\varepsilon\partial_1{U}^{\pm} \pm{A}_1\partial_1{U}^{\pm}+A_2\partial_2{U}^{\pm} =f^{\pm}\qquad \mbox{in}\ \Omega_T,\label{mod1}
\end{equation}
\begin{equation}
[p]= a\varphi,\quad [v]=0,\quad [H_2]=b\varphi,\quad \partial_0\varphi =v_1^+ \qquad \mbox{on}\ \partial\Omega_T,
\label{mod2}
\end{equation}
\begin{equation}
({U}^+,U^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{mod3}
\end{equation}
where $A_{\alpha}$ are the matrices of the MHD system \eqref{4} calculated on a constant vector with $\hat{v}_1=0$, $v_2=\hat{v}_2$, $H_1=\widehat{H}_1$, etc. ($\hat{v}_2=\hat{v}_2^\pm ={\rm const}$, $\widehat{H}_1 =\widehat{H}_1^\pm ={\rm const}$, etc.); $a=-[\partial_1\hat{p}]={\rm const}$, $b=-[\partial_1\widehat{H}_2]={\rm const}$, $\partial_0=\partial_t+\hat{v}_2\partial_2$, and we consider yet homogeneous boundary conditions.
The dual operators have the form $(L^\pm )^*=-L^\pm$ (recall that we consider constant coefficients) and the boundary conditions for the dual problem are defined from the requirement
\begin{multline*}
\sum_{\pm}\Big\{
(L^{\pm}U^{\pm},\overline{U}^{\pm})_{L^2(\Omega_T)}-(U^{\pm},(L^{\pm})^*\overline{U}^{\pm})_{L^2(\Omega_T)}\Big\} \\= -\big[(A_1U,\overline{U})_{L^2(\Omega_T)}\big]+\varepsilon \sum_{\pm}(U^\pm ,\overline{U}^\pm )_{L^2(\Omega_T)}=0,
\end{multline*}
with $\overline{U}^{\pm}|_{t=T}=0$ and $U^\pm$ satisfying the boundary conditions \eqref{mod2}. Omitting simple calculations, we write down these boundary conditions:
\begin{eqnarray*}
\partial_0w=(a+b\widehat{H}_2)\overline{v}_1^+
-\varepsilon a \overline{p}^+-b( \widehat{H}_1\overline{v}_2^+ +\varepsilon \overline{H}_2^+), & \\[3pt]
[\overline{v}_1]=\varepsilon \langle \overline{p}\rangle , & \\[3pt]
\widehat{H}_1 [\overline{H}_2]=-\varepsilon \langle \overline{v}_1\rangle , & \\[3pt]
[\widehat{H}_2 \overline{v}_1 -\widehat{H}_1\overline{v}_2]=\varepsilon \langle \overline{H}_2\rangle , & \\[3pt]
\overline{H}_1^+ = \overline{H}_1^- =\overline{S}^+=\overline{S}^- =0& \qquad \mbox{at}\ x_1=0,
\end{eqnarray*}
where $\langle z\rangle :=(z^+ +z^-)|_{x_1=0}$ and
\begin{equation}
w=[\overline{p}+\widehat{H}_2\overline{H}_2]-\varepsilon \langle \overline{v}_1\rangle . \label{ww}
\end{equation}
Changing the time $t$ to $-t$ and then shifting it to the value $T$, we obtain the dual problem in the same form as the original problem \eqref{mod1}--\eqref{mod3}:
\begin{equation}
A_0\partial_t\overline{U}^{\pm} +\varepsilon\partial_1\overline{U}^{\pm} \mp {A}_1\partial_1\overline{U}^{\pm}-A_2\partial_2\overline{U}^{\pm} =\overline{f}^{\pm}\qquad \mbox{in}\ \Omega_T,\label{mod4}
\end{equation}
\begin{eqnarray}
\overline{\partial}_0w=\varepsilon a \overline{p}^+-(a+b\widehat{H}_2)\overline{v}_1^+
+b( \widehat{H}_1\overline{v}_2^+ +\varepsilon \overline{H}_2^+), & \label{mod5.1} \\[3pt]
[\overline{v}_1]=\varepsilon \langle \overline{p}\rangle , & \label{mod5.2} \\[3pt]
\widehat{H}_1 [\overline{H}_2]=-\varepsilon \langle \overline{v}_1\rangle , & \label{mod5.3} \\[3pt]
[\widehat{H}_2 \overline{v}_1 -\widehat{H}_1\overline{v}_2]=\varepsilon \langle \overline{H}_2\rangle , & \label{mod5.4} \\[3pt]
\overline{H}_1^+ = \overline{H}_1^- =\overline{S}^+=\overline{S}^- =0& \qquad \mbox{on}\ \partial\Omega_T, \label{mod5.5}
\end{eqnarray}
\begin{equation}
(\overline{U}^+,\overline{U}^-)=0\qquad \mbox{for}\ t<0,\label{mod6}
\end{equation}
where $\overline{\partial}_0=\partial_t-\hat{v}_2\partial_2$. The hyperbolic systems \eqref{mod4} together have four additional incoming characteristics compared to systems \eqref{mod1}. That is, the number of boundary conditions in \eqref{mod5.1}--\eqref{mod5.5} (they are eight together) coincides with the number of incoming characteristics of systems \eqref{mod4}.
We introduce the notation
\[
V^\pm =(\overline{p}^\pm ,\overline{v}_1^\pm , \overline{v}_2^\pm ,\overline{H}_2^\pm ).
\]
Then, in view of \eqref{ww} and the boundary conditions \eqref{mod5.2}--\eqref{mod5.4}, using the fact that $\langle z\rangle =2z^+_{|x_1=0} -[z]$, we obtain
\begin{equation}
[V]=2\varepsilon
\begin{pmatrix}
\overline{v}_1^+ +(\widehat{H}_2/\widehat{H}_1)\,\overline{v}_2^+ \\[3pt]
\overline{p}^+ \\[3pt]
(\widehat{H}_2/\widehat{H}_1)\,\overline{p}^+ - (1/\widehat{H}_1)\,\overline{H}_2^+\\[3pt]
-(1/\widehat{H}_1)\,\overline{v}_2^+
\end{pmatrix}
+\varepsilon^2 B_1V^+ + w\,B_2V^+,
\label{du1}
\end{equation}
where the coefficient matrix $B$ has elements of order ${\cal O}(1)$ as $\varepsilon \rightarrow 0$ whose exact form is of no interest and the exact form of the coefficient matrix $B_2$ is also unimportant for subsequent calculations.
As usual, the quadratic form
\[
Q=[( A_1\overline{U} , \overline{U} )] -\varepsilon \langle |\overline{U}|^2 \rangle = 2[\overline{p}\,\overline{v}_1]+2\widehat{H}_2[\overline{v}_1\overline{H}_2]-2\widehat{H}_1[\overline{v}_2\overline{H}_2] -\varepsilon \langle |V|^2 \rangle
\]
with the boundary matrix plays the crucial role for deriving a priori estimates by the energy method. Here we have already used the boundary conditions \eqref{mod5.5}. Applying then \eqref{du1} and omitting simple algebraic calculations, we get
\begin{multline}
Q=2\varepsilon |V^+_{|x_1=0}|^2 +\varepsilon^2 (MV^+,V^+)|_{x_1=0}\,+ w\, (q,V^+)|_{x_1=0} \\
= \varepsilon \langle |V|^2 \rangle +\varepsilon^2 (\widetilde{M}V^+,V^+)|_{x_1=0}\,+ w\, (\tilde{q},V^+)|_{x_1=0},
\label{QQQ}
\end{multline}
where the coefficient matrices $M$ and $\widetilde{M}$ as well the coefficient vectors $q$ and $\tilde{q}$ are of no interest (the elements of these matrices are of order ${\cal O}(1)$ for small $\varepsilon$, i.e., between the first two terms in the right-hand side of \eqref{QQQ} the first positive one is leading).
Using \eqref{QQQ} and the Young inequality, by standard arguments of the energy method we obtain for systems \eqref{mod4} (for $\varepsilon$ small enough) the energy inequality
\begin{multline}
\sum_{\pm}\big(\|\overline{U}^\pm (t)\|^2_{L^{2}(\mathbb{R}^2_+)} +\|\overline{U}^\pm _{|x_1=0} \|^2_{L^{2}(\partial\Omega_t)}\big) \\ \leq {C}({\varepsilon})
\Big( \|\overline{f}\|^2_{L^{2}(\Omega_T)} + \sum_{\pm}\|\overline{U}^\pm \|^2_{L^{2}(\Omega_t)}+ \|w|_{x_1=0} \|^2_{L^{2}(\partial\Omega_t)}
\Big)
\label{mod8}
\end{multline}
(note that, in view of \eqref{mod5.5}, $\|\overline{V}^\pm _{|x_1=0} \|^2_{L^{2}(\partial\Omega_t)}=\|\overline{U}^\pm _{|x_1=0} \|^2_{L^{2}(\partial\Omega_t)}$). At the same time, from the boundary condition \eqref{mod5.1} we get
\begin{equation}
\|w|_{x_1=0} (t)\|^2_{L^{2}(\mathbb{R})}\leq
C\Big( \delta\|V^+|_{x_1=0} \|^2_{L^{2}(\partial\Omega_t)} +\frac{1}{\delta}\|w|_{x_1=0} \|^2_{L^{2}(\partial\Omega_t)} \Big),
\label{mod9}
\end{equation}
with a positive constant $\delta$. By choosing $\delta$ small enough and combining inequalities \eqref{mod8} and \eqref{mod9}, one gets
\begin{multline*}
\sum_{\pm}\|\overline{U}^\pm (t)\|^2_{L^{2}(\mathbb{R}^2_+)} +\|w |_{x_1=0}(t)\|^2_{L^{2}(\mathbb{R})} \\ \leq {C}({\varepsilon})
\Big( \|\overline{f}\|^2_{L^{2}(\Omega_T)} + \sum_{\pm}\|\overline{U}^\pm \|^2_{L^{2}(\Omega_t)}+ \|w|_{x_1=0} \|^2_{L^{2}(\partial\Omega_t)}
\Big).
\end{multline*}
Applying Gronwall's lemma to the last inequality gives the desired a priori estimate
\begin{equation}
\sum_{\pm}\|\overline{U}^\pm \|_{L^{2}(\Omega_T)} \leq C({\varepsilon}) \sum_{\pm}\|\overline{f}^\pm \|_{L^{2}(\Omega_T)}
\label{estdu}
\end{equation}
for the dual problem.
Thanks to the control on the trace of solution (``strict dissipativity'') we can easily generalize estimate \eqref{estdu} to the case of inhomogeneous boundary conditions in \eqref{mod5.1}--\eqref{mod5.5}. Thus, for the problem adjoint to \eqref{r48^}--\eqref{r51^} we can derive the a priori $L^2$ estimate
\begin{equation}
\sum_{\pm}\|\overline{U}^{\pm \varepsilon} \|_{L^{2}(\Omega_T)} \leq {C}({\varepsilon}) \biggl\{\sum_{\pm}\|\overline{f}^\pm \|_{L^{2}(\Omega_T)} + \|\overline{g}\|_{L^{2}(\partial\Omega_T)}\biggr\}
\label{est_reg_b}
\end{equation}
with no loss of derivatives from the data $\overline{f}^\pm$ and $\overline{g}$. In view of the a priori estimates \eqref{est_reg_a} and \eqref{est_reg_b}, the classical duality argument gives the existence of a weak $L^2$ solution to problem \eqref{r48^}--\eqref{r51^}. Then, tangential differentiation and the fact the boundary $x_1=0$ is not characteristic for $\varepsilon \neq 0$ gives the existence of an $H^1$ solution for any fixed sufficiently small parameter $\varepsilon >0$. Its uniqueness follows from the a priori estimate \eqref{est_reg_a}.
\end{proof}
The a priori estimate \eqref{est_reg'} is not uniform in $\varepsilon$ and not suitable to pass to the limit as $\varepsilon\rightarrow 0$. However, thanks to Lemma \ref{lem2} we have now the existence of solutions of the regularized problem for any fixed sufficiently small parameter $\varepsilon >0$. Estimate \eqref{est_reg'} as well as Lemma \ref{lem2} holds true even if the stability condition \eqref{RTL} is violated. Below, assuming the fulfilment of \eqref{RTL} and a more regularity for the source term $g$ (as in Theorem \ref{t1}, $g\in H^{3/2}(\partial\Omega_T)$), we will get for problem \eqref{r48^}--\eqref{r51^} an a priori estimate uniform in $\varepsilon$. Actually, this estimate is nothing else than our basic a priori estimate \eqref{main_est}.
\subsection{Uniform in $\varepsilon$ estimate and passage to the limit}
We now revisit several places in the arguments of Section \ref{sec:4} and adapt them for problem \eqref{r48^}--\eqref{r51^}. First of all, as for problem \eqref{b48^}--\eqref{b51^}, it is enough to prove the a priori estimate \eqref{main_est'} for a corresponding reduced problem with homogeneous boundary conditions. Passing to the new unknown
\[
U^{\pm \varepsilon}=\dot{U}^{\pm \varepsilon}-\widetilde{U}^\pm ,
\]
where $\widetilde{U}^\pm$ is the same as in \eqref{a87'}, we obtain the following problem (for the sake of notational convenience, we below omit the index $\varepsilon$ in $U^{\pm \varepsilon}$, $F^{\pm \varepsilon}$, etc.):
\begin{multline}
A_0(\widehat{U}^{\pm})\partial_t{U}^{\pm} +\bigl(\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})-\varepsilon I_6\bigr)\partial_1{U}^{\pm}\\ +A_2(\widehat{U}^{\pm} )\partial_2{U}^{\pm}
+\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm}){U}^{\pm} =F^{\pm}\qquad \mbox{in}\ \Omega_T,\label{rb48b}
\end{multline}
\begin{eqnarray}
{[}p{]}=- \varphi {[}\partial_1\hat{p}{]}, & \label{rb50b.1}\\[3pt]
{[}v{]}=0, & \label{rb50b.2}\\[3pt]
{[}H_{\tau}{]}=-\varphi {[}\partial_1\widehat{H}_{\tau}{]}, & \label{rb50b.3} \\[3pt]
{v}_{N}^+ = \partial_0\varphi - \varphi \partial_1\hat{v}_N^+ & \qquad \mbox{on}\ \partial\Omega_T
\label{rb50b.4}
\end{eqnarray}
\begin{equation}
({U}^+,{U}^-,\varphi )=0\qquad \mbox{for}\ t<0,\label{rb51b}
\end{equation}
where
\[
F^\pm =f^\pm-A_0(\widehat{U}^{\pm})\partial_t\widetilde{U}^{\pm} -\left(\widetilde{A}_1(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})-\varepsilon I_6\right)\partial_1\widetilde{U}^{\pm} \\ -A_2(\widehat{U}^{\pm} )\partial_2\widetilde{U}^{\pm}-\mathcal{C}(\widehat{U}^{\pm},\widehat{\Psi}^{\pm})\widetilde{U}^\pm
\]
obeys the uniform in $\varepsilon$ estimate \eqref{nonhom_F'}.
Solutions to problem \eqref{rb48b}--\eqref{rb51b} satisfy \eqref{b43b} and the counterpart of \eqref{aa1^} reads:
\begin{equation}
\partial_t a^{\pm}-\varepsilon\partial_1a^{\pm}+ \frac{1}{\partial_1\widehat{\Phi}^{\pm}}\left\{ (\hat{w}^{\pm} ,\nabla a^{\pm}) + a^{\pm}\bigl({\rm div}\,\hat{u}^{\pm}-\varepsilon\partial_1^2\widehat{\Phi}^{\pm}\bigr)\right\}={\mathcal F}^{\pm}\quad \mbox{in}\ \Omega_T,
\label{raa1^}
\end{equation}
where $a^{\pm}=F_7^{\pm}/\partial_1\widehat{\Phi}^{\pm},\quad {\mathcal F}^{\pm}=({\rm div}\,{F}_{h}^{\pm})/\partial_1\widehat{\Phi}^{\pm}$,
${F}_{h}^{\pm}=(F_{N}^{\pm} ,\partial_1\widehat{\Phi}^{\pm}F_5^{\pm})$ and $F_{N}^{\pm}=F_4^{\pm}-F_5^{\pm}\partial_2\widehat{\Psi}^{\pm}$. As \eqref{aa1^}, equations \eqref{raa1^} do not still need any boundary conditions and we get estimates \eqref{f7^} and \eqref{f7"}. Moreover, below we will also need a uniform in $\varepsilon$ estimate for the trace $\varepsilon F_7^+|_{x_1=0}$. Exactly as for the case of strictly dissipative boundary conditions, from \eqref{raa1^} we easily obtain the inequality
\[
\|a^+(t)\|^2_{L^2(\mathbb{R}^2_+)} +\varepsilon\|a^+_{|x_1=0}\|^2_{L^2(\partial\Omega_t)}\leq C\left\{\|F^+ \|^2_{H^1(\Omega_T)} + \|a^+\|^2_{L^2(\Omega_T)}\right\}
\]
giving, together with \eqref{f7^}, the estimate
\[
\sqrt{\varepsilon}\|a^+_{|x_1=0}\|_{L^2(\partial\Omega_T)}\leq C\|F^+ \|_{H^1(\Omega_T)}
\]
and
\begin{equation}
\|\varepsilon F_7^+|_{x_1=0}\|_{L^2(\partial\Omega_T)}\leq \sqrt{\varepsilon}C\|a^+_{|x_1=0}\|_{L^2(\partial\Omega_T)}\leq C\|F^+ \|_{H^1(\Omega_T)}.
\label{F7tr}
\end{equation}
In fact, in the proof of estimate \eqref{main_est'} for problem \eqref{rb48b}--\eqref{rb51b} only \eqref{eqH} needs a special care whereas the rest arguments are the same as in Section \ref{sec:4} for problem \eqref{b48b}--\eqref{b51b}. Indeed, taking into account \eqref{b43b}, for sufficiently small $\varepsilon$ we still can resolve systems \eqref{rb48b} for the normal derivatives $\partial_1V^\pm$ without the appearance of the unwanted multiplier $1/\varepsilon$. The rest of the arguments towards the proof of estimates \eqref{d1U} and \eqref{1Vt} are the same as in the proof of Proposition \ref{p5}. In particular, the counterparts of equations \eqref{entropy} will contain the additional terms $-\varepsilon\partial_1S^\pm$ in the left-hand sides of the equations. But, such equations for $S^\pm$ do not still need any boundary conditions and we get \eqref{ent_est}.
In the left-hand side of the counterpart of the energy inequality \eqref{q0} for problem \eqref{rb48b}--\eqref{rb51b} we have the additional positive terms
\[
\varepsilon\sum_{\pm}\|\partial_0U^\pm _{|x_1=0}\|^2_{L^2(\partial\Omega_t)}
\]
which can be just thrown away to make the energy inequality rougher. The rest of the arguments towards the proof of estimate \eqref{estd0} are again the same as in Section \ref{sec:4}, and it is important that the constant $C$ in \eqref{estd0} for \eqref{rb48b}--\eqref{rb51b} does not depend on $\varepsilon$.
Throwing away most of the positive terms containing the multiplier $\varepsilon$ in the counterpart of \eqref{q2} we obtain inequality \eqref{q2} with the quadratic form $Q_2$ replaced by $\widetilde{Q}_2$, where
\begin{equation}
2\widetilde{Q}_2 =2Q_2 +\varepsilon |\partial_2H_2^+|_{x_1=0}|^2.
\label{tildQ2}
\end{equation}
Let us now consider the counterpart of equation \eqref{eqH} for \eqref{rb48b}:
\begin{equation}
R = -\partial_0H_N^+ +\underline{\varepsilon\partial_1H_N^+ }\\ +{\rm coeff}\,v_N^+ +{\rm coeff}\,v_2^+ +{\rm coeff}\,H_N^+ +F_N^+\qquad \mbox{on}\ \partial\Omega_T,
\label{reqH}
\end{equation}
where we have the additional underlined term. In view of \eqref{b43b}, \eqref{reqH} implies
\begin{equation}
R = -\partial_0H_N^+ -\underline{\varepsilon\partial_2H_2^+ } +{\rm coeff}\,v_N^+ +{\rm coeff}\,v_2^+ +{\rm coeff}\,H_N^+ +F_N^+ +\underline{\varepsilon F_7^+}\qquad \mbox{on}\ \partial\Omega_T,
\label{reqH'}
\end{equation}
where we have the two additional underlined terms compared to \eqref{eqH}.
By virtue of \eqref{tildQ2} and \eqref{reqH'},
\[
2\widetilde{Q}_2 =2Q_2 + \varepsilon P+ \varepsilon\, {\rm coeff}\,F_7^+|_{x_1=0}\partial_2\varphi + \varepsilon\, {\rm coeff}\,F_7^+|_{x_1=0}\varphi ,
\]
where $Q_2$ is now given in \eqref{Q2"} (it is not the same as in \eqref{tildQ2}) and
\[
P= |\partial_2H_2^+|_{x_1=0}|^2+ {\rm coeff}\,\partial_2H_2^+|_{x_1=0}\partial_2\varphi + {\rm coeff}\,\partial_2H_2^+|_{x_1=0}\varphi .
\]
By the Young inequality we estimate:
\[
P \geq \frac{1}{2} |\partial_2H_2^+|_{x_1=0}|^2 - C((\partial_2\varphi )^2 + \varphi ^2)\geq - C((\partial_2\varphi )^2 + \varphi ^2).
\]
Using then the last inequality and estimate \eqref{F7tr} we get \eqref{IntQ2} and \eqref{Mt} with $\widetilde{Q}_2$ instead of $Q_2$ in the left-hand side of \eqref{IntQ2}, where the constant $C$ in \eqref{Mt} does not depend on $\varepsilon$. The rest of the arguments are the same as in Section \ref{sec:4} and for problem \eqref{rb48b}--\eqref{rb51b} we obtain the uniform in $\varepsilon$ estimate \eqref{main_est'}.
Returning to the original regularized problem \eqref{r48^}--\eqref{r51^} with nonhomogeneous boundary conditions, we obtain for it the uniform in $\varepsilon$ a priori estimate \eqref{main_est}.
\begin{lemma}
Let the basic state \eqref{a21} satisfies assumptions \eqref{a5}--\eqref{avn} (in the 2D planar case) together with condition \eqref{RTL}. Then for any sufficiently small constant $\varepsilon >0$ and for all $f^{\pm} \in H^1(\Omega_T)$ and $g\in H^{3/2}(\partial\Omega_T)$ which vanish in the past the a priori estimate
\begin{equation}
\sum_{\pm}\|\dot{U}^{\pm \varepsilon}\|_{H^{1}(\Omega_T)}+\|\varphi^{\varepsilon}\|_{H^1(\partial\Omega_T)} \leq C\biggl\{\sum_{\pm}\|f^\pm \|_{H^{1}(\Omega_T)}+ \|g\|_{H^{3/2}(\partial\Omega_T)}\biggr\}
\label{rmain_est}
\end{equation}
holds for problem \eqref{r48^}--\eqref{r51^}, where $C=C(K,\bar{\rho}_0,\bar{p}_0,\bar{\kappa}_0,\epsilon,T)>0$ is a constant independent of $\varepsilon$ and the data $f^\pm$ and $g$.
\label{lem3}
\end{lemma}
We are now in a position to pass to the limit as $\varepsilon \rightarrow 0$. In view of Lemma \ref{lem2}, for all sufficiently small $\varepsilon$ problem \eqref{r48^}--\eqref{r51^} admits a unique solution with the regularity described in Theorem \ref{t1}. Due to the uniform a priori estimate \eqref{rmain_est} we can extract a subsequence weakly convergent to $((\dot{U}^+, \dot{U}^-) ,\varphi )\in H^1(\Omega_T)\times H^1(\partial\Omega_T)$ with $(\dot{U}^+, \dot{U}^-)|_{x_1=0}\in H^{1/2}(\partial\Omega_T)$. Passing to the limit in \eqref{r48^}--\eqref{r51^} as $\varepsilon\to 0$ immediately gives that
$(\dot{U}^+, \dot{U}^- ,\varphi )$ is a solution to problem \eqref{b48^}--\eqref{b51^}. The a priori estimate \eqref{main_est} implies its uniqueness. The proof of Theorem \ref{t1} is complete. | {"config": "arxiv", "file": "1311.6373.tex"} |
TITLE: Find $k_{max}$ if such $\frac{2(a^2+kab+b^2)}{(k+2)(a+b)}\ge \sqrt{ab}$
QUESTION [1 upvotes]: Let $a,b>0$ then we have
$$\color{crimson}{\dfrac{2(a^2+kab+b^2)}{(k+2)(a+b)}\ge \sqrt{ab}}$$ Find $k_{\max}$
Everything I tried has failed so far.
Here is one thing I tried, but obviously didn't work.
Consider the Special case $a=b$
then
$$\color{crimson}{\dfrac{2(a^2+kab+b^2)}{(k+2)(a+b)}=\dfrac{2(2a^2+ka^2)}{2(k+2)a}=a=RHS}$$Thanks in advance
REPLY [3 votes]: The inequality can be rewritten as $$2(a-b)^2\ge (k+2)(\sqrt{a}-\sqrt{b})^2$$ which is equivalent when $a\neq b$ to $$2(\sqrt{a}+\sqrt{b})^2\ge (k+2)\sqrt{ab}$$ or $$2(\sqrt{a}-\sqrt{b})^2\ge (k-6)\sqrt{ab}$$ This shows that the inequality always holds for $k=6$. For any $k>6$, we may take $a=1, b=1+\epsilon$ for sufficiently small $\epsilon>0$, and the inequality becomes false. So $k=6$ is best. | {"set_name": "stack_exchange", "score": 1, "question_id": 1788931} |
TITLE: $\lim_{x\to 0}(1+\sin(x)-x)^{1/x^3}$
QUESTION [2 upvotes]: Evaluate$$\lim_{x\to 0}\left(1+\sin(x)-x\right)^{{1}/{x^3}}$$
i was able to solve it using a taylor exp' for $\sin(x)$ but id like to know if there is a "simpler " way. something along the lines of $e^{\log}$-ing it, or L'hopital-ing it...
REPLY [1 votes]: You know that $\lim_{x\to a}f(x)^{g(x)}$ exists if and only if $\lim_{x\to a}g(x)\log f(x)=l$ exists and, in this case, the original limit is $e^l$ (with the extended rule that, if $l=-\infty$, the original limit is $0$ and, if $l=\infty$, the original limit is $\infty$).
So let's compute
$$
\lim_{x\to0}\frac{\log(1+\sin x-x)}{x^3}
$$
The Taylor expansion of $\log(1+t)$ is $t+o(t)$; moreover the Taylor expansion of $\sin x$ is $x-x^3/6+o(x^3)$. Thus
\begin{align}
\log(1+\sin x-x)
&=\log\left(1+x-\frac{x^3}{6}+o(x^3)-x\right) \\
&=\log\left(1-\frac{x^3}{6}+o(x^3)\right) \\
&=-\frac{x^3}{6}+o(x^3)
\end{align}
So we have
$$
\lim_{x\to0}\frac{\log(1+\sin x-x)}{x^3}=
\lim_{x\to0}\frac{-x^3/6+o(x^3)}{x^3}=-\frac{1}{6}
$$
Thus your limit is
$$
\lim_{x\to0}(1+\sin x-x)^{1/x^3}=e^{-1/6}
$$ | {"set_name": "stack_exchange", "score": 2, "question_id": 1823766} |
TITLE: Why do $\int_{0}^{\pi}d\theta\cos k\theta \cos n\theta$ and $\int_{0}^{\pi}d\theta\sin k\theta \sin n\theta$ equal zero, except when $|k|=n$?
QUESTION [1 upvotes]: I do not understand why for the two integrals below, the result is always $0$ unless $|k|=n$, in which case the result is $\pi/2$.
$$\int_{0}^\pi d\theta\cos k\theta \cos n\theta \qquad\qquad
\int_{0}^\pi d\theta \sin k \theta \sin n \theta$$
I have tried using trigonometric identities to get a general solution but I had no luck understanding the nature of the integrals. If anyone can point out any hints or patterns, it would be much appreciated. Thanks
REPLY [3 votes]: Let $$I_1 = \int_0^\pi \sin(kx)\sin(nx)dx $$
$$I_2 = \int_0^\pi \cos(kx)\cos(nx)dx $$
$$ I_3 = I_1 + I_2 = \int_0^\pi \cos((k-n)x)dx$$
$$I_4 = I_2 - I_1 = \int_0^\pi \cos((k+n)x)dx$$
Provided that $n$ and $k$ are both integers , both $I_3$ and $I_4$ will vanish unless $k=\pm n$, in which case one still vanishes and the other equals $\pi$.
REPLY [1 votes]: The cause is something referred to as the orthogonality of the functions sine and cosine. We define two functions, $f,g$ to be orthogonal on an interval $[a,b]$ if $\int_a^b fg = 0$. The cosine and sine functions exhibit this property over an interval whose length is an integer multiple of their period.
I'll cover showing this in the cosine case. The sine case is fairly similar.
If you try getting the antiderivative in the cosine case for $k \ne n$, we begin with
$$\int \cos(kx)\cos(nx) dx$$
Using one of the product to sum formulas (here's a handy trig reference sheet by the way), we see
$$\int \cos(kx)\cos(nx) dx = \frac 1 2 \left( \int \cos((k-n)x)dx + \int \cos((k+n)x)dx \right)$$
Evaluating the antiderivatives, we see then
$$\int \cos(kx)\cos(nx) dx = \frac 1 2 \left( \color{blue}{\frac{1}{k-n}} \sin((k-n)x) + \frac{1}{k+n}\sin((k+n)x) \right) + C$$
for constant of integration $C$. Notice how this is undefined if $k=n$ because of the fraction in blue. As a result, we have to derive the $k=n$ case separately. Thus, if $k=n$,
$$\int \cos(kx)\cos(nx) dx = \int \cos^2(kx) dx$$
Using one of the alternate forms of the half-angle formulas on that reference sheet, we see that
$$\int \cos^2(kx) dx = \frac 1 2 \int dx + \frac 1 2 \int \cos(2kx)dx$$
Evaluating the antiderivative, we get
$$\int \cos^2(kx) dx = \frac 1 2 x + \frac 1 2 \cdot \frac{\sin(2x)}{2k} = \frac 1 2 x + \frac 1 {4k} \sin(2x) + C$$
Thus we conclude:
$$\int \cos(kx)\cos(nx) dx = \left\{\begin{matrix}
\frac 1 2 \left( \frac{1}{k-n} \sin((k-n)x) + \frac{1}{k+n}\sin((k+n)x) \right) + C & k \neq n\\
\frac 1 2 x + \frac 1 {4k} \sin(2x) + C & k=n
\end{matrix}\right.$$
If we try to then evaluate the antiderivative over an interval $[a,a+2\pi]$, we get an interesting result. Note that there is a proof out there that the integral from $[a,a+p]$ for a function of period $p$ is equal to that of the integral over $[0,p]$, so it is sufficient to look at the integral over $[0,2\pi]$.
Evaluating at the respective bounds in the antiderivative, if $k,n$ are integers, you'll immediately realize that the sine of a multiple of $\pi$ is zero, and thus the first case is $0$. In the $k=n$ case, we simply get $\pi/2$ by a similar process. Thus,
$$\int_a^{a+2\pi} \cos(kx)\cos(nx) dx = \int_0^{2\pi} \cos(kx)\cos(nx) dx =\left\{ \begin{matrix}
0 & k \neq n \\
\pi/2 & k=n
\end{matrix}\right. $$
This also gives us that $\cos(kx)$ is orthogonal to $\cos(nx)$ whenever $k \neq n$. This proves to be useful in some higher mathematical settings, e.g. Fourier analysis. | {"set_name": "stack_exchange", "score": 1, "question_id": 3178758} |
TITLE: Uniformly scale a hollow cylinder, but keep the hole radius constant
QUESTION [1 upvotes]: I'd like to linearly scale a hollow cylinder so that I get some specific volume while keeping the original proportions (i.e., outer radius & height). The problem is that I'd like the size of the hole to stay the same, so I can't just use the cube root of the old-to-new-volume ratio as the scaling factor for all dimensions.
How can this be solved?
Background: I'm making a set of molds for concrete weight plates. The heaviest plate has a standardized diameter and the height (thickness) will be determined by the amount of concrete poured. Lighter weights are arbitrarily sized, and using the same diameter would make them too thin (fragile), while using the same thickness and smaller diameter would make them too fat to stack them all on the end of a barbell. So proportionally scaling down the largest mold seems like the best option.
REPLY [0 votes]: Let's say our disks have inner radius $r$ (which is constant under scaling), outer radius $R$ (so $r < R$), and thickness $h$. The volume is
$$
V = \pi R^{2}h - \pi r^{2}h = \pi(R^{2} - r^{2})h.
$$
Our goal is to pick a sequence of $R$s and $h$s to get specified volumes/weights.
If we scale the (outer) radius and thickness together, with scale factors $s_{0} = 1$, $s_{1}$, $s_{2}$, $s_{3}$, ..., we get the sequence $R$, $s_{1}R$, $s_{2}R$, $s_{3}R$, ..., and similarly for $h$. The resulting volumes are
\begin{align*}
V_{0} &= \pi(R^{2} - r^{2})h, \\
V_{1} &= \pi(s_{1}^{2}R^{2} - r^{2})s_{1}h = \pi [s_{1}^{3}R^{2} - s_{1}r^{2}]h, \\
V_{2} &= \pi(s_{2}^{2}R^{2} - r^{2})s_{2}h = \pi [s_{2}^{3}R^{2} - s_{2}r^{2}]h, \\
V_{3} &= \pi(s_{3}^{2}R^{2} - r^{2})s_{3}h = \pi [s_{3}^{3}R^{2} - s_{3}r^{2}]h,
\end{align*}
and so forth.
Note that it might be desirable (aesthetically and/or mechanically) instead to scale the dimensions so that the area and thickness scale together. For example, making a disk half as thick wouldn't make the radius half as large, but about $70$ percent as large, because $\sqrt{1/2} \approx 0.707\dots$. In this case, the sequence of radii is $R$, $\sqrt{s_{1}}R$, $\sqrt{s_{2}}R$, $\sqrt{s_{3}}R$, ..., and the resulting volumes are
\begin{align*}
V_{0} &= \pi(R^{2} - r^{2})h, \\
V_{1} &= \pi(s_{1}R^{2} - r^{2})s_{1}h = \pi [s_{1}^{2}R^{2} - s_{1}r^{2}]h, \\
V_{2} &= \pi(s_{2}R^{2} - r^{2})s_{2}h = \pi [s_{2}^{2}R^{2} - s_{2}r^{2}]h, \\
V_{3} &= \pi(s_{3}R^{2} - r^{2})s_{3}h = \pi [s_{3}^{2}R^{2} - s_{3}r^{2}]h,
\end{align*}
and so forth. The respective results look like this:
As you note, volume does not scale proportionally to the dimensions because the inner radius is constant. But no matter which strategy we use, one of the above or some other, the next step is to take the desired volumes (weights) and the standarized largest radius $R$, calculate $h$, and then successively calculate $s_{1}$, $s_{2}$, $s_{3}$, ... either numerically or by algebra.
A fringe benefit of the second scheme is that we only need to solve a quadratic equation to get the scale factors. For the first method we need to solve a cubic. This can be done algebraically, but is messier.
The next time you hear someone make a crack about the uselessness of algebra, I hope you'll set them straight. ;) | {"set_name": "stack_exchange", "score": 1, "question_id": 4273512} |
TITLE: continuous extension of $P_r*F(θ)$
QUESTION [0 upvotes]: Let $F:[-π,π]→[0,∞]$ be $2π$_ periodic, integrable and "continuous" function. For $-π≤θ≤π$ and $0≤r<1$.
Show that $$ P_r*F(θ)=\dfrac{1}{2\pi}\int_{-\pi}^{\pi}\dfrac{1-r^2 }{1-2r \cos (θ-t)+r^2}F(t) dt$$
can be continuously extended to the unit circle, as mapping again to$ [0,∞]$?
I know that $ P_r*F(θ)→ F(\theta)$ as $r→ 1$, when $F$ is continous on the unit circle, but I'm not sure how to prove this 'continuous extension' when $F$ takes $∞$, If any one can help!
Thanks.
REPLY [0 votes]: The proof for the usual continuous (finite) case is local, so in other words, showing that $P_r*F(e^{i\theta})→ F(e^{i\theta})$ depends only on a small neighborhood of $e^{i\theta}$ on the circle (and moreover one can show "full" continuity in the sense that if $u(re^{i\theta})=P_r*F(e^{i\theta})$ we actually have $u(z) \to F(e^{i\theta}), z \to e^{i\theta}, |z| \le 1$.
In particular, for finite points of continuity, $\theta$ nothing changes as long as we can show that away from $e^{i\theta}$ the Poisson integral converges to zero with $r$. Fixing $e^{i\theta}=1$ for notational convenience, this would mean to show that
$\dfrac{1}{2\pi}\int_{\delta}^{2\pi-\delta}\dfrac{1-r^2 }{1-2r \cos (t)+r^2}F(t) dt \to 0, r \to 1, \delta >0$ arbitrary; in the usual case this follows immediately from the Poisson Kernel properties and the boundness of $F$, while here we use that $F \ge 0$ integrable so we can apply the Monotone convergence theorem since $g(r,t)=\dfrac{1-r^2 }{1-2r \cos (t)+r^2}F(t)$ is decreasing in $r$ to zero (at least where $F(t)$ is finite hence ae) for $r \ge |\cos \delta|$ - as the derivative in $r$ is clearly negative for $r \ge |\cos t|$ and $|\cos t| \le \cos \delta$
Hence the only thing you need to do is to prove the result at points $\theta$ where $F$ is infinite and assume for notational convenience $e^{i\theta}=1$ - so we need to prove that $u(z) \to \infty, z \to 1$; but for each $N$ there there is a neighborhood of $1$ where $F(e^{i\theta}) \ge N, |\theta| \le 2\alpha_N$ for some small $\alpha_N >0$ so
$u(re^{i\theta})=\dfrac{1}{2\pi}\int_{-\pi}^{\pi}\dfrac{1-r^2 }{1-2r \cos (θ-t)+r^2}F(t) dt=\dfrac{1}{2\pi}\int_{-\pi}^{\pi}\dfrac{1-r^2 }{1-2r \cos t+r^2}F(t+\theta) dt \ge$
$\ge \dfrac{1}{2\pi}\int_{-\alpha_N}^{\alpha_N}\dfrac{1-r^2 }{1-2r \cos t+r^2}F(t+\theta) dt \ge \dfrac{N}{2\pi} \int_{-\alpha_N}^{\alpha_N}\dfrac{1-r^2 }{1-2r \cos t+r^2}dt$, for $|\theta| \le \alpha_N$
But now as before $\dfrac{1}{2\pi}(\int_{-\pi}^{-\alpha_N}+\int_{\alpha_N}^{\pi})\dfrac{1-r^2 }{1-2r \cos (t)+r^2} dt \to 0, r \to 1$ so one can pick $r_N$ for which the integral above is at most $1/N, r \ge r_N$ and putting things together we get:
$u(re^{i\theta}) \ge \dfrac{N}{2\pi} \int_{-\alpha_N}^{\alpha_N}\dfrac{1-r^2 }{1-2r \cos t+r^2}dt \ge \dfrac{N}{2\pi} (\int_{-\pi}^{\pi}\dfrac{1-r^2 }{1-2r \cos t+r^2}dt-1/N)=\dfrac{N-1}{2\pi}$ for $|\theta| \le \alpha_N, r \ge r_N$ hence indeed $u(z) \to \infty, z \to 1$ and we are done! | {"set_name": "stack_exchange", "score": 0, "question_id": 3891702} |
TITLE: Can a cylindrical chamber geared into a larger one on the outside decrease the size need for artificial gravity?
QUESTION [0 upvotes]: I’m designing a interplanetary ship and I’m trying to figure out how to fix the large size required for a ship with artificial gravity given budget. My question is if there was a main cylindrical body of a space ship which was spinning at x1 velocity and inside that was another cylindrical chamber geared into it being spun at x1 by the outer cylinder. Could the inside chamber be spun more by gearing it in on the inside of the main chamber to a motor of sorts? Basically it would be spinning by X1 plus the more rotation that x2 would spin it. I’m hoping this might reduce the requirements of such a large ship. This could be completely wrong I’m just wondering is it and if so could someone point me in the right direction.
Basically to the inside chamber it’s like it’s only spinning x2 but from POV of outer space it’s spinning X1 and X2 because it’s being rotated by the outside cylindrical body 1X and a motor spinning itself x2.
REPLY [0 votes]: You can do this, although it may actually make it harder to develop such a ship rather than easier.
The key equation is the equation for centripetal acceleration: $a_c=r\omega^2$, where $r$ is the radius of the rotation and $\omega$ is the angular velocity (how fast it is spinning). There's a few variants of that (some use velocity rather than angular velocity), but this one is in a convenient form to see what matters here.
If you create a smaller inner rotation, you inherently decrease the centripetal acceleration at the same angular velocity, and thus are forced to spin it faster to achieve the same pseudo-gravity. So you have to watch out for that.
Your approach would be useful in situations where you had a reason to keep the outside "spun down." Perhaps you were looking to observe the stars. However, the price you pay is the coupling. That gear train your describe is a mechanical linkage that has to operate very reliably for very long amounts of time. It may be easier to simply spin up the whole craft, just to avoid such mechanics. You might need exotic tricks to minimize wear, like magnetic gears.
In general, its easier to spin up the craft because you can use thrusters to do so. However, what you describe is really just a twisted version of reaction wheels, which are used on many spacecraft like the ISS to control orientation. So in that sense it works.
I would be careful making the inner cylinder too small. One of the major reasons to make these rings big is that the human brain is not used to rotating frames. That's why happy-go-pukey rides like the Tilt-a-wheel are such a thrill. In particular, the Coriolis effect really messes with our brains. If we make the rings large, we can achieve the same pseudogravity with a smaller angular rate, which is better for our minds.
I don't have the book on hand, but Rendezvous with Rama, by Arthur C. Clarke, had a brilliant example of this. In that book, there is a 1km diameter cylinder. You dock in the center, and walk down a spiral staircase to the bottom. In the book they describe it as a treacherous descent. It seems so easy at first in the low pseudogravity that you might try skipping a step or two to speed the climb down to the bottom. But one had to constantly remember that the stairs were rotating away from you. One misstep and you may float away from the stairs, unable to stop until colliding with the spinning surface 1km below. | {"set_name": "stack_exchange", "score": 0, "question_id": 632138} |
{\bf Problem.} A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different?
{\bf Level.} Level 3
{\bf Type.} Prealgebra
{\bf Solution.} There are 10 possibilities for the first digit. After the first digit has been chosen, there are 9 possibilities for the second digit, and after the first two digits have been chosen there are 8 possibilities for the last digit. The total number of possible settings is $10\cdot 9\cdot 8=\boxed{720}$. | {"set_name": "MATH"} |
TITLE: Proving existence of a square-free sequence
QUESTION [11 upvotes]: I found this problem and a solution sketch in a MathOverflow answer, and I thought it was nice enough to deserve more attention and a properly written solution.
Problem:
Prove that for each natural number $n$, there is some natural number $r$ for which the $n$ integers $r+1^2,r+2^2,…r+n^2$ are all square-free.
REPLY [2 votes]: Let $p_1, p_2, \cdots , p_k$ be all the prime numbers less than $n^2$ (for some $k$). For any $K > k$, let $p_{k+1}, p_{k+2}, \cdots , p_K$ be the next prime numbers after that (larger than $n^2$).
For $1 \le i \le k$, there is at least one value of $r$ $\pmod {p_i^2}$ that satisfies $r + 1^2, r+2^2, \cdots , r+n^2 \not \equiv 0$ (i.e. there is at least one viable possible remainder for $r$.) To see this, note that the squares modulo $p_i^2$ do not include $p_i$ itself, thus setting $r \equiv -p_i$ is sufficient.
For $k+1 \le i \le K$, the numbers $1^2, 2^2, \cdots , n^2$ are all distinct $\pmod {p_i}$, because $p_i > p_{k+1} > n^2$. Therefore, there are exactly $(p_i^2 - n)$ viable possibilities for $r \pmod {p_i}$.
Now let $N = p_1^2p_2^2p_3^2\cdots p_K^2$.
By the Chinese remainder theorem, there are at least
$$(1)(1) \cdots (1)(p_{k+1}^2 - n)(p_{k+2}^2 - n)\cdots (p_{K}^2 - n)$$
viable options for $r$ modulo $N$ (by which we mean values of $r$ modulo $N$ that cause $r + 1^2, r + 2^2, \cdots r+n^2$ to be free from squares of all the primes $p_1, \cdots ,p_K$).
For any $K$, we define
$$
x_K := \frac{\text{Number of viable options for } r \text{ mod } N}{N}
$$
which evaluates to
$$
\frac{(p_{k+1}^2 - n)(p_{k+2}^2 - n)\cdots (p_{K}^2 - n)}{p_1^2p_2^2p_3^2\cdots p_K^2}
=
\frac{1}{p_1}\frac{1}{p_2} \cdots \frac{1}{p_k} \left(1 - \frac{n}{p_{k+1}^2}\right)\cdots \left(1 - \frac{n}{p_K^2}\right)
$$
The infinite product $\lim_{K \to \infty} \left(x_K \right)$ converges to a positive number, and the viable options for $r$ modulo $N$ for the first $K+1$ primes are a subset of the viable options for $r$ modulo $N$ for the first $K$ primes for any $K$.
Therefore, there are infinitely many values of $r$ which work for every $K$.
In particular, there is at least one positive integer $r$ such that $r + 1^2, r+2^2, \cdots r+n^2$ are squarefree. | {"set_name": "stack_exchange", "score": 11, "question_id": 368513} |
TITLE: Integrate a rational function with a denominator in the form a - cos(x)
QUESTION [3 upvotes]: I am preparing for a basic level calculus test and came across this problem:
$$ \int_0^{\pi/2} \dfrac{1}{2-\cos{x}} dx$$
Which appears to be simple enough before I realize that I can't multiply the top and bottom by $ 2+\cos{x} $ and get anywhere fast. The prompt and Wolfram Alpha suggest a u-substitution of $ u = \tan{\frac{x}{2}} $ but I really have no idea how to implement that. How is this substitution used? Is this even an appropriate problem for my current level of math knowledge?
REPLY [2 votes]: We start with
$$\int_0^{\pi/2} \dfrac{1}{2-\cos{x}} dx$$
Let's start as you suggest and see where it takes us, multiplying top and bottom by $2 + \cos x$.
$$ \int_0^{\pi/2} \frac{2 + \cos x}{4 - \cos^2 x} dx = \int_0^{\pi/2} \frac{2 + \cos x}{3 + \sin^2 x} dx.$$
The mathematics of wishful thinking suggests that we split this integral and handle the easy part first.
$$ \int_0^{\pi/2} \frac{\cos x}{3 + \sin^2 x} dx = \int_0^1 \frac{1}{3 + u^2} du$$
when we perform the substitution $u = \sin x$. This is now a classic inverse trigonometric integral, and will lead you to $\arctan$.
What did we leave behind? We left
$$ \int_0^{\pi/2} \frac{2}{3 + \sin^2 x} dx.$$
This looks a bit tricky. One way to proceed is to recognize $3 = 3\sin^2 x + 3 \cos^2 x$, so that we have
$$ \int_0^{\pi/2} \frac{2}{3 \cos^2 x + 4\sin^2 x} dx.$$
Now multiplying by $\sec^2(x)/\sec^2(x)$ gives
$$\int_0^{\pi/2} \frac{2 \sec^2 x}{3 \cot^2 x + 4} dx.$$
Now you can finish with the substition $u = 3\cot^2 x + 4$.
Is this what I'd recommend? It's hard to say. The substitution you mention is a well-known substitution. But it's important to recognize that with these problems, as long as you follow your nose and charge forward, you'll probably reach a resolution. | {"set_name": "stack_exchange", "score": 3, "question_id": 1173023} |
TITLE: I can't find the critical points for this function. I showed my work :)
QUESTION [3 upvotes]: So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$
I know the derivative.
Derivative = $(3x^2-1)/(x^3-x)^2$
To find Critical Points I equal to $0$.
$x=1/\sqrt3$ and $x=-1/\sqrt3 $
But Critical points are the Max and Min value of your graph... and the graph is a little tricky... I don't know what to do... Because in my opinion, there are no critical points. It goes to infinity and -infinity.
Opinions? Help please!
REPLY [1 votes]: The points you found are just local extrema. Regarding the + and - $\infty$, you can simply note that those are local maxima of the function. | {"set_name": "stack_exchange", "score": 3, "question_id": 1267338} |
TITLE: which statement best describes the linear correlation
QUESTION [1 upvotes]: Over the course of a given week, 39 members of a reading club went to the library
The table below shoes the ages of these 39 members and the number of time they visited the library that week
Which one of the following statements best describes the linear correlation between the ages of the members and the number of library visits?
a) the correlation is positive and high
b) the correlation is positive and low
c) the correlation is negative and high
d) the correlation is negative and low
I don't understand how to solve this question just by looking at the table.
Do i draw a scatter plot? but we don't have ordered pairs (x,y)
REPLY [1 votes]: You have $39$ ordered pairs of the form $\langle\text{age},\text{number of visits}\rangle$; $2$ of them are $\langle 13,3\rangle$, $6$ of them are $\langle 13,5\rangle$, and so on. Thus, you could indeed draw a scatter-plot. But you probably don’t really need to, if you look carefully at the table. Where are the largest entries? $6$ of the seven $16$-year-olds went just once, $8$ of the ten $15$-year-olds went two or three times, $4$ of the nine $14$-year-olds went four times, and $6$ of the ten $13$-year-olds went $5$ times. What kind of correlation does that look like? | {"set_name": "stack_exchange", "score": 1, "question_id": 509473} |
TITLE: Proving the locus of a Fourier series is a system of perpendicular lines
QUESTION [0 upvotes]: From "Fourier's series and integrals" by H.S. Carslaw, there is the following question:
Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is represented by two systems of lines at right angles dividing the $(x,y)$-plane into squares of area $\pi^2$.
Really I have no idea how to prove it. First of all, the $(-1)^{n-1}$ means the sign in front of the sines is changing from positive to negative and back, yes? I don't understand how if $n$ is not changing the sum can be zero? What if all of the terms are positive, or all of the terms are negative? I think so - am I wrong? Also how to prove the claim in question would be interesting. Please help.
REPLY [3 votes]: My answer is not really straightforward anyway... Let's rewrite your equation as $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{2 n^2} \left(\cos(n (x-y)) - \cos(n (x+y))\right) = 0$.
This is equivalent to (convergence of the $f(t)$ series being clear) :
$$f(x-y)=f(x+y)\ \ \mathrm{for}\ f(t)=\sum_{n=1}^\infty \frac{(-1)^{n-1}\cos(n t)}{n^2}$$
But $\displaystyle f(t)=-\sum_{n=1}^\infty \frac{\cos(n (t+\pi))}{n^2}$ and this last sum is well known (or may be obtained by integration of the classical 'Sawtooth Wave' $\sum_{n=1}^\infty \frac{\sin(n (u))}{n}=\frac{\pi-u}{2}$ ) : $$ \sum_{n=1}^\infty \frac{\cos(n u)}{n^2}=\frac{(\pi-u)^2}{4}-\frac{\pi^2}{12}\ \ \mathrm{for}\ u \in (0,2\pi)$$
So that $f(t)=\frac{\pi^2}{12}-\frac{t^2}{4}$ for $t \in (-\pi,\pi)$ and $f(t+2k\pi)=f(t)$ (of course $f$ is even).
At this point $f(x-y)=f(x+y)$ is possible only for $x-y=x+y \mod (2 \pi)$ or $y-x=x+y \mod (2 \pi)$ and I'll let you conclude (and reverify all this of course! :-)). | {"set_name": "stack_exchange", "score": 0, "question_id": 98273} |
\subsection{Hecke eigenvalues of the Yoshida lift}Hereafter,
we denote by $\pi$ the automorphic representation of $D^\times_\A$ associated with $\bff$.
If $F=\Q\times \Q$,
the automorphic form $\bff$ is a pair $(\bff_1, \bff_2)$ of automorphic forms on $D^\times_{0,\A}$
associated with $\bff_i$ for $i=1,2$.
We also define $\Pi$ to be the automorphic representation of ${\rm GSp}_4(\A)$
associated with $\theta(\varphi, \bff)$.
We write the decomposition to the restricted tensor product of $\pi$
as $\pi=\ot^\prime_v\pi_v$.
We use the similar notation for $\pi_1, \pi_2$ and $\Pi$. It is well known that $\theta(\test,\bff^\dag)$ is an eigenform of Hecke operators $T_1(p)$ and $T_2(p)$ at $p\ndivides N_F$.
In this subsection, we show that $\theta(\test,\bff^\dag)$ is also an eigenform of the $U_1(p)$-operators for $p|N_F$ and compute their eigenvalues. First we call the definition of certain Hecke operators.\begin{align*}
T_1(p) =& {\rm GSp}_4(\Z_p) {\rm diag}(p,p,1,1) {\rm GSp}_4(\Z_p),
\quad T_2(p) = {\rm GSp}_4(\Z_p) {\rm diag}(p,p^2,p,1) {\rm GSp}_4(\Z_p), \\
U_1(p) =& \Gamma_0(p) {\rm diag}(p,p,1,1) \Gamma_0(p).
\end{align*}
For $K={\rm GSp}_4(\Z_p)$ or $\Gamma_0(p)$ and $h\in M_4(\Z_p)\cap {\rm GSp}_4(\Q_p)$, we decompose
\begin{align*}
K h K = \coprod_i h_i K.
\end{align*}
Then, we define an action of
a double coset $KhK$ on
a cusp form $F$ on ${\rm GSp}_4({\A})$ of level $\Gamma_0(N)$ to be
\begin{align*}
[KhK] (F) (g) = \sum_i F(g h_i),
\end{align*}
where if $p\nmid N$ (resp. $p\mid N$), then $K={\rm GSp}_4(\Z_p)$ (resp. $\Gamma_0(p)$).
For $p\mid N$, we also define the Atkin-Lehner operator $W_p$ to be
\begin{align*}
W_p F(g) = F(g \pMX{}{-\bfone_2}{p^{e_p}\bfone_2}{} ),
\quad e_p:={\rm ord}_p(N).
\end{align*}
In the rest of this subsection, we describe the action of $T_1(p), T_2(p), U_1(p)$ and $W_p$
on Yoshida lifts $\theta(\varphi, \bff ^\dag)$.
Firstly, we compute the action of double cosets.
\begin{lem}\label{heckedecomp}
We have the following decompositions:
\begin{align*}
U_1(p) =& \coprod_{X\in {\rm Sym}_2(\Z/p\Z)}
\pMX{p\bfone_2}{X}{}{\bfone_2}
\Gamma_0(p).
\end{align*}
\end{lem}
\begin{proof}
The decomposition in the statement follows from a similar arguments
given in \cite[Section 6.1]{rs07}.
\end{proof}
\begin{lem}\label{thetazero}
Let $p$ be a prime dviding $N^+$ and split in $F/{\Q}$.
For $t\in H^0(\Q_p)$, let $\varphi^t_{p}$ be
the characteristic function of $t {\rm M}_2(\Z_p)^{\oplus 2} t^{-1}$.
Put $\varphi^t= \otimes_{v\neq p}\test_v \otimes \varphi^t_p$.
Then, $\theta(\varphi^t, \bff ^\dag)$ is vanishing.
\end{lem}
\begin{proof}
We may assume that $g \in {\rm Sp}_4({\A})$.
By the definition of theta series, we find that
\begin{align*}
\theta(\varphi^t, \bff ^\dag) (g)
= & \int_{[H^0_1]} \sum_{x \in \bfX}
\langle \omega(h, g)\varphi^t(x), \bff ^\dag(h) \rangle_\cW dh \\
= & \sum_{x \in H^0_1({\Q})\backslash\bfX}
\sum_{\gamma \in H^0_1({\Q})}
\int_{[H^0_1]}
\langle \omega(\gamma h, g)\varphi^t(x),
\bff ^\dag( \gamma h) \rangle_\cW dh \\
= & \sum_{x \in H^0_1({\Q})\backslash\bfX}
\int_{H^0_1({\A})}
\langle \omega( h, g)\varphi^t(x),
\bff ^\dag( h) \rangle_\cW dh.
\end{align*}
We denote $\wh{R}^\times_{\frakN^+}$ by $\cU$
and write $\cU=\prod_v \cU_v$.
Define $\wtd{\cU}_p = {\rm GL}_2(\Z_p) \times_{\Z^\times_p}
{\rm GL}_2(\Z_p) \subset H^0(\Q_p)$.
We also define
$\wtd{\cU}=\wtd{\cU}_p\times \prod_{v\neq p} \cU_v$.
Then, for each $u \in t^{-1} \wtd{\cU} t$, we find that $\varphi^t(u^{-1} x) = \varphi^t(x)$.
Hence, we obtain
\begin{align*}
& \int_{H^0_1(\A)}
\langle \omega( h, g)\varphi^t(x),
\bff ^\dag( h) \rangle_\cW dh \\
=& {\rm vol}(dh, \cU \cap t^{-1} \cU t)
\int_{H^0_1(\A) / t^{-1 } \wtd{\cU} t }
\sum_{u \in t^{-1} \wtd{\cU} t / \cU \cap t^{-1} \cU t}
\langle \omega( h u, g)\varphi^t(x),
\bff ^\dag( h u) \rangle_\cW dh \\
=& {\rm vol}(dh, \cU \cap t^{-1} \cU t)
\int_{H^0_1(\A) / t^{-1 } \wtd{\cU} t }
\langle \omega( h , g)\varphi^t(x),
\sum_{u \in \wtd{\cU} / t \cU t^{-1} \cap \cU }
\bff ^\dag( h t^{-1} u t) \rangle_\cW dh.
\end{align*}
Define
$\widetilde\bff ^\dag (h)
= \sum_{u \in \wtd{\cU} / t \cU t^{-1} \cap \cU }
\bff ^\dag( h u t)$ for $h\in H^0(\A)$.
Then, we have
$\widetilde\bff ^\dag(h\wtd{u}) = \wtd{\bff} ^\dag(h)$
for each $\wtd{\cU}$.
Since $\pi$ has no non-trivial $\wtd{\cU}$-invariant vector,
$\wtd{\bff} ^\dag $ have to be zero.
This proves $\theta( \varphi^t, \bff ^\dag) =0$.
\end{proof}
\begin{prop}\label{thetaeigen}
We denote by $\bar{\ast}$ the complex conjugate of $\ast$.
Let $c(f_i)_{p}$ be the Hecke eigenvalue of $f$ at $p$ $(i=1,2)$.
\begin{enumerate}
\item \label{thetaeigen(1)}
For $i=1,2$, the theta series $\theta(\test, \bff ^\dag)$ is an eigen form of $T_i(p)$
for $p\nmid N$.
In particular, we have
\begin{align*}
T_1(p) ( \theta(\test, \bff ^\dag) )
=& \theta(\test, \bff ^\dag) \times
\begin{cases} p(\bar{ c(f_1)_p} + \bar{c(f_2)_p }), & p: \text{split in } F/\Q, \\
0, & p:\text{non-split in } F/\Q, \end{cases} \\
T_2(p) (\theta(\test, \bff ^\dag) )
=& \theta(\test, \bff ^\dag) \times
\begin{cases} ((p^2-1)+ p\bar{ c(f_1)_p} \bar{c(f_2)_p }), & p: \text{split in } F/\Q, \\
-(p^2+1 + p c(f)_p ), & p:\text{non-split in } F/\Q. \end{cases}
\end{align*}
\item \label{thetaeigen(2)}
Assume that $p$ is split in $F/\Q$ and $p$ divides {\rm l.c.m.}($N_1, N_2$).
If $p \mid N_i$ for $i=1$ or $2$,
we write $\pi=\sigma(\mu_i |\cdot|^\frac{1}{2}, \mu_i |\cdot|^{-\frac{1}{2}})$.
Then, we have
\begin{align*}
U_1(p) ( \theta(\test, \bff ^\dag) )
=& \theta(\test, \bff ^\dag) \times
\begin{cases} p\mu_1(p) = p\mu_2(p), & p\mid {\rm g.c.d.}(N_1, N_2), \pi_{1,p}\cong \pi_{2,p}, \\
p\mu_1(p), & p\mid N_1, p\nmid N_2, \\
p\mu_2(p), & p\nmid N_1, p\mid N_2. \end{cases}
\end{align*}
\item \label{thetaeigen(3)}
If $p$ is non-split in $F/\Q$, we have
\begin{align*}
U_1(p) (\theta(\test, \bff ^\dag)) =& pW_p( \theta(\test, \bff ^\dag)),
\quad U_1(p) (W_p(\theta(\test, \bff ^\dag) ) ) = p \theta(\test, \bff ^\dag).
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof}
The first statement follows from
\cite[Theorem 5.1]{yo80} (resp. \cite[Theorem 5.2, Remark 5.3]{yo80})
if $p$ is split (resp. non-split and odd) in $F/\Q_p$.
In the case that $p=2$ is non-split in $F/\Q_p$,
the statement for $T_i(p)$ is \cite[p.543, Main Theorem]{jlr12}.
We prove the second statement.
We suppose that $p$ divides $N$, since the other case is prove in the same way.
Let $\eta_p=\pMX{0}{1}{-p}{0}$ and $h_0 = (\bfone_2,\eta_p^{-1}) \in H^0(\Q_p)$.
Note that we have $\nu(h_0)=p$.
By the definition of theta series and Lemma \ref{heckedecomp},
we find that
\begin{align*}
U_1(p)\theta(\test,\bff^\dag)(g)&=\int_{[H^0_1]}\sum_{B\in {\rm S}_2(\Zp/p\Zp)}\pair{\theta(hh'h_0;g\pMX{p\bfone_2}{B}{0}{\bfone_2}\test)}{\bff^\dag(hh'h_0}_\cW\rmd h\\
&=\theta(\varphi',\rho(h_0)\bff^\dag)(g),
\end{align*}
where $\rho(h_0)\bff^\dag(h)=\bff^\dag(hh_0)$ and
\[\varphi':=\sum_{B\in {\rm Sym}_2(\Z/p\Z)}\om_p(h_0,\pMX{p\bfone_2}{B}{0}{\bfone_2})\test.\]
where $h^\prime \in H({\A})$ such that $\nu(h^\prime) = \nu(g)$.
If $B=\pMX{a}{b}{b}{c}\in{\rm Sym_2}(\Zp)$, then we have \begin{align}
\omega_p(h_0, \pMX{p\bfone_2}{B}{0}{\bfone_2})
\test_p(x)
=& p^2 \cdot \omega_{V_p} ( \pMX{p\bfone_2}{}{}{p^{-1}\bfone_2}
\pMX{\bfone_2}{p^{-1}B }{}{ \bfone_2 } )
\test_p(h^{-1}_0 x) \notag \\
=& p^{-2}\psi_p(a {\rm n}(x_1) + \rpair{x_1}{x_2} + c {\rm n}(x_2) )
\test_p(x_1\eta_p,x_2\eta_p ). \label{weilactp}
\end{align}
We find that
\[\varphi'(x_1,x_2)=\sum_{a,b,c}\psi_p(a{\rm n}(x_1)+b\rpair{x_1}{x_2}+c{\rm n}(x_2))\test(x_1\eta_p,x_2\eta_p).\]
Then $(x_1,x_2)\in\supp\varphi'$ if and only if $x_1,x_2\in R_0\eta_p^{-1}$, ${\rm n}(x_1)\in\Zp$,
$\rpair{x_1}{x_2}\in\Zp$ and ${\rm n}(x_2)\in\Zp$. Recall that $\test_p$ is the characteristic function of
$L_p:=R_0\oplus R_0$, where $R_0=\pMX{\Zp}{\Zp}{p\Zp}{\Zp}$. For $x\in {\rm M}_2(\Qp)$, $x\eta_p\in R_0$ if and only if \begin{align*}
x \in L':=\pMX{\Z_p}{p^{-1}\Z_p}{\Z_p}{\Z_p}.
\end{align*}
Define lattices $L_1,L_2$ by
\[L_1=\pMX{\Z_p}{\Z_p}{\Z_p}{\Z_p};\quad L_2=\pMX{\Z_p}{p^{-1}\Z_p}{p\Z_p}{\Z_p}.\]
Then for $(x_1,x_2)\in L'\oplus L'$,
\begin{align*}{\rm n}(x_1),\,{\rm n}(x_2)\in\Zp\iff &x_1,x_2\in L_1\cup L_2;\\
(x_1,x_2)\in \Zp\iff& (x_1,x_2)\in L_1\oplus L_1\text{ or }L_2\oplus L_2.
\end{align*}
Therefore, we find that
\[\varphi'(x)=\bbI_{L_1\oplus L_2}+\bbI_{L_2\oplus L_2}-\bbI_{R_0\oplus R_0}.\]
Hence, we find that
\begin{align*}
\sum_{ \substack{ a, b, c \\ {\rm mod} \ p } }
p^{-2} \psi_p(a {\rm n}(x_1) + b(x_1, x_2) + c {\rm n}(x_2) )
\test_p( h_0x_1, h_0x_2 )
= p({\mathbb I}_{\tilde{L}^{\oplus 2}_1}(x) + {\mathbb I}_{\tilde{L}^{\oplus 2}_2}(x)
- {\mathbb I}_{(\tilde{L}_1 \cap \tilde{L}_2 )^{\oplus 2}}(x) ).
\end{align*}
Put $\varphi^{(p)} = \otimes_{v\neq p} \test_v$.
Then, by Lemma \ref{thetazero}, we find that, for $i=1,2$,
\begin{align*}
\theta(g; \varphi^{(p)} {\mathbb I}_{L^{\oplus 2}_i}, \bff ^\dag) =0.
\end{align*}
Since ${\mathbb I}_{(\tilde{L}_1 \cap \tilde{L}_2 )^{\oplus 2}} = \test_p$, we obtain
\begin{align*}
U_1(p) (\theta(g; \test, \bff ^\dag))
= -p (g; \test, \bff ^\dag|_{h_0}),
\end{align*}
where we define $\bff ^\dag|_{h_0}(h) = \bff ^\dag(hh_0)$.
By \cite[3.1.2 Proposition]{sc02}, we find that
$\bff ^\dag|_{h_0} = -\mu_1(p) \bff ^\dag$.
This proves the second statement.
We prove the third statement.
In the similar way as above,
we compute
\begin{align*}
U_1(p) ( \theta(g; \test, \bff ^\dag) )
=& \sum_{ X\in S_2(\Z/p\Z)}
\int_{[H^0_1]} \sum_{x \in \bfX}
\langle \omega(hh^\prime,
g \pMX{\bfone_2}{ X}{}{\bfone_2}
\pMX{p\bfone_2}{}{}{\bfone_2} )
\test(x),
\bff ^\dag(hh^\prime)
\rangle_\cW dh,
\end{align*}
where $h^\prime \in H^0({\A})$ such that $\nu(h^\prime) = \nu(g)$.
We note the following identity:
\begin{align*}
\pMX{\bfone_2}{ \pMX{a}{b}{b}{c} }{}{\bfone_2}
\pMX{p\bfone_2}{}{}{\bfone_2}
= \pMX{}{-\bfone_2}{p\bfone_2}{}
\pMX{}{\bfone_2}{-\bfone_2}{}
\pMX{p\bfone_2}{}{}{p^{-1}\bfone_2}
\pMX{\bfone_2}{ p^{-1} \pMX{a}{b}{b}{c} }{}{\bfone_2}.
\end{align*}
We compute the following action of the Weil representation:
\begin{align*}
& \omega_{V_p}(
\pMX{}{\bfone_2}{-\bfone_2}{}
\pMX{p\bfone_2}{}{}{p^{-1}\bfone_2}
\pMX{\bfone_2}{ p^{-1} \pMX{a}{b}{b}{c} }{}{ \bfone_2 } )
\test_p(x_1, x_2) \\
=&p^{-4} \omega_{V_p}( \pMX{}{\bfone_2}{-\bfone_2}{} )
\psi_p( p ( a{\rm n}(x_1) + b(x_1, x_2) + c{\rm n}(x_2) ) )
\test_p(px_1, px_2).
\end{align*}
By the similar argument in the proof of the second statement,
we find that
\begin{align*}
\sum_{ \substack{ a, b, c \\ {\rm mod}\ p } }
p^{-4} \psi_p( p ( a{\rm n}(x_1) + b(x_1, x_2) + c{\rm n}(x_2) ) )
\test_p(px_1, px_2)
= p^{-1} {\mathbb I}_{\tilde{L}^{\oplus 2}}(x),
\end{align*}
where $\tilde{L} =\pMX{\cO_p}{p^{-1}\Z_p}{\Z_p}{\cO_p} \cap V_p$.
It is easy to see that
\begin{align*}
\omega_{V_p}( \pMX{}{\bfone_2}{-\bfone_2}{} )({\mathbb I}_{\tilde{L}^{\oplus 2}})
= p^2 \test_p.
\end{align*}
Hence, we obtain
\begin{align*}
U_1(p) ( \theta(g; \test, \bff ^\dag) )
= p \sum_{ \substack{ a, b, c \\ {\rm mod}\ p } }
\int_{[H^0_1]} \sum_{x \in \bfX}
\langle \omega(hh^\prime,
g \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x),
\bff ^\dag(hh^\prime) \rangle_\cW dh
= pW_p ( \theta(g; \test, \bff ^\dag) ).
\end{align*}
This proves the third statement.
\end{proof}
\begin{rem}\label{U(p)evrem}
Let $\Pi\cong \otimes^\prime_v \Pi_v$
be the cuspidal automorphic representation of ${\rm GSp}_4({\A})$
attached to $\theta(\test, \newform )$.
We compare Proposition \ref{thetaeigen} with
some representation theoretic properties of $\Pi_p$
which are described in \cite{rs07} and \cite{sc05}.
We use notation of \cite[Table A.1]{rs07} and \cite[Table 1]{jlr12}
for irreducible admissible representations of ${\rm GSp}_4(\Q_p)$.
\begin{enumerate}
\item If $p\mid N$ is split, then by \cite[Table 1]{jlr12} and \cite[Theorem A.9, Theorem A.10]{gi11},
we find that $\Pi_p$ is isomorphic
to either (IIa), (IVc), (Va), (Vb$^\ast$), (VIa) or (VIb).
By \cite[Table A. 15]{rs07} and \cite[2.6 Proposition]{ca80}, we find that
\begin{align*}
\dim_{\C} \Pi^{\Gamma_0(p)}_p
= \begin{cases} 1, & \text{if } \Pi_p \text{ is type (IIa), (IVc), (VIa), (VIb)}, \\
0, & \text{if } \Pi_p \text{ is type (Va), (Vb$^\ast$)}. \end{cases}
\end{align*}
Hence, if $\Pi_p$ is either type (IIa), (IVc), (VIa) or (VIb),
then we can also deduce Proposition \ref{thetaeigen}(\ref{thetaeigen(2)}) by
\cite[(24)]{sc05}.
If $\Pi_p$ is either type (Va) or (Vb$^\ast$),
then $p\mid {\rm g.c.d}(N_1, N_2)$ and $\pi_{1,p} \not\cong \pi_{2,p}$.
Hence, it is necessary to consider this case in Proposition \ref{thetaeigen}(\ref{thetaeigen(2)}).
\item If $p\mid N$ is inert, then by \cite[Table 1]{jlr12} and \cite[Theorem A.11]{gi11},
we find that $\Pi_p$ is isomorphic
to either (IIIa) or (Va).
By \cite[Table A. 15]{rs07}, we find that
\begin{align*}
\dim_{\C} \Pi^{\Gamma_0(p)}_p
= \begin{cases} 2, & \text{if } \Pi_p \text{ is type (IIIa)}, \\
0, & \text{if } \Pi_p \text{ is type (Va)}. \end{cases}
\end{align*}
By \cite[(24)]{sc05}, we find that, if $\Pi_p$ is type (IIIa), then
the characteristic polynomial of $U_1(p)$ is given by
$X^2-p^2$, where $X$ is an indeterminate.
This is compatible with Proposition \ref{thetaeigen}(\ref{thetaeigen(3)}).
\end{enumerate}
\end{rem}
We compute the action of Atkin-Lehner operator $W_p$ on Yoshida lifts as follows:
\begin{prop}\label{ALTheta}
Suppose that $p\mid N_D$ is split in $F/{\Q}$.
We denote by $\ep(\pi_{i,p}) \in \{\pm 1\}$ the local root number of $\pi_{i,p}$ .
Then, there exists $\varepsilon_p(\theta(\test, \bff ^\dag) ) \in \{\pm 1\}$ such that
\begin{align*}
W_p \theta(\test, \bff ^\dag) = \varepsilon_p(\theta(\test, \bff ^\dag)) \theta(\test, \bff ^\dag).
\end{align*}
Furthermore,the values $\varepsilon_p(\theta(\test, \bff ^\dag) )$ are given as follows
\begin{itemize}
\item if $p$ divides only one of $ N^+_1$ or $N^+_2$,
then $\varepsilon_p(\theta(\test, \bff ^\dag) )=\ep(\pi_{1,p})\ep(\pi_{2,p})$;
\item if $p\mid {\rm l.c.m.}(N^+_1, N^+_2)$ and $\ep(\pi_{1,p})=\ep(\pi_{2,p})$,
then $\varepsilon_p(\theta(\test, \bff ^\dag) )=\ep(\pi_{1,p})$;
\item if $p\mid N^-$ and $\ep(\pi_{1,p})=\ep(\pi_{2,p})$,
then $\varepsilon_p(\theta(\test, \bff ^\dag) )=-\ep(\pi_{1,p})$.
\end{itemize}
\end{prop}
\begin{proof}
Assume that $p\mid N_1$ and $p\nmid N_2$.
In this case, we have $\ep(\pi_{2,p}) = 1$.
Put $h_0 = \varrho( \pMX{0}{-1}{p}{0} ,1) \in H^0(\Q_p)$.
Then, by the definition of $W_p$, we have
\begin{align*}
W_p \theta(g; \test, \bff ^\dag)
= \int_{[H^0_1]} \sum_{x\in \bfX}
\langle \omega(hh^\prime h_0, g \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x),
\bff (hh^\prime h_0)
\rangle_\cW
dh,
\quad (\nu(h^\prime) = \nu(g).)
\end{align*}
We compute
$ \omega( h_0, \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x)
= p^2 \widehat{\test}(h^{-1}_0 x)$,
where the Fourier transform $\widehat{\test}$ at $p$ of $\test$
with respect to the self-dual Haar measure on $V^{\oplus 2}_p$,
which is explicitly given in \cite[Section 3.4]{hn15}.
It is easy to see that
\begin{align*}
\widehat{\test}(x)
= p^{-2} {\mathbb I}_{(R^\ast_p)^{\oplus 2}}(x), \quad
(R^\ast_p := \begin{pmatrix} \Z_p & p^{-1} \Z_p \\ \Z_p & \Z_p \end{pmatrix}).
\end{align*}
Hence, we find that
\begin{align*}
\omega( h_0, \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x)
= {\mathbb I}_{(R^\ast_p)^{\oplus 2}}(h^{-1}_1x)
= \test(x).
\end{align*}
By \cite[3.2.2 Theorem]{sc02}, we have
$\bff ^\dag (hh^\prime h_0) = \ep(\pi_{1,p})\bff ^\dag (hh^\prime)$.
This proves the statement.
Both cases that $p\nmid N_1, p \mid N_2$ and that $p$ divides ${\rm l. c. m.} (N^+_1, N^+_2)$
are proved in the same way with the above argument.
Assume that $p$ divides $N^-$.
Since $N^-$ is square-free,
there exist an unramified quadratic character $\mu_i : {\Q}^\times_p \to \C^\times$
such that $\pi_{i,p} \cong \mu_i \circ {\rm n}$.
Let $h_1$ (resp. $h_2$) be an element of $D^\times_p$
such that ${\rm n}(h_1) = p$ (resp. ${\rm n}(h_2) =1$).
Put $h_0=\varrho( (h_1,h_2) ,1)$.
In the same way as above, we compute
$ \omega( h_0, \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x)
= p^2 \widehat{\test}(h^{-1}_0 x)$,
where $\widehat{\test}$ is the Fourier transform at $p$ with respect to
the self-dual Haar measure on $V^{\oplus 2}_p$ of $\test$.
Put $R^\ast_{0,p} = \{ y \in D_0 : (x, y) \in \Z_p, \text{ for } x \in R_{0,p} \}$.
Then, it is known that $R^\ast_{0,p} = R_{0,p} h_1$
(see \cite[(25.7) Theorem]{re75}).
Hence, we find that
\begin{align*}
\widehat{\test}(x)
= {\rm vol}(d\mu, R_{0,p} h_1 )^2 {\mathbb I}_{(R_{0,p} h_1)^{\oplus 2}}(x)
= p^{-2}\test(h_0 x).
\end{align*}
This proves
$ \omega( h_0, \pMX{}{-\bfone_2}{p\bfone_2}{} )\test(x)
= \test(x)$.
Since $\pi_{i,p} \cong \mu_i \circ {\rm n}$, we see that
$\bff ^\dag(hh_0) = \mu_1(p) \bff ^\dag(h)$.
Since the local root number $\ep(\pi_{i,p})$ is given by $-\mu_i(p)$,
we obtain the identity in the statement.
\end{proof}
\begin{rem}
\begin{enumerate}
\item
Suppose that $p\mid N$ is split.
As already seen in Remark \ref{U(p)evrem},
if $\Pi^{\Gamma_0(p)}_p\neq 0$,
then $\Pi_p$ is either type (IIa), (IVc), (VIa) or (VIb)
and $\dim_\C \Pi^{\Gamma_0(p)}_p =1$.
In these cases, the eigen values of $W_p$ are also given in \cite[Table A.15]{rs07}.
\item If $\ep(\pi_{1,p}) \neq \ep(\pi_{2,p})$ for $p\mid{\rm l.c.m.}(N_1, N_2)$ which is split,
then $\Pi^{\Gamma_0(p)}_p=0$ by \cite[Table 1]{jlr12} and \cite[Table A.15]{rs07}.
Hence, it is not necessary to consider this case in Proposition \ref{ALTheta}.
\item Suppose that $p\mid N^+$ is inert.
In this case, if $\Pi_p\neq 0$,
then $\dim_{\C} \Pi^{\Gamma_0(p)}_p=2$.
By \cite[Table A.15]{rs07}, we find that
there exists $v_\varepsilon \in \Pi_p$ such that $W_pv_\varepsilon = \varepsilon v_\varepsilon$
for each $\varepsilon = \pm 1$.
This vector $v_\varepsilon$ is explicitly given by
$\theta(\test, \bff ^\dag) + \varepsilon W_p\theta(\test, \bff ^\dag)$.
We also note that, by Proposition \ref{thetaeigen} (\ref{thetaeigen(3)}),
\begin{align*}
U_1(p) ( \theta(\test, \bff ^\dag) + \varepsilon W_p\theta(\test, \bff ^\dag) )
= \varepsilon p ( \theta(\test, \bff ^\dag) + \varepsilon W_p\theta(\test, \bff ^\dag) ).
\end{align*}
\end{enumerate}
\end{rem}
For reader's convenience, we summarize admissible representations $\Pi_p$
of ${\rm GSp}_4(\Q_p)$ attached to Yoshida lifts which are discussed in this paper, as follows:
\begin{prop}\label{ThetaRepProp}
For the case that $F$ is a real quadratic field, we put $N^+_1=N^+_2=N^+$.
We assume that
Then, $\Pi_v$ is described as follows:
\begin{enumerate}
\item Assume that $v=w_1w_2$ is split in $F/\Q$.
\begin{enumerate}
\item\label{Theta1a} If $v\nmid N$, write $\pi_i=\pi(\mu_i,\nu_i)$ for $i=1,2$.
\begin{enumerate}
\item\label{Theta1ai} If $\mu_2\nu^{-1}_1\neq |\cdot |^\pm$ and $\nu_2\mu^{-1}_1\neq |\cdot |^\pm$, $\Pi_v$ is type (I);
\item\label{Theta1aii} If either $\mu_2\nu^{-1}_1= |\cdot |^\pm$ or $\nu_2\mu^{-1}_1= |\cdot |^\pm$, $\Pi_v$ is type (IIIb).
\end{enumerate}
\item\label{Theta1b} If $v\mid N^-$, write $\pi_i=\sigma(\mu_i|\cdot|^{\frac{1}{2}}, \mu_i|\cdot|^{-\frac{1}{2}})$ for $i=1,2$.
\begin{enumerate}
\item\label{Theta1bi} If $\mu_1=\mu_2$, $\Pi_v$ is type (VIb);
\item\label{Theta1bii} If $\mu_1\neq\mu_2$, $\Pi_v$ is type (Vb$^\ast$).
\end{enumerate}
\item\label{Theta1c} If either $v\mid N^+_1, v\nmid N^+_2$ or $v\nmid N^+_1, v\mid N^+_2$,
we may assume that $\pi_1=\pi(\mu_1,\nu_1)$ and $\pi_2=\sigma(\mu_2|\cdot|^{\frac{1}{2}}, \mu_2|\cdot|^{-\frac{1}{2}})$.
\begin{enumerate}
\item\label{Theta1ci} If $\mu_2\nu^{-1}_1\neq|\cdot|^{\pm\frac{3}{2}}$, $\Pi_v$ is type (IIa);
\item\label{Theta1cii} If $\mu_2\nu^{-1}_1=|\cdot|^{\pm\frac{3}{2}}$, $\Pi_v$ is type (IVc).
\end{enumerate}
\item\label{Theta1d} If $v\mid N^+_i$ for $i=1,2$, write $\pi_i=\sigma(\mu_i|\cdot|^{\frac{1}{2}},\mu_i|\cdot|^{-\frac{1}{2}})$.
\begin{enumerate}
\item\label{Theta1di} If $\mu_1=\mu_2$, $\Pi_v$ is type (VIa);
\item\label{Theta1dii} If $\mu_1\neq \mu_2$, $\Pi_v$ is type (Va).
\end{enumerate}
\end{enumerate}
\item Assume that $v$ is non-split in $F/\Q$.
If $\mu$ is a Galois invariant character, then $\widehat{\mu}$ denotes a character of $\Q^\times_v$ such that $\mu = \widehat{\mu}\circ{\rm N}_{F/\Q}$.
\begin{enumerate}
\item\label{Theta2a} If $v\nmid N$, $\Pi_v$ is type (I).
\item\label{Theta2b} If $v\mid N^+$, write $\pi=\sigma(\mu|\cdot|^{\frac{1}{2}},\mu|\cdot|^{-\frac{1}{2}})$ with $\mu=\mu^\sigma$.
\begin{enumerate}
\item\label{Theta2bi} If $\widehat{\mu}^2$ is trivial, $\Pi_v$ is type (Va);
\item\label{Theta2bii} If $\widehat{\mu}^2=\eta_{F/\Q}$, $\Pi_v$ is type (IIIa).
\end{enumerate}
\end{enumerate}
\end{enumerate}
In particular, type (I), (IIa), (IIIa), (Va), (VIa), (IXa) are generic.
Type (IIIb), (IVc), (Vb$^\ast$), (VIb) are non-generic.
Only type (Vb$^\ast$) is supercuspidal.
\end{prop}
\begin{proof}
We note that characters appearing here are all unramified, since our initial automorphic representations have square-free level with the trivial central character.
In particular, if $F/\Q$ is non-split, these characters are Galois invariant.
We also note that, by \cite[Theorem A.9, Theorem A.10, Theorem A.11]{gi11},
$\Pi_v$ is supercuspidal if and only if $v\mid N^-$ is split
and the local root numbers $\ep(\pi_{1, v})$ and $\ep(\pi_{2,v})$
of $\pi_{1,v}$ and $\pi_{2,v}$ respectively does not coincide.
Then, the statement follows from \cite[Table 1]{jlr12}.
\end{proof} | {"config": "arxiv", "file": "1609.07669/Heckop.tex"} |
TITLE: Line integral along the curve $\gamma(t)=(4 \cos t, 4 \sin^2 t) $
QUESTION [1 upvotes]: Let us consider the vector field in the plane:
$$\vec{F}=\left(x \frac{e^{x^2+y^2} - e}{x^2+y^2},y\frac{e^{x^2+y^2} - e}{x^2+y^2}\right)$$
calculate the line integral along the curve defined by:
$$\gamma:
\begin{cases}
x=4 \cos t \\
y=4 \sin^2 t\\
\end{cases}
$$
with $t\in[0,\pi/2]$.
Any suggestions please?
REPLY [1 votes]: Here is a direct evaluation using this link
$\begin{align}
dx&=-4\sin t \\
dy&=8\cos t \sin t \\
F&=\Big(\frac{2[\exp{(2\cos 4t +14)}-e] \cos t }{7+\cos 4t},\frac{2[\exp{(2\cos 4t +14})-e] \sin^2 t }{7+\cos 4t}\Big)
\end{align}$
Your integral will hence be
$\begin{align}I&=-\int_0^{\frac{\pi}{2}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=-\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=-\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt+\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=0.
\end{align}$
where at last I have changed the variable for the second integral from $t$ to $\frac{\pi}{2}-t$. | {"set_name": "stack_exchange", "score": 1, "question_id": 1133312} |
TITLE: length of modules in arbitrary exact sequences
QUESTION [3 upvotes]: Let $R$ be a commutative, noetherian ring. Given the exact sequence of $R$-modules of finite length $ 0 \rightarrow M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0 $. Is there an equation, connecting the lengths of the modules like in the case of short exact sequences?
REPLY [3 votes]: For a short exact sequence of finite length modules $0\to A\to B\to C\to 0$, it holds that $l(B)=l(A)+l(C)$.
Split your sequence into two short exact ones:
\begin{gather}
0\to M_0\to M_1\to K\to 0\\
0\to K\to M_2\to M_3\to 0
\end{gather}
Then $l(K)=l(M_1)-l(M_0)$ and $l(K)=l(M_2)-l(M_3)$, so we get that
$$
l(M_0)-l(M_1)+l(M_2)-l(M_3)=0
$$
which generalizes the relation for short exact sequences.
This goes on by easy induction.
REPLY [0 votes]: In these situations, the alternating sum of the lengths is zero. Here,
$$l(M_0)-l(M_1)+l(M_2)-l(M_3)=0.$$
One can split this into two exact sequences
$$0\to M_0\to M_1\to N\to 0$$
and
$$0\to N\to M_2\to M_3\to 0$$
with the same $N$, also of finite length, and
$$l(N)=l(M_1)-l(M_0)=l(M_2)-l(M_3)$$
etc. | {"set_name": "stack_exchange", "score": 3, "question_id": 2959644} |
TITLE: Is it possible to predict number of edges in a strongly connected directed graph?
QUESTION [0 upvotes]: I know that
A graph that is complete and undirected with $n$ nodes will have $n(n-1)/2$ edges
A graph that is complete and directed with $n$ nodes will have $n(n-1)$ edges
A graph that is connected (without cycles) and undirected with $n$ nodes will have $(n-1)$ edges
My question is
1) A graph that is strongly connected (without cycles) and directed with $n$ nodes will have $?$ edges
2) A graph that is strongly connected (with cycles) and directed with $n$ nodes will have $?$ edges
I am not sure about the last two. Could someone help shed some light and provide an analytic form for them as well as provide an explanation why this is the case?
My attempt is
1) is impossible because we need cycles in a directed graph to make it strongly connected. (Just a gut feeling that I am failing to articulate into actual logic)
2) is simply $2(n-1)$ (Just a gut feeling that I am failing to articulate into actual logic)
REPLY [1 votes]: Your gut feeling is right:
A cycle is a sequence $v_1, v_2, ... v_k$ of vertices such that: $v_1 = v_k$, and there is a directed edge between $v_i$ and $v_{i+1}$ for $i \in [1, k-2]$.
A graph is said to be strongly connected or diconnected if every vertex is reachable from every other vertex. In particular, $v_2$ is reachable from $v_1$ and $v_1$ is reachable from $v_2$, which means that there is a cycle in the graph.
A (directed) cycle graph with $n$ vertices and $n$ edges is strongly connected. Conversely, any graph with $n-1$ edges is not strongly connected. Furthermore, the complete graph is strongly connected. So a strongly connected graph with $n$ nodes will have between $n$ and $n(n-1)$ edges. | {"set_name": "stack_exchange", "score": 0, "question_id": 2689757} |
\chapter{Killing Horizons and Near-Horizon Geometry}
\label{ch:background}
In this section we describe the properties of Killing Horizons and introduce Gaussian Null Coordinates which are particularly well adapted for such geometries, illustrated by certain important examples in $D=4$.\footnote{We use natural units with $G = c = \hbar = 1$} As the purpose of this thesis is to investigate the geometric properties of supersymmetric near-horizon geometries, it will be particularly advantageous to work in a co-ordinate system
which is specially adapted to describe Killing Horizons.
In what follows, we will assume that the black hole event horizon is a Killing horizon. Rigidity theorems imply that the black hole horizon is Killing
for both non-extremal and extremal black holes, under certain assumptions, have been constructed, e.g. \cite{rigidity1, gnull, axi1, axi2}. The assumption that the event horizon is Killing
enables the introduction of Gaussian Null co-ordinates \cite{isen, gnull} in a neighbourhood
of the horizon. The analysis of the near-horizon geometry is significantly simpler than that of the
full black hole solution, as the near-horizon limit reduces the system to a set of
equations on a co-dimension 2 surface, ${\cal{S}}$, which is the spatial section of
the event horizon.
\section{Killing horizons}
\theoremstyle{definition}
\begin{definition}{A null hypersurface $\cal{H}$ is a {\bf Killing Horizon} of a Killing vector field $\xi$ if it is normal to $\cal{H}$ i.e $\exists$ a Killing vector field $\xi$ everywhere on the spacetime $\cal{M}$ which becomes null only on the horizon $\cal{H}$.}
\end{definition}
If $\xi$ is a Killing vector then the Killing horizon $\cal{H}$ can be identified with the surface given by $g(\xi,\xi) = 0$. A Killing horizon is a more local description of a horizon since it can be formulated in terms of local coordinates.
For an example, consider the Schwarzchild metric,
\bea
ds^2 = -\bigg(1-\frac{2M}{r}\bigg)dt^2 + \bigg(1-\frac{2M}{r}\bigg)^{-1}dr^2 + r^2 (d\theta^2 + \sin^2{\theta} d\phi^2) \ .
\label{scwarzsol}
\ee
The Killing horizon is generated by the timelike Killing vector $\xi = \partial_{t}$ which becomes null on the horizon $r=2M$ since $g(\xi, \xi) = -(1-\frac{2M}{r})$.
Associated to a Killing horizon is a geometrical quantity known as the surface gravity $\kappa$. If the surface gravity vanishes, then the Killing horizon is said to be extreme or degenerate. The surface gravity $\kappa$ is defined as,
\begin{eqnarray}
\xi^{\nu}\nabla_{\nu}\xi^{\mu}\big|_{\cal{H}} = \kappa \xi^{\mu} \ .
\end{eqnarray}
This can be rewritten as
\begin{eqnarray}
\nabla_{\mu}(\xi^2)\big|_{\cal{H}} = -2\kappa \xi_{\mu} \ .
\end{eqnarray}
By Frobenius theorem, a vector $\xi^\mu$ is hyperspace orthogonal if,
\begin{eqnarray}
\xi_{[\mu}\nabla_{\nu}\xi_{\rho]} = 0 \ .
\end{eqnarray}
Since $\xi$ is Killing, we can rewrite this as,
\begin{eqnarray}
\xi_{\rho} \nabla_{\mu}{\xi_{\nu}} = - 2\xi_{[\mu}\nabla_{\nu]}{\xi_{\rho}} \ .
\end{eqnarray}
By contracting with $\nabla^{\mu}{\xi^{\nu}}$ and evaluating on $\cal{H}$ we get,
\begin{eqnarray}
\xi_{\rho} (\nabla_{\mu}{\xi_{\nu}})(\nabla^{\mu}{\xi^{\nu}}) = -2 (\nabla^{\mu}{\xi^{\nu}})(\xi_{[\mu}\nabla_{\nu]}\xi_{\rho}) = - 2\kappa \xi^{\mu}\nabla_{\mu}{\xi_{\rho}} = -2\kappa^2 \xi_{\rho} \ .
\end{eqnarray}
Thus we can write,
\begin{eqnarray}
\kappa^2 = -\frac{1}{2}(\nabla_{\mu}\xi_{\nu})(\nabla^{\mu}\xi^{\nu})\big|_{\cal{H}} \ .
\end{eqnarray}
The surface gravity of a static Killing horizon can be interpreted as the acceleration, as exerted at infinity, needed to keep an object on the horizon. For the Schwarzchild metric the surface gravity is $\kappa = \frac{1}{4M}$ which is non-vanishing.
\section{Gaussian null coordinates}
In order to study near-horizon geometries we need to introduce a coordinate system which is regular and adapted to the horizon. We will consider a $D$-dimensional stationary black hole metric,
for which the horizon is a Killing horizon, and the metric is regular at the horizon. A set of Gaussian Null coordinates \cite{isen, gnull} $\{u, r, y^{I}\}$
will be used to describe the metric, where $r$ denotes the coordinate transverse to the horizon as the radial distance away from the event horizon which is located at $r=0$ and $y^I,~ I=1, \dots, D-2$ are local co-ordinates on ${\cal S}$. The metric components have no dependence on $u$, and the timelike isometry $\xi = {\partial \over \partial u}$ is null on the horizon at $r=0$. As shown in \cite{gnull} (see Appendix A) the black hole metric in a patch containing the horizon is given by,
\bea
\label{gncmetric}
ds^2 = 2du dr + 2r h_I(y, r) du dy^I - r f(y, r) du^2 + ds_{\cal S}^2 \ .
\ee
The spatial horizon section ${\cal S}$ is given by $u=const,~ r=0$ with the metric
\bea
ds_{\cal S}^2 = \gamma_{I J}(y, r)dy^I dy^J \ .
\ee
where $\gamma_{I J}$ is the metric on the spatial horizon section $\cal{S}$ and $f, h_{I}$ and $\gamma_{I J}$ are smooth functions of $(r, y^I)$ so that the spacetime is smooth.
We assume that ${\cal{S}}$, when restricted to $r=const.$ for sufficiently small values of $r$, is compact and without boundary. The
1-form $h$, scalar $\Delta$ and metric $\gamma$ are functions of $r$ and $y^{I}$; they are smooth in $r$ and regular at the horizon.
The surface gravity associated with the Killing vector $\xi$ can be computed from this metic, to obtain $\kappa = \frac{1}{2}f(y,0)$.
It is instructive to consider a number of important 4-dimensional examples. In each
case the co-ordinate transformation used to write the metric in regular coordinates around the horizon, GNC and Kerr coordinates, which removes the co-ordinate singularity at the horizon.
\vspace{2mm}
\newtheorem{eg}{Example}
\begin{eg}
Consider the Schwarzschild solution ({\ref{scwarzsol}}) and
make the change of coordinates $(t, r, \theta, \phi) \rightarrow (u, r, \theta, \phi)$ with $t \rightarrow u + \lambda(r)$ and
\bea
\lambda(r) = -r-2M\ln(r-2M) \ .
\ee
Thus in GNC the metric can be written as,
\bea
ds^2 = -r(r+2M)^{-1}du^2 + 2du dr + (r+2M)^2 (d\theta^2 + \sin^2{\theta} d\phi^2) \ .
\ee
where we have also made shift $r \rightarrow 2M + r$ so that the horizon is now located at $r=0$. We remark that the derivation of the Gaussian null co-ordinates for the Schwarzschild solution is identical to that of the standard Eddington-Finkelstein co-ordinates.
\end{eg}
\vspace{2mm}
\begin{eg}It is also straightforward to consider Reisser-Nordstr\"om solution
({\ref{rnsolution}}). We make the same co-ordinate transformation as for the Schwarzschild analysis,
but take
\bea
\lambda(r) = -r - M\ln(r^2 -2Mr + Q^2) + \bigg(\frac{Q^2 - 2M^2}{\sqrt{Q^2 - M^2}}\bigg)\arctan{\bigg(\frac{r-M}{\sqrt{Q^2 - M^2}}\bigg)} \ .
\ee
This produces the following metric
\bea
\label{rngncoord}
ds^2 = -\bigg(1-\frac{2M}{r} + \frac{Q^2}{r^2}\bigg)du^2 + 2du dr + r^2 (d\theta^2 + \sin^2{\theta} d\phi^2) \ .
\ee
The event horizon is located at the outer horizon $r = r_+ \equiv M + \sqrt{M^2 - Q^2}$. We can also make the shift $r \rightarrow r_+ + r$ so that the horizon is now located at $r=0$.
\end{eg}
\vspace{2mm}
\begin{eg}For the Kerr metric, given in ({\ref{kerrsol}}), we make the change of co-ordinates $(t, r, \theta, \phi) \rightarrow (u, r, \theta, \tilde{\phi})$ with $t \rightarrow u + \lambda_1(r),~ \phi \rightarrow \tilde{\phi} + \lambda_2(r)$ and
take
\bea
\lambda_1(r) &=&
-r - M\ln(r^2 -2Mr + a^2) - \bigg(\frac{2M^2}{\sqrt{a^2 - m^2}}\bigg)\arctan{\bigg(\frac{r-M}{\sqrt{a^2 - M^2}}\bigg)} \ ,
\nonumber \\
\lambda_2(r) &=& - \bigg(\frac{a}{\sqrt{a^2 - m^2}}\bigg)\arctan{\bigg(\frac{r-M}{\sqrt{a^2 - M^2}}\bigg)} \ .
\ee
which produces the metric
\bea
ds^2 &=& -\bigg(\frac{r^2 - 2Mr + a^2\cos^2 \theta}{r^2 + a^2 \cos^2 \theta}\bigg)du^2 + 2du dr - \underbrace{a\sin^2 \theta dr d\phi}_{(*)} - \bigg(\frac{2aMr \sin^2 \theta}{r^2 + a^2\cos^2 \theta}\bigg) du d\phi
\nonumber \\
&+& (r^2+a^2 \cos^2 \theta)d\theta^2 + \bigg(\frac{\sin^2 \theta (a^2 (a^2 - 2Mr + r^2) \cos^2 \theta + (2Mr + r^2)a^2 + r^4)}{r^2 + a^2 \cos^2 \theta} \bigg)d\phi^2 \ ,
\nonumber \\
\ee
where the tildes have been dropped.
The event horizon is located at the outer horizon $r = r_+ \equiv M + \sqrt{M^2 - a^2}$. We can also make the shift $r \rightarrow r_+ + r$ so that the horizon is now located at $r=0$.
\end{eg}
\vspace{2mm}
\begin{eg}
For the Kerr-Newman metric, given in ({\ref{kerrnewsol}}), on making the change of coordinates $(t, r, \theta, \phi) \rightarrow (u, r, \theta, \tilde{\phi})$ with $t \rightarrow u + \lambda_1(r),~ \phi \rightarrow \tilde{\phi} + \lambda_2(r)$ and taking
\bea
\lambda_1(r) &=&
-r - M\ln(r^2 -2Mr + Q^2 + a^2) + \bigg(\frac{Q^2 - 2M^2}{\sqrt{Q^2 - M^2 + a^2}}\bigg)\arctan{\bigg(\frac{r-M}{\sqrt{Q^2 - M^2 + a^2}}\bigg)} \ ,
\nonumber \\
\lambda_2(r) &=& - \bigg(\frac{a}{\sqrt{Q^2 - M^2 + a^2}}\bigg)\arctan{\bigg(\frac{r-M}{\sqrt{Q^2 - M^2 + a^2}}\bigg)} \ ,
\ee
the following metric is found
\bea
ds^2 &=& -\bigg(\frac{r^2 - 2Mr + a^2\cos^2 \theta +Q^2}{r^2 + a^2 \cos^2 \theta}\bigg)du^2 + 2du dr - \underbrace{a\sin^2 \theta dr d\phi}_{(*)}
\nonumber \\
&+& \bigg(\frac{a \sin^2 \theta(Q^2 - 2Mr)}{r^2 + a^2\cos^2 \theta}\bigg) du d\phi
+ (r^2+a^2 \cos^2 \theta)d\theta^2
\nonumber \\
&+& \bigg(\frac{\sin^2 \theta (a^2 (a^2 + Q^2 - 2Mr + r^2) \cos^2 \theta + (2Mr + r^2 -Q^2)a^2 + r^4)}{r^2 + a^2 \cos^2 \theta} \bigg)d\phi^2 \ ,
\label{kngncoords}
\ee
where the tildes have been dropped.
The event horizon is located at the outer horizon $r = r_+ \equiv M + \sqrt{M^2 - Q^2 -a^2}$. We can also make the shift $r \rightarrow r_+ + r$ so that the horizon is now located at $r=0$.
\end{eg}
The resulting metric for both Kerr and Kerr-Newman are expressed in terms of regular coordinates around the horizon, known as Kerr coordinates. These are evidently different from the usual coordinates in GNC since it contains a non-zero $dr d\phi$ term (*) and the Killing vector $\partial_u$ is not null on the horizon. Nonetheless, this extra term will disappear in the near-horizon limit for extreme horizons as we shall see (see Appendix A).
\subsection{Extremal horizons}
Since the near-horizon geometry is only well defined for extremal black holes, with vanishing surface gravity, it will be useful to consider some examples. The two examples of particular interest are the extremal Reisser-Nordstr\"om solution and the extremal Kerr-Newman solution.
\vspace{2mm}
\begin{eg}
For the case of Reisser-Nordstr\"om, the extremal solution is obtained by setting
$Q=M$. On taking the metric given in ({\ref{rngncoord}}) and setting $Q=M$, and also
shifting $r \rightarrow M + r$ so that the horizon is now located at $r=0$, we find the metric in GNC,
\bea
ds^2 = - r^2 \big(M+r\big)^{-2} du^2 + 2 du dr + \big(M+r\big)^2 (d\theta^2 + \sin^2{\theta} d\phi^2) \ .
\ee
\end{eg}
\vspace{2mm}
\begin{eg}
For the case of the Kerr-Newman, the extremal solution is obtained by setting $Q^2 = M^2 - a^2$ in the metric ({\ref{kngncoords}}), and also shifting $r \rightarrow M + r$ so that the horizon is now located at $r=0$, we find the metric the following metric in Kerr coordinates,
\bea
ds^2 &=& -\bigg(\frac{r^2 -a^2 + a^2\cos^2 \theta}{(r+M)^2 + a^2 \cos^2 \theta}\bigg)du^2 + 2du dr - a\sin^2 \theta dr d\phi
\nonumber \\
&-& \bigg(\frac{a\sin^2 \theta (a^2 +M^2 + 2Mr)}{(r+M)^2 + a^2\cos^2 \theta}\bigg) du d\phi
+ ((r+M)^2+a^2 \cos^2 \theta)d\theta^2
\nonumber \\
&+& \bigg(\frac{\sin^2 \theta (a^2 r^2 \cos^2 \theta + a^4 + (r^2+4Mr + 2M^2 )a^2 + (r+M)^4)}{(r+M)^2 + a^2 \cos^2 \theta}\bigg) d\phi^2 \ .
\ee
\end{eg}
The extremal Kerr ($Q=0,~ a=M$) and the extreme RN is obtained by the extreme Kerr-Newman by setting $a=M$ and $a=0$ respectively.
\section{The near-horizon limit}
Having constructed the Gaussian null co-ordinates, we shall consider a particular type of limit which exists for extremal solutions, called the near-horizon limit \cite{reall}. This limit can be thought of as a decoupling limit in which
the asymptotic data at infinity is scaled away, however the geometric structure
in a neighbourhood very close to the horizon is retained.
We begin by considering the Gaussian null co-ordinates adapted to a Killing horizon
$\cal{H}$ associated with the Killing vector $\xi = \partial_u$, identified with the hypersurface given by $r=0$. The Killing vector becomes null on the horizon, since $g(\xi, \xi) = -r f(y, r)$.
\begin{eqnarray}
ds^2 = 2(dr + r h_{I}(y,r)dy^{I} - \frac{1}{2}r f(y,r) du) du + \gamma_{I J}(y,r)dy^{I}dy^{J} \ .
\end{eqnarray}
As we have mentioned earlier, the surface gravity associated to the Killing vector $\xi$ is given by $\kappa = \frac{1}{2}f(y,0)$. To take the near-horizon limit we first make the rescalings
\begin{eqnarray}
r\rightarrow \epsilon \hat{r}, \, u\rightarrow \epsilon^{-1} \hat{u}, \, y^{I}\rightarrow y^{I} \ ,
\end{eqnarray}
which produces the metric (after dropping the hats),
\begin{eqnarray}
ds^2 = 2(dr + r h_{I}(y,\epsilon r) dy^{I} - \frac{1}{2}r \epsilon^{-1} f(y, \epsilon r) du) du + \gamma_{I J}(y,\epsilon r)dy^{I}dy^{J} \ .
\end{eqnarray}
Since $f$ is analytic in $r$ we have an expansion
\begin{eqnarray}
f(y,r) = \sum_{n=0}^{\infty}\frac{r^n}{n!}\partial^{n}_{r}f\big|_{r=0} \ ,
\end{eqnarray}
and a similar expansion for $h_I$ and $\gamma_{I J}$. Therefore,
\begin{eqnarray}
\epsilon^{-1}f(y,\epsilon r) &=& \sum_{n=0}^{\infty} \epsilon^{n-1}\frac{r^n}{n!}\partial^{n}_{r}f\big|_{r=0} \nonumber \\
&=& \frac{f(y,0)}{\epsilon} + r \, \partial_{r}f\big|_{r=0} + \sum_{n=2}^{\infty} \epsilon^{n-1}\frac{r^n}{n!}\partial^{n}_{r}f\big|_{r=0} \ .
\end{eqnarray}
The near-horizon limit then corresponds to taking the limit $\epsilon \rightarrow 0$.
This limit is clearly only well-defined when $f(y,0)=0$,
corresponding to vanishing surface gravity. Hence the near-horizon limit is only well defined for extreme black holes. Thus, for extremal black holes, after taking the near-horizon limit we have the metric,
\begin{eqnarray}
\label{nhmetricf}
ds_{NH}^2 = 2(dr + r h_{I}dy^{I} - \frac{1}{2}r^2 \Delta du) du + \gamma_{I J}dy^{I}dy^{J} \ ,
\label{nhmmx}
\end{eqnarray}
where we have defined $\Delta = \partial_{r}f\big|_{r=0}$ and $h_I, \gamma_{I J}$ are evaluated at $r=0$ so that the $r$-dependence is fixed on $\cal{H}$. $\{\Delta, h_I, \gamma_{I J}\}$ are collectively known as the near-horizon data and depend only on the coordinates $y^I$. In Appendix A consider an arbitrary metric written in the coordinates $(u,r,y^I)$ which is regular around the horizon $r=0$ generated by a Killing vector $\partial_u$. We consider the conditions on the metric components for the near-horizon limit to be well defined and show that the metric under a certain condition can be written as (\ref{nhmetricf}) upon identification of the near-horizon data.
The near-horizon metric (\ref{nhmmx}) also has a new scale symmetry, $r \rightarrow \lambda r,~ u \rightarrow \lambda^{-1}u$ generated by the Killing vector $L=u\partial_{u} - r\partial_{r}$. This, together with the Killing vector $V=\partial_u$ satisfy the algebra $[V, L] = V$ and they form a 2-dimensional non-abelian symmetry group ${\cal{G}}_2$. We shall show that for a very large class of supersymmetric near-horizon geometries, this further enhances into a larger symmetry algebra, which will include a $\mathfrak{sl}(2,\mathbb{R})$ subalgebra. This has previously been shown for non-supersymmetric extremal black hole horizons \cite{genextrsl}.
Supersymmetric black holes in four and five dimensions are necessarily extreme. To see why this is to be expected, we recall that Killing spinors are the parameters of preserved supersymmetry of a solution, so a supersymmetric solution to any supergravity theory necessarily admits a Killing spinor $\epsilon$. The bilinear $K^{\mu} = \bar{\epsilon}\Gamma^{\mu}\epsilon$ is
a non-spacelike Killing vector field i.e. $K^2 \leq 0$. Suppose a supersymmetric Killing horizon $\cal{H}$ is invariant under the action of $K$, then $K$ must be null and $dK^2 = -2\kappa K$ on the horizon. It follows that $K^2$ attains a maximum
on the horizon, and therefore $dK^2=0$ which implies that the horizon is extremal. It is also known in five dimensions that there exists a real scalar spinor bilinear $f$, with the property that $K^2=-f^2$. Assuming that the Killing spinor is analytic in $r$ in a neighbourhood of the horizon, this implies that $K^2 \sim -r^2$ in a neighbourhood of the horizon and this also implies that the horizon is extremal. A similar argument holds in four dimensions.
A near-extremal black hole is a black hole which is not far from the extremality. The calculations of the properties of near-extremal black holes are usually performed using perturbation theory around the extremal black hole; the expansion parameter known as non-extremality \cite{nextr1,nextr2}. In supersymmetric theories, near-extremal black holes are often small perturbations of supersymmetric black holes. Such black holes have a very small surface gravity and Hawking temperature, which consequently emit a small amount of Hawking radiation. Their black hole entropy can often be calculated in string theory, much like in the case of extremal black holes, at least to the first order in non-extremality.
To extend the horizon into the bulk away from the near-horizon limit, one has to consider the full $r$-dependence of the near-horizon data \cite{tdef, moduli}, which are evaluated at $r=0$ and thus depend only the coordinates $y^I$ of the spatial horizon section ${\cal S}$ in the near-horizon decoupling limit. We thus extend the data $\{\Delta(y), h_I(y), \gamma_{I J}(y)\} \rightarrow \{{\hat{\Delta}}(y,r), {\hat{h}_I}(y,r), {\hat{\gamma}_{I J}(y,r)}\}$, taylor expand around $r=0$ and consider the first order deformation of the horizon fields, where the usual near-horizon data is given by,
\bea
{\hat{\Delta}}(y,0) = \Delta,~ {\hat{h}}_{I}(y,0) = h_I,~ {\hat{\gamma}_{I J}(y,0)} = \gamma_{I J} \ .
\ee
\subsection{Examples of near-horizon geometries}
Now we will give examples of near-horizon geometries for the extremal Reisser-Nordstr\"om, Kerr and Kerr-Newman solution to illustrate the emergence of an extra isometry which forms the ${\mathfrak{sl}}(2,\bR)$ algebra \cite{genextrsl},
\vspace{2mm}
\begin{eg}
It is instructive to consider the case of the extremal Reisser-Nordstr\"om solution with metric written in Gaussian null co-ordinates as:
\bea
ds^2 = - r^2 \big(M+r\big)^{-2} du^2 + 2 du dr + \big(M+r\big)^2 (d\theta^2 + \sin^2{\theta} d\phi^2) \ .
\ee
On taking the near-horizon limit as described previously, the metric becomes
\begin{eqnarray}
ds^2 = 2(dr - \frac{1}{2}r^2 \Delta du) du + \gamma_{1 1}d\theta^2 + \gamma_{2 2}d\phi^2
\end{eqnarray}
with the near-horizon data,
\bea
\Delta = \frac{1}{M^2},~~ \gamma_{1 1} = M^2, ~~\gamma_{2 2} = M^2\sin^2 \theta \ ,
\ee
which is the metric of $AdS_2 \times S^2$.
The isometries of $AdS_2$, denoted by $\{ K_1, K_2, K_3 \}$ are given by
\bea
K_1 = \partial_{u},~K_2 = -u\partial_{u} + r\partial_{r},~ K_3 = -\frac{u^2}{2}\partial_{u} + (M^2 + u r)\partial_{r} \ ,
\ee
which satisfy the $\mathfrak{sl}(2,\mathbb{R})$ algebra
\bea
[K_1, K_2] = -K_1,~[K_1, K_3] = K_2,~[K_2,K_3] = -K_3 \ ,
\ee
and the isometries of the $S^2$ are given by $\{ K_4, K_5, K_6 \}$, with
\bea
K_4 &=& \partial_{\phi},~~ K_5 =\sin{\phi}\partial_{\theta} + \cos{\phi}\cot{\theta}\partial_{\phi},~~
K_6 = \cos{\phi}\partial_{\theta} - \sin{\phi}\cot{\theta}\partial_{\phi} \ ,
\ee
which satisfy the Lie algebra ${\mathfrak{so}}(3)$,
\bea
[K_4, K_5] = K_6,~ [K_4, K_6] = -K_5,~ [K_5, K_6] = K_4 \ .
\ee
\end{eg}
\vspace{2mm}
\begin{eg}
Now let us consider the extremal Kerr metric. In the usual NHL we first take the extremal limit ($a=M$) in Kerr coordinates,
\bea
ds^2 &=& -\bigg(\frac{r^2 -M^2 +M^2\cos^2 \theta}{(r+M)^2 + M^2 \cos^2 \theta}\bigg)du^2 + 2du dr - M\sin^2 \theta dr d\phi
\nonumber \\
&-& \bigg(\frac{2M^2(r+M)\sin^2 \theta }{(r+M)^2 + M^2\cos^2 \theta}\bigg) du d\phi
+ ((r+M)^2+M^2 \cos^2 \theta)d\theta^2
\nonumber \\
&+& \bigg(\frac{(M^2 r^2 \cos^2 \theta + 4M^4 + 8M^3 r + 7M^2 r^2 + 4M r^3 + r^4)\sin^2 \theta }{(r+M)^2 + M^2 \cos^2 \theta}\bigg) d\phi^2 \ ,
\ee
and then the near-horizon limit
\bea
r\rightarrow \epsilon \hat{r},~ u\rightarrow \epsilon^{-1} \hat{u},~ \phi \rightarrow \hat{\phi} + \frac{\hat{u}}{2M}\epsilon^{-1},~ \epsilon \rightarrow 0 \ ,
\ee
and subsequently drop the hats and repeat this after we make the change,
\bea
r \rightarrow \bigg(\frac{2}{\cos^2 \theta + 1}\bigg) \hat{r} \ ,
\ee
to get the metric into the form,
\begin{eqnarray}
ds^2 = 2(dr + r h_1 d\theta + r h_2 d\phi - \frac{1}{2}r^2 \Delta du) du + \gamma_{1 1}d\theta^2 + \gamma_{2 2}d\phi^2 \ ,
\end{eqnarray}
and the near-horizon data given by,
\bea
\Delta &=& \frac{(\cos^4 \theta + 6\cos^2 \theta - 3)}{M^2 (\cos^2 \theta + 1)^3 } \ ,
\nonumber \\
h_1 &=& \frac{2\cos \theta\sin \theta}{\cos^2 \theta + 1},~~ h_2 = \frac{4 \sin^2 \theta}{(\cos^2 \theta + 1)^2} \ ,
\nonumber \\
\gamma_{1 1} &=& M^2(\cos^2 \theta + 1),~~ \gamma_{2 2} = \frac{4M^2 \sin^2 \theta}{\cos^2 \theta + 1} \ .
\ee
The Killing vectors ${K_1, K_2, K_3, K_4}$ of this near-horizon metric are given by,
\bea
K_1 &=& \partial_{u},~ K_2 = -u\partial_{u} + r\partial_{r} - \partial_\phi \ ,
\nonumber \\
K_3 &=& -\frac{u^2}{2}\partial_{u} + (2M^2 + u r)\partial_{r} - u\partial_{\phi} \ ,
\nonumber \\
K_4 &=& \partial_{\phi} \ ,
\ee
with the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \times \mathfrak{u}(1)$,
\bea
[K_1,K_2]=-K_1,~ [K_1,K_3]= K_2,~[K_2,K_3]=-K_3 \ .
\ee
\end{eg}
\vspace{2mm}
\begin{eg}
Finally, we consider the Kerr-Newman metric in Kerr coordinates. We take the extremal limit ($Q^2 = M^2 - a^2$)
\bea
ds^2 &=& -\bigg(\frac{r^2 -a^2 + a^2\cos^2 \theta}{(r+M)^2 + a^2 \cos^2 \theta}\bigg)du^2 + 2du dr - a\sin^2 \theta dr d\phi
\nonumber \\
&-& \bigg(\frac{a (a^2 +M^2 + 2Mr)\sin^2 \theta}{(r+M)^2 + a^2\cos^2 \theta}\bigg) du d\phi
+ ((r+M)^2+a^2 \cos^2 \theta)d\theta^2
\nonumber \\
&+& \bigg(\frac{ (a^2 r^2 \cos^2 \theta + a^4 + (r^2+4Mr + 2M^2 )a^2 + (r+M)^4)\sin^2 \theta}{(r+M)^2 + a^2 \cos^2 \theta}\bigg) d\phi^2 \ ,
\ee
and then the near-horizon limit,
\bea
r\rightarrow \epsilon \hat{r},~ u\rightarrow \epsilon^{-1} \hat{u},~ \phi \rightarrow \hat{\phi} + \frac{a \hat{u}}{(a^2 + M^2)}\epsilon^{-1},~ \epsilon \rightarrow 0 \ ,
\ee
after which we also make the change
\bea
r \rightarrow \bigg(\frac{(a^2 + M^2)}{a^2\cos^2 \theta + M^2}\bigg)\hat{r} \ ,
\ee
and dropping the hats after each coordinate transformation to get the metric into the form,
\begin{eqnarray}
ds^2 = 2(dr + r h_1 d\theta + r h_2 d\phi - \frac{1}{2}r^2 \Delta du) du + \gamma_{1 1}d\theta^2 + \gamma_{2 2}d\phi^2 \ ,
\end{eqnarray}
with the near-horizon data,
\bea
\Delta &=& \frac{(a^4 \cos^4 \theta + 6a^2 M^2\cos^2 \theta - 4a^2 M^2 + M^4)}{ (a^2\cos^2 \theta + M^2)^3 } \ ,
\nonumber \\
h_1 &=& \frac{2a^2\cos \theta\sin \theta}{a^2\cos^2 \theta + M^2},~~ h_2 = \frac{2 a M(a^2 + M^2) \sin^2 \theta}{(a^2\cos^2 \theta + M^2)^2} \ ,
\nonumber \\
\gamma_{1 1} &=& a^2\cos^2 \theta + M^2,~~ \gamma_{2 2} = \frac{(a^2 +M^2)^2 \sin^2 \theta}{a^2\cos^2 \theta + M^2} \ .
\ee
The Killing vectors ${K_1, K_2, K_3, K_4}$ of this near-horizon metric are given by,
\bea
K_1 &=& \partial_{u},~K_{2} = -u\partial_{u} + r\partial_{r}- \bigg(\frac{2aM}{a^2 + M^2}\bigg) \partial_\phi \ ,
\nonumber \\
K_3 &=& -\frac{u^2}{2}\partial_{u} + (a^2 + M^2 + u r)\partial_{r} - \bigg(\frac{2aM u}{a^2 + M^2}\bigg)\partial_{\phi} \ ,
\nonumber \\
K_4 &=& \partial_{\phi} \ ,
\ee
with the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \times \mathfrak{u}(1)$,
\bea
[K_1,K_2]=-K_1,~ [K_1,K_3]= K_2,~[K_2,K_3]=-K_3 \ .
\ee
\end{eg}
As we have previously remarked, the isometries $K_1$ and $K_2$ are generic for all
near-horizon geometries. In these cases, an additional isometry $K_3$ is present; which also follow from known near-horizon symmetry theorems \cite{genextrsl} for non-supersymmetric extremal horizons.
We shall show that the emergence of such an extra isometry, in the near-horizon limit, which forms the $\mathfrak{sl}(2,\mathbb{R})$ algebra is generic for {\it supersymmetric} black holes.
\subsection{Curvature of the near-horizon geometry}
As we will see, geometric equations (such as Einstein's equations) for a near-horizon geometry can be equivalently written as geometric equations defined purely on a $(D-2)$-dimensional spatial cross section manifold ${\cal S}$ of the horizon. It is convenient to introduce a null-orthonormal frame for the near-horizon metric, denoted by $(\bbe^A)$, where $A=(+,-, i)$, $i=1, \dots, D-2$ and
\be
\label{basis1}
\bbe^+ = du, \qquad \bbe^- = dr + rh -{1 \over 2} r^2 \Delta du, \qquad \bbe^i = \bbe^i{}_I dy^I \ ,
\ee
so that $ds^2 = g_{A B} \bbe^A \bbe^B = 2 \bbe^+ \bbe^- + \delta_{ij} \bbe^i \bbe^j~,$ where $\bbe^i$ are vielbeins for the horizon metric $\delta_{i j}$. The dual basis vectors are frame derivatives which are expressed in terms of co-ordinate derivatives as
\begin{eqnarray}
\label{frco}
\bbe_+ = \partial_+ = \partial_u +{1 \over 2} r^2 \Delta \partial_r ~,~~
\bbe_- = \partial_- = \partial_r ~,~~
\bbe_i = \partial_i = {\tilde{\partial}}_i -r h_i \partial_r \ .
\end{eqnarray}
The spin-connection 1-forms satisfy $d\bbe^A= -\Omega^A_{\phantom{A}B} \wedge \bbe^B$ and are given by
\begin{eqnarray}
\Omega_{+-} &=& -r\Delta \bbe^+ + \frac{1}{2}h_i \bbe^i \,,
\nonumber \\
\Omega_{+i} &=& -\frac{1}{2}r^2(\partial_i \Delta-\Delta h_i)\bbe^+-\frac{1}{2}h_i\bbe^- +{1 \over 2} r dh_{ij} \bbe^j \,,~~~
\nonumber \\
\Omega_{-i} &=& -\frac{1}{2}h_i \bbe^+ \,, \quad \Omega_{i j} = \tilde{\Omega}_{i j}- {1 \over 2} r dh_{ij} \bbe^+ \ ,
\end{eqnarray}
where $\tilde{\Omega}_{i j}$ are the spin-connection 1-forms of the $(D-2)$-manifold ${{\cal{S}}}$ with metric $\delta_{i j}$ and basis ${\bf{e}}^i$. Here we have made use of the following identities:
\bea
d{\bf{e^+}} = 0, \; \; \; d{\bf{e^-}} = {\bf{e^-}} \wedge h + r dh + \frac{1}{2}{\bf{e^+}}\wedge(-r^2\Delta h + r^2 d\Delta + 2r\Delta{\bf{e^-}}) \ .
\ee
The non-vanishing components of the spin connection are
\begin{eqnarray}
\label{spin}
&&\Omega_{-,+i} = -{1 \over 2} h_i~,~~~
\Omega_{+,+-} = -r \Delta, \quad \Omega_{+,+i} ={1 \over 2} r^2( \Delta h_i - \partial_i \Delta),
\cr
&&\Omega_{+,-i} = -{1 \over 2} h_i, \quad \Omega_{+,ij} = -{1 \over 2} r dh_{ij}~,~~~
\Omega_{i,+-} = {1 \over 2} h_i, \quad \Omega_{i,+j} = -{1 \over 2} r dh_{ij},
\cr
&&\Omega_{i,jk}= \tilde\Omega_{i,jk} \ .
\end{eqnarray}
The curvature two-forms defined by $\rho_{AB}= d\Omega_{AB}+ \Omega_{AC} \wedge \Omega^{C}_{\phantom{C}B}$ give the Riemann tensor in this basis using $\rho_{AB}= \frac{1}{2} R_{ABCD} \bbe^C \wedge \bbe^D$ and are given in Appendix A.
The non-vanishing components of the Ricci tensor with respect to the basis ({\ref{basis1}}) are
\bea
R_{+-} &=& {1 \over 2} \tilde{\nabla}^i h_i - \Delta -{1 \over 2} h^2~,~~~
R_{ij} = {\hat{R}}_{ij} + \tilde{\nabla}_{(i} h_{j)} -{1 \over 2} h_i h_j \ ,
\nonumber \\
R_{++} &=& r^2 \bigg( {1 \over 2} \tilde{\nabla}^2 \Delta -{3 \over 2} h^i \tilde{\nabla}_i \Delta -{1 \over 2} \Delta \tilde{\nabla}^i h_i + \Delta h^2
+{1 \over 4} (dh)_{ij} (dh)^{ij} \bigg) \ ,
\nonumber \\
R_{+i} &=& r \bigg( {1 \over 2} \tilde{\nabla}^j (dh)_{ij} - (dh)_{ij} h^j - \tilde{\nabla}_i \Delta + \Delta h_i \bigg) \ ,
\ee
where ${\hat{R}}$ is the Ricci tensor of the metric $\delta_{i j}$ on the horizon section ${\cal S}$ in the $\bbe^i$ frame. The spacetime contracted Bianchi identity implies the following identities \cite{genextrsl} on ${\cal S}$:
\begin{eqnarray}
\label{Sid}
R_{++} &=& - \frac{1}{2} r (\tilde{\nabla}^i-2h^i) R_{+ i} \,,
\cr
R_{+ i} &=& r\bigg(-\tilde{\nabla}^j [R_{j i} - \frac{1}{2}\delta_{j i}( R^k_{~k}+2R_{+-})] + h^jR_{j i} - h_i R_{+-}\bigg) \ ,
\end{eqnarray}
which may also be verified by computing this directly from the above expressions.
\subsection{The supercovariant derivative}
We can also decompose the supercovariant derivative of the spinor $\epsilon$ given by\footnote{We use the Clifford algebra conventions with mostly positive signature and $\{ \Gamma_{\mu}, \Gamma_{\nu} \} = 2g_{\mu \nu}$},
\bea
\label{covepsilon}
\nabla_{\mu} \epsilon = \partial_{\mu} \epsilon + \frac{1}{4}\Omega_{\mu, \nu \rho}\Gamma^{\nu \rho} \epsilon \ ,
\ee
with respect to the basis ({\ref{basis1}}),
which will be useful later for the analysis of KSEs. After expanding each term and evaluating the components of the spin connection with ({\ref{spin}}) and the frame derivatives with ({\ref{frco}}) we have,
\bea
\nabla_{+} \epsilon &=& \partial_u \epsilon + \frac{1}{2}r^2 \Delta \partial_r \epsilon + \frac{1}{4}r^2(\Delta h_i - \partial_i \Delta)\Gamma^{+i}\epsilon - \frac{1}{4}h_i\Gamma^{-i}\epsilon
- \frac{1}{2}r\Delta\Gamma^{+-}\epsilon - \frac{1}{8}r(dh)_{ij}\Gamma^{ij}\epsilon \ ,
\cr
\nabla_{-} \epsilon &=& \partial_{r} \epsilon - \frac{1}{4}h_i \Gamma^{+i}\epsilon \ ,
\cr
\nabla_{i} \epsilon &=& \tilde{\nabla}_{i}\epsilon - r\partial_{r} \epsilon h_i - \frac{1}{4}r(dh)_{ij}\Gamma^{+j}\epsilon + \frac{1}{4}h_i \Gamma^{+-}\epsilon \ .
\ee
The integrability condition for ({\ref{covepsilon}) can be written in terms of the Riemann and Ricci tensor as,
\bea
[\nabla_{\mu}, \nabla_{\nu}]\epsilon &=& \frac{1}{4} R_{\mu \nu, \rho \sigma} \Gamma^{\rho \sigma} \epsilon,~\Gamma^{\nu}[\nabla_{\mu}, \nabla_{\nu}]\epsilon = -\frac{1}{2}R_{\mu \sigma} \Gamma^{\sigma}\epsilon \ .
\ee
Similarly, the covariant derivative of a vector $\xi^{\rho}$ can be written in terms of the spin connection as,
\bea
\nabla_{\mu}{\xi^{\rho}} = \partial_{\mu}{\xi^{\rho}} + \Omega_{\mu,}{}^{\rho}{}_{\lambda} \xi^{\lambda} \ ,
\ee
and for a Killing vector $\xi$ we can write the integrability condition associated with the covariant derivative in terms of the Riemann and Ricci tensor as,
\bea
[\nabla_{\mu}, \nabla_{\nu}]\xi^{\rho} = R^{\rho}{}_{\lambda, \mu \nu}\xi^{\lambda},~\nabla_{\mu}\nabla_{\nu} \xi^{\mu} = R_{\rho \nu}\xi^{\rho} \ .
\ee
\section{Field strengths}
Consider a $p$-form field strength, $F_{(p)}$. Suppose that
the components of this field strength, when written in the
Gaussian null co-ordinates are independent of $u$ and smooth (or at least $C^2$) in $r$,
and furthermore that it admits a well-defined near-horizon limit. Such a field strength, after taking the near-horizon limit, can always be decomposed with respect to the basis ({\ref{basis1}) as follows:
\bea
F_{(p)} = {\bf{e^+}} \wedge {\bf{e^-}} \wedge L_{(p-2)} + r{\bf{e^+}}\wedge M_{(p-1)} + N_{(p)},~~ p>1 \ .
\ee
where $L_{(p-2)}$, $M_{(p-1)}$ and $N_{(p)}$ are $p-2$, $p-1$ and $p$-forms
on the horizon spatial cross-section which are independent of $u$ and $r$.
On taking the exterior derivative one finds\footnote{$d_h \alpha = d\alpha - h \wedge \alpha$}
\bea
dF_{(p)} = {\bf{e}^+} \wedge {\bf{e}^-} \wedge (d_h L_{(p-2)} - M_{(p-1)}) + r{\bf{e}^+}\wedge (-d_h M_{(p-1)} - dh \wedge L_{(p-2)}) + dN_{(p)} \ .
\ee
If $F_{(p)} = dA_{(p-1)}$ with gauge potential $A_{(p-1)}$ then $dF_{(p)} = 0$ as with the common Bianchi identities, we get the following conditions;
\bea
M_{(p-1)} = d_h L_{(p-2)},~~d_h M_{(p-1)} = -dh \wedge L_{(p-2)},~~dN_{(p)} = 0 \ .
\ee
The third implies $N_{(p)}$ is a closed form on the spatial section $\cal{S}$. The second condition is not independent as it is implied by the first.
We will now give a reminder of the maximum principle and the classical Lichnerozicz theorem, which are crucial in establishing the results of (super)symmetry enhancement.
\section{The maximum principle}
In the analysis of the global properties of near-horizon geometries, we shall
obtain various equations involving the Laplacian of a non-negative scalar $f$.
Typically $f$ will be associated with the modulus of a particular spinor.
Such equations will be analysed either by application of integration by parts, or
by the Hopf maximum principle. The background manifold ${\cal{N}}$
is assumed to be smooth and compact without boundary, and all tensors are also
assumed to be smooth.
In the former case, we shall obtain second order PDEs on $\cal{N}$ given by,
\bea
\nabla^i \nabla_i f + \lambda^i \nabla_i f + \nabla_{i}(\lambda^i)f = \alpha^2 \ ,
\ee
where $\lambda^i$ is a smooth vector and $\alpha \in \bR$. This can be rewritten as,
\bea
\nabla^{i} V_i = \alpha^2 \ ,
\ee
with $V_i = \nabla_i f + \lambda_i f$. By partial integration over $\cal{N}$, the LHS vanishes since it is a total derivative and we have,
\bea
\alpha = 0,~~ \nabla^i V_i = 0 \ .
\ee
In the latter case, we shall obtain PDEs of the form
\bea
\nabla^{i}\nabla_{i}f + \lambda^i \nabla_{i}f = \alpha^2 \ ,
\ee
and an application of the Hopf maximum principle, which states that if $f \geq 0$ is a $C^2$-function which attains a maximum value in $\cal{N}$ then,
\bea
f = const,~~ \alpha = 0, \ .
\ee
\section{The classical Lichnerowicz theorem}
A particularly important aspect of the analysis of the Killing spinor equations associated with the near-horizon geometries of black holes is the proof of certain types of generalized Lichnerowicz theorems. These state that if a spinor is a zero mode of a certain class of near-horizon Dirac operators, then it is also parallel with respect to a particular class of supercovariant derivatives, and also satisfies various algebraic conditions. These Dirac operators and supercovariant connections depend linearly on certain types of $p$-form fluxes which appear
in the supergravity theories under consideration. Before attempting to derive these results it is instructive to recall how the classical Lichnerowicz theorem arises, in the case when the fluxes are absent.
On any spin compact manifold ${\cal{N}}$, without boundary, one can establish the equality
\bea
\int_{\cal{N}} \langle \Gamma^i \nabla_i \epsilon, \Gamma^j \nabla_j \epsilon \rangle= \int_{\cal{N}} \langle \nabla_i \epsilon , \nabla^i \epsilon \rangle+\int_{\cal{N}} {R\over 4} \langle \epsilon , \epsilon \rangle \ .
\label{classlich}
\ee
To show this, we let
\bea
{\cal{I}} = \int_{\cal{N}} \langle \nabla_i \epsilon ,\nabla^i \epsilon \rangle
- \langle \Gamma^i \nabla_i \epsilon ,
\Gamma^j \nabla_j \epsilon \rangle \ .
\ee
This can be rewritten as\footnote{The gamma matrices are Hermitian $(\Gamma^i)^{\dagger} = \Gamma^i$ with respect to this inner product.},
\bea
{\cal{I}} = \int_{\cal{N}} -\nabla_{i}\langle \epsilon, \Gamma^{i j}\nabla_{j}\epsilon \rangle + \int_{\cal{N}} \langle \epsilon, \Gamma^{i j}\nabla_i \nabla_j \epsilon\rangle \ .
\ee
The first term vanishes since the integrand is a total derivative and for the second term we use $\Gamma^{i j}\nabla_i \nabla_j \epsilon = - \frac{1}{4}R \epsilon$, thus we have
\bea
{\cal{I}} = -\int_{\cal{N}} {R\over 4} \langle \epsilon , \epsilon \rangle \ ,
\ee
where $\nabla$ is the Levi-Civita connection, $\langle \cdot, \cdot\rangle$ is the real and positive definite $Spin$-invariant Dirac inner product (see Appendix B) identified with the standard Hermitian inner product and $R$ is the Ricci scalar.
On considering the identity ({\ref{classlich}}), it is clear that if $R>0$ then
the Dirac operator has no zero modes. Moreover, if $R=0$, then the zero modes of the Dirac operator are parallel with respect to the Levi-Civita connection.
An alternative derivation of this result can be obtained by noting that if $\epsilon$ satisfies the Dirac equation $\Gamma^{i}\nabla_{i}\epsilon = 0$, then
\bea
\nabla^{i}\nabla_{i} \parallel \epsilon \parallel^2 = \frac{1}{2}R \parallel \epsilon \parallel^2 + 2\langle \nabla_{i}\epsilon, \nabla^{i}\epsilon \rangle \ .
\label{classlich2}
\ee
This identity is obtained by writing
\bea
\nabla^i \nabla_i \parallel \epsilon \parallel^2 = 2\langle\epsilon ,\nabla^i \nabla_i\epsilon\rangle + 2 \langle\nabla^i \epsilon, \nabla_i \epsilon\rangle \ .
\ee
To evaluate this expression note that
\bea
\nabla^i \nabla_i \epsilon = \Gamma^{i}\nabla_{i}(\Gamma^{j}\nabla_j \epsilon) -\Gamma^{i j}\nabla_i \nabla_j \epsilon = \frac{1}{4}R\epsilon \ .
\ee
On considering the identity ({\ref{classlich2}}), if $R \geq 0$, then the RHS of
({\ref{classlich2}}) is non-negative. An application of the Hopf maximum principle then implies that $ \parallel \epsilon \parallel^2$ is constant, and moreover
that $R=0$ and $\nabla \epsilon=0$.
\begin{comment}
\section{Vacuum}
\bea
S = \int d^{D}x \sqrt{-g} R
\ee
with $R_{\mu \nu} = 0$
\bea
\label{vGSE}
{\cal D}_{\mu}\epsilon = \nabla_{\mu}\epsilon = 0
\ee
\bea
[{\cal D}_{\mu}, {\cal D}_{\nu}]\epsilon = [\nabla_{\mu}, \nabla_{\nu}]\epsilon = \frac{1}{4}R_{\mu \nu, \rho \sigma}\Gamma^{\rho \sigma}
\ee
\bea
\Gamma^{\nu}[{\cal D}_{\mu}, {\cal D}_{\nu}]\epsilon = \frac{1}{4}\Gamma^{\nu} R_{\mu \nu, \rho \sigma} \Gamma^{\rho \sigma} = \frac{1}{4}R_{\mu \nu, \rho \sigma}(\Gamma^{\nu \rho \sigma} + 2g^{\nu [ \rho}\Gamma^{\sigma]}) = -\frac{1}{2}R_{\mu \sigma} \Gamma^{\sigma}
\ee
\bea
\nabla_{\mu} \epsilon &=& \partial_{\mu} \epsilon + \frac{1}{4}\Omega_{\mu, \nu \rho}\Gamma^{\nu \rho} \epsilon
\nonumber \\
&=& \partial_{\mu} \epsilon + \frac{1}{4}(2\Omega_{\mu, +i}\Gamma^{+i} + 2\Omega_{\mu, -i}\Gamma^{-i} + 2\Omega_{\mu, +-}\Gamma^{+-} + \Omega_{\mu, jk}\Gamma^{jk}) \epsilon
\nonumber \\
&=& \partial_{\mu} \epsilon + \frac{1}{2}\Omega_{\mu, +i}\Gamma^{+i}\epsilon + \frac{1}{2}\Omega_{\mu, -i}\Gamma^{-i}\epsilon + \frac{1}{2}\Omega_{\mu, +-}\Gamma^{+-}\epsilon + \frac{1}{4}\Omega_{\mu, jk}\Gamma^{jk}\epsilon
\nonumber \\
\ee
We can expand along the $\mu = +, -, i$ directions to obtain,
\bea
\nabla_+ \epsilon &=& \partial_+ \epsilon + \frac{1}{4}r^2(\Delta h_i - \partial_i \Delta)\Gamma^{+i}\epsilon - \frac{1}{4}h_i\Gamma^{-i}\epsilon - \frac{1}{2}r\Delta\Gamma^{+-}\epsilon - \frac{1}{8}r(dh)_{ij}\Gamma^{ij}\epsilon
\nonumber \\
\ee
\bea
\nabla_{-} \epsilon = \partial_{-} \epsilon - \frac{1}{4}h_i \Gamma^{+i}\epsilon
\ee
\bea
\nabla_{i} \epsilon &=& \partial_i \epsilon - \frac{1}{4}r(dh)_{ij}\Gamma^{+j}\epsilon + \frac{1}{4}h_i \Gamma^{+-}\epsilon + \frac{1}{4}\tilde{\Omega}_{i,jk}\Gamma^{jk}\epsilon
\nonumber \\
&=& \nabla_{i}\epsilon - r\partial_{r} \epsilon h_i - \frac{1}{4}r(dh)_{ij}\Gamma^{+j}\epsilon + \frac{1}{4}h_i \Gamma^{+-}\epsilon
\ee
The description of the metric near extreme Killing horizons as expressed in Gaussian null coordinates can be written as,
\be
ds^2 &=&2 \bbe^+ \bbe^- + \delta_{ij} \bbe^i \bbe^j~,
\ee
where we have introduced the frame
\be
\label{basis1}
\bbe^+ = du, \qquad \bbe^- = dr + rh -{1 \over 2} r^2 \Delta du, \qquad \bbe^i = e^i_I dy^I~,
\ee
and the dependence on the coordinates $u$ and $r$ is explicitly given. The spatial horizon section ${\cal S}$, is the co-dimension 2 submanifold given by the equation $r=u=0$, i.e.~all these
components of the fields depend only on the coordinates of ${\cal S}$.
We can evaluate the Einstein field equation along the lightcone. This gives
\bea
\label{veq1}
{1 \over 2} \tn^i h_i -\Delta -{1 \over 2}h^2
&=& 0~,
\ee
and
\bea
\label{veq2}
{\tilde{R}}_{ij} &=& -\tn_{(i} h_{j)}
+{1 \over 2} h_i h_j
\ee
where $\tilde \nabla$ is the Levi-Civita connection of the metric on ${\cal S}$. Note also that the $++$ and $+i$ components of the
Einstein equation, which are
\bea
\label{veq3}
{1 \over 2} \tn^i \tn_i \Delta -{3 \over 2} h^i \tn_i \Delta-{1 \over 2} \Delta \tn^i h_i
+ \Delta h^2 +{1 \over 4} dh_{ij} dh^{ij} = 0
\ee
and
\bea
\label{veq4}
{1 \over 2} \tn^j dh_{ij}-dh_{ij} h^j - \tn_i \Delta + \Delta h_i = 0
\ee
are implied by ({\ref{veq1}}), ({\ref{veq2}}) as shown in the previous section. We can integrate (\ref{veq1}) over the horizon section $\cal{S}$ to find
\bea
\int_{\cal{S}} \bigg(\Delta +{1 \over 2}h^2\bigg) = 0,
\ee
Assuming $\Delta \geq 0$ we have $\Delta = 0, h=0$ thus the horizon section is ${\cal{S}} = T^{d-2}$ and the near-horizon geometry is locally isometric to $\R^{1,d-1}$. Here we don't need to consider the supersymmetry variations to classify the geometry, since the bosonic field equations are enough to establish this. Nonetheless, we can find the conditions required for such a solution to exist as the steps will be similar when we consider the near-horizon geometries of the various supergravity theories considered in this thesis. We can first integrate along the two lightcone directions, i.e.~we integrate the KSEs
along the $u$ and $r$ coordinates. To do this, we decompose $\epsilon$ as
\bea
\e=\e_++\e_-~,
\label{vksp1}
\ee
where $\Gamma_\pm\epsilon_\pm=0$, and find that
\bea\label{vlightconesol}
\e_+=\phi_+(u,y)~,~~~\e_-=\phi_- + \frac{1}{4} r \Gamma_- h_{i}\Gamma^{i} \phi_+~,
\ee
and
\bea
\phi_-=\eta_-~,~~~\phi_+=\eta_+ + \frac{1}{4} u \Gamma_+ h_{i}\Gamma^{i} \eta_-~,
\ee
and $\eta_\pm$ depend only on the coordinates of the spatial horizon section ${\cal S}$. As spinors on ${\cal S}$, $\eta_\pm$ are sections of the $Spin(D-2)$ bundle on ${\cal S}$
associated with the appropriate representation. Equivalently, the $Spin(D-1,1)$ bundle $S$ on the spacetime when restricted to ${\cal S}$ decomposes
as $S=S_-\oplus S_+$ according to the lightcone projections $\Gamma_\pm$. Although $S_\pm$ are distinguished by the lightcone chirality, they are isomorphic
as $Spin(D-2)$ bundles over ${\cal S}$. We shall use this in the counting of supersymmetries horizons.
Substituting the solution of the KSEs along the lightcone directions (\ref{vlightconesol}) back into the gravitino KSE (\ref{vGSE}) and appropriately expanding in the $r,u$ coordinates, we find that
for the $\mu = \pm$ components, one obtains the additional conditions
\bea
\label{vint1}
&&\bigg({1\over2}\Delta - {1\over8}(dh)_{ij}\Gamma^{ij} + {1\over 8}h^2 \bigg)\phi_{+} = 0
\ee
\bea
\label{vint2}
&&\bigg(\frac{1}{4}\Delta h_i \Gamma^{i} - \frac{1}{4}\partial_{i}\Delta \Gamma^{i} -\frac{1}{32}(dh)_{ij}h_{k}\Gamma^{i j k} + \frac{1}{32}(dh)_{i j}h^{j}\Gamma^{i} \bigg) \phi_{+} = 0,
\nonumber \\
\ee
\bea
\label{vint3}
&&\bigg(-\frac{1}{2}\Delta - \frac{1}{8}(dh)_{ij}\Gamma^{ij} - {1\over 8} h^2 \bigg)\phi_{-} = 0 \ .
\ee
Where $\Theta = \frac{1}{4}h_{i}\Gamma^{i}$. Similarly the $\mu=i$ component gives
\bea
\label{vint4}
&&\tilde \nabla_i \phi_\pm\mp {1\over 4} h_i \phi_\pm = 0
\ee
and
\bea
\label{vint5}
\tilde \nabla_i \tau_{+} -\frac{3}{4} h_i\tau_{+} -\frac{1}{4}(dh)_{ij}\Gamma^{j} \phi_{+} = 0
\ee
where we have set
\bea
\label{iiaint6}
\tau_{+} = \Theta \phi_{+} \ .
\ee
All the conditions above can be viewed as integrability conditions along the lightcone and mixed lightcone and ${\cal S}$ directions.
The substitution of the spinor (\ref{vksp1}) into the KSEs produces a large number of additional conditions. These can be seen
either as integrability conditions along the lightcone directions, as well as integrability conditions along the mixed lightcone and
${\cal S}$ directions, or as KSEs along ${\cal S}$. A detailed analysis, of the formulae obtained reveals
that the independent KSEs are those that are obtained from the naive restriction of the KSE to ${\cal S}$. In particular,
the independent KSEs are
\bea
\label{vcovr}
\nabla_{i}^{(\pm)}\eta_{\pm} = 0~,
\ee
where
\bea
\nabla_{i}^{(\pm)}&=& \nabla_{i} \mp \frac{1}{4}h_{i}~,
\ee
with
$\nabla^{(\pm)}$ arise from the supercovariant connection as restricted to ${\cal S}$ .
One can also show that if $\eta_{-}$ solves $(\ref{vcovr})$ then,
\bea
\eta_+ = \Gamma_{+}\Theta \eta_{-}~,
\label{vepfem}
\ee
also solves $(\ref{vcovr})$.
To prove that such horizons always admit an even number of supersymmetries, one has to prove that there are as many $\eta_+$ Killing spinors as there are $\eta_-$ Killing spinors,
i.e.~that the $\eta_+$ and $\eta_-$ Killing spinors come in pairs. For the vacuum case this is not possible since we have $\Theta = 0$ from the analysis of (\ref{veq1}) or the integrability condition (\ref{vint1}) as we shall show in what follows.
For this, we shall identify the Killing spinors with the zero modes of Dirac-like operators
which depend on the fluxes and then use the index theorem to count their modes.
We define horizon Dirac operators associated with the supercovariant derivatives following from the gravitino KSE as
\bea
{\cal D}^{(\pm)} \equiv \Gamma^{i}\nabla_{i}^{(\pm)} = \Gamma^{i}\nabla_{i} \mp\frac{1}{4}h_{i}\Gamma^{i}~,
\ee
First let us establish that the $\eta_+$ Killing spinors can be identified with the zero modes of a ${\cal D}^{(+)}$. It is straightforward to see
that if $\eta_+$ is a Killing spinor, then $\eta_+$ is a zero mode of ${\cal D}^{(+)}$. So it remains to demonstrate the converse.
For this assume that $\eta_+$ is a zero mode of ${\cal D}^{(+)}$, i.e.~${\cal D}^{(+)}\eta_+=0$, one can then establish the equality
\bea
{\nabla}^{i}{\nabla}_{i}\parallel\eta_+\parallel^2 - h^i {\nabla}_{i}\parallel\eta_+\parallel^2 = 2\parallel{\nabla^{(+)}}\eta_{+}\parallel^2,
\label{vmaxprin}
\ee
It is clear that if the last term on the right-hand-side of the above identity is positive semi-definite, then one can apply the maximum principle on $\parallel\eta_+\parallel^2$
as the fields are assumed to be smooth, and ${\cal S}$ compact. The maximum principle implies that $\eta_+$ are Killing spinors and $\parallel\eta_+\parallel=\mathrm{const}$.
To summarize we have established that,
\bea
\nabla_{i}^{(+)}\eta_+=0~,\Longleftrightarrow~ {\cal D}^{(+)}\eta_+=0~.
\ee
Moreover $\parallel\eta_+\parallel^2$ is constant on ${\cal S}$.
Next we shall establish that the $\eta_-$ Killing spinors can also be identified with the zero modes of a modified horizon Dirac operator ${\cal D}^{(-)}$.
It is clear that all Killing spinors $\eta_-$ are zero modes of ${\cal D}^{(-)}$. To prove the converse,
suppose that $\eta_-$ satisfies ${\cal D}^{(-)} \eta_-=0$.
The proof proceeds by calculating the Laplacian of $\parallel \eta_-
\parallel^2$. One can then establish the formula
\bea
\label{vl2b}
{\nabla}^{i} \big( \nabla_{i}\parallel \eta_- \parallel^2 + \parallel \eta_- \parallel^2 h_{i} \big)
= 2 \parallel{\nabla^{(-)}}\eta_{-}\parallel^2~,
\ee
The last term on the RHS of ({\ref{vl2b}}) is negative semi-definite and on integrating ({\ref{vl2b}}) over
${\cal{S}}$ and assuming that ${\cal{S}}$ is compact and without boundary, one finds that ${\nabla^{(-)}}\eta_{-}=0$.
Therefore, we have shown that
\bea
\nabla_{i}^{(-)}\eta_-=0~,\Longleftrightarrow~ {\cal D}^{(-)}\eta_-=0~ \ .
\ee
This concludes the relationship between Killing spinors and zero modes of modified horizon Dirac operators.
From the integrability condition (\ref{vint1}) one can also show
\bea
\bigg({1\over2}\Delta + {1\over 8}h^2 \bigg)\parallel\eta_+\parallel^2 = 0,
\ee
since $\parallel\eta_+\parallel^2$ is constant, we have find $h=0, \Delta =0$ which is consistent with the result we obtained from the decomposition of the vacuum einstein equation (\ref{veq1}).
In the investigation of the integrability conditions, one can show in the analysis of the independent conditions that if $\eta_-$ is a Killing spinor, then
$\eta_+ = \Gamma_+ \Theta \eta_-$ is also a Killing spinor. Since we know that the $\eta_+$ and $\eta_-$ Killing spinors
appear in pairs, the formula (\ref{vepfem}) provides a way to construct the $\eta_+$ Killing spinors from the $\eta_-$ ones.
However, this is only the case provided that $\eta_+=\Gamma_+ \Theta \eta_-\not=0$. For the vacuum case, this does not hold since $h=0$ implies $\Theta = 0$.
\end{comment} | {"config": "arxiv", "file": "1910.01080/background/background.tex"} |
TITLE: A supposedly "trivial" logic question
QUESTION [3 upvotes]: The professor told me that the solution is trivial, but I must have missed something because I don't even see anyway to start.
Consider an arbitrary language $L$ (which can contains function, constant and relation symbols) and a first-order theory $T$ that can be axiomatized by sentences that have no $\neg$ at all. Prove or find a counterexample that $T$ is always consistent.
REPLY [3 votes]: To show consistency, it is enough to show that there is a model. Consider the $1$-element structure $M$ in which all constant symbols and function symbols have the only possible interpretation, and all relations are true.
We need to show that all the negation-free sentences of our language are true in $M$. In principle this is done by induction on complexity. Start from atomic formulas, and observe that when an atomic formula has its free variables replaced by the one element of $M$, the atomic formula is true in $M$. | {"set_name": "stack_exchange", "score": 3, "question_id": 1703962} |
TITLE: First step with the equation $\sigma(n)=\phi(n)+\operatorname{rad}(n)$: a first statement
QUESTION [2 upvotes]: I am wondering about the equation $$\sigma(n)=\phi(n)+\operatorname{rad}(n),\tag{1}$$ where for integers $n\geq 1$ we denote with $\sigma(n)$ the sum of divisors function, $\phi(n)$ is Euler's totient function and with $\operatorname{rad}(n)$ we denote the radical of the integer $n$, see in Wikipedia this definition. I am inspired in the equation by professor Iannucci, see the nice [1].
Currently (I've no implemented the function $\operatorname{rad}(n)$) I don't know if such equation $(1)$ was in the literature, and if it is possible deduce some interesting fact. I've dilucidated that the only integer less than 74 that satisifies $(1)$, is $n=2$. And I've searched in OEIS several strings like than these sigma(n)-phi(n), rad(n)... I am searched the equation in OEIS, in about 32 pages.
Question. If you can identify the sequence associated to $$\sigma(n)=\phi(n)+\operatorname{rad}(n)$$ and if it was in the literature, please refer it. Additionally, if you can compute more terms of such sequence it is good. I am interested about what should be the first step to study the specific equation $(1)$ by means of mathematical reasoning with the purpose to set some statement related with it (not so professional as papers in a journal, I say the first interesting statement about this equation). Many thanks.
I know that is very important have more terms of our sequence, also the size of each side in the equation, as calculated Iannucci in the first paragraph, but I have no the average value of $$\phi(n)+\operatorname{rad}(n).$$ Finally I believe that it should be very imporant the parity of each of our summands.
Then, imagine that a professor ask you about how set some mathematics about this equation, what is your reasoning and statement/conjecture? I hope that my question isn't too broad. I am asking about the first and more important fact about this equation. After that I have some answer I should choose an answer.
References:
[1] Douglas E. Iannucci, On the Equation $\sigma(n)=n+\phi(n)$, Journal of Integer Sequences, Vol. 20 (2017), article 17.6.2.
Also could be interesting check Hasler's sequence A228947, also from the On-line Encyclopedia of integer sequences.
REPLY [1 votes]: Using results from the Iannucci paper referenced in the problem statement, we can show that $n=2$ is the only solution to the problem. (Note: Eq (1) refers to the equation so-numbered above in the problem statement.)
Claim: Any solution to Eq (1) is square-free.
Proof: Suppose $n$ is not square-free. Say $n=p^k\cdot b$ for some prime $p$, with $k\ge 2$ and $p\nmid b$. We will show that $n$ is not a solution to Eq (1), and thus establish the Claim.
We have
$$\begin{array}{cccr}
\mbox{rad}(n)+\phi(n)&=&p\cdot\mbox{rad}(b)+(p^k-p^{k-1})\phi(b) \\
&\le& p\cdot b+(p^k-p^{k-1})b\\
&=& n+(p-p^{k-1})b& \\
&\le&n &(\mbox{because }k\ge 2) \\
&<&\sigma(n)
\end{array}$$
Thus Eq(1) cannot hold for such $n$. So the Claim is established.
Now let $n$ be a solution of Eq(1). By the claim, $n$ is square-free. Hence $\mbox{rad}(n)=n$. And so in this situation, Eq (1) becomes $n+\phi(n)=\sigma(n)$.
But Iannucci shows (in Theorem 2 of the above referenced paper) that any integer $n\ge 2$ satisfying $n+\phi(n)=\sigma(n)$ must be an odd perfect square. But our candidate $n$ is square-free. We conclude that $n=2$ is the only solution to Eq (1). | {"set_name": "stack_exchange", "score": 2, "question_id": 2341985} |
TITLE: units of $\mathbb Z[\sqrt{-5}]$
QUESTION [3 upvotes]: I'm trying to find units of $\mathbb Z[\sqrt{-5}]$.
So let $a,b\in \mathbb Z[\sqrt{-5}]$ s.t. $ab=1$. if $a=a_1+\sqrt{-5}a_2$ and $b=b_1+\sqrt{-5}b_2$, then we get $$\begin{cases}a_1b_1-5a_2b_2=1\\ a_1b_2+b_1a_2=0\end{cases}.$$
First equation give us $a_1,b_1\in \mathbb Z/5\mathbb Z$ and thus $$(a_1,b_1)\in \{(1,1),(2,3),(4,4)\},
$$
but how can I continue ? I was wondering to solve the second equation in $\mathbb Z/5\mathbb Z$, but I get $a_1b_2=4b_1a_2$, but it's unfortunately not conclusive.
REPLY [3 votes]: If you equip $\mathbb Z[\sqrt{-5}]$ with the field norm $N$ defined by $N(a+b\sqrt{-5}) = a^2+5b^2$, then I leave it to you to check that the norm is multiplicative: $N(\alpha\beta)=N(\alpha)N(\beta)$. Thus, if $\alpha\in\mathbb Z[\sqrt{-5}]$ is a unit, $N(\alpha) = 1$, since there exists $\beta\in \mathbb Z[\sqrt{-5}]$ such that $\alpha\beta=1$, and the norm is multiplicative, so $N(\alpha\beta) = N(1) = 1 = N(\alpha)N(\beta)$.
Thus, the only units to be had are when $\alpha = \pm 1$.
REPLY [1 votes]: There is a method in commutative algebra to solve such problems. Let $N:\mathbb{Z}[\sqrt{-5}]\to\mathbb{Z}_+ $ is given by $N(a+b\sqrt{-5})=a^2+5b^2 $, called norm function. Its an ordinary theorem that $\alpha$ is invertible iff $N(\alpha)=1$.
In this case, $a+b\sqrt{-5} $ is invertible iff $a^2+5b^2=1$. Is it enough? | {"set_name": "stack_exchange", "score": 3, "question_id": 2164039} |
TITLE: PDE on a Damped Wave Equation
QUESTION [0 upvotes]: What are the eigenvalues and eigenfunctions of $$ X''+X'-\sigma X=0 \\ \text{ with boundary condition } X(0)=X(l)=0$$
I know that for $X''-\sigma X=0 $, the eigenvalues would be $ -\left ( \frac{n \pi}{l} \right )^2$ and the eigenfunction would be $ X= \ sin(n \pi x/l)$
Just not sure how to deal with the additional X' in my problem.
REPLY [2 votes]: A way to solve for the eigenvalues $\sigma$ is to first solve
$$
X''+X'-\sigma X =0\\
X(0)=0,\;\; X'(0)=1.
$$
The solution of this equation is unique. You should have studied such equations by now. You can factor the equation into operator form with $D=\frac{d}{dx}$:
$$
\{(D+1/2)^{2}-(1/4+\sigma)\}X = 0 \\
(D+1/2-\sqrt{1/4+\sigma})(D+1/2+\sqrt{1/4+\sigma})X=0.
$$
The solutions are
$$
X(x)=A\exp(\{-1/2+\sqrt{1/4+\sigma}\}x)+B\exp(\{-1/2-\sqrt{1/4+\sigma}\}x)
$$
You get $X(0)=0$ with $A+B=0$. You get $X'(0)=1$ with
$$
A(-1/2+\sqrt{1/4+\sigma})+B(-1/2-\sqrt{1/4+\sigma})=1.
$$
Assuming $A+B=0$ then gives $(A-B)\sqrt{1/4+\sigma}=1$.
The desired constants are
$$
A=\frac{1}{2\sqrt{1/4+\sigma}},\;\; B=-\frac{1}{2\sqrt{1/4+\sigma}}.
$$
Therefore,
$$
X(x) = \frac{e^{-x/2}}{\sqrt{1/4+\sigma}}\sinh(\sqrt{1/4+\sigma}x),\;\; \sigma \ne -1/4.
$$
An advantage of using a normalized $X$ is that the special case where $\sigma=-1/4$ is handled by taking a limit of the above as $\sigma\rightarrow-1/4$: this gives $xe^{-x/2}$, which is the proper solution for $\sigma=-1/4$. (The special case being handled by a limit is guaranteed by the general theory when you used normalized functions.) This special case where $\sigma=-1/4$ is not $0$ at $l$, which allows us to rule out $\sigma=-1/4$ as a possible eigenvalue.
Any function $Y$ for which $Y''+Y'-\sigma Y=0$ and $Y(0)=0$ must be a constant multiple of the above solution. So finding a non-zero solution of $Y(0)=Y(l)=0$ is equivalent to finding $\sigma$ so that $X(l)=0$ for the above $X$ because $Y$ must be a non-zero constant multiple of $X$ by uniqueness of $X$. $\sinh$ is not $0$ at $l$, but it switches to $\sin$ if the argument is complex. So $\sigma < -1/4$ must hold, and the possible values of $\sigma$ must satisfy
$$
\sqrt{1/4+\sigma} = in\pi/l \\
\implies 1/4+\sigma = -n^{2}\pi^{2}/l^{2} \\
\implies \sigma = -\frac{1}{4}-\frac{n^{2}\pi^{2}}{l^{2}},\;\;\; n=1,2,3,\cdots.
$$
This is what Juan had posted and deleted. It's worth seeing how to do this kind of thing at least once. | {"set_name": "stack_exchange", "score": 0, "question_id": 1346494} |
\section{Preliminaries}
In this section, we review the well known belief propagation (BP) method for error correcting codes and the deep neural network model that is constructed by deep unfolding this BP method.
Here we summarize the message passing belief propagation sum-product and min-sum algorithms along with their deep unfolding neural network models.
We consider the discrete-time additive white Gaussian noise (AWGN) channel model. In transmission time $i \in [1:n]$, the channel output is $Y_i = gX_i + Z_i$. where $g$ is the channel gain, $X_i \in \{ \pm 1\}$, $\{ Z_i \}$ is a AWGN ($\sigma^2$) process, independent of the channel input $X^n = x^n(m)$, $m \in [1:2^{k}]$, and $k$ is the message length.
\subsection{Belief Propagation (BP) Algorithms}
Given a linear error correcting code $C$ of block-length $n$, characterized by a parity check matrix $H$ of size $n \times (n-k)$, every codeword $x \in C$ satisfies $H^T x = \0$, where $H^T$ denotes the transpose of $H$ and $0$ the vector of $n-k$ zeros.
The BP decoder for a linear code can be constructed from the Tanner graph of the parity check matrix.
The Tanner graph contains $n-k$ check nodes and $n$ variable nodes arranged as a bipartite graph with edges between check and variable nodes, which represent messages passing between these nodes.
If $H_{vc} = 1$, there is an edge between variable node $v$ and check node $c$.
A simple example is given in Figure~\ref{fig:bp}(a) with $n = 3$ variable nodes and $(n-k) = 2$ check nodes.
The BP decoder consists of multiple iterations.
Each iteration corresponds to one round of messaging passing between variable and check nodes.
\begin{figure}[!t]
\centering
\vspace{-0.1in}
\includegraphics[width=\linewidth]{fig/bp_illustrate2.pdf}
\caption{Tanner graph of a code and BP unfolding architecture.}
\vspace{-0.15in}
\label{fig:bp}
\end{figure}
In iteration $i$, on edge $e = (v,c)$, the message $x_{i, (v,c)}$ passed from variable node $v$ to the check node $c$ is given by
\begin{align}
\label{eq:m_vc_bp}
x_{i, e = (v,c)} = l_v + \sum_{e' = (c',v), c' \neq c} x_{i-1, e'},
\end{align}
where $l_v$ is the log-likelihood ratio of the channel output corresponding to the $v$th codeword bit $C_v$, i.e.,
\begin{align}
l_v = \log\frac{\Pr(C_v = 1 | y_v)}{\Pr(C_v = 0 | y_v)}. \nonumber
\end{align}
In the sum-product algorithm, the message $x_{i, (c,v)}$ from check node $c$ to variable node $v$ in iteration $i$ is computed as
\begin{align}
\label{eq:m_cv_bp}
x_{i, e = (c,v)} = 2 \tanh^{-1}\Big( \prod_{e' = (v',c), v' \neq v} \tanh \frac{x_{i,e'}}{2} \Big).
\end{align}
For numerical stability, the messages going into a node in both Eq.~(\ref{eq:m_vc_bp}) and Eq.~(\ref{eq:m_cv_bp}) are normalized at every iteration and all the messages are also truncated within a fixed range.
Note that the computation in Eq.~(\ref{eq:m_cv_bp}) involves repeated multiplications and hyperbolic functions, which may lead to numerical problems in actual implementations. Even with truncation of message values within a certain range, e.g., commonly $[-10, 10]$, we still observe numerical problems when running the sum-product algorithm.
The min-sum algorithm uses a ``min-sum'' approximation of the above message as follows:
\begin{align}
x_{i, e = (c,v)} = \min_{e' = (v',c), v' \neq v} |x_{i,e'}| \prod_{e' = (v',c), v' \neq v} \sign(x_{i,e'}). \nonumber
\end{align}
Suppose the BP decoder runs $L$ iterations. Then the $v$th output after final iteration is given by
\begin{align}
\label{eq:output}
o_v = l_v + \sum_{e' = (c',v)} x_{L,e'}.
\end{align}
\subsection{Unfolding BP Models}
The BP decoder has an equivalent Trellis representation that motivates the unfolding neural network architecture.
A simple example is provided in Figure~\ref{fig:bp}(b).
Each BP iteration $i$ unfolds into 2 hidden layers, $2\times i-1$ and $2\times i$, corresponding to two passes of messages from variable to check nodes and from check to variable nodes.
The number of nodes in each hidden layer is the same as the number of edges in the Tanner graph and, respectively, each node computes the message sent through each edge.
Thus, there are $2\times L$ hidden layers in this Trellis representation in addition to an input and output layer.
The input takes log-likelihood ratios of channel received signals and the output computes Eq.~\ref{eq:output}.
The unfolding neural network model shares the same architecture as the Trellis representation and adds trainable weights to the edges.
In other words, in odd layer $i$, the computation at node $i, e = (v,c)$, performs the following operation (activation function):
\begin{align}
\label{eq:m_vc}
x_{i, e = (v,c)} = w_{i,v}l_v + \sum_{e' = (c',v), c' \neq c} w_{i,e,e'}x_{i-1, e'}.
\end{align}
Note that trainable weights $w_{i,v}$ and $w_{i,e,e'}$ are introduced in the computation.
These weights can be trained and address various challenging obstacles with the BP method: 1) reduce the large number of iterations needed in BP, leading to more efficiency; 2) improve decoding performance and possibly approach optimal ML criterion, thanks to the ability to possibly minimize the effects of short cycles known to cause difficulties for BP decoders.
In even layer $i$, the computation at node $i, e = (v,c)$ is similar to that in the regular BP algorithm:
\begin{align}
\label{eq:m_cv}
x_{i, e = (c,v)} = 2 \tanh^{-1}\Big( \prod_{e' = (v',c), v' \neq v} \tanh \frac{x_{i-1,e'}}{2} \Big).
\end{align}
For numerical stability, we also apply normalization and truncation of messages after every layer.
Nodes at the output layer perform
\begin{align}
\label{eq:o_v}
o_v = \sigma \Big(w_{2L + 1,v}l_v + \sum_{e' = (c',v)} w_{2L+1,v,e'} x_{2L,e'} \Big),
\end{align}
where $\sigma(.)$ is the sigmoid function to obtain probability from the log-likelihood ratio representation.
Note that trainable weights $w_{2L + 1,v}$ and $w_{2L+1,v,e'}$ are also introduced here. | {"config": "arxiv", "file": "2112.11491/ECC_Globecom 2021_submit/preliminary3.tex"} |
\section{Numerical experiments} \label{Sec_results}
The methodology developed in the previous sections is now demonstrated on a set of examples. Different aspects of the global solution method are illustrated on a 1-D stochastic diffusion equation. A more computationally involved example is next considered with a Shallow Water problem with multiple sources of uncertainty.
\subsection{Approximating a random variable}
We consider the 1-D stochastic diffusion equation presented in section~\ref{Good_basis}, briefly recalled here for sake of convenience:
\be
\nabla_x \, \left(\nu\left(x, \bxi\right) \, \nabla_x u\left(x,\bxi\right)\right) = F\left(x, \bxi\right), \qquad u\left(x_-,\bxi\right) = u_-, \: u\left(x_+,\bxi\right) = u_+. \label{diff_eq2}
\ee
The stochastic approximation basis relies on a HDMR format with a maximum interaction order $\Ninter = 3$ and 1-D Legendre polynomials $\left\{\psi_\alpha\right\}_{\alpha=1}^{\No}$ of maximum degree $\No = 8$.
In this section, the focus is on approximating a purely random quantity, \ie, disregarding its spatial dependence. We then rely on samples of the solution $u\left(x, \bxi\right)$ taken at a given spatial location $x^\star$: $\left\{\uq := u\left(x^\star, \bxiq\right)\right\}_{q=1}^\Nq$.
\subsubsection{Influence of the number of samples}
We first focus on the achieved accuracy in the approximation with a given budget $\Nq + \Nqhat$ samples. The number of test points $\Nqt$ to estimate the approximation error $\varepsilon$, Eq.~\eqref{sto_error_def}, is chosen sufficiently large so that $\varepsilon$ is well estimated, $\Nqt = 10,000$. In Fig.~\ref{L2_MM_schemes_Nd8}, the performance of the present gLARS-ALS methodology is compared with both a plain HDMR approximation, \ie, with no subset selection hence considering the whole {\apriori} approximation basis, and a Polynomial Chaos (PC) approximation with a sparse grid technique. The Smolyak scheme associated with a Gauss-Patterson quadrature rule is used as the sparse grid, with varying number of points in the 1-D quadrature rule and varying levels. The dimensionality of the stochastic space is $\Nd = 8$.
The sparse grid is seen to require a large number of samples to reach a given approximation accuracy.\footnote{Note that the plain Smolyak scheme is used here, which does not exploit anisotropy in the response surface. More sophisticated Smolyak-based approximations have been developed, see \cite{Nobile_al_07}, and are expected to provide better results.
} The HDMR-format approximation, with various interaction orders $\Ninter$, provides a better performance than PC/Smolyak but still requires more points to reach a given accuracy than the present methodology which performs significantly better in approximating the QoI from a given dataset. The present gLARS-ALS approximation error is also seen to be smooth and monotonic when the amount of information varies. When $\Nq$ is large enough, the subset selection step becomes useless as all $\cardPprior$ terms of the proposal basis can be estimated from the large amount of information and the present gLARS-ALS performance should then be similar as that of the HDMR. This is indeed what is observed for $\Nq \gtrapprox 5000$. Note that the benefit of a subset selection step in terms of accuracy improvement increases with the dimension $\Nd$ as the size $\cardPprior$ of the potential dictionary then grows.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.65\textwidth, draft = false]{L2_Npts_pres_Nd8_2.eps}
\caption{Convergence of the approximation with the number of samples $\Nq + \Nqhat$. Different approximation methods are compared: plain HDMR, PC/Smolyak scheme sparse grid spectral decomposition and the present gLARS-ALS. The convergence is plotted in terms of $\varepsilon$. $\Nd = 8$, $\No = 8$, $\Ninter = 3$, $\NinterPC = 3$.}
\label{L2_MM_schemes_Nd8}
\end{center}
\end{figure}
\subsubsection{Influence of the stochastic dimension}
The approximation accuracy of the present method is now studied when the dimension of the stochastic space varies. The same problem as above is considered but with various truncation orders of the source $F$ and the diffusion coefficient $\nu$ definitions, see Eqs.~\eqref{nu_F_definition}. The solution of the diffusion problem \eqref{diff_eq2} is of dimension $\Nd = \Nd_F + \Nd_\nu$ and the dimensions $\Nd_F$ and $\Nd_\nu$ are varied together, $\Nd_F = \Nd_\nu$. The resulting approximation error is plotted in Fig.~\ref{L2_MM_schemes_Nds} for different $\Nd$ when the number of available samples varies. From $\Nd = 8$ to $\Nd = 40$, the required number of points for a given accuracy is seen to increase significantly, between a 2- and a 10-fold factor. However this is much milder than the increase in the potential approximation basis cardinality, \ie, if not subset selection was done, as $\cardPprior$ shifts from $10,565$ $\left(\Nd = 8\right)$ to $1.7 \times 10^6$ $\left(\Nd = 40\right)$, demonstrating the efficiency of the subset selection step which activates only a small fraction of the dictionary. When $\Nd$ further increases from 40 to 100, the performance remains essentially the same with hardly any loss of accuracy for a fixed $\Nq$: the solution method is able to capture the low-dimensional manifold onto which the solution essentially lies and an increase in the size of the solution space hardly affects the number of samples it requires. This capability is a crucial feature when available data is scarce and the solution space is very large.
As an illustration, when $\Nd = 100$, and with the parameters retained, the potential cardinality of the approximation basis is about $27 \times 10^6$ while the number of available samples is $\mathcal{O}\left(100-10,000\right)$. It clearly illustrates the pivotal importance of the subset selection step. Note that if one substitutes a Polynomial Chaos approximation to the present HDMR format, about $352 \times 10^9$ terms need be evaluated with the present settings, a clearly daunting task.
For sake of completeness, the approximation given by a CP-format, Eq.~\eqref{CP-like}, is also considered. The univariate functions $\left\{f_{i,r}\right\}$ are approximated with the same polynomial approximation as in the present gLARS-ALS approach and a Tikhonov-based regularized ALS technique is used to determine each $f_{i,r}$ in turn given the others. Upon convergence, the next set of modes $\left\{f_{1,r+1}, \ldots, f_{\Nd,r+1}\right\}$ is evaluated until a maximum rank $\nr$ set by cross-validation. At each rank $r$, the best approximation, as estimated by cross-validation, is retained from a set of initial conditions and regularization parameter values. As can be appreciated from Fig.~\ref{L2_MM_schemes_Nds}, the number of samples required for a given approximation error is significantly larger than with the present gLARS-HDMR method.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.65\textwidth, draft = false]{L2_Npts_pres_Nds_2.eps}
\caption{Convergence of the approximation with the number of samples $\Nq + \Nqhat$ and for different dimensionality of the QoI. The present gLARS-ALS approach is compared with a CANDECOMP-PARAFAC-type technique (labeled `CP').}
\label{L2_MM_schemes_Nds}
\end{center}
\end{figure}
\subsubsection{Subset selection}
To further illustrate the subset selection step, the set of second and third order interaction retained modes $\left\{f_{\bgamma}\right\}_{\bgamma \in \basissetfpost}$ are plotted in Fig.~\ref{duo_tree} in the $\Nd = 40$ case. Each bullet represents one of the $\Nd$ stochastic dimensions and each line connects two (2-nd order, left plot) or three (3-rd order, right plot) dimensions, denoting a retained mode. The first $\Nd_F = 20$ of the 40 dimensions are associated with the source term $F$ in the stochastic equation and are represented as the solid bullets of the first two quadrants, $d \in \left[1, 20\right]$. The other $\Nd_\nu = 20$ dimensions are associated with the uncertain diffusion coefficient $\nu$ and are plotted as open bullets in the 3-rd and 4-th quadrants, $d \in \left[21, 40\right]$. The dimensions introduced by these two quantities are sorted with the associated magnitude of the eigenvalues $\sigma_F$ and $\sigma_\nu$ of their kernel, see Eqs.~\eqref{KL_eigen}, which decreases along the counter-clockwise direction. Hence, the norm of the eigenvalues of the kernel associated with $F$ decreases when one goes counter-clockwise from the first to the second quadrant. Likewise, the norm of the eigenvalues associated with dimensions introduced by $\nu$ decreases from the third to the fourth quadrant. Dominant dimensions of the stochastic space for the output $u$ approximation are thus expected to lie at the beginning of the first and/or third quadrant.
From the plot of second order modes (left), the subset selection process is seen to retain interaction modes mainly associated with dominant eigenvalues of both $F$ and $\nu$: they mainly link bullets from the first (dominant) dimensions associated with $F$ to the first (dominant) dimensions associated with $\nu$, as one might expect. Further, modes associated with two dimensions both introduced by $\nu$ are seen to be selected while two dimensions both associated with $F$ are rarely connected: the subset selection procedure is able to capture the nonlinearity associated with $\nu$ in the QoI and retains corresponding interaction modes. Indeed, note from Eq.~\eqref{diff_eq2} that the source term $F$ interacts linearly with the solution $u$ while the diffusion coefficient is nonlinearly coupled with $u$ and hence, interaction modes between two dimensions introduced by $F$ do not contribute to the approximation. The third order modes (right plot) also illustrate the nonlinearity associated with $\nu$: the retained modes either connect dimensions associated with $\nu$ only or with one $F$-related and two $\nu$-related dimensions. Again, no two dimensions of $F$ are connected, consistently with the linear dependence of $u$ with $F$. These results illustrate the effectiveness of the procedure to unveil the dominant dependence structure and to discard unnecessary approximation basis functions.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = 0, width = 0.49\textwidth, draft = false]{duo_wobox.eps}
\includegraphics[angle = 0, width = 0.49\textwidth, draft = false]{triad_wobox.eps}
\caption{Graphical representation of the interaction modes retained by the subset selection procedure. Left: second order modes are plotted as a line linking two dimensions (bullets). Right: third order modes are represented as 3-branch stars and connect three dimensions.}
\label{duo_tree}
\end{center}
\end{figure}
\subsubsection{Robustness} \label{Robust_results}
The robustness of the approximation against measurement noise is now investigated. The dataset $\left\{\bxiq, \uq\right\}_{q=1}^\Nq$ is corrupted with noise, mimicking variability introduced in practice in measurements and coordinates estimation. Denoting the nominal value with a star as superscript, noise in the coordinates is modeled as
\be
\bxiq = {\bxiq}^\star + s \, \bzetaq, \qquad \forall \, 1 \le q \le \Nq,
\ee
with $s > 0$. The noise is modeled as an additive $\Nd$-dimensional, zero-centered, unit variance, Gaussian random vector $\bzeta$ biased so that $\bxiq \in \left[-1, 1\right]^\Nd$, $\forall \, q$. It is independent from one sample $q$ to another. Without loss of generality, measurements are here modeled as being corrupted with a multiplicative noise: $\uq = {\uq}^\star \, \left(1 + s_u \, \zeta_u^{(q)}\right)$, with $s_u = 0.2$ and $\zeta_u \sim \mathcal{N}\left(0, 1\right)$.
The evolution of the approximation accuracy when the noise intensity $s$ in the coordinates varies is plotted in Fig.~\ref{Noise_fig} in terms of error estimation $\varepsilon$. We compare gLARS-ALS using standard least squares (LS) with its `robust' counterpart relying on weighted total least squares (wTLS) as discussed in section~\ref{Noise_section}.
When the noise intensity increases, the error exponentially increases, quickly deteriorating the quality of the approximation with a noise standard deviation here as low as $s = 3 \times 10^{-5}$. When the noise is strong (low SNR), both the LS and the wTLS methods achieve poor accuracy. However, if the dataset is only mildly corrupted with noise, the wTLS approach is seen to achieve a significantly better accuracy than the standard least squares. While it is useful only on a range of SNR and somehow computationally costly, this feature is deemed important for a successful solution method in an experimental context where noise is naturally present.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.65\textwidth, draft = false]{SNR.eps}
\caption{Robustness of the approximation w.r.t. noise in the data: approximation error $\varepsilon$ from the standard least squares (LS) and weighted total least squares (wTLS). $\Nd = 5$, $\Nq = 500$.}
\label{Noise_fig}
\end{center}
\end{figure}
\subsubsection{Scaling of the solution method} \label{Num_cost_example}
In this section, the numerical complexity associated with the different steps of the solution method is illustrated in terms of computational time. Numerical experiments are carried-out with varying number of samples $\Nq$ and solution space dimensions $\Nd$. When one is varying, the other remains constant. The nominal parameters are $\Nd = 40$ (dimension of the stochastic space), $\No = 6$ (maximum total order of the Legendre polynomials), $\Nq = 1000$ (number of samples), $\Ninter = 3$ (maximum interaction order of the truncated HDMR approximation), $\NoLARS = 5$ (maximum total polynomial order in the subset selection step).
Numerical results are gathered in Fig.~\ref{Num_cost}.
The asymptotic behavior of the number $\nJ$ of required subset selection iterations as introduced in section \ref{Complexity_sec} might be different according to which limit is considered. For the present stochastic diffusion problem, first and second interaction order modes tend to be selected first. Assuming the active set $\basissetfpost$ is dominated by first and second interaction order modes, it can easily be shown that the number of retained groups then satisfies
\be
\nJ \le 1 + \nJzero + \textrm{min}\left[\frac{\Nd \, \left(\Nd-1\right)}{2}, 2 \, \frac{\Nq - \nJzero \, \NoLARS}{\NoLARS \, \left(\NoLARS+1\right)}\right], \qquad \nJzero \le \textrm{min}\left[\Nd, \frac{\Nq}{\NoLARS}\right].
\ee
In the present example, second order interaction groups dominate the retained set so that the number of retained groups tends to scale as $\nJ \propto \Nq / \NoLARS^2$.
From Eq.~\eqref{theo_cost_subset} and for the present nominal parameters, it results in the following limit behavior for the subset selection step:
\bea
\lim_{\Nd \rightarrow +\infty} \mathscr{J}_\mathrm{subsel} & \propto & \Nq^2 \, \cardPprior / \NoLARS^2 \qquad \longrightarrow {\rm here:} \quad \propto \Nq^2 \, \Nd^\Ninter \, \NoLARS^{\Ninter - 2}, \nonumber \\
\lim_{\Nq \rightarrow +\infty} \mathscr{J}_\mathrm{subsel} & \propto & \Nq^2 \, \cardPprior / \NoLARS^2 \qquad \longrightarrow {\rm here:} \quad \propto \Nq^2 \, \Nd^\Ninter \, \NoLARS^{\Ninter - 2}.
\eea
Similarly, the cost associated with the coefficients evaluation is considered. The number of interaction modes $\basissetfcur$ effectively varies between $1$ and $\mathcal{O}\left(\Nq / \No^2\right)$ along the solution procedure, and, since the cost associated with solving the least squares problem dominates that of the matrix assembly, the cost of their evaluation finally simplifies in $\mathscr{J}_\mathrm{coef} \propto \mathcal{O}\left(\Nq^2 \, \No\right)$ or $\mathscr{J}_\mathrm{coef} \propto \mathcal{O}\left(\Nq^3 / \No\right)$ depending on whether the coefficients are updated whenever an additional group is considered or not, see section~\ref{comp_lambda} and second bullet of step~(\ref{solvestep}) in Algorithm~\ref{Algo_sumup}. In the present regime, the cost is found not to depend on $\Nd$.
These asymptotic behaviors are consistent with the numerical experiments as can be appreciated from Fig.~\ref{Num_cost}. The coefficients are here updated whenever a new mode from the selected set is considered, hence $\mathscr{J}_\mathrm{coef} \propto \mathcal{O}\left(\Nq^3 / \No\right)$.
It is seen that the subset selection step scales less favorably than the coefficients evaluation step with the dimensionality of the random variable. This stresses the benefit of a carefully chosen {\apriori} approximation basis to reduce as much as possible the cardinality $\cardPprior$ of the potential dictionary the final set is determined from by the subset selection step.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{Scale_Ndim.eps}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{Scale_M.eps}
\caption{Numerical cost of the subset selection and coefficients evaluation steps as a function the stochastic dimension $\Nd$ and size of the dataset $\Nq$. Approximation coefficients are fully updated for each new mode. Nominal parameters are $\Nd = 40$, $\No = 6$, $\Nq = 1000$, $\Ninter = 3$, $\NoLARS = 5$, $\NinterPC = 3$.}
\label{Num_cost}
\end{center}
\end{figure}
\subsection{Stochastic process approximation}
We now consider the approximation of the space-dependent random solution $u\left(x, \bxi\right)$ under the form \eqref{KL_approx_format} using Algorithm~\ref{Algo_separated}. The approximation obtained from different number of samples $\left\{\xq, \bxiq, \uq\right\}$ is compared with the Karhunen-Lo\`eve modes, computed from a full knowledge of the QoI, which henceforth constitute the reference solution. The simulation relies on the following parameters: $\cardx = 32$, $\No = 10$, $\Nd = 6$, $\Ninter = 3$, $\NinterPC = 3$. The potential approximation basis cardinality is about $\cardx \, \cardPprior \simeq 10^5$.
Fig.~\ref{KL_1D} shows the first and second spatial modes, $\modex_1\left(x\right)$ and $\modex_2\left(x\right)$ for different sizes of the dataset, $\Nq = 1000$, $3000$, $9000$ and $26,000$. The mean mode $\modex_0\left(x\right)$ is virtually indistinguishable from the reference solution mean mode for any of the dataset sizes and is not plotted. On the left plot $\left(\modex_1\left(x\right)\right)$, it is seen that the approximation is decent, even with as low as $\Nq = 1000$ samples. For $\Nq = 3000$, the approximation is good. This $\left(1 + \Nd\right) = 7$-dimensional case corresponds to $\Nq^{1 / (1 + 6)} \simeq 3.1$ samples per solution space dimension only and about $\Nq / \left(\cardx \, \cardPprior\right) \simeq 3 \%$ of the potentially required information.
For approximating the second spatial mode (Fig.~\ref{KL_1D}, right plot), more points are needed to reach a good accuracy but $\Nq = 26,000$ is seen to already deliver a good performance. Quantitative approximation error results are gathered in Table~\ref{epstab_1D} for various separation ranks $\KLrank$ and number of samples $\Nq$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{Mode_1_1D.eps}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{Mode_2_1D.eps}
\caption{First ($\modex_1\left(x\right)$, left) and second ($\modex_2\left(x\right)$, right) spatial approximation modes of the stochastic diffusion solution. The reference (Karhunen-Lo\`eve) solution is plotted for comparison (thick line).}
\label{KL_1D}
\end{center}
\end{figure}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{rccc}
\hline \hline
$\Nq$ $\backslash$ $\KLrank$ & 0 & 1 & 2 \\
\hline
1000 & $5.5 \times 10^{-3}$ & $7.4 \times 10^{-4}$ & $7.4 \times 10^{-4}$ \\
3000 & $5.5 \times 10^{-3}$ & $4.2 \times 10^{-4}$ & $2.7 \times 10^{-4}$ \\
9000 & $5.5 \times 10^{-3}$ & $3.1 \times 10^{-4}$ & $1.0 \times 10^{-4}$ \\
26,000 & $5.4 \times 10^{-3}$ & $2.8 \times 10^{-4}$ & $6.2 \times 10^{-5}$ \\
\hline
\end{tabular}
\caption{Evolution of the approximation error $\varepsilon$ with the decomposition rank $\KLrank$ and the number of samples $\Nq$.}
\label{epstab_1D}
\end{center}
\end{table}
The satisfactory performance of the present method can be understood from the upper part of the Karhunen-Lo\`eve approximation (normalized) spectrum plotted in Fig.~\ref{KLspectrum_1D}. The norm of the eigenvalues decays quickly so that the first two modes contribute more than 90 \% of the QoI $L^2$-norm, showing that this problem efficiently lends itself to the present separation of variables-based methodology.
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.6\textwidth, draft = false]{Lambda_1D.eps}
\caption{Normalized upper spectrum of the Karhunen-Lo\`eve approximation.}
\label{KLspectrum_1D}
\end{center}
\end{figure}
\subsection{A Shallow Water flow example}
The methodology is now applied to the approximation of the stochastic solution of a Shallow Water flow simulation with multiple sources of uncertainty. It is a simple model for the simulation of wave propagation on the ocean surface. Waves are here produced by the sudden displacement of the sea bottom at a given magnitude, extension and location, all uncertain. Further, the propagation of the waves also depends on the topography of the ocean floor, which is also uncertain.
\subsubsection{Model}
The problem is governed by the following set of equations:
\bea
\frac{D \, \vx}{D \: t} & = & f_C \: \vy - g \: \frac{\partial h}{\partial x_1} - b \: \vx + \Su, \label{SWE_ori_a}\\
\frac{D \: \vy}{D \: t} & = & - f_C \: \vx - g \: \frac{\partial h}{\partial x_2} - b \: \vy + \Sv, \\
\frac{\partial h}{\partial t} & = & - \frac{\partial \left( \vx \: \left( H + h \right) \right)}{\partial x_1} - \frac{\partial \left( \vy \: \left( H + h \right) \right)}{\partial x_2} + \Sh, \label{SWE_ori_c}
\eea
where $\left(\vx\left(\bx,\bxi,t\right) \, \vy\left(\bx,\bxi,t\right)\right)$ is the velocity vector at the surface, $\bx = \left(x_1 \, x_2\right) \in \Omega \subset \mathbb{R}^2$, $h\left(\bx,\bxi,t\right)$ the elevation of the surface from its position at rest, $H\left(\bx,\bxi\right)$ the sea depth, $f_C$ models the Coriolis force, $b$ is the viscous drag coefficient, $g$ the gravity constant and $\Su\left(\bx,\bxi,t\right)$, $\Sv\left(\bx,\bxi,t\right)$, $\Sh\left(\bx,\bxi,t\right)$ are the source fields. Without loss of generality, the drag $b$ and the Coriolis force $f_C$ are neglected. No slip boundary conditions apply for the velocity. The sources are modeled as acting on $h$ only, $\Su \equiv 0$ and $\Sv \equiv 0$. $\Sh$ models the source term acting on $h$ due to, say, an underwater seismic event. The fluid density and the free surface pressure are implicitly assumed constant.
\subsubsection{Sources of uncertainty}
Letting $\bxi = \left( \bxi' \, \bxi''\right)$, the ocean floor topography $H\left(\bx,\bxi\right)$ is described with a $\NH$-term expansion:
\be
H \left( \bx,\bxi' \right) = \overline{H} \left( \bx \right) + \sum_{i=1}^{\NH}{\sqrt{\lambda_i} \: \xi_i' \left( \theta \right) \, \varphi_i^H(\bx)}, \label{H_eq}
\ee
with $\bxi' = \left(\xi'_1 \, \ldots \, \xi'_\NH\right)$ the stochastic germ associated to the uncertainty in $H$. Random variables $\left\{\xi'_i\right\}_{i=1}^{\NH}$ are \textit{iid}, uniformly distributed. The source $\Sh$ is also uncertain and is modeled as a time-dependent, spatially distributed, quantity:
\be
\Sh\left(\bx,\bxi'',t\right) = a_t\left(t\right) \, a_\xi\left(\xi''_1\right) \, \exp\left({-\frac{\left(\bx - \bx_\Sh\left(\xi''_3\right)\right)^T \, \left(\bx - \bx_\Sh\left(\xi''_3\right)\right)}{\sigma_\Sh\left(\xi''_2\right)^2}}\right),
\ee
where $a_t\left(t\right)$ is a given time envelop, $a_\xi\left(\xi''_1\right)$ the uncertain source magnitude, $\sigma_\Sh\left(\xi''_2\right)$ drives the uncertain source spatial extension and $\bx_\Sh\left(\xi''_3\right)$ is the uncertain spatial location.
The solution of the Shallow Water problem then lies in a $\left(\Nd = \NH + 3\right)$-dimensional stochastic space. Full details on the problem and the numerical implementation of the simulation are given in \cite{Mathelin_al_CompMech}.
\subsubsection{Approximation from an available database}
As an illustration of the methodology, we aim at approximating the sea surface field at a fixed amount of time $t^\star$ after a seismic event. The QoI is then a random field $u\left(\bx, \bxi\right) = h\left(\bx, \bxi, t^\star\right)$. An accurate description of this field is of importance for emergency plans in case of a seaquake. Sea level measurements of the surface at various spatial locations from past events constitute the experimental dataset $\left\{\bxq, \bxiq, h\left(\bxq, \bxiq, t^\star\right)\right\}_{q=1}^{\Nq}$ used to derive an approximation of $u$ under a separated form: $u\left(\bx, \bxi\right) \approx \left<u\right>_{\Nq}\left(\bx\right) + \sum_{n=1}^\KLrank{\modex_n\left(\bx\right) \, \modesto_n\left(\bxi\right)}$.
The solution method here relies on a $\Nq = 37,000$-sample dataset complemented with $\Nqhat = 5000$ cross-validation samples and a $\Nqt = 5000$ set for error estimation. We consider a $\NH=5$ expansion for the topography, leading to a stochastic dimension of $\Nd = 5+3=8$. The effective number of samples per dimension is then about $\Nq^{1/\left(\Ndx + \Nd\right)} \simeq 2.9$. The approximation is determined based on a $\cardx = 484$ spatial discretization DOFs (spectral elements) at the deterministic level and $\No=6$-th order Legendre polynomials $\left\{\psi_\alpha\right\}$, $\Ninter = 3$, $\NinterPC = 3$, for the stochastic modes. The cardinality of this {\apriori} basis is then $\cardx \, \cardPprior \simeq 770 \times 10^3 \gg \Nq$, again relying on an efficient subset selection step to make the approximation problem well-posed.
The approximation error when the rank $\KLrank$ varies is shown in Table~\ref{Norme_x_SWE}. It is seen that estimating the mean spatial mode $\modex_0$ leads to a relative error of about $0.12$ while adding the first $\left(\modex_1, \modesto_1\right)$ and second $\left(\modex_2, \modesto_2\right)$ pair drops it to about $0.05$. Further adding pairs does not lower the approximation error with this dataset and more samples would be needed to accurately estimate them. Spatial modes $\modex_0$ and $\modex_1$ of the separated approximation are plotted in Fig.~\ref{KL_SWE} for illustration.
\begin{table}[ht!]
\begin{center}
\begin{tabular}{c|cccc}
$\KLrank$ & 0 & 1 & 2 & 3\\
\hline
$\varepsilon$ & 0.117 & 0.056 & 0.046 & 0.044 \\
\end{tabular}
\caption{Relative approximation error $\varepsilon$ evolution with the decomposition rank $\KLrank$. $\Nq = 37,000$.}
\label{Norme_x_SWE}
\end{center}
\end{table}
\begin{figure}[!ht]
\begin{center}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{SWE_Modex_1.eps}
\includegraphics[angle = -90, width = 0.49\textwidth, draft = false]{SWE_Modex_2.eps}
\caption{Mean ($\modex_0\left(\bx\right) \equiv \left<u\right>_{\Nq}\left(\bx\right)$, left) and first ($\modex_1\left(\bx\right)$, right) spatial modes.}
\label{KL_SWE}
\end{center}
\end{figure} | {"config": "arxiv", "file": "1302.7083/results.tex"} |
\begin{document}
\title{Large deviations for method-of-quantiles estimators of one-dimensional parameters\thanks{The support of
Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`{a} e le
loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM) is acknowledged.}}
\author{Valeria Bignozzi\thanks{Dipartimento di Statistica e Metodi
Quantitativi, Universit\`{a} di Milano Bicocca, Via Bicocca degli
Arcimboldi 8, I-20126 Milano, Italia. e-mail:
\texttt{[email protected]}}\and Claudio
Macci\thanks{Dipartimento di Matematica, Universit\`a di Roma Tor
Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italia.
e-mail: \texttt{[email protected]}}\and Lea
Petrella\thanks{Dipartimento di Metodi e Modelli per l'Economia,
il Territorio e la Finanza, Sapienza Universit\`{a} di Roma, Via
del Castro Laurenziano 9, I-00161 Roma, Italia. e-mail:
\texttt{[email protected]}}}
\date{}
\maketitle
\begin{abstract}
\noindent We consider method-of-quantiles estimators of unknown
parameters, namely the analogue of method-of-moments estimators
obtained by matching empirical and theoretical quantiles at some
probability level $\lambda\in(0,1)$. The aim is to present large
deviation results for these estimators as the sample size tends to
infinity. We study in detail several examples; for specific models
we discuss the choice of the optimal value of $\lambda$ and we
compare the convergence of the method-of-quantiles and
method-of-moments estimators.\\
\ \\
\emph{AMS Subject Classification:} 60F10; 62F10; 62F12.\\
\emph{Keywords:} location parameter; methods of moments; order
statistics; scale parameter; skewness parameter.
\end{abstract}
\section{Introduction}\label{sec:introduction}
Estimation of parameters of statistical or econometric models is
one of the main concerns in the parametric inference framework.
When the probability law is specified (up to unknown parameters),
the main tool to solve this problem is the Maximum Likelihood (ML)
technique; on the other hand, whenever the assumption of a
particular distribution is too restrictive, different solutions
may be considered. For instance the Method of Moments (MM) and the
Generalized Method of Moments (GMM) provide valuable alternative
procedures; in fact the application of these methods only requires
the knowledge of some moments.
A different approach is to consider the Method of Quantiles (MQ),
that is the analogue of MM with quantiles; MQ estimators are
obtained by matching the empirical percentiles with their
theoretical counterparts at one or more probability levels.
Inference via quantiles goes back to \cite{AitchinsonBrown} where
the authors consider an estimation problem for a three-parameter
log-normal distribution; their approach consists in minimizing a
suitable distance between the theoretical and empirical quantiles,
see for instance \cite{Koenker2005}. Successive papers deal with
the estimation of parameters of extreme value (see
\cite{Hassanein69a} and \cite{Hassanein72}), logistic (see
\cite{Hassanein69b}) and Weibull (see \cite{Hassanein71})
distributions. A more recent reference is \cite{CastilloHadi}
where several other distributions are studied. We also recall
\cite{DominicyVeredas} where the authors consider an indirect
inference method based on the simulation of theoretical quantiles,
or a function of them, when they are not available in a closed
form. In \cite{SgouropoulosYaoYastremiz}, an iterative procedure
based on ordinary least-squares estimation is proposed to compute
MQ estimators; such estimators can be easily modified by adding a
LASSO penalty term if a sparse representation is desired, or by
restricting the matching within a given range of quantiles to
match a part of the target distribution. Quantiles and empirical
quantiles represent a key tool also in quantitative risk
management, where they are studied under the name of Value-at-Risk
(see for instance \cite{McNeil}).
In our opinion, MQ estimators deserve a deeper investigation
because of several advantages. They allow to estimate parameters
when the moments are not available and they are invariant with
respect to increasing transformations; moreover they have less
computational problems, and behave better when distributions are
heavy-tailed or their supports vary with the parameters.
The aim of this paper is to present large deviation results for MQ
estimators (as the sample size tends to infinity) for statistical
models with one-dimensional unknown parameter $\theta\in\Theta$,
where the parameter space $\Theta$ is a subset of the real line;
thus we match empirical and theoretical quantiles at one
probability level $\lambda\in(0,1)$. The theory of large
deviations is a collection of techniques which gives an asymptotic
computation of small probabilities on an exponential scale (see
e.g. \cite{DemboZeitouni} as a reference on this topic). Several
examples of statistical models are considered throughout the
paper, and some particular distributions are studied in detail.
For most of the examples considered, we are able to find an
explicit expression for the rate function which governs the large
deviation principle of the MQ estimators and, when possible, our
investigation provides the optimal $\lambda$ that guarantees a
faster convergence to the true parameter (see Definition
\ref{def:optimal-lambda}). Further we compare MQ and MM estimators
in terms of the local behavior of the rate functions around the
true value of the parameter in the spirit of Remark
\ref{rem:comparison-between-rfs}. Which one of the estimators
behaves better strictly depends on the type of parameter we have
to estimate and varies upon distributions. However, we provide
explicit examples (a part from the obvious ones where the MM
estimators are not available) where MQ estimators are preferable.
We conclude with the outline of the paper. In Section
\ref{sec:Preliminaries} we recall some preliminaries. Sections
\ref{sec:results-for-MQestimators} and
\ref{sec:results-for-MMestimators} are devoted to the results for
MQ and MM estimators, respectively. In Section \ref{sec:examples}
we present examples for different kind of parameters (e.g. scale,
location, skewness, etc.), and for each example specific
distributions are discussed in Section
\ref{sec:local-comparison-for-examples}.
\section{Preliminaries}\label{sec:Preliminaries}
In this section we present
some preliminaries on large deviations and we provide a rigorous
definition of the MQ estimators studied in this paper (see
Definition \ref{def:MQ-estimators} below).
\subsection{Large deviations}\label{sub:LD-preliminaries}
We start with the concept of large deviation principle (LDP for
short). A sequence of random variables $\{W_n:n\geq 1\}$ taking
values on a topological space $\mathcal{W}$ satisfies the LDP with
rate function $I:\mathcal{W}\to[0,\infty]$ if $I$ is a lower
semi-continuous function,
$$\liminf_{n\to\infty}\frac{1}{n}\log P(W_n\in O)\geq-\inf_{w\in O}I(w)\ \mbox{for all open sets}\ O$$
and
$$\limsup_{n\to\infty}\frac{1}{n}\log P(W_n\in C)\leq-\inf_{w\in C}I(w)\ \mbox{for all closed sets}\ C.$$
We also recall that a rate function $I$ is said to be good if all
its level sets \mbox{$\{\{w\in\mathcal{W}:I(w)\leq\eta\}:\eta\geq 0\}$}
are compact.
\begin{remark}[Local comparison between rate functions around the unique common zero]\label{rem:comparison-between-rfs}
It is known that, if $I(w)$ uniquely vanish at some
$w_0\in\mathcal{W}$, then the sequence of random variables
converges weakly to $w_0$. Moreover, if we have two rate functions
$I_1$ and $I_2$ which uniquely vanish at the same point
$w_0\in\mathcal{W}$, and if $I_1(w)>I_2(w)>0$ in a neighborhood of
$w_0$ (except $w_0$) then any sequence which satisfies the LDP
with rate function $I_1$ converges to $w_0$ faster than any
sequence which satisfies the LDP with rate function $I_2$.
\end{remark}
We also recall a recent large deviation result on order statistics
of i.i.d. random variables (see Proposition
\ref{prop:Theorem3.2-in-HMP-restricted} below) which plays a
crucial role in this paper. We start with the following condition.
\begin{condition}\label{cond:(*)inHMP-with-restriction}
Let $\{X_n:n\geq 1\}$ be a sequence of i.i.d. real valued random
variables with distribution function $F$, and assume that $F$ is
continuous and strictly increasing on $(\alpha,\omega)$, where
$-\infty\leq\alpha<\omega\leq\infty$. Moreover let $\{k_n:n\geq
1\}$ be such that $k_n\in\{1,\ldots,n\}$ for all $n\geq 1$ and
$\lim_{n\to\infty}\frac{k_n}{n}=\lambda\in(0,1)$.
\end{condition}
We introduce the following notation: for all $k\geq 1$,
$X_{1:k}\leq\cdots\leq X_{k:k}$ are the order statistics of the
sample $X_1,\ldots,X_k$; for $p,q\in(0,1)$ we set
\begin{equation}\label{eq:def-function-H}
H(p|q):=p\log\frac{p}{q}+(1-p)\log\frac{1-p}{1-q},
\end{equation}
that is the relative entropy of the Bernoulli distribution with
parameter $p$ with respect to the Bernoulli distribution with
parameter $q$.
\begin{proposition}[Theorem 3.2 in \cite{HashorvaMacciPacchiarotti} for $\lambda\in(0,1)$]\label{prop:Theorem3.2-in-HMP-restricted}
Assume that Condition \ref{cond:(*)inHMP-with-restriction} holds.
Then $\{X_{k_n:n}:n\geq 1\}$ satisfies the LDP with good rate
function $I_{\lambda,F}$ defined by
$$I_{\lambda,F}(x):=\left\{\begin{array}{ll}
H(\lambda|F(x))&\ \mbox{for}\ x\in(\alpha,\omega)\\
\infty&\ \mbox{otherwise}.
\end{array}\right.$$
\end{proposition}
\begin{remark}[$I_{\lambda,F}^{\prime\prime}(F^{-1}(\lambda))$ as the inverse of an asymptotic variance]\label{rem:Dasgupta-connection}
Theorem 7.1(c) in \cite{Dasgupta} states that, under suitable
conditions, $\{\sqrt{n}(X_{k_n:n}-F^{-1}(\lambda)):n\geq 1\}$
converges weakly to the centered Normal distribution with variance
$\sigma^2:=\frac{\lambda(1-\lambda)}{(F^\prime(F^{-1}(\lambda)))^2}$.
Then, if we assume that $F$ is twice differentiable, we can check
that
$I_{\lambda,F}^{\prime\prime}(F^{-1}(\lambda))=\frac{1}{\sigma^2}$
with some computations.
\end{remark}
A more general formulation of Proposition
\ref{prop:Theorem3.2-in-HMP-restricted} could be given also for
$\lambda\in\{0,1\}$ but, in view of the applications presented in
this paper, we prefer to consider a restricted version of the
result with $\lambda\in(0,1)$ only. This restriction allows to
have the goodness of the rate function $I_{\lambda,F}$ (see Remark
1 in \cite{HashorvaMacciPacchiarotti}) which is needed to
apply the contraction principle (see e.g. Theorem 4.2.1 in
\cite{DemboZeitouni}).
\subsection{MQ estimators}\label{sub:MQ-preliminaries}
Here we present a rigorous definition of MQ estimators. In view of
this, the next Condition \ref{cond:forMQestimator} plays a crucial
role.
\begin{condition}\label{cond:forMQestimator}
Let $\{F_\theta:\theta\in\Theta\}$ be a family of distribution
functions where $\Theta\subset\mathbb{R}$ and, for all
$\theta\in\Theta$, $F_\theta$ satisfies the same hypotheses of the
distribution function $F$ in Condition
\ref{cond:(*)inHMP-with-restriction}, for some
$(\alpha_\theta,\omega_\theta)$. Moreover, for $\lambda\in(0,1)$,
consider the function
$F_{(\bullet)}^{-1}(\lambda):\Theta\to\mathcal{M}$, where
$\mathcal{M}:=\bigcup_{\theta\in\Theta}(\alpha_\theta,\omega_\theta)$,
defined by
$$[F_{(\bullet)}^{-1}(\lambda)](\theta):=F_\theta^{-1}(\lambda).$$
We assume that, for all $m\in\mathcal{M}$, the equation
$[F_{(\bullet)}^{-1}(\lambda)](\theta)=m$ admits a unique solution
(with respect to $\theta\in\Theta$) which will be denoted by
$(F_{(\bullet)}^{-1}(\lambda))^{-1}(m)$.
\end{condition}
Now we are ready to present the definition.
\begin{definition}\label{def:MQ-estimators}
Assume that Condition \ref{cond:forMQestimator} holds. Then
$\left\{(F_{(\bullet)}^{-1}(\lambda))^{-1}(X_{[\lambda n]:n}):n\geq
1\right\}$ is a sequence of MQ estimators (for the level
$\lambda\in(0,1)$).
\end{definition}
Proposition \ref{prop:LD-for-MQ-estimators} below provides the LDP
for the sequence of estimators in Definition
\ref{def:MQ-estimators} (as the sample size $n$ goes to infinity)
when the true value of the parameter is $\theta_0\in\Theta$.
Actually we give a more general formulation in terms of
$\left\{(F_{(\bullet)}^{-1}(\lambda))^{-1}(X_{k_n:n}):n\geq
1\right\}$, where $\{k_n:n\geq 1\}$ is a sequence as in Condition
\ref{cond:(*)inHMP-with-restriction}.
\section{Results for MQ estimators}\label{sec:results-for-MQestimators}
In this section we prove the LDP for the sequence of estimators in
Definition \ref{def:MQ-estimators}. Moreover we discuss some
properties of the rate function; in particular Proposition
\ref{prop:second-derivative-I} (combined with Remark
\ref{rem:comparison-between-rfs} above) leads us to define a
concept of optimal $\lambda$ presented in Definition
\ref{def:optimal-lambda} below.
We start with our main result and, in view of this, we present the
following notation:
\begin{equation}\label{eq:def-function-h}
h_{\lambda,\theta_0}(\theta):=F_{\theta_0}(F_\theta^{-1}(\lambda))\
(\mbox{for}\
F_\theta^{-1}(\lambda)\in(\alpha_{\theta_0},\omega_{\theta_0})).
\end{equation}
\begin{proposition}[LD for MQ estimators]\label{prop:LD-for-MQ-estimators}
Assume that $\{k_n:n\geq 1\}$ is as in Condition
\ref{cond:(*)inHMP-with-restriction} and that Condition
\ref{cond:forMQestimator} holds. Moreover assume that, for some
$\theta_0\in\Theta$, $\{X_n:n\geq 1\}$ are i.i.d. random variables
with distribution function $F_{\theta_0}$. Then, if the
restriction of $(F_{(\bullet)}^{-1}(\lambda))^{-1}$ on
$(\alpha_{\theta_0},\omega_{\theta_0})$ is continuous,
$\left\{(F_{(\bullet)}^{-1}(\lambda))^{-1}(X_{k_n:n}):n\geq
1\right\}$ satisfies the LDP with good rate function
$I_{\lambda,\theta_0}$ defined by
$$I_{\lambda,\theta_0}(\theta):=\left\{\begin{array}{ll}
\lambda\log\frac{\lambda}{h_{\lambda,\theta_0}(\theta)}
+(1-\lambda)\log\frac{1-\lambda}{1-h_{\lambda,\theta_0}(\theta)}&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ F_\theta^{-1}(\lambda)\in(\alpha_{\theta_0},\omega_{\theta_0})\\
\infty&\ \mbox{otherwise},
\end{array}\right.$$
where $h_{\lambda,\theta_0}(\theta)$ is defined by
\eqref{eq:def-function-h}.
\end{proposition}
\begin{proof}
Since the restriction of $(F_{(\bullet)}^{-1}(\lambda))^{-1}$ on
$(\alpha_{\theta_0},\omega_{\theta_0})$ is continuous, a
straightforward application of the contraction principle yields
the LDP of
$\left\{(F_{(\bullet)}^{-1}(\lambda))^{-1}(X_{k_n:n}):n\geq
1\right\}$ with good rate function $I_{\lambda,\theta_0}$ defined
by
$$I_{\lambda,\theta_0}(\theta):=\inf\left\{I_{\lambda,F_{\theta_0}}(x):x\in(\alpha_{\theta_0},\omega_{\theta_0}),
(F_{(\bullet)}^{-1}(\lambda))^{-1}(x)=\theta\right\},$$ where
$I_{\lambda,F_{\theta_0}}$ is the good rate function in
Proposition \ref{prop:Theorem3.2-in-HMP-restricted}, namely the
good rate function defined by
$I_{\lambda,F_{\theta_0}}(x):=H(\lambda|F_{\theta_0}(x))$, for
$x\in(\alpha_{\theta_0},\omega_{\theta_0})$. Moreover the set
$\left\{x\in(\alpha_{\theta_0},\omega_{\theta_0}):(F_{(\bullet)}^{-1}(\lambda))^{-1}(x)=\theta\right\}$
has at most one element, namely
$$\left\{x\in(\alpha_{\theta_0},\omega_{\theta_0}):(F_{(\bullet)}^{-1}(\lambda))^{-1}(x)=\theta\right\}=
\left\{\begin{array}{ll}
\{F_\theta^{-1}(\lambda)\}&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ F_\theta^{-1}(\lambda)\in(\alpha_{\theta_0},\omega_{\theta_0})\\
\emptyset&\ \mbox{otherwise};
\end{array}\right.$$
thus we have
$I_{\lambda,\theta_0}(\theta)=H(\lambda|F_{\theta_0}(F_\theta^{-1}(\lambda)))=H(\lambda|h_{\lambda,\theta_0}(\theta))$
for $\theta\in\Theta$ such that
$F_\theta^{-1}(\lambda)\in(\alpha_{\theta_0},\omega_{\theta_0})$,
and $I_{\lambda,\theta_0}(\theta)=\infty$ otherwise. The proof is
completed by taking into account the definition of the function $H$
in \eqref{eq:def-function-H}.
\end{proof}
\begin{remark}[Rate function invariance with respect to increasing transformations]\label{rem:invariance-wrt-increasing-transformations}
Let $\{F_\theta:\theta\in\Theta\}$ be a family of distribution
functions as in Condition \ref{cond:forMQestimator} and assume
that there exists an interval $(\alpha,\omega)$ such that
$(\alpha_\theta,\omega_\theta)=(\alpha,\omega)$ for all
$\theta\in\Theta$. Moreover let
$\psi:(\alpha,\omega)\to\mathbb{R}$ be a strictly increasing
function. Then, if we consider the MQ estimators based on the
sequence $\{\psi(X_n):n\geq 1\}$ instead of $\{X_n:n\geq 1\}$, we
can consider an adapted version of Proposition
\ref{prop:LD-for-MQ-estimators} with $(\psi(\alpha),\psi(\omega))$
in place of $(\alpha,\omega)$, $F_\theta\circ\psi^{-1}$ in place
of $F_\theta$ and, as stated in Property 1.5.16 in
\cite{DenuitDhaeneGoovaertsKaas}, $\psi\circ F_\theta^{-1}$ in
place of $F_\theta^{-1}$. The LDP provided by this adapted version
of Proposition \ref{prop:LD-for-MQ-estimators} is governed by the
rate function $I_{\lambda,\theta_0;\psi}$ defined by
$$I_{\lambda,\theta_0;\psi}(\theta):=\left\{\begin{array}{ll}
H(\lambda|F_{\theta_0}\circ\psi^{-1}(\psi\circ
F_\theta^{-1}(\lambda)))
&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ \psi\circ F_\theta^{-1}(\lambda)\in(\psi(\alpha),\psi(\omega))\\
\infty&\ \mbox{otherwise}
\end{array}\right.$$
instead of
$$I_{\lambda,\theta_0}(\theta):=\left\{\begin{array}{ll}
H(\lambda|F_{\theta_0}(F_\theta^{-1}(\lambda)))
&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ F_\theta^{-1}(\lambda)\in(\alpha,\omega)\\
\infty&\ \mbox{otherwise}.
\end{array}\right.$$
One can easily realize that $I_{\lambda,\theta_0;\psi}$ and
$I_{\lambda,\theta_0}$ coincide.
\end{remark}
By taking into account the rate function in Proposition
\ref{prop:LD-for-MQ-estimators}, it would be interesting to
compare two rate functions $I_{\lambda_1,\theta_0}$ and
$I_{\lambda_2,\theta_0}$ in the spirit of Remark
\ref{rem:comparison-between-rfs} for a given pair
$\lambda_1,\lambda_2\in (0,1)$; namely it would be interesting to
have a strict inequality between $I_{\lambda_1,\theta_0}$ and
$I_{\lambda_2,\theta_0}$ in a neighborhood of $\theta_0$ (except
$\theta_0$).
Thus, if both rate functions are twice differentiable,
$I_{\lambda_1,\theta_0}$ is locally larger (resp. smaller) than
$I_{\lambda_2,\theta_0}$ around $\theta_0$ if we have
$I_{\lambda_1,\theta_0}^{\prime\prime}(\theta_0)>I_{\lambda_2,\theta_0}^{\prime\prime}(\theta_0)$
(resp.
$I_{\lambda_1,\theta_0}^{\prime\prime}(\theta_0)<I_{\lambda_2,\theta_0}^{\prime\prime}(\theta_0)$).
So it is natural to give an expression of
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ under suitable
hypotheses.
\begin{proposition}[An expression for $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$]\label{prop:second-derivative-I}
Let $I_{\lambda,\theta_0}$ be the rate function in Proposition
\ref{prop:LD-for-MQ-estimators}. Assume that $F_{\theta_0}(\cdot)$
and $F_{(\cdot)}^{-1}(\lambda)$ are twice differentiable. Then
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{(h_{\lambda,\theta_0}^\prime(\theta_0))^2}{\lambda(1-\lambda)}
=\frac{\{F_{\theta_0}^\prime(F_{\theta_0}^{-1}(\lambda))\}^2}{\lambda(1-\lambda)}
\left(\left.\frac{d}{d\theta}F_\theta^{-1}(\lambda)\right|_{\theta=\theta_0}\right)^2,$$
where $h_{\lambda,\theta_0}(\theta)$ is defined by
\eqref{eq:def-function-h}.
\end{proposition}
\begin{proof}
One can easily check that
$$h_{\lambda,\theta_0}(\theta_0)=\lambda\ \mbox{and}\
h_{\lambda,\theta_0}^\prime(\theta_0)=F_{\theta_0}^\prime(F_{\theta_0}^{-1}(\lambda))\cdot
\left.\frac{d}{d\theta}F_\theta^{-1}(\lambda)\right|_{\theta=\theta_0}.$$
Moreover after some computations we get
$$I_{\lambda,\theta_0}^\prime(\theta)=h_{\lambda,\theta_0}^\prime(\theta)\left(\frac{1-\lambda}{1-h_{\lambda,\theta_0}(\theta)}
-\frac{\lambda}{h_{\lambda,\theta_0}(\theta)}\right)$$ and
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta)=
h_{\lambda,\theta_0}^{\prime\prime}(\theta)\left(\frac{1-\lambda}{1-h_{\lambda,\theta_0}(\theta)}-\frac{\lambda}{h_{\lambda,\theta_0}(\theta)}\right)
+(h_{\lambda,\theta_0}^\prime(\theta))^2\left(\frac{\lambda}{h_{\lambda,\theta_0}^2(\theta)}+\frac{1-\lambda}{(1-h_{\lambda,\theta_0}(\theta))^2}\right).$$
Thus $I_{\lambda,\theta_0}^\prime(\theta_0)=0$ and
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=(h_{\lambda,\theta_0}^\prime(\theta_0))^2\left(\frac{1}{\lambda}+\frac{1}{1-\lambda}\right)=
\frac{(h_{\lambda,\theta_0}^\prime(\theta))^2}{\lambda(1-\lambda)}$.
The proof is completed by taking into account the expression of
$h_{\lambda,\theta_0}^\prime(\theta_0)$ above.
\end{proof}
Finally, by taking into account Proposition
\ref{prop:second-derivative-I} (and what we said before it), it is
natural to consider the following
\begin{definition}\label{def:optimal-lambda}
A value $\lambda_{\mathrm{max}}\in(0,1)$ is said to be
\emph{optimal} if it maximizes
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$, namely if we have
$I_{\lambda_{\mathrm{max}},\theta_0}^{\prime\prime}(\theta_0)=\sup_{\lambda\in(0,1)}I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$.
\end{definition}
\section{Results for MM estimators}\label{sec:results-for-MMestimators}
The aim of this section is to present a version of the above
results for MM estimators; namely the LDP and an expression of
$J_{\theta_0}^{\prime\prime}(\theta_0)$, where $J_{\theta_0}$ is
the rate function which governs the LDP of MM estimators. In
particular, when we compare MM and MQ estimators in terms of speed
of convergence by referring to Remark
\ref{rem:comparison-between-rfs}, the value
$J_{\theta_0}^{\prime\prime}(\theta_0)$ will be compared with
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ in Proposition
\ref{prop:second-derivative-I}.
We start with the following condition which allows us to define the
MM estimators.
\begin{condition}\label{cond:forMMestimator}
Let $\{F_\theta:\theta\in\Theta\}$ be a family of distribution
functions as in Condition \ref{cond:forMQestimator}, and consider
the function $\mu:\Theta\to\mathcal{M}$, where
$\mathcal{M}:=\bigcup_{\theta\in\Theta}(\alpha_\theta,\omega_\theta)$, defined by
$$\mu(\theta):=\int_{\alpha_\theta}^{\omega_\theta}xdF_\theta(x).$$
We assume that, for all $m\in\mathcal{M}$, the equation
$\mu(\theta)=m$ admits a unique solution (with respect to
$\theta\in\Theta$) which will be denoted by $\mu^{-1}(m)$.
\end{condition}
From now on, in connection with this condition, we introduce the
following function:
\begin{equation}\label{eq:rf-cramer-theorem}
\Lambda_\theta^*(x):=\sup_{\gamma\in\mathbb{R}}\left\{\gamma
x-\Lambda_\theta(\gamma)\right\},\ \mbox{where}\
\Lambda_\theta(\gamma):=\log\int_{\alpha_\theta}^{\omega_\theta}e^{\gamma
x}dF_\theta(x).
\end{equation}
It is well-known that, if $\{X_n:n\geq 1\}$ are i.i.d. random
variables with distribution function $F_\theta$, and if we set
$\bar{X}_n:=\frac{X_1+\cdots+X_n}{n}$ for all $n\geq 1$, then
$\{\bar{X}_n:n\geq 1\}$ satisfies the LDP with rate function
$\Lambda_\theta^*$ in \eqref{eq:rf-cramer-theorem} by Cram\'{e}r
Theorem on $\mathbb{R}$ (see e.g. Theorem 2.2.3 in
\cite{DemboZeitouni}).
Then we have the following result.
\begin{proposition}[LD for MM estimators]\label{prop:LD-for-MM-estimators}
Assume that Condition \ref{cond:forMMestimator} holds. Moreover
assume that, for some $\theta_0\in\Theta$, $\{X_n:n\geq 1\}$ are
i.i.d. random variables with distribution function
$F_{\theta_0}$.\\
(i) If $\mu^{-1}(m):=c_1m+c_0$ for some $c_1,c_0\in\mathbb{R}$
such that $c_1\neq 0$, then $\{\mu^{-1}(\bar{X}_n):n\geq 1\}$
satisfies the LDP with rate function $J_{\theta_0}$ defined by
$$J_{\theta_0}(\theta):=\left\{\begin{array}{ll}
\Lambda_{\theta_0}^*(\mu(\theta))&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ \mu(\theta)\in(\alpha_{\theta_0},\omega_{\theta_0})\\
\infty&\ \mbox{otherwise}.
\end{array}\right.$$
(ii) If the restriction of $\mu^{-1}$ on
$(\alpha_{\theta_0},\omega_{\theta_0})$ is continuous and if
$\Lambda_{\theta_0}^*$ is a good rate function, the same LDP holds
and $J_{\theta_0}$ is a good rate function.
\end{proposition}
\begin{proof}
(i) In this case $\mu(\theta):=\frac{\theta-c_0}{c_1}$ and
$\{\mu^{-1}(\bar{X}_n):n\geq 1\}$ is again a sequence of empirical
means of i.i.d. random variables. Then the LDP still holds by
Cram\'{e}r Theorem on $\mathbb{R}$, and the rate function
$J_{\theta_0}$ is defined by
$$J_{\theta_0}(\theta):=\sup_{\gamma\in\mathbb{R}}\left\{\gamma\theta-\Lambda_{\theta_0}(c_1\gamma)-\gamma c_0\right\},$$
which yields
$$J_{\theta_0}(\theta)=\sup_{\gamma\in\mathbb{R}}\left\{c_1\gamma\frac{\theta-c_0}{c_1}-\Lambda_{\theta_0}(c_1\gamma)\right\}
=\sup_{\gamma\in\mathbb{R}}\left\{c_1\gamma\mu(\theta)-\Lambda_{\theta_0}(c_1\gamma)\right\}=\Lambda_{\theta_0}^*(\mu(\theta)),$$
as desired.\\
(ii) Since the restriction of the function $\mu^{-1}$ on
$(\alpha_{\theta_0},\omega_{\theta_0})$ is continuous and
$\Lambda_{\theta_0}^*$ is a good rate function, a straightforward
application of the contraction principle yields the LDP of
$\left\{\mu^{-1}(\bar{X}_n):n\geq 1\right\}$ with good rate
function $J_{\theta_0}$ defined by
$$J_{\theta_0}(\theta):=\inf\left\{\Lambda_{\theta_0}^*(x):x\in(\alpha_{\theta_0},\omega_{\theta_0}),\mu^{-1}(x)=\theta\right\}.$$
Moreover the set
$\left\{x\in(\alpha_{\theta_0},\omega_{\theta_0}):\mu^{-1}(x)=\theta\right\}$
has at most one element, namely
$$\left\{x\in(\alpha_{\theta_0},\omega_{\theta_0}):\mu^{-1}(x)=\theta\right\}=
\left\{\begin{array}{ll}
\{\mu(\theta)\}&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ \mu(\theta)\in(\alpha_{\theta_0},\omega_{\theta_0})\\
\emptyset&\ \mbox{otherwise};
\end{array}\right.$$
thus we have
$J_{\theta_0}(\theta)=\Lambda_{\theta_0}^*(\mu(\theta))$ for
$\theta\in\Theta$ such that
$\mu(\theta)\in(\alpha_{\theta_0},\omega_{\theta_0})$, and
$J_{\theta_0}(\theta)=\infty$ otherwise.
\end{proof}
Now, in the spirit of Remark \ref{rem:comparison-between-rfs}, it
would be interesting to have a local strict inequality between the
rate function $I_{\lambda,\theta_0}(\theta)$ in Proposition
\ref{prop:LD-for-MQ-estimators} for MQ estimators (for some
$\lambda\in(0,1)$), and the rate function $J_{\theta_0}(\theta)$
in Proposition \ref{prop:LD-for-MM-estimators} for MM estimators.
Then we can repeat the same arguments which led us to Proposition
\ref{prop:second-derivative-I}. Namely, if both rate functions
$J_{\theta_0}$ and $I_{\lambda,\theta_0}$ (for some
$\lambda\in(0,1)$) are twice differentiable, $J_{\theta_0}$ is
locally larger (resp. smaller) than $I_{\lambda,\theta_0}$ around
$\theta_0$ if
$J_{\theta_0}^{\prime\prime}(\theta_0)>I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$
(resp.
$J_{\theta_0}^{\prime\prime}(\theta_0)<I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$).
So it is natural to give an expression of
$J_{\theta_0}^{\prime\prime}(\theta_0)$ under suitable hypotheses.
\begin{proposition}[An expression for $J_{\theta_0}^{\prime\prime}(\theta_0)$]\label{prop:second-derivative-J}
Let $J_{\theta_0}$ be the rate function in Proposition
\ref{prop:LD-for-MM-estimators}. Assume that, for all
$\theta\in\Theta$, the function $\Lambda_\theta$ in
\eqref{eq:rf-cramer-theorem} is finite in a neighborhood of the
origin $\gamma=0$ and that $\mu(\cdot)$ is twice differentiable.
Then
$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{(\mu^\prime(\theta_0))^2}{\sigma^2(\theta_0)}$,
where $\sigma^2(\cdot)$ is the variance function.
\end{proposition}
\begin{proof}
One can easily check that
$$J_{\theta_0}^\prime(\theta)=(\Lambda_{\theta_0}^*)^\prime(\mu(\theta))\mu^\prime(\theta)\
\mbox{and}\
J_{\theta_0}^{\prime\prime}(\theta)=(\Lambda_{\theta_0}^*)^{\prime\prime}(\mu(\theta))(\mu^\prime(\theta))^2+\mu^{\prime\prime}(\theta)
(\Lambda_{\theta_0}^*)^\prime(\mu(\theta)).$$ Then we can conclude
noting that
$(\Lambda_{\theta_0}^*)^{\prime\prime}(\mu(\theta))=\frac{1}{\sigma^2(\theta)}$
and $(\Lambda_{\theta_0}^*)^\prime(\mu(\theta))=0$.
\end{proof}
\begin{remark}[On the functions $\Lambda_\theta$ and $\Lambda_\theta^*$ in \eqref{eq:rf-cramer-theorem}]\label{rem:goodness-hypothesis}
The function $\Lambda_\theta$ is finite in a neighborhood of the
origin $\gamma=0$ when we deal with empirical means (of i.i.d.
random variables) with light-tailed distribution. Typically
$\Lambda_\theta^*$ is a good rate function only in this case.
\end{remark}
\section{Examples}\label{sec:examples}
The aim of this section is to present several examples of
statistical models with unknown parameter $\theta\in\Theta$, where
$\Theta\subset\mathbb{R}$; in all the examples we always deal with
one-dimensional parameters assuming all the others to be known.
Let us briefly introduce the examples presented below. We
investigate distributions with scale parameter in Example
\ref{ex:scale-parameter}, with location parameter in Example
\ref{ex:location-parameter}, and with skewness parameter in
Example \ref{ex:skew-parameter}. We remark that in Example
\ref{ex:skew-parameter} we use the epsilon-Skew-Normal
distribution defined in \cite{MudholkarHutson}; this choice is
motivated by the
availability of an explicit expression of the inverse of the
distribution function giving us the possibility of obtaining
explicit formulas. Moreover we present Example
\ref{ex:Pareto-distributions} with Pareto distributions, which
allows to give a concrete illustration of the content of Remark
\ref{rem:invariance-wrt-increasing-transformations}. In all these
statistical models the intervals
$\{(\alpha_\theta,\omega_\theta):\theta\in\Theta\}$ do not depend
on $\theta$ and we simply write $(\alpha,\omega)$. Finally we
present Example \ref{ex:right-endpoint-parameter} where we have
$(\alpha_\theta,\omega_\theta)=(0,\theta)$ for
$\theta\in\Theta:=(0,\infty)$; namely for this example $\theta$ is
a right-endpoint parameter.
In all examples (except Example \ref{ex:Pareto-distributions}) we
give a formula for $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$
(as a consequence of Proposition \ref{prop:second-derivative-I})
which will be used for the local comparisons between rate
functions (in the spirit of Remark
\ref{rem:comparison-between-rfs}) analyzed in Section
\ref{sec:local-comparison-for-examples}.
In what follows we say that a distribution function $F$ on
$\mathbb{R}$ has the \emph{symmetry property} if it is a
distribution function of a symmetric random variable, i.e. if
$F(x)=1-F(-x)$ for all $x\in\mathbb{R}$. In such a case we have
$F^{-1}(\lambda)=-F^{-1}(1-\lambda)$ for all $\lambda\in(0,1)$.
\begin{example}[Statistical model with a scale parameter $\theta\in\Theta:=(0,\infty)$]\label{ex:scale-parameter}
Let $F_\theta$ be defined by
$$F_\theta(x):=G\left(\frac{x}{\theta}\right)\ \mbox{for}\ x\in(\alpha,\omega),$$
where $G$ is a strictly increasing distribution function on
$(\alpha,\omega)=(0,\infty)$ or
$(\alpha,\omega)=(-\infty,\infty)$. Then
$$F_\theta^{-1}(\lambda):=\theta G^{-1}(\lambda)\ \mbox{and}\ h_{\lambda,\theta_0}(\theta)=F_{\theta_0}(F_\theta^{-1}(\lambda))
=G\left(\frac{\theta}{\theta_0}\cdot G^{-1}(\lambda)\right);$$ it
is important to remark that, when
$(\alpha,\omega)=(-\infty,\infty)$, the value $\lambda=G(0)$
(which yields $G^{-1}(\lambda)=0$) is not allowed. Now we give a
list of some specific examples studied in this paper.
For the case $(\alpha,\omega)=(0,\infty)$ we consider the Weibull
distribution:
\begin{equation}\label{eq:df-Weibull}
G(x):=1-\exp(-x^{\rho})\ (\mbox{where}\ \rho>0)\ \mbox{and}\
G^{-1}(\lambda):=\left(-\log(1-\lambda)\right)^{1/\rho}.
\end{equation}
We also give some specific examples where
$(\alpha,\omega)=(-\infty,\infty)$ and, in each case,
$\eta\in\mathbb{R}$ is a known location parameter (and the
not-allowed value $\lambda=G(0)$ depends on $\eta$): the Normal
distribution
\begin{equation}\label{eq:df-Normal-scale-parameter}
G(x):=\Phi(x-\eta)\ \mbox{and}\
G^{-1}(\lambda):=\eta+\Phi^{-1}(\lambda),
\end{equation}
where $\Phi$ is the standard Normal distribution function;
the Cauchy distribution
\begin{equation}\label{eq:df-Cauchy-scale-parameter}
G(x):=\frac{1}{\pi}\left(\arctan(x-\eta)+\frac{\pi}{2}\right)\
\mbox{and}\
G^{-1}(\lambda):=\eta+\tan\left(\left(\lambda-\frac{1}{2}\right)\pi\right);
\end{equation}
the logistic distribution
\begin{equation}\label{eq:df-logistic-scale-parameter}
G(x):=\frac{1}{1+e^{-(x-\eta)}}\ \mbox{and}\
G^{-1}(\lambda):=\eta-\log\left(\frac{1}{\lambda}-1\right);
\end{equation}
the Gumbel distribution
\begin{equation}\label{eq:df-Gumbel-scale-parameter}
G(x):=\exp(-e^{-(x-\eta)})\ \mbox{and}\
G^{-1}(\lambda):=\eta-\log(-\log\lambda).
\end{equation}
If $G$ is twice differentiable we have
\begin{equation}\label{eq:second-derivative-scale-parameter}
I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\{G^\prime(G^{-1}(\lambda))G^{-1}(\lambda)\}^2}{\lambda(1-\lambda)\theta_0^2}
\end{equation}
by Proposition \ref{prop:second-derivative-I}; so, if it is
possible to find an optimal $\lambda_{\mathrm{max}}$, such a value
does not depend on $\theta_0$ (on the contrary it could depend on
the known location parameter $\eta$ as we shall see in Section
\ref{sec:local-comparison-for-examples}). Moreover one can check
that $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=0$ if we
consider the not-allowed value $\lambda=G(0)\in(0,1)$ (when
$(\alpha,\omega)=(-\infty,\infty)$) because $G^{-1}(\lambda)=0$,
and that
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=I_{1-\lambda,\theta_0}^{\prime\prime}(\theta_0)$
(for all $\lambda\in(0,1)$) if $G$ is symmetric as it happens, for
instance, in \eqref{eq:df-Normal-scale-parameter},
\eqref{eq:df-Cauchy-scale-parameter} and
\eqref{eq:df-logistic-scale-parameter} with $\eta=0$.
\end{example}
\begin{example}[Statistical model with a location parameter $\theta\in\Theta:=(-\infty,\infty)$]\label{ex:location-parameter}
Let $F_\theta$ be defined by
$$F_\theta(x):=G(x-\theta)\ \mbox{for}\ x\in(\alpha,\omega)=(-\infty,\infty),$$
where $G$ is a strictly increasing distribution function on
$(\alpha,\omega)=(-\infty,\infty)$. Then
$$F_\theta^{-1}(\lambda):=\theta+G^{-1}(\lambda)\ \mbox{and}\ h_{\lambda,\theta_0}(\theta)=F_{\theta_0}(F_\theta^{-1}(\lambda))
=G\left(\theta+G^{-1}(\lambda)-\theta_0\right).$$ We give some
specific examples studied in this paper and, in each case, $s>0$
is a known scale parameter: the Normal distribution
\begin{equation}\label{eq:df-Normal-location-parameter}
G(x):=\Phi\left(\frac{x}{s}\right)\ \mbox{and}\
G^{-1}(\lambda):=s\cdot\Phi^{-1}(\lambda);
\end{equation}
the Cauchy distribution
\begin{equation}\label{eq:df-Cauchy-location-parameter}
G(x):=\frac{1}{\pi}\left(\arctan\frac{x}{s}+\frac{\pi}{2}\right)\
\mbox{and}\
G^{-1}(\lambda):=s\cdot\tan\left(\left(\lambda-\frac{1}{2}\right)\pi\right);
\end{equation}
the logistic distribution
\begin{equation}\label{eq:df-logistic-location-parameter}
G(x):=\frac{1}{1+e^{-x/s}}\ \mbox{and}\
G^{-1}(\lambda):=-s\cdot\log\left(\frac{1}{\lambda}-1\right);
\end{equation}
the Gumbel distribution
\begin{equation}\label{eq:df-Gumbel-location-parameter}
G(x):=\exp(-e^{-x/s})\ \mbox{and}\
G^{-1}(\lambda):=-s\cdot\log(-\log\lambda).
\end{equation}
If $G$ is twice differentiable we have
\begin{equation}\label{eq:second-derivative-location-parameter}
I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\{G^\prime(G^{-1}(\lambda))\}^2}{\lambda(1-\lambda)}
\end{equation}
by Proposition \ref{prop:second-derivative-I}; so, if it is
possible to find an optimal $\lambda_{\mathrm{max}}$, such a value
does not depend on $\theta_0$ and on the known scale parameter
$s$. Moreover one can check that
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=I_{1-\lambda,\theta_0}^{\prime\prime}(\theta_0)$
(for all $\lambda\in(0,1)$) if $G$ has the symmetry property (as
happens for $G$ in \eqref{eq:df-Normal-location-parameter},
\eqref{eq:df-Cauchy-location-parameter} and
\eqref{eq:df-logistic-location-parameter}, and not for $G$ in
\eqref{eq:df-Gumbel-location-parameter}).
\end{example}
\begin{example}[Statistical model with a skewness parameter $\theta\in\Theta:=(-1,1)$]\label{ex:skew-parameter}
Let $F_\theta$ be defined by
$$F_\theta(x):=\left\{\begin{array}{ll}
(1+\theta)G(\frac{x}{1+\theta})&\ \mbox{for}\ x\leq 0\\
\theta+(1-\theta)G(\frac{x}{1-\theta})&\ \mbox{for}\ x>0,
\end{array}\right.\ \mbox{with}\ x\in(\alpha,\omega)=(-\infty,\infty),$$
where $G$ is a strictly increasing distribution function on
$(\alpha,\omega)=(-\infty,\infty)$ with the symmetry property.
Then
$$F_\theta^{-1}(\lambda):=\left\{\begin{array}{ll}
(1+\theta)G^{-1}(\frac{\lambda}{1+\theta})&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta}{2}]\\
(1-\theta)G^{-1}(\frac{\lambda-\theta}{1-\theta})&\ \mbox{for}\
\lambda\in(\frac{1+\theta}{2},1)
\end{array}\right.$$
and
$$h_{\lambda,\theta_0}(\theta)=F_{\theta_0}(F_\theta^{-1}(\lambda))=\left\{\begin{array}{ll}
(1+\theta_0)G\left(\frac{1+\theta}{1+\theta_0}G^{-1}(\frac{\lambda}{1+\theta})\right)&\ \mbox{for}\ \theta\geq 2\lambda-1\\
\theta_0+(1-\theta_0)G\left(\frac{1-\theta}{1-\theta_0}G^{-1}(\frac{\lambda-\theta}{1-\theta})\right)&\
\mbox{for}\ \theta<2\lambda-1.
\end{array}\right.$$
We can consider the same specific examples presented in Example
\ref{ex:location-parameter}, i.e. the functions $G$ in
\eqref{eq:df-Normal-location-parameter},
\eqref{eq:df-Cauchy-location-parameter} and
\eqref{eq:df-logistic-location-parameter} for some known scale
parameter $s>0$.
If $G$ is twice differentiable and $G^{\prime\prime}(0)=0$ we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\left\{\begin{array}{ll}
\frac{1}{\lambda(1-\lambda)}\left[G^\prime(G^{-1}(\frac{\lambda}{1+\theta_0}))
\left(G^{-1}(\frac{\lambda}{1+\theta_0})-\frac{\lambda}{1+\theta_0}(G^{-1})^\prime(\frac{\lambda}{1+\theta_0})\right)\right]^2
&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta_0}{2}]\\
\frac{1}{\lambda(1-\lambda)}\left[G^\prime(G^{-1}(\frac{\lambda-\theta_0}{1-\theta_0}))
\left(-G^{-1}(\frac{\lambda-\theta_0}{1-\theta_0})+\frac{\lambda-1}{1-\theta_0}(G^{-1})^\prime(\frac{\lambda-\theta_0}{1-\theta_0})\right)\right]^2
&\ \mbox{for}\ \lambda\in(\frac{1+\theta_0}{2},1)
\end{array}\right.$$
by Proposition \ref{prop:second-derivative-I}, and therefore
\begin{equation}\label{eq:second-derivative-skew-parameter}
I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\left\{\begin{array}{ll}
\frac{1}{\lambda(1-\lambda)}\left[G^\prime(G^{-1}(\frac{\lambda}{1+\theta_0}))G^{-1}(\frac{\lambda}{1+\theta_0})-\frac{\lambda}{1+\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta_0}{2}]\\
\frac{1}{\lambda(1-\lambda)}\left[-G^\prime(G^{-1}(\frac{\lambda-\theta_0}{1-\theta_0}))
G^{-1}(\frac{\lambda-\theta_0}{1-\theta_0})+\frac{\lambda-1}{1-\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(\frac{1+\theta_0}{2},1);
\end{array}\right.
\end{equation}
so one can expect that, if it is possible to find an optimal
$\lambda_{\mathrm{max}}$, such a value depends on $\theta_0$ (this
is what happens in Section
\ref{sec:local-comparison-for-examples}). Moreover one can check
that $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ in
\eqref{eq:second-derivative-skew-parameter} does not depend on
$s$, $I_{1/2,0}^{\prime\prime}(0)=1$ and
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=I_{1-\lambda,-\theta_0}^{\prime\prime}(-\theta_0)$
(for all $\lambda\in(0,1)$).
\end{example}
\begin{example}[Statistical model with Pareto distributions with $\theta\in\Theta:=(0,\infty)$]\label{ex:Pareto-distributions}
Let $F_\theta$ be defined by
$$F_\theta(x):=1-x^{-1/\theta}\ \mbox{for}\ x\in(\alpha,\omega)=(1,\infty).$$
Then
\begin{equation}\label{eq:basic-positions-Pareto-distributions}
F_\theta^{-1}(\lambda):=e^{-\theta\log(1-\lambda)}=(1-\lambda)^{-\theta}\
\mbox{and}\
h_{\lambda,\theta_0}(\theta)=F_{\theta_0}(F_\theta^{-1}(\lambda))
=1-(1-\lambda)^{\theta/\theta_0}.
\end{equation}
We remark that, if we consider Example \ref{ex:scale-parameter}
with $G$ as in \eqref{eq:df-Weibull} with $\rho=1$, namely
$$\tilde{F}_\theta(x)=1-e^{-x}\ \mbox{for}\ (\tilde{\alpha},\tilde{\omega})=(0,\infty),$$
we can refer to Remark
\ref{rem:invariance-wrt-increasing-transformations} with
$$\psi(x):=e^x\ \mbox{for}\ x\in(\tilde{\alpha},\tilde{\omega}):=(0,\infty)$$
(note that
$(\psi(\tilde{\alpha}),\psi(\tilde{\omega}))=(1,\infty)=(\alpha,\omega)$).
Then, as pointed out in Remark
\ref{rem:invariance-wrt-increasing-transformations},
$I_{\lambda,\theta_0;\psi}$ and $I_{\lambda,\theta_0}$ coincide;
in fact $h_{\lambda,\theta_0}(\theta)$ in
\eqref{eq:basic-positions-Pareto-distributions} meets the analogue
expression in Example \ref{ex:scale-parameter} with $G$ as in
\eqref{eq:df-Weibull} with $\rho=1$, namely
$$\tilde{F}_{\theta_0}(\tilde{F}_\theta^{-1}(\lambda))=G\left(\frac{\theta}{\theta_0}\cdot G^{-1}(\lambda)\right)
=1-\exp\left(-\frac{\theta}{\theta_0}\cdot(-\log(1-\lambda))\right)=1-(1-\lambda)^{\theta/\theta_0}.$$
\end{example}
\begin{example}[Statistical model with a \lq\lq right endpoint\rq\rq parameter $\theta\in\Theta:=(0,\infty)$]\label{ex:right-endpoint-parameter}
Let $F_\theta$ be defined by
$$F_\theta(x):=\frac{G(x)}{G(\theta)}\ \mbox{for}\ x\in(\alpha_\theta,\omega_\theta):=(0,\theta),$$
where $G:[0,\infty)\to[0,\infty)$ is a strictly increasing
function such that $G(0)=0$. Then
$$F_\theta^{-1}(\lambda):=G^{-1}(\lambda G(\theta))\ \mbox{and}\ h_{\lambda,\theta_0}(\theta)=F_{\theta_0}(F_\theta^{-1}(\lambda))
=\frac{\lambda G(\theta)}{G(\theta_0)}\ (\mbox{for}\ \lambda
G(\theta)\in(0,G(\theta_0))).$$ Moreover, after some computations,
we get
$$I_{\lambda,\theta_0}(\theta):=\left\{\begin{array}{ll}
\lambda\log\frac{G(\theta_0)}{G(\theta)}
+(1-\lambda)\log\frac{(1-\lambda)G(\theta_0)}{G(\theta_0)-\lambda
G(\theta)}
&\ \mbox{for}\ 0<\theta<G^{-1}\left(\frac{G(\theta_0)}{\lambda}\right)\\
\infty&\ \mbox{otherwise}.
\end{array}\right.$$
As a specific example we can consider $G(x)=x$ (for all $x$); in
such a case $F_\theta$ is the distribution function concerning the
uniform distribution on $(0,\theta)$. Finally, if $G$ is twice
differentiable, we have
\begin{equation}\label{eq:second-derivative-right-endpoint-parameter}
I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\lambda(G^\prime(\theta_0))^2}{(1-\lambda)G^2(\theta_0)}
\end{equation}
by Proposition \ref{prop:second-derivative-I}.
\end{example}
\section{Local comparisons between rate functions for some examples}\label{sec:local-comparison-for-examples}
In this section we analyze the examples presented in Section
\ref{sec:examples}. We consider local comparisons
between rate functions in the spirit of Remark
\ref{rem:comparison-between-rfs} and, more precisely, the
following two issues.
\begin{itemize}
\item A discussion on the choice of optimal values of $\lambda$
in order to get the best rate of convergence. More precisely we
study the behavior of
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ (varying
$\lambda\in(0,1)$) in order to find an optimal
$\lambda_{\mathrm{max}}$ in the sense of Definition
\ref{def:optimal-lambda}.
\item The comparison of the convergence of the MQ estimators and
of the MM estimators. More precisely, when we deal with MM
estimators, we compare
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ in Proposition
\ref{prop:second-derivative-I} (for some $\lambda\in(0,1)$) and
$J_{\theta_0}^{\prime\prime}(\theta_0)$ in Proposition
\ref{prop:second-derivative-J}; obviously, when we have an optimal
$\lambda_{\mathrm{max}}$, we take
$\lambda=\lambda_{\mathrm{max}}$. In the single case presented
below where MM estimators are not defined, we compare the
convergence of the MQ estimators and of suitable GMM estimators.
\end{itemize}
We find at least an optimal $\lambda_{\mathrm{max}}$ for all
examples except for Example \ref{ex:right-endpoint-parameter}
(where we should consider $\lambda=1$). For several particular
examples where $(\alpha,\omega)=(-\infty,\infty)$ we have
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=I_{1-\lambda,\theta_0}^{\prime\prime}(\theta_0)$
for all $\lambda\in(0,1)$; then, in those cases, we have the
optimal value $\lambda_{\mathrm{max}}=1/2$ or, by symmetry, two
distinct optimal values $\lambda_{\mathrm{max},1}$ and
$\lambda_{\mathrm{max},2}=1-\lambda_{\mathrm{max},1}$.
In view of what follows it is useful to consider two suitable
values $\tilde{\lambda}_1,\tilde{\lambda}_2\in(0,1)$ presented in
the next Lemma \ref{lem:particular-values}. The value
$\tilde{\lambda}_1$ appears in the computations for the Weibull
distribution in Example \ref{ex:scale-parameter} (and also in the
computations for Example \ref{ex:Pareto-distributions} as a
trivial consequence), while the $\tilde{\lambda}_2$ appears in the
computations for the Gumbel distribution in both Examples
\ref{ex:scale-parameter} and \ref{ex:location-parameter}; however,
interestingly, Lemma \ref{lem:particular-values}(iii) shows the
close relationship between $\tilde{\lambda}_1$ and
$\tilde{\lambda}_2$.
\begin{lemma}[The values $\tilde{\lambda}_1$ and $\tilde{\lambda}_2$]\label{lem:particular-values}
The following statements hold.\\
(i) Let $f_1$ be the function defined by
$f_1(\lambda):=\frac{(1-\lambda)(\log(1-\lambda))^2}{\lambda}$.
Then $\sup_{\lambda\in(0,1)}f_1(\lambda)=f_1(\tilde{\lambda}_1)$,
where $\tilde{\lambda}_1\simeq 0.7968$ is the unique value
$(1/2,1)$ such that
$-2\tilde{\lambda}_1-\log(1-\tilde{\lambda}_1)=0$.\\
(ii) Let $f_2$ be the function defined by
$f_2(\lambda):=\frac{\lambda(\log\lambda)^2}{1-\lambda}$. Then
$\sup_{\lambda\in(0,1)}f_2(\lambda)=f_2(\tilde{\lambda}_2)$, where
$\tilde{\lambda}_2\simeq 0.2032$ is the unique value $(0,1/2)$
such that $\log\tilde{\lambda}_2+2-2\tilde{\lambda}_2=0$.\\
(iii) We have $\tilde{\lambda}_1+\tilde{\lambda}_2=1$.
\end{lemma}
\begin{proof}
(i) The derivative of $f_1$ is
$$\frac{d}{d\lambda}f_1(\lambda)=\frac{\log(1-\lambda)[-2\lambda-\log(1-\lambda)]}{\lambda^2}$$
and $-2\lambda-\log(1-\lambda)<0$ (resp.
$-2\lambda-\log(1-\lambda)>0$) if $\lambda<\tilde{\lambda}_1$
(resp. $\lambda>\tilde{\lambda}_1$); therefore we have
$\frac{d}{d\lambda}f_1(\lambda)>0$ for
$\lambda\in(0,\tilde{\lambda}_1)$ and
$\frac{d}{d\lambda}f_1(\lambda)<0$ for
$\lambda\in(\tilde{\lambda}_1,1)$.\\
(ii) The derivative of $f_2$ is
$$\frac{d}{d\lambda}f_2(\lambda)=\frac{\log\lambda\cdot[\log\lambda+2-2\lambda]}{(1-\lambda)^2}$$
and $\log\lambda+2-2\lambda<0$ (resp. $\log\lambda+2-2\lambda>0$)
if $\lambda<\tilde{\lambda}_2$ (resp.
$\lambda>\tilde{\lambda}_2$); therefore we have
$\frac{d}{d\lambda}f_2(\lambda)>0$ for
$\lambda\in(0,\tilde{\lambda}_2)$ and
$\frac{d}{d\lambda}f_2(\lambda)<0$ for
$\lambda\in(\tilde{\lambda}_2,1)$.\\
(iii) The equality $\tilde{\lambda}_1+\tilde{\lambda}_2=1$ can be
easily checked by taking into account how the values
$\tilde{\lambda}_1$ and $\tilde{\lambda}_2$ are defined in
statements (i) and (ii).
\end{proof}
\subsection{Example \ref{ex:scale-parameter}}\label{sub:example1}
In this section we consider the particular example of the Weibull
distribution with $(\alpha,\omega)=(0,\infty)$, and the particular
examples with $(\alpha,\omega)=(-\infty,\infty)$. In each part we
analyze the MQ estimators, and we conclude with the MM estimators.
\paragraph{Analysis of MQ estimators for Weibull distribution.}
Here we consider $G$ in \eqref{eq:df-Weibull}. By
\eqref{eq:second-derivative-scale-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\rho^2(1-\lambda)(\log(1-\lambda))^2}{\lambda\theta_0^2}.$$
Then we have a unique optimal value
$\lambda_{\mathrm{max}}=\tilde{\lambda}_1$ (for every $\rho$ and
$\theta_0$), where $\tilde{\lambda}_1$ is the value in Lemma
\ref{lem:particular-values}(i); in fact, if we consider the
function $f_1$ in that lemma, we have
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\rho^2f_1(\lambda)}{\theta_0^2}$.
\paragraph{MM versus MQ estimators for Weibull distribution.}
We start with the analysis of MM estimators. We have
$\mu(\theta):=\theta\Gamma(1+1/\rho)$, and therefore
$\mu^{-1}(m):=\frac{m}{\Gamma(1+1/\rho)}$, where $\Gamma$ is the usual Gamma function. Thus, by Proposition
\ref{prop:LD-for-MM-estimators}(i), $\{\mu^{-1}(\bar{X}_n):n\geq
1\}$ satisfies the LDP with rate function $J_{\theta_0}$ with
$c_1=\frac{1}{\Gamma(1+1/\rho)}$ and $c_0=0$. In what follows we
consider the light-tailed case $\rho\geq 1$ and the heavy-tailed
case $\rho\in(0,1)$; some more details on the exponential
distribution case $\rho=1$ are given in Remark \ref{rem:rho=1}.\\
\underline{Light-tailed case} (namely $\rho\geq 1$). The rate
functions $J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ are good and
we can refer to the comparison between
$$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{\Gamma^2(1+1/\rho)}{\theta_0^2\left[\Gamma(1+2/\rho)-\Gamma^2(1+1/\rho)\right]}$$
(this value is a consequence of Proposition
\ref{prop:second-derivative-J} noting that
$\sigma^2(\theta)=\theta^2\left[\Gamma(1+2/\rho)-\Gamma^2(1+1/\rho)\right]$)
and, for the optimal value
$\lambda_{\mathrm{max}}=\tilde{\lambda}_1$ in Lemma
\ref{lem:particular-values}(i),
$$I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)=\frac{(1-\tilde{\lambda}_1)(\log(1-\tilde{\lambda}_1))^2}{\tilde{\lambda}_1\theta_0^2}
=\frac{(1-\tilde{\lambda}_1)(2\tilde{\lambda}_1)^2}{\tilde{\lambda}_1\theta_0^2}=\frac{4\tilde{\lambda}_1(1-\tilde{\lambda}_1)}{\theta_0^2}.$$
We remark that $J_{\theta_0}^{\prime\prime}(\theta_0)$ is an
increasing function of $\rho\in(0,\infty)$; in fact, if we set
$$a(\rho):=\Gamma^\prime(1+2/\rho)\Gamma(1+1/\rho)-\Gamma(1+2/\rho)\Gamma^\prime(1+1/\rho),$$
we have $a(\rho)>0$ (noting that
$\frac{\Gamma^\prime(1+2/\rho)}{\Gamma(1+2/\rho)}>\frac{\Gamma^\prime(1+1/\rho)}{\Gamma(1+1/\rho)}$
because the digamma function
$x\mapsto\frac{\Gamma^\prime(x)}{\Gamma(x)}$ is increasing on
$(0,\infty)$) and therefore
$$\frac{d}{d\rho}\left(\frac{\Gamma^2(1+1/\rho)}{\theta_0^2\left[\Gamma(1+2/\rho)-\Gamma^2(1+1/\rho)\right]}\right)
=\frac{2\Gamma(1+1/\rho)a(\rho)}
{\rho^2\theta_0^2\left(\frac{\Gamma(1+2/\rho)}{\Gamma^2(1+1/\rho)}-1\right)^2\Gamma^4(1+1/\rho)}>0.$$
Thus, for all $\rho\geq 1$, MM estimators converge faster than
every MQ estimators because
$$J_{\theta_0}^{\prime\prime}(\theta_0)\geq\frac{1}{\theta_0^2}>
\frac{4\tilde{\lambda}_1(1-\tilde{\lambda}_1)}{\theta_0^2}=I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)$$
noting that $\inf_{\rho\geq
1}J_{\theta_0}^{\prime\prime}(\theta_0)=\left.J_{\theta_0}^{\prime\prime}(\theta_0)\right|_{\rho=1}=\frac{1}{\theta_0^2}$
and $4\tilde{\lambda}_1(1-\tilde{\lambda}_1)\simeq 0.6476$.\\
\underline{Heavy-tailed case} (namely $\rho\in(0,1)$). The rate
functions $J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ are not good
(we recall Remark \ref{rem:goodness-hypothesis} presented above).
We can say that $J_{\theta_0}(\theta)=0$ for $\theta\geq\theta_0$;
thus $I_{\lambda,\theta_0}(\theta)>J_{\theta_0}(\theta)$ for
$\theta>\theta_0$ (for all $\lambda\in(0,1)$). Then we have to
compare $I_{\tilde{\lambda}_1,\theta_0}(\theta)$ and
$J_{\theta_0}(\theta)$ in a left neighborhood of $\theta_0$,
namely when $\theta\in(\theta_0-\delta,\theta_0)$ for $\delta>0$
small enough. Therefore it suffices to compare
$I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)$ and the
left second derivative
$\left.\frac{d^2}{d\theta^2}J_{\theta_0}(\theta-)\right|_{\theta=\theta_0}$
which coincides with $J_{\theta_0}^{\prime\prime}(\theta_0)$
presented above for the light-tailed case. We already explained
that
$\left.\frac{d^2}{d\theta^2}J_{\theta_0}(\theta-)\right|_{\theta=\theta_0}$
is an increasing function of $\rho\in(0,\infty)$; moreover
$$\left.\frac{d^2}{d\theta^2}J_{\theta_0}(\theta-)\right|_{\theta=\theta_0}
=\frac{1}{\theta_0^2\left[\frac{\Gamma(1+2/\rho)}{\Gamma^2(1+1/\rho)}-1\right]}\to
0\ \mbox{as}\ \rho\to 0$$ by taking into account the asymptotic
behavior of Gamma function. In conclusion there exists
$\rho_0\simeq 0.81068$ (computed numerically) such that:
\begin{enumerate}
\item
$I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)>\left.\frac{d^2}{d\theta^2}J_{\theta_0}(\theta-)\right|_{\theta=\theta_0}$
for $\rho\in (0,\rho_0)$;
\item
$I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)<\left.\frac{d^2}{d\theta^2}J_{\theta_0}(\theta-)\right|_{\theta=\theta_0}$
for $\rho\in (\rho_0,1)$.
\end{enumerate}
Thus, by taking into account Remark
\ref{rem:comparison-between-rfs}, MQ estimators (with
$\lambda=\tilde{\lambda}_1$) converge faster than MM estimators in
the first case while, in the second case, the convergence of MQ
and MM estimators cannot be compared because we cannot find
$\delta>0$ such that
$I_{\tilde{\lambda}_1,\theta_0}(\theta)>J_{\theta_0}(\theta)$ or
$I_{\tilde{\lambda}_1,\theta_0}(\theta_)<J_{\theta_0}(\theta)$ for
$0<|\theta-\theta_0|<\delta$.
\begin{remark}[The case $\rho=1$, namely the exponential distribution]\label{rem:rho=1}
If $\rho=1$ the rate functions $J_{\theta_0}$ and
$\Lambda_{\theta_0}^*$ coincide (in fact $\Gamma(1+1/\rho)=1$) and
an explicit expression of $\Lambda_{\theta_0}^*$ is available,
namely
$$J_{\theta_0}(\theta)=\Lambda_{\theta_0}^*(\theta)=\left\{\begin{array}{ll}
\frac{\theta}{\theta_0}-1-\log\left(\frac{\theta}{\theta_0}\right)&\ \mbox{for}\ \theta\in(0,\infty)\\
\infty&\ \mbox{otherwise};
\end{array}\right.$$
then we can directly compute
$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{1}{\theta_0^2}$,
which meets the above expression. In this case the MM estimators
coincide with the ML estimators, and we already expected that they
converge faster than the MQ estimators.
\end{remark}
\paragraph{Analysis of MQ estimators for particular examples with $(\alpha,\omega)=(-\infty,\infty)$.}
Here we present the results concerning the specific examples
listed above. For all cases except the one with the Gumbel
distribution we choose $\eta$ in order to have
$G(0)\in\{0.25,0.5,0.75\}$, and we have some common features:
$G(0)=1/2$ for $\eta=0$ (actually the symmetry property holds); we
obtain symmetric values with respect $\lambda=1/2$ such that the
more the tails of the distributions are light, the more the
numerical values of $\lambda_{\mathrm{max}}$ are distant from
$\lambda=1/2$. The case with the Gumbel distribution behaves
differently because the symmetry property fails for each fixed
value of $\eta$. In all cases we can only give numerical values.\\
\underline{Normal distribution} (namely $G$ in
\eqref{eq:df-Normal-scale-parameter}). We have $G(0)=\Phi(-\eta)$
and, by \eqref{eq:second-derivative-scale-parameter},
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{(\varphi(\Phi^{-1}(\lambda))\{\eta+\Phi^{-1}(\lambda)\})^2}{\lambda(1-\lambda)\theta_0^2},$$
where $\varphi$ is the standard Normal probability density function.
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\eta=0&\ 0.06\ (\mbox{and}\ 0.94\ \mbox{by symmetry})\\
\eta=-\Phi^{-1}(1/4)&\ 0.90\\
\eta=-\Phi^{-1}(3/4)&\ 1-0.90=0.10\ (\mbox{by symmetry})
\end{array}\right.$$
\underline{Cauchy distribution} (namely $G$ in
\eqref{eq:df-Cauchy-scale-parameter}). We have
$G(0)=\frac{1}{\pi}\left(\arctan(-\eta)+\frac{\pi}{2}\right)$ and,
by \eqref{eq:second-derivative-scale-parameter},
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\left\{\eta+\tan\left(\left(\lambda-\frac{1}{2}\right)\pi\right)\right\}^2}
{\pi^2\left\{1+\tan^2\left(\left(\lambda-\frac{1}{2}\right)\pi\right)\right\}^2\lambda(1-\lambda)\theta_0^2}.$$
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\eta=0&\ 0.21\ (\mbox{and}\ 0.79\ \mbox{by symmetry})\\
\eta=1&\ 0.65\\
\eta=-1&\ 1-0.65=0.35\ (\mbox{by symmetry})
\end{array}\right.$$
\underline{Logistic distribution} (namely $G$ in
\eqref{eq:df-logistic-scale-parameter}). We have
$G(0)=\frac{1}{1+e^\eta}$ and, by
\eqref{eq:second-derivative-scale-parameter},
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\lambda(1-\lambda)\left(\eta-\log(\frac{1}{\lambda}-1)\right)^2}{\theta_0^2}.$$
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\eta=0&\ 0.08\ (\mbox{and}\ 0.92\ \mbox{by symmetry})\\
\eta=\log 3&\ 0.85\\
\eta=-\log 3&\ 1-0.85=0.15\ (\mbox{by symmetry})
\end{array}\right.$$
\underline{Gumbel distribution} (namely $G$ in
\eqref{eq:df-Gumbel-scale-parameter}). We have
$G(0)=\exp(-e^\eta)$ and, by
\eqref{eq:second-derivative-scale-parameter},
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\lambda(\log\lambda)^2(\eta-\log(-\log\lambda))^2}{(1-\lambda)\theta_0^2}.$$
Some numerical inspections reveal that in general, for each fixed
value of $\eta$, we can find an optimal value
$\lambda_{\mathrm{max}}=\lambda_{\mathrm{max}}(\eta)$. Then, if we
consider the value $\tilde{\lambda}_2$ and the function $f_2$ in
Lemma \ref{lem:particular-values}(ii), we can say that
$$\lambda_{\mathrm{max}}=\lambda_{\mathrm{max}}(\eta)\to\tilde{\lambda}_2\ \mbox{as}\ |\eta|\to\infty$$
because $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ behaves
like $\frac{f_2(\lambda)\eta^2}{\theta_0^2}$ when $|\eta|$ is
large.
\paragraph{MM versus MQ estimators for particular examples with $(\alpha,\omega)=(-\infty,\infty)$.}
The MM estimators are well-defined only for the case with Gumbel
distribution and, in the spirit of Remark
\ref{rem:comparison-between-rfs}, we can compare
$J_{\theta_0}^{\prime\prime}(\theta_0)$ and
$I_{\lambda_{\mathrm{max}},\theta_0}^{\prime\prime}(\theta_0)$.
However, for Normal and logistic distributions, it is possible to
consider suitable GMM estimators $\{\tilde{\Theta}_n:n\geq 1\}$ by
matching empirical and theoretical variances; so we present the
rate function $\tilde{J}_{\theta_0}$ which governs the LDP of
$\{\tilde{\Theta}_n:n\geq 1\}$ and, at least for the case of
Normal distribution, we can give an expression of
$\tilde{J}_{\theta_0}^{\prime\prime}(\theta_0)$ and we can compare
the convergence of MQ and GMM estimators.\\
\underline{Gumbel distribution}. We have
$\mu(\theta):=\eta+\theta\gamma_*$, where $\gamma_*$ is the
Euler's constant, and therefore
$\mu^{-1}(m):=\frac{m-\eta}{\gamma_*}$. Thus, by Proposition
\ref{prop:LD-for-MM-estimators}(i), $\{\mu^{-1}(\bar{X}_n):n\geq
1\}$ satisfies the LDP with rate function $J_{\theta_0}$ with
$c_1=\frac{1}{\gamma_*}$ and $c_0=\frac{-\eta}{\gamma_*}$. The
rate functions $J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ are good
(we take into account Remark \ref{rem:goodness-hypothesis}) and we
can refer to the comparison between
$I_{\lambda_{\mathrm{max}},\theta_0}^{\prime\prime}(\theta_0)$ and
$J_{\theta_0}^{\prime\prime}(\theta_0)$. We remark that, by
Proposition \ref{prop:second-derivative-J}, we have
$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{6\gamma_*^2}{\theta_0^2\pi^2}$
for each fixed value of $\eta$ (in fact we have
$\sigma^2(\theta)=\frac{\theta^2\pi^2}{6}$). Then, for all
$\eta\in\mathbb{R}$, we have
$I_{\lambda_{\mathrm{max}},\theta_0}^{\prime\prime}(\theta_0)>J_{\theta_0}^{\prime\prime}(\theta_0)$
noting that, for $\tilde{\lambda}_1$ and $\tilde{\lambda}_2$ as in
Lemma \ref{lem:particular-values}, we have
$\max\{I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0),I_{\tilde{\lambda}_2,\theta_0}^{\prime\prime}(\theta_0)\}>J_{\theta_0}^{\prime\prime}(\theta_0)$
(see Figure \ref{fig1}).
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=0.5\textwidth]{Gumbel_eta.pdf}
\caption{The second derivatives
$I_{\tilde{\lambda}_2,\theta_0}^{\prime\prime}(\theta_0)$ (broken
line) and
$I_{\tilde{\lambda}_1,\theta_0}^{\prime\prime}(\theta_0)$ (dotted
line) as functions of $\eta$. The solid line represents the value
of $J_{\theta_0}^{\prime\prime}(\theta_0)$ which does not depend
on $\eta$.}\label{fig1}
\end{center}
\end{figure}\\
\underline{Normal (and logistic) distribution}. The MM estimators
are not well-defined because $\mu(\theta):=\eta$. So it is natural
to match empirical and theoretical variances, i.e.
$$\sigma^2(\theta)=\frac{1}{n}\sum_{i=1}^n(x_i-\eta)^2,\ \mbox{where}\ \sigma^2(\theta)=c\theta^2\ \mbox{and}\ c=\left\{\begin{array}{ll}
1&\ \mbox{for the Normal distribution}\\
\frac{\pi^2}{3}&\ \mbox{for the logistic distribution},
\end{array}\right.$$
and we obtain the GMM estimators $\{\tilde{\Theta}_n:n\geq 1\}$
defined by
$$\tilde{\Theta}_n:=\left(\frac{1}{cn}\sum_{i=1}^n(x_i-\eta)^2\right)^{1/2}.$$
Then, by adapting the proof of Proposition
\ref{prop:LD-for-MM-estimators}, we can consider the function
$$\tilde{\Lambda}_{\theta_0}^*(y):=\sup_{\gamma\in\mathbb{R}}\left\{\gamma
y-\tilde{\Lambda}_{\theta_0}(\gamma)\right\},\ \mbox{where}\
\tilde{\Lambda}_{\theta_0}(\gamma):=\log\int_{\alpha_\theta}^{\omega_\theta}e^{\gamma
(x-\eta)^2}dF_{\theta_0}(x),$$ and we can say
$\{\tilde{\Theta}_n:n\geq 1\}$ satisfies the LDP with good rate
function $\tilde{J}_{\theta_0}$ defined by
$$\tilde{J}_{\theta_0}(\theta):=\inf\{\tilde{\Lambda}_{\theta_0}^*(y):(y/c)^{1/2}=\theta\}.$$
From now on we restrict the attention to the case with Normal
distribution because we can give explicit formulas. We have
$$\tilde{\Lambda}_{\theta_0}(\gamma)=\left\{\begin{array}{ll}
\frac{1}{2}\log\left(\frac{\theta_0^2/2}{\theta_0^2/2-\gamma}\right)&\ \mbox{if}\ \gamma<\frac{\theta_0^2}{2}\\
\infty&\ \mbox{if}\ \gamma\geq\frac{\theta_0^2}{2},
\end{array}\right.\ \tilde{\Lambda}_{\theta_0}^*(y)=\left\{\begin{array}{ll}
\frac{1}{2}\left[\frac{y}{\theta_0^2}-1-\log\left(\frac{y}{\theta_0^2}\right)\right]&\ \mbox{if}\ y>0\\
\infty&\ \mbox{if}\ y\leq 0,
\end{array}\right.$$
and
$$\tilde{J}_{\theta_0}(\theta)=\left\{\begin{array}{ll}
\frac{1}{2}\left[\frac{\theta^2}{\theta_0^2}-1-\log\left(\frac{\theta^2}{\theta_0^2}\right)\right]&\ \mbox{if}\ \theta>0\\
\infty&\ \mbox{if}\ \theta\leq 0;
\end{array}\right.$$
thus, after some computations, we get
$\tilde{J}_{\theta_0}^{\prime\prime}(\theta_0)=\frac{2}{\theta_0^2}$
for all value of $\eta\in\mathbb{R}$. We conclude with the
comparison between MQ and GMM estimators. Some numerical
inspections reveal that in general, for each fixed value of
$\eta$, we can find an optimal value
$\lambda_{\mathrm{max}}=\lambda_{\mathrm{max}}(\eta)$ (their
numerical values for $\eta=0$, $\eta=\Phi^{-1}(1/4)$ and
$\eta=\Phi^{-1}(3/4)$ were presented above); moreover
$I_{\lambda_{\mathrm{max}}(\eta),\theta_0}^{\prime\prime}(\theta_0)>\tilde{J}_{\theta_0}^{\prime\prime}(\theta_0)$
for $|\eta|$ large enough because, for each fixed
$\lambda\in(0,1)$,
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)\to\infty\ \mbox{as}\ |\eta|\to\infty.$$
On the other hand we cannot say that
$I_{\lambda_{\mathrm{max}}(\eta),\theta_0}^{\prime\prime}(\theta_0)>\tilde{J}_{\theta_0}^{\prime\prime}(\theta_0)$
for all $\eta\in\mathbb{R}$; in fact, for $\eta=0$, we have
$$I_{\lambda_{\mathrm{max}}(0),\theta_0}^{\prime\prime}(\theta_0)\simeq\frac{0.6085}{\theta_0^2}<\frac{2}{\theta_0^2}=\tilde{J}_{\theta_0}^{\prime\prime}(\theta_0)$$
(where $\lambda_{\mathrm{max}}(0)\simeq 0.06$ or
$\lambda_{\mathrm{max}}(0)\simeq 0.94$). For completeness,
following the same lines of the particular case with Gumbel
distribution, we remark that
$$\lambda_{\mathrm{max}}=\lambda_{\mathrm{max}}(\eta)\to 1/2\ \mbox{as}\ |\eta|\to\infty$$
because, if we consider the function
$f(\lambda):=\frac{(\varphi(\Phi^{-1}(\lambda)))^2}{\lambda(1-\lambda)}$,
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ behaves like
$\frac{f(\lambda)\eta^2}{\theta_0^2}$ when $|\eta|$ is large, and
$\sup_{\lambda\in(0,1)}f(\lambda)=f(1/2)$.
\subsection{Example \ref{ex:location-parameter}}\label{sub:example2}
We start with the analysis of MQ estimators. We conclude with the
MM estimators, and their comparison with the MQ estimators.
\paragraph{Analysis of MQ estimators.}
Here we present the results concerning the specific examples
listed above. In all cases, except the one with Gumbel
distribution, we can conclude that $\lambda=1/2$ is optimal;
however we can find counterexamples (see Appendix \ref{appendix}).
A further common feature (for all cases except the one with Gumbel
distribution) is that
$\left.\frac{d}{d\lambda}I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)\right|_{\lambda=1/2}=0$
(and obviously this does not guarantee that $\lambda=1/2$ is an
optimal; this will be explained in Appendix \ref{appendix}); in
fact we have
\begin{multline*}
\left.\frac{d}{d\lambda}I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)\right|_{\lambda=1/2}=
\left.\frac{2G^\prime(G^{-1}(\lambda))G^{\prime\prime}(G^{-1}(\lambda))(G^{-1})^\prime(\lambda)\lambda(1-\lambda)
-(1-2\lambda)\{G^\prime(G^{-1}(\lambda))\}^2}{\lambda^2(1-\lambda)^2}\right|_{\lambda=1/2}\\
=\left.\frac{2G^{\prime\prime}(G^{-1}(\lambda))\lambda(1-\lambda)
-(1-2\lambda)\{G^\prime(G^{-1}(\lambda))\}^2}{\lambda^2(1-\lambda)^2}\right|_{\lambda=1/2}=0
\end{multline*}
noting that $G^{-1}(1/2)=0$ (because the distribution function $G$
has the symmetry property) and $G^{\prime\prime}(0)=0$ (because
the probability density function $G^\prime(x)$ has a maximum at
$x=0$).\\
\underline{Normal distribution} (namely $G$ in
\eqref{eq:df-Normal-location-parameter}). By
\eqref{eq:second-derivative-location-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\varphi^2(\Phi^{-1}(\lambda))}{s^2\lambda(1-\lambda)}.$$
One can check numerically that we have a unique optimal
$\lambda_{\mathrm{max}}$ (for every $s$), namely
$\lambda_{\mathrm{max}}=0.5$.\\
\underline{Cauchy distribution} (namely $G$ in
\eqref{eq:df-Cauchy-location-parameter}). By
\eqref{eq:second-derivative-location-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{1}{s^2\pi^2\left\{1+\tan^2\left(\left(\lambda-\frac{1}{2}\right)\pi\right)\right\}^2\lambda(1-\lambda)}.$$
One can check numerically that we have a unique optimal
$\lambda_{\mathrm{max}}$ (for every $s$), namely
$\lambda_{\mathrm{max}}=0.5$.\\
\underline{Logistic distribution} (namely $G$ in
\eqref{eq:df-logistic-location-parameter}). By
\eqref{eq:second-derivative-location-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\lambda(1-\lambda)}{s^2}$$
One can immediately check (we have a polynomial with degree 2)
that we have a unique optimal $\lambda_{\mathrm{max}}$ (for every
$s$), namely $\lambda_{\mathrm{max}}=0.5$.\\
\underline{Gumbel distribution} (namely $G$ in
\eqref{eq:df-Gumbel-location-parameter}). By
\eqref{eq:second-derivative-location-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{\lambda(\log\lambda)^2}{s^2(1-\lambda)}$$
Then we have a unique optimal value
$\lambda_{\mathrm{max}}=\tilde{\lambda}_2$ (for every $s$), where
$\tilde{\lambda}_2$ is the value in Lemma
\ref{lem:particular-values}(ii); in fact, if we consider the
function $f_2$ in that lemma, we have
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{f_2(\lambda)}{s^2}$.
\paragraph{MM versus MQ estimators.}
The MM estimators are well-defined in all cases except the one
with Cauchy distribution. Moreover, by taking into account Remark
\ref{rem:goodness-hypothesis}, we can always refer to the
comparison between $J_{\theta_0}^{\prime\prime}(\theta_0)$ and
$I_{\lambda_{\mathrm{max}},\theta_0}^{\prime\prime}(\theta_0)$.\\
\underline{Normal distribution}. In this case
$\mu(\theta):=\theta$. Thus, by Proposition
\ref{prop:LD-for-MM-estimators}(i), $\{\mu^{-1}(\bar{X}_n):n\geq
1\}=\{\bar{X}_n:n\geq 1\}$ satisfies the LDP with rate function
$J_{\theta_0}$ defined by
$$J_{\theta_0}(\theta)=\Lambda_{\theta_0}^*(\theta)=\frac{(\theta-\theta_0)^2}{2s^2}$$
($J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ coincide noting that
$c_1=1$ and $c_0=0$). Then, since $\lambda_{\mathrm{max}}=1/2$ we
have to compare
$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{1}{s^2}$ (which meets
the expression provided by Proposition
\ref{prop:second-derivative-J} noting that $\sigma^2(\theta)=s^2$)
and $I_{1/2,\theta_0}^{\prime\prime}(\theta_0)=\frac{2}{\pi s^2}$
and, obviously, we have
$J_{\theta_0}^{\prime\prime}(\theta_0)>I_{1/2,\theta_0}^{\prime\prime}(\theta_0)$
(for every $s$). Thus MM estimators converge faster than every MQ
estimators; in some sense we already expected this noting that the
MM estimators coincide with the ML estimators.\\
\underline{Logistic distribution}. In this case
$\mu(\theta):=\theta$. Thus, by Proposition
\ref{prop:LD-for-MM-estimators}(i), $\{\mu^{-1}(\bar{X}_n):n\geq
1\}=\{\bar{X}_n:n\geq 1\}$ satisfies the LDP with rate function
$J_{\theta_0}$, which coincides with $\Lambda_{\theta_0}^*$ (we
have again $c_1=1$ and $c_0=0$); in this case we cannot provide an
explicit expression of $\Lambda_{\theta_0}^*$. Then, since
$\lambda_{\mathrm{max}}=1/2$ we have to compare
$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{3}{\pi^2s^2}$ (this
is a consequence of Proposition \ref{prop:second-derivative-J}
noting that $\sigma^2(\theta)=\frac{\pi^2s^2}{3}$) and
$I_{1/2,\theta_0}^{\prime\prime}(\theta_0)=\frac{1}{4s^2}$ and,
obviously, we have
$J_{\theta_0}^{\prime\prime}(\theta_0)>I_{1/2,\theta_0}^{\prime\prime}(\theta_0)$
(for every $s$). Thus MM estimators converge faster than every MQ
estimators but, differently from what happens for the case with
Normal distribution, they do not coincide with ML estimators.\\
\underline{Gumbel distribution}. In this case
$\mu(\theta):=\theta+s\gamma_*$, where $\gamma_*$ is the Euler's
constant. Thus, by Proposition \ref{prop:LD-for-MM-estimators}(i),
$\{\mu^{-1}(\bar{X}_n):n\geq 1\}=\{\bar{X}_n:n\geq 1\}$ satisfies
the LDP with rate function $J_{\theta_0}$ (with $c_1=1$ and
$c_0=-s\gamma_*$); in this case we cannot provide an explicit
expression of $\Lambda_{\theta_0}^*$. The rate functions
$J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ are good and we can
refer to the comparison between
$$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{6}{\pi^2s^2}$$
(this value is a consequence of Proposition
\ref{prop:second-derivative-J} noting that
$\sigma^2(\theta)=\frac{\pi^2s^2}{6}$) and, for the optimal value
$\lambda_{\mathrm{max}}=\tilde{\lambda}_2$ defined in Lemma
\ref{lem:particular-values}(ii),
$$I_{\tilde{\lambda}_2,\theta_0}^{\prime\prime}(\theta_0)=\frac{\tilde{\lambda}_2(\log\tilde{\lambda}_2)^2}{s^2(1-\tilde{\lambda}_2)}
=\frac{\tilde{\lambda}_2(2\tilde{\lambda}_2-2)^2}{s^2(1-\tilde{\lambda}_2)}=\frac{4\tilde{\lambda}_2(1-\tilde{\lambda}_2)}{s^2}.$$
We can check numerically that
$I_{\tilde{\lambda}_2,\theta_0}^{\prime\prime}(\theta_0)>J_{\theta_0}^{\prime\prime}(\theta_0)$
(for every $s$); in fact we have
$4\tilde{\lambda}_2(1-\tilde{\lambda}_2)\simeq 0.6476$ (we get a
numerical value obtained for the statistical model with Weibull
distributions because
$4\tilde{\lambda}_2(1-\tilde{\lambda}_2)=4\tilde{\lambda}_1(1-\tilde{\lambda}_1)$
by Lemma \ref{lem:particular-values}(iii)) and
$\frac{6}{\pi^2}\simeq 0.6079$. Thus MQ estimators with the
optimal value $\lambda_{\mathrm{max}}$ converge faster than MM
estimators.
\subsection{Example \ref{ex:skew-parameter}}\label{sub:example3}
Here we analyze the MQ estimators for the specific examples listed
above. In all cases we can only give numerical values; such values
depend on the unknown parameter $\theta_0$, and therefore we do
not discuss the comparison with the MM estimators (as we do for
the other examples). We have the same feature highlighted for
Example \ref{ex:scale-parameter} with
$(\alpha,\omega)=(-\infty,\infty)$, namely the more the tail of
the distributions are light, the more the numerical values of
$\lambda_{\mathrm{max}}$ are distant from $\lambda=1/2$.\\
\underline{Normal distribution} (namely $G$ in
\eqref{eq:df-Normal-location-parameter}). By
\eqref{eq:second-derivative-skew-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\left\{\begin{array}{ll}
\frac{1}{\lambda(1-\lambda)}\left[\varphi(\Phi^{-1}(\frac{\lambda}{1+\theta_0}))\Phi^{-1}(\frac{\lambda}{1+\theta_0})-\frac{\lambda}{1+\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta_0}{2}]\\
\frac{1}{\lambda(1-\lambda)}\left[-\varphi(\Phi^{-1}(\frac{\lambda-\theta_0}{1-\theta_0}))\Phi^{-1}(\frac{\lambda-\theta_0}{1-\theta_0})
+\frac{\lambda-1}{1-\theta_0}\right]^2 &\ \mbox{for}\
\lambda\in(\frac{1+\theta_0}{2},1).
\end{array}\right.$$
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\theta_0=0&\ 0.15\ (\mbox{and}\ 0.85\ \mbox{by symmetry})\\
\theta_0=1/2&\ 0.94\\
\theta_0=-1/2&\ 1-0.94=0.06\ (\mbox{by symmetry})
\end{array}\right.$$
\underline{Cauchy distribution} (namely $G$ in
\eqref{eq:df-Cauchy-location-parameter}). By
\eqref{eq:second-derivative-skew-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\left\{\begin{array}{ll}
\frac{1}{\lambda(1-\lambda)}\left[\frac{\tan((\frac{\lambda}{1+\theta_0}-\frac{1}{2})\pi)}
{\pi\left(1+\tan^2((\frac{\lambda}{1+\theta_0}-\frac{1}{2})\pi)\right)}-\frac{\lambda}{1+\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta_0}{2}]\\
\frac{1}{\lambda(1-\lambda)}\left[-\frac{\tan((\frac{\lambda-\theta_0}{1-\theta_0}-\frac{1}{2})\pi)}
{\pi\left(1+\tan^2((\frac{\lambda-\theta_0}{1-\theta_0}-\frac{1}{2})\pi)\right)}+\frac{\lambda-1}{1-\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(\frac{1+\theta_0}{2},1).
\end{array}\right.$$
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\theta_0=0&\ 0.39\ (\mbox{and}\ 0.61\ \mbox{by symmetry})\\
\theta_0=1/2&\ 0.84\\
\theta_0=-1/2&\ 1-0.84=0.16\ (\mbox{by symmetry})
\end{array}\right.$$
\underline{Logistic distribution} (namely $G$ in
\eqref{eq:df-logistic-location-parameter}). By
\eqref{eq:second-derivative-skew-parameter} we have
$$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\left\{\begin{array}{ll}
\frac{1}{\lambda(1-\lambda)}\left[-\frac{\lambda}{1+\theta_0}(1-\frac{\lambda}{1+\theta_0})\log(\frac{1+\theta_0}{\lambda}-1)-\frac{\lambda}{1+\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(0,\frac{1+\theta_0}{2}]\\
\frac{1}{\lambda(1-\lambda)}\left[\frac{\lambda-\theta_0}{1-\theta_0}(1-\frac{\lambda-\theta_0}{1-\theta_0})
\log(\frac{1-\theta_0}{\lambda-\theta_0}-1)+\frac{\lambda-1}{1-\theta_0}\right]^2
&\ \mbox{for}\ \lambda\in(\frac{1+\theta_0}{2},1).
\end{array}\right.$$
Moreover
$$\left.\begin{array}{lc}
&\ \mbox{numerical values for}\ \lambda_{\mathrm{max}}\\
\theta_0=0&\ 0.22\ (\mbox{and}\ 0.78\ \mbox{by symmetry})\\
\theta_0=1/2&\ 0.92\\
\theta_0=-1/2&\ 1-0.92=0.08\ (\mbox{by symmetry})
\end{array}\right.$$
\subsection{Example \ref{ex:Pareto-distributions}}\label{sub:example4}
Here we analyze Example \ref{ex:Pareto-distributions}. For MQ
estimators we have the same rate function presented in Example
\ref{ex:scale-parameter} with $(\alpha,\omega)=(0,\infty)$ when
$G$ is as in \eqref{eq:df-Weibull} and $\rho=1$. Thus we have a
unique optimal $\lambda_{\mathrm{max}}$ which does not depend on
$\theta_0$, namely $\lambda_{\mathrm{max}}=\tilde{\lambda}_1$
where $\tilde{\lambda}_1$ is defined in Lemma
\ref{lem:particular-values}(i).
Now we briefly discuss the MM estimators for Example
\ref{ex:Pareto-distributions}. We recall that $\mu(\theta)$ is
finite only if $\theta\in\tilde{\Theta}:=(0,1)$, where
$\tilde{\Theta}\subset\Theta=(0,\infty)$. So we could consider the
mean function on the restricted parameter space $\tilde{\Theta}$,
i.e.
$$\mu(\theta)=\frac{1/\theta}{1/\theta-1}=\frac{1}{1-\theta}\ \mbox{for}\ \theta\in\tilde{\Theta}.$$
Then, if we consider the restricted parameter space
$\tilde{\Theta}$, the MM estimators
$\left\{\mu^{-1}(\bar{X}_n):n\geq 1\right\}$ are defined by
$\mu^{-1}(\bar{X}_n)=1-\bar{X}_n^{-1}$, and the function
$\mu^{-1}(\cdot)$ is continuous on $(\alpha,\omega)=(1,\infty)$.
Unfortunately we cannot apply Proposition
\ref{prop:LD-for-MM-estimators} because we cannot consider neither
the hypotheses of Proposition \ref{prop:LD-for-MM-estimators}(i)
(obvious) nor the hypotheses of Proposition
\ref{prop:LD-for-MM-estimators}(ii) because Pareto distributions
are heavy-tailed and $\Lambda_{\theta_0}^*$ is not good (see
Remark \ref{rem:goodness-hypothesis}).
\subsection{Example \ref{ex:right-endpoint-parameter}}\label{sub:example5}
Here we analyze Example \ref{ex:right-endpoint-parameter}. As far
as the MQ estimators are concerned, we can say that we cannot find
an optimal $\lambda$ because
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ is an increasing
function; in fact, by
\eqref{eq:second-derivative-right-endpoint-parameter}, the
derivative of $I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ with
respect to $\lambda$ is
$$\frac{d}{d\lambda}I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{(G^\prime(\theta_0))^2}{(1-\lambda)^2G^2(\theta_0)}.$$
We can also say that the larger is $\lambda$ the faster is the
convergence of the MQ estimators.
In the remaining part we deal with as the MM estimators, and their
comparison with the MQ estimators. Obviously the rate functions
$J_{\theta_0}$ and $\Lambda_{\theta_0}^*$ are good (we take into
account Remark \ref{rem:goodness-hypothesis}) and we can refer to
the comparison between $J_{\theta_0}^{\prime\prime}(\theta_0)$ and
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ (for
$\lambda\in(0,1)$); for completeness we remark that we cannot
obtain an explicit expression of $\Lambda_{\theta_0}^*$ (even for
the simplest case with the uniform distributions, i.e. the case
$G(x)=x$ for all $x\in(0,\infty)$). It is easy to check that, if
we consider $\lambda_0$ defined by
\begin{equation}\label{eq:threshold-value-lambda-righ-endpoint}
\lambda_0:=\frac{(\mu^\prime(\theta_0))^2G^2(\theta_0)}
{(\mu^\prime(\theta_0))^2G^2(\theta_0)+\sigma^2(\theta_0)(G^\prime(\theta_0))^2},
\end{equation}
we have
$$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{(\mu^\prime(\theta_0))^2}{\sigma^2(\theta_0)}
>\frac{\lambda(G^\prime(\theta_0))^2}{(1-\lambda)G^2(\theta_0)}=I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)\ \mbox{for}\ \lambda<\lambda_0$$
and
$$J_{\theta_0}^{\prime\prime}(\theta_0)=\frac{(\mu^\prime(\theta_0))^2}{\sigma^2(\theta_0)}
<\frac{\lambda(G^\prime(\theta_0))^2}{(1-\lambda)G^2(\theta_0)}=I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)\
\mbox{for}\ \lambda>\lambda_0$$ by Proposition
\ref{prop:second-derivative-J} and
\eqref{eq:second-derivative-right-endpoint-parameter}.
Thus the MQ estimators converge faster than MM estimators if
$\lambda$ is close to 1; this is not surprising because the case
$\lambda=1$ concerns the case of ML estimators $\{X_{n:n}:n\geq
1\}$. For completeness we can say that, if $\{X_n:n\geq 1\}$ are
i.i.d. with distribution function $F_{\theta_0}$ as in Example
\ref{ex:right-endpoint-parameter}, the LDP in Proposition
\ref{prop:Theorem3.2-in-HMP-restricted} with $\lambda=1$ is
governed by a good rate function; thus we can consider a version
of Proposition \ref{prop:LD-for-MQ-estimators} with $\lambda=1$,
and we have the LDP of $\{X_{n:n}:n\geq 1\}$ with good rate
function $I_{1,\theta_0}$ defined by
$$I_{1,\theta_0}(\theta):=\left\{\begin{array}{ll}
\log\frac{1}{h_{\lambda,\theta_0}(\theta)}=\log\frac{G(\theta_0)}{G(\theta)}
&\ \mbox{for}\ \theta\in\Theta\ \mbox{such that}\ \theta\in(0,\theta_0)\\
\infty&\ \mbox{otherwise}.
\end{array}\right.$$
We also remark that in general the threshold value $\lambda_0$ in
\eqref{eq:threshold-value-lambda-righ-endpoint} depends on
$\theta_0$. In fact, for $G(x):=e^x-1$, after some computations we
have $\mu(\theta)=\frac{\theta e^\theta}{e^\theta-1}-1$,
$\sigma^2(\theta)=\frac{(e^\theta-1)^2-\theta^2e^\theta}{(e^\theta-1)^2}$,
and therefore
$$\lambda_0=\frac{e^{2\theta_0}-2e^{\theta_0}(1+\theta_0)+(1+\theta_0)^2}
{2e^{2\theta_0}-e^{\theta_0}(4+2\theta_0+\theta_0^2)+2+2\theta_0+\theta_0^2}.$$
Interestingly we can say that $\lambda_0$ does not depend on
$\theta_0$ if $G(x):=x^y$ for some $y>0$; in fact we have
$\mu(\theta)=\frac{y\theta}{y+1}$,
$\sigma^2(\theta)=\frac{y\theta^2}{(y+2)(y+1)^2}$, and therefore
$$\lambda_0=\frac{(\frac{y}{y+1})^2(\theta_0^y)^2}{(\frac{y}{y+1})^2(\theta_0^y)^2+\frac{y\theta_0^2}{(y+2)(y+1)^2}(y\theta_0^{y-1})^2}
=\frac{1}{1+\frac{y}{y+2}}=\frac{y+2}{2y+2}.$$ For instance, for
the specific case of uniform distributions cited in Example
\ref{ex:right-endpoint-parameter} (for which we have
$\mu(\theta)=\frac{\theta}{2}$ and
$\sigma^2(\theta)=\frac{\theta^2}{12}$ for all
$\theta\in(0,\infty)$; so the sequence
$\left\{\mu^{-1}(\bar{X}_n):n\geq 1\right\}$ in Proposition
\ref{prop:LD-for-MM-estimators} is defined by
$\mu^{-1}(\bar{X}_n)=2\bar{X}_n$) we have $G(x):=x$, and therefore
we get $\lambda_0=3/4$ by setting $y=1$.
Finally we remark that, in general, we cannot find $\delta>0$ such
that $I_{\lambda_0,\theta_0}(\theta)>J_{\theta_0}(\theta)$ or
$I_{\lambda_0,\theta_0}(\theta)<J_{\theta_0}(\theta)$ for
$0<|\theta-\theta_0|<\delta$; for instance (see Figure \ref{fig2}
where $\theta_0=1$) this happens for the statistical model with
uniform distributions cited above (where $G(x):=x$ and
$\lambda_0=3/4$).
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=0.5\textwidth]{Uniform_theta.pdf}
\caption{The rate functions $I_{3/4,\theta_0}(\theta)$ (broken
line) and $J_{\theta_0}(\theta)$ (solid line) in a neighborhood of
$\theta_0=1$ for the statistical model with uniform
distributions.}\label{fig2}
\end{center}
\end{figure}
\appendix
\section{A class of counterexamples}\label{appendix}
In Section \ref{sub:example2}, for all the examples where the
distribution $G$ is symmetric, we find that
$G^{\prime\prime}(0)=0$ and that there is a unique optimal value
$\lambda_{\mathrm{max}}$, namely $\lambda_{\mathrm{max}}=0.5$.
Here we show that this is not necessarily the case, indeed we
present a procedure to construct another function $\tilde{G}$ with
the symmetry property and such that
$\tilde{G}^{\prime\prime}(0)=0$; this function will be determined
starting from a function $G$ with the properties cited above (for
instance it could be one of the choices illustrated in Example
\ref{ex:location-parameter} except the one concerning Gumbel
distribution). The aim is to illustrate that, for such a function
$\tilde{G}$, $\lambda=0.5$ cannot be an optimal value.
The function $\tilde{G}$ is defined by
$$\tilde{G}(x):=\left\{\begin{array}{ll}
\frac{G(1+x)}{2G^\prime(0)+1}&\ \mbox{for}\ x\leq -1\\
\frac{1}{2}+\frac{G^\prime(0)x}{2G^\prime(0)+1}&\ \mbox{for}\ |x|<1\\
\frac{2G^\prime(0)+G(x-1)}{2G^\prime(0)+1}&\ \mbox{for}\ x\geq 1.
\end{array}\right.$$
One can check that, if $G$ is twice differentiable, then
$\tilde{G}$ is also twice differentiable (and in particular the
condition $G^{\prime\prime}(0)=0$ is needed to say that
$\tilde{G}$ is twice differentiable); the details are omitted.
Moreover, if we consider
$$G(-1)=\frac{1}{2(2G^\prime(0)+1)}\ \mbox{and}\ G(1)=\frac{4G^\prime(0)+1}{2(2G^\prime(0)+1)}$$
(we recall that $G(0)=\frac{1}{2}$ by the symmetry property of
$G$), we have
$$\tilde{G}^{-1}(\lambda):=\left\{\begin{array}{ll}
G^{-1}(\lambda(2G^\prime(0)+1))-1&\ \mbox{for}\ \lambda\leq G(-1)\\
(\lambda-\frac{1}{2})\frac{2G^\prime(0)+1}{G^\prime(0)}&\ \mbox{for}\ G(-1)<\lambda<G(1)\\
G^{-1}(\lambda(2G^\prime(0)+1)-2G^\prime(0))+1&\ \mbox{for}\
\lambda\geq G(1).
\end{array}\right.$$
Then, around $\lambda=1/2$ (more precisely for
$\lambda\in(G(-1),G(1))$ because $G(-1)\in(0,\frac{1}{2})$ and
$G(1)\in(\frac{1}{2},1)$), we have
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)=\frac{1}{\lambda(1-\lambda)}\left(\frac{G^\prime(0)}{2G^\prime(0)+1}\right)^2$
by \eqref{eq:second-derivative-location-parameter}; thus
$\lambda=0.5$ cannot be an optimal value because
$I_{\lambda,\theta_0}^{\prime\prime}(\theta_0)$ is locally
minimized at $\lambda=1/2$ (in fact $\lambda=1/2$ maximizes the
denominator $\lambda(1-\lambda)$). | {"config": "arxiv", "file": "1611.04765/ldmq_vbcmlp.tex"} |
TITLE: Evaluating closed form of $I_n=\int_0^{\pi/2} \underbrace{\cos(\cos(\dots(\cos}_{n \text{ times}}(x))\dots))~dx$ for all $n\in \mathbb{N}$.
QUESTION [28 upvotes]: I was wondering if there is any way to evaluate a general closed form solution to the following integral for all $n\in \mathbb{N}$.
$$I_n=\int_0^{\pi/2} \underbrace{\cos(\cos(\cos(\dots(\cos}_{n \text{ times}}(x))\dots)))~dx \tag{1}$$
I have already evaluated closed forms to this integral for certain values of $n$, however I am still missing a closed form for a large number of values of $n$. Those in $\color{red}{\text{red}}$ I numerically evaluated, meaning that I currently do not have a closed form for them.
$$\begin{array}{c|c}n&I_n\\\hline0&\dfrac{\pi^2}{8}\\1&1\\2&\dfrac{\pi J_0(1)}{2}\\\color{red}{3}&\color{red}{\approx 1.11805}\\\color{red}{4}&\color{red}{\approx 1.18186} \\ \color{red}{5} & \color{red}{\approx 1.14376} \\ \color{red}{6}&\color{red}{\approx 1.17102}\\\color{red}{\vdots}&\color{red}{\vdots}\\\infty&\alpha\cdot \dfrac{\pi}{2} \approx 1.16095\end{array}$$
Where $J_p(\cdot )$ is the Bessel function of the first kind, and $\alpha$ is the Dottie Number. The cases $n=0$ and $n=1$ are trivial, hence I will not show how I derived these solutions. Hence, I will show how I derived the case where $n=2$ and $n\to \infty$.
Evaluating $I_2$: i.e $\int_0^{\pi/2} \cos(\cos(x))~dx$.
Introducing the definition of the Bessel Function of the first kind:
$$J_{\beta}(z)=\frac{1}{\pi}\int_0^{\pi} \cos(z\sin{\theta}-\beta \theta)~d\theta$$
We can use the substitution $\theta=u+\frac{\pi}{2}$ to obtain:
$$\begin{align} J_{\beta}(z)&=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(z\sin\left(u+\frac{\pi}{2}\right)-\beta\left(u+\frac{\pi}{2}\right)\right)~du\\&=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(z\cos(u)-\beta\left(u+\frac{\pi}{2}\right)\right)~du \end{align}$$
To get it into a similar form to our case, notice that we can let $\beta=0$ and $z=1$. Therefore:
$$J_0(1)=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos(\cos(u))~du$$
At first sight, it may seem like the bounds are problematic. However, note that $f(x)=\cos(\cos(x))$ is an even function, hence we know that:
$$J_0(1)=\frac{2}{\pi}\int_0^{\pi/2} \cos(\cos(u))~du \iff \int_0^{\pi/2} \cos(\cos(u))~du=\frac{\pi J_0(1)}{2}$$
Evaluating $\lim\limits_{n\to \infty} I_n$:
I realized that as $n\to \infty$, the integrand will converge to a constant function for all $x\in \mathbb{R}$, as shown below. The blue, yellow, green and red curves is when $n=1,2,5,10$ respectively.
I figured that we can represent the repeated composition of functions by the following recurrence $x_{n+1}=\cos(x_n)$. Using the principles of fixed point iteration, we know thus know that the value it tends to is the unique solution to $x=\cos(x)$. This turns out to be the Dottie Number, which I evaluated numerically using the Newton-Raphson Method and denoted this value by $\alpha$. I obtained:
$$\alpha\approx 0.739085133215161$$
Hence:
$$\lim_{n\to \infty} I_n=\int_0^{\pi/2}\cos(\cos(\cos(\dots(\cos(x))\dots)))~dx=\alpha\cdot \frac{\pi}{2} \approx 1.160952212443092$$
As mentioned, I am unsure how to evaluate closed forms for the cases when $n\geq 3$. I've checked some other definitions such as the Struve function $\mathbf{H}_{\gamma}(\cdot)$, though it only seems to be useful when evaluating $\int_0^{\pi/2} \sin(\sin(x))~dx$, which is not the integral we are considering. Hence, I would appreciate some guidance on how to evaluate a general closed form for $(1)$ for all $n\in \mathbb{N}$, if possible.
REPLY [8 votes]: The mere fact that even the simple case $n=2$ ceases to possess a meaningful closed form in terms of elementary functions, and an entirely new function had to be invented from scratch in order to express its value, should be enough to settle all questions one might have concerning the possibility of finding such a form for larger values of the argument. Indeed, the very next case, $n=3,$ is not known to be expressible even in terms of special functions. That $\cos^{[\infty]}(x)$ is a constant $($since the function is both bound and monotone$)$ certainly constitutes a blessing, but such clearly does not hold for finite values of the iterator. $($ See also Liouville's theorem and the Risch algorithm $).$ | {"set_name": "stack_exchange", "score": 28, "question_id": 2295034} |
TITLE: PDE Using Fourier Series
QUESTION [0 upvotes]: I'm trying to find the solution to(I don't need to find the coefficient):
$v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$
$v(0, t) = 0$
$v_x(l, t) = 0$
$v(x, 0) = −U$
Where U is a constant
This the answer that I get:
$$v(x, t) = \sum_{n=0}^{\infty}A_n e^{-(\frac{n\pi}{l})^2kt}
sin(\frac{nπx}{l})$$
which comes directly from separating the variables of $v(x,t)$
But the correct answer is this:
$$v(x, t) = \sum_{n=0}^{\infty}A_n e^{-(\frac{(n+1/2)\pi}{l})^2kt}
sin(\frac{(n + 1/2)πx}{l})$$
why do I have to add the $\frac{1}{2}$ to $n$?
REPLY [0 votes]: Your solution doesn't satisfy the second boundary condition. Your solution satisfies the boundary conditions $$v(0,t) = 0 \,\,\,\,\, \text{ and } \,\,\,\,\, v(l,t) = 0, $$ whereas you need it to satisfy $$v(0,t) = 0 \,\,\,\,\, \text{ and } \,\,\,\,\, \frac{dv}{dx}(l,t) = 0.$$ Adding the $1/2$ accomplishes that. | {"set_name": "stack_exchange", "score": 0, "question_id": 1691207} |
TITLE: The sequence $\left\{ \frac{5}{n} \right\}_{n=1}^\infty$ is not monotonic, or not convergent, or bounded. Why?
QUESTION [1 upvotes]: I recently watched a video where a Calculus instructor asked the viewers to try to come up with a monotonic, unbounded, convergent sequence, if they could, as an exercise to understand these concepts better.
I came up with $$\{a_n\}_{n=1}^\infty = \left\{ \frac{5}{n} \right\}_{n=1}^\infty$$
I claim it is monotonic because because the sequence is decreasing.
I claim it is convergent because as $n \to \infty, \, a_n \to 0$.
I claim it is unbounded because for every one of its members, we can always find a smaller one.
However, right after the exercise, we were presented with the following theorem:
If a sequence is convergent, then it is bounded.
Hence my solution must be wrong, and at least one of my claims must be false. But I'm not sure which one, or why. Is anyone able to shed some light on this?
REPLY [1 votes]: Your 3rd claim is wrong as all above have suggested. Also, the reasoning would follow this order:
the sequence is monotonically decreasing
the sequence is bounded above by $5$ and below by $0$
hence by monotone-bounded theorem,the sequence converges to a real number,in this case to $0$. | {"set_name": "stack_exchange", "score": 1, "question_id": 3307912} |
\begin{document}
\begin{center}
{\LARGE {Large Deviations Theorems in Nonparametric Regression on Functional Data}}
\end{center}
\begin{center}
\large{Mohamed Cherfi}\vspace{3mm}
{\small \it L.S.T.A., Universit\'e Pierre et Marie Curie. 175, rue du Chevaleret,
8{\`e}me
{\'e}tage, b\^atiment A,\\
75013
PARIS FRANCE.}
\end{center}
\renewcommand{\thefootnote}{}
\footnote{ \vspace*{-4mm} \noindent{{\it E-mail address:}
[email protected]} }
\vspace{3mm} \hrule \vspace{3mm} {\small \noindent{\bf Abstract.}
\\
In this paper we prove large deviations principles for the
Nadaraya-Watson estimator of the regression of a real-valued variable with a functional covariate. Under suitable conditions, we show pointwise and uniform large deviations theorems with good rate functions.\\
\vspace{0.2cm}
\noindent{\small {\it AMS Subject Classifications}: primary 60F10; 62G08}
\noindent{\small {\it Keywords}: Large deviation; nonparametric regression ; functional data. }
\vspace{4mm}\hrule
\section{Introduction}
Let $\{(Y_i,\,X_i),~i\geq 1\}$ be a sequence of independent and identically distributed
random vectors. The random variables $Y_i$ are real, with $\mathds{E}|Y|<\infty$, and the $X_i$ are random vectors with values in a semi-metric space $(\mathcal{X},d(\cdot,\cdot))$.
Consider now the functional regression model,
\begin{equation}\label{eqReg}
Y_i:=\mathds{E}(Y|X_i)+\varepsilon_i=r(X_i)+\varepsilon_i \quad i=1,\ldots,n,
\end{equation}
where $r$ is the regression operator mapping $\mathcal{X}$ onto $\mathbb{R}$, and the $\varepsilon_i$ are real variables such that, for all
$i$, $\mathds{E}(\varepsilon_i|X_i ) = 0$ and $\mathds{E}(\varepsilon_i^2|X_i ) =\sigma^2_{\varepsilon}(X_i ) < \infty$. Note that in practice $\mathcal{X}$ is a normed
space which can be of infinite dimension (e.g., Hilbert or Banach space) with norm $\|\cdot\|$ so that
$d(x, x') = \|x - x'\|$, which is the case in this paper.
\noindent \cite{FV2004} provided a consistent estimate for the nonlinear regression operator $r$, based on the usual finite-dimensional smoothing ideas, that is
\begin{equation}\label{eqNW}
\displaystyle{\widehat{r}_n(x):=\frac{\sum_{i=1}^n Y_iK\bigg(\frac{\|x-X_i\|}{h_n}\bigg)}{\sum_{i=1}^nK\bigg(\frac{\|X_i-x\|}{h_n}\bigg)}},
\end{equation}
where $K(\cdot)$ is a real-valued kernel and $h_n$ the bandwidth, is a sequence of positive real numbers converging to $0$ as $n\longrightarrow\infty$. Note that the bandwidth $h_n$ depends on $n$, but we drop this index for simplicity. In what follows $K_{h}(u)$ stands for $\displaystyle{K\bigg(\frac{u}{h}\bigg)}$. The estimator defined in (\ref{eqNW}) is a generalization to the functional framework of the classical Nadaraya-Watson regression estimator. The asymptotic properties of this estimate have been studied extensively by several authors, we cite among others \cite{FMV2007}, for a complete survey see the monograph by \cite{FV2006}.
\noindent The large deviations behavior of the Nadaraya-Watson estimate of the regression function, have been studied at first by
\cite{Louanib}, sharp results have been obtained by \cite{joutard} in the univariate framework. In the multidimensional case \cite{Mok2} obtained pointwise large and moderate deviations results for the Nadaraya-Watson and recursive kernel estimators of the regression.
In this work, we are interested in the problem of establishing large deviations principles of the regression operator estimate $\widehat{r}_n(\cdot)$. The results stated in the paper deal with pointwise and uniform large deviations probabilities of $\widehat{r}_n(\cdot)$ from $r(\cdot)$. The organization of the paper is as follows, in Section 2 we will state pointwise and uniform large deviations results. The proofs are given in section 3.
\section{Results}
Let $F_x(h)=P[\|X_i-x\|\leq h]$, be the cumulative distribution of the real variable $W_i=\|X_i-x\|$. As in \cite{FMV2007}, let $\varphi$ be the real valued function defined by
\begin{equation}\label{varphi}
\varphi(u)=\mathds{E}\big\{r(X)-r(x)\big|\|X-x\|=u\big\}.
\end{equation}
Before stating our results, we will consider the following conditions.
\begin{enumerate}
\item[(C.1)]The kernel $K$ is positive, with compact support $[0,1]$ of class $\mathcal{C}^1$ on $[0,1)$, $K(1)>0$ and its derivative $K'$ exists on $[0,1)$ and $K'(u)<0$.
\item[(C.2)]$K$ is Lipschitz.
\item[(C.3)]The operator $r$ verifies the following Lipschitz property:
\begin{equation}\label{LR}
\forall (u,v)\in\mathcal{X}^2,\,\exists C,\,|r(u)-r(v)|\leq C \|u-v\|^{\beta}
\end{equation}
\item[(C.4)]There exist three functions $\ell(\cdot)$, $\phi(\cdot)$ (supposed increasing and strictly positive and tending to zero as $h$ goes to zero) and $\zeta_0(\cdot)$ such that
\begin{enumerate}
\item[(i)]$F_x(h)=\ell(x)\phi(h)+o(\phi(h)),$
\item[(ii)]for all $u\in[0,1],$ $\lim_{h\rightarrow0}\frac{\phi(uh)}{\phi(h)}=:\lim_{h\rightarrow0}\zeta_h(u)=\zeta_0(u).$
\end{enumerate}
\item[C.5] $\varphi'(0)$ exists.
\end{enumerate}
There exist many examples fulfilling the decomposition mentioned in condition ({\rm{C.4}}), see for instance Proposition 1 in \cite{FMV2007}. The conditions stated above are classical in nonparametric estimation for functional data, see for instance \cite{FMV2007} and references cited therein.
\noindent Let now introduce the following functions,
\begin{equation}\label{I}
I(t)=\exp\{-t\lambda K(1)\}-1+t\lambda \int_0^1K'(u)\exp\{ -t\lambda K(u)\}\zeta_0(u)~{\rm{d}}u;
\end{equation}
$\displaystyle{\Gamma_x^+(\lambda)=\inf_{t>0}\{\ell(x)I(t)\}}$; $\displaystyle{\Gamma_x^-(\lambda)=\inf_{t>0}\{\ell(x)I(-t)\}}$ and $\displaystyle{\Gamma_x(\lambda)=\max\{\Gamma_x^+(\lambda);\Gamma_x^-(\lambda)\}}$.
\noindent Let $x$ be a an element of the functional space $\mathcal{X}$ and $\lambda>0$. Our first theorem deals with pointwise large deviations probabilities.
\newtheorem{theorem1}{Theorem}
\begin{theorem1}\label{theorem1}
Assume that the conditions {\rm{(C.1)--(C.5)}} are satisfied. If $n\phi(h)\longrightarrow\infty$, then for any $\lambda>0$ and any $x\in\mathcal{X}$, we have
\begin{enumerate}
\item[{\rm{(a)}}]
\begin{equation}\label{eq1Th1}
\lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\widehat{r}_n(x)-r(x)>\lambda\big)=\Gamma_x^+(\lambda)
\end{equation}
\item[{\rm{(b)}}]\begin{equation}\label{eq2Th1}
\lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\widehat{r}_n(x)-r(x)<-\lambda\big)=\Gamma_x^-(\lambda)
\end{equation}
\item[{\rm{(c)}}]\begin{equation}\label{eq3Th1}
\lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(|\widehat{r}_n(x)-r(x)|>\lambda\big)=\Gamma_x(\lambda)
\end{equation}
\end{enumerate}
\end{theorem1}
To establish uniform large deviations principles for the regression estimator we need the following assumptions.
Let $\mathcal{C}$ be some compact subset of $\mathcal{X}$ and $B(z_k,\xi)$ a ball centered at $z_k\in\mathcal{X}$ with radius $\xi$, such that
for any $\xi>0$,
\begin{subequations}\label{Topo}
\begin{equation}
\mathcal{C}\subset\bigcup_{k=1}^{\tau}B(z_k,\xi),
\label{subeqnpart}
\end{equation}
\begin{equation}
\exists \alpha>0,\quad \exists C>0,\quad \tau\xi^{\alpha}=C.
\label{geometric}
\end{equation}
\end{subequations}
The above conditions on the covering of the compact set $\mathcal{C}$ by a finite number of balls, the geometric link between the number of balls $\tau$ and the radius $\xi$ are necessary to prove uniform convergence in the context of functional non-parametric regression and many functional non-parametric settings, see the discussion in \cite{FV2008}.
\noindent Before stating the Theorem about the uniform version of our result, we introduce the following function
\begin{equation}\label{g}
g(\lambda)=\sup_{x\in\mathcal{C}}\Gamma_x(\lambda).
\end{equation}
\newtheorem{theorem2}[theorem1]{Theorem}
\begin{theorem2}\label{theorem2}
Assume that the conditions {\rm{(C.1)--(C.5)}} are satisfied. If $n\phi(h)\longrightarrow\infty$, then for any compact set $\mathcal{C}\subset\mathcal{X}$ satisfying conditions (\ref{Topo}) and for any $\lambda>0$,
\begin{equation}\label{eq1Th2}
\lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)=g(\lambda)
\end{equation}
\end{theorem2}
\section{Proofs}
\subsection{Proof of Theorem \ref{theorem1}}
We only prove the statement (\ref{eq1Th1}), (\ref{eq2Th1}) is derived in the same way.
\noindent{\rm{(a)}} Write
$$Z_{n}= \sum_{i=1}^n \big\{Y_i-r(x)-\lambda\big\}K_h(\|X_i-x\|)$$
Define $\Phi_x^{(n)}(t):=\mathds{E}\exp(tZ_{n})$ to be the moment generating function of $Z_{n}$. To prove the large deviations principles, we seek the limit of $\displaystyle{\frac{1}{n\phi(h)}\log\Phi_x^{(n)}(t)}$ as $n\longrightarrow\infty$.
\noindent Observe that
\begin{equation*}
\Phi_x^{(n)}(t)=\bigg\{1+\mathds{E}\bigg(\exp\{t[r(X_1)-r(x)-\lambda]K_h(\|X_1-x\|)\}-1\bigg)\bigg\}^n.
\end{equation*}
Using the definition of the function $\varphi$ in (\ref{varphi}), we can write
\begin{eqnarray*}
\Phi_x^{(n)}(t)&=&\bigg\{1+\mathds{E}\bigg(\exp\{t[\varphi(\|X_1-x\|)-\lambda]K_h(\|X_1-x\|)\}-1\bigg)\bigg\}^n,\\
&=&\bigg\{1+\int_0^h\bigg(\exp\{ t[\varphi(u)-\lambda]K_h(u)\}-1\bigg)~{\rm{d}}F_x(u)\bigg\}^n.\\
&=&\bigg\{1+\int_0^1\bigg(\exp\{ t[\varphi(hu)-\lambda]K(u)\}-1\bigg)~{\rm{d}}F_x(hu)\bigg\}^n.\\
\end{eqnarray*}
By {\rm{(C.5)}}, using a first order Taylor expansion of $\varphi$ about zero, we obtain
\begin{eqnarray*}
\Phi_x^{(n)}(t)&=&\bigg\{1+\int_0^1\bigg(\exp\{ t[hu\varphi'(0)-\lambda+o(1)]K(u)\}-1\bigg)~{\rm{d}}F_x(hu)\bigg\}^n,\\
\end{eqnarray*}
Integrating by parts and by {\rm{(C.1)}}, we have
\begin{eqnarray*}
\Phi_x^{(n)}(t)&=&\bigg\{1+\big(\exp\{ t[h\varphi'(0)-\lambda]K(1)\}-1\big)F_x(h)\\
&-&\int_0^1t[h\varphi'(0)K(u)+K'(u)(uh\varphi'(0)-\lambda)]\\
&&\exp\{ t[hu\varphi'(0)-\lambda]K(u)\}F_x(hu)~{\rm{d}}u\bigg\}^n.\\
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\log\Phi_x^{(n)}(t)&=&n\log\bigg\{1+\big(\exp\{ t[h\varphi'(0)-\lambda]K(1)\}-1\big)F_x(h)\\
&-&\int_0^1t[h\varphi'(0)K(u)+K'(u)(uh\varphi'(0)-\lambda)]\\
&&\exp\{ t[hu\varphi'(0)-\lambda]K(u)\}F_x(hu)~{\rm{d}}u\bigg\}.\\
\end{eqnarray*}
Using Taylor expansion of $\log(1+v)$ about $v=0$, we obtain
\begin{eqnarray*}
\log\Phi_x^{(n)}(t)&=&n\bigg\{\big(\exp\{ t[h\varphi'(0)-\lambda]K(1)\}-1\big)F_x(h)\\
&-&\int_0^1t[h\varphi'(0)K(u)+K'(u)(uh\varphi'(0)-\lambda)]\\
&&\exp\{ t[hu\varphi'(0)-\lambda]K(u)\}F_x(hu)~{\rm{d}}u+O(h)\bigg\}.\\
\end{eqnarray*}
Hence, from Assumption {\rm{(C.4)}} (ii) it follows that
\begin{eqnarray*}
\lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log\Phi_x^{(n)}(t)&=&\ell(x)\bigg\{\exp\{-t\lambda K(1)\}-1+t\lambda \int_0^1K'(u)\exp\{ -t\lambda K(u)\}\zeta_0(u)~{\rm{d}}u\bigg\}\\
&=&\ell(x)I(t).
\end{eqnarray*}
Using Theorem of \cite{plachkysteinebach}, the proof of the theorem can be completed as in \cite{Louania}.
\noindent {\rm{(c)}} Observe that for any $x\in\mathcal{X}$,
\begin{equation*}
\max\{P(\widehat{r}_n(x)-r(x)>\lambda);\,P(\widehat{r}_n(x)-r(x)<-\lambda)\}\leq P(|\widehat{r}_n(x)-r(x)|>\lambda)
\end{equation*}
and
\begin{equation*}
P(|\widehat{r}_n(x)-r(x)|>\lambda)\leq 2\max\{P(\widehat{r}_n(x)-r(x)>\lambda);\,P(\widehat{r}_n(x)-r(x)<-\lambda)\}.
\end{equation*}
Hence,
\begin{equation*}
\Gamma_x(\lambda)\leq \lim_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P(|\widehat{r}_n(x)-r(x)|>\lambda)\leq \max\{\Gamma_x^{+}(\lambda);\,\Gamma_x^{-}(\lambda)\}=\Gamma_x(\lambda),
\end{equation*}
which complete the proof.
\subsection{Proof of Theorem \ref{theorem2}}
First for any $x_0\in\mathcal{C}$, by Theorem \ref{theorem1} we have
\begin{eqnarray*}
\liminf_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)&\geq & \liminf_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(|\widehat{r}_n(x_0)-r(x_0)|>\lambda\big)\\
&\geq& \Gamma_{x_0}(\lambda).
\end{eqnarray*}
Hence
\begin{equation}\label{borneinf}
\liminf_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)\geq g(\lambda).
\end{equation}
To prove the reverse inequality, we note that by conditions (\ref{Topo}) it follows
\begin{equation}\label{sup1}
\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|\leq\max_{1\leq k \leq \tau}\sup_{x\in B(z_k,\xi)}|\widehat{r}_n(x)-r(x)|.
\end{equation}
Hence,
\begin{equation}\label{sup2}
\sup_{x\in B(z_k,\xi)}|\widehat{r}_n(x)-r(x)|\leq \sup_{x\in B(z_k,\xi)}|\widehat{r}_n(x)-\widehat{r}_n(z_k)|+ \sup_{x\in B(z_k,\xi)}|r(z_k)-r(x)|+ |\widehat{r}_n(z_k)-r(z_k)|.
\end{equation}
Using the fact that
$K$ is Lipschitz by condition {\rm{(C.2)}}, there exists $C>0$ so that
$$\sup_{x\in B(z_k,\xi)}|\widehat{r}_n(x)-\widehat{r}_n(z_k)|\leq \frac{C\xi}{n\phi(h)h}\sum_{i=1}^n|Y_i|.$$
For $n$ sufficiently large, we choose $\xi$ according to the preassigned $\epsilon>0$, so that
\begin{equation}\label{sup3}
\sup_{x\in B(z_k,\xi)}|\widehat{r}_n(x)-\widehat{r}_n(z_k)|\leq \epsilon.
\end{equation}
Moreover, $r$ is Lipschitz, hence for a suitable choice of $\xi$
\begin{equation}\label{sup4}
\sup_{x\in B(z_k,\xi)}|r(z_k)-r(x)|\leq \epsilon .
\end{equation}
Finally, (\ref{sup1})-(\ref{sup4}) yield
\begin{equation}\label{sup5}
\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|\leq\max_{1\leq k \leq \tau}\big\{ |\widehat{r}_n(z_k)-r(z_k)|+2\epsilon\big\},
\end{equation}
which implies
$$P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)\leq\sum_{k=1}^{ \tau}P\big(|\widehat{r}_n(z_k)-r(z_k)|>\lambda-2\epsilon\big).$$
Thus,
$$P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)\leq\tau \max_{1\leq k\leq \tau}P\big(|\widehat{r}_n(z_k)-r(z_k)|>\lambda-2\epsilon\big).$$
It follows, by Theorem \ref{theorem1}, that
$$\limsup_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)\leq \inf_{t>0} \sup_{x\in\mathcal{C}}\ell(x)I_{\epsilon}(t),$$
where
$$I_{\epsilon}(t)=\exp\{-(t\lambda-2\epsilon) K(1)\}-1+t(\lambda-2\epsilon)\int_0^1K'(u)\exp\{ -t(\lambda-2\epsilon) K(u)\}\zeta_0(u)~{\rm{d}}u.$$
By continuity arguments, and the fact that
$$\inf_{t>0}\sup_{x\in\mathcal{C}}\ell(x)I(t)=\sup_{x\in\mathcal{C}}\inf_{t>0}\ell(x)I(t),$$
we obtain
\begin{equation}\label{bornesup}
\limsup_{n\rightarrow\infty}\frac{1}{n\phi(h)}\log P\big(\sup_{x\in\mathcal{C}}|\widehat{r}_n(x)-r(x)|>\lambda\big)\leq g(\lambda).
\end{equation}
\noindent Combining (\ref{borneinf}) and (\ref{bornesup}), we see that the limit exists which is $g(\lambda)$. | {"config": "arxiv", "file": "0902.3137.tex"} |
TITLE: When Should I Use Taylor Series for Limits?
QUESTION [3 upvotes]: I get confused between when to apply L'Hospital Rule and Taylor Series.
Is there any set of trigger points in the questions, that would be easier to solve with Taylor Series?
For Example, If the denominator is in terms of a large power of $x (>3)$, then L'Hospital Rule usually becomes complicated and is not advised.
Edit:
Solve
$\lim _{x\to 0}\left(\left(\sin x\right)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right)$
The options given are: $0, 1, -1, \infty$
REPLY [1 votes]: When you see, as $f\to0$, the following functions, it is often a good way to use Taylor series rather than L'Hospital rule.
$\log(1+f) = f+o(f)$
$\sin f = f+o(f)$
$\cos f = 1+ o(1)$
$e^f = 1+o(1)$
$\tan f = f+o(f)$
$\arctan f = f+o(f)$
(More terms of the series are sometimes necessary). | {"set_name": "stack_exchange", "score": 3, "question_id": 3032030} |
TITLE: Discretization of Newton's Equation with velocity perpendicular to gradient
QUESTION [2 upvotes]: Suppose I have a function $f:\mathbb{R}^n\to\mathbb{R}$ and $g(x)=\nabla_xf(x)$ and $H(x) = \nabla^2 f(x)$ are its gradient and Hessian matrix. I have the following ODE system
$$
\begin{align}
\dot x(t) &= v(t) \\
\dot v(t) &= - \frac{v(t)^\top H(x_t) v(t)}{||g(x(t))||^2} g(x(t))
\end{align}
$$
with initial condition $(x_0, v_0)$ with $v_0\perp g(x_0)$. Clearly $\frac{d}{dt} v(t)^\top g(x(t)) = 0$ so that the velocity stays perpendicular to the gradient $v_t\perp g(x(t))$. I would like to discretize this ODE. I have read about Euler, Verlet and Leapfrog methods. Usually, when you have the system
$$
\dot x(t) = v(t) \\
\dot v(t) = a(t)
$$
the Leapfrog method workds well. However everyone seems to say this is not a good method to use when the force/acceleration depends on the velocity, as in this case. The lecture notes say that this system of equations could be discretized as
$$
\begin{align}
x_{t + \delta/2} &= x_t + \frac{\delta}{2}v_t \\
v_{t+\delta} &=v_t - \delta \frac{v(t)^\top H(x_{t + \delta/2}) v(t)}{||g(x_{t + \delta/2})||^2} g(x_{t + \delta/2}) \\
x_{t+\delta} &= x_{t + \delta/2} + \frac{\delta}{2} v_{t + \delta}
\end{align}
$$
but doesn't explain why this is the correct scheme and what scheme this even is. To me it looks like a wrong Leapfrog method where they are doing things in reverse: doing half a position step, a full velocity step and half a position step. Can someone help me understand this?
REPLY [2 votes]: As I mentioned, your system is the system of differential equations, written in the ambient space $\mathbb{R}^m$, of the geodesics on the equipotential surface $f(x) = c$.
So let us have a smooth function $$f \, : \, \mathbb{R}^m \, \longrightarrow \, \mathbb{R}$$ which we call 'the potential', and let us fix a real number (a value) $c \,\in \, \mathbb{R}$ which gives us the smooth level hyper-surface
$$M_c \, = \, \{ \, x \, \in \, \mathbb{R}^m \,\, | \,\,\, f(x) = c\,\,\}$$
If you restrict the standard flat Euclidean metric on $M_c$ you get a Riemannian metric on $M_c$. Now, the geodesics of $M_c$ are not geodesics of $\mathbb{R}^m$, the latter are just straight lines, but the geodesics of $M_c$ try to follow the curved surface of $M_c$ with as little deviation from the ambient geodesics as possible. Which means that their acceleration, that causes them to curve and follow the surface instead of staying straight, when orthogonally projected onto the tangent space of $M_c$ at each point on the geodesic, should be zero (so no acceleration should be visible for creatures on the surface, that do not look into the ambient space). What is that mean. Let $x = x(t)$ be a geodesic on $M_c$. Then the acceleration should be its second derivative, i.e.
$$\text{acceleration} \, = \, \frac{d^2x}{dt^2}$$
Furthermore, since the orthogonal projection of $\frac{d^2x}{dt^2}$ onto the tangent space of $M_c$ at the point $x$ should be zero, that means that
$$\frac{d^2x}{dt^2} \,\,\text{ should be aligned with the normal vector to $M_c$ at $x$ }$$
But a normal vector to $M_c$ at $x$ is the gradient $g(x) \, = \, \nabla f(x)$ so
$$\frac{d^2x}{dt^2} \,\,\text{ should be aligned with the normal vector $\nabla f(x)$ }$$
And there you have the first hint, that the equations of the geodesic should look like
$$\frac{d^2x}{dt^2} \, = \, \lambda \, \nabla f(x)$$ where you can expect
$$\lambda \, = \, \lambda\Big(x, \, \frac{dx}{dt}\Big)$$
Combine the latter system of equation with the restriction $$f(x) \, = \, c$$
When you differentiate the latter restriction with respect to $t$, you get the first new restriction
$$\nabla f(x)^T \, \frac{dx}{dt} \, = \, 0$$
i.e. the velocity $\frac{dx}{dt}(t)$ of $x(t)$ is always perpendicular to $\nabla f(x)$, which means the velocity is tangent to the hyper-surface $M_c$ (which is not surprising, it is expected and required). Now, we go even further and differentiate the latter dot product identity once more with respect to $t$ and get
$$\nabla f(x)^T \, \frac{d^2x}{dt^2} \, + \, \frac{dx}{dt}^T H_f(x) \, \frac{dx}{dt} \, = \, 0$$
Plug $\frac{d^2x}{dt^2} \, = \, \lambda \, \nabla f(x)$ into the latter identity and you get
$$\nabla f(x)^T \, \big(\lambda \, \nabla f(x)\,\big) \, + \, \frac{dx}{dt}^T H_f(x) \, \frac{dx}{dt} \, = \, 0$$
$$\lambda \, \nabla f(x)^T \, \, \nabla f(x) \, + \, \frac{dx}{dt}^T H_f(x) \, \frac{dx}{dt} \, = \, 0$$
$$\lambda \, \big| \nabla f(x) \big|^2 \, + \, \frac{dx}{dt}^T H_f(x) \, \frac{dx}{dt} \, = \, 0$$ so when you solve for $\lambda$ you get
$$\lambda \, = \, - \, \frac{\, \frac{dx}{dt}^T H_f(x) \, \frac{dx}{dt} \,}{\big| \nabla f(x) \big|^2}$$
And now you see that the system of differential equations for the geodesics on $M_c$, written with variables from the ambient space $\mathbb{R}^m$, is
\begin{align}
&\frac{dx}{dt} \, = \, v\\
&\frac{dv}{dt} \, = \, -\, \frac{\, v^T H_f(x) \, v \,}{\big| \nabla f(x) \big|^2}\,\,\nabla f(x)
\end{align}
with initial conditions and restrictions
\begin{align}
&x(0) \, = \, x_0\\
&v(0) \, = \, v_0\\
&f(x_0) \, = \, c\\
&\nabla f(x_0)^T\,v_0 \, = \, 0
\end{align}
Then, any solution to the system of equations that satisfies the initial restrictions will keep satisfying them for all times $t$, i.e. for all $t$
\begin{align}
&f\big(x(t)\big) \, = \, c\\
&\nabla f\big(\,x(t)\,\big)^T\,v(t) \, = \, 0
\end{align}
That's because the equations have been constructed to satisfy these conditions in the first place. By differentiating these identities once and twice and combine with the initial restrictions, you can verify the identities for all $t$, which we actually already did in order to determine the exact formula for $\lambda$.
One more observation. The magnitude of the velocity $v(t)$ is constant for all $t$. Indeed
\begin{align}
\frac{d}{dt}\,
\big(\,\, |v|^2 \,\big) \, =& \, 2 \, v^T \, \frac{dv}{dt} \, = \, 2\, v^T \left( \, -\, \frac{\, v^T H_f(x) \, v \,}{\big| \nabla f(x) \big|^2}\,\,\nabla f(x) \, \right)\\
=& \, 2\, \left( \, -\, \frac{\, v^T H_f(x) \, v \,}{\big| \nabla f(x) \big|^2} \right) \Big(\, v^T\, \nabla f(x) \, \Big)\\
=& \, 2\, \left( \, -\, \frac{\, v^T H_f(x) \, v \,}{\big| \nabla f(x) \big|^2} \right) \Big(\, \nabla f(x)^T v \, \Big)\\
=& \, 2\, \left( \, -\, \frac{\, v^T H_f(x) \, v \,}{\big| \nabla f(x) \big|^2} \right)\cdot 0\\
=& \, 0
\end{align}
so $|v(t)| = |v_0| = v_0$ is constant for all $t$ (another feature of the geodesic flow).
In the light of this analysis, I would try to develop and implement a time-discrete version of the geodesic flow on $M_c$, replicating as many of the properties above as possible, instead of trying direct blind numerics for the differential equations.
Here is my suggestion:
A discrete geodesic flow on the hyper-equipotential surface $M_c$ of the function $f(x)$ with fixed small step $h$.
Assume you have generated the following pair of position and velocity
$$\big(\,x_n,\, v_n\,\big)$$
satisfying the restrictions
\begin{align}
&f(x_n) \, = \, c\\
&\nabla f(x_n)^T\,v_n =\, 0\\
&|v_n| \, = \, v_0
\end{align}
Step 1. Generate the new intermediate position $$\tilde{x}_n \, = \, x_n \, + \, h\,v_n$$
Step 2. Solve the non-linear system of $n+1$ equations for the unknown $n+1$ variables
$\big(\,x_{n+1}, \,\, \lambda_{n+1}\,\big)$
\begin{align}
&\lambda_{n+1}\nabla f(x_{n+1}) \, + \, x_{n+1} \, = \, \tilde{x}_n\\
&f(x_{n+1}) \, = \, c
\end{align}
Geometrically, this system tells you that $x_{n+1}$ is the orthogonal projection of $\tilde{x}_{n}$ onto the hyper-surface $M_c$ along the normal vector $\nabla f(x_{n+1})$ of $M_c$, calculated at the projected point $x_{n+1}$. In other words, $x_{n+1}$ is chosen on the hyper-surface $M_c$ so that the $\lambda-$parametrized line $$x_{n+1} \, + \, \lambda \nabla f(x_{n+1})\, , $$ which is orthogonal to $M_c$, passes through the point $\tilde{x}_n$. As you can see, here we have $\lambda_{n+1}\nabla f(x_{n+1}) $ which is the discrete analogue of the orthogonal force that keeps the geodesic on the hyper-surface $M_c$ preventing it from escaping into the ambient space $\mathbb{R}^m$.
You can try to solve this system of non-linear equation by say Newton's method, for which you will need the hessian $H_f(x)$ and that's where the hessian appears in this discrete scheme. So, start with initial conditions $x^0_{n+1} \, = \,\tilde{x}_n,\,\, \lambda^0_{n+1} = 0$. Then keep iterating over $k$, with $n$ fixed, and keep solving the linear system for the new variables $\big(\,x_{n+1}^{k+1}, \,\, v_{n+1}^{k+1}\, \big)$ form the previous already known variables $\big(\,x_{n+1}^{k}, \,\, v_{n+1}^{k}\, \big)$ calculate the Jacobian
$$
\Big[\,J_{n+1}^k\,\Big] \, =
\,\begin{bmatrix}
\lambda_{n+1}^k H_f({x}_{n+1}^k) + I_{m\times m} & \nabla f({x}_{n+1}^k)\\
\nabla f({x}_{n+1}^k)^T & 0
\end{bmatrix}
$$
and then use it as the matrix for the system of linear equations for the unknown variables $\big(\,x_{n+1}^{k+1}, \,\, v_{n+1}^{k+1}\, \big)$
$$
\Big[\,J_{n+1}^k\,\Big] \begin{bmatrix} x_{n+1}^{k+1}\\ \lambda_{n+1}^{k+1}\end{bmatrix} \, = \, \Big[\,J_{n+1}^k\,\Big] \begin{bmatrix} x_{n+1}^{k}\\ \lambda_{n+1}^{k}\end{bmatrix} \, - \, \begin{bmatrix} \lambda_{n+1}\nabla f(x_{n+1}^{k}) \, + \, x_{n+1}^{k} \, - \, \tilde{x}_n\\ f(x_{n+1}^k) \, - \, c\end{bmatrix}
$$
$$k = k + 1$$
until $$|f(x_{n+1}^{k}) \, - \, c|^2 \,<\, \varepsilon$$ for some fixed $\varepsilon$ threshold error.
In attempt to save computational time and memory (hopefully it works), maybe you can try to cheat a bit and calculate the Jacobian for Newton's method only at the initial conditions $x^0_{n+1} \, = \,\tilde{x}_n,\,\, \lambda^0_{n+1} = \lambda_{n}$ (or $\lambda_{n+1}^0 = 0$ which allows you to skip the hessian altogether, but I do not know if it works) and keep reusing it. Here $\lambda_n$ is the result from the calculation from the previous point $x_n$.
So first, calculate and invert the following Jacobian at $\tilde{x}_n$
$$
\Big[\,\text{J inv} \, \Big] \, =
\,\begin{bmatrix}
\lambda_{n+1}^0 H_f(\tilde{x}_n) + I_{m \times m} & \nabla f(\tilde{x}_{n})\\
\nabla f(\tilde{x}_{n})^T & 0
\end{bmatrix}^{-1}
$$
Second, again starting from initial conditions $x^0_{n+1} \, = \,\tilde{x}_n,\,\, \lambda^0_{n+1} = 0$ (or $\lambda^0_{n+1} = \lambda_n$ from the preceding calculation for $x_n$), and while $$|f(x_{n+1}^{k}) \, - \, c|^2 \,\geq \, \varepsilon$$ keep iterating over $k$ the discrete dynamical system
$$\begin{bmatrix} x_{n+1}^{k+1}\\ \lambda_{n+1}^{k+1}\end{bmatrix} \, = \, \begin{bmatrix} x_{n+1}^{k}\\ \lambda_{n+1}^{k}\end{bmatrix} \, - \, \Big[\,\text{J inv} \, \Big] \begin{bmatrix} \lambda_{n+1}\nabla f(x_{n+1}^{k}) \, + \, x_{n+1}^{k} \, - \, \tilde{x}_n\\ f(x_{n+1}^k) \, - \, c\end{bmatrix}$$
$$k = k + 1$$
The result is the new position $x_{n+1}$, which is on the surface $M_c$ (numerically :) )
Step 3. Finally, we calculate the new velocity $v_{n+1}$. We project orthogonally the old velocity $v_n$ onto the tangent hyper-plane of $M_c$ at the new position $x_{n+1}$ and then we rescale it to have magnitude $v_0$
\begin{align}
&\tilde{v}_n \, = \, v_n \, - \, \left(\,\frac{\,\nabla f(x_{n+1})^T \, v_n\,}{|\nabla f(x_{n+1})|^2}\,\right)\, \nabla f(x_{n+1})\\
&v_{n+1} \, = \, v_0 \, \frac{\tilde{v}_n}{|\tilde{v}_n|}
\end{align}
As you can see here, just like in the smooth case, the velocities $v_{n+1}$ and $v_n$ are coplanar with the normal gradient vector $\nabla f(x_{n+1})$. And again, you can see that the velocity evolves only along the normal gradient vector $\nabla f(x_{n+1})$, which is the discrete analogue of the normal force redirecting the velocity in the smooth case.
By executing steps 1, 2 and 3 we obtain the new position and velocity of the geodesic flow
$$\big(\,x_{n+1},\, v_{n+1}\,\big)$$
By construction, the new pair also satisfies the geodesic restrictions
\begin{align}
&f(x_{n+1}) \, = \, c\\
&\nabla f(x_{n+1})^T\,v_{n+1} =\, 0\\
&|v_{n+1}| \, = \, v_0
\end{align}
By iterating steps 1, 2, 3 you get a discrete analogue of the geodesic flow on $M_c$. And I think the result will have a fairly good behaviour and will emulate many of the properties of the smooth geodesic flow.
Edit. As a test, I implemented this method for the case of the geodesic flow on a 3D ellipsoid. I chose an ellipsoid whose axes are aligned with the coordinate axes. I implemented the method using a fixed Jacobian for Newton's method, when generating the orthogonal projection of the intermediate point onto the surface of the ellipsoid. It works quite well, so for nice surfaces probably there is no need to calculate a hessian, which is good news.
import numpy as np
import matplotlib.pyplot as plt
def Jacobian_inv(z_, A):
grad = A.dot(z_[0:3])
J = np.empty((4,4), dtype=float)
J[0:3,0:3] = np.diag([1,1,1])
J[0:3, 3] = grad
J[3, 0:3] = 2*grad
J[3,3] = 0.
return np.linalg.inv(J)
def project_position(z_med, A, accuracy):
z = z_med
J_1 = Jacobian_inv(z, A)
while True:
Ax = A.dot(z[0:3])
xAx_1 = z[0:3].dot(Ax) - 1.
if abs(xAx_1) < accuracy:
return z
F = np.concatenate( ( z[3]*Ax + z[0:3] - z_med[0:3], np.array([xAx_1]) ) )
z = z - J_1.dot(F)
def project_velocity(x_, v_, norm_v, A):
Ax = A.dot(x_)
v = v_ - (Ax.dot(v_))*Ax / Ax.dot(Ax)
return norm_v * v / np.sqrt(v.dot(v))
def geod_flow_step(z_, v_, norm_v, A, accuracy, step):
z = np.concatenate( ( z_[0:3] + step * v_, np.array([0.]) ) )
z = project_position(z, A, accuracy)
v = project_velocity(z[0:3], v_, norm_v, A)
return z, v
def geod_flow(x_in, v_in, A, norm_v, accuracy, step, n_steps):
n = n_steps #int(arc_length / step)
x = np.empty((n, 3), dtype=float)
v = np.empty((n, 3), dtype=float)
x[0,:] = x_in
v[0,:] = v_in
z = np.array([x_in[0], x_in[1], x_in[2], 0.0])
for m in range(n-1):
z, v[m+1,:] = geod_flow_step(z, v[m,:], norm_v, A, accuracy, step)
x[m+1,:] = z[0:3]
return x, v
x0 = np.array([3., 0., 0.])
v0 = np.array([0., 1., 2.])
semi_axes = np.array([3.*3., 2.*2, 1.*1.])
D = np.diag(1. / semi_axes)
norm_v0 = 1.
accuracy = 1e-7
step = 0.05
v0 = v0 / np.linalg.norm(v0)
v0 = norm_v0 * v0
x, v = geod_flow(x0, v0, D, norm_v0, accuracy, step, 1500)
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.set_xlim((-4, 4))
ax.set_ylim((-4, 4))
ax.set_zlim((-4, 4))
ax.plot(x[:,0], x[:,1], x[:,2], 'r-')
ax.plot(x[0,0], x[0,1], x[0,2], 'bo')
ax.plot(x[-1,0], x[-1,1], x[-1,2], 'go')
plt.show()
Starting point $[3, 0, 0]$ and direction vector $[0, 1, 1]$
Starting point $[3, 0, 0]$ and direction vector $[0, 1, 2]$
Starting point $[3, 0, 0]$ and direction vector $[0, 1, 3]$
The blue point is the starting point, the green one is the endpoint.
Geodesic on a submanifold of a Euclidean space. A geodesic on a manifold is a curve whose covariant derivative (i.e. derivative within the framework of the manifold's geometry) is zero. This means that there is no acceleration on the manifold that makes the geodesic turn, so the geodesic is straight within the manifold's geometry. However, in our case, the manifold is a hyper-surface in some Euclidean space. So the geometry of this hyper-surface is inherited from the ambient Euclidean geometry. A geodesic on the hyper-surface doesn't curve as seen from the hyper-surface, but it definitely curves in the ambient Euclidean space, because the hyper-surface itself is not straight (unless it is hyper-plane) and the geodesic itself is not a straight line. So, the geodesic should curve in the ambient space but should not curve from the point of view of the hyper-surface. That means that the acceleration vector of the geodesic, which exists in the Euclidean space, should not be visible on the hyper-surface. This last statement simply means that the orthogonal projection of this acceleration on the tangent hyper-plane of the hyper-surface at each point on the geodesic should not exist, i.e. it should be zero. This is true exactly when the acceleration vector at each point of the geodesic in the Euclidean space is perpendicular to the tangent plane at that point. The gradient of the hype-surface is also perpendicular to the tangent hyper-plane at each point from the hyper-surface. Hence, the acceleration vector and the gradient must be colinear, i.e. aligned. | {"set_name": "stack_exchange", "score": 2, "question_id": 649959} |
TITLE: What is Cauchy Schwarz in 8th grade terms?
QUESTION [65 upvotes]: I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman's terms, as if you are explaining it to someone who has just started geo in 8th grade?
REPLY [0 votes]: The most easy example I know, even useful in everyday life:
2 same rectangles each made out of edge length of a and b
are smaller or equal to
2 squares, each made of a and b separately.
Or in ever day terms:
Squares make the best use of edge length to gain area.
General:
|⟨X,Y⟩|≤∥X∥⋅∥Y∥
in a more concrete form (case):
|ab| ≤ |a||b|
... more concrete (rectangle vs square - case):
2 *(a * b) ≤ a * a + b * b | {"set_name": "stack_exchange", "score": 65, "question_id": 1519714} |
TITLE: One-loop effective potential of Standard Model
QUESTION [5 upvotes]: The one loop Coleman-Weinberg contribution of a scalar field to the
effective potential (in $\overline{MS}$-scheme) is:
\begin{equation}
\mathrm{const.} \times m^4(\phi_c) \left( \log \left( \frac{m^2(\phi_c)}{\mu^2}\right) -\frac{3}{2} \right)
\end{equation}
Now, I have a problem with this formula. In theories with spontaneous symmetry breaking, like the standard model, the background field dependent mass will actually be negative. For the SM effective potential (usually calculated in Landau gauge) we have the Higgs field and the Goldstone fields, with:
\begin{eqnarray}
m_H(\phi_c) = 3 \lambda \phi_c^2 - m^2 \\
m_G(\phi_c) = \lambda \phi_c^2 - m^2
\end{eqnarray}
At the Higgs VEV $m_H(v)=2m^2$ and $m_G(v)=0$. Where $m$ and $\lambda$ are
the renormalized parameters of the tree level Higgs potential.
My problem is that this implies, for example, for $\phi_c < v$ the
Goldstone boson $\phi_c$ dependent mass is negative, and the logarithm is
complex, also as $\phi_c \to v$ the real part of the log goes to $-\infty$.
Am I doing something silly or does the formula really break down? Or is there a way of making sense of the imaginary potential?
REPLY [2 votes]: You are correct in observing that the effective potential is complex for certain values of the background field. This is somewhat of a thorny issue. In principle, when deriving the effective potential one writes it as a Legendre transformation of the generating functional of connected diagrams $ W[J] $. This assumes that the effective potential is convex in the variable $\phi$. When there is spontaneous symmetry breaking present at the classical level this is obviously not true, as evidenced by the negative curvature at the origin of field space. The conventional wisdom is that our whole derivation does not breakdown into nonsense. Instead, when a field dependent mass turns negative then the corresponding field value does not represent a stable state. This is signaled through the effective potential by it acquiring a non-zero imaginary part that plays the role of a decay rate of this state. However, see [Precision decay rate calculations in quantum field theory] for how to calculate the actual physical decay rate properly.
Your second observation, that the logarithm diverges as $ \phi \rightarrow v $ is correct but also nothing to worry about. This is because the prefactor of the logarithm, $ \left(m_G^2\right)^2 $, also goes to zero in this limit (faster than the logarithm diverges).
On a more general note, there is nothing inconsistent about evaluating the effective potential for general field values. However, the effective potential does not in general represent a physical quantity. Only when evaluated at extrema.
I recommend [Consistent Use of Effective Potentials] for a deep treatment of the effective potential and how to derive physical quantities from it. | {"set_name": "stack_exchange", "score": 5, "question_id": 185039} |
TITLE: About a norm : $p(uv)=p(u)p(v)$ all the time?
QUESTION [0 upvotes]: Say, $p$ is a norm on a vector space.
Then can we say that
$$p(uv)=p(u)p(v)$$ all the time?
Thanks.
REPLY [0 votes]: Norm p is a map from a vector space V to the reals R. Assuming uv is the inner product (which maps VxV to some set of scalars S) then p(uv) is only defined when V = S. In that case you would have something like p(uv)=|uv| and p(u)p(v)=|u||v|. Otherwise p(uv) is not really defined for more general vector spaces. | {"set_name": "stack_exchange", "score": 0, "question_id": 1189758} |
\begin{document}
\title{Domains of existence for finely holomorphic functions}
\author{Bent Fuglede}
\address{Department of Mathematical Sciences\\
University of Copenhagen
\\Universitetsparken 5
\\2100 K\o benhavn, \\Denmark}
\email{[email protected]}
\author{Alan Groot}
\author{Jan Wiegerinck}
\address{KdV Institute for Mathematics
\\University of Amsterdam
\\Science Park 105-107
\\P.O. box 94248, 1090 GE Amsterdam
\\The Netherlands}\email{[email protected]}
\email{[email protected]}
\thanks{We are grateful to Jan van Mill for many enlightening discussions.}
\subjclass[2010]{30G12, 30A14, 31C40}
\keywords{finely holomorphic function, domain of existence}
\begin{abstract} We show that fine domains in $\CC$ with the property that they are Euclidean $F_\sigma$ and $G_\delta$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\CC\setminus \QQ$ or $\CC\setminus (\QQ\times i\QQ)$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are \emph{not} fine domains of existence.
\end{abstract}
\maketitle
\section*{Introduction}
It was already known to Weierstrass that every domain $\Omega$ in $\CC$ is a \emph{domain of existence}, roughly speaking, it admits a holomorphic function $f$ that cannot be extended analytically at any boundary point of $\Omega$. In his thesis \cite{Bo1892} Borel showed, however, that it may be that $f$ can be (uniquely) extended to a strictly larger set $X$ in an "analytic" way, albeit that $X$ is no longer Euclidean open. This eventually led Borel to the introduction of his Cauchy domains and monogenic functions, cf.~\cite{Bo1917}. Finely holomorphic functions on fine domains in $\CC$ as introduced by the first named author, are the natural extension and setting for Borel's ideas, see \cite{Fu81}, also for some historic remarks on Borel's work. In this paper we will study \emph{fine domains of existence}, roughly speaking, fine domains in $\CC$ that admit a finely holomorphic function that cannot be extended as a finely holomorphic function at any fine boundary point. Definition \ref{Def.existence} contains a precise definition. Here the results are different from both the classical Weierstrass case of one variable and the classical several variable case, where Hartogs showed that there exist domains $D\subsetneq D^*$ in $\CC^n$ with the property that every holomorphic function on $D$ extends to the larger domain $D^*$.
Our results are as follows. In Section \ref{sec2} we show that every fine domain that is a Euclidean $F_\sigma$ as well as a Euclidean $G_\delta$ is a fine domain of existence. Euclidean domains are of this form and therefore are fine domains of existence. We also show that \emph{regular} fine domains are fine domains of existence. Noting that on Euclidean domains holomorphic and finely holomorphic functions are the same, our Theorem \ref{Ufinecompactexhaustion} includes Weierstrass' theorem. In Section \ref{sec3}, however, we show that fine domains $V$ with the property that their complement contains a non-empty polar set $E$ that is of the first Baire category \emph{in its Euclidean closure} $K$ (in particular, $E$ has no Euclidean isolated points) and $(K\setminus E)\subset V$, are \emph{not} fine domains of existence.
Pyrih showed in \cite{pyrih} that the unit disc is a fine domain of existence, and that as a corollary of the proof, the same holds for simply connected Euclidean domains by the Riemann mapping theorem.
The starting point of our research was the following observation. Edlund \cite{Edl} showed that every closed set $F\subset \CC$ admits a continuous function $f:F\to \CC$ such that the graph of $f$ is completely pluripolar in $\CC^2$. It is easy to see that by construction this $f$ is finely holomorphic on the fine interior of $F$. In \cite{EEW} the main result can be phrased as follows: \emph{If a finely holomorphic function on a fine domain $D$ admits a finely holomorphic extension to a strictly larger fine domain $D'$, then the graph of $f$ over $D$ is not completely pluripolar. } Hence, if $D$ is the fine interior of a Euclidean closed set, every fine component of $D$ is a fine domain of existence. This is extended by our Theorem \ref{Ufinecompactexhaustion} because the fine interior of any finely closed set is regular.
In the next section we recall relevant results about the fine topology and fine holomorphy.
\section{Preliminaries on the fine topology and finely holomorphic functions}
Recall that a set $E\subset \CC$ is \emph{thin} at $a\in \CC$ if $a\notin \overline E$ or else if there exists a subharmonic function $u$ on an open neighborhood of $a$ such that
\[\limsup_{z\to a, z\in E\setminus \{a\}}u(z)<u(a).\]
The fine topology was introduced by H. Cartan in a letter to Brelot as the weakest topology that makes all subharmonic functions continuous. He pointed out that $E$ is thin at $a$ if and only if $a$ is not in the fine closure of $E$. The fine topology has the following known features. Finite sets are the only compact sets in the fine topology, which follows easily from the fact that every polar set $X\subset\CC$ is discrete in the fine topology, it consists of finely isolated points. The fine topology is Hausdorff, completely regular, Baire, and quasi--Lindel\"of, i.e. a union of finely open sets equals the union of a countable subfamily and a polar set, cf.~e.g.~\cite[Lemma71.2]{ArGa} . For our purposes fine connectedness is important.
Recall the following
\begin{theorem}[Fuglede]\label{locallyconnected}
The fine topology on $\CC$ is \emph{locally connected}. That is, for any $a \in \CC$ and any fine neighborhood $U$ of $a$, there exists a finely connected finely open neighborhood $V$ of $a$ with $V \subset U$.
\end{theorem}
See \cite[p.92]{Fu} (or see \cite{ElMarzguioui2006} for a proof using only elementary properties of subharmonic functions).
In fact, the fine topology is even \emph{locally polygonally arcwise connected} as was shown in \cite{Fu}, but we need something stronger.
\begin{definition}
A \emph{wedge} is a polygonal path consisting of two line segments $[a,w]$ and $[w,b]$ of equal length.
\end{definition}
The result we need is as follows,
\begin{theorem}[\cite{Fu80}] \label{wedgetheorem}Let $U$ be a finely open set in $\CC$ and $\alpha>1$. Then for every $w \in U$, there exists a fine neighborhood $V$ of $w$ such that any two distinct points $a, b \in V$ can be connected by a wedge contained in $U$ of total length less than $\alpha|a-b|$.
\end{theorem}
Lyons, see \cite[p. 16]{Lyons1980} had already proven that $a$ and $b$ can be connected by a polygonal path consisting of two line segments of total length less than $\alpha |a-b|$ and his proof essentially contained Theorem \ref{wedgetheorem}.
For an even stronger result see Gardiner, \cite[Theorem A]{Gard}.
We will also need the following elementary lemma. Let $B(x,r)$ denote the open disc with radius $r$ about $x\in \CC$, $\overline B(x,r)$ its closure, and let $C(x,r)$ denote its boundary.
\begin{lemma} \label{Lem1} Let $V$ be a fine neighborhood of $a\in\CC$. Then there exists $C_0>1$ such that for $C>C_0$ and every $n\in \NN$ there exists $t\in [C^{-n-1}, C^{-n}]$ with $C(a, t)\subset V$.
\end{lemma}
\begin{proof} We can assume $a=0$. A local basis at $0$ for the fine topology consists of the sets $B(0,r, h)= \{z\in B(0,r) : h(z)>0\}$,
where $r>0$, $h$ is subharmonic on $B(0,r)$ and $h(0)=1$. See \cite[Lemma 3.1]{ElMarzguioui2006} for a proof. Thus for some $h$ and $r$ we have $V\supset W:=\{z\in B(0,r): h(z)\ge 1/2 \}$ and $B(0,r)\setminus V$ is contained in $F:=B(0,r)\setminus W$, which is Euclidean open, hence an $F_\sigma$, that is thin at $0$, because $W$ contains the finely open set $B(0,r,h)$ which contains $0$.
Theorem 5.4.2 in \cite{Ran} states for given $r>0$ that there is a constant $\Gamma>0$ such that
\[\int_E \frac1x dx<\Gamma<\infty,\]
where
\[E:=\{s:0<s<r\mid \ \exists \theta \text{\ with\ } s e^{i\theta}\in F\}.\]
Let $C_0=\max\{e^{\Gamma}, 1/r\}$. For $C>C_0$ each interval of the form $[C^{-n-1},C^{-n}]$ is contained in $(0,r)$, but can not be contained in $E$. Hence there exists $t\in [C^{-n-1},C^{-n}]$ such that $C(0,t)\cap F=\emptyset$, that is, $C(0,t)\subset W\subset V$.
\end{proof}
For more information on the fine topology see \cite[Part I, Chapter XI]{Doob},
\cite[Chapter VII]{ArGa}.
\medbreak
There are several equivalent definitions for finely holomorphic functions on a fine domain in $D\subset\CC$, cf.~\cite{Fu8081, Fu81, Fu88, Lyons1980}.
\begin{definition} A function $f$ on a fine domain $D$ is called finely differentiable at $z_0\in D$ with fine complex derivative $f'(z_0)$ if there exists $f'(z_0)\in\CC$ such that for every $\eps>0$ there exists a fine neighborhood $V\subset D$ of $z_0$ such that
\[\left|\frac{f(z)-f(z_0)}{z-z_0}-f'(z_0)\right|<\eps \quad\text{for all } z\in V.\]
In other words, the limit $(f(z)-f(z_0))/(z-z_0)$ exists as $z-z_0\to 0$ finely.
The function $f$ is finely holomorphic on $D$ if and only if $f$ is finely differentiable at every point of $D$ and $f'$ is finely continuous on $D$.
\end{definition}
We will use the following characterizations of fine holomorphy.
\begin{theorem}[\cite{Fu81}] \label{finehol} Let $f$ be a complex valued function on a fine domain $D$. The following are equivalent
\begin{enumerate}
\item The function $f$ is finely holomorphic on $D$.
\item Every point $z_0\in D$ admits a fine neighborhood $V\subset D$ such that $f$ is a uniform limit of rational functions on $V$.
\item The functions $f$ and $z\mapsto zf(z)$ are both (complex valued) finely harmonic functions on $D$.
\end{enumerate}
\end{theorem}
In the following theorem we collect properties of finely holomorphic functions that indicate how much this theory resembles classical function theory.
\begin{theorem}[\cite{Fu8081}] Let $f: D\to \CC$ be finely holomorphic. Then
\begin{enumerate}
\item The function $f$ has fine derivatives $f^{(k)}$ of all orders $k$, and these are finely holomorphic on $D$.
\item Every point $z_0\in D$ admits a fine neighborhood $V\subset D$ such that for every $m=0,1,2,\ldots$
\begin{equation} \label{Tay}
\left|f(w)-\sum_{k=0}^{m-1}\frac{f^{(k)}(z)}{k!}(w-z)^k\right|\big/ |w-z|^m
\end{equation} is bounded on $V\times V$ for $z\ne w$.
\item At any point $z$ of $D$ the Taylor expansion \eqref{Tay} uniquely determines $f$ on $D$. (If all coefficients equal 0, then $f$ is identically $0$.)
\end{enumerate}
\end{theorem}
We now introduce finely isolated singularities.
\begin{definition} Let $D$ be a fine domain, $a\in D$, and $f$ finely holomorphic on $D\setminus \{a\}$.
\begin{itemize}
\item If $f$ extends as a finely holomorphic function to all of $D$, then $f$ has a removable singularity at $a$.
\item If $f(z)\to\infty$ for $z\to a$ finely, $f$ has a pole at $a$.
\item If $f$ has no pole at $a$ nor a removable singularity, then $f$ has an essential singularity at $a$.
\end{itemize}
\end{definition}
\begin{theorem}\label{Riemann} Let $D$ be a fine domain and $a\in D$. Let $f$ be finely holomorphic on $D\setminus\{a\}$ and suppose that $f$ is bounded on a fine neighborhood $V$ of $a$. Then $a$ is a removable singularity of $f$.
\end{theorem}
\begin{proof} (As indicated in \cite{Fu88}.) We apply Theorem \ref{finehol}, no. 3. Clearly $f$ and $z\mapsto zf(z)$ are bounded, finely harmonic functions on $V\setminus \{a\}$. \cite[Corollary 9.15]{Fu} states that these functions extend to be finely harmonic on all of $D$ and again by Theorem \ref{finehol} $f$ is finely holomorphic on $D$.
\end{proof}
\section{Fine domains of existence}\label{sec2}
For a set $A$ in $\CC$, we denote its fine interior by $\fint A$, its fine closure by $\fcl{A}$ and its fine boundary by $\partial_f A$.
\begin{definition}\label{Def.existence} A fine domain $U$ is called a \emph{fine domain of existence} if there exists a finely holomorphic function $f$ on $U$ with the property that for every fine domain $V$ that intersects $\partial_f U$ and for every fine component $\Omega$ of $V\cap U$, the restriction $f|_\Omega$ admits no finely holomorphic extension to $V$.
\end{definition}
\begin{definition} Let $U$ be a fine domain in $\CC$. If there exist compact sets $K_n\subset U$ such that $U = \bigcup_{n=1}^\infty K_n$ with $K_1 \subset \fint K_2 \subset K_2 \subset \fint K_3 \subset \cdots$, then the sequence $\{K_n\}$ is called a \emph{fine exhaustion} of $U$. If, moreover, the $K_n$ have the property that every bounded component of $\CC \setminus K_n$ contains a point from $\CC \setminus U$, we call $\{K_n\}$ a \emph{special fine exhaustion} of $U$.
\end{definition}
We have the following lemma, which is a consequence of the Lusin--Menchov property of the fine topology.
\begin{lemma}[{\cite[Corollary 13.92]{MR2589994}}]\label{LuMe}
Let $U$ be a finely open set and let $K$ be a compact subset of $U$. Then there exists a Borel finely open set $V$ such that $K \subset V \subset \overline{V} \subset U$.
\end{lemma}
\begin{prop}\label{generalUfinecompactexhaustion}
Let $U$ be a fine domain that is a Euclidean $F_\sigma$. Then $U$ admits a special fine exhaustion.
\end{prop}
\begin{proof}
Let $U = \bigcup_{n=1}^\infty F_n$ for compact sets $F_1 \subset F_2 \subset \cdots$. Fix a strictly increasing sequence $(r_n)_{n \geq 1}$ tending to infinity such that $F_n \subset B(0,r_n)$ for all $n \geq 1$. We put $K_1 := F_1$. For the construction of $K_2$, note that $F_2 \cup K_1 (=F_2)$ is a compact subset of $U$. By Lemma \ref{LuMe}, there exists a finely open set $V_2$ such that $F_2 \cup K_1 \subset V_2 \subset \overline{V_2} \subset U$. Then the set $K_2 := \overline{V_2} \cap \overline{B(0,r_2)}$ is compact and
\[
\fint K_2
= \fint\overline{V_2} \cap \fint\overline{B(0,r_2)} \supset V_2 \cap B(0,r_2) \supset F_2 \supset F_1 = K_1.
\]
As induction hypothesis, suppose that for some $n \geq 2$, we have found compact sets $K_1, \ldots, K_n$ such that $K_j \subset \fint K_{j+1}$ for all $1 \leq j \leq n-1$, $F_j \subset K_j$ for all $1 \leq j \leq n$ and $K_j \subset B(0,r_{j+1})$ for all $1 \leq j \leq n$ and that we have found finely open sets $V_2, \ldots, V_n$ such that $F_j \cup K_{j-1} \subset V_j \subset \overline{V_j} \subset U$ for all $2 \leq j \leq n$.
We now prove the induction step. Note that the set $F_{n+1} \cup K_n$ is a compact subset of $U$. By the previous lemma, there exists a finely open set $V_{n+1}$ such that $F_{n+1} \cup K_n \subset V_{n+1} \subset \overline{V_{n+1}} \subset U$. The set $K_{n+1} := \overline{V_{n+1}} \cap \overline{B(0,r_{n+1})}$ is compact, contained in $U$ and in $B(0,r_{n+2})$ and
\[
\fint K_{n+1} = \fint\overline{V_{n+1}} \cap \fint\overline{B(0,r_{n+1})} \supset V_{n+1} \cap B(0,r_{n+1}) \supset F_{n+1} \cup K_n
\]
and therefore $\fint K_{n+1} \supset K_n$ and $K_{n+1} \supset F_{n+1}$. This proves the induction step.
Consequently, we can find compact subsets $K_1, K_2, \ldots$ of $U$ such that $K_1 \subset\fint K_2 \subset K_2 \subset\fint K_3 \subset \cdots$ and $F_n \subset K_n$ for all $n \geq 1$. It follows that $U = \bigcup_{n=1}^\infty K_n$, which proves that $U$ admits a fine exhaustion.
We will next adapt this fine exhaustion so that every bounded component of $\CC \setminus K_n$ contains a point from $\CC \setminus U$.
Observe that $K_n=\CC\setminus\bigcup_j D^n_j$, where for each $n$ the $D^n_j$ are the countably, possibly infinitely, many mutually disjoint open components of $\CC\setminus K_n$. Let $D^n_0$ be the unbounded component.
Then for any $n$ and any finite or infinite sequence $1\le j_1\le j_2\le \cdots$ the set $K_n\cup \bigcup_k D^n_{j_k}$ is compact too.
For every $n$ we set $K^*_n=K_n\cup\bigcup_k D^n_{j_k}$, where the $D^n_{j_k}$ are those components, if any, of $\CC\setminus K_n$ that are completely contained in $U$. We claim that $K_n^*\subset \fint(K_{n+1}^*)$.
Indeed, if $x\in K_n$ then $x\in \fint K_{n+1}\subset\fint (K_{n+1}^*)$. Now let $x\in D_{j_k}^n(\subset U)$. For the proof that $x\in K^*_{n+1}$ we may suppose that $x\notin K_{n+1}$. Then $x$ belongs to a (necessarily bounded) component of $ \CC\setminus K_{n+1}$ that is completely contained in $D_{j_k}^n\subset U$, hence $x\in K^*_{n+1}$. It follows that $D_{j_k}^n\subset K_{n+1}^*$ and as $D_{j_k}^n$ is Euclidean open, $D_{j_k}^n\subset \fint(K_{n+1}^*)$. This proves the claim.
\end{proof}
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.02]
\clip (0,0) circle (100.5);
\fill (90:0) circle (0.1);
\draw (0,0) circle (100);
\draw (0,-100) circle (100);
\filldraw[white] (0,0) -- ([shift=(20:20)]0:0) arc (20:340:20) -- cycle;
\draw ([shift=(22.5:20)]90:0) arc (22.5:337.5:20);
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (10,10) -- (0,20) -- (-10,10) -- cycle;
\draw (0,0) -- (5,10) -- (0,20);
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (-10,-10) -- (0,-20) -- (10,-10) -- cycle;
\draw (0,0) -- (-7,-10) -- (0,-20);
\end{tikzpicture}
\end{subfigure}
\qquad
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.07]
\clip (-30,-30) rectangle (30,30);
\draw (0,0) circle (100);
\draw (0,-100) circle (100);
\filldraw[white] (0,0) -- ([shift=(20:20)]0:0) arc (20:340:20) -- cycle;
\draw ([shift=(22.5:20)]90:0) arc (22.5:337.5:20);
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (10,10) -- (0,20) -- (-10,10) node[above]{\tiny $S_1$} -- cycle;
\draw (0,0) -- (5,10) node[pos=1,left]{\tiny $W_1$} -- (0,20);
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (-10,-10) node[below]{\tiny $S_2$} -- (0,-20) -- (10,-10) -- cycle;
\draw (0,0) -- (-7,-10) node[pos=0.9,right]{\tiny $W_2$} -- (0,-20);
\filldraw[fill=red!20, draw=red!50, opacity=0.4] (-50,-7.6536686473) rectangle (50,7.6536686473);
\draw (-16,-3) node[above] {\tiny $b$} -- (40,12) node[pos=0.8,above] {\tiny $q$};
\filldraw (-16,-3) circle (0.1);
\end{tikzpicture}
\end{subfigure}
\qquad
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.02]
\clip (0,0) circle (100.5);
\draw (0,0) circle (100);
\draw (0,-100) circle (100);
\filldraw[white] (0,0) -- ([shift=(20:20)]0:0) arc (20:340:20) -- cycle;
\draw ([shift=(22.5:20)]90:0) arc (22.5:337.5:20);
\draw (0,0) -- (3.82683432365,7.6536686473);
\draw (0,0) -- (-5.35756805311,-7.6536686473);
\end{tikzpicture}
\end{subfigure}
\qquad
\begin{subfigure}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.08]
\clip (-30,-30) rectangle (30,30);
\draw (0,0) circle (100);
\draw (0,-100) circle (100);
\filldraw[white] (0,0) -- ([shift=(20:20)]0:0) arc (20:340:20) -- cycle;
\draw ([shift=(22.5:20)]90:0) arc (22.5:337.5:20);
\draw (0,0) -- (3.82683432365,7.6536686473);
\draw (0,0) -- (-5.35756805311,-7.6536686473);
\end{tikzpicture}
\end{subfigure}
\caption{\label{boogsnijconstructie}Replacing a small circular arc.}
\end{figure}
We will need the following lemma. Figure \ref{boogsnijconstructie} illustrates its content.
\begin{lemma}\label{boogsnijlemma} Let $D$ be an open disc, $U$ a finely open subset of $D$, and let $a\in C\cap U$ where $C$ is an open circular arc contained in $D$ with the property that $D\setminus C$ has two components.
Then there exists a sequence $(r_j)_j$ of positive numbers decreasing to 0, such that for every $r_j$ there exists a compact set $C'(=C'_j)\subset U$, which is the union of four arcs, such that
\begin{enumerate}
\item $D\setminus C'$ is connected;
\item $C'\setminus \overline{B(a,r_j)}=C\setminus \overline{B(a,r_j)}$;
\item Every wedge $\ell = [p,b] \cup [b,q]$ with $|p-b|=|b-q|>r_j$
that meets $C$ also meets $C'$.
\end{enumerate}
\end{lemma}
\begin{proof} By Theorem \ref{wedgetheorem} there exists a fine neighborhood $V\subset U$ of $a$ such that any two distinct points $p,q\in V$ can be connected by a wedge $L\subset U$ of length less than $\sqrt 2 |p-q|$. Lemma \ref{Lem1} provides us with a sequence $(r_j)_j$ such that $C(a,r_j)\subset V$.
After scaling and rotating we may assume that $C=C(-i,1)\cap D$ and that $a=0$.
We can also assume all $r_j$ less than $1$ and so small that $\overline{B(0, r_j)}\subset D$. Now we fix $r=r_j$ and set
\[C_1=\left(C\setminus\left( B(0,r)\cap\{\Re z <0\}\right)\right)\cup\left(\{re^{i\theta}: \pi/6\le \theta\le 11\pi/6\} \right).\]
$C_1$ is contained in $V$ along with $C(0,r)$, therefore $i r$, respectively $-i r$, (both on $C_1$) can be connected to $0(=a)\in V\subset U$ by a wedge $W_1$, respectively $W_2$, of length less than $\sqrt 2r$ contained inside $U$. Observe that the wedges $W_1, W_2$ are contained in $\overline{B(0,r)}$, and in fact $W_1$ is contained in the closed square $S_1$ with diagonal $[0,ir]$ and similarly $W_2$ is contained in the closed square $S_2$ with diagonal $[0,-ir]$, see Figure \ref{boogsnijconstructie}.
Let $\ell= [p,b] \cup [b,q]$ be any wedge with $|p-b|=|b-q|>2r$ that meets $C$ and assume that $\ell$ does not meet $C_1$. Then one of its constituting segments, say $[b,q]$, meets $C\cap B(0,r)\cap\{\Re z<0\}$ and because $|q-b|>2r$, it meets $C(0,r)$ in a point $re^{i\theta_0}$ with $-\pi/6<\theta_0<\pi/6$. Therefore, as $r<1$, $[b,q]\cap(S_1\cup S_2)$ is contained in $\{z: |\Im z|<r\sin(\pi/6)=r/2\} \cap(S_1\cup S_2)$, as indicated in Figure \ref{boogsnijconstructie}.
Now we define $L_j=W_j\cap \{z: |\Im z|\le r/2
\}\subset U$, $j=1,2$ and $C'=C_1\cup L_1\cup L_2\subset U$ is the the union of three circular arcs and one arc consisting of two straight segments, all contained in $U$. Clearly $\ell$ meets $C'$ and $D\setminus C'$ is connected.
\end{proof}
\begin{prop}\label{prop2}
Let $U \subsetneq \CC$ be a nonempty fine domain that is a Euclidean $F_\sigma$ and let $\emptyset\ne F_1\subset F_2\subset \cdots$ be an increasing sequence of compact sets in $\CC\setminus U$.
Then there exist sequences $(K_n)_{n \geq 1}$ and $(L_n)_{n \geq 1}$ of nonempty compact sets such that \begin{enumerate}
\label{alph}
\item[(a)] $K_1 \subset \fint K_2 \subset K_2 \subset\fint K_3 \subset \cdots$ and $\bigcup_{n=1}^\infty K_n=U$,
\item[(b)] $d(K_n, L_n) > 0$ for all $n \geq 1$,
\item[(c)] $L_n \subset K_{n+1}$ for all $n \geq 1$,
\item[(d)] for any $n \geq 1$, every two points $w,z \in \CC \setminus (K_n \cup L_n)$ in the same component of $\CC \setminus K_n$ also lie in the same component of $\CC \setminus (K_n \cup L_n)$,
\item[(e)] every bounded component of $\CC \setminus (K_n \cup L_n)$ contains a point from $\CC \setminus U$,
\item[(f)] every wedge of length at least $1/n$ that meets $F_n$ and $K_n$, also meets $L_n$.
\end{enumerate}
Moreover, there exists a sequence $(f_\nu)_{\nu \geq 1}$ of rational functions with poles in $\CC \setminus U$ such that
\begin{enumerate}
\item the sup-norm $\|f_n - f_{n-1}\|_{K_n} < 1/2^n$ for all $n \geq 2$ and
\item $|f_n| > n$ on $L_n$ for all $n \geq 1$.
\end{enumerate}
The sequence $(f_\nu)_\nu$ converges uniformly on any $K_n$ to a function $f$ that is finely holomorphic on $U$ and has the property that $|f| > n-1/2^{n}$ on $L_n$ for any $n \geq 1$.
\end{prop}
\begin{proof} For $(K_n)$ we take a special fine exhaustion of $U$, $U=\bigcup_{n=1}^\infty K_n$, which exists in view of Proposition \ref{generalUfinecompactexhaustion}.
We next construct a sequence $(L_n)_{n \geq 1}$ of compact sets such that $d(L_n, K_n) > 0$ and $L_n \subset K_{n+1}$ for all $n \geq 1$.
Let $n \geq 1$. The set $F_n\subset\CC\setminus U$ is compact and is therefore covered (uniquely) by those finitely many components of $\CC \setminus K_n$ that meet $F_n$. We denote these components by $D_1^n, D_2^n, \ldots, D_{k_n}^n$. Since $K_n$ and $F_n$ are compact and $K_n \cap F_n = \emptyset$, it follows that $\delta_n := d(K_n, F_n) > 0$. Consequently,
\begin{equation}\label{eq2.5}
\{z \in \CC : d(z, K_n) < \delta_n\} \cap F_n = \emptyset.
\end{equation}
Since $K_n \subset \fint K_{n+1}$, by Lemma \ref{Lem1} each $z \in \partial K_n$ is the center of a circle $C(z,\epsilon_z)$ contained in $\fint K_{n+1}$ of radius $\epsilon_z$ less than $\delta_n$.
We will now construct $L_n$ by constructing the intersections $L_n \cap D_k^n$ (with some abuse of notation) in each of the components $D_1^n, D_2^n, \ldots, D_{k_n}^n$ and taking the union of these sets.
Fix a component $D_k^n$. Since $\partial D_k^n \subset \partial K_n$ is compact and is covered by the open disks $B(z,\epsilon_z)$ where $z \in \partial D_k^n$, it follows that there are finitely many points $z_1, \ldots, z_{m} \in \partial D_k^n$ such that $\partial D_k^n \subset \bigcup_{j=1}^{m} B(z_j, \epsilon_{z_j})$. Note that $(\partial \bigcup_{j=1}^{m} B(z_j, \epsilon_{z_j})) \cap D_k^n$ is a finite union of closed circular arcs $\mathcal{C}_l$ --and possibly finitely many points disjoint from these arcs-- of the components of $\bigcup_{j=1}^m\partial B(z_j, \epsilon_{z_j})\setminus\bigcup_{k=1
}^{m} B(z_k, \epsilon_{z_k})$. We define a compact set $L^*_{n,k}:=\bigcup_{l=1}^M \mathcal{C}_l\subset D_k^n$. Also note that every path $\gamma$, in particular every wedge, that connects a point $z \in F_n \cap D_k^n$ and a point $w \in K_n$ must intersect one of the arcs $\mathcal{C}_l$ in $L^*_{n,k}$, because $d(z,K_n)\ge\delta_n>\epsilon_z$ in view of (\ref{eq2.5}), and $w\in K_n\subset\mathbb C\setminus D^n_k$. The open set $\CC \setminus L_{n,k}^*$ has only finitely many, say $N\ge 2$, components as each of the $\mathcal{C}_l$ belongs to the boundary of two of these components. We will replace $L_{n,k}^* $ by $L_n\cap D_k^n$ in the following way to achieve that $\CC \setminus (L_n \cap D_k^n)$ is connected and the properties (a), (b) and (c) are kept.
To do this, note that $\mathcal{C}_1$ belongs to the boundary of two components of $\CC \setminus L_{n,k}^* $. Pick a point $a$ of $\mathcal{C}_1$ that is not an endpoint of this closed arc. Since $a\in \fint K_{n+1}\setminus K_n$, clearly $a \notin F_{n+1}\supset F_n$, and again since $a$ lies in the relative interior of the arc $\mathcal{C}_1$, we see that for all $r > 0$ small enough, $\overline{B (a,r)} \cap K_n = \emptyset = \overline{B(a,r)} \cap F_n$ and $\overline{B(a,r)}\cap L_{n,k}^*$ is contained in the relative interior of $\mathcal{C}_1$.
We apply Lemma \ref{boogsnijlemma} with $D=B(a,r)$, $U=\fint K_{n+1}\cap B(a,r)$, $r_j<1/n$ and obtain a compact $\mathcal{C}_1'$. Then $\CC\setminus (\mathcal{C}_1'\cup\bigcup_{l=2}^M\mathcal{C}_l)$ has $N-1$ components.
If $\mathcal{C}_2$ belongs to the boundary of two different components of $\CC\setminus (\mathcal{C}_1'\cup\bigcup_{l=2}^M\mathcal{C}_l)$ we replace it by $\mathcal{C}'_2$ likewise, reducing the number of components of the complement by one again. If $C_2$ belongs to the boundary of only one component, we just put $\mathcal{C}'_2= \mathcal{C}_2$. Proceeding in this way, we end with $L_n\cap D_k^n:=\bigcup_{l=1}^M \mathcal{C}'_l$, the complement of which is connected.
By setting $L_n := \bigcup_{k=1}^n (L_n \cap D_k^n)$, we now find that any two points $w, z \in \CC \setminus (K_n \cup L_n)$ that are in the same component of $\CC \setminus K_n$ are in the same component of $\CC \setminus (K_n \cup L_n)$ as well, while Lemma \ref{boogsnijlemma} guarantees that (f) is satisfied. Also, note that $L_n \subset\fint K_{n+1} \setminus K_n$, so that $K_n \cap L_n = \emptyset$. It follows that $d(K_n, L_n) > 0$ for every $n \geq 1$.
We have obtained sequences $(K_n)_{n \geq 1}$ and $(L_n)_{n \geq 1}$ of compact sets that satisfy the properties (a) -- (f).
Because of property (b), we can
apply Runge's theorem recursively for each $n$ to the holomorphic function $g_n$ that equals $0$ on an open set containing $K_n$ and is equal to $\sum_{j=1}^{n-1}\max_{z\in L_n}\{|R_j(z)|\}+n+1$ on an open set containing $L_n$. Thus there exists for each $n$ a rational function $R_n$ with
\begin{align}
\label{r1} |R_n|&<1/2^n,\quad\text{on $K_n$};\\
\label{r2} |R_n|&>\sum_{j=1}^{n-1}\max_{z\in L_n}\{|R_j(z)|\}+n,\quad\text{on $L_n$},
\end{align}
while $R_n$ has at most one pole in a preassigned point in each of the bounded components of $\mathbb C\setminus (K_n\cup L_n)$. Because of property (e) these poles may be taken from $\mathbb C\setminus U$. Set
\[f_k= \sum_{j=1}^k R_j. \]
Then $f_k$ clearly satisfies (1) and (2).
It now follows from (1) and (a) that there exists a function $f$ on $U$ such that $f_k \to f$ uniformly on any $K_n$. Note that $f$ is finely holomorphic on $U$, because if $z \in U$, then $z \in K_n$ for some $n \geq 1$ and, by property (a), $K_{n+1}$ is a compact fine neighborhood of $z$ on which $f_k \to f$ uniformly. Furthermore, it follows from (2) and (c) above that for any $n \geq 1$ and $z\in L_n$
\[|f_n(z)| \geq |R_n(z)|-\sum_{j=1}^{n-1}|R_j(z)|> n.\]
Finally, $|f|> n-1/2^n$ follows immediately from (1) and (2).
\end{proof}
We shall prove in Theorem \ref{Ufinecompactexhaustion} that under suitable conditions on the fine domain, the function that we have just constructed admits no finely holomorphic extension outside the domain. We need some facts about (regular) fine domains.
It is known that every regular fine domain $U$ is an $F_\sigma$ because $\mathbb C\setminus U$ is a base in the sense of Brelot, and hence a $G_\delta$. Moreover, we need the following Lemma, which is (iv) of \cite[Theorem 7.3.11]{ArGa}.
\medskip\noindent
\begin{lemma} \label{lemma1.1}Every finely closed set $A\subset\mathbb C$ is the disjoint union of a (Euclidean) $F_\sigma$ and a polar set.\end{lemma}
\begin{lemma}\label{lem2.7} Let $\emptyset\ne U\subset V$ be fine domains and let $E$ be a polar subset of $\CC\setminus U$. Then either there exists a point $a\in (\partial_f U\cap V)\setminus E$, or $V=U\cup e$ for the polar set $e:=(V\setminus U)\cap E$, whereby $e$ is empty and hence $U=V$ in case $U$ is a regular fine domain.
\end{lemma}
\begin{proof}
If $(\partial_fU\cap V)\setminus E$ is empty then $(\partial_fU\cap V)\subset E\cap(V\setminus U)=e$, hence the fine boundary $\partial_fU\cap(V\setminus e)$ of $U$ relative to $V\setminus e$ is empty. Since $V\setminus e$ is finely open and finely connected along with $V$, by \cite[Theorem 12.2]{Fu}, and since $U\ne\emptyset$ this means that $U=V\setminus e$, that is, $V=U\cup e$ as claimed. If $U$ is regular then $e$ is empty because $e$ and $U$ are disjoint and $V$ is finely open.
\end{proof}
\begin{theorem}\label{Ufinecompactexhaustion}
Let $U \subset \CC$ be a non-empty fine domain that is either (1) both Euclidean $F_\sigma$ and Euclidean $G_\delta$, or (2) a regular fine domain. Then $U$ is a fine domain of existence.
\end{theorem}
\begin{proof}
\medbreak In both cases $U$ is an $F_\sigma$, and we can write $U=\bigcup_{n=1}^\infty K_n$ with $K_j\subset K_{j+1}$ compact subsets of $U$. Because of Lemma \ref{lemma1.1}, $\CC \setminus U=F\cup E$, where $F=\bigcup_{n=1}^\infty F_n$ with $F_j\subset F_{j+1}$ compact in $\CC\setminus U$, is a Euclidean $F_\sigma$ and $E$ is a polar set which we can assume to be empty in Case (1). Let $L_j\subset K_{j+1}$ be compact sets and $f$ a finely holomorphic function on $U$ as constructed in Proposition \ref{prop2}.
Let $V$ be a fine domain that meets $\partial_fU$ and let $\Omega$ be a fine component of $U\cap V$. Then $\partial_f\Omega\cap V\subset \partial_f U\cap V$, because if $x\in \partial_f\Omega\cap V\cap U$ there would be a finely connected fine neighborhood of $x$ contained in $V\cap U$, contradicting that $\Omega$ is a fine component of $V\cap U$.
Suppose that $f|_\Omega$ admits a finely holomorphic extension $\tilde{f}$ defined on $V$.
By Lemma \ref{lem2.7} applied to $\Omega$ in place of $U$, there is a point $z_0\in(\partial_f\Omega\cap V)\setminus E$, whereby in Case (1) $E=\emptyset$ and hence $z_0\in(\partial_f\Omega\cap V)\subset\partial_f U\cap V\subset F$. In Case (2), if $V=\Omega\cup e$, then $U\cup V=U\cup e$, which is impossible because $U$ is a regular domain. Hence $z_0\in(\partial_f\Omega\cap V)\setminus E=(\partial_fU\cap V)\setminus E\subset F$.
Again by Proposition \ref{prop2} there is a compact fine neighborhood $V_0$ of $z_0$ in $V$ on which $\tilde{f}$ is the uniform limit of rational functions with poles off $V_0$. In particular, $\tilde{f}$ is continuous and bounded on $V_0$, say $|\tilde{f}| \leq M$ on $V_0$. Choose a fine domain $V_1\subset V_0$ containing $z_0$ and such that, for given $\alpha>1$, any two distinct points $a,b$ of $V_1$ can be joined by a wedge of total length less than $\alpha|a-b|$ contained in $V_0$, see Theorem \ref{wedgetheorem}.
Choose $z_1\in\Omega\setminus\{z_0\}$. By the above and by Lemma \ref{Lem1}, there exists for every $\epsilon>0$ an $0<r_0 <\epsilon$ with $r_0<|z_1-z_0|$ such that $C(z_0, r_0) \subset V_1$.
Choose $z_2\in\Omega$ with $|z_2-z_0|<r_0$ (possible since $z_0\in\partial_f\Omega\subset\partial\Omega$). There is a path $\gamma\subset\Omega$ joining $z_1$ and $z_2$ and meeting $C(z_0,r_0)$ at a point $w_0$ of $\Omega\cap V_1$. Thus $C(z_0,r_0)\cap\Omega\ne\emptyset$. Recall that also $z_0\in V_1$.
Let $l_0 = [w_0, b_0] \cup [b_0, z_0]$ with $|w_0 - b_0| = |b_0 - z_0|$ be a wedge of total length less than $\alpha|w_0 - z_0|$ contained in $V_0$. Since $w_0 \in U$, we have $w_0 \in K_n$ for all $n$ large enough. Also, $z_0 \in F$, hence $z_0 \in F_n$ for all $n$ large enough. Therefore we can take an $n \geq 1$ large enough such that $w_0 \in K_n$ and $z_0 \in F_n$ and such that $n-1/2^{n-1}>M$ and $4/n < |w_0 - b_0| + |b_0 - z_0|$ hold simultaneously. Then by Proposition \ref{prop2} (f), $l_0$ meets $L_n\cap\Omega$.
Let $z\in l_0 \cap L_n\cap\Omega$. Since $z \in l_0 \cap\Omega\subset V_0$, we have $|\tilde{f}(z)| \leq M$. On the other hand, since $z \in L_n \cap\Omega$, we have $\tilde{f}(z) = f(z)$ and by (2) in Proposition \ref{prop2} $|\tilde{f}(z)| = |f(z)| \geq n-1/2^{n-1} > M$, which is a contradiction. We conclude that $f$ does not admit a finely holomorphic extension and that $U$ is a fine domain of existence.
\end{proof}
In the rest of this section we extend the above theorem a little.
\begin{lemma}
Let $U_1$ and $U_2$ be fine domains of existence, and suppose that $(\partial_fU_1)\cap(\partial_fU_2)=\emptyset$. Then every fine component of $U_1\cap U_2$ is a fine domain of existence.
\end{lemma}
\begin{proof}
The assertion amounts to $U_1\cap U_2$ being a ``finely open set of existence'' (if nonvoid) in the obvious sense. Because $(\partial_fU_1)\cap(\partial_fU_2)=\emptyset$ we have, writing $U_1\cap U_2=U_0$,
\[\partial_f U_0=(U_1\cap\partial_fU_2)\cup(U_2\cap\partial_fU_1).\]
By hypothesis there exists for $i=1,2$ a finely holomorphic function $h_i$ on $U_i$ such that for any fine domain $V$ that intersects $\partial_f U_i$ and any fine component $\Omega_i$ of $U_i\cap V$, $h_i|\Omega_i$ does not extend finely holomorphically to $V$.
Let $V$ be any fine domain that intersects $\partial_fU_0$ and let $\Omega$ be a fine component of $V\cap U_0$. Without loss of generality we may assume that $V$ intersects $U_2\cap\partial_fU_1$, and by shrinking $V$ that $V\subset U_2$. Then $\Omega$ is a fine component of $U_1\cap V=U_1\cap(U_2\cap V)=U_0\cap V$.
The function $h_0:=h_1|U_0+h_2|U_0$ is then finely holomorphic on $U_0$. Because $h_2$ is finely holomorphic on $V\subset U_2$, the function $h_0|_\Omega$ is extendible to $V$
if and only if $h_1|_\Omega$ is extendible over $V$,
and that is not the case. Therefore $U_0$ is a finely open set of existence.
\end{proof}
\begin{theorem}\label{Thm2}
Every fine domain $U\subset\mathbb C$ such that the set $I$ of irregular fine boundary points for $U$ is both an $F_\sigma$ and a $G_\delta$ is a fine domain of existence.
\end{theorem}
\begin{proof}
In the above lemma take $U_1=U_r$ (the regularization of $U$) and $U_2=\CC\setminus I$.
Since $I$ is polar, $U_2$ is a fine domain along with $\mathbb C$. Since $U$ is finely connected so is $U_r$. In fact, $U_r=U\cup E$, where $E$ denotes the polar set of finely isolated points of $\mathbb C\setminus U$. If $U_r=V_1\cup V_2$ with $V_1,V_2$ finely open and disjoint then
\[U=U_r\setminus E=(V_1\setminus E)\cup(V_2\setminus E)\]
with $V_1\setminus E$ and $V_2\setminus E$ finely open and disjoint. It follows that for example $V_1\setminus E=\emptyset$ and hence $V_1=\emptyset$, showing that $U_r$ indeed is finely connected.
By hypothesis, the complement of $U_2$ and hence $U_2$ itself is an $F_\sigma$ and a $G_\delta$. Then by Theorem \ref{Ufinecompactexhaustion} $U_1$ and $U_2$ are fine domains of existence. By Theorem \ref{Ufinecompactexhaustion}, $U_1\cap U_2=U_r\setminus I=U$ is indeed a fine domain of existence.
\end{proof}
\section{Fine domains that are not domains of existence}\label{sec3}
\begin{prop}\label{prop1} Let $(a_n)$ be a sequence in $\CC$ that converges to $a$. Suppose that $V_n$ is a fine neighborhood of $a_n$ of the form \[V_n=V_n(a_n,h_n,r)=\{z\in B(a_n,r): h_n(z)>0\},\]
where $h_n$ is a subharmonic function on $B(a_n,r)$ such that $h_n(a_n)=1/2$ and $h_n<1$ on $B(a_n,r)$.
Then $\bigcup_{n=1}^\infty V_n$ is a (possibly deleted) fine neighborhood of $a$.
\end{prop}
\begin{proof} Let $r_1<r$. Then there exists $n_0>0$ such that for $n\ge n_0$ the function $h_n$ is defined on $B(a,r_1)$. Let
\[h=(\sup_{n\ge n_0}\{h_n|_{B(a, r_1)}: n\ge n_0\}),\]
on $B(a, r_1)$ and let $h^*$ denote its upper semi-continuous regularization. Then $h^*$ is subharmonic, $h^*\le 1$, and $h^*(a)\ge\limsup_{n\to \infty}h(a_n)\ge 1/2$.
Let $V_0=\{z\in B(a,r_1): h^*(z)>0\}$. Then $V_0$ is a fine neighborhood of $a$, since $h^*$ is finely continuous. Because the set $X=\{h<h^*\}\setminus\{a\}$ is polar, it is finely closed, therefore the set $V=V_0\setminus X$ is a fine neighborhood of $a$.
We claim that $V\setminus\{a\}\subset\bigcup_{n\ge n_0} V_n$. Indeed, if $z\in V\setminus \{a\}$ then $h(z)=h^*(z)>0$ hence $h_n(z)>0$ for some $n\ge n_0$, and $z\in V_n$.
\end{proof}
\begin{prop}\label{mainprop} Let $E$ be a non-empty
polar set in $\CC$ and suppose that $E$ is of the first Baire category in its Euclidean closure $K$. Suppose that $f$ is finely holomorphic on a finely open set $V$ such that $K\setminus E\subset V$. Then there exist a Euclidean open ball $B$ that meets $K$, and a finely open fine neighborhood $V_1$ of $K\cap B$ such that $f$ is bounded on $V_1\setminus E$.
\end{prop}
\begin{proof} Let $x\in V\setminus E$. Denote by $U_x$ a finely open subset of $V$ containing $x$ and having the property stated in Theorem \ref{wedgetheorem} applied to $V$. By shrinking we may arrange that $U_x$ has the form
\[U_x=U_x(x,h_x,r_x)=\{z\in B(x,r_x): h_x(z)>0\},\]
where $h_x$ is a subharmonic function on $B(x,r_x)$ such that $h_x(x)=1/2$ and $h_x<1$ on $B(x, r_x)$.
Let $X_j$ be the set of $x\in K\setminus E$ such that $r_x>1/j$, and $|f|\le j$ on $U_x$ and that $U_x$ satisfies Lemma \ref{Lem1} with $C_0=j$.
Then $K\setminus E=\bigcup_j X_j$, hence by the Baire category theorem, there exist $j_0$ and an open Euclidean ball which we may assume to be the unit disc $\DD$, such that $\emptyset\ne K \cap \DD\subset \overline X_{j_0}$ (the Euclidean closure of $X_{j_0}$).
\smallskip
Let $w\in K\cap \DD$. Then $w=\lim_{n\to\infty} x_n$ for a sequence $(x_n)_n$ in $X_{j_0}$. As $r_{x_n}>1/j_0$, Proposition \ref{prop1} gives us that $\bigcup_{n}U_{x_n}$ is a (possibly deleted) fine neighborhood of $w$ on which $|f|\le j_0$ and $W_w:=\{w\}\cup\bigcup_{n}U_{x_n}$ is a fine neighborhood of $w$. If $w\in (K\setminus E)\cap \DD$, $|f|\le j_0$ on $W_w$. In fact, $f$ is finely holomorphic on $V$, hence $|f|$ is finely continuous on the finely open set $V\cap W_w$ which contains $w$ because $V\supset K\setminus E$. If $|f(w)|>j_0$ then $|f|>j_0$ on some fine neighborhood $Z$ of $w$, which contradicts that $Z$ meets $W_w\setminus\{w\}$ because $\mathbb C$ has no finely isolated points.
The set $V_1=\bigcup_{w\in K\cap \DD}W_w$ is a finely open fine neighborhood of $K\cap\DD$ with the property that on $|f|\le j_0$ on $V_1\setminus E$.
\end{proof}
\begin{remark} The reader should be aware that the condition \emph{$E$ is of the first Baire category in its Euclidean closure $K$} prevents sets like $E=\{1/n,n=1,2,\ldots \}$. Indeed, $E$ can not be written as countable union of nowhere dense subsets in its Euclidean closure $K=E\cup\{0\}$, because this set $E$ is relatively open in $K$.
\end{remark}
\begin{theorem}\label{MT} Suppose that $E$ and $K$ are as in Proposition \ref{mainprop}, and that $V$ is a fine domain such that $V\cap E=\emptyset$ and $K\setminus E\subset V$. Then (1) $E\subset \partial_fV$ and (2) $V$ is not a fine domain of existence.
\end{theorem}
\begin{proof}
For (1) observe that $\partial_f V=\partial V$, because $V$ is connected and therefore not thin at any of its Euclidean boundary points.
Thus it suffices to show that $E\subset \partial V$. We have $E=\bigcup_{n=1}^\infty F_n$, a countable union of nowhere dense subsets of $K$, which is of the second category in itself. Therefore, $E$ is contained in the closure of $K\setminus E$. If not, suppose to reach a contradiction, that $O$ were an open set in $K$ not meeting $K\setminus E$, then $O= \bigcup_{n=1}^\infty (F_n\cap O)$, a countable union of nowhere dense sets, contradicting that $K$ is of the second category. As $K\setminus E$ is contained in the domain $V$, we find that $E$ must be contained in the Euclidean closure of $V$ and hence in $\partial_fV=\partial V$ because $V\cap E=\emptyset$.
For (2) let $f$ be finely holomorphic on $V$.
By Proposition \ref{mainprop} there exist an open ball $B$ that meets $K$ (and hence $E$) and a finely open set $V_1$ containing $K\cap B$ such that $f$ is bounded on $V_1\cap V\cap B\subset V_1\setminus E$.
Because $E$ is polar, $V_1\cap V$ is a deleted finely open fine neighborhood of every point in $E\cap B$, hence each point $a$ of $E\cap B$ is a finely isolated singularity of $f$. Theorem \ref{Riemann} implies that $f$ extends over each $a\in E\cap B$ and hence has a finely holomorphic continuation to the finely open set $V\cup[(V\cup E)\cap B]=V\cup(E\cap B)$, which contains $V$ properly because $(E\cap B)\subset \partial_fV$.
\end{proof}
\begin{example}\label{exa3.5} Let $V=\CC\setminus \QQ$, $E=\QQ$, and $K=\RR$. Then $V$, $E$, and $K$ satisfy the conditions of Theorem \ref{MT}, therefore $\CC\setminus \QQ$ is not a domain of existence for finely holomorphic functions.
Similarly $\CC\setminus (\QQ\oplus i\QQ)$ is not a domain of existence for finely holomorphic functions.
\end{example}
\begin{question}From Theorem \ref{MT} and Example \ref{exa3.5} it is clear that the requirement that $U$ be an $F_\sigma$ in Theorem \ref{Ufinecompactexhaustion} can not be dropped.
But is it perhaps true that every fine domain that is a Euclidean $F_\sigma$ is a fine domain of existence? It would suffice to prove this for $U=\mathbb C\setminus I$ with $I$ a polar $G_\delta$, see the proof of Theorem \ref{Thm2}.
\end{question}
\bibliographystyle{amsplain} | {"config": "arxiv", "file": "1706.02498.tex"} |
TITLE: Can the empty-set be used for a model that satisfies an axiom system?
QUESTION [1 upvotes]: I'm reading some notes on Model Theory and wondering if axiom systems that make no existential claims are trivially satisfied by the empty set. For instance, if you just have the axiom that all points are on some line, would taking the model $\mathbb{P}=\emptyset=\mathbb{L}$ prove the consistency of the axioms?
REPLY [3 votes]: No, usually an interpretation is not allowed to be empty.
In some sense this is just an arbitrary choice, but it turns out that proof systems that are sound and complete with respect to model theory without empty models can be simpler than when one needs to account for empty models.
For example, if $p$ is some predicate, then $p(x)\lor\neg p(x)$ is usually provable as an instance of a tautology, and the most natural rule for introducing existential quantifiers would then allow one to prove $\exists x.p(x)\lor\neg p(x)$. This rule would not be sound if empty domains were allowed.
Since the empty structure is generally a quite uninteresting case, a priori we might either restrict the model theory to forbid empty structures, or change our proof theory to be sound with respect to the empty structure. In practice it turns out that assuming that structures are non-empty is a lot less cumbersome than the necessary changes to the proof system would be, so this is what is usually done. | {"set_name": "stack_exchange", "score": 1, "question_id": 1165331} |
TITLE: Can anyone explain to me why $(34)(123) = (124)$?
QUESTION [1 upvotes]: $$\begin{align}
(34)H&= \{(34)(1),(34)(123),(34)(132)\}\\
& = \{(34),(124),(1432)\}.
\end{align} $$
Can anyone explain to me why $(34)(123) = (124)$?
I don't understand coset multiplying can you help me with this
REPLY [1 votes]: If $\sigma=(123)$, $\tau=(34)$, and we compose right-to-left (as is consistent with your other compositions), then
$$\begin{align}
1 &\stackrel{\sigma}{\mapsto} 2 \stackrel{\tau}{\mapsto}2, \\
2&\stackrel{\sigma}{\mapsto} 3 \stackrel{\tau}{\mapsto} 4, \\
4 &\stackrel{\sigma}{\mapsto} 4 \stackrel{\tau}{\mapsto} 3, \\
3 &\stackrel{\sigma}{\mapsto} 1 \stackrel{\tau}{\mapsto} 1,
\end{align}$$
so $\tau\sigma=(34)(123)=(1243)$. | {"set_name": "stack_exchange", "score": 1, "question_id": 3446529} |
TITLE: Induced maps between spectra
QUESTION [3 upvotes]: Let $f:A\to B$ be a ring homomorphism. If $f^{-1}$ induces a bijection between the ideals of $B$ and a set $U$ of ideals of $A$, and this bijection reflects the inclusion (and reflects the property of being prime), can I conclude that $\operatorname {Spec}f$ is an homeomorphism between $\operatorname {Spec} B$ and $U\cap\operatorname {Spec} A$?
From the hypothesis is immediate that $\operatorname {Spec} f$ induces a (continuous) bijection between $\operatorname {Spec} B$ and $U\cap\operatorname {Spec} A$; to prove that is closed in the image one can use that, for an ideal $I\subseteq B$ and a closed $V(I)$, $$\mathfrak p \in \operatorname{Spec}f (V(I))\iff f(\mathfrak p)\subseteq I, \mathfrak p\in U \iff \mathfrak p \subseteq f^{-1} (I),\mathfrak p\in U\iff \mathfrak p \in V(f^{-1}(I))\cap U.$$ Is my arguement correct? If yes, are there notable examples of homomorphisms $f$ with the mentioned properties, other than the projection $A\to A/I$ and the canonical homomorphism $A\to S^{-1} A$?
REPLY [2 votes]: One correct claim that can be proven is that if $f:A\to B$ is a homomorphism of rings so that $f^{-1}$ reflects containment of ideals (ie $f^{-1}(I)\subset f^{-1}(J)$ implies $I\subset J$), then the induced map on spectra $\def\Spec{\operatorname{Spec}}\Spec B\to\Spec A$ is a homeomorphism on to its image. Here's how:
The map $\varphi:\Spec B\to \Spec A$ is continuous: the preimage of some $V(I)$ for $I\subset A$ is $V(f(I)B)$, so the preimage of a closed set is closed (this is general for any map of rings).
The map $\varphi$ is injective: if $\mathfrak{q}_1,\mathfrak{q}_2\subset B$ are prime ideals with $f^{-1}(\mathfrak{q}_1)=f^{-1}(\mathfrak{q}_2)=\mathfrak{p}$ a prime ideal of $A$, then since $f^{-1}$ reflects containment we see that $\mathfrak{q}_1=\mathfrak{q}_2$. Therefore $\varphi$ is a bijection on to its image.
The map $\varphi:\Spec B\to \varphi(\Spec B)$ is closed. Suppose $\mathfrak{p}\subset A$ is in the image of $\varphi$, that is, there exists a prime ideal $\mathfrak{q}\subset B$ with $f^{-1}(\mathfrak{q})=\mathfrak{p}$. Further, suppose that $\mathfrak{p}$ is in the closure of $\varphi(V(I))$ for some ideal $I\subset B$. As $\overline{\varphi(V(I))}=V(f^{-1}(I))$ for any map of rings (where the closure is taken in $\Spec A$), we see that $f^{-1}(I)\subset f^{-1}(\mathfrak{q})$, and as $f^{-1}$ reflects containment we must have $I\subset\mathfrak{q}$. Therefore $\mathfrak{q}\in V(I)$ and $\varphi(V(I))$ is closed in $U$. (This looks to be approximately what you've written, but I don't find your presentation to be so clear - maybe that's my fault.)
A continuous closed bijection is a homeomorphism, so $\varphi:\Spec B\to \varphi(\Spec B)$ is a homeomorphism.
Note that if we drop this assumption about reflecting containment, we get counterexamples: if $R$ is a discrete valuation ring with maximal ideal $\mathfrak{m}$, then the map $f:R\to R/\mathfrak{m}\times \operatorname{Frac}(R)$ is a continuous bijection which is not a homeomorphism and $f^{-1}$ does not reflect containment. (This is noted in the other answer and its comments.)
As far as a characterization of what sort of maps of rings have this property other than (compositions of) localizations and quotients, I'm not sure that there is a well-known description. Any such map will be an epimorphism of rings, and there's a seminar by Samuel about these that I'm aware of but have not dug in to much. Some of the examples here should satisfy your assumptions and also give you an idea that this is maybe not such a simple condition to classify. | {"set_name": "stack_exchange", "score": 3, "question_id": 4300359} |
\begin{document}
\maketitle
\begin{abstract}
This article is concerned with a spectral optimization problem: in a smooth bounded domain $\O$, for a bounded function $m$ and a nonnegative parameter $\alpha$, consider the first eigenvalue $\lambda_\alpha(m)$ of the operator $\mathcal L_m$ given by $\mathcal L_m(u)= -\operatorname{div} \left((1+\alpha m)\nabla u\right)-mu$. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $\lambda_\alpha(m)$ with respect to $m$. Such a problem is related to the so-called ``two phase extremal eigenvalue problem'' and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain.
We prove that unless the domain is a ball, this problem has no ``regular'' solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.
\end{abstract}
\noindent\textbf{Keywords:} shape derivatives, drifted Laplacian, bang-bang functions, spectral optimization, homogenization, reaction-diffusion equations.
\medskip
\noindent\textbf{AMS classification:} 35K57, 35P99, 49J20, 49J50, 49Q10
\tableofcontents
\section{Introduction and main results}
In recent decades, much attention has been paid to extremal problems involving eigenvalues, and in particular to shape optimization problems in which the unknown is the domain where the eigenvalue problem is solved (see e.g. \cite{Henrot2006,henrot2017shape} for a survey).
The study of these last problems is motivated by stability issues of vibrating bodies, wave propagation in composite environments, or also on conductor thermal insulation.
In this article, we are interested in studying a particular extremal eigenvalues problem, involving a drift term, and which comes from the study of mathematical biology problems; here, we can show that the problem then boils down to a "two-phase" type problem, meaning that the differential operator whose eigenvalues we are trying to optimise has $-\nabla \cdot(A\nabla)$ as a principal part, and that $A$ is an optimisation variable, see Section \ref{BiologicalMotivations}. The influence of drift terms on optimal design problems is not so well understood. Such problems naturally arise for instance when looking for optimal shape design for two-phase composite materials \cite{DambrineKateb,Laurain,MuratTartar}. We expand on the bibliography in Section \ref{SSE:BackgroundH} of this paper, but let us briefly recall that, for composite materials, a possible formulation reads: given $\O$, a bounded connected open subset of $\R^n$ and a set of admissible non-negative densities $\mathcal{M}$ in $\O$, solve the optimal design problem
\begin{equation}\tag{$\hat P_\alpha$}\label{Pg:TwoPhase}
\inf_{m\in \mathcal M} \hat\lambda_\alpha(m)
\end{equation}
where $\hat \lambda_\alpha(m)$ denotes the first eigenvalue of the elliptic operator
$$
\hat{ \mathcal L}_\alpha^m:W^{1,2}_0(\O)\ni u\mapsto -\n \cdot \left((1+\alpha m)\n u\right).
$$
Restricting the set of admissible densities to {\it bang-bang} ones (in other words to functions taking only two different values) is known to be relevant for the research of structures optimizing the compliance. We refer to Section \ref{Pr:NonExistence} for detailed bibliographical comments.
Mathematically, the main issues regarding Problem \eqref{Pg:TwoPhase} concern the existence of optimal densities in $\mathcal M$, possibly the existence of optimal {\it bang-bang} densities (i.e characteristic functions). In this case, it is interesting to try to describe minimizers in a qualitative way.
In what follows, we will consider a refined version of Problem \eqref{Pg:TwoPhase}, where the operator $\hat{ \mathcal L}_\alpha^m$ is replaced with
\begin{equation}\label{def:Lmalpha}
\mathcal L_m^\alpha:W^{1,2}_0(\O)\ni u\mapsto -\n \cdot \left((1+\alpha m)\n u\right)-mu.
\end{equation}
Besides its intrinsic mathematical interest, the issue of minimizing the first eigenvalue of $ \mathcal L_m^\alpha$ with respect to densities $m$ is motivated by a model of population dynamics (see Section~\ref{BiologicalMotivations}).
Before providing a precise mathematical frame of the questions we raise in what follows, let us roughly describe the main results and contributions of this article:
\begin{itemize}
\item by adapting the methods developed by Murat and Tartar, \cite{MuratTartar}, and Cox and Lipton, \cite{CoxLipton}, we show that the first eigenvalue of $ \mathcal L_m^\alpha$ has no regular minimizer in $\mathcal M$ unless $\Omega$ is a ball;
\item if $\Omega$ is a ball, denoting by $m_0^*$ a minimizer of $ \mathcal L_m^0$ over $\mathcal M$ (known to be {\it bang-bang} and radially symmetric), we show the following stationarity result: $m_0^*$ still minimizes $ \mathcal L_m^\alpha$ over radially symmetric distributions of $\mathcal M$ whenever $\alpha$ is small enough and in small dimension ($n=1,2,3$). Such a result appears unexpectedly difficult to prove. Our approach is based on the use of a well chosen path of quasi-minimizers and on a new type of local argument.
\item if $\Omega$ is a ball, we investigate the local optimality of ball centered distributions among all distributions and prove a quantitative estimate on the second order shape derivative by using a new approach relying on a kind of comparison principle for second order shape derivatives.
\end{itemize}
Precise statements of these results are given in Section \ref{sec:mainRes}.
\subsection{Mathematical setup}
Throughout this article, $m_0$, $\kappa$ are fixed positive parameters.
Since in our work we want to extend the results of \cite{LamboleyLaurainNadinPrivat}, let us define the set of admissible functions
$$
\mathcal M_{m_0,\kappa}(\Omega)=\left\{m\in L^\infty(\O)\, , 0\leq m\leq \kappa\, ,\fint_\O m=m_0\right\},
$$
where $\fint_\O m$ denotes the average value of $m$ (see Section~\ref{sec:notations}) and assume that $m_0<\kappa$ so that $\mathcal M_{m_0,\kappa}(\O)$ is non-empty.
Given $\alpha\geq 0$ and $m\in \mathcal M_{m_0,\kappa}(\O)$, the operator $\mathcal L_m^\alpha$ is symmetric and compact. According to the spectral theorem, it is diagonalizable in $L^2(\O)$. In what follows, let $\lambda_\alpha(m)$ be the first eigenvalue for this problem. According to the Krein-Rutman theorem, $\lambda_\alpha(m)$ is simple and its associated $L^2(\O)$-normalized eigenfunction $u_{\alpha,m}$ has a constant sign, say $u_{\alpha,m}\geq 0$. Let $R_{\alpha,m}$ be the associated Rayleigh quotient given by
\begin{equation}
R_{\alpha,m}:W^{1,2}_0(\O)\ni u\mapsto \frac{\int_\O (1+\alpha m)|\n u|^2-\int_\O mu^2}{\int_\O u^2}.\end{equation}
We recall that $\lambda_\alpha(m)$ can also be defined through the variational formulation
\begin{equation}\label{Eq:Rayleigh} \lambda_{\alpha}(m):=\inf_{u\in W^{1,2}_0(\O)\, , u\neq 0} R_{\alpha,m}(u)=R_{\alpha,m}(u_{\alpha,m}).
\end{equation}
and that $u_{\alpha,m}$ solves
\begin{equation}\label{Eq:EigenFunction}
\left\{
\begin{array}{ll}
-\n \cdot \Big((1+\alpha m)\n u_{\alpha,m}\Big)-mu_{\alpha,m}=\lambda_\alpha(m)u_{\alpha,m}&\text{ in }\O,
\\u_{\alpha,m}=0\text{ on }\partial \O.
\end{array}
\right.
\end{equation}
in a weak $W^{1,2}_0(\O)$ sense.
In this article, we address the optimization problem
\begin{equation}\label{Pb:OptimizationEigenvalue}\tag{$P_\alpha$}\fbox{$\displaystyle
\inf_{m\in \mathcal M_{m_0,\kappa}(\O)}\lambda_\alpha(m).$}
\end{equation}
This problem is a modified version of the standard two-phase problem studied in \cite{MuratTartar,CoxLipton}; we detail the bibliography associated with this problem in Subsection \ref{SSE:BackgroundH}. It is notable that it is relevant in the framework of population dynamics,
when looking for optimal resources configurations in a heterogeneous environment for species survival, see Section \ref{BiologicalMotivations}.
\subsection{Main results}\label{sec:mainRes}
Before providing the main results of this article, we state a first fundamental property of the investigated model, reducing in some sense the research of general minimizers to the one of {\it bang-bang} densities. It is notable that, although the set of {\it bang-bang} densities is known to be dense in the set of all densities for the weak-star topology, such a result is not obvious since it rests upon continuity properties of $\lambda_\alpha$ for this topology. We overcome this difficulty by exploiting a convexity-like property of $\lambda_\alpha$.
\begin{proposition}[weak {\it bang-bang} property]\label{Th:BangBang}
Let $\O$ be a bounded connected subset of $\R^n$ with a Lipschitz boundary and let $\alpha>0$ be given. For every $m\in \mathcal M_{m_0,\kappa}(\O)$, there exists a {\it bang-bang} function $\tilde m\in \mathcal M_{m_0,\kappa}(\O)$ such that
$$\lambda_\alpha(m)\geq \lambda_\alpha(\tilde m).$$
Moreover, if $m$ is not {\it bang-bang}, then we can choose $\tilde m$ so that the previous inequality is strict.
\end{proposition}
In other words, given any resources distribution $m$, it is always possible to construct a {\it bang-bang} function $\tilde m$ that improves the criterion.
\paragraph{Non-existence for general domains.}
In a series of paper, \cite{CasadoDiaz1,CasadoDiaz2,CasadoDiaz3}, Casado-Diaz proved that the problem of minimizing the first eigenvalue of the operator $u\mapsto -\n \cdot(1+\alpha m)\n u$ with respect to $m$ does not have a solution when $\partial \O$ is connected. His proof relies on a study of the regularity for this minimization problem, on homogenization and on a Serrin type argument. The following result is in the same vein, with two differences: it is weaker than his in the sense that it needs to assume higher regularity of the optimal set, but stronger in the sense that we do not make any strong assumption on $\partial \O$. For further details regarding this literature, we refer to Section \ref{TwoPhase}.
\begin{theorem}\label{Th:NonExistence}
Let $\O$ be a bounded connected subset of $\R^n$ with a Lipschitz connected boundary, let $\alpha>0$ and $n\geq 2$. If the optimization problem \eqref{Pb:OptimizationEigenvalue} has a solution $\hat m \in \mathcal M_{m_0,\kappa}(\O)$, then this solution writes $\hat m=\kappa \mathds{1}_{\hat E}$, where $\hat E$ is a measurable subset of $\O$. Moreover, if $\partial \hat E$ is a $\mathscr C^2$ hypersurface and if $\O$ is connected, then $\O$ is a ball.
\end{theorem}
The proof of this Theorem relies on methods developed by Murat and Tartar, \cite{MuratTartar}, Cox and Lipton, \cite{CoxLipton}, and on a Theorem by Serrin \cite{Serrin1971}.
\paragraph{Analysis of optimal configurations in a ball.}
According to Theorem \ref{Th:NonExistence}, existence of regular solutions fail when $\O$ is not a ball. This suggest to investigate the case $\O=\B(0,R)$, which is the main goal of what follows.
Let us stress that proving the existence of a minimizer in this setting and characterizing it is a hard task. Indeed, to underline the difficulty, notice in particular that none of the usual rearrangement techniques (the Schwarz rearrangement or the Alvino-Trombetti one, see Section~\ref{TwoPhase}), that enable in general to reduce the research of solutions to radially symmetric densities, and thus to get compactness properties, can be applied here.
\begin{center}
\textsf{\large{The case of radially symmetric distributions}}
\end{center}
Here, we assume that $\O$ denotes the ball $\mathbb B(0,R)$ with $R>0$. We define $r_0^*>0$ as the unique positive real number such that
$$\kappa \frac{\left|\mathbb B(0,r_0^*)\right|}{\left|\mathbb B(0,R)\right|}=V_0.
$$
Let
{\begin{equation} \label{eq:m0*}m_0^*=\kappa \mathds{1}_{\mathbb B(0,r_0^*)}=\kappa \mathds{1}_{E^*_0}\end{equation} }
be the centered distribution known to be the unique minimizer of $\lambda_0$ in $\mathcal M_{m_0,\kappa}(\O)$ (see e.g. \cite{LamboleyLaurainNadinPrivat}).
In what follows, we restrict ourselves to the case of radially symmetric resources distributions.
\begin{theorem}\label{Th:RadialStability}
Let $\O=\mathbb B(0,R)$ and let $\mathcal M_{rad}$ be the subset of radially symmetric distributions of $ \mathcal M_{m_0,\kappa}(\O)$. The optimization problem
$$\inf_{m\in \mathcal M_{rad}}\lambda_\alpha(m)$$ has a solution. Furthermore, when $n=1,2,3$, there exists $\alpha^*>0$ such that, for any $\alpha<\alpha^*$, there holds
\begin{equation}
\min_{m\in \mathcal M_{rad}}\lambda_\alpha(m)=\lambda_\alpha(m^*_0).\end{equation}\end{theorem}
The proof of the existence part of the theorem relies on rearrangement techniques that were first introduced by Alvino and Trombetti in \cite{AlvinoTrombetti} and then refined in \cite{ConcaMahadevanSanz}.
The stationarity result, i.e the fact that $m_0^*$ is a minimizer among radially symmetric distributions, was proved in the one-dimensional case in \cite{CaubetDeheuvelsPrivat}. To extend this result to higher dimensions, we developed an approach involving a homogenized version of the problem under consideration. The small dimensions hypothesis is due to a technical reason, which arises when dealing with elliptic regularity for this equation.
Restricting ourselves to radially symmetric distributions might appear surprising since one could expect this result to be true without restriction, in $ \mathcal M_{m_0,\kappa}(\O)$. For instance, a similar result has been shown in the framework of two-phase eigenvalues \cite{ConcaMahadevanSanz}, as a consequence of the Alvino-Trombetti rearrangement. Unfortunately, regarding Problem \eqref{Pb:OptimizationEigenvalue}, no standard rearrangement technique leads to the { conclusion}, because of the specific form of the involved Rayleigh quotient. A first attempt in the investigation of the ball case is then to consider the case of radially symmetric distributions. It is notable that, even in this case, the proof appears unexpectedly difficult.
Finally, we note that, as a consequence of the methods developed to prove Theorem~\ref{Th:RadialStability}, when a small amount of resources is available, the centered distribution $m^*_0$ is optimal among all resources distributions, regardless of radial symmetry assumptions.
\begin{corollary}\label{Th:Sketch}
Let $\O=\mathbb B(0,R)$ and $m_0^*$ be defined {by (\ref{eq:m0*})} There exist $\underline m>0$, $\underline \alpha>0$ such that, if
$m_0\leq \underline m$ and $\alpha<\underline\alpha$, then the unique solution of \eqref{Pb:OptimizationEigenvalue} is $m^*_0=\kappa \mathds 1_{E^*_0}.$
\end{corollary}
\begin{center}
\textsf{\large{Local stability of the ball distribution with respect to Hadamard perturbations of resources sets}}
\end{center}
In what follows, we tackle the issue of the local minimality of $m^*_0$ in $\mathcal M_{m_0,\kappa}(\O)$ with the help of a shape derivative approach. We obtain partial results in dimension $n=2$.
Let $\O$ be a bounded connected domain with a Lipschitz boundary, and consider a {\it bang-bang} function $m\in \mathcal M_{m_0,\kappa}(\O)$ writing $m=\kappa \mathds{1}_E$, for a measurable subset $E$ of $\O$ such that
$\kappa |E|=m_0|\O|$. Let us write $\lambda_\alpha(E):=\lambda_\alpha\left(\mathds{1}_E\right)$, with a slight abuse of notation.
Let us assume that $E$ has a $\mathscr C^2$ boundary. Let $V:\O\rightarrow \R^n$ be a $W^{3,\infty}$ vector field with compact support, and define for every $t$ small enough, $E_t:=\left(\operatorname{Id}+tV\right)E$.
For $t$ small enough, $\phi_t:=\operatorname{Id}+tV$ is a smooth diffeomorphism from $E$ to $E_t$, and $E_t$ is an open connected set with a $\mathscr C^2$ boundary.
If $\mathcal F:E\mapsto \mathcal F(E)$ denotes a shape functional, the first (resp. second) order shape derivative of $\mathcal F$ at $E$ in the direction $V$ is
$$
\mathcal F'(E)[V]:=\left.\frac{d}{dt}\right|_{t=0}\mathcal F(E_t) \quad \left(\text{ resp. }\left.\frac{d^2}{d t^2}\right|_{t=0} \mathcal F(E_t)\right)
$$
whenever these quantities exist.
For further details regarding the notion of shape derivative, we refer to \cite[Chapter 5]{HenrotPierre}.
It is standard to write optimality conditions in terms of a sort of tangent space for the measure constraint: indeed, since the volume constraint $\operatorname{Vol}(E_t)=m_0\operatorname{Vol}(\O)/\kappa$ is imposed, we will deal with vector fields $V$ satisfying the linearized volume condition $\int_E \n \cdot V=0$. We thus call \textit{admissible at $E$} such vector fields and introduce
\begin{equation}\label{Eq:X}\mathcal X(E):=\left\{V\in W^{3,\infty}(\R^n;\R^n)\, , \int_E \n \cdot V=0\, , \Vert V\Vert _{W^{3,\infty}}\leq 1\right\}.
\end{equation}
A shape $E\subset \O$ with a $\mathscr C^2$ boundary such that $\kappa |E|=m_0|\O|$ is said to be critical if
\begin{equation}\label{Eq:FOO}
\forall V \in \mathcal X(E),\quad \lambda_\alpha'(E)[V]=0.
\end{equation}
or equivalently, if there exist a Lagrange multiplier $\Lambda_\alpha$ such that $ \left(\lambda_\alpha'-\Lambda_\alpha \operatorname{Vol}'\right)(E)[V]= 0$ for all $V \in W^{1,\infty}(\O)$, where $\operatorname{Vol}:\O\mapsto |\O|$ denotes the volume functional.
Furthermore, if $E$ is a local minimizer for Problem \eqref{Pb:OptimizationEigenvalue}, then one has
\begin{equation}\label{Eq:SOO}
\forall V \in \mathcal X(E), \quad \left(\lambda_\alpha''-\Lambda_\alpha \operatorname{Vol}''\right)(E)[V,V]\geq 0.
\end{equation}
In what follows, we will still assume that $\O$ denotes the ball $\mathbb B(0,R)$ with $R>0$.
\begin{theorem}\label{Th:ShapeStability}
Let us assume that $n=2$ and that $\O=\mathbb B(0,R)$. The ball $E=\mathbb B(0,r_0^*)=\mathbb B^*$ satisfies the shape optimality conditions \eqref{Eq:FOO}-\eqref{Eq:SOO}. Furthermore, if $\Lambda_\alpha$ is the Lagrange multiplier associated with the volume constraint, there exist two constants $\overline\alpha>0$ and $C>0$ such that, for any $\alpha\in [0,\overline \alpha)$ and any vector field $V\in \mathcal X(\B^*)$, there holds
$$\left(\lambda_\alpha''-\Lambda_\alpha \operatorname{Vol}''\right)(\B^*)[V,V]\geq C\Vert V\cdot \nu \Vert _{L^2(\mathbb S^*)}^2.$$
\end{theorem}
\begin{remark}The proof requires explicit computation of the shape derivative of the eigenfunction. We note that in \cite{DambrineKateb} such computations are carried out for the two-phase problem and that in \cite{KaoLouYanagida} such an approach is undertaken to investigate the stability of certain configurations for a weighted Neumann eigenvalue problem.
The main contribution of this result is to shed light on a monotonicity principle that enables one
to lead a careful asymptotic analysis of the second order shape derivative of the functional as $\alpha \to 0$. It is important to note that, although this allows us to deeply analyze the second order optimality conditions, it is expected that the optimal coercivity norm in the right-hand side above is expected to be $H^{\frac12}$ whenever $\alpha>0$, which we do not recover with our method. The reason why we believe that in this context the optimal coercivity norm is $H^{\frac12}$ is that in \cite{DambrineKateb}, precise computations for the two-phase problem \eqref{Pg:TwoPhase} are carried out and a $H^{\frac12}$ coercivity norm is obtained for certain classes of parameters. On the other hand, when $\alpha =0$, it was shown in \cite{MazariQuantitative} that the optimal coercivity norm is $L^2$, and a quantitative inequality was then derived.
\end{remark}
\begin{remark}
We believe that our strategy of proof may be used to obtain the same kind of coercivity norm in the three-dimensional case. However, we believe that such a generalization would be non-trivial and need technicalities. Since the main contribution is to introduce a methodology to study the positivity of second-order shape derivative, we simply provide a possible strategy to prove the result in the three dimensional case in the concluding section of the proof of Theorem~\ref{Th:ShapeStability}, see Section~\ref{Se:CclSha}.
\end{remark}
The rest of this article is dedicated to proofs of the results we have just outlined.
\subsection{A biological application of the problem}\label{BiologicalMotivations}
Equation \eqref{Eq:EigenFunction} arises naturally when dealing with simple population dynamics in heterogeneous spaces.
Let $\e\geq 0$ be a parameter of the model. We consider a population density whose flux is given by
$$
\mathcal J_\e=-\n u+\e u \n m.
$$
Since $\n m$ might not make sense if $m$ is assumed to be only measurable, we temporarily omit this difficulty by assuming it smooth enough so that the expression above makes sense. The term $u \n m$ appears as a drift term and stands for a bias in the population movement, modeling a tendency of the population to disperse along the gradient of resources and hence move to favorable regions. The parameter $\e$ quantifies the influence of the resources distribution on the movement of the species.
The complete associated reaction diffusion equation, called ``logistic diffusive equation'', reads
$$
\frac{\partial u}{\partial t}=\n \cdot \Big(\n u-\e u \n m\Big)+mu-u^2\quad \text{in }\O,
$$
completed with suitable boundary conditions. In what follows, we will focus on {Dirichlet boundary conditions} meaning that the boundary of $\O$ is lethal for the population living inside.
Plugging the change of variable $v=e^{-\e m}u$ in this equation leads to
$$
\frac{\partial v}{\partial t}=\Delta v+\e \n m\cdot \n u +mv-e^{\e m}v\quad \text{in }\O.
$$
It is known (see e.g. \cite{BelgacemCosner,Belgacem,Murray}) that the asymptotic behavior of this equation is driven by
the principal eigenvalue of the operator $\tilde {\mathcal L}:u\mapsto -\Delta u-\e \n m\cdot \n u -mu$.
The associated principal eigenfunction $\psi$ satisfies
$$
-\n \cdot \left(e^{\e m}\n \psi\right)-me^{\e m}\psi=\tilde \lambda_\e \psi e^{\e m}\quad \text{in }\O.
$$
Following the approach developed in \cite{LamboleyLaurainNadinPrivat}, optimal configurations of resources correspond to the ones ensuring the fastest convergence to the steady-states of the PDE above, which comes to
minimizing $\tilde \lambda_\e(m)$ with respect to $m$.
By using Proposition \ref{Th:BangBang}, which enables us to only deal with {\it bang-bang} densities $m$, one shows easily that minimizing $\tilde \lambda_\e(m)$ over $\mathcal M_{m_0,\kappa}(\O)$ is equivalent to minimizing $\lambda_\e(m)$ over $\mathcal M_{m_0,\kappa}(\O)$, in other words to
Problem \eqref{Pb:OptimizationEigenvalue} with $\alpha=\e$. Theorem \ref{Th:NonExistence} can thus be interpreted as follows in this framework: assuming that the population density moves along the gradient of the resources, it is not possible to lay the resources in an optimal way.
Note that the conclusion is completely different in the case $\alpha=0$ (see \cite{LamboleyLaurainNadinPrivat}) or in the one-dimensional case (i.e. $\O=(0,1)$) with $\alpha >0$ (see \cite{CaubetDeheuvelsPrivat}), where minimizers exist.
In the last case, optimal configurations for three kinds boundary conditions (Dirichlet, Neumann, Robin) have been obtained, by using a new rearrangement technique.
Finally, let us mention the related result \cite[Theorem 2.1]{Hamel2011}, dealing with Faber-Krahn type inequalities for general operators of the form
$$
\mathcal K:u\mapsto -\n \cdot(A\n u)- V\cdot \n u-mu
$$
where $A$ is a positive symmetric matrix.
Let us denote the first eigenvalue of $\mathcal K$ by $E(A,V,m)$. It is shown, by using new rearrangements, that there exist radially symmetric elements $A^*,V^*,m^*$ such that
$$
0< \inf A\leq A^*\leq \Vert A\Vert _\infty,\quad \Vert A^{-1}\Vert _{L^1}=\Vert (A^*)^{-1}\Vert _{L^1},\quad
\Vert V^*\Vert _{L^\infty}\leq \Vert V\Vert _{L^\infty}
$$
and $E(A,V,m)\geq E(A^*,V^*,m^*)$.
We note that applying this result directly to our problem would not allow us to conclude. Indeed, we would get that for every $\O$ of volume $V_1$ and every $m\in \mathcal M_{m_0,\kappa}(\O)$, if $\O^*$ is the ball of volume $V_1$, there exist two radially symmetric functions $m_1$ and $m_2$ satisfying $m_1$, $m_2$ in $\mathcal M_{m_0,\kappa}(\O)$ such that $\lambda_\alpha(m)\geq \mu_\alpha(m_1,m_2)$, where $\mu_\alpha(m_1,m_2)$ is the first eigenvalue of the operator $-\n \cdot((1+\alpha m_1)\n )-m_2$.
We note that this result could also be obtained by using the symmetrization techniques of \cite{AlvinoTrombetti}.
Finally, let us mention optimal control problems involving a similar model but a different cost functional, related to:
\begin{itemize}
\item the total size of the population for a logistic diffusive equation in \cite{Mazari2020,MNPChapter,Mazari2021}.
\item optimal harvesting of a marine resource, investigated in the series of articles \cite{MR3035462,MR2476432,MR3628307}.
\end{itemize}
\subsection{Notations and notational conventions, technical properties of the eigenfunctions}\label{sec:notations}
Let us {sum up} the notations used throughout this article.
\begin{itemize}[label=\textbullet]
\item $\R_+$ is the set of non-negative real numbers. $\R_+^*$ is the set of positive real numbers.
\item $n$ is a fixed positive integer and $\O$ is a bounded connected domain in $\R^n$.
\item if $E$ denotes a subset of $\O$, the notation $\mathds{1}_E$ stands for the characteristic function of $E$, equal to 1 in $E$ and 0 elsewhere.
\item the notation $\Vert \cdot\Vert $ used without subscript refers to the standard Euclidean norm in $\R^n$. When referring to the norm of a Banach space $\mathcal X$, we write it $\Vert \cdot\Vert _{\mathcal X}$.
\item The average of every $f\in L^1(\O)$ is denoted by $\fint_\O f:=\frac1{|\O|}\int_\O f$.
\item $\nu$ stands for the outward unit normal vector on $\partial \O$.
\item if $m(\cdot)$ is a given function in $L^\infty(\O)$ and $\alpha$ a positive real number, we will use the notation $\sigma_{\alpha,m}$ to denote the function $1+\alpha m$. {When there is no ambiguity, we sometimes use the notation $\sigma_{\alpha}$ to alleviate notations.}
\item If $E$ denotes a subset of $\O$ with $\mathscr{C}^2$ boundary, we will use the notations
$$
f|_{int}(y)=\lim_{x\in E,x\to y}f(x)\quad \text{and}\quad f|_{ext}(y):=\lim_{x \in (\O\backslash E),x\to y}f(y)
$$
so that $\llbracket f\rrbracket =f|_{ext}-f|_{int}$ denotes the jump of $f$ at $x\in \partial E$.
\end{itemize}
\section{Preliminaries}\label{Pr:BangBang}
\def\dlamh{{\dot \lambda_\alpha(m)[h]}}
\def\ddlamh{{\ddot \lambda_\alpha(m)[h]}}
\def\dduamh{{\ddot u_{\alpha,m}[h]}}
\subsection{Switching function}
To derive optimality conditions for Problem~\eqref{Pb:OptimizationEigenvalue}, we introduce the tangent cone to $\mathcal{M}_{m_0,\kappa}(\O)$ at any point of this set.
\begin{definition} (\cite[chapter 7]{HenrotPierre})\label{def:tgtcone}
For every $m\in \mathcal M_{m_0,\kappa}(\O)$, the tangent cone to the set $\mathcal{M}_{m_0,\kappa}(\O)$ at $m$, also called the \textit{admissible cone} to the set $\mathcal{M}_{m_0,\kappa}(\O)$ at $m$, denoted by $\mathcal{T}_{m}$ is the set of functions $h\in L^\infty(\O)$ such that, for any sequence of positive real numbers $\varepsilon_n$ decreasing to $0$, there exists a sequence of functions $h_n\in L^\infty(\O)$ converging to $h$ as $n\rightarrow +\infty$, and $m+\varepsilon_nh_n\in\mathcal{M}_{m_0,\kappa}(\O)$ for every $n\in\N$.\label{footnote:cone}
\end{definition}
Notice that, as a consequence of this definition, any $h \in \mathcal T_m$ satisfies $\fint_\O h=0$.
\begin{lemma}\label{diff:valPvecP}
Let $m\in \mathcal M_{m_0,\kappa}(\O)$ and $h\in \mathcal T_m$. The mapping $ \mathcal M_{m_0,\kappa}(\O)\ni m\mapsto u_{\alpha,m}$ is twice differentiable at $m$ in direction $h$ in a strong $L^2(\O)$ sense and in a weak $W^{1,2}_0(\O)$ sense, and the mapping $ \mathcal M_{m_0,\kappa}(\O)\ni m\mapsto \lambda_\alpha$ is twice differentiable in a strong $L^2(\O)$ sense.
\end{lemma}
The proof of this lemma is technical and is postponed to Appendix \ref{Ap:Differentiability}.
For $t$ small enough, let us introduce the mapping $g_h:t\mapsto \lambda_\alpha \left([m+th]\right)$. Hence, $g_h$ is twice differentiable.
The first and second order derivatives of $\lambda_\alpha$ at $m$ in direction $h$, denoted by $\dot\lambda_\alpha(m)[h]$ and $\ddot\lambda_\alpha(m)[h]$, are defined by
$$
\dot \lambda_\alpha(m)[h]:=g_h'(0)\quad\text{and}\quad \ddot\lambda_\alpha(m)[h]:=g_h''(0).
$$
\begin{lemma}\label{Le:DeriveeL21}
Let $m\in \mathcal{M}_{m_0,\kappa}(\O)$ and $h\in \mathcal{T}_m$. The mapping $m\mapsto \lambda_\alpha(m)$ is differentiable at $m$ in direction $h$ in $L^2$ and its differential reads
\begin{equation}\label{Eq:Derivee21}
\dot \lambda_\alpha(m)[h]=\int_\O h\psi_{\alpha,m},\quad \text{with}\quad \psi_{\alpha,m}:=\alpha |\n u_{\alpha,m}|^2-u_{\alpha,m}^2.
\end{equation}
The function $\psi_{\alpha,m}$ is called {\it switching function}.
\end{lemma}
\begin{proof}
According to Lemma~\ref{diff:valPvecP}, we can differentiate the variational formulation associated to \eqref{Eq:EigenFunction} and get that the differential $\duamh$ of $m\mapsto u_{\alpha,m}$ at $m$ in direction $h$ satisfies, with $\sigma_\alpha :=1+\alpha m$,
\begin{equation}\label{Eq:L2Derivative1}\left\{\begin{array}{lll}
-\n \cdot\Big(\sigma_\alpha\n \duamh\Big)-\alpha \n \cdot\Big(h\n \uam\Big)=&\dlamh \uam +\lambda_\alpha(m)\duamh & \\
& +m\duamh+h\uam & \text{ in }\O,\\
\duamh=0 {\text{ on } \partial \O},\\
\int_\O \uam \duamh=0.&
\end{array}\right.
\end{equation}
Multiplying {(\ref{Eq:L2Derivative1})} by $\uam$, integrating by parts and using that $\uam$ is normalized in $L^2(\O)$ leads to
\begin{align*}
\dlamh =&\underbrace{\int_\O \sigma_\alpha \n \duamh\cdot \n \uam -\lambda_{\alpha}(m)\int_\O \uam \duamh-\int_\O m\uam \duamh }_{=0\text{ according to \eqref{Eq:EigenFunction}}}\\
&+\int_\O \alpha h |\n \uam|^2-\int_\O h u_{\alpha,m}^2.
\end{align*}
\end{proof}
\subsection{Proof of Proposition~\ref{Th:BangBang}}
The proof relies on concavity properties of the functional $\lambda_\alpha$. More precisely, let $m_1,m_2\in \mathcal M_{m_0,\kappa}(\O)$. We will show that the map $f:[0,1]\ni t\mapsto \lambda_\alpha \left((1-t)m_1+tm_2\right)$ is strictly concave, i.e that $ f''< 0$ on $[0,1]$.
Note that the characterization of the concavity in terms of second order derivatives makes sense, according to Lemma~\ref{diff:valPvecP}, since $\lambda_\alpha$ is twice differentiable. Before showing this concavity property, let us first explain why it implies the conclusion of Proposition~\ref{Th:BangBang} (the weak {\it bang-bang} property).
{Suppose that $m\in \mathcal M_{m_0,\kappa}(\O)$ is not { bang-bang}}. The set $\mathcal{I}=\{0<m<\kappa\}$ is then of positive Lebesgue measure and $m$ is therefore not extremal in $ \mathcal M_{m_0,\kappa}(\O)$, according to \cite[Prop.~7.2.14]{HenrotPierre}.
We then infer the existence of $t \in (0,1)$ as well as two distinct elements $m_1$ and $m_2$ of $ \mathcal M_{m_0,\kappa}(\O)$ such that $m=(1-t) m_1+tm_2$.
Because of the strict concavity of $\lambda_\alpha$, the solution of the optimization problem $\min \{\lambda_\alpha((1-t) m_1+tm_2)\}$ is either $m_1$ or $m_2$, and moreover, $m$ cannot solve this problem.
Assume that $m_1$ solves this problem without loss of generality. One thus has $\lambda_\alpha(m_1)<\lambda_\alpha (m)$. Since the subset of {\it bang-bang} functions of $\mathcal M_{m_0,\kappa}(\O)$ is dense in $\mathcal M_{m_0,\kappa}(\O)$ for the weak-star topology of $L^\infty(\O)$, there exists a sequence of {\it bang-bang} functions $(m^k)_{k\in \N}$ of $\mathcal M_{m_0,\kappa}(\O)$ converging weakly-star to $m_1$ in $L^\infty(\O)$. Furthermore, $\lambda_\alpha$ is upper semicontinuous for the weak-star topology of $L^\infty(\O)$, since it reads as the infimum of continuous linear functionals for this topology. Let $\varepsilon>0$. We infer the existence of $k_\varepsilon\in \N$ such that $\lambda_\alpha(m^{k_\varepsilon})\leq \lambda_\alpha(m_1)+\varepsilon$. By choosing $\varepsilon$ small enough, we get that $\lambda_\alpha(m^{k_\varepsilon})<\lambda_\alpha (m)$, whence the result.
\medskip
It now remains to prove that $f$ is strictly concave. Let $m\in \mathcal M_{m_0,\kappa}(\O)$, and set $m_1=m$, $h=m_2-m_1$, we observe that $f''(t)=\ddot\lambda_{\alpha}((1-t)m_1+tm_2)[h]$ for all $t\in [0,1]$.
The differential $\duamh$ of $m\mapsto u_{\alpha,m}$ at $m$ in direction $h$, denoted $\duamh$, satisfies \eqref{Eq:L2Derivative1} and the second order {Gateaux derivatives} $\dduamh$ and $\ddlamh$ solve, with $\sigma_\alpha:=1+\alpha m$,
\begin{equation}\label{Eq:L2Derivative2}
\left\{\begin{array}{lll}
-\n \cdot\Big(\sigma_\alpha\n \dduamh\Big)-2\alpha \n \cdot\Big(h\n \duamh\Big)=&\ddlamh \uam +2\dlamh \duamh \\
&+\lambda_\alpha(m)\dduamh+m\dduamh+2h\duamh & \text{ in }\O,\\
\dduamh=0 \quad {\text{ on } \partial \O.}&&
\end{array}\right.\end{equation}
Multiplying {(\ref{Eq:L2Derivative2})} by $\uam$, using that $\uam$ is normalized in $L^2(\O)$ and integrating by parts yields
\begin{align*}
\ddlamh&=\underbrace{\int_\O \sigma_\alpha \n \dduamh\cdot \n \uam-\lambda_{\alpha}(m)\int_\O \uam \dduamh-\int_\O m\uam \dduamh }_{=0\text{ according to \eqref{Eq:EigenFunction}}}
\\&+2\alpha \int_\O h \n \duamh,\n \uam-2\int_\O h\uam\duamh
\\&=2\left(-\int_\O \sigma_\alpha |\n \duamh|^2+\int_\O m\duamh^2+\lambda_\alpha(m)\int_\O \duamh^2\right)+2\underbrace{\dlamh \int_\O u_{\alpha,m} \duamh}_{=0\text{ since $\int_\O \uam \duamh=0$}}
\\&=2\int_\O \duamh^2\left(-R_{\alpha,m}[\duamh]+\lambda_\alpha(m)\right)< 0,
\end{align*}
where the last inequality comes from the observation that, whenever $h\neq 0$, one has $\duamh\neq 0$ and $\duamh$ is in the orthogonal space to the first eigenfunction $\uam$ in $L^2(\O)$. Since the first eigenvalue is simple, the Rayleigh quotient of $\duamh$ is greater than $\lambda_{\alpha}(m)$.
\section{Proof of Theorem \ref{Th:NonExistence}}\label{Pr:NonExistence}
This proof is based on a homogenization argument, inspired from the notions and techniques introduced in \cite{MuratTartar}. In the next section, we gather the preliminary tools and material involved in what follows.\subsection{Background material on homogenization and bibliographical comments}\label{SSE:BackgroundH}
Let us recall several usual definitions and results in homogenization theory we will need hereafter.
\begin{definition}[$H$-convergence]
Let $(m_k)_{k\in \N}\in \mathcal M_{m_0,\kappa}(\O)^\N$ and for every $k\in \N$, define respectively $\sigma_k$ and $u_k(f)$ by $\sigma_k=1+\alpha m_k$ and as the unique solution of
$$
\left\{\begin{array}{ll}-\n \cdot(\sigma_k \n u_k(f))=f&\text{ in }\O\\
u_k(f)=0 \text{ on }\partial \O
\end{array}
\right.
$$
where $f\in L^2(\O)$ is given. We say that the sequence $(\sigma_k)_{k\in \N}$ $H$-converges to $A:\O\to M_{n}(\R)$ if, for every $f\in L^2(\O)$, the sequence $(u_k(f))_{k\in \N}$ converges weakly to $u_\infty$ in $\wo$ and the sequence $(\sigma_k\n u_k)_{k\in \N}$ converges weakly to $A\n u_\infty $ in $ L^2(\O)$, where $u_\infty$ solves
$$\left\{\begin{array}{ll}-\n \cdot(A\n u_\infty)=f&\text{ in }\O\, ,
\\u_\infty=0 \text{ on }\partial \O\end{array}
\right.$$
In that case, we will write $\sigma_k \overset{H}{\underset{k\to \infty}\longrightarrow} A$.
\end{definition}
\begin{definition}[arithmetic and geometric means]\label{De:Homo}
Let $m\in \mathcal M_{m_0,\kappa}(\O)$ and $\sigma=1+\alpha m$. We define the arithmetic mean of $\sigma$ by $\Lambda_+(m)=\sigma$, and its harmonic mean by $\Lambda_-(m)=\frac{1+\alpha \kappa}{1+\alpha(\kappa-m)}$.
One has $\Lambda_-(m)\leq \Lambda_+(m)$, according to the arithmetic-harmonic inequality, with equality if and only if
$m$ is a {\it bang-bang} function.
\end{definition}
\begin{prnonumbering}\cite[Proposition 10]{MuratTartar}
Let $(m_k)_{k\in \N}\in \mathcal M_{m_0,\kappa}(\O)^\N$ and $(\sigma_k)_{k\in \N}$ given by $\sigma_k=1+\alpha m_k$. Up to a subsequence, there exists $m\in \mathcal M_{m_0,\kappa}(\O)$ such that $(m_k)_{k\in \N}\in \mathcal M_{m_0,\kappa}(\O)^\N$ converges to $m$ for the weak-star topology of $L^\infty$.
Assume moreover that the sequence $(\sigma_k)_{k\in \N}$ $H$-converges to a matrix $A$. Then, $A$ is a symmetric matrix, its spectrum $\Sigma(A)=\{\lambda_1,\dots,\lambda_n\}$ is real, and
\begin{equation} \label{Spec}\tag{$J_1$} \Lambda_-(m)\leq \min \Sigma(A)\leq \max \Sigma(A)\leq \Lambda_+(m).\end{equation}
\begin{equation}\label{Spec1}\tag{$J_2$}\sum_{j=1}^n \frac1{\lambda_j-1}\leq \frac1{\Lambda_-(m)-1}+\frac{n-1}{\Lambda_+(m)-1},\end{equation}
\begin{equation} \label{Spec2}\tag{$J_3$}\sum_{j=1}^n \frac1{1+\alpha\kappa-\lambda_j}\leq \frac1{1+\alpha \kappa-\Lambda_-(m)}+\frac{n-1}{1+\alpha\kappa -\Lambda_+(m)}.\end{equation}
\end{prnonumbering}
For a given $m\in \mathcal M_{m_0,\kappa}(\O)$, we introduce
$$
M_m^\alpha=\{A:\O\rightarrow S_n(\R)\, , A \text{ satisfies \eqref{Spec}-\eqref{Spec1}-\eqref{Spec2}}\}.
$$
For a matrix-valued application $A\in M_m^\alpha$ for some $m\in \mathcal M_{m_0,\kappa}(\O)$, it is possible to define the principal eigenvalue of $A$ via Rayleigh quotients as
\begin{equation}\label{Eq:Zeta}
\zeta_\alpha(m,A):=\inf_{u\in \wo\, , \int_\O u^2=1}\int_\O A\n u\cdot \n u-\int_\O mu^2.
\end{equation}
Note that the dependence of $\zeta_\alpha$ on the parameter $\alpha$ is implicitly contained in the condition $A\in M_m^\alpha$. We henceforth focus on the following relaxed version of the optimization problem:
\begin{equation}\label{Pb:OptimizationRelax}
\inf_{m\in \mathcal M_{m_0,\kappa}(\O)\, , A\in M_m^\alpha} \zeta_\alpha(m,A).
\end{equation}
for which we have the following result.
\begin{thmnonumbering}\cite[Proposition 10]{MuratTartar}
\begin{itemize}
\item[(i)] For every $m\in \mathcal M_{m_0,\kappa}(\O)$ and $A \in M_m^\alpha$, there exists a sequence $(m_k)_{k\in \N}\in \mathcal M_{m_0,\kappa}(\O)$ such that $(m_k)_{k\in \N}$ converges to $m$ for the weak-star topology of $L^\infty$, and the sequence $(\sigma_k)_{k\in \N}$ defined by $\sigma_k=1+\alpha m_k$ $H$-converges to $A$, as $k\to +\infty$.
\item[(ii)] The mapping $(m,A)\mapsto \lambda_\alpha(m,A)$ is continuous with respect to the $H$-convergence (see in particular \cite{Oleinik}).
\item[(iii)] The variational problem \eqref{Pb:OptimizationRelax}
has a solution $(\hat{m},\hat{A})$; by definition, $\hat{A}\in M_{\hat{m}}^\alpha$. Furthermore, if $\hat{u}$ is the associated eigenfunction, then $\hat{A} \n \hat{u} =\Lambda_-(\hat{m})\n \hat{u}$.
\end{itemize}
\end{thmnonumbering}
This theorem allows us to solve Problem~\eqref{Pb:OptimizationRelax}.
\begin{corollary}\cite{MuratTartar}\label{Co:Equivalence}
If Problem~\eqref{Pb:OptimizationEigenvalue} has a solution $\hat{m}$, then the couple $(\hat{m},1+\alpha \hat{m})$ solves Problem~\eqref{Pb:OptimizationRelax}.
\end{corollary}
\begin{proof}[Proof of Corollary \ref{Co:Equivalence}]
Assume that the solution of \eqref{Pb:OptimizationRelax} is $(\hat{m}, \hat{A})$ and that $\hat{A}\neq 1+\alpha \hat{m}$. Then there exists a sequence $(m_k)_{k\in \N}$ converging
weak-star in $L^\infty$ to $\hat{m}$ and such that the sequence $(1+\alpha m_k)_{k\in \N}$ $H$-converges to $\hat{A}$. This means that
$$\lambda_\alpha(\hat{m})=\zeta_\alpha(\hat{m},1+\alpha \hat{m})>\zeta_\alpha(\hat{m},\hat{A})=\underset{k\to \infty}\lim \lambda_\alpha(m_k)$$ which immediately yields a contradiction.
\end{proof}
Let us end this section with several bibliographical comments on such problems.
\paragraph{Bibliographical comments on the two-phase conductors problem.}\label{TwoPhase}
Problem \eqref{Pg:TwoPhase} {with $\mathcal M:=\mathcal M_{m_0,\kappa}(\Omega)$} has drawn a lot of attention in the last decades, since the seminal works by Murat and Tartar, \cite{Murat,MuratTartar}
Roughly speaking, this optimal design problem is, in general, ill-posed and one needs to introduce a relaxed formulation to get existence.
We refer to \cite{Allaire,CoxLipton,MuratTartar,Oleinik}.
Let us provide the main lines strategy to investigate existence issues for Problem \eqref{Pg:TwoPhase}, according to \cite{Murat,MuratTartar}.
If the solution $(\hat{m},1+\alpha\hat{m})$ to the relaxed problem \eqref{Pb:OptimizationRelax} is a solution to the original problem \eqref{Pg:TwoPhase}, then there exists a measurable subset $\hat{E}$ of $\O$ such that $\hat{m}=\kappa \mathds{1}_{\hat{E}}$. If furthermore $\hat{E}$ is assumed to be smooth enough, then, denoting by $\hat{u}$ the principal eigenfunction associated with $(\hat{m},\lambda_\alpha(m))=(\hat{m},\zeta_\alpha(\hat{m}, 1+\alpha \hat{m}))$, we get that $\hat{u}$ and $(1+\alpha \hat{m}) \frac{\partial \hat{u}}{\partial \nu}$ must be constant on $\partial \hat{E}$.
The function $1+\alpha \hat{m}$ being discontinuous across $\partial E$, the optimality condition above has to be understood in the following sense: the function $(1+\alpha \hat{m}) \frac{\partial \hat{u}}{\partial \nu}$, a priori discontinuous, is in fact continuous across $\partial E$ and even constant on it.
Note that these arguments have been generalized in \cite{CoxLipton}. These optimality conditions, combined with Serrin's Theorem \cite{Serrin1971}, suggest that Problem \eqref{Pg:TwoPhase} could have a solution if, and only if $\O$ is a ball. The best results known to date are the following ones.
\begin{thmnonumbering}
\begin{itemize}
\item[(i)] Let $\O$ be an open set such that $\partial \O$ is $\mathscr C^2$ and connected. Problem \eqref{Pg:TwoPhase} has a solution if and only if $\O$ is a ball \cite{CasadoDiaz3}.
\item[(ii)] If $\O$ is a ball, then Problem \eqref{Pg:TwoPhase} has a solution which is moreover radially symmetric \cite{ConcaMahadevanSanz}.
\end{itemize}
\end{thmnonumbering}
Regarding the second part of the theorem, the authors used a particular rearrangement coming to replace $1+\alpha m$ by its harmonic mean on each level-set of the eigenfunction. Such a rearrangement has been first introduced by Alvino and Trombetti \cite{AlvinoTrombetti}. This drives the author to reduce the class of admissible functions to radially symmetric ones, which allow them to conclude thanks to a compactness argument \cite{AlvinoTrombettiLions}. These arguments are mimicked to derive the existence part of Theorem~\ref{Th:RadialStability}.
Finally, let us mention \cite{ConcaLaurainMahadevan,Laurain}, where the optimality of annular configurations in the ball is investigated.
A complete picture of the situation is then depicted in the case where $\alpha$ is small, which is often referred to as the "low contrast regime".
We also mention \cite{DambrineKateb} , where a shape derivative approach is undertaken to characterize minimizers when $\O$ is a ball.
\subsection{Proof of Theorem \ref{Th:NonExistence}}
Let us assume the existence of a solution to Problem \eqref{Pb:OptimizationEigenvalue}, denoted $\hat{m}$. According to Proposition~\ref{Th:BangBang}, there exists a measurable subset $\hat{E}$ of $\O$ such that
$\hat{m}=\kappa \mathds{1}_{\hat{E}}$. Let us introduce $\hat{\sigma}:=1+\alpha \hat{m}$ and $\hat{u}$, the $L^2$-normalized eigenfunction associated to $\hat{m}$.
Let us now assume that $\partial \hat{E}$ is $\mathscr C^2$.
\paragraph{Step 1: derivation of optimality conditions.}
What follows is an adaptation of \cite{CoxLipton}. For this reason, we only recall the main lines.
Let us write the optimality condition
for the problem
$$\min_{m\in \mathcal M_{m_0,\kappa}(\O)}\min_{A\in M_m^\alpha}\zeta_\alpha(m,A)=\lambda_\alpha(\hat{m}),$$
where $\zeta_\alpha$ is given by \eqref{Eq:Zeta}. Let $h$ be an admissible perturbation at $\hat{m}$. In \cite{MuratTartar} it is is proved that for every $\e>0$ small enough, there exists a matrix-valued application $A_\e \in M_{\hat{m}+\e h}$ such that
$$
A_\e\n \hat{u}=\Lambda_-(\hat{m}+\e h)\n \hat{u}\quad \text{in }\O,
$$
where $\Lambda_-$ has been introduced in Definition \ref{De:Homo}.
Fix $\e$ as above. Since $(\hat{m},1+\alpha \hat{m})$ is a solution of the Problem \eqref{Pb:OptimizationRelax}, one has
\begin{align*}
\int_\O A_\e \n \hat{u}\cdot \n \hat{u}-\int_\O (m+\e h)\hat{u}^2&\geq \zeta_\alpha(m+\e h,A_\e) \\
&\geq \zeta_\alpha(\hat{m},1+\alpha \hat{m})=\int_\O \hat{\sigma} |\n \hat{u}|^2-\int_\O \hat{m}\, \hat{u}^2.
\end{align*}
where one used the Rayleigh quotient definition of $\zeta_\alpha$ as well as the minimality of $(\hat{m},1+\alpha \hat{m})$.
Dividing the last inequality by $\e$ and passing to the limit yields
$$\int_\O h\left.\frac{d\Lambda_-(m)}{dm}\right|_{m=\hat{m}}|\n \hat{u}|^2-h\hat{u}^2\geq 0.$$
Using that $d\Lambda_-/dm=\alpha \Lambda_-(m)^2/(1+\alpha\kappa)$, and that $\hat{m}$ is a {\it bang-bang} function (so that $\Lambda_-(\hat{m})=\hat{\sigma}$),
we infer that the first order optimality conditions read: there exists $\mu\in \R$ such that
\begin{equation}\label{Eq:OptimalityH}
\left\{\Psi_\alpha< \mu\right\}\subset \hat{E}\subset \left\{\Psi_\alpha\leq \mu\right\}\quad \text{where}\quad \Psi_\alpha:=\frac{\alpha}{1+\alpha\kappa}{\hat{\sigma}}^2|\n \hat{u}|^2-{\hat{u}}^2.
\end{equation}
Since the flux $\hat{\sigma} \frac{\partial \hat{u}}{\partial \nu}$ is continuous across $\partial \hat{E}$, one has necessarily $\Psi_\alpha=\mu$ on $\partial \hat{E}\backslash \partial \Omega$.
Now, let us follow the approach used in \cite{MuratTartar} and \cite{CasadoDiaz3} to simplify the writing of the optimality conditions. Notice first that $\hat{u}$ and $\hat{\sigma}^2 \left|\frac{\partial \hat{u}}{\partial \nu_{\hat{E}}}\right|^2$ are continuous across $\partial \hat{E}$. Let $\n_\tau \hat{u}$ denote the tangential gradient of $\hat{u}$ on $\partial \hat{E}$.
For the sake of clarity, the quantities computed on $\partial \hat{E}$ seen as the boundary of $\hat{E}$ will be denoted with the subscript $int$, whereas the ones computed on $\partial \hat{E}$ seen as part of the boundary of $\hat{E}^c$ will be denoted with the subscript $ext$. According to the optimality conditions \eqref{Eq:OptimalityH}, one has
\begin{align*}
\begin{split}
\left.\frac\alpha{1+\alpha\kappa} \hat{\sigma}^2 \left|\n_\tau \hat{u}\right|^2+\frac\alpha{1+\alpha\kappa} \hat{\sigma}^2\left(\frac{\partial \hat{u}}{\partial \nu}\right)^2-\hat{u}^2\right|_{int}
\leq \left.\frac\alpha{1+\alpha\kappa} \hat{\sigma}^2 \left|\n_\tau \hat{u}\right|^2+\frac\alpha{1+\alpha\kappa} \hat{\sigma}^2\left(\frac{\partial \hat{u}}{\partial \nu}\right)^2-\hat{u}^2\right|_{ext}
\end{split}\end{align*}
on $\partial\hat{E}\backslash \partial \Omega$. By continuity of the flux $\hat{\sigma} \frac{\partial \hat{u}}{\partial \nu}$, we infer that
$\alpha \hat{\sigma}^2 \left.\left|\n_\tau \hat{u}\right|^2\right|_{int}\leq \alpha \hat{\sigma}^2 \left.\left|\n_\tau \hat{u}\right|^2\right|_{ext}$ which comes to $(1+\alpha \kappa) \left.\left|\n_\tau \hat{u}\right|^2\right|_{int}\leq \left.\left|\n_\tau \hat{u}\right|^2\right|_{ext}$. Since $ \left.\left|\n_\tau \hat{u}\right|^2\right|_{int}= \left.\left|\n_\tau \hat{u}\right|^2\right|_{ext}$, we have $\left.\n_\tau \hat{u}\right|_{\partial \hat{E}}=0$. Therefore, $\hat{u}$ is constant on $\partial \hat{E}\backslash \partial \Omega$ and since $\Psi_\alpha$ is constant on $\partial \hat{E}\backslash \partial \Omega$, it follows that $\left|\frac{\partial \hat{u}}{\partial \nu}\right|^2_{int}$ is constant as well on $\partial \hat{E}\backslash \partial \Omega$.
To {sum up}, the first order necessary conditions drive to the following condition:
\begin{equation}\label{Eq:Opt}
\text{The functions }\hat{u}\text{ and } |\n \hat{u}|\text{ are constant on $\partial \hat{E}\backslash \partial \Omega$.}
\end{equation}
\paragraph{Step 2: proof that $\O$ is necessarily a ball.}
To prove that $\O$ is a ball, we will use Serrin's Theorem, that we recall hereafter.
\begin{thmnonumbering}\cite[Theorem 2]{Serrin1971}
Let $\mathscr E$ be a connected domain with a $\mathscr C^2$ boundary, $h$ a $\mathscr C^1(\R;\R)$ function and let $f\in \mathscr C^2\left(\overline{\mathscr E}\right)$ be a function satisfying
$$
-\Delta f= h(f)\,, \quad f>0 \text{ in }\mathscr E,\quad f=0\text{ on }\partial \mathscr E,\quad \frac{\partial f}{\partial \nu}\text{ is constant on }\partial \mathscr E.$$
Then $\mathscr E$ is a ball and $f$ is radially symmetric.
\end{thmnonumbering}
According to \eqref{Eq:Opt}, let us introduce $\hat{\mu}=\left.\hat{u}\right|_{\partial \hat{E}}$. One has $\hat{\mu}>0$ by using the maximum principle. Let us set $f=\hat{u}-\hat{\mu}$, $h(z)=\left(\lambda_\alpha(\hat{m})+\kappa\right)z$ and call $\mathscr E$ a given connected component of $\hat{E}$. By assumption, $\hat{E}$ is a $\mathscr C^2$ set, and, according to \eqref{Eq:Opt}, the function $\partial \left(\hat{u}-\hat{\mu}\right)/\partial \nu$ is constant on $\partial \hat{E}$.
The next result allows us to verify the last assumption of Serrin's theorem.
\begin{lemma}\label{Eq:Opt2}
There holds $\hat{u}>\hat{\mu}$ in $\hat{E}$.
\end{lemma}
For the sake of clarity, the proof of this lemma is postponed to the end of this section.\\
Let us now come back to the proof that $\O$ is necessarily a ball. Take $x\in\partial \O$. Then $x$ belongs either to the closure of $\Omega \backslash \hat E$ or to the closure of $\hat{E}$. In the first case, there exists a connected component $\Gamma$ of $\partial \big(\O\backslash \hat{E}\big)$ which contains $x$.
Let us assume by contradiction that this connected component also intersects $\O$, then according to \eqref{Eq:Opt}, one has $\hat{u}$ constant on $\Gamma \cap \O$ and according to the Dirichlet boundary conditions, one has $\hat{u}=0$ on $\Gamma$, and hence $\hat{u}$ reaches its minimal value in the open set $\O$. According to the strong maximum principle, one gets that $\hat{u}(\cdot)=0$, and we have reached a contradiction.
Hence, $\Gamma \subset \partial\O$ and $\Gamma$ is connected. Similarly, if $x$ belongs to the closure of $\hat{E}$, then there exists a connected component of $\partial\hat{E}$ containing $x$, which is included in $\partial\O$. Hence, $\partial \O$ is the union of closed connected components of the boundaries of $\hat{E}$ and $\O\backslash \hat{E}$. As $\partial\O$ is connected by hypothesis, there only exists one such connected component, that we denote $\Gamma$. This implies in particular that if $\partial \hat E\cap \partial \O\neq \emptyset$, then $\partial (\O\backslash \hat E)\cap\partial \O=\emptyset$, and conversely, if $\partial\hat{E}^c \cap\partial \O\neq \emptyset$, then $\partial \hat E\cap \partial \O=\emptyset$.
{Assume first that the closure of $\O\backslash \hat{E}$ meets $\partial \O$. As $\hat{E}$ does not intersect $\partial \O$ in this case, one has $\hat{u}$ and $|\nabla \hat{u}|$ constant over the whole boundary of $\hat{E}$ by (\ref{Eq:Opt}) and thus Serrin's theorem applies: any connected component of $\hat{E}$ is a ball and $\hat{u}$ is radially symmetric over it. We can hence fix a ball $\mathbb B(x_0;a_1)\subset \hat E$ such that $\hat u$ is radially symmetric in it.}
Let us now consider the largest ball $\mathbb{B}(x_{0};a)\subset \O$ such that $\hat u$ is radially symmetric in $\mathbb B(x_0;a)$. If $\partial \mathbb B(x_0;a)\cap \partial \O\neq \emptyset$, then $\hat u=0$ on $\partial \mathbb B(x_0;a)$ which, by the maximum principle, implies that $\O=\mathbb B(x_0;a)$. We can hence assume that $\hat u>0$ on $\partial \mathbb B(x_0;a)$. Let us define
$$\mu_a:=\left.\frac{\alpha}{1+\alpha\kappa}{\hat{\sigma}}^2|\n \hat{u}|^2-{\hat{u}}^2\right|_{\partial \mathbb B(x_0;a)}.$$ If $\mu_a<\mu$ then by continuity of $\frac{\alpha}{1+\alpha\kappa}{\hat{\sigma}}^2|\n \hat{u}|^2-{\hat{u}}^2$, it follows that $m=0$ on $\mathbb B(x_0;a+\delta)\backslash \mathbb B(x_0;a-\delta)$. Thus, applying the Cauchy-Kovalevskaya theorem to all the tangential derivatives of $\hat{u}$ yields that $\hat{u}$ is radially symmetric in $\mathbb B(x_0;a+\delta)$ which is a contradiction with the definition of $a$. Indeed, the same arguments as \cite[Proof of Theorem~1, Part~2]{LamboleyLaurainNadinPrivat} would yield that if one writes $\hat{u}(x)=U(r)$, with $r:=|x-x_{0}|$, then there exists $\delta>0$ such that $U'(r)=0$ for all $r\in [a,a+\delta)$. Hence $u$ would remain constant on the annulus $\{a<|x-x_{0}<a+\delta\}$, yielding a contradiction.
The same reasoning yields the same contradiction if $\mu_a>\mu$, in which case $m=\kappa$ on $\mathbb B(x_0;a+\delta)\backslash \mathbb B(x_0;a-\delta)$. Finally, if $\mu_a=\mu$ then if follows from Lemma \ref{Eq:Opt2} that either $m=0$ in $\mathbb B(x_0;a)\backslash \mathbb B(x_0;a-\delta)$ and $m=\kappa$ in $\mathbb B(x_0;a+\delta)\backslash \mathbb B(x_0;a)$ or $m=\kappa$ in $\mathbb B(x_0;a)\backslash \mathbb B(x_0;a-\delta)$ and $m=0$ in $\mathbb B(x_0;a+\delta)\backslash \mathbb B(x_0;a)$. In both case, one can apply Carleman's unique continuation Theorem as done in \cite[Proof of Theorem 2.1, Step 3]{CasadoDiaz3} to conclude that $\hat u$ is radially symmetryc in $\mathbb B(x_0;\delta)$.
{In the case where $\partial\hat{E}$ meets $\partial \O$, then the boundary of $\O\backslash \hat{E}$ does not, and we conclude by using the same reasoning on $\O\backslash \hat{E}$ instead of $\hat{E}$, showing that $\hat{u}<\hat{\mu}$ on $\O\backslash \hat{E}$ and applying Serrin's theorem to $\hat{\mu}-\hat{u}$. }
\begin{proof}[Proof of Lemma \ref{Eq:Opt2}]
Let us set $v=\hat{u}-\hat{\mu}$, hence $v$ solves
\begin{equation}
\left\{\begin{array}{ll}
-\Delta v=\left(\lambda_\alpha(\hat{m})+\kappa\right)v+\left(\lambda_\alpha(\hat{m})+\kappa\right)\hat{\mu} & \text{ in }\hat{E},\\
v=0 \text{ on }\partial\hat{E}.
\end{array}\right.
\end{equation}
and we are led to show that $v>0$ in $\hat{E}$. Let $\lambda^D(\O)$ be the first Dirichlet eigenvalue\footnote{In other words
\begin{equation}\label{Eq:LambdaD}\lambda^D(\O)=\inf_{u\in W^{1,2}_0( \O)\,,\int_{\O}u^2=1}\int_\O|\n u|^2>0.\end{equation}} of the Laplace operator in $E$.
By using the Rayleigh quotient \eqref{Eq:Rayleigh} we have
\begin{align*}
\lambda_\alpha(\hat{m})&= \min_{u\in W^{1,2}_0(\O)\,,u\neq 0}R_{\alpha,m}(u)
> \min_{u\in W^{1,2}_0(\O)\,,u\neq 0}\frac{\frac12\int_\O|\n u|^2-\kappa\int_\O u^2}{\int_\O u^2}
=\lambda^D(\O)-\kappa,
\end{align*}
so that
$\lambda_\alpha(\hat{m})+\kappa> \lambda^D(\O)>0$.
Now, since $v=0$ on $\partial \hat{E}$ and that $\hat{E}$ is a $\mathscr C^2$ open subset of $\O$, the extension $\tilde v$ of $v$ by zero outside $\hat{E}$ belongs to $W^{1,2}_0(\O)$. Since $(\lambda_\alpha(\hat{m})+\kappa)$ and $\hat{\mu}$ are non-negative, we get
$$
-\Delta \tilde v\geq (\lambda_\alpha(\hat{m})+\kappa)\tilde v\text{ in }\hat{E}
$$
Splitting $\tilde v$ into its positive and negative parts as $\tilde v=\tilde v_+-\tilde v_-$ and multiplying the equation by $\tilde v_-$ we get after an integration by parts
$$-\int_{\O} |\n \tilde v_-|^2=-\int_{\hat{E}} |\n \tilde v_-|^2\geq -(\lambda_\alpha(\hat{m})+\kappa)\int_{\hat{E}} \tilde v_-^2=-(\lambda_\alpha(\hat{m})+\kappa)\int_{\O} \tilde v_-^2.$$
Using that $\lambda_\alpha(\hat{m})+\kappa> \lambda^D(\O)>0$, we get
$$\int_\O |\n \tilde v_-|^2\leq (\lambda_\alpha(\hat{m})+\kappa)\int_\O \tilde v_-^2<\lambda^D(\O)\int_\O \tilde v_-^2,
$$
which, combined with the Rayleigh quotient formulation of $\lambda^D(\O)$ yields $\tilde v_-=0$. Hence $v$ is nonnegative in $\hat{E}$.
Using moreover that $(\lambda_\alpha(\hat{m})+\kappa)\geq 0$ and $\hat{\mu}\geq 0$ yields that $-\Delta v\geq 0$ in $\hat{E}$
Notice that $v$ does not vanish identically in $\hat{E}$. Indeed, $u$ would otherwise be constant in $\hat{E}$ which cannot arise because of \eqref{Eq:EigenFunction}. According to the strong maximum principle, we infer that $v>0$ in $\hat{E}$.
\end{proof}
\begin{remark}
Following the arguments by Casado-Diaz in \cite{CasadoDiaz3}, it would be possible to weaken the regularity assumption on $E$ provided that we assume the stronger hypothesis that $\partial\O$ is simply connected. Indeed, in that case, assuming that $E$ is only of class $\mathscr C^1$ leads to the same conclusion.
\end{remark}
\section{Proof of Theorem \ref{Th:RadialStability}}
Throughout this section, $\O$ will denote the ball $\mathbb B(0,R)$, which will also be denoted $\B$ for the sake of simplicity. Let $r^*_0\in (0,R)$ be chosen in such a way that
$m^*_0=\kappa \mathds 1_{\mathbb B(0,r^*_0)}$ belongs to $\mathcal M_{m_0,\kappa}(\O).$ Let us introduce the notation $E_0^*=\B(0,r_0^*)$.
The existence part of { Theorem \ref{Th:RadialStability}} follows from a straightforward adaptation of \cite{ConcaMahadevanSanz}. In what follows, we focus on the second part of this theorem, that is, the stationarity of minimizers provided $\alpha$ is small enough.
\subsection{Steps of the proof for the stationarity } \label{sec:stepproofstationarity}
We argue by contradiction, assuming that, for any $\alpha>0$, there exists a radially symmetric distribution $\tilde m_{\alpha}$ such that $\lambda_\alpha(\tilde m_{\alpha})<\lambda_\alpha(m^*_0)$. Consider the resulting sequence $\{\tilde m_\alpha\}_{\alpha > 0}$.
\begin{itemize}
\item \textbf{Step 1:} we prove that $\{\tilde m_\alpha\}_{\alpha \to 0}$ converges strongly to $m^*_0$ in $L^1$, as $\alpha \to 0$. Regarding the associated eigenfunction, we prove that $\{u_{\alpha,m_\alpha}\}_{\alpha> 0}$ converges strongly to $u_{0,m^*_0}$ in $\mathscr C^0$ and that $\alpha \n u_{\alpha, m_\alpha}$ converges to 0 in $L^\infty(\B)$, as $\alpha\to 0$.
\item \textbf{Step 2:} by adapting \cite[Theorem 3.7]{Laurain}, we prove that we can {restrict} ourselves to considering {\it bang-bang} radially symmetric distributions of resources $\tilde m_{\alpha}=\kappa \mathds 1_{\tilde E}$ such that the Hausdorff distance $d_H(\tilde E,E_0^*)$ is arbitrarily small.
\item \textbf{Step 3:} this is the main innovation of the proof.
Introduce $h_\alpha=\tilde m_\alpha-m^*_0$, and consider the path $\{m_t\}_{t\in [0,1]}$ from $m_\alpha$ to $m^*_0$ defined by $m_t=m_0^*+th_\alpha$. We then consider the mapping
$$
f_\alpha:t\mapsto \zeta_\alpha(m_t,\Lambda_-(m_t))
$$
where $\zeta_\alpha$ and $\Lambda_-(m_t)$ are respectively given by { equation~\eqref{Eq:Zeta} and definition~\ref{De:Homo}}. Notice that, since $m^*_0$ and $\tilde m_\alpha$ are {\it bang-bang}, $f_\alpha(0)=\lambda_\alpha(m^*_0)$ and $f_\alpha(1)=\lambda_\alpha(m_\alpha)$ according to Def. \ref{De:Homo}. Let $u_t$ be a $L^2$ normalized eigenfunction associated with $(m_t,\Lambda_-(m_t))$, in other words a solution to the equation
\begin{equation}\label{Eq:Ut}
\left\{\begin{array}{ll}
-\n\cdot\left( \Lambda_-(m_t)\n u_t\right)=\zeta_\alpha(m_t,\Lambda_-(m_t)) u_t+m_tu_t & \text{ in }\B\\
u_t=0 & \text{ on }\partial \B\\
\int_\B u_t^2=1. &
\end{array}\right.
\end{equation}
According to the proof of the optimality conditions \eqref{Eq:OptimalityH}, one has
$$
f_\alpha'(t)=\int_\B h_\alpha\left(\frac\alpha{1+\alpha\kappa} \Lambda_-(m_t)^2|\n u_t|^2-u_t^2 \right).
$$
Applying the mean value theorem yields the existence of $t_1\in (0,1)$ such that $\lambda_\alpha(\tilde m_\alpha)-\lambda_\alpha(m^*_0)={f_{\alpha}'(t_1)}$. This enables us to show that, for $t\in [0,1]$ and $\alpha$ small enough, one has
$$
f_\alpha'(t)\geq C \int_\B |h_\alpha| \operatorname{dist}(\cdot, \S)
$$ for some $C>0$, giving in turn $\lambda_\alpha(m_\alpha)-\lambda_\alpha(m^*_0)\geq C \int_\B |h_\alpha| \operatorname{dist}(\cdot, \S).$ (we note that the same quantity is obtained in \cite{Laurain}. Nevertheless, we obtain it in a more straightforward manner which bypasses the exact decomposition of eigenfunctions and eigenvalues used there.).
\end{itemize}
Let us now provide the details of each step.
\subsection{Step 1: convergence of quasi-minimizers and of sequences of eigenfunctions}
We first investigate the convergence of quasi-minimizers.
\begin{lemma}\label{Le:Convergence}
Let $\{m_\alpha\}_{\alpha>0}$ be a sequence in $\mathcal M_{m_0,\kappa}(\O)$ such that,
\begin{equation}\label{eq:lem:lambalphaineg}
\forall \alpha>0, \quad \lambda_\alpha(m_\alpha)\leq \lambda_\alpha(m^*_0).
\end{equation}
Then, $\{m_\alpha\}_{\alpha>0}$ converges strongly to $m^*_0$ in $L^1(\O)$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Le:Convergence}]
The sequence $(\lambda_\alpha(m_\alpha))_{\alpha>0}$ is bounded from above. Indeed, choosing any test function $\psi\in W^{1,2}_0(\O)$ such that $\int_\O \psi^2=1$, it follows from \eqref{Eq:Rayleigh} that $\lambda_\alpha(m_\alpha)\leq (1+\alpha \kappa)\Vert \n \psi\Vert_2^2+\kappa \Vert \psi\Vert_2^2.$
Similarly, using once again \eqref{Eq:Rayleigh}, we get that if $\xi_\alpha$ is the first eigenvalue associated to the operator $-(1+\alpha \kappa)\Delta-\kappa $, then $\lambda_\alpha(m_\alpha)\geq \xi_\alpha$.
Since $(\xi_\alpha)_{\alpha>0}$ converges to the first eigenvalue of $-\Delta -\kappa$ as $\alpha \to0$, $(\xi_\alpha)_{\alpha>0}$ is bounded from below whenever $\alpha$ is small enough. Combining these facts yields that the sequence $(\lambda_\alpha(m_\alpha))_{\alpha> 0}$ is bounded by some positive constant $M$ and converges, up to a subfamily, to $\tilde \lambda$.
For any $\alpha>0$, let us denote by $u_\alpha$ the associated $L^2$-normalized eigenfunction associated to $\lambda_\alpha(m_\alpha)$. From the weak formulation of equation \eqref{Eq:EigenFunction} and the normalization condition $\int_\O u_\alpha^2=1$, we infer that
\begin{align*}
\Vert \n u_\alpha\Vert _2^2&=\int_\O |\n u_\alpha|^2\leq \int_\O (1+\alpha m)|\n u_\alpha|^2=\int_\O m_\alpha u_\alpha^2+\lambda_\alpha(m_\alpha)\int_\O u_\alpha^2 \leq (M+\kappa).
\end{align*}
According to the Poincar\'e inequality and the Rellich-Kondrachov Theorem, the sequence $(u_\alpha)_{\alpha >0}$ is uniformly bounded in $W^{1,2}_0(\O)$ and converges, up to subfamily, to $\tilde u\in W^{1,2}_0(\O)$ weakly in $W^{1,2}_0(\O)$ and strongly in $L^2(\O)$, and moreover $\tilde u$ is also normalized in $L^2(\O)$.
Furthermore, since $L^2$ convergence implies pointwise convergence (up to a subfamily), $\tilde u$ is necessarily nonnegative in $\O$. Let $\tilde m$ be a closure point of $(m_\alpha)_{\alpha >0}$ for the weak-star topology of $L^\infty$. Passing to the weak limit in the weak formulation of the equation solved by $u_\alpha$, namely Eq.~\eqref{Eq:EigenFunction}, one gets
$$
-\Delta \tilde u-\tilde m\tilde u=\tilde \lambda \tilde u\quad \text{in }\O.
$$
Since $\tilde u\geq 0$ and $\int_{\B(0,R)}\tilde u^2=1$, it follows that $\tilde u$ is the principal eigenfunction of $-\Delta -\tilde m$, so that $\tilde \lambda=\lambda_0(m^*_0)$.
Mimicking this reasoning enables us to show in a similar way that, up to a subfamily, $(\lambda_\alpha(m^*_0))_{\alpha > 0}$ converges to $\lambda_0(m^*_0)$ and $(u_{\alpha,m^*_0})_{\alpha > 0}$ converges to $u_{0,m^*_0}$ as $\alpha \to 0$. Passing to the limit in the inequality \eqref{eq:lem:lambalphaineg} and since $m^*_0$ is the only minimizer of $\lambda_0$ in $\mathcal M_{m_0,\kappa}(\O)$ according to the Faber-Krahn inequality, we infer that necessarily, $\tilde m=m^*_0$. Moreover, $m^*_0$ being an extreme point of $\mathcal M_{m_0,\kappa}(\O)$, the subfamily $(m_\alpha)_{\alpha > 0}$ converges to $\tilde m=m^*_0$ (see \cite[Proposition 2.2.1]{HenrotPierre}), strongly in $L^1(\O)$.
\end{proof}
A straightforward adaptation of the proof of Lemma \ref{Le:Convergence} yields that both sets $\{\lambda_\alpha(m)\}_{m\in \mathcal M_{m_0,\kappa}(\O)}$ and $\{\Vert u_{\alpha,m}\Vert_{W^{1,2}(\O)}\}_{m\in \mathcal M_{m_0,\kappa}(\O)}$ are uniformly bounded whenever $\alpha \leq 1$.
Let us hence introduce $M_0>0$ such that
\begin{equation}\label{metz1946}
\forall \alpha\in [0,1], \quad \max \{|\lambda_\alpha(m)|, \Vert \uam\Vert _{W^{1,2}_0(\O)}\}\leq M_0.
\end{equation}
The next result is the only ingredient of the proof of Theorem~\ref{Th:RadialStability} where the low dimension assumption on $n$ is needed.
\begin{lemma}\label{Le:Bounds}
Let us assume that $n=1,2,3$. There exists $M_1>0$ such that, for every radially symmetric distribution $m\in \mathcal M_{m_0,\kappa}(\O)$ and every $\alpha\in [0,1]$, there holds
$$
\Vert u_{\alpha,m}\Vert _{W^{1,\infty}(\B)}\leq M_1.
$$
Furthermore, define $\tsm$, $\tm$ and $\pam:(0,R)\to \R$ by
$$\forall x \in \B,\quad \uam(x)=\pam\left(|x|\right),\quad \sigma_{\alpha,m}(x)=\tsm(|x|),\quad m(x)=\tm(|x|),$$
then $\tsm(\pam)'$ belongs to $W^{1,\infty}(0,R)$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Le:Bounds}]
This proof is inspired by \cite[Proof of Theorem 3.3]{Laurain}. It is standard that for every $\alpha\in [0,1]$ and every radially symmetric distribution $m\in \mathcal M_{m_0,\kappa}(\O)$, the eigenfunction $u_{\alpha,m}$ is itself radially symmetric.
By rewriting the equation \eqref{Eq:EigenFunction} on $ \uam$ in {spherical} coordinates, on sees that $\pam$ solves
\begin{equation}\label{Eq:EFRadial}
\left\{
\begin{array}{ll}
-\frac{d}{dr}\left(r^{n-1}\tsm \frac{d}{dr}\pam\right)=\left(\lambda_\alpha(m)\pam+\tm \pam\right)r^{n-1} & \text{in }(0,R)\\
\pam(R)=0. &
\end{array}
\right.
\end{equation}
By applying the Hardy Inequality\footnote{This inequality reads (see e.g. \cite[Lemma 1.3]{AloisKufner} or \cite{Hardy}): for any non-negative $f$,
$$
\int_0^\infty f(x)^2dx\leq 4\int_0^\infty x^2f'(x)^2dx.
$$}
to $f=\pam$, we get
{\begin{eqnarray*}
\int_0^R \p^{2}_{\alpha,m}(r)\, dr & \leq & 4\int_0^R r^2\p'_{\alpha,m}(r)^2\, dr \\
& \leq & 4R^2\int_0^R \left(\frac{r}{R^{2}}\right)^{n-1}\p'_{\alpha,m}(r)^2dr=4R^{4-2n}\Vert\n \uam\Vert_{L^2(\B)}^2\leq M,
\end{eqnarray*}}
since $n\in \{1,2,3\}$. Hence, there exists $C>0$ such that
\begin{equation}\label{Borne1D}
\Vert \pam\Vert _{L^2(0,R)}^2\leq C.
\end{equation}
We will successively prove that $\pam$ is uniformly bounded in $W^{1,2}_0(0,R)$, then in $L^\infty(0,R)$ to infer that $\p'_{\alpha,m}$ is bounded in $L^\infty(0,R)$. This proves in particular that $\sigma_{\alpha,m}\p_{\alpha,m}'\in L^{\infty}(0,R)$.
We will then conclude that $\sigma_{\alpha,m}\p_{\alpha,m}'\in W^{1,\infty}(0,R)$ by using that it is a continuous function whose derivative is uniformly bounded in $L^\infty$ by the equation on $\pam$.\
According to \eqref{metz1946}, {one sees that} $\Vert r^{\frac{n-1}2}\p'_{\alpha,m}\Vert _{L^2(0,R)}=\Vert\n \uam\Vert_{L^2(\B)}$ is bounded and therefore, $r^{n-1}\p'_{\alpha,m}(r)$ converges to 0 as $r\to 0$. {If $n=1$, such a convergence holds since $\p'_{\alpha,m}(0)=0$ by radial symmetry}. Hence, integrating Eq. \eqref{Eq:EFRadial} between $0$ and $r>0$ yields
$$
\tsm(r)\p'_{\alpha,m}(r)=-\frac1{r^{n-1}}\int_0^r t^{n-1}\left(\lambda_\alpha(m)\pam(t)+\tm(t) \pam(t)\right) dt.
$$
By using the Cauchy-Schwarz inequality and \eqref{Borne1D}, we get the existence of $\tilde M>0$ such that
\begin{eqnarray*}
\Vert \p'_{\alpha,m}\Vert _{L^2(0,R)}^2 &\leq &\int_0^1 \left(\tsm \p'_{\alpha,m}\right)^2(r)\, dr\\
&=&\int_0^1\frac1{r^{2(n-1)}}\left(\int_0^r t^{n-1}\left(\lambda_\alpha(m)\pam(t)+\tm(t) \pam(t)\right) dt\right)^2\, dr\\
&\leq &\int_0^1\frac1{r^{2(n-1)}}(\lambda_\alpha(m)+\kappa)^2\Vert \pam\Vert _{L^2(0,R)}^2\Vert {t\mapsto t^{n-1}}\Vert _{L^2(0,r)}^2\, dr\\
&\leq &\frac{\tilde M}{4n-2}\Vert \pam\Vert _{L^2(0,R)}^2\leq \tilde M\Vert \pam\Vert _{L^2(0,R)}^2\leq \tilde M C,
\end{eqnarray*}
Hence, $\pam$ is uniformly bounded in $W^{1,2}_0(0,R)$.
It follows from standard Sobolev embedding's theorems that there exists a constant $M_2>0$, such that
$\Vert \pam\Vert _{L^\infty(0,R)}\leq M_2$.
Finally, plugging this estimate in the equality
$$
\tsm(r)\p'_{\alpha,m}(r)=-\frac1{r^{n-1}}\int_0^r t^{n-1}\left(\lambda_\alpha(m)\pam(t)+\tm(t) \pam(t)\right) dt
$$
and since $t^{n-1}\leq r^{n-1}$ on $(0,r)$, we get that $\p'_{\alpha,m}$ is uniformly bounded in $L^\infty(0,R)$.
\end{proof}
The next lemma is a direct corollary of Lemma \ref{Le:Convergence}, Lemma \ref{Le:Bounds} and the Arzela-Ascoli Theorem.
\begin{lemma}\label{Le:ConvergenceFP}
Let $(m_\alpha)_{\alpha>0}$ be a sequence of radially symmetric functions of $\mathcal M_{m_0,\kappa}(\O)$ such that, for every $\alpha\in [0,1]$, $\lambda_{\alpha}(m_\alpha)\leq \lambda_{\alpha}(m^*_0)$. Then, up to a subfamily,
$u_{\alpha,m_\alpha}$ converges to $u_{0,m^*_0}$ for the strong topology of $\mathscr C^0(\overline\O)$ as $\alpha\to 0$.
\end{lemma}
\subsection{Step 2: reduction to particular resource distributions close to $m_0^*$}
Let us consider a sequence of radially symmetric distributions $(m_\alpha)_{\alpha > 0}$ such that, for every $\alpha\in [0,1]$, $\lambda_\alpha(m_\alpha)\leq \lambda_\alpha(m^*_0)$.
According to Proposition~\ref{Th:BangBang}, we can assume that each $m_\alpha$ is a {\it bang-bang}, in other words that $m_\alpha =\kappa \mathds{1}_{E_\alpha}$ where $E_\alpha$ is a measurable subset of $\B(0,R)$. For every $\alpha \in [0,1]$, one introduces $d_\alpha=d_H(E_\alpha,E^*_0)$, the Hausdorff distance of $E_\alpha$ to $E^*_0$.
\begin{lemma}\label{Le:Hausdorff}
For every $\e>0$ small enough, there exists $\overline \alpha>0$ such that, for every $\alpha\in [0, \overline \alpha]$, there exists a {radially symmetric } measurable subset $\tilde{E}_\alpha$ of $\O$ such that
$$
\lambda_\alpha(\kappa\mathds{1}_{E_\alpha})\geq \lambda_\alpha(\kappa\mathds{1}_{\tilde{E}_\alpha}), \quad |E_\alpha|=|\tilde{E}_\alpha|\quad \text{and}\quad d_H(\tilde{E}_\alpha,E^*_0) \leq \e.
$$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Le:Hausdorff}]
Let $\alpha\in [0,1]$. Observe first that
$\lambda_\alpha(m)=\int_\B|\n \uam|^2+\alpha \int_\B m|\n \uam|^2-\int_\B mu_{\alpha,m}^2=\int_\B |\n \uam|^2+\int_\B m\psi_{\alpha,m}$, where $\psi_{\alpha,m}$ has been introduced in Lemma~\ref{Le:DeriveeL21}.
We will first construct $ \tilde m_\alpha$ in such a way that
$$
\lambda_\alpha(m_\alpha)\geq \int_\B |\n u_{\alpha,m_\alpha}|^2+\int_\O \psi_{\alpha,m_\alpha} \tilde m_\alpha\geq \lambda_\alpha(\tilde m_\alpha),
$$
and, to this aim, we will define $\tilde m_\alpha$ as a suitable level set of $\psi_{\alpha,m_\alpha}$. Thus, we will evaluate the Hausdorff distance of these level sets to $E^*_0$. The main difficulty here rests upon the lack of regularity of the switching function $\psi_{\alpha,m_\alpha}$, which is { not even continuous (see Figure (\ref{fig:phi})).}
According to Lemmas~\ref{Le:Bounds} and \ref{Le:ConvergenceFP}, $\psi_{\alpha,m_\alpha}$ converges to $-u_{0,m^*_0}^2$ for the strong topology of $L^\infty(\B)$.
Recall that $m^*_0=\kappa \mathds{1}_{\mathbb B(0,r^*_0)}$ and let $V_0$ be defined by $V_0=|\mathbb B(0,r^*_0)|$.
Let us define $\mu_\alpha^*$ by dichotomy, as the only real number such that
$$
|\underline \omega_\alpha|\leq V_0\leq |\overline \omega_\alpha|,
$$
where $\underline \omega_\alpha=\{\psi_{\alpha,m_\alpha}<\mu_\alpha^*\}$ and $\overline \omega_\alpha=\{\psi_{\alpha,m_\alpha}\leq \mu_\alpha^*\}$.
Since $\left|\{\psi_{0,m^*_0}<-\varphi_{0,m_0^*}^2(r_0^*)\}\right|=V_0$, we deduce that $(\mu_\alpha^*)$ converges to $-\varphi_{0,m^*_0}^2(r^*_0)$ as $\alpha\to 0$.
Since $\varphi_{0,m^*_0}$ is decreasing, we infer that for any $\e>0$ small enough, there exists $\overline \alpha>0$ such that: for every $\alpha\in [0, \overline \alpha]$, $\mathbb B(0,r^*_0-\e)\subset \underline \omega_\alpha\subset \overline \omega_\alpha\subset \mathbb B(0,r^*_0+\e)$.
Therefore, there exists a radially symmetric set $B_\e^\alpha$ such that
$$
\underline \omega_\alpha \subset B_\e^\alpha\subset \overline \omega_\alpha,\quad |B_\e^\alpha|=V_0 ,\quad d_H(B_\e^\alpha,E^*_0)\leq \e.
$$
Since $E_\alpha$ and $B_\e^\alpha$ have the same measure, one has $|(E_\alpha)^c\cap B_\e^\alpha|=|E_\alpha\cap (B_\e^\alpha)^c|$, we introduce $\tilde m_\alpha=\kappa \mathds{1}_{B_\e^\alpha}$ so that $\tilde m_\alpha$ belongs to $\mathcal M_{m_0,\kappa}(\O)$.
\begin{figure}[H]\label{fig:phi}
\begin{center}
\includegraphics[width=5.2cm]{graphPsialpha.png}
\caption{Possible graph of the discontinuous function $\psi_{\alpha,m_\alpha}$. The bold intervals on the $x$ axis correspond to $\{m_\alpha=0\}$.
}
\end{center}
\end{figure}
By construction, one has
\begin{align*}
\lambda_\alpha(m_\alpha)&=\int_\B (1+\alpha m_\alpha)|\n u_{\alpha,m_\alpha}|^2-\int_\B m_\alpha u_{\alpha,m}^2
=\int_\B |\n u_{\alpha,m_\alpha}|^2+\int_\B \psi_{\alpha,m_\alpha}m_\alpha
\\&=\int_\B |\n u_{\alpha,m_\alpha}|^2+\kappa \int_{E_\alpha}\psi_{\alpha,m_\alpha}
=\int_\B |\n u_{\alpha,m_\alpha}|^2+\kappa \int_{E_\alpha\cap (B_\e^\alpha)^c}\psi_{\alpha,m_\alpha}+\kappa \int_{E_\alpha\cap B_\e^\alpha}\psi_{\alpha,m_\alpha}
\\&\geq \int_\B |\n u_{\alpha,m_\alpha}|^2+\kappa \mu_\alpha^* |E_\alpha \cap( B_\e^\alpha)^c|+\kappa \int_{E_\alpha\cap B_\e^\alpha}\psi_{\alpha,m_\alpha}
\\&=\int_\B |\n u_{\alpha,m_\alpha}|^2+\kappa \mu_\alpha^* |(E_\alpha)^c \cap B_\e^\alpha|+\kappa \int_{E_\alpha\cap B_\e^\alpha}\psi_{\alpha,m_\alpha}
\\&\geq \int_\B |\n u_{\alpha,m_\alpha}|^2+\kappa \int_{(E_\alpha)^c\cap B_\e^\alpha}\psi_{\alpha,m_\alpha}+\kappa \int_{E_\alpha\cap B_\e^\alpha}\psi_{\alpha,m_\alpha}
=\int_\B |\n u_{\alpha,m_\alpha}|^2+\int_\B \tilde m_\alpha \psi_{\alpha,m_\alpha}
\\&=\int_\B\sigma_{\alpha,\tilde m_\alpha} |\n u_{\alpha,m_\alpha}|^2-\int_\B \tilde m_\alpha u_{\alpha,m}^2
\geq \lambda_\alpha(\tilde m_\alpha),
\end{align*}
the last inequality coming from the variational formulation \eqref{Eq:Rayleigh}.
The expected conclusion thus follows by taking $\tilde E_\alpha:=B_\e^\alpha$.
\end{proof}
From now on we will replace $m_\alpha$ by $\kappa\mathds{1}_{\tilde{E}_\alpha}$ and still denote this function by $m_\alpha$ with a slight abuse of notation.
\subsection{Step 3: conclusion, by the mean value theorem}
Recall that, according to Section~\ref{sec:stepproofstationarity}, for every $\alpha\in [0,1]$, the mapping $f_\alpha$ is defined by $f_\alpha(t):=\zeta_\alpha(m_t,\Lambda_-(m_t))$ for all $t\in [0,1]$
We claim that $f_\alpha$ belongs to $\mathscr C^1$. This follows from similar arguments to those of the $L^2$ differentiability of $m\mapsto \lambda_\alpha(m)$ in Appendix \ref{Ap:Differentiability}. Following the proof of \eqref{Eq:OptimalityH}, it is also straightforward that for every $t\in [0,1]$, one has
\begin{equation}\label{Eq:DeriveeFAlpha}
f_\alpha'(t)=\int_\B \left(\frac\alpha{1+\alpha\kappa} \Lambda_-(m_t)^2 |\n u_t|^2-u_t^2\right)h_\alpha.\end{equation}
Finally, since $m^*_0$ and $m_\alpha$ are {\it bang-bang}, it follows from Definition~\ref{De:Homo} that
$f_\alpha(0)=\lambda_\alpha(m^*_0)$ and $f_\alpha(1)=\lambda_\alpha(m_\alpha)$.
Since $m_\alpha$ is assumed to be radially symmetric, so is $m_t$ for every $t\in [0,1]$ thanks to a standard reasoning, and, therefore, so is $u_t$. With a slight abuse of notation, we identify $m_t$, $u_t$ and $\Lambda_-(m_t)$ with their radially symmetric part $\tilde m_t$, $\tilde u_t$, $\tilde \Lambda_-(m_t)$ defined on $[0,R]$ by
$$
u_t(x)=\tilde u_t(|x|), \quad m_t(x)=\tilde m_t(|x|),\quad \Lambda_-(m_t)(x)=\tilde \Lambda_-(\tilde m_t)(|x|).
$$
Then the function $u_t$ (defined on $[0,R]$) solves the equation
\begin{equation}\label{Eq:UtRadial}
\left\{\begin{array}{ll}-\frac{d}{dr}\left(r^{n-1}\Lambda_-(m_t)\frac{du_t}{dr}\right)=\left(\zetat u_t+m_t u_t\right)r^{n-1}& r\in [0,R]\\
u_t(R)=0 &\\
\int_0^R r^{n-1}u_t(r)^2dr=\frac1{c_n},&
\end{array}\right.
\end{equation}
where $c_n=|\mathbb S(0,1)|$. As a consequence, an immediate adaptation of the proof of Lemma~\ref{Le:Bounds} yields:
\begin{lemma}\label{Le:BoundsUt}
There exists $M>0$ such that
$$
\max \left\{\Vert u_t\Vert _{W^{1,\infty}},\Big\Vert \Lambda_-(m_t)u_t'\Big\Vert_{W^{1,\infty}}\right\}\leq M.
$$
Furthermore, $\Lambda_-(m_t)u_t'$ converges to $u_{0,m^*_0}'$ in $L^\infty(0,R)$ and uniformly with respect to $t\in [0,1]$, as $\alpha\to 0$.
\end{lemma}
According to the mean value theorem, there exists $t_1=t_1(\alpha)\in [0,1]$ such that
$$
\lambda_\alpha(m_\alpha)-\lambda_\alpha(m^*_0)=f_\alpha(1)-f_\alpha(0)=f_\alpha'(t_1)
$$
and by using Eq. \eqref{Eq:DeriveeFAlpha}, one has
$$
f_\alpha'(t_1)=\int_\B \left(\frac\alpha{1+\alpha\kappa} \Lambda_-(m_{t_1})^2 |\n u_{t_1}|^2-u_{t_1}^2\right)h_\alpha,
$$
where $h_\alpha=m_\alpha-m^*_0$.
Let us introduce $I_\alpha^\pm$ as the two subsets of $ [0,R]$ given by $I_\alpha^\pm=\{h_\alpha=\pm1\}$. Let $\e>0$. According to Lemma~\ref{Le:Hausdorff}, we have, for $\alpha$ small enough,
$I_\alpha^+\subset [r^*_0,r^*_0+\e]$ and $ I_\alpha^-\subset [r^*_0-\e;r^*_0]$.
Finally, let us introduce
$$
\mathfrak F_1:=\frac\alpha{1+\alpha\kappa} \Lambda_-(m_{t_1})^2 |\n u_{t_1}|^2-u_{t_1}^2.
$$
According to Lemma~\ref{Le:BoundsUt}, $\mathfrak F_1$ belongs to $W^{1,\infty}$ and $ \mathfrak F_1+u_{\alpha,m^*_0}^2$ converges to $0$ as $\alpha\to 0$, for the strong topology of $W^{1,\infty}(0,R)$.
Moreover, there exists $M>0$ independent of $\alpha$ such that for $\e>0$ small enough,
$$
-M\leq 2 u_{\alpha,m^*_0}\frac{du_{\alpha,m^*_0}}{dr}\leq-M\quad \text{ in } [r^*_0-\e;r^*_0+\e]
$$
and it follows that
$$
\frac{M}2\leq \frac{d\mathfrak F_1}{dr}\leq 2M\quad \text{ in }[r^*_0-\e;r^*_0+\e]$$ for $\alpha$ small enough.
Hence, since $\mathfrak F_1$ is Lipschitz continuous and thus absolutely continuous, one has for every $y\in [0,\e]$,
\begin{align*}
\mathfrak F_1(r^*_0+y)&=\mathfrak F_1(r^*_0)+\int_{r^*_0}^{r^*_0+y} \mathfrak F_1'(s)\, ds\geq \mathfrak F_1(r^*_0)+\frac{M}2 y \\
\text{and}\quad \mathfrak F_1(r^*_0-y)&=\mathfrak F_1(r^*_0)+\int_{r^*_0-y}^{r^*_0} (-\mathfrak F_1'(s))\, ds\leq \mathfrak F_1(r^*_0)-\frac{M}{2} y.
\end{align*}
Since $h_\alpha \leq 0 $ in $[r^*_0-\e;r^*_0]$ and $h_\alpha\geq 0$ in $ [r^*_0,r^*_0+\e]$, we have
\begin{eqnarray*}
h_\alpha(r^*_0+y)\mathfrak F_1(r^*_0+y)&\geq &h_\alpha(r^*_0+y)\mathfrak F_1(r^*)+\frac{|h_\alpha| (r^*_0+y)M}2 y\\
\text{and}\quad h_\alpha(r^*_0-y)\mathfrak F_1(r^*_0-y)&\geq& h_\alpha(r^*_0-y)\mathfrak F_1(r^*)+\frac{|h_\alpha|(r^*_0-y) M}2 y.
\end{eqnarray*}
for every $y\in [0,\e]$. Hence, using that $\int_\B h_\alpha=0$, we infer that
\begin{align*}
f_\alpha'(t_1)&=\int_\B \left(\frac\alpha{1+\alpha\kappa}\Lambda_-(m_{t_1})^2 |\n u_{t_1}|^2-u_{t_1}^2\right)h_\alpha =c_{n}\int_0^R h_\alpha(s) \mathfrak F_1(s) s^{n-1}\, ds
\\&=c_n \left(\int_{r^*_0-\e}^{r^*_0}h_\alpha\mathfrak F_1(s)s^{n-1}ds+\int_{r^*_0}^{r^*_0+\e} h_\alpha\mathfrak F_1(s)s^{n-1}\, ds\right)
\\&\geq c_n\left(\int_{r^*_0-\e}^{r^*_0}h_\alpha(s)\mathfrak F_1(r^*)s^{n-1}ds+\int_{r^*_0}^{r^*_0+\e} h_\alpha(s)\mathfrak F_1(r^*)s^{n-1}\, ds\right)&
\\&+\frac{c_nM}2 \left(\int_{r^*_0-\e}^{r^*_0}|h_\alpha|(s) |r^*_0-s|s^{n-1}ds+\int_{r^*_0}^{r^*_0+\e} |h_\alpha|(s)|r^*_0-s|s^{n-1}\, ds\right)
\\&=\frac{c_nM}2\int_\B |h_\alpha|\operatorname{dist}(\cdot,\S),
\end{align*}
which concludes Step 3. Theorem \ref{Th:RadialStability} is thus proved.
\begin{remark}\label{Susu:Concluding}
Regarding the proof of Theorem \ref{Th:RadialStability}, it would have been more natural to consider the path $t\mapsto \left(\lambda_\alpha(m_t),m_t\right)$ rather than $t\mapsto \left( \zetat,m_t\right)$. However, we would have been led to consider {$\mathfrak G_1=\alpha \kappa |\n u_{\alpha,m_{t_{1}}}|^2-u_{\alpha,m_{t_{1}}}^2$ instead of $\mathfrak F_1$}. Unfortunately, this would have been more intricate because of the regularity of $\mathfrak G_1$, which is discontinuous and thus, no longer a $W^{1,\infty}$ function, so that a Lemma analogous to Lemma \ref{Le:BoundsUt} would not be true. Adapting step by step the arguments of \cite{Laurain} would nevertheless be possible although much more technical.
\end{remark}
\section{Sketch of the proof of Corollary \ref{Th:Sketch}}\label{Se:Sketch}
We do not give all details since the proof is then very similar to the ones written previously. We only underline the slight differences in every step.
To prove this result, we consider the following relaxation of our problem, which is reminiscent of the problems considered in \cite{Hamel2011}. Let us consider, for any pair $(m_1,m_2)\in \mathcal M_{m_0,\kappa}(\O)^2$, the first eigenvalue of the operator $\mathscr N:u\mapsto -\n \cdot\left((1+\alpha m_1)\n u\right)-m_2 u$, and write it $\eta_\alpha(m_1,m_2)$. Let $m^*_0:=\kappa \mathds 1_{\mathbb B(0,R)}$.
By using the results of \cite{Hamel2011} or alternatively, applying the rearrangement of Alvino and Trombetti, \cite{AlvinoTrombetti} as it has been done in \cite{ConcaMahadevanSanz}, one proves the existence of a radially symmetric function $\tilde m_1$ such that
$$\eta_\alpha(m_1,m_2)\geq \eta_\alpha(\tilde m_1,m^*_0),$$
so that we are done if we can prove that, for any $m\in \mathcal M(\O)$ there holds
\begin{equation}\label{Eq:Sk}
\eta_\alpha(m,m^*_0)\geq \eta_\alpha(m^*_0,m^*_0).
\end{equation}
We claim that \eqref{Eq:Sk} holds for any $m\in \mathcal M_{m_0,\kappa}$, provided that $m_0$ and $\alpha$ be small enough. Let us describe the main steps of the proof:
\begin{itemize}
\item {\bf Step 1:} mimicking the compactness argument used in \cite{ConcaMahadevanSanz}, one shows that there exists a solution $m_\alpha$ to the problem
$$\inf_{m\in \mathcal M_{m_0,\kappa}(\O)}\eta_\alpha(m,m^*_0),$$
which is radially symmetric and {\it bang-bang}. We write it $m_\alpha=\kappa \mathds 1_{E_\alpha}$.
\item {\bf Step 2:} let $\mu_0$ and $r_0^*$ be the unique real numbers such that
$$
\left|\left\{|\n u_{0,m^*_0}|^2\leq \mu_0\right\}\right|=V_0=|\mathbb B(0,r^*_0)|.
$$
Introducing $E_0=\left\{|\n u_{0,m^*_0}|^2\leq \mu_0\right\}$, we prove that
$m_\alpha$ converges in $L^1(\O)$ to $\kappa \mathds 1_{E_0}$ as $\alpha \to 0$.
\item {\bf Step 3:} we establish that if $m_0$ is small enough, then $E_0=\mathbb B(0,r^*_0)$. This is done by proving that $u_{0,m^*_0}$ converges in $\mathscr C^1$ to the first Dirichlet eigenfunction of the ball as $r^*_0\to 0$ and by determining the level-sets of this first eigenfunction, as done in \cite[Section 2.2]{ConcaLaurainMahadevan}.
\item {\bf Step 4:} once this limit identified, we mimick the steps of the proof of Theorem \ref{Th:RadialStability} (reduction to a small Hausdorff distance and mean value theorem for a well-chosen auxiliary function) to conclude that one necessarily has $m_\alpha=m^*_0$ for $\alpha$ small enough.
\end{itemize}
\section{Proof of Theorem \ref{Th:ShapeStability}}
Throughout this section, we will denote by $\mathbb B^*$ the ball $\mathbb B(0,r^*_0)$, where $r^*_0$ is chosen so that $m^*_0=\kappa \mathds{1}_{\mathbb B^*}$ belongs to $\mathcal M_{m_0,\kappa}(\O)$.
When it makes sense, we will write $f|_{int}(y)=\lim_{x\in \B^*,x\to y}f(x)$, $f|_{ext}(y):=\lim_{x \in (\B^*)^c,x\to y}f(y)$, so that $\llbracket f\rrbracket =f|_{ext}-f|_{int}$ denotes the jump of $f$ at the boundary $\S$.
\subsection{Preliminaries}
\allowdisplaybreaks
\def\dri{{\frac{\partial}{\partial r_i}}}
\def\uk{{u_{1,\alpha}^{(k)}}}
For $\e>0$, let us introduce $\mathbb B^*_\e:=(\operatorname{Id}+\e V)\mathbb B^*$ and define $u_\e$ as the $L^2$-normalized first eigenfunction associated with $m_\e=\kappa \mathds{1}_{\mathbb B^*_\e}$.
It is well known (see e.g. \cite{Henrot2006,HenrotPierre}) that $u_\e$ expands as
\begin{equation}\label{An:EigenFunction}
u_\e=u_{0,\alpha}+\e u_{1,\alpha}+\e^2 \frac{u_{2,\alpha}}2+\underset{\e \to 0}{\operatorname{o}} (\e^2)\quad \text{in }H^1(\B^*)\text{ and in }H^1(\O \backslash \B^*),
\end{equation}
where, in particular, $u_{0,\alpha}=u_{\alpha,m^*_0}$, whereas $\lambda_\alpha(\mathbb B^*_\e)$ expands as
\begin{eqnarray}
\lambda_\alpha(\mathbb B^*_\e)&=&\lambda_{0,\alpha}+\e\lambda_\alpha'(\mathbb B^*)[V]+\frac{\e^2}{2}\lambda_\alpha''(\mathbb B^*)[V]+\underset{\e \to 0}{\operatorname{o}}(\e^2)\nonumber \\
&=&\lambda_{0,\alpha}+\e\lambda_{1,\alpha}+\frac{\e^2}{2}\lambda_{2,\alpha}+\underset{\e \to 0}{\operatorname{o}}(\e^2).\label{An:EigenValue}
\end{eqnarray}
By mimicking the proof of Lemma \ref{Le:Bounds}, one shows the following symmetry result.
\begin{lemma}\label{Le:Rad}
The function $u_{\alpha,m^*_0}$ is radially symmetric. Let $\p_{\alpha,m^*_0}$, $\tm$ and $\tilde \sigma_{\alpha,\tilde m}$ be such that $u_{\alpha,m^*_0}=\p_{\alpha,m^*_0}(|\cdot|)$, $m^*_0=\tm(|\cdot|)$ and $\tilde \sigma_{\alpha,\tilde m}=1+\alpha \tm$. Then $\p_{\alpha,m^*_0}$ satisfies the ODE
\begin{equation}\label{Eq:RadU0}\left\{
\begin{array}{ll}
-\frac{d}{dr}\left(r^{n-1}\tilde \sigma_{\alpha,\tilde m} \varphi_{\alpha,m_0^*}'\right)=\left(\lambda_\alpha(m_0^*)+\tm \right)\varphi_{\alpha,m_0^*} r^{n-1} & \text{in }(0,R)\\
\varphi_{\alpha,m_0^*}(R)=0 &
\end{array}
\right.
\end{equation}
complemented by the following jump conditions
\begin{equation}\label{Eq:U0}
\llbracket \p_{\alpha,m^*_0}\rrbracket (r^*_0)=\llbracket \tilde \sigma_{\alpha,m^*_0}\p_{\alpha,m^*_0}'\rrbracket (r^*_0)=0,\quad \llbracket \tilde \sigma_{\alpha,\tilde m}\p_{\alpha,m_0^*}''\rrbracket (r^*_0)=\kappa \p_{\alpha,m^*_0}(r^*_0).
\end{equation}
Furthermore, $\p_{\alpha,m^*_0}$ converges to $\p_{0,m^*_0}$ for the strong topology of $\mathscr C^1$ as $\alpha\to 0$.
\end{lemma}
\subsection{Computation of the first and second order shape derivatives}
\paragraph{A remark on the type of vector fields we consider}
Hadamard's structure theorem (see for instance \cite[Theorem~5.9.2 and the remark below]{HenrotPierre}) ensures that the first order derivative in the direction of a vector field $V$ only depends on the normal trace of $V$. This allows us to work with only normal vector fields $V$ to compute the first order derivative.
Once it is established that $\mathbb B^*$ is a critical shape, we can use Hadamard's structure theorem \cite[Theorem~5.9.2 and the remark below]{HenrotPierre} which states that the second order shape derivative, when computed at a critical shape only depends on the normal trace, hence we will also, for second order shape derivatives, work with normal vector fields.
Since we are working in {two dimensions}, this means that one can deal with vector fields $V$ given in polar coordinates by
$$
V(r^*_0,\theta)=g(\theta)\begin{pmatrix}\cos\theta\\\sin\theta \end{pmatrix}.
$$
The proof of the shape differentiability at the first and second order of $\lambda_\alpha$, based on an implicit function argument according to the method of \cite{MignotMuratPuel}, is exactly similar to \cite[Proof of Theorem 2.2]{DambrineKateb}. For this reason, we admit it. Nevertheless, in what follows, we provide some details on the computation of these derivatives for the sake of completeness, since some steps differ a bit from those done in the references above.
\paragraph{Computation and analysis of the first order shape derivative.}
Let us prove that $\B^*$ is a critical shape in the sense of \eqref{Eq:FOO}.
\begin{lemma}\label{Le:Cr}
The first order shape derivative of $\lambda_\alpha$ at $\B^*$ in direction $V$ reads
\begin{equation}
\lambda_{1,\alpha}=\lambda_\alpha'(\B^*)[V]=\int_{\S} V\cdot \nu .
\end{equation}
For all $V \in \mathcal X(\B^*)$ (defined by \eqref{Eq:X}), one has $\lambda_{1,\alpha}=0$ meaning that $\B^*$ satisfies \eqref{Eq:FOO}.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Le:Cr}]
First, elementary computations show that $u_{1,\alpha}$ solves
\begin{equation}\label{Eq:EigenDerivative1}
\left\{
\begin{array}{ll}
-\n \cdot \Big(\sigma_\alpha \n u_{1,\alpha}\Big)=\lambda_{1,\alpha}u_{0,\alpha}+\lambda_{0,\alpha}u_{1,\alpha}+m^*_0u_{1,\alpha}&\text{ in }\B(0,R),\\
\left\llbracket \sigma_\alpha\frac{\partial u_{1,\alpha}}{\partial \nu}\right\rrbracket (r^*_0\cos\theta ,r^*_0\sin\theta )=-\kappa g(\theta)u_{0,\alpha},& \\
\left\llbracket u_{1,\alpha}\right\rrbracket (r^*_0\cos\theta ,r^*_0\sin\theta )=-g(\theta)\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket (r^*_0\cos\theta ,r^*_0\sin\theta ),&
\end{array}
\right.
\end{equation}
where $\sigma_\alpha=1+\alpha m_0^*$ and the notation $\llbracket\cdot\rrbracket$ denote the jumps of the functions at $\S$.
The derivation of the main equation of \eqref{Eq:EigenDerivative1} is an adaptation of the computations in \cite{DambrineKateb}.
To derive the jump on $u_{1,\alpha}$, we follow \cite{DambrineKateb} and differentiate the continuity equation
$\llbracket u_\e\rrbracket _{\partial \B_\e^*}=0$.
Formally plugging \eqref{An:EigenFunction} in this equation yields
$$u_{1,\alpha}\vert_{int}(r^*_0,\theta)+\left.g(\theta)\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int}=u_{1,\alpha}\vert_{ext}(r^*_0,\theta)+\left.g(\theta)\frac{\partial u_{0,\alpha}}{\partial r}\right|_{ext},$$
and hence
$$
\llbracket u_{1,\alpha}\rrbracket = u_{1,\alpha}|_{ext}-u_{1,\alpha}|_{int}=-g(\theta)\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket.
$$
Note that the same goes for the normal derivative: we differentiate the continuity equation
$$\left\llbracket (1+\alpha m_\e)\frac{\partial u_{\e,\alpha}}{\partial \nu}\right\rrbracket_{\partial \B_\e^*}=0,$$
yielding
$$\left\llbracket \sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket=-g(\theta)\left\llbracket\sigma_\alpha \frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right\rrbracket.
$$
According to the equation $-\sigma_\alpha \Delta u_{0,\alpha}=\lambda_\alpha(m^*_0)u_{0,\alpha}+m^*_0u_{0,\alpha}$ in $\B^*$, this rewrites
\begin{equation}\label{Eq:Jump1}
\left\llbracket \sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket =-\kappa g(\theta)u_{0,\alpha}.
\end{equation}
Now, using $u_{0,\alpha}$ as a test function in \eqref{Eq:EigenDerivative1}, we get
\begin{align*}
\lambda_{1,\alpha}&=-\int_{\mathbb B(0,R)} u_{0,\alpha} \nabla\cdot (\sigma_\alpha\nabla u_{1,\alpha})-\int_{\mathbb B(0,R)} m^*_0 u_{1,\alpha}u_{0,\alpha}\\
&=-\int_{\mathbb B(0,R)} u_{0,\alpha} \nabla\cdot (\sigma_\alpha\nabla u_{1,\alpha})+\int_{\mathbb B(0,R)} u_{1,\alpha} \nabla \cdot (\sigma_\alpha u_{0,\alpha})
\\&=\int_{\S}u_{0,\alpha}\left\llbracket \sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial \nu} \right\rrbracket -\int_{\S}\left\llbracket \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}u_{1,\alpha}\right\rrbracket \\
&=-r^*_0\int_{0}^{2\pi} \kappa g(\theta)u_{0,\alpha}(r^*_0)^2\, d\theta+r^*_0\int_0^{2\pi}g(\theta)\left(\sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\right)\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket\, d\theta\\
&=r^*_0\int_0^{2\pi}g(\theta)\left(-\kappa u_{0,\alpha}(r^*_0)^2+\left\llbracket \sigma_\alpha\left( \frac{\partial u_{0,\alpha}}{\partial r}\right)^2\right\rrbracket \right)\, d\theta.
\end{align*}
by using that $\int_{\B(0,R)}u_{\e}^2=1$, so that $\int_{\B(0,R)} u_{0,\alpha}u_{1,\alpha}=0$ by differentiation.
Since $u_{0,\alpha}$ is radially symmetric according to Lemma \ref{Le:Rad}, we introduce the two real numbers
{\begin{equation}\label{LM}
\eta_\alpha:=-\kappa u_{0,\alpha}(r^*_0)^2+\left\llbracket \sigma_\alpha\left( \frac{\partial u_{0,\alpha}}{\partial r}\right)^2\right\rrbracket
\quad \text{and}\quad \lambda_{1,\alpha}:=r^*_0\eta_\alpha \int_0^{2\pi}g(\theta)\, d\theta.
\end{equation}}
It is easy to see that $V$ belongs to $\mathcal X(\B^*)$ if, and only if $\int_0^{2\pi}g=0$ so that we finally have $\lambda_{1,\alpha}=0.$
\end{proof}
\paragraph{Computation of the Lagrange multiplier.}
The existence of a Lagrange multiplier $\Lambda_\alpha\in \R$ related to the volume constraint $\operatorname{Vol}(E)=m_0\operatorname{Vol}(\O)/\kappa$ is standard, and one has
$$
\forall V \in \mathcal X(\B^*), \quad \left(\lambda_\alpha'-\Lambda_\alpha\operatorname{Vol}'\right)(\B^*)[V]=0.
$$
Since
$$\operatorname{Vol}'(\B^*)[V]=\int_{\S} V\cdot \nu =r^*_0\int_0^{2\pi}g(\theta)d\theta.$$
(see e.g. \cite[chapitre 5]{HenrotPierre}) and since
$$\lambda_\alpha'(\B^*)[V]=r^*_0\eta_\alpha\int_0^{2\pi}g(\theta)d\theta,$$
where $\eta_\alpha$ is defined by \eqref{LM}, the Lagrange multiplier reads
$$\Lambda_\alpha=\eta_\alpha=-\kappa u_{0,\alpha}(r^*_0)^2+\left\llbracket \sigma_\alpha\left( \frac{\partial u_{0,\alpha}}{\partial r}\right)^2\right\rrbracket .$$
\paragraph{Computation of the second order derivative and second order optimality conditions.}
Let us compute the second order derivative of $\lambda_\alpha$.
By using the Hadamard structure Theorem (see \cite[Theorem~5.9.2 and the following remark]{HenrotPierre}), since $\B^*$ is a critical shape in the sense of \eqref{Eq:FOO}, it is not restrictive to deal with vector fields that are normal to the $\partial \B^*=\mathbb S^*$, according to the so-called structure theorem which provides the generic structure of second order shape derivatives. This allow us to identify any such $V\in \mathcal X(\B^*)$ with a periodic function $g:[0,2\pi]\rightarrow \R$ such that
$$
V(r^*_0\cos\theta ,r^*_0\sin\theta )=g(\theta)\begin{pmatrix}\cos\theta \\\sin\theta \end{pmatrix}.
$$
\begin{lemma}\label{Le:SecondDerivative}
For every $V\in \mathcal X(\B^*)$, one has for the coefficient $\lambda_{2,\alpha}= \lambda_\alpha''(\B^*)[V,V]$ introduced in \eqref{An:EigenValue} the expression
\begin{eqnarray*}
\lambda_{2,\alpha}&=& 2\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{int}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu -2\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu \\
&& +\int_{\S}\left(-\frac1{r^*_0}\left\llbracket \sigma_\alpha|\n u_{0,\alpha}|^2\right\rrbracket -\frac{\kappa}{r^*_0}u_{0,\alpha}^2\right) (V\cdot \nu)^2
-2\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2.
\end{eqnarray*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Le:SecondDerivative}]
In the computations below, we do not need to make the equation satisfied by $u_{2,\alpha}$ explicit, but we nevertheless will need several times the knowledge of $\llbracket u_{2,\alpha}\rrbracket $ at $\S$.
In the same fashion that we obtained the jump conditions on $u_{1,\alpha}$
Let us differentiate two times the continuity equation $\llbracket u_\e\rrbracket _{\partial \B^*_\e}=0$. We obtain
\begin{equation}\label{Eq:Jump2}
\llbracket u_{2,\alpha}\rrbracket _{\partial \B_\e^*}=-2g(\theta)\left\llbracket \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket -g(\theta)^2 \left\llbracket \frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right\rrbracket .
\end{equation}
Now, according to Hadamard second variation formula (see \cite[Chapitre 5, page 227]{HenrotPierre} for a proof), if $\O$ is a $\mathscr C^2$ domain and $f$ is two times differentiable at 0 and taking values in $W^{2,2}(\O)$, then one has
\begin{equation}\label{Eq:Hada2}
\left.\frac{d^2}{dt^2}\right|_{t=0}\int_{(\operatorname{Id}+tV)\O} f(t)=\int_{\O} f''(0)+2\int_{\partial \O} f'(0) V\cdot \nu +\int_{\partial \O}\left(H f(0)+\frac{\partial f(0)}{\partial \nu}\right) (V\cdot \nu)^2,
\end{equation}
where $H$ denotes the mean curvature. We apply it to $f(\e)=\sigma_{\alpha,\e}|\n u_\e|^2-m_\e u_\e^2$ on $\mathbb B(0,R)$, since $\lambda_\alpha(m_\e)=\int_{\mathbb B(0,R)}{ f(\e)}$.
Let us distinguish between the two subdomains $\B_\e^*$ and $(\B_\e^*)^c$. We introduce
$$
D_1 = \left. \frac{d^2}{d\e^2}\right|_{\e=0}\int_{\B_\e^*}\left(\sigma_{\alpha,\e}|\n u_\e|^2-\kappa u_\e^2\right) \quad \text{and}\quad
D_2 = \left. \frac{d^2}{d\e^2}\right|_{\e=0}\int_{(\B_\e^*)^c}\left(\sigma_{\alpha,\e}|\n u_\e|^2\right),
$$
so that $ \lambda_\alpha''(\B^*)[V,V]=D_1+D_2$.
One has
\begin{eqnarray*}
D_1 &=&\int_{\B^*}2(1+\alpha \kappa) \n u_{2,\alpha}\cdot \n u_{1,\alpha}+2\int_{\B_\e^*}(1+\alpha \kappa)|\n u_{1,\alpha}|^2
\\&&-2\kappa\int_{\B^*}u_{2,\alpha}u_{0,\alpha}-2\kappa\int_{\B^*}u_{1,\alpha}u_{0,\alpha}
\\&&+4\int_{\S}(1+\alpha \kappa) (\n u_{1,\alpha}|_{int}\cdot \n u_{0,\alpha}|_{int}) V\cdot \nu -4\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu
\\&&+\int_{\S}\left(\frac1{r^*_0}(1+\alpha \kappa)|\n u_{0,\alpha}|_{int}^2-\frac\kappa{r^*}u_{0,\alpha}^2+2(1+\alpha\kappa)\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int}\left.\frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right|_{int}\right.
\\&&\left.-2\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int}\right) (V\cdot \nu)^2,
\end{eqnarray*}
and taking into account that the mean curvature has a sign on $(\B_\e^*)^c$, one has
\begin{eqnarray*}
D_2 &=&\int_{(\B^*)^c}2 \n u_{2,\alpha}\cdot \n u_{1,\alpha}+2\int_{(\B^*)^c}|\n u_{1,\alpha}|^2
\\&&-4\int_{\S} (\n u_{1,\alpha}|_{ext}\cdot \n u_{0,\alpha}|_{ext}) V\cdot \nu
\\&&+\int_{\S}\left(-\frac1{r^*_0}|\n u_{0,\alpha}|_{ext}^2-2\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{ext}\left.\frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right|_{ext}\right) (V\cdot \nu)^2.
\end{eqnarray*}
Summing these two quantities, we get
\begin{eqnarray*}
\lambda_{2,\alpha}&=&2\int_{\B(0,R)}\sigma_\alpha \n u_{0,\alpha}\cdot \n u_{2,\alpha}-2\int_{\B(0,R)}m^*_0 u_{0,\alpha}u_{2,\alpha}
+2\int_{\B(0,R)}\sigma_\alpha|\n u_{1,\alpha}|^2-2\int_{\mathbb B(0,R)}m^*_0 u_{1,\alpha}^2
\\&&-4\int_{\S}\sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket V\cdot \nu -4\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu
\\&&+\int_{\S}\left(-\frac1{r^*_0}\left\llbracket \sigma_\alpha|\n u_{0,\alpha}|^2\right\rrbracket -\frac{\kappa}{r^*_0}u_{0,\alpha}^2\right) (V\cdot \nu)^2
\\&&
-2\int_{\S}\left\llbracket \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right\rrbracket (V\cdot \nu)^2-2\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2.
\end{eqnarray*}
To simplify this expression, let us use Eq. \eqref{Eq:U0}. Introducing
$$
D_3= \int_{\B(0,R)}\sigma_\alpha \n u_{0,\alpha}\cdot \n u_{2,\alpha}-\int_{\mathbb B(0,R)} u_{0,\alpha}u_{2,\alpha}-\lambda_\alpha(\B^*)\int_{\B(0,R)} u_{0,\alpha}u_{2,\alpha},
$$
one has
$$
D_3=\int_{\S}\llbracket u_{2,\alpha}\rrbracket \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r},
$$
and hence, by using Equation \eqref{Eq:Jump2}, one has
\begin{align*}
D_3 &=-2\int_{\S}\llbracket u_{2,\alpha}\rrbracket \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\\
&= 4\int_{\S} \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket V\cdot \nu +2\int_{\S} \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right\rrbracket (V\cdot \nu)^2.
\end{align*}
Similarly, let
$$
D_4=\int_{\B(0,R)}\sigma_\alpha |\n u_{1,\alpha}|^2-\int_{\B(0,R)}m^*_0u_{1,\alpha}^2.
$$
By using Eq. \eqref{Eq:EigenDerivative1} and the fact that $\lambda_{1,\alpha}=0$, one has
\begin{eqnarray*}
D_4 &=& \lambda_\alpha(\B^*)\int_{\mathbb B(0,R)} u_{1,\alpha}^2-\int_{\S}\left\llbracket u_{1,\alpha}\sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket
\\&=&\lambda_\alpha(\B^*)\int_{\mathbb B(0,R)} u_{1,\alpha}^2-\int_{\S}\left\llbracket u_{1,\alpha}\right\rrbracket\left.\left(\sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial r}\right)\right|_{ext}-\int_{\S}u_{1,\alpha}|_{int}\left\llbracket \sigma_\alpha \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket
\\&=&\lambda_\alpha(\B^*)\int_{\mathbb B(0,R)} u_{1,\alpha}^2+\int_{\S} \left(\sigma_\alpha\partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket+\kappa u_{1,\alpha}|_{int} u_{0,\alpha}\right) V\cdot \nu.
\end{eqnarray*}
Finally, by differentiating the normalization condition $\int_{\B(0,R)}u_\e^2=1$, we get
\begin{equation}\label{normal}
\int_{\B(0,R)}u_{0,\alpha}u_{2,\alpha}+\int_{\B(0,R)}u_{1,\alpha}^2=0.\end{equation}
Combining the equalities above, one gets
\begin{eqnarray*}
\lambda_{2,\alpha}&=&2\lambda_\alpha(\B^*)\left(\int_{\B(0,R)}u_{0,\alpha}u_{2,\alpha}+\int_{\B(0,R)}u_{1,\alpha}^2\right) +4\int_{\S} \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket V\cdot \nu
\\&&+2\int_{\S} \sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right\rrbracket (V\cdot \nu)^2+2\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu
\\&&+2\kappa \int_{\S} u_{1,\alpha}|_{int} u_{0,\alpha}V\cdot \nu-4\int_{\S}\sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\left\llbracket \frac{\partial u_{1,\alpha}}{\partial r}\right\rrbracket V\cdot \nu
\\&&-4\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu -\int_{\S}\left(\frac1{r^*_0}\left\llbracket \sigma_\alpha|\n u_{0,\alpha}|^2\right\rrbracket +\frac{\kappa}{r^*_0}u_{0,\alpha}^2\right) (V\cdot \nu)^2
\\&&-2\int_{\S}\left[\sigma_\alpha \frac{\partial u_{0,\alpha}}{\partial r}\frac{\partial^2 u_{0,\alpha}}{\partial r^2}\right] (V\cdot \nu)^2-2\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2
\\&=&2\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu
-2\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu
\\&& -\int_{\S}\left(\frac1{r^*_0}\left[\sigma_\alpha|\n u_{0,\alpha}|^2\right]+\frac{\kappa}{r^*_0}u_{0,\alpha}^2\right) (V\cdot \nu)^2-2\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2.
\end{eqnarray*}
We have then obtained the desired expression.
\end{proof}
\paragraph{Strong stability.}
Recall here that, as mentioned before, since we are dealing with a critical point of the functional $\lambda_\alpha$, it is enough to consider perturbation $V$ normal to the boundary of $\B^*$, in other words such that $V=(V\cdot \nu)\nu$.
Under such an assumption, the second derivative of the volume is known to be (see e.g. \cite[Section 5.9.6]{HenrotPierre})
\begin{equation}\label{Eq:VolumeDerivative}
\operatorname{Vol}''(\B^*)[V,V]=\int_{\S}H (V\cdot \nu)^2.
\end{equation}
Hence, introducing $D_5=(\lambda_\alpha''-\eta_\alpha\operatorname{Vol}'')(\B^*)[V,V]$ and taking into account Lemma \ref{Le:SecondDerivative}, \eqref{LM} and \eqref{Eq:VolumeDerivative}, we have
\begin{eqnarray*}
D_5&=&2\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu-2\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2
\\ &&-2\kappa\int_{\S}u_{1,\alpha}|_{ext}u_{0,\alpha} V\cdot \nu
+\int_{\S}\left(-\frac1{r^*_0}\left[\sigma_\alpha|\n u_{0,\alpha}|^2\right]-\frac{\kappa}{r^*_0}u_{0,\alpha}^2\right) (V\cdot \nu)^2
\\&&+\kappa \int_{\S}\frac1{r^*_0}u_{0,\alpha}^2 (V\cdot \nu)^2-\int_{\S}\frac1{r^*_0}\left[\sigma_\alpha|\n u_{0,\alpha}|^2\right] (V\cdot \nu)^2
\\&=&2\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu-2\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2
\\ &&-2\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu
-\int_{\S}\frac2{r^*_0}\left[\sigma_\alpha|\n u_{0,\alpha}|^2\right] (V\cdot \nu)^2.
\end{eqnarray*}
We are then led to determine the signature of the quadratic form
\begin{eqnarray}
\mathcal F_\alpha[V,V]&=&\frac12(\lambda_\alpha''-\Lambda_\alpha\operatorname{Vol}'')(\B^*)[V,V] \label{Eq:Lambda2}\\
&=&\int_{\S} \sigma_\alpha \partial_r u_{1,\alpha}|_{ext}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket V\cdot \nu -\kappa\int_{\S}u_{1,\alpha}|_{int}u_{0,\alpha} V\cdot \nu \nonumber
\\&&+\int_{\S}\left(-\frac2{r^*}\left\llbracket \sigma_\alpha|\n u_{0,\alpha}|^2\right\rrbracket \right) (V\cdot \nu)^2-\int_{\S}\kappa u_{0,\alpha}\left.\frac{\partial u_{0,\alpha}}{\partial r}\right|_{int} (V\cdot \nu)^2.\nonumber
\end{eqnarray}
\subsection{Analysis of the quadratic form $\mathcal{F}_\alpha$}
\paragraph{Separation of variables and first simplification.}
Each perturbation $g\in L^2(0,2\pi)$ such that $\int_0^{2\pi} g=0$ expands as
$$
g=\sum_{k=1}^\infty \left( \gamma_k \cos(k\cdot)+\beta_k\sin(k\cdot)\right), \quad \text{with }\gamma_0=0.
$$
For every $k\in \N^*$, let us introduce $g_k:=\cos(k\cdot)$ and $\tilde g_k:=\sin(k\cdot)$. For any $k\in \N^*$, let $u_{1,\alpha}^{(k)}$ be the solution of Eq.~\eqref{Eq:EigenDerivative1} associated with the perturbation $g_k$.
It is readily checked that there exists a function $\varphi_{k,\alpha}:[0,R]\rightarrow \R$ such that
$$
\forall (r,\theta)\in [0,R]\times [0,2\pi], \quad \uk(r,\theta)=g_k(\theta)z_{k,\alpha}(r).
$$
Furthermore, $\varphi_{k,\alpha}$ solves the ODE
\begin{equation}\label{Eq:PsiK}
\left\{
\begin{array}{ll}
-\sigma_\alpha z_{k,\alpha}''-\frac{\sigma_{\alpha}}rz_{k,\alpha}'(r) =\left(\lambda_{0,\alpha}-\frac{k^2}{r^2}\right)z_{k,\alpha}+m^*_0z_{k,\alpha}&\text{ in }(0,r_0^*)\cup (r_0^*,R),\\
\left\llbracket \sigma_\alpha z_{k,\alpha}'\right\rrbracket (r^*_0)=-\kappa u_{0,\alpha}(r^*_0)& \\
\left\llbracket z_{k,\alpha}\right\rrbracket (r^*_0)=-\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket(r^*_0),
\\z_{k,\alpha}'(0)=z_{k,\alpha}(R)=0.&
\end{array}
\right.
\end{equation}
Regarding $\tilde g_k$, if we define $\tilde u_{1,\alpha}^{(k)}$ in a similar fashion, it is readily checked that
$$\forall (r,\theta)\in [0,R]\times [0,2\pi]\, ,\tilde u_{1,\alpha}^{(k)}(r,\theta)=\tilde g_k(\theta) z_{k,\alpha}(r).$$
Therefore, any admissible perturbation $g$ writes
$$
g=\sum_{k=1}^\infty \left\{\gamma_kg_k+\beta_k\tilde g_k\right\}\quad \text{with } \gamma_0=0,
$$
and the solution $u_{1,\alpha}$ associated with $g$ writes
$$u_{1,\alpha}=\sum_{k=1}^\infty \left\{\gamma_k \uk+\beta_k \tilde u_{1,\alpha}^{(k)}\right\}.$$
Using the orthogonality properties of the family $\{g_k\}_{k\in \N^*}\cup \{\tilde g_k\}_{k\in \N}$, it follows that $\mathcal F_\alpha[V,V]$ given by \eqref{Eq:Lambda2} reads
\begin{eqnarray}
\mathcal F_\alpha[V,V] &=&\frac{r^*_0}2\sum_{k=1}^\infty\left(\sigma_\alpha z_{k,\alpha}'(r^*_0)|_{ext}\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket -\kappa z_{k,\alpha}|_{int}u_{0,\alpha}(r^*_0)\right)\left(\gamma_k^2+\beta_k^2\right)\nonumber
\\&&-\frac{r^*_0}2\sum_{k=0}^\infty \kappa u_{0,\alpha}(r^*_0)\frac{\partial u_{0,\alpha}}{\partial r}(r^*_0)\left(\gamma_k^2+\beta_k^2\right)-\sum_{k=1}^\infty2 \left\llbracket \sigma_\alpha |\n u_{0,\alpha}|^2\right\rrbracket \left(\gamma_k^2+\beta_k^2\right)\nonumber
\\&=&\frac{r^*_0\kappa u_{0,\alpha}(r^*_0)}2\sum_{k=1}^\infty\left(-\left.\frac{\partial u_{0,\alpha}}{\partial r}(r^*_0)\right|_{int}- z_{k,\alpha}|_{int}\right)\left(\gamma_k^2+\beta_k^2\right)\nonumber
\\&&+\sum_{k=1}^\infty \left(-2\left[\sigma_\alpha |\n u_{0,\alpha}|^2\right]+\sigma_\alpha z_{k,\alpha}'(r^*_0)|_{ext}\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket \right)\left(\gamma_k^2+\beta_k^2\right).\label{Eq:Ptn}
\end{eqnarray}
Define, for any $k\in \N$,
$$
\omega_{k,\alpha}:=\frac{r_0^*\kappa u_{0,\alpha}(r_0^*)}{2}\left(\left.-\frac{\partial u_{0,\alpha}}{\partial r}(r^*_0)\right|_{int}- z_{k,\alpha}|_{int}(r_0^*)\right)$$ and $$ \zeta_{k,\alpha}:=-2\left\llbracket \sigma_\alpha |\n u_{0,\alpha}|^2\right\rrbracket +\sigma_\alpha z_{k,\alpha}'(r^*_0)|_{ext}\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket .
$$
Thus,
$$
\mathcal F_\alpha[V,V]=\sum_{k=1}^\infty\left(\omega_{k,\alpha}+\zeta_{k,\alpha}\right)\left(\gamma_k^2+\beta_k^2\right).
$$
The end of the proof is devoted to proving the local shape minimality of the centered ball, which relies on an asymptotic analysis of the sequences $\{\omega_{k,\alpha}\}_{k\in \N}$ and $\{\zeta_{k,\alpha}\}_{k\in \N}$ as $\alpha$ converges to 0.
\begin{proposition}\label{Le:AAO}
There exists $C>0$ and $\overline \alpha>0$, there exists $M\in \R$ such that for any $\alpha \leq \overline \alpha$ and any $k\in \N$, one has
\begin{equation}\label{Eq:OmegaK}
\omega_{k,\alpha}\geq C>0,
\quad \text{and}\quad
\zeta_{k,\alpha}\geq - M\alpha.
\end{equation}
\end{proposition}
The last claim of Theorem \ref{Th:ShapeStability} is then an easy consequence of this proposition. The rest of the proof is devoted to the proof of Proposition~\ref{Le:AAO}, which follows from the combination of the following series of lemmas.
\begin{lemma}\label{Le:Positivity}
There exists $\overline \alpha>0$ such that, for every $\alpha\in [0, \overline \alpha]$, $z_{1,\alpha}$ is nonnegative on $(0,R)$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Le:Positivity}]
For the sake of notational simplicity, we temporarily drop the dependence on $\alpha$ and denote $z_{1,\alpha}$ by $z_\alpha$. The function $z_\alpha$ solves the ODE
$$
\left\{
\begin{array}{ll}
-\sigma_\alpha z_\alpha''-\frac{\sigma_{\alpha}}rz_\alpha'(r) =\left(\lambda_{0,\alpha}-\frac{1}{r^2}\right)z_\alpha+m^*z_\alpha&\text{ in }(0,r_0^*)\cup (r_0^*,R),
\\\left\llbracket \sigma_\alpha z_\alpha'\right\rrbracket (r^*_0)=-\kappa u_{0,\alpha}(r^*_0)&
\\\left\llbracket z_\alpha\right\rrbracket (r^*_0)=-\left\llbracket \frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket (r^*_0),
\\z_{\alpha}(R)=0.&
\end{array}
\right.
$$
Let us introduce $p_\alpha=z_\alpha/u_{0,\alpha}$. One checks easily that $p_\alpha$ solves the ODE
$$
-\sigma_\alpha p_\alpha''-\frac{\sigma_{\alpha}}rp_\alpha' =-\frac{1}{r^2}p_\alpha-2p_\alpha'\frac{u_{0,\alpha}'}{u_{0,\alpha}}\quad \text{in }(0,R).
$$
Furthermore, $p_\alpha$ satisfies the jump conditions
$$\llbracket p_\alpha\rrbracket (r^*_0)=-\frac{\left\llbracket \partial_r u_{0,\alpha}\right\rrbracket (r^*_0)}{u_{0,\alpha}(r^*_0)} =\frac{-\alpha \kappa \partial_r u_{0,\alpha}|_{int}}{u_{0,\alpha}(r^*_0)}>0
\quad \text{and}\quad
\llbracket \sigma_\alpha p_\alpha'\rrbracket (r^*_0)=-\kappa+\frac{\sigma_\alpha \partial_r u_{0,\alpha}}{u_{0,\alpha}(r^*_0)^2}\left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket.$$
To show that $z_\alpha$ is nonnegative, we argue by contradiction and consider first the case where a negative minimum is reached at an interior point $r_-\neq r^*_0$. Then, $p_\alpha$ is $\mathscr C^2$ in a neighborhood of $r_-$ and we have
$$0\geq -p_\alpha''(r_-)=-\frac{p_\alpha(r_-)}{\sigma_\alpha r_-^2}>0,$$
whence the contradiction.
To exclude the case $r_-=R$, let us notice that, according to L'Hospital's rule, one has $p_\alpha(R)=z_\alpha'(R)/u_{0,\alpha}'(R)$. According to the Hopf lemma applied to $u_{0,\alpha}$, this quotient is well-defined. If $p_\alpha(R)<0$ then it follows that $z_\alpha'(R)>0$. However, one has $p_\alpha'(r)\sim z_\alpha'(r)/(2u_{0,\alpha}(r))>0$ as $r\to R$, which contradicts the fact that a minimum is reached at $R$.
Let us finally exclude the case where $r_-=r^*_0$. Mimicking the elliptic regularity arguments used in the proofs of Lemmas \ref{Le:Bounds} and \ref{Le:ConvergenceFP}, we get that $p_\alpha$ converges to $p_0$ as $\alpha \to 0$ for the strong topologies of $\mathscr C^0([0,r^*_0])$ and $\mathscr C^0([r^*_0,R])$.
To conclude, it suffices hence to prove that $p_0$ is positive in a neighborhood of $r^*_0$. We once again argue by contradiction and assume that $p_0$ reaches a negative minimum at $r_-\in [0,R]$.
Notice that $r_-\neq r_0^*$ since $\llbracket p_0\rrbracket (r^*_0)=0$ and $\llbracket p_0'\rrbracket (r^*_0)=-\kappa<0$.
If $r_-\in (0,R)$, since $r_-\neq r_0^*$, we claim that $p_0$ is $\mathscr C^2$ in a neighborhood of $r_-$ and, if $p_0(r_-)<0$, the contradiction follows from
$$0\geq -p_0''(r_-)=-\frac{p_0(r_-)}{(r_-)^2}>0.$$ For the same reason, a negative minimum cannot be reached at $r=0$.
If $r_-=R$, we observe that $p_0(R)=z_0'(R)/u_{0,0}'(R)$.
According to the Hopf lemma applied to $u_{0,0}$, this quantity is well-defined. If $p_0(R)<0$, then it follows that $z_0'(R)>0$. However, $p_0'(r)\sim z_0'(r)/(2u(r))>0$ as $r\to R$, which contradicts the fact that $R$ is a minimizer.
Therefore $p_0$ is positive in a neighborhood of $r^*_0$ and we infer that $p_\alpha$ is non-negative, so that, in turn, $z_\alpha \geq 0$ in $[0,R]$.
\end{proof}
\begin{lemma}\label{Cl:Comp}
Let $\overline \alpha$ be defined as in Lemma \ref{Le:Positivity}. Then, for every $ \alpha\in [0, \overline \alpha]$ and every $k\in \N$,
\begin{equation}\label{Eq:CompPs}z_{k,\alpha}\leq z_{1,\alpha}.\end{equation}
As a consequence, for any $\alpha \leq \overline \alpha$ and any $k\in \N$, there holds $\omega_{k,\alpha}\geq \omega_{1,\alpha}$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Cl:Comp}]
Since $\omega_{k,\alpha}-\omega_{1,\alpha}=\frac{r_0^*\kappa u_{0,\alpha}(r_0^*)}{2}\left(- z_{k,\alpha}|_{int}(r^*_0)+z_{1,\alpha}|_{int}(r^*_0)\right)$, and since we further have $\frac{r_0^*\kappa u_{0,\alpha}(r_0^*)}{2}>0$,
the fact that $\omega_{k,\alpha}\geq \omega_{1,\alpha}$ will follow from \eqref{Eq:CompPs}, on which we now focus. Let us set $\Psi_k=z_{1,\alpha}-z_{k,\alpha}$. From the jump conditions on $z_{1,\alpha}$ and $z_{k,\alpha}$, one has $\llbracket \Psi_k\rrbracket (r_0)=\llbracket \sigma_\alpha\Psi_k'\rrbracket (r_0)=0$.
The function $\Psi_k$ satisfies
\begin{eqnarray*}
-\sigma_\alpha\Psi_k''-\sigma_\alpha\frac{\Psi_k'}r&=&-\left(\lambda_{0,\alpha}-\frac{k^2}{r^2}\right)z_{k,\alpha}-m^*_0z_{k,\alpha}+\left(\lambda_{0,\alpha}-\frac1{r^2}\right)z_{1,\alpha}+m^*_0\psi_{1,\alpha}
\\&>&\left(\lambda_{0,\alpha}-\frac{k^2}{r^2}\right)z_{k,\alpha}-m^*_0z_{k,\alpha}+\left(\lambda_{0,\alpha}-\frac{k^2}{r^2}\right)z_{1,\alpha}+m^*_0\psi_{1,\alpha}
\\&>&\left(\lambda_{0,\alpha}-\frac{k^2}{r^2}\right)\Psi_{k}+m^*_0\Psi_{k}.
\end{eqnarray*}
since $z_{1,\alpha}\geq0$, according to Lemma~\ref{Le:Positivity}. Since $\Psi_k$ satisfies Dirichlet boundary conditions, $\Psi_k\geq0$ in $(0,R)$.
\end{proof}
\begin{lemma}\label{Cl:Final}
There exists $C>0$ such that, for every $\alpha \in [0, \overline \alpha]$, where $\overline \alpha$ is introduced on Lemma~\eqref{Le:Positivity}, one has $\omega_{1,\alpha}\geq C$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Cl:Final}]
Let us introduce $\Psi=-{\partial u_{0,\alpha}}/{\partial r}- z_{1,\alpha}$. Since
$$\omega_{1,\alpha}=\frac{r_0^*\kappa u_{0,\alpha}(r_0^*)}{2}\Psi(r_0^*)$$ and since $\frac{r_0^*\kappa u_{0,\alpha}(r_0^*)}{2}$ converges, as $\alpha \to0$, to $\frac{r_0^*\kappa u_{0,0}(r_0^*)}{2}>0$, it suffices to prove that $\Psi(r_0^*)\geq C>0$ for some $C$ when $\alpha \to 0$.
According to \eqref{Eq:RadU0}, we have $\llbracket \Psi\rrbracket (r^*_0)=\llbracket \Psi'\rrbracket (r^*_0)=0$. Furthermore,
$\Psi(R)=-\frac{\partial u_{0,\alpha}}{\partial r}(R)>0$ according to the Hopf Lemma and $\Psi(0)=0$. Finally, since $\Psi$ solves the ODE, one has
$$
-\frac1r(\sigma_\alpha \Psi')'=\left(\lambda_{0,\alpha}-\frac1{r^2}\right)\Psi+m^*\Psi\quad \text{in }(0,R),
$$
it follows that $\Psi$ is positive in $(0,R]$. Furthermore, $\Psi$ converges to $\Psi_0$ for the strong topology of $\mathscr C^0([0,R])$ and $\Psi_0$ solves the ODE
$$
\left\{\begin{array}{ll}
-\frac1r( \Psi_0')'=\left(\lambda_{0,0}-\frac1{r^2}\right)\Psi_0+m^*_0\Psi_0 & \text{in }(0,R)\\
\Psi_0(R)=-\frac{\partial u_{0,0}}{\partial r}(R)>0. &
\end{array}\right.
$$
Hence there exists $C>0$ such that, for every $\alpha \in [0, \overline \alpha]$, one has $\Psi(r^*_0)\geq C>0$.
\end{proof}
It remains to prove the second inequality of\eqref{Eq:OmegaK}.
As a consequence of the convergence result stated in Lemma~\ref{Le:Rad}, one has
\begin{equation}\label{bus:1904}
\left\llbracket \sigma_\alpha |\n u_{0,\alpha}|^2\right\rrbracket =\operatorname{O}(\alpha), \quad \left\llbracket\frac{\partial u_{0,\alpha}}{\partial r}\right\rrbracket=\left.\alpha \kappa \frac{\partial u_{0,\alpha}}{\partial r}\right|_{int}<0.
\end{equation}
It follows that we only need to prove that there exists a constant $M>0$ such that, for any $\alpha\in [0, \overline \alpha]$, and any $k\in \N^*$,
\begin{equation}\label{Eq:Tain3}
M\geq \sigma_\alpha z_{k,\alpha}'|_{ext}(r^*_0)
\end{equation}
so that
$$
\zeta_{k,\alpha}=\operatorname{O}(\alpha)+\sigma_\alpha z_{k,\alpha}'\vert_{ext}(r^*_0)\left.\alpha \kappa \frac{\partial u_{0,\alpha}}{\partial r}(r^*_0)\right|_{int}\geq \operatorname{O}(\alpha)-M\alpha\kappa\left|\left.\frac{\partial u_{0,\alpha}}{\partial r}(r^*_0)\right|_{int}\right|
$$
To show the estimate \eqref{Eq:Tain3}, let us distinguish between small and large values of $k$.
To this aim, we introduce $N\in \N$ as he smallest integer such that
\begin{equation}\label{tain}
\lambda_{0,\alpha}+m^*_0-\frac{k^2}{r^2}<0 \text{ in }(0,R)
\end{equation}
for every $k\geq N$ and $\alpha \in [0, \overline \alpha]$. The existence of such an integer follows immediately from the convergence of $(\lambda_{0,\alpha})_{\alpha>0}$ to $\lambda_0(m^*_0)$ as $\alpha\to 0$.
\medskip
First, we will prove that, for every $k\geq N$,
\begin{equation}\label{Eq:DerPos}
z_{k,\alpha}'(r^*_0)|_{ext}<0
\end{equation}
and that there exists $M>0$ such that, for every $k\leq N$,
\begin{equation}\label{Eq:DerPos2}
|z_{k,\alpha}'(r^*_0)|_{ext}|\leq M
\end{equation}
which will lead to \eqref{Eq:Tain3} and thus yield the desired conclusion.
To show \eqref{Eq:DerPos}, let us argue by contradiction, assuming that $z_{k,\alpha}'(r^*_0)|_{ext}>0$.
Since the jump $\llbracket \sigma_\alpha z_{k,\alpha}'\rrbracket =-\kappa u_{0,\alpha}(r^*_0)$ is negative, it follows that
$$
(1+\alpha \kappa)z_{k,\alpha}'(r^*_0)|_{int}=z_{k,\alpha}'(r^*_0)|_{ext}-\llbracket \sigma_\alpha z_{k,\alpha}'\rrbracket >0.
$$
By mimicking the reasonings in the proof of Lemma~\ref{Le:Positivity}, $z_{k,\alpha}$ cannot reach a negative minimum on $(0,r^*_0)$ since \eqref{tain} holds true. Therefore, since $z_{k,\alpha}(0)=0$ and $z_{k,\alpha}'(r^*_0)|_{int}>0$, one has necessarily $z_{k,\alpha}(r^*_0)|_{int}>0$, which in turn gives
$z_{k,\alpha}(r^*_0)|_{ext}>0$ since $\llbracket z_{k,\alpha}\rrbracket =-\alpha \kappa \frac{\partial u_{0,\alpha}}{\partial r}>0$.
Furthermore, $z_{k,\alpha}(R)=0$. Since $z_{k,\alpha}(r^*_0)|_{ext}>0$ and $z_{k,\alpha}'(r^*_0)|_{ext}>0$, it follows that $z_{k,\alpha}$ reaches a positive maximum at some interior point $r_1$, satisfying hence
$$
0\leq -z_{k,\alpha}''(r_-)=\left(\lambda_{0,\alpha}+m^*-\frac{k^2}{r^2}\right)z_{k,\alpha}(r_-)<0,
$$
leading to a contradiction.
\medskip
Let us now deal with small values of $k$, by assuming $k\leq N$. We will prove that \eqref{Eq:DerPos2} holds true.
To this aim, we will compute $z_{k,\alpha}$. Let $J_k$ (resp. $Y_k$) be the $k$-th Bessel function of the first (resp. the second) kind. One has
$$z_{k,\alpha}(r)=\left\{\begin{array}{ll}
A_{k,\alpha}J_k(\sqrt{\frac{\lambda_{0,\alpha}+\kappa}{1+\alpha\kappa}} \frac{r}R) & \text{ if }r\leq r^*_0, \\
B_{k,\alpha}J_k(\sqrt{\lambda_{0,\alpha}}\frac{r}R)+C_{k,\alpha} Y_k(\sqrt{\lambda_{0,\alpha}}\frac{r}R) & \text{ if }r^*_0\leq r\leq R,
\end{array}\right.$$
where $X_{k,\alpha}=(B_{k,\alpha},C_{k,\alpha},A_{k,\alpha})$ solves the linear system
$$
\mathcal A_{k,\alpha}X_{k,\alpha}=b_\alpha
$$
where
$$
b_\alpha=\begin{pmatrix}0\\-\kappa u_{0,\alpha}(r^*_0)\\-\left[\partial_R u_{0,\alpha}\right]\\\end{pmatrix}
$$
and
$$
\mathcal A_{k,\alpha}=\begin{pmatrix}
J_k\left(\sqrt{\lambda_0}\right)&Y_k\left(\sqrt{\lambda_0}\right)&0
\\\sqrt{\lambda_{0,\alpha}}J_k'(\sqrt{\lambda_{0,\alpha}+\kappa}\r)&\sqrt{\lambda_{0,\alpha}}Y_k'(\sqrt{\lambda_{0,\alpha}+\kappa}\r)&-\sqrt{\frac{\lambda_{0,\alpha}+\kappa}{1+\alpha\kappa}}J_k'(\sqrt{\frac{\lambda_{0,\alpha}+\kappa}{1+\alpha\kappa}}\r)
\\J_k(\sqrt{\lambda_{0,\alpha}}\r)&Y_k(\sqrt{\lambda_{0,\alpha}}\r)&-J_k(\sqrt{\frac{\lambda_{0,\alpha}+\kappa}{1+\alpha\kappa}}\r)
\end{pmatrix}.
$$
It is easy to check that
$$
\Vert \mathcal A_{k,\alpha}-\mathcal A_{k,0}\Vert \leq M\alpha
$$
where $M$ only depends\footnote{Indeed, $\{J_k,Y_k\}_{k\leq N}$ are uniformly bounded in $\mathscr C^2([r_0^*/R-\e,R])$ for every $\e>0$ small enough. Since we consider a finite number of indices $k$, there exists $\delta>0$ (depending only on $N$) such that
$$
\forall k\in \{0, \dots,N\}, \quad \det(\mathcal A_{k,\alpha})\geq \delta>0.
$$
Then, since $\Vert X_\alpha-X_0\Vert \leq M\alpha$, it follows from the Cramer formula that there exists $M$ (depending only on $N$) such that
$\Vert
X_{k,\alpha}-X_{k,0}\Vert_{L^\infty}\leq M\alpha.
$
} on $N$.
Hence it is enough to prove that
$|\psi_{k,0}'(r^*_0)|\leq M$ for some $M>0$ depending only on $N$, which is straightforward since the set of indices is finite. The expected conclusion follows.
\subsection{Conclusion}
From Eq. \eqref{Eq:OmegaK} and Lemma \ref{Cl:Final}, there exists $C>0$ and $M>0$ such that $\omega_{k,\alpha}\geq C>0$ and $\zeta_{k,\alpha}\geq -M\alpha$ for every $\alpha\in [0, \overline \alpha]$ and $k\in \N$, from which we infer that
\begin{align*}
\mathcal F_\alpha[V,V]&\geq \left(C-M\alpha\right)\sum_{k=1}^\infty \left(\gamma_k^2+\beta_k^2\right)\geq \frac{C}2 \Vert V\cdot \nu \Vert_{L^2}^2.
\end{align*}
according to Eq. \eqref{Eq:Ptn}.
\subsection{Concluding remark: possible extension to higher dimensions}\label{Se:CclSha}
Let us briefly comment on possible extensions of this method to higher dimensions. Indeed, although we do not tackle this issue in this article, we believe that the coercivity norm obtained in Theorem~\ref{Th:ShapeStability} could also be obtained in the three-dimensional case. Nevertheless, we believe that such an extension would need tedious and technical computations.
Since our objective here was to introduce a methodology to investigate stability issues for the shape optimization problems we deal with, we slightly comment on this claim and explain how we believe that our proof can be adapted to the case $d=3$.
Let $\O$ denote the ball $\mathbb B(0,R)$ in $\R^3$ and $\B^*$ be the centered three-dimensional ball $\mathbb B(0,R)$ of volume $m_0|\O|/\kappa$. Let us assume without loss of generality that $R=1$, so that $\partial \B^*$ is the euclidean unit sphere $\mathbb S^2$.
As a preliminary result, one first has to show that the principal eigenfunction $u_{\alpha,m^*_0}$ is radially symmetric and that $\B^*$ is a critical shape by the same arguments as in the proof of Theorem~\ref{Th:ShapeStability}, which allows us to compute the Lagrange multiplier $\Lambda_\rho$ associated to the volume constraint. Let $\mathcal L_{\Lambda_\rho}$ be the associated shape Lagrangian.
For an integer $k$, we define $H_k$ as the space of spherical harmonics of degree $k$ i.e as the eigenspace associated with the eigenvalue $-k(k+1)$ of the Laplace-Beltrami operator $\Delta_{\mathbb S^2}$. $H_k$ has finite dimension $d_k$, and we furthermore have
$$L^2(\mathbb S^2)=\bigoplus_{k=1}^\infty H_k.$$
Let us consider a Hilbert basis $\{y_{k,\ell}\}_{\ell=1,\dots,d_k}$ of $H_k$.
For an admissible vector field $V,$ one must then expand $ V\cdot \nu$ in the basis of spherical harmonics as
\begin{equation}\label{eq:fin}
V\cdot \nu=\sum_{k=1}^\infty\sum_{\ell=1}^{d_k} \alpha_{k,\ell}(V\cdot \nu) y_{k,\ell}.\end{equation}
Then, one has to diagonalise the second-order shape derivative of $\mathcal L_{\Lambda_\rho}$ and prove that there exists a sequence of coefficients $\{\omega_{k,\ell,\rho}\}_{k\in \N\,, 0\leq \ell\leq d_k}$ such that for every $V\cdot \nu$ expanding as \eqref{eq:fin}, the second order derivative of the shape Lagrangian in direction $V$ reads
$$
\mathcal L_{\Lambda_\rho}''=\sum_{k=1}^\infty\sum_{\ell=1}^{d_k}\left(\alpha_{k,\ell}(V\cdot\nu)\right)^2 \omega_{k,\ell}.
$$
We believe this diagonalization can be proved using separation of variables and the orthogonality properties of the family $\{y_{k,\ell}\}_{k\in \N^*,\ell=1,\dots,d_k}$.
Using the separation of variables, each coefficient $\omega_{k,\ell}$ can be written in terms of derivatives of a family of solutions of one dimensional differential equations. The main difference with the proof of Theorem~\ref{Th:ShapeStability} comes from the fact that the main part of the ODE is not $-\frac1r\frac{d}{dr}(r(1+\alpha m_0^*)\frac{d}{dr})$ anymore, but $-\frac1{r^2}\frac{d}{dr}(r^2(1+\alpha m_0^*)\frac{d}{dr})$. The important fact is that maximum principle arguments may still be used to analyze the diagonalized expression of $\mathcal L_{\Lambda_\rho}''$ and to obtain a uniform bound from below for the sequence $\{\alpha_{k,\ell}\}_{k\in \N^*\,, \ell=1,\dots,d_k}$.
\appendix
\section{Proof of Lemma~\ref{diff:valPvecP}}\label{Ap:Differentiability}
We prove hereafter that the mapping $m\mapsto(u_{\alpha,m}, \lambda_\alpha(m))$ is twice differentiable (and even $\mathscr C^\infty$) in the $L^2$ sense, the proof of the differentiability in the weak $W^{1,2}(\O)$ sense being similar.
Let $m^*\in \mathcal M_{m_0,\kappa}(\O)$, $\sigma_\alpha:=1+\alpha m^*$, and $(u_0,\lambda_0)$ be the eigenpair associated with $m^*$. Let $ h\in \mathcal{T}_{m^*}$ (see Def.~\ref{def:tgtcone}).
Let $m^*_h:=m^*+h$ and $\sigma_{m^*+h}:=1+\alpha(m^*+h).$ Let $(u_{h},\lambda_h)$ be the eigenpair associated with $m^*_h$.
Let us introduce the mapping $G$ defined by
$$
G:\left\{\begin{array}{ll}
\mathcal{T}_{m^*}\times W^{1,2}_0(\O)\times \R\to W^{-1,2}(\O)\times \R,&
\\(h, v, \lambda)\mapsto \left(-\n \cdot (\sigma_{m^*+h}\n v))-\lambda v-m^*_hv, \int_\O v^2-1\right).&\end{array}
\right.$$
From the definition of the eigenvalue, one has $G(0,u_{0},\lambda_{0})=0$.
Moreover, $G$ is $\mathscr C^\infty$ in $\mathcal{T}_{m^*}\cap B\times W^{1,2}_0(\O)\times \R$, where $B$ is an open ball centered at $ 0$. The differential of $G$ at $(0,u_0,\lambda_0)$ reads
$$
D_{v,\lambda}G(0,u_0,\lambda_0)[w,\mu]=\left(-\n \cdot (\sigma_\alpha \n w)-\mu u_0-\lambda_0 w-m^*w, \int_\O 2u_0w\right).
$$
Let us show that this differential is invertible. We will show that, if $(z,k)\in W^{-1,2}(\O)\times \R$, then there exists a unique pair $(w,\mu)$ such that $D_{v,\lambda}G(0,u_0,\lambda_0)[w,\mu]=(z,k)$.
According to the Fredholm alternative, one has necessarily $\mu=-\langle z,u_0\rangle_{L^2(\O)}$ and for this choice of $\mu$, there exists a solution $w_1$ to the equation
$$
-\n \cdot (\sigma_\alpha \n w)-\mu u_0-\lambda_0 w-m^*w=z\quad \text{in }\O .
$$
Moreover, since $\lambda_0$ is simple, any other solution is of the form $w=w_1+tu_0$ with $t\in \R$. From the equation $2\int_\O u_0w=k$, we get $t=k/2-\int_\O w_1u_0$. Hence, the pair $(w,\mu)$ is uniquely determined. According to the implicit function theorem, the mapping $h\mapsto (u_h,\lambda_h)$ is $\mathscr C^\infty$ in a neighbourhood of $\vec 0$.
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\end{document} | {"config": "arxiv", "file": "2001.02958/ArticleDrifted_MNP.tex"} |
TITLE: Demiclosedness at a point instead on a set
QUESTION [1 upvotes]: I have been reading a few research papers for iterative techniques of approximating fixed points of nonexpansive maps. In one of the papers, I found a lemma which is as follows:-
Theorem 1: Let $C$ be a closed convex subset of a uniformly convex Banach space $E$, and $T$ a nonexpansive map on $C$. Then, $I - T$ is demiclosed at zero.
This lemma was referenced to a theorem in the book "Topics in metric fixed point theory" by Goebel and Kirk. However, in the book, the theorem is stated as follows:-
Theorem 2: Let $E$ be a uniformly convex Banach space and $C$ be a nonempty closed and convex subset of $E$. Let $T: C \rightarrow E$ be a non-expansive mapping. Then, $T$ is demiclosed on $C$.
There are two questions that come to my mind:-
What do we mean by demiclosedness at a point? All the definitions that I could find deal with demiclosedness on a set. In particular, we say that a map $T: C \rightarrow E$ is demiclosed if for any sequence $\left( x_n \right)$ in $C$ which converges weakly to $x \in E$ and $T \left( x_n \right) \rightarrow y$ in $E$, we have $x \in C$ and $f \left( x \right) = y$. How do we extend (or restrict) this definition to define demiclosedness at a point?
Are these two theorems equivalent? If so, does it mean that demiclosedness of a map on a closed and convex subset is same as saying that the demiclosed at a point?
Edit:-
I did find out the meaning of demiclosedness at a point. It is defined as follows:-
Definition: Let $X$ be a Banach space and $T$ be a mapping with domain $D \left( T \right)$ and range $R \left( T \right)$. Then, $T$ is said to be demiclosed at a point $p \in R \left( T \right)$ if for every sequence $\left( x_n \right)_{n \in \mathbb{N}}$ in $D \left( T \right)$ that converges weakly to a point $x \in D \left( T \right)$ and the sequence $\left( Tx_n \right)_{n \in \mathbb{Z}}$ in $R \left( T \right)$ converges strongly to $p$, we have $Tx = p$.
Now, Theorem $1$ makes sense to me. However, there is still a problem in proving the theorem. While exploring more on the subject, I came across the book "Topics in Fixed Point Theory" by M. A. Khamsi. In that book, a theorem similar to Theorem $1$ is proved. However, it uses Opial's condition. Thus, now the question is: "Is the Theorem $1$ stated here still true for Banach spaces not satisfying Opial's condition?"
REPLY [1 votes]: Yes this is true. Not all spaces with Opial’s property are uniformly convex and vice versa is true. The sequential space $\ell^1$ is not uniformly convex but has Opial’s property; whereas $\ell^p$ (for $p\in(1,\infty)$) has both Opial’s and uniform convexity property; however, $L^p[0,1]$ (for $p\in(1,\infty)$) has uniform convexity but does not have Opial’s property. | {"set_name": "stack_exchange", "score": 1, "question_id": 3527233} |
TITLE: $ (1 + x)^{1/x} $ when $ x \to \infty$ without L’Hospital’s Rule
QUESTION [4 upvotes]: Title is the question itself.
How can I show below without L’Hospital’s Rule
$$ \lim_{x\to \infty} ( 1 + x )^\frac{1}{x} = 1 $$
REPLY [0 votes]: HINT:
Let $x\ge 1$ and $n = [x]$ so $n\le x < n+1$. We have
$$\left(1+\frac{1}{n+1}\right)^n< (1+\frac{1}{x})^x < \left(1+\frac{1}{n}\right)^{n+1}$$ | {"set_name": "stack_exchange", "score": 4, "question_id": 2402447} |
\begin{document}
\maketitle
\pagenumbering{arabic}
\begin{abstract}
Despite the superior empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks.
Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime.
We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bounds. With such sharp characterizations, we further explore how various learning parameters impact the generalization capability of overparameterized MAML, including explicitly identifying typical data and task distributions that can achieve diminishing generalization error with overparameterization, and characterizing the impact of adaptation learning rate on both excess risk and the early stopping time. Our theoretical findings are further validated by experiments.
\end{abstract}
\section{Introduction}
Meta-learning~\cite{hospedales2020meta} is a learning paradigm which aims to design algorithms that are capable of gaining knowledge from many previous tasks and then using it to improve the performance on future tasks efficiently. It has exhibited great power in various machine learning applications spanning over few-shot image classification~\cite{ren2018meta,rusu2018meta}, reinforcement learning~\cite{gupta2018meta} and intelligent medicine~\cite{gu2018meta}.
One prominent type of meta-learning approaches is an optimization-based method, Model-Agnostic Meta-Learning (MAML)~\cite{finn2017model}, which achieves impressive results in different tasks~\cite{obamuyide2019model,bao2019few,antoniou2018train}. The idea of MAML is to learn a good initialization $\boldsymbol{\omega}^{*}$, such that for a new task we can adapt quickly to
a good task parameter starting from $\boldsymbol{\omega}^{*}$. MAML takes a bi-level implementation: the inner-level initializes at the meta parameter and takes task-specific updates using a few steps of gradient descent (GD), and the outer-level optimizes the meta parameter across all tasks.
With the superior empirical success, theoretical justifications have been provided for MAML and its variants over the past few years from both optimization~\cite{finn2019online,wang2020globala,fallah2020convergence,ji2022theoretical} and generalization perspectives~\cite{amit2018meta,denevi2019learning,fallah2021generalization,chen2021generalization}. However, most existing analyses
did not take overparameterization into consideration, which we deem as crucial to demystify the remarkable generalization ability of deep meta-learning~\cite{zhang2021understanding, hospedales2020meta}. More recently, \cite{wang2020globalb} studied the MAML with overparameterized deep neural nets and derived a complexity-based bound to quantify the difference between the empirical and population loss functions at their optimal solutions. However, complexity-based generalization bounds tend to be weak in the high dimensional, especially in the overparameterized regime. Recent works~\cite{bernacchia2021meta,zou2021unraveling} developed more precise bounds for overparameterized setting under a mixed linear regression model, and identified the effect of adaptation learning rate on the generalization. Yet, they considered only the simple isotropic covariance for data and tasks, and did not explicitly capture how the generalization performance of MAML depends on the data and task distributions.
Therefore, the following important problem still remains largely open:
\begin{center}
\emph{ Can \textbf{overparameterized} MAML generalize well to a new task, under general data and task distributions?}
\end{center}
In this work, we utilize the mixed linear regression, which is widely adopted in theoretical studies for meta-learning~\cite{kong2020meta,bernacchia2021meta,denevi2018learning,bai2021important}, as a proxy to address the above question. In particular, we assume that each task $\tau$ is a noisy linear regression and the associated weight vector is sampled from a common distribution. Under this model, we consider one-step MAML meta-trained with stochastic gradient descent (SGD), where
we minimize the loss evaluated at single GD step further ahead for each task.
Such settings correspond to real-world implementations of MAML~\cite{finn2017meta,li2017meta,hospedales2020meta} and are extensively considered in theoretical analysis~\cite{fallah2020convergence,chen2022bayesian,fallah2021generalization}.
The focus of this work is the overparameterized regime, i.e., the data dimension $d$ is far larger than the meta-training iterations $T$ ($d\gg T$).
\subsection{Our Contributions}
Our goal is to characterize the generalization behaviours of the MAML output in the overparameterized regime, and to explore how different problem parameters, such as data and task distributions, the adaptation learning rate $\btr$, affect the test error. The main contributions are highlighted below.
\begin{itemize}
\item Our first contribution is a sharp characterization (both upper and lower bounds) of the excess risk of MAML trained by SGD. The results are presented in a general manner, which depend on a new notion of effective meta weight, data spectrum, task covariance matrix, and other hyperparameters such as training and test learning rates.
In particular, the {\bf effective meta weight} captures an essential property of MAML, where the inner-loop gradient updates have distinctive effects on different dimensions of data eigenspace, i.e., the importance of "leading" space will be magnified whereas the "tail" space will be suppressed.
\item We investigate the influence of data and task distributions on the excess risk of MAML. For $\log$-decay data spectrum, our upper and lower bounds establish a sharp phase transition of the generalization. Namely, the excess risk vanishes for large $T$ (where benign fitting occurs) if the data spectrum decay rate is faster than the task diversity rate, and non-vanishing risk occurs otherwise. In contrast, for polynomial or exponential data spectrum decays, excess risk always vanishes for large $T$ irrespective of the task diversity spectrum.
\item We showcase the important role the adaptation learning rate $\btr$ plays in the excess risk and the early stopping time of MAML. We provably identify a novel tradeoff between the different impacts of $\btr$ on the "leading" and "tail" data spectrum spaces as the main reason behind the phenomena that the excess risk will first increase then decrease as $\btr$ changes from negative to positive values under general data settings. This complements the explanation based only on the "leading" data spectrum space given in~\cite{bernacchia2021meta} for the isotropic case.
We further theoretically illustrate that $\btr$ plays a similar role in determining the early stopping time, i.e., the iteration at which MAML achieves steady generalization error.
\end{itemize}
\textbf{Notations.}
We will use bold lowercase and capital letters for vectors and matrices respectively. $\mathcal{N}\left(0, \sigma^{2}\right)$ denotes the Gaussian distribution with mean $0$ and variance $\sigma^2$. We use $f(x) \lesssim g(x)$ to denote the case $f(x) \leq c g(x)$ for some constant $c>0$. We use the standard big-O notation and its variants: $\mathcal{O}(\cdot), \Omega(\cdot)$, where $T$ is the problem parameter that becomes large. Occasionally, we use the symbol $\widetilde{\mathcal{O}}(\cdot)$
to hide $\polylog(T)$ factors. $\mathbf{1}_{(\cdot)}$ denotes the indicator function. Let $x^{+}=\max\{x,0\}$.
\section{Related Work}
\label{sec-related}
\paragraph{Optimization theory for MAML-type approaches} Theoretical guarantee of MAML was initially provided in~\cite{finn2017meta} by proving a universal approximation property under certain conditions. One line of theoretical works have focused on the optimization perspective. \cite{fallah2020convergence} established the convergence guarantee of one-step MAML for general nonconvex functions, and \cite{ji2022theoretical} extended such results to the multi-step setting. \cite{finn2019online} analyzed the regret bound for online MAML. \cite{wang2020globala,wang2020globalb} studied the global optimality of MAML with sufficiently wide deep neural nets (DNN). Recently, \cite{collins2022maml} studied MAML from a representation point of view, and showed that MAML can provably recover the ground-truth subspace. h
\paragraph{Statistical theory for MAML-type approaches.}
One line of theoretical analyses lie in the statistical aspect. \cite{fallah2021generalization} studied the generalization of MAML
on recurring and unseen tasks. Information theory-type generalization bounds for MAML were developed in~\cite{jose2021information,chen2021generalization}.
\cite{chen2022bayesian} characterized the gap of generalization error between MAML and Bayes MAML. \cite{wang2020globalb} provided the statistical error bound for MAML with overparameterized DNN. Our work falls into this category,
where the overparameterization has been rarely considered in previous works. Note that \cite{wang2020globalb} only derived the generalization bound from the complexity-based perspective to study the difference between the empirical and population losses for the obtained optimization solutions. Such complexity bound is typically related to the data dimension~\cite{neyshabur2018towards} and may yield vacuous bound in the high dimensional regime. However, our work show that the generalization error of MAML can be small even the data dimension is sufficiently large.
\paragraph{Overparamterized meta-learning.}
\cite{du2020few,sun2021towards} studied overparameterized meta-learning from a representation learning perspective.
The most relevant papers to our work are~\cite{zou2021unraveling,bernacchia2021meta}, where they derived the population risk
in overparameterized settings to show the effect of the adaptation learning rate for MAML. Our analysis differs from these works from two essential perspectives: \romannum{1}).\ we analyze the excess risk of MAML based on the optimization trajectory of SGD in non-asymptotic regime, highlighting the dependence of iterations $T$, while they directly solved the MAML objective asymptotically; \romannum{2}). \cite{zou2021unraveling,bernacchia2021meta} mainly focused on the simple isotropic case for data and task covariance, while
we explicitly explore the role of data and task distributions under general settings.
\paragraph{Overparameterized linear model.} There has been several recent progress in theoretical understanding of overparameterized linear model under different scenarios, where the main goal is to provide non-asymptotic generalization guarantees, such as studies of linear regression ~\cite{bartlett2020benign}, ridge regression~\cite{tsigler2020benign}, constant-stepsize SGD~\cite{zou2021benign}, decaying-stepsize SGD~\cite{wu2021last}, GD~\cite{xu2022relaxing}, Gaussian Mixture models~\cite{wang2021benign}. This paper aims to derive the non-asymptotic excess risk bound for MAML under mixed linear model, which can be independent of data dimension $d$ and still converge as the iteration $T$ enlarges.
\section{Preliminary}\label{sec-form}
\subsection{Meta Learning Formulation}
In this work, we consider a standard meta-learning setting~\cite{fallah2021generalization}, where a number of tasks share some similarities, and the learner aims to find a good model prior by leveraging task similarities, so that the learner can quickly find a desirable model for a new task by adapting from such an initial prior.
{\bf Learning a proper initialization.}
Suppose we are given a collection of tasks $\textstyle\{\tau_t\}^{T}_{t=1}$ sampled from some distribution $\mathcal{T}$. For each task $\tau_t$, we observe $N$ samples $\textstyle\mathcal{D}_{t}\triangleq (\mathbf{X}_t,\mathbf{y}_{t})=\left\{\left(\mathbf{x}_{t, j}, y_{t, j}\right) \in \mathbb{R}^{d} \times \mathbb{R}\right\}_{j \in\left[N\right]}\stackrel{i.i.d.}{\sim} \mathbb{P}_{\phi_{t}}(y|\mathbf{x}) \mathbb{P}(\mathbf{x})$, where $\phi_t$ is the model parameter for the $t$-th task. The collection of $\{\mathcal{D}_{t}\}^{T}_{t=1}$ is denoted as $\mathcal{D}$. Suppose that $\mathcal{D}_{t}$ is randomly split into training and validation sets, denoted respectively as $\mathcal{D}^{\text{in}}_{t}\triangleq (\Xb^{\text{in}}_t,\yb_t^{\text{in}})$ and $\mathcal{D}^{\text{out}}_{t}\triangleq (\Xb^{\text{out}}_t,\yb_t^{\text{out}})$, correspondingly containing $n_{1}$ and $n_2$ samples (i.e., $N=n_1+n_2$).
We let $\boldsymbol{\omega}\in\mathbb{R}^{d}$ denote the initialization variable. Each task $\tau_t$ applies an inner algorithm $\mathcal{A}$ with such an initial and obtains an output $\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t})$. Thus, the adaptation performance of $\boldsymbol{\omega}$ for task $\tau_t$ can be measured by the mean squared loss over the validation set given by $\textstyle\ell(\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t}):= \frac{1}{2n_2}\sum^{n_2}_{j=1} \left(\left\langle \mathbf{x}^{\text{out}}_{t,j}, \mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t})\right\rangle-y^{\text{out}}_{t,j}\right)^{2}$. The goal of meta-learning is to find an optimal
initialization $\hat{\boldsymbol{\omega}}^{*}\in\mathbb{R}^{d}$ by minimizing the following empirical meta-training loss:
\begin{align}
\min_{\boldsymbol{\omega}\in\mathbb{R}^{d}} \widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega};\mathcal{D}) \quad \text{ where }\;
\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega};\mathcal{D})&=\frac{1}{T}\sum^{T}_{t=1}\ell(\mathcal{A}(\boldsymbol{\omega};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})\label{emp_loss}.
\end{align}
In the testing process, suppose a new task $\tau$ sampled from $\mathcal{T}$ is given, which is associated with the dataset $\mathcal{Z}$ consisting of $m$ points with the task. We apply the learned initial $\hat{\boldsymbol{\omega}}^{*}$
as well as the inner algorithm
$\mathcal{A}$ on $\mathcal{Z}$ to produce a task predictor. Then the test performance can be evaluated via the following population loss:
\begin{align}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega})=\mathbb{E}_{\tau \sim \mathcal{T}} \mathbb{E}_{\mathcal{Z},(\mathbf{x}, y)\sim \mathbb{P}_{\phi}(y \mid \mathbf{x}) \mathbb{P}(\mathbf{x})} \left[\ell\left(\mathcal{A}\left(\boldsymbol{\omega}; \mathcal{Z}\right);(\mathbf{x},y)\right)\right].
\label{obj}
\end{align}
\paragraph{Inner Loop with one-step GD.}
Our focus of this paper is the popular meta-learning algorithm MAML~\cite{finn2017model}, where inner stage takes a few steps of GD update initialized from $\boldsymbol{\omega}$. We consider one step for simplicity, which is commonly adopted in the previous studies~\cite{bernacchia2021meta,collins2022maml,gao2020modeling}. Formally, for any $\boldsymbol{\omega}\in\mathbb{R}^d$, and any dataset $(\mathbf{X},\mathbf{y})$ with $n$ samples, the inner loop algorithm for MAML with a learning rate $\beta$ is given by
\begin{align}
\mathcal{A}(\boldsymbol{\omega};(\mathbf{X},\mathbf{y})):= \boldsymbol{\omega}-\beta \nabla_{\boldsymbol{\omega}} \ell\left(\boldsymbol{\omega};(\mathbf{X},\mathbf{y})\right)=
(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\boldsymbol{\omega}+\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{y}.
\end{align}
We allow the learning rate to differ at the meta-training and testing stages, denoted as $\beta^{\text{tr}}$ and $\beta^{\text{te}}$ respectively. Moreover, in subsequent analysis, we will include the dependence on the learning rate to the inner loop algorithm and loss functions as
$\mathcal{A}(\boldsymbol{\omega},\beta;(\mathbf{X},\mathbf{y}))$, $\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta;\mathcal{D})$ and $\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta)$.
\paragraph{Outer Loop with SGD.}
We adopt SGD to iteratively update the meta initialization variable $\boldsymbol{\omega}$ based on the empirical meta-training loss \cref{emp_loss}, which is how MAML is implemented in practice~\cite{finn2017meta}.
Specifically, we use the constant stepsize SGD with iterative averaging~\cite{fallah2021generalization, denevi2018learning,denevi2019learning},
and the algorithm is summarized in \Cref{alg-meta-sgd}. Note that at each iteration, we use one task for updating the meta parameter, which can be easily generalized to the case with a mini-batch tasks for each iteration.
\begin{algorithm}[ht]
\caption{MAML with SGD}\label{alg-meta-sgd}
\begin{algorithmic}
\REQUIRE Stepsize $\alpha>0$, meta learning rate $\beta^{\text{tr}}>0$
\ENSURE $\boldsymbol{\omega}_{0}$
\FOR{$t=1$ to $T$}
\STATE Receive task $\tau_t$ with data $\mathcal{D}_t$
\STATE Randomly divided into training and validation set: $\mathcal{D}^{in}_{t}=(\mathbf{X}^{in}_t, \mathbf{y}^{in}_t)$, $\mathcal{D}^{out}_{t}=(\mathbf{X}^{out}_t, \mathbf{y}^{out}_t)$
\STATE Update $\boldsymbol{\omega}_{t+1} =\boldsymbol{\omega}_{t}-\alpha \nabla \ell(\mathcal{A}(\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})$
\ENDFOR
\RETURN $\overline{\boldsymbol{\omega}}_T=\frac{1}{T}\sum^{T-1}_{t=0} \boldsymbol{\omega}_{t}$
\end{algorithmic}
\end{algorithm}
\paragraph{Meta Excess Risk of SGD.} Let $\boldsymbol{\omega}^{*}$ denote the optimal solution to the population meta-test error
\cref{obj}.
We define the following excess risk
for the output $\overline{\boldsymbol{\omega}}_T$ of SGD:
\begin{align}
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\triangleq \mathbb{E}\left[\mathcal{L}(\mathcal{A},\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\right]-\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})\label{excess}
\end{align}
which identifies the difference between adapting from the SGD output $\overline{\boldsymbol{\omega}}_T$ and from the optimal initialization $\boldsymbol{\omega}^{*}$. Assuming that each task contains a fixed constant number of samples, the total number of samples over all tasks is $\mathcal{O}(T)$. Hence, the overparameterized regime can be identified as $d\gg T$, which is the focus of this paper, and is in contrast to the well studied underparameterized setting with finite dimension $d$ $(d\ll T)$.
The goal of this work is to characterize the impact of SGD dynamics, demonstrating how the iteration $T$ affects the excess risk, which has not been considered in the previous overparameterized MAML analysis~\cite{bernacchia2021meta,zou2021unraveling}.
\subsection{Task and Data Distributions}
To gain more explicit knowledge of MAML, we specify the task and data distributions in this section.
{\bf Mixed Linear Regression.}
We consider a canonical case in which the tasks are linear regressions. This setting has been commonly adopted recently in~\cite{bernacchia2021meta,bai2021important,kong2020meta}.
Given a task $\tau$, its model parameter $\phi$ is determined by
$\boldsymbol{\theta}\in\mathbb{R}^{d}$,
and the output response is generated as follows:
\begin{align}
y=\boldsymbol{\theta}^{\top} \mathbf{x}+z, \quad \xb\sim\mathcal{P}_{\xb},\quad z\sim \mathcal{P}_{z}
\end{align}
where $\xb$ is the input feature, which follows the same distribution $\mathcal{P}_{\xb}$ across different tasks, and $z$ is the i.i.d.\ Gaussian noise sampled from $\mathcal{N}(0,\sigma^2)$. The task signal $\boldsymbol{\theta}$ has the mean $\boldsymbol{\theta}^{*}$ and the covariance $\Sigma_{\boldsymbol{\theta}}\triangleq \mathbb{E}[\boldsymbol{\theta}\boldsymbol{\theta}^{\top}]$. Denote the distribution of $\boldsymbol{\theta}$ as $\mathcal{P}_{\boldsymbol{\theta}}$. We do not make any additional assumptions on $\mathcal{P}_{\boldsymbol{\theta}}$, whereas recent studies on MAML~\cite{bernacchia2021meta,zou2021unraveling} assume it to be Gaussian and isotropic.
{\bf Data distribution.} For the data distribution $\mathcal{P}_{\xb}$, we
first introduce some mild regularity conditions:
\begin{enumerate}
\item $\xb\in\mathbb{R}^d$ is mean zero with covariance operator $\bSigma=\mathbb{E}[\xb \xb^{\top}]$;
\item The spectral decomposition of $\bSigma$ is $\boldsymbol{V} \boldsymbol{\Lambda} \boldsymbol{V}^{\top}=\sum_{i>0} \lambda_{i} \boldsymbol{v}_{i} \boldsymbol{v}_{i}^{\top}$, with decreasing eigenvalues $\lambda_1\geq \cdots\geq\lambda_d>0$, and suppose $\sum_{i>0}\lambda_{i} <\infty $.
\item $\bSigma^{-\frac{1}{2}} \mathbf{x}$ is $\sigma_{\xb}$-subGaussian.
\end{enumerate}
To analyze the stochastic approximation method SGD
, we take the following standard fourth moment condition~\cite{zou2021benign, jain2017markov, berthier2020tight}.
\begin{assumption}[Fourth moment condition] There exist positive constants $c_1,b_1>0$, such that for any positive semidefinite (PSD)
matrix $\mathbf{A}$, it holds that
\begin{align*}
b_1 \operatorname{tr}(\bSigma \mathbf{A}) \Sigma+\Sigma \mathbf{A }\bSigma \preceq\mathbb{E}_{\mathbf{x} \sim \mathcal{P}_{\mathbf{x}}}\left[\mathbf{x x}^{\top} \mathbf{A} \mathbf{x} \mathbf{x}^{\top}\right] \preceq c_1 \operatorname{tr}(\bSigma \mathbf{A}) \Sigma
\end{align*}
For the Gaussian distribution, it suffices to take $c_1=3,b_1=2.$
\end{assumption}
\subsection{Connection to a Meta Least Square Problem.} After instantiating our study on the task and data distributions in the last section, note that $\textstyle\nabla\ell(\mathcal{A}(\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D}^{\text{in}}_{t});\mathcal{D}^{\text{out}}_{t})$ is linear
with respect to $\boldsymbol{\omega}$. Hence, we can reformulate the problem \cref{emp_loss} as a least square (LS) problem with transformed meta inputs and output responses.
\begin{proposition}[Meta LS Problem]\label{prop1} Under the mixed linear regression model,
the expectation of the meta-training loss \cref{emp_loss} taken over task and data distributions can be rewritten as:
\begin{align}\label{linear-loss}
\mathbb{E}\left[\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D})\right]= \mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})=
\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align}
The meta data are given by
\begin{align}\Bb =& \frac{1}{\sqrt{n_2}}\Xb^{out}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\nonumber\\ \boldsymbol{\gamma} =& \frac{1}{\sqrt{n_2}}\Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big)\label{meta-data} \end{align}
where $\Xb^{\text{in}}\in \mathbb{R}^{n_1\times d}$,$\zb^{\text{in}}\in \mathbb{R}^{n_1}$,$\Xb^{\text{out}}\in \mathbb{R}^{n_2\times d}$ and $\zb^{\text{out}}\in \mathbb{R}^{n_2}$ denote the inputs and noise for training and validation.
Furthermore,
we have
\begin{align}
\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ with meta noise } \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0. \label{linear}
\end{align}
\end{proposition}
Therefore, the meta-training objective
is equivalent to searching for a $\boldsymbol{\omega}$, which is close to the task mean $\boldsymbol{\theta}^{*}$.
Moreover,
with the specified data and task model, the optimal solution for meta-test loss \cref{obj} can be directly calculated~\cite{gao2020modeling}, and we obtain $\boldsymbol{\omega}^{*}=\mathbb{E}[\btheta] =\btheta^{*}$. Hence, the meta excess risk~\cref{excess} is identical to the standard excess risk~\cite{bartlett2020benign} for the linear model \cref{linear}, i.e., $ R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]$, but with more complicated input and output data expressions. The following analysis will focus on this transformed linear model.
Furthermore, we can calculate the statistical properties of the reformed input $\Bb$, and obtain the meta-covariance: $$ E[\Bb^{\top}\Bb]=(\Ib-\beta^{\text{tr}}\bSigma)^2\bSigma+\frac{{\beta^{\text{tr}}}^2}{n_1}(F-\bSigma^3)$$ where $F=E[\xb\xb^{\top}\Sigma \xb\xb^{\top}]$. Let $\Xb\in\mathbb{R}^{n\times d}$ denote the collection of $n$ i.i.d.\ samples from $\mathcal{P}_{\xb}$, and denote
$$
\Hb_{n,\beta}=\mathbb{E}[(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)
]=(\Ib-\beta\bSigma)^2\bSigma+\frac{\beta^2}{n}(F-\bSigma^3).
$$
We can then write $E[\Bb^{\top}\Bb]=\Hb_{n_1,\beta^{\text{tr}}}$.
Regarding the form of $\Bb$ and $\Hb_{n_1,\beta^{\text{tr}}}$, we need some
further conditions on the higher order moments of the data distribution.
\begin{assumption}[Commutity
]\label{ass-comm}
$F=E[\xb\xb^{\top}\bSigma \xb\xb^{\top}]$ commutes with the data covariance $\bSigma$.
\end{assumption}
\Cref{ass-comm} holds for Gaussian data. Such commutity of $\bSigma$ has also been considered in ~\cite{zou2021benign}.
\begin{assumption}[Higher order moment condition]\label{ass:higherorder}
Given $|\beta|<\frac{1}{\lambda_1}$ and $\bSigma$, there exists a constant $C(\beta,\bSigma)>0$, for large $n>0$, s.t. for any unit vector $\vb\in\mathbb{R}^d$, we have:
\begin{align}\label{hoc}
\mathbb{E}[\|\vb^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\vb\|^2]< C(\beta,\bSigma).
\end{align}
\end{assumption}
In \Cref{ass:higherorder}, the analytical form of $C(\beta,\bSigma)$ can be derived if $\bSigma^{-\frac{1}{2}}\mathbf{x}$ is Gaussian.
Moreover, if $\beta=0$, then we obtain $C(\beta,\bSigma)=1$. Further technical discussions
are presented in Appendix.
\section{Main Results}\label{sec-main}
In this section, we present our analyses on generalization properties of MAML optimized by average SGD and derive insights on the effect of various parameters. Specifically, our results consist of three parts. First, we characterize the meta excess risk of MAML trained with SGD. Then, we establish the generalization error bound for various types of data and task distributions, to reveal which kind of overparameterization regarding data and task is essential for diminishing meta excess risk. Finally, we explore how the adaptation learning rate $\btr$ affects the excess risk and the training dynamics.
\subsection{Performance Bounds}
Before starting our results, we first introduce relevant notations and concepts. We define the following rates of interest (See \Cref{remark-f} for further discussions)
\begin{align*} c(\beta,\bSigma) &:= c_1(1+8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2))\\
f(\beta,n,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&:=c(\beta,\bSigma)\operatorname{tr}({\bSigma_{\boldsymbol{\theta}}\bSigma})+4c_1\sigma^2\sigma_x^2\beta^2\sqrt{C(\beta,\bSigma)}\operatorname{tr}(\bSigma^2)+\sigma^2/n\\
g(\beta,n, \sigma,\bSigma, \bSigma_{\btheta}) & :={\sigma^2+b_1\operatorname{tr}(\bSigma_{\btheta}\Hb_{n,\beta})+\beta^2\mathbf{1}_{\beta\leq 0} b_1 \operatorname{tr}(\bSigma^2)/{n}}.
\end{align*}
Moreover, for a positive semi-definite matrix $\Hb$, s.t. $\Hb$ and $\bSigma$ can be diagonalized simultaneously, let $\mu_i(\Hb)$ denote its corresponding eigenvalues for $\vb_i$, i.e. $\Hb = \sum_{i}\mu_i(\Hb)\vb_i\vb_i^{\top}$ (Recall $\vb_i$ is the $i$-th eigenvector of $\bSigma$).
We next introduce the following new notion of the \emph{effective meta weight}, which will serve as an important quantity for capturing the generalization of MAML.
\begin{definition}[Effective Meta Weights]\label{meta-weight}
For $|\btr|,|\bte|<1/\lambda_1$, given step size $\alpha $ and iteration $T$, define
\begin{equation}
\Xi_i (\bSigma
,\alpha,T)=\begin{cases}
\mu_i(\Hb_{m,\beta^{\text{te}}})/\left(T \mu_i(\Hb_{n_1,\beta^{\text{tr}}})\right) & \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}; \\
T\alpha^2 \mu_i(\Hb_{n_1,\btr})\mu_i(\Hb_{m,\beta^{\text{te}}})& \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T}.
\end{cases}
\end{equation}
We call $ \mu_i(\Hb_{m,\beta^{\text{te}}})/ \mu_i(\Hb_{n_1,\beta^{\text{tr}}})$ and $ \mu_i(\Hb_{m,\beta^{\text{te}}})\mu_i(\Hb_{n_1,\beta^{\text{tr}}})$ the \textbf{meta ratio} (See \Cref{remark-weight}).
\end{definition}
We omit the arguments of the effective meta weight $\Xi_i$ for simplicity in the following analysis.
Our first results characterize matching upper and lower bounds on the meta excess risk of MAML in terms of the effective meta weight.
\begin{theorem}[Upper Bound]\label{thm-upper} Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$, then the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded above as follows
\[R(\wl_{T},\bte)\leq \text{Bias}+ \text{Var} \]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [\underbrace{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}_{V_1}+\underbrace{ 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}_{V_2}]
\end{align*}
\end{theorem}
\begin{remark}\label{remark-1}
The primary error source of the upper bound are two folds. The bias term corresponds to the error if we directly implement GD updates towards the meta objective~\cref{linear-loss}. The variance error is composed of the disturbance of meta
noise $\boldsymbol{\xi}$ (the $V_1$ term), and the randomness of SGD itself (the $V_2$ term). Regardless of data or task distributions, for proper stepsize $\alpha$, we can easily derive that the bias term is $\mathcal{O}(\frac{1}{T})$, and the $V_2$ term is also $\mathcal{O}(\frac{1}{T})$, which is dominated by $V_1$ term ($\Omega(1)$). Hence, to achieve the vanishing risk, we need to understand the roles of $\Xi_i$ and $f(\cdot)$
\end{remark}
\begin{remark}[Effective Meta Weights]\label{remark-weight}
By \Cref{meta-weight}, we separate the data eigenspace into “\textbf{leading}” $(\geq \frac{1}{\alpha T})$ and “{\bf tail}” $(< \frac{1}{\alpha T})$ spectrum spaces with different meta weights.
The meta ratios
indicate the impact of one-step gradient update. For large $n$,
$\mu_i(\Hb_{n,\beta})\approx (1-\beta\lambda_i)^2\lambda_i$, and hence a larger $\btr$ in training will increase the weight for “leading” space and decrease the weight for “tail” space, while a larger $\bte$ always decreases the weight.
\end{remark}
\begin{remark}[Role of $f(\cdot)$]\label{remark-f} $f(\cdot)$ in variance term consists of various sources of meta noise $\boldsymbol{\xi}$, including inner gradient updates ($\beta$), task diversity ($\bSigma_{\btheta}$) and noise from regression tasks ($\sigma$). As mentioned in \Cref{remark-1}, understanding $f(\cdot)$ is critical in our analysis. Yet,
due to the multiple randomness origins, techniques for classic linear regression~\cite{zou2021benign,jain2017markov} cannot be directly applied here.
Our analysis overcomes such non-trivial challenges. $g(\cdot)$ in \Cref{thm-lower} plays a similar role to $f(\cdot)$.
\end{remark}
Therefore, \Cref{thm-upper} implies that overparameterization is crucial for diminishing risk under the following conditions:
\begin{itemize}[itemsep=2pt,topsep=0pt,parsep=0pt]
\item For $f(\cdot)$: $\operatorname{tr}(\bSigma\bSigma_{\btheta})$ and $\operatorname{tr}(\bSigma^2)$ is small compared to $T$;
\item For $\Xi_i$: the dimension of "leading" space is $o(T)$, and the summation of meta ratio over "tail" space is $o(\frac{1}{T})$.
\end{itemize}
We next provide a lower bound on the meta excess risk, which matches the upper bound in order.
\begin{theorem}[Lower Bound]~\label{thm-lower}
Following the similar notations in ~\Cref{thm-upper}, Then
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\bte) \ge & \frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} +\frac{1}{n_2}\cdot \frac{1}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
\times & [\frac{1}{100} g(\btr,n_1, \bSigma, \bSigma_{\btheta})+\frac{b_1}{1000}
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i].
\end{align*}
\end{theorem}
Our lower bound can also be decomposed into bias and variance terms as the upper bound. The bias term well matches the upper bound up to absolute constants. The variance term differs from the upper bound only by $\frac{1}{n_2}$, where $n_2$ is the batch size of each task, and is treated as a constant (i.e., does not scale with $T$)~\cite{jain2018parallelizing,shalev2013accelerated} in practice. Hence, in the overparameterized regime where $d\gg T$ and $T$ tends to be sufficiently large, the variance term also matches that in the upper bound w.r.t.\ $T$.
\subsection{The Effects of Task Diversity}\label{sec-main-task}
From \Cref{thm-upper} and \Cref{thm-lower}, we observe that the task diversity $\bSigma_{\btheta}$ in $f(\cdot)$ and $g(\cdot)$ plays a crucial role in the performance guarantees for MAML. In this section,
we explore several types of data distributions to further characterize the effects of the task diversity.
We take the single task setting as a comparison with meta-learning, where the task diversity diminishes (tentatively say $\bSigma_{\btheta}\rightarrow\mathbf{0}$), i.e., each task parameter $\boldsymbol{\theta}=\boldsymbol{\theta}^{*}$. In such a case, it is unnecessary to do one-step gradient in the inner loop and we set $\btr=0$, which is equivalent to directly running SGD. Formally, the {\bf single task setting} can be described as outputting $\overline{\boldsymbol{\omega}}^{\text{sin}}_{T}$ with iterative SGD that minimizes $\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},0;\mathcal{D})$ with meta linear model as $\boldsymbol{\gamma} = \frac{1}{\sqrt{n_2}}(\Xb^{out}\boldsymbol{\theta}^{*}+\zb^{\text{out}})$.
\Cref{thm-upper} implies that the data spectrum should decay fast, which leads to a small dimension of "leading" space and small meta ratio summation over "tail" space. Let us first consider a relatively slow decaying case: $\lambda_k=k^{-1}\log^{-p}(k+1)$ for some $p>1$. Applying \Cref{thm-upper}, we immediately derive the theoretical guarantees for single task:
\begin{lemma}[Single Task]\label{lem-single}
If $|\beta^{\text{te}}|<\frac{1}{\lambda_1}$ and if the spectrum of $\bSigma$ satisfies $\lambda_k=k^{-1}\log^{-p}(k+1)$, then $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\mathcal{O}(\frac{1}{\log^{p}(T)})$
\end{lemma}
At the test stage, if we set $\beta^{\text{te}}=0$, then the meta excess risk for the single task setting, i.e., $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$, is exactly the excess risk in classical linear regression~\cite{zou2021benign}. \Cref{lem-single} can be regarded as a generalized version of Corollary 2.3 in \cite{zou2021benign}, where they provide the upper bound for $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$, while we allow a one-step fine-tuning for testing.
Lemma~\ref{lem-single} suggests that the $\log$-decay is
sufficient to assure that $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,0)$ is diminishing when $d\gg T$.
However, in meta-learning with multi-tasks, the task diversity captured by the task spectral distribution can highly affect the meta excess risk. In the following, our \Cref{thm-upper} and \Cref{thm-lower} (i.e., upper and lower bounds) establish a sharp phase transition of the generalization for MAML for the same data spectrum considered in Lemma~\ref{lem-single}, which is in contrast to the single task setting (see \Cref{lem-single}), where $\log$-decay data spectrum always yields vanishing excess risk.
\begin{proposition}[MAML, $\log$-Decay Data Spectrum]\label{prop-hard}
Given $|\btr|, |\beta^{\text{te}}|<\frac{1}{\lambda_1}$, under the same data distribution as in \Cref{lem-single} with $d=\mathcal{O}(\operatorname{poly}(T))$ and the spectrum of $\bSigma_{\btheta}$, denoted as $\nu_i$, satisfies $\nu_k=\log^{r}(k+1)$ for some $r>0$, then
$$R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\begin{cases}
\Omega(\log^{r-2p+1}(T))&{r\geq 2p-1}\\ \mathcal{O}(\frac{1}{\log^{p-(r-p+1)^{+}}(T)}) &{r<2p-1}
\end{cases}$$
\end{proposition}
\Cref{prop-hard} implies that under $\log$-decay data spectrum parameterized by $p$, the meta excess risk of MAML experiences a phase transition determined by the spectrum parameter $r$ of task diversity. While slower task diversity rate $r < 2p-1$ guarantees vanishing excess risk, faster task diversity rate $r \ge 2p-1$ necessarily results in non-vanishing excess risk.
\Cref{prop-hard} and \Cref{lem-single} together indicate that while $\log$-decay data spectrum always yields benign fitting (vanishing risk) in the single task setting, it can yield non-vanishing risk in meta learning due to fast task diversity rate.
We further validate our theoretical results in \Cref{prop-hard} by experiments.
We consider the case $p=2$. As shown in \Cref{fig:subfig:1}, when $r<2p-1$, the test error quickly converges to the Bayes error. When $r>2p-1$, \Cref{fig:subfig:2} illustrates that MAML already
converges on the training samples, but the test error (which is further zoomed in \Cref{fig:subfig:3}) levels off and does not vanish, showing MAML generalizes poorly when $r>2p-1$.
\begin{figure}[ht]
\centering
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task1-te.png}
\caption{\small$\nu_i=0.25\log^{1.5}(i+1)$}
\label{fig:subfig:1}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task3.png}
\caption{\small$\nu_i=0.25\log^{8}(i+1)$}
\label{fig:subfig:2}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{task3-te.png}
\caption{\small $\nu_i=0.25\log^{8}(i+1)$}
\label{fig:subfig:3}
\end{subfigure}
\caption{The effects of task diversity. $d=500$, $T=300$, $\lambda_i = \frac{1}{i\log(i+1)^2}$, $\btr=0.02$, $\bte=0.2$
}
\label{fig:twopicture}
\vspace{-0.3cm}
\end{figure}
Furthermore, we show that the above phase transition that occurs for $\log$-decay data distributions no longer exists for data distributions with faster decaying spectrum.
\begin{proposition}[MAML, Fast-Decay Data Spectrum]\label{prop-fast} Under the same task distribution as in \Cref{prop-hard}, i.e., the spectrum of $\bSigma_{\btheta}$, denoted as $\nu_i$, satisfies $\nu_k=\log^{r}(k+1)=\widetilde{O}(1)$ for some $r>0$, and the data distribution with $d=\mathcal{O}(\operatorname{poly}(T))$ satisfies:
\begin{enumerate}[itemsep=2pt,topsep=0pt,parsep=0pt]
\item $\lambda_k=k^{-q}$ for some $q>1$, $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\mathcal{O}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$ and $R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item $\lambda_k=e^{-k}$, $R(\overline{\boldsymbol{\omega}}^{\text{sin}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}(\frac{1}{T})$ and $R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})=\widetilde{\mathcal{O}}(\frac{1}{T})$.
\end{enumerate}
\end{proposition}
\subsection{On the Role of Adaptation Learning Rate}\label{sec-main-stopping}
The analysis in \cite{bernacchia2021meta} suggests a surprising observation that a negative learning rate (i.e., when $\beta^{\text{tr}}$ takes a negative value) optimizes the generalization for MAML under mixed linear regression models. Their results indicate that the testing risk initially increases and then decreases as $\beta^{\text{tr}}$ varies from negative to positive values around zero for Gaussian isotropic input data and tasks. Our following proposition supports such a trend, but with a novel tradeoff in SGD dynamics as a new reason for the trend, under more general data distributions.
Denote $\overline{\boldsymbol{\omega}}^{\beta}_T$ as the average SGD solution of MAML after $T$ iterations that uses $\beta$ as the inner loop learning rate.
\begin{proposition}\label{prop-tradeoff}
Let $s=T\log^{-p}(T)$ and $d=T\log^{q}(T)$, where $p,q>0$.
If the spectrum of $\bSigma$ satisfies
$$\lambda_{k}= \begin{cases}1 / s, & k \leq s \\ 1 /(d-s), & s+1 \leq k \leq d. \end{cases}
$$
Suppose the spectral parameter $\nu_i$ of $\bSigma_{\btheta}$ is $O(1)$, and let the step size $\alpha=\frac{1}{2 c(\btr, \bSigma) \operatorname{tr}(\bSigma)}$. Then for large $n_1$, $|\beta^{\text{tr}}|, |\beta^{\text{te}}|<\frac{1}{\lambda_1}$, we have
\begin{align}\label{eq-tradeoff}
R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_T,\bte)\lesssim
& \mathcal{O}\Big(\frac{1}{\log^{p}(T)}\Big) \frac{1}{(1-\btr \lambda_{1})^{2}}+\mathcal{O}\Big(\frac{1}{\log^{q} (T)}\Big)\Big(1-\btr \lambda_{d}\Big)^{2}
+\widetilde{\mathcal{O}}(\frac{1}{T}).
\end{align}
\end{proposition}
The first two terms in the bound of \cref{eq-tradeoff} correspond to the impact of effective meta weights $\Xi_i$ on the "leading" and "tail" spaces, respectively, as we discuss in \Cref{remark-weight}. Clearly, the learning rate $\btr$ plays a tradeoff role in these two terms, particularly when $p$ is close to $q$. This explains the fact that the test error first increases and then decreases as $\btr$ varies from negative to positive values around zero. Such a tradeoff also serves as the reason for the first-increase-then-decrease trend of the test error under more general data distributions as we demonstrate in \Cref{fig:tradeoff}. This complements the reason suggested in \cite{bernacchia2021meta}, which captures only the quadratic form $\frac{1}{\left(1-\btr \lambda_{1}\right)^{2}}$ of $\btr$ for isotropic $\bSigma$, where there exists only the "leading" space without "tail" space.
\begin{figure}[ht]
\centering
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_1.pdf}
\caption{$\lambda_i=\frac{1}{i\log(i+1)^2}$}
\label{fig:tradeoff:1}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_3.pdf}
\caption{$\lambda_i=\frac{1}{i\log(i+1)^3}$}
\label{fig:tradeoff:2}
\end{subfigure}
\begin{subfigure}{.31\textwidth}
\centering
\includegraphics[width=\linewidth]{beta_2.pdf}
\caption{$\lambda_i=\frac{1}{i^2}$}
\label{fig:tradeoff:3}
\end{subfigure}
\caption{$R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_T,\bte)$ as a function of $\beta^{\text{tr}}$. $d=200$, $T=100$, $\bSigma_{\btheta}=\frac{0.8^2}{d}\mathbf{I}$, $\bte=0.2$
}
\label{fig:tradeoff}
\end{figure}
Based on the above results, incorporating with our dynamics analysis, we surprisingly find that $\btr$ not only affects the final risk, but also plays a pivot role towards the early iteration that the testing error tends to be steady. To formally study such a property, we define the stopping time as follows.
\begin{definition}[Stopping time]
Given $\btr,\bte$, for any $\epsilon>0$, the corresponding stopping time $t_{\epsilon}(\btr,\bte)$ is defined as:
\[
t_{\epsilon}(\btr,\bte) = \min t\quad
\text{s.t. }\; R(\wl^{\btr}_t;\bte)<\epsilon.
\]
\end{definition}
In the sequel, we may omit the arguments in $t_{\epsilon}$ for simplicity. We consider the similar data distribution in \Cref{prop-tradeoff} but parameterized by $K$, i.e., $s=K\log^{-p}(K)$ and $d=K\log^{q}(K)$, where $p,q> 0$. Then we can derive the following characterization for $t_{\epsilon}$.
\begin{corollary}~\label{col-stop}
If the assumptions in \Cref{prop-tradeoff} hold and $p=q$. Further, let $\bSigma_{\btheta}=\eta^2\mathbf{I}$, and $|\beta^{\text{tr}}|<\frac{1}{\lambda_1}$. Then for $t_{\epsilon}(\btr,\bte)\in (s, K]$, we have:
\begin{align}
\exp\Big(\epsilon^{-\frac{1}{p}} \Big[\frac{L_{l}}{(1-\btr\lambda_1)^2}+ L_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big)
\leq t_{\epsilon}\leq \exp\Big(\epsilon^{-\frac{1}{p}}\Big[\frac{U_{l}}{(1-\btr\lambda_1)^2}+ U_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big) \label{eq-stopping}
\end{align}
where $L_l$, $L_t$, $U_l$, $U_t>0$ are factors for "leading" and "tail" spaces that are independent of $K$\footnote{Such terms have been suppressed for clarity. Details are presented in the appendix.}.
\end{corollary}
\Cref{eq-stopping} suggests that the early stopping time $t_{\epsilon}$ is also controlled by the tradeoff role that $\btr$ plays in the "leading" ($U_l,L_l$) and "tail" spaces ($U_t,L_t$), which takes a similar form as the bound in \Cref{prop-tradeoff}. Therefore, the trend for $t_{\epsilon}$ in terms of $\btr$ will exhibit similar behaviours as the final excess risk, and hence the optimal $\btr$ for the final excess risk will lead to an earliest stopping time. We plot the training and test errors for different $\btr$ in Figure~\ref{fig:stopping}, under the same data distributions as \Cref{fig:tradeoff:1} to validate our theoretical findings. As shown in \Cref{fig:stopping:1}, $\btr$ does not make much difference in the training stage (the process converges for all $\btr$ when $T$ is larger than $100$). However, in \Cref{fig:stopping:2} at test stage, $\btr$ significantly affects the iteration when the test error starts to become relatively flat.
Such an early stopping time first increases then decreases as $\btr$ varies from $-0.5$ to $0.7$, which resembles the change of final excess risk in \Cref{fig:tradeoff:1}.
\begin{figure}[H]
\centering
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=\linewidth]{stopping-tr.pdf}
\caption{Training Risk}
\label{fig:stopping:1}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=\linewidth]{stopping-te.pdf}
\caption{Test Error}
\label{fig:stopping:2}
\end{subfigure}
\caption{Training and test curves for different $\beta^{\text{tr}}$. $d=500$, $\lambda_i=\frac{1}{i\log^2(i+1)}$,$\bSigma_{\btheta}=\frac{0.8^2}{d}\mathbf{I}$, $\bte=0.2$
}
\label{fig:stopping}
\end{figure}
\section{Conclusions }\label{sec-conclusion}
In this work, we give the theoretical treatment towards the generalization property of MAML based on their optimization trajectory in non-asymptotic and overparameterized regime.
We provide both upper and lower bounds on the excess risk of MAML trained with average SGD. Furthermore, we explore which type of data and task distributions are
crucial for diminishing error with overparameterization, and discover the influence of adaption learning rate both on the generalization error and the dynamics, which brings novel insights towards the distinct effects of MAML's one-step gradient updates on "leading" and "tail" parts of data eigenspace.
\bibliographystyle{plain}
\bibliography{main}
\appendix
\newpage
\renewcommand{\appendixpagename}{\centering \sffamily \LARGE Appendix}
\appendixpage
\vspace{5mm}
\section{Proof of Proposition~\ref{prop1}}
We first show how to connect the loss function associated with MAML to a Meta Least Square Problem.
\begin{proposition}[\Cref{prop1} Restated] Under the mixed linear regression model,
the expectation of the meta-training loss
taken over task and data distributions can be rewritten as:
\begin{align}
\mathbb{E}\left[\widehat{\mathcal{L}}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}};\mathcal{D})\right]= \mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})=
\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align}
The meta data are given by
\begin{align}\Bb =& \frac{1}{\sqrt{n_2}}\Xb^{out}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\label{ap-eq-data}\\ \boldsymbol{\gamma} =& \frac{1}{\sqrt{n_2}}\Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big) \end{align}
where $\Xb^{\text{in}}\in \mathbb{R}^{n_1\times d}$,$\zb^{\text{in}}\in \mathbb{R}^{n_1}$,$\Xb^{\text{out}}\in \mathbb{R}^{n_2\times d}$ and $\zb^{\text{out}}\in \mathbb{R}^{n_2}$ denote the inputs and noise for training and validation.
Furthermore,
we have
\begin{align}
\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ with meta noise } \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0. \label{ap-linear}
\end{align}
\end{proposition}
\begin{proof}
We first rewrite $\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})$ as follows:
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})
&=\mathbb{E}\left[ \ell(\mathcal{A}(\boldsymbol{\omega},\btr;\mathcal{D}^{\text{in}});\mathcal{D}^{\text{out}})\right]\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\sum^{n_2}_{j=1} \left(\left\langle \mathbf{x}^{\text{out}}_{j}, \mathcal{A}(\boldsymbol{\omega},\btr;\mathcal{D}^{\text{in}})\right\rangle-y^{\text{out}}_{j}\right)^{2}\right]\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\|\Xb^{\text{out}}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\bomega+\frac{\btr}{n_1}{\Xb^{\text{in}}}^{T}\yb^{\text{in}}-\yb^{\text{out}} \|^2\right].
\end{align*}
Using the mixed linear model:
\begin{align}
\yb^{\text{in}}= \mathbf{X}^{\text{in}}\boldsymbol{\theta}+\zb^{\text{in}},\quad \yb^{\text{out}}= \mathbf{X}^{\text{out}}\boldsymbol{\theta}+\zb^{\text{out}},
\end{align}
we have
\begin{align*}
\mathcal{L}&(\mathcal{A},\boldsymbol{\omega},\beta^{\text{tr}})\\
&=\mathbb{E}\left[ \frac{1}{2n_2}\|\Xb^{\text{out}}\Big(\mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\bomega\right.\\
&-\left. \Big( \Xb^{\text{out}}\Big( \mathbf{I}-\frac{\beta^{\text{tr}}}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{out}-\frac{\beta^{\text{tr}}}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}\Big)\|^2\right]\\
&= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\boldsymbol{\omega}-\boldsymbol{\gamma}\right\|^{2}\right].
\end{align*}
Moreover, note that $\btheta-\btheta^{*}$ has mean zero and is independent of data and noise, and define
\begin{align}
\boldsymbol{\xi}=\frac{1}{\sqrt{n_2}}\left( \Xb^{\text{out}}\left( \mathbf{I}-\frac{\btr}{n_1} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\right)(\btheta-\btheta^{*})+\zb^{\text{out}}-\frac{\btr}{n_1} \Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\zb^{in}\right).\label{ap-eq-xi}
\end{align}
We call $\boldsymbol{\xi}$ as meta noise, and then we have
\begin{align*}
&\boldsymbol{\gamma} = \Bb\boldsymbol{\theta}^{*}+\boldsymbol{\xi}\quad \text{ and }\quad \mathbb{E}[\boldsymbol{\xi}\mid\Bb]=0.
\end{align*}
\end{proof}
\begin{lemma}[Meta Excess Risk]\label{ap-lemma-excess}
Under the mixed linear regression model, the meta excess risk can be rewritten as follows:
\begin{align*}
R(\wl_T, \bte)=\frac{1}{2}\mathbb{E}\|\wl_T-\btheta^{*}\|^2_{\Hte}
\end{align*}
where $\|\ab\|_{\Ab}^{2}=\ab^{T} \Ab \ab$. Moreover, the Bayes error is given by
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})=\frac{1}{2}\operatorname{tr}(\bSigma_{\btheta}\Hte)+\frac{\sigma^2{\bte} ^2}{2m}+\frac{\sigma^2}{2}.
\end{align*}
\end{lemma}
\begin{proof}
Recall that
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\triangleq \mathbb{E}\left[\mathcal{L}(\mathcal{A},\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})\right]-\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})
\end{align*}
where $\boldsymbol{\omega}^{*}$ denotes the optimal solution to the population meta-test error. Under the mixed linear model, such a solution can be directly calculated~\cite{gao2020modeling}, and we obtain $\boldsymbol{\omega}^{*}=\mathbb{E}[\btheta] =\btheta^{*}$. Hence,
\[R(\wl_T, \bte)=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right],\]
where \begin{align}\Bb =& {\xb^{\text{out}}}^{\top}\Big(\mathbf{I}-\frac{\beta^{\text{te}}}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\nonumber\\ \boldsymbol{\gamma} =& {\xb^{\text{out}}}^{\top}\Big( \mathbf{I}-\frac{\beta^{\text{te}}}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\Big)\boldsymbol{\theta}+\zb^{\text{out}}-\frac{\beta^{\text{te}}}{m} {\xb^{\text{out}}}^{\top}{\Xb^{\text{in}}}^{\top}\zb^{\text{in}}, \end{align}
and
$\xb^{\text{out}}\in\mathbb{R}^{d}$, $\zb^{\text{out}}\in\mathbb{R}^{d}$, $\Xb^{\text{in}}\in\mathbb{R}^{m\times d}$ and $\zb^{\text{in}}\in\mathbb{R}^{m}$. The forms of $\Bb$ and $\bgamma$ are slightly different since we allow a new adaptation rate $\bte$ and the inner loop has $m$ samples at test stage. Similarly
\begin{align}
\xi=\left(\underbrace{ {\xb^{\text{out}}}^{\top}\left( \mathbf{I}-\frac{\bte}{m} {\Xb^{\text{in}}}^{T} {\Xb^{\text{in}}}\right)(\btheta-\btheta^{*})}_{\xi_1}+\underbrace{\zb^{\text{out}}}_{\xi_2}\underbrace{-\frac{\btr}{m} {\xb^{\text{out}}}^{\top}{\Xb^{\text{in}}}^{\top}\zb^{in}}_{\xi_3}\right).
\end{align}
Then we have
\begin{align*}
R(\wl_T, \bte)&=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\wl_{T}-\boldsymbol{\gamma}\right\|^{2}-\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]\\
&=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\| \Bb(\wl_{T}-\btheta^{*})\|^2\right]\\
&=\frac{1}{2}\mathbb{E}\|\wl_T-\btheta^{*}\|^2_{\Hte}
\end{align*}
where the last equality follows because $\mathbb{E}\left[\Bb^{\top}\Bb\right]=\Hte$ at the test stage.
The Bayes error can be calculated as follows:
\begin{align*}
\mathcal{L}(\mathcal{A},\boldsymbol{\omega}^{*},\beta^{\text{te}})&= \mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\left\|\Bb\btheta^{*}-\boldsymbol{\gamma}\right\|^{2}\right]=\mathbb{E}_{\Bb , \boldsymbol{\gamma}} \frac{1}{2}\left[\xi^2\right]\\
&\overset{(a)}{=}\frac{1}{2}\left(\mathbb{E}\left[\xi_1^{2}\right]+\mathbb{E}\left[\xi_2^{2}\right]+\mathbb{E}\left[\xi_3^{2}\right]\right)\\
&=\frac{1}{2}(\operatorname{tr}(\bSigma_{\btheta}\Hte)+\frac{{\bte}^2\sigma^2}{m}+\sigma^2)
\end{align*}
where $(a)$ follows because $\xi_1,\xi_2,\xi_3$ are independent and have zero mean conditioned on $\Xb^{\text{in}}$ and $\xb^{\text{out}}$.
\end{proof}
\section{Analysis for Upper Bound (Theorem~\ref{thm-upper}) }
\subsection{Preliminaries}
We first introduce some additional notations.
\begin{definition}[Inner product of matrices]
For any two matrices $\Cb,\Db$, the inner product of them is defined as
$$
\langle \Cb,\Db \rangle = \operatorname{tr}(\Cb^{\top}\Db).
$$
\end{definition}
We will use the following property about the inner product of matrices throughout our proof.
\begin{property}
If $\mathbf{C} \succeq 0$ and $\mathbf{D} \succeq \mathbf{D}^{\prime}$, then we have $\langle\mathbf{C}, \mathbf{D}\rangle \geq\left\langle\mathbf{C}, \mathbf{D}^{\prime}\right\rangle$.
\end{property}
\begin{definition}[Linear operator]
Let $\otimes$ denote the tensor product. Define the following linear operators on symmetric matrices:
$$
\begin{gathered}
\mathcal{M}=\mathbb{E}\left[ \Bb^{\top}\otimes\Bb^{\top}\otimes \Bb\otimes\Bb\right]\quad
\widetilde{\mathcal{M}}:= \Htr\otimes \Htr
\quad
\mathcal{I}:= \mathbf{I} \otimes \mathbf{I}
\\ \mathcal{T}:=\Hb_{n_1,\btr} \otimes \mathbf{I}+\mathbf{I} \otimes \Hb_{n_1,\btr}-\alpha \mathcal{M}, \quad \widetilde{\mathcal{T}}=\Hb_{n_1,\btr} \otimes \mathbf{I}+\mathbf{I} \otimes \Hb_{n_1,\btr}-\alpha \Hb_{n_1,\btr} \otimes \Hb_{n_1,\btr}.
\end{gathered}
$$
\end{definition}
We next define the operation of the above linear operators on a symmetric matrix $\Ab$ as follows.
$$
\begin{gathered}
\mathcal{M} \circ \mathbf{A}=\mathbb{E}\left[\mathbf{B}^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb\right], \quad \widetilde{\mathcal{M}} \circ \mathbf{A}=\Htr \mathbf{A} \Htr, \quad \mathcal{I} \circ \mathbf{A}=\mathbf{A},
\\
\mathcal{T}\circ \Ab = \Htr\Ab+\Ab\Htr -\alpha \mathbb{E}\left[\mathbf{B}^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb\right]\\
\tilde{\mathcal{T}}\circ \Ab = \Htr\Ab+\Ab\Htr -\alpha\Htr\Ab\Htr.
\end{gathered}
$$
Based on the above definitions, we have the following equations hold.
$$
\begin{gathered}
(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{A}=\mathbb{E}\left[\left(\mathbf{I}-\alpha \Bb^{\top}\Bb\right) \mathbf{A}\left(\mathbf{I}-\alpha \Bb^{\top}\Bb\right)\right]\\(\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{A}=(\mathbf{I}-\alpha \Htr) \mathbf{A}(\mathbf{I}-\alpha \Htr).
\end{gathered}
$$
For the linear operators, we have the following technical lemma.
\begin{lemma}\label{lemma-linearop} We call the linear operator $\mathcal{O}$ a PSD mapping, if for every symmetric PSD matrix $\mathbf{A}$, $\mathcal{O}\circ \Ab$ is also PSD matrix. Then we have:
\begin{enumerate}[label=\roman*]
\item[(i)] $ \mathcal{M}$, $\widetilde{\mathcal{M}}$ and
$(\mathcal{M}-\widetilde{\mathcal{M}}) $ are all PSD mappings.
\item[(ii)] $\tilde{\mathcal{T}}-\mathcal{T}$, $\mathcal{I}-\alpha \mathcal{T}$ and $\mathcal{I}-\alpha \tilde{\mathcal{T}}$ are all PSD mappings.
\item[(iii)] If $0<\alpha< \frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$, then $\tilde{\mathcal{T}}^{-1}$ exists, and is a PSD mapping.
\item[(iv)] If $0<\alpha<\frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$, $\tilde{\mathcal{T}}^{-1} \circ \Htr\preceq \mathbf{I}$.
\item[(v)] If $0<\alpha<\frac{1}{c(\btr,\bSigma) \operatorname{tr}(\bSigma
)}$, then $\mathcal{T}^{-1} \circ \mathbf{A}$ exists for PSD matrix $\mathbf{A}$, and $\mathcal{T}^{-1}$ is a PSD mapping.
\end{enumerate}
\end{lemma}
\begin{proof}
Items (i) and (iii) directly follow from the proofs in \cite{jain2017markov,zou2021benign}.
For $(\romannum{4})$, by the existence of $\tilde{\mathcal{T}}^{-1}$, we have
\begin{align*}
\tilde{\mathcal{T}}^{-1} \circ \Htr&=\sum_{t=0}^{\infty} \alpha (\mathcal{I}- \alpha\tilde{\mathcal{T}})^{t}\circ \Htr\\
&=\sum_{t=0}^{\infty} \alpha (\mathbf{I}-\alpha \Htr)^{t}\Htr(\mathbf{I}-\alpha \Htr)^{t}\\
&\preceq \sum_{t=0}^{\infty} \alpha (\mathbf{I}-\alpha \Htr)^{t}\Htr=\mathbf{I}.
\end{align*}
For $(\romannum{5})$, for any PSD matrix $\mathbf{A}$, consider
$$
\mathcal{T}^{-1} \circ \mathbf{A}=\alpha \sum_{k=0}^{\infty}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}.
$$
We first show that $\sum_{k=0}^{\infty}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}$ is finite, and then it suffices to show that the trace is finite, i.e.,
\begin{align}
\sum_{k=0}^{\infty} \operatorname{tr}\left((\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}\right)<\infty. \label{b1-fin}
\end{align}
Let $\mathbf{A}_k=(\mathcal{I}-\gamma \mathcal{T})^{k} \circ \mathbf{A}$. Combining with the definition of $\mathcal{T}$, we obtain
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right) &=\operatorname{tr}\left(\mathbf{A}_{k-1}\right)-2\alpha \operatorname{tr}\left(\Htr \mathbf{A}_{k-1}\right)+\alpha^{2} \operatorname{tr}\left(\mathbf{A} \mathbb{E}\left[\Bb^{\top}\Bb \Bb^{\top}\Bb \right]\right).
\end{aligned}
$$
Letting $\Ab=\Ib$ in \Cref{prop-4}
, we have $\mathbb{E}\left[\Bb^{\top}\Bb \Bb^{\top}\Bb \right]\preceq c(\btr,\bSigma
) \operatorname{tr}(\bSigma) \Htr$. Hence
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right) & \leq \operatorname{tr}\left(\mathbf{A}_{k-1}\right)-\left(2 \alpha-\alpha^{2} c(\btr,\bSigma
) \operatorname{tr}(\bSigma)\right) \operatorname{tr}\left(\Htr \mathbf{A}_{k-1}\right) \\
& \leq \operatorname{tr}\left((\mathbf{I}-\alpha \Htr) \mathbf{A}_{k-1}\right)\quad \text{ by } \alpha< \frac{1}{c(\btr,\bSigma) \operatorname{tr}(\bSigma
)}\\
& \leq\left(1-\alpha \min_{i}\{\mu_{i}(\Htr)\}\right) \operatorname{tr}\left(\mathbf{A}_{k-1}\right).
\end{aligned}
$$
If $\alpha<\frac{1}{\min_{i}\{\mu_{i}(\Htr)\}}$, then we substitute it into \cref{b1-fin} and obtain
$$
\sum_{k=0}^{\infty} \operatorname{tr}\left((\mathcal{I}-\alpha \mathcal{T})^{k} \circ \mathbf{A}\right)=\sum_{k=0}^{\infty} \operatorname{tr}\left(\mathbf{A}_{k}\right) \leq \frac{\operatorname{tr}(\mathbf{A})}{\alpha\min_{i}\{\mu_{i}(\Htr)\}}<\infty
$$
which guarantees the existence of $\mathcal{T}^{-1}$. Moreover, $\Ab_k$ is a PSD matrix for every $k$ since $\mathcal{I}-\alpha \mathcal{T}$ is a PSD mapping. The $\mathcal{T}^{-1} \circ \mathbf{A}=\alpha \sum_{k=0}^{\infty} \Ab_k$ must be a PSD matrix, which implies that $\mathcal{T}^{-1}$ is PSD mapping.
\end{proof}
\begin{property}[Commutity]
Suppose Assumption $2$ holds, then for all $n>0$, $|\beta|<1/\lambda_1$, $\mathbf{H}_{n,\beta}$ with different $n$ and $\beta$ commute with each other.
\end{property}
\subsection{Fourth Moment Upper Bound for Meta Data}
In this section, we provide a technical result for the fourth moment of meta data $\Bb$, which is essential throughout the proof of our upper bound.
\begin{proposition}\label{prop-4}
Suppose Assumptions 1-3 hold. Given $|\beta|<\frac{1}{\lambda_1}$, for any PSD matrix $\Ab$, we have
\begin{align*}
\mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\preceq c(\btr,\bSigma)\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \Htr
\end{align*}
where $c(\beta,\bSigma):= c_1\left(1+ 8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)\right)$.
\end{proposition}
\begin{proof}
Recall that $\Bb=\frac{1}{\sqrt{n_2}} \Xb^{\text{out}} (\mathbf{I}-\frac{\beta}{n_1}{\mathbf{X}^{\text{in}}}^{\top}\mathbf{X}^{\text{in}})$. With a slight abuse of notations, we write $\btr$ as $\beta$, $\mathbf{X}^{\text{in}}$ as $\mathbf{X}$ in this proof. First consider the case $\beta\geq 0$. By the definition of $\Bb$, we have
\begin{align*}
\mathbb{E}&\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\ &= \mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\frac{1}{n_2}{\Xb^{\text{out}}}^{\top} \Xb^{\text{out}}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\frac{1}{n_2}{\Xb^{\text{out}}}^{\top} \Xb^{\text{out}}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}\mathbf{X})\right] \\
&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left((\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\\&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma+ \frac{\beta^2}{n_1^2} \mathbf{X}^{\top}\mathbf{X}\bSigma \mathbf{X}^{\top}\mathbf{X})\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the second inequality follows from Assumption 1. Let $\xb_i$ denote the $i$-th row of $\Xb$. Note that $\xb_i =\Sigma^{\frac{1}{2}}\zb_i$, where $\zb_i$ is independent $\sigma_x$-sub-gaussian vector.
For any $\xb_{i_1},\xb_{i_2},\xb_{i_3},\xb_{i_4}$, where $1\leq i_1,i_2,i_3,i_4\leq n_1 $, we have:
\begin{align*}
&\mathbb{E}\left[\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&= \mathbb{E}\left[\operatorname{tr}(\bSigma^{\frac{1}{2}}\Ab\bSigma^{\frac{1}{2}}\zb_{i_1}\zb_{i_2}^{\top}\bSigma^{2}\zb_{i_3}\zb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n _1}\mathbf{X}^{\top}\mathbf{X})\right]\\
& = \sum_{k,j}\mu_k\lambda^2_j \mathbb{E}\left[(\zb_{i_4}^{\top}\ub_k)(\zb_{i_1}^{\top}\ub_k)(\zb_{i_4}^{\top}\vb_j)(\zb_{i_1}^{\top}\vb_j)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Sigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the SVD of $\bSigma^{\frac{1}{2}}\Ab\bSigma^{\frac{1}{2}}$ is $\sum_{j} \mu_{j} \ub_{j}\ub^{\top}_{j}$, the SVD of $\bSigma$ is $\sum_{j} \lambda_{j} \vb_{j}\vb^{\top}_{j}$. For any unit vector $\wb\in\mathbb{R}^{d}$, we have:
\begin{align*}
\wb^{\top}&\mathbb{E}\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n,\beta}\right]\wb\\
& \leq \sum_{k,j}\mu_k\lambda^2_j \sqrt{\mathbb{E}\left[\left((\zb_{i_4}^{\top}\ub_k)(\zb_{i_1}^{\top}\ub_k)(\zb_{i_4}^{\top}\vb_j)(\zb_{i_1}^{\top}\vb_j)^2\right)\right] } \\
&\quad\times \sqrt{\mathbb{E}\left[\|\wb^{\top}\Hb^{-\frac{1}{2}}_{n_1,\beta}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\Sigma (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n_1,\beta}\wb\|^2\right]}\\
&\leq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma)\operatorname{tr}(\bSigma^2)
\end{align*}
where the first inequality follows from the Cauchy Schwarz inequality; the last inequality is due to Assumption 3 and the property of sub-Gaussian distributions~\cite{vershynin2018high}.
Therefore,
\begin{align*}
\mathbb{E}&\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\right]\\
&\preceq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma^2)\mathbf{I}
\end{align*}
which implies
$$\mathbb{E}\left[\operatorname{tr}(\Ab\xb_{i_1}\xb_{i_2}^{\top}\bSigma \xb_{i_3}\xb_{i_4}^{\top})(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\preceq 64\sqrt{C(\beta,\bSigma)}\sigma_x^4 \operatorname{tr}(A\bSigma^2) \Hb_{n_1,\beta}.$$
Hence,
\begin{align*}
& \mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\
&\preceq c_1\mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma+ 64\sqrt{C}\sigma_x^4\beta^2 \bSigma\operatorname{tr}(\bSigma^2))\right) (\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n}\mathbf{X}^{\top}\mathbf{X})\right]\\
&\preceq c_1(1+64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2))\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \mathbf{H}_{n_1,\beta}.
\end{align*}
Now we turn to $\beta<0$, and derive
\begin{align*}
\mathbb{E}&\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right] \\
&\preceq c_1 \mathbb{E}\left[\operatorname{tr}\left((\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\mathbf{A} (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\\&= c_1 \mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma-\underbrace{\frac{\beta}{n_1}(\mathbf{X}^{\top}\mathbf{X}\bSigma +\bSigma \mathbf{X}^{\top}\mathbf{X})}_{\Jb_1}+ \frac{\beta^2}{n_1^2} \mathbf{X}^{\top}\mathbf{X}\bSigma \mathbf{X}^{\top}\mathbf{X})\right)\right.\\
&\quad\cdot \left. (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right].
\end{align*}
We can bound the extra term $\Jb_1$ in the similar way as $\beta>0$.
For any $\xb_{i}$, $1\leq i\leq n_1$, we have
\begin{align*}
\mathbb{E}&\left[\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma\right)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&= \mathbb{E}\left[\operatorname{tr}\left(\zb_{i}^{\top}\bSigma^{\frac{3}{2}}\Ab\bSigma^{\frac{1}{2}}\zb_{i}\right)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
& = \sum_{k}\iota_k \mathbb{E}\left[(\zb_{i}^{\top}\boldsymbol{\kappa}_k)^2(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]
\end{align*}
where the SVD of $\bSigma^{\frac{3}{2}}\Ab\bSigma^{\frac{1}{2}}$ is $\sum_{k} \iota_{k} \boldsymbol{\kappa}_{k}\boldsymbol{\kappa}^{\top}_{k}$. Similarly, for any unit vector $\wb\in\mathbb{R}^{d}$, we can obtain
\begin{align*}
\wb^{\top}& \mathbb{E}\left[\Hb^{-\frac{1}{2}}_{n_1,\beta}\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X}\right)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\right]\wb\\
&\leq \sum_{k}\iota_k \sqrt{\mathbb{E}[(\zb_{i}^{\top}\boldsymbol{\kappa}_k)^4]}\sqrt{\mathbb{E}[\|\wb^{\top}\Hb^{-\frac{1}{2}}_{n_1,\beta}(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\Hb^{-\frac{1}{2}}_{n_1,\beta}\wb\|^2]}\\
&\leq 4 \sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(A\bSigma^2)
\end{align*}
which implies:
\begin{align*}
\mathbb{E}\left[\operatorname{tr}\left(\Ab\xb_{i}\xb_{i}^{\top}\bSigma)(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X}\right)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\preceq 4\sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(\Ab\bSigma^2) \Hb_{n_1,\beta}.
\end{align*}
Hence,
\begin{align*}
& \mathbb{E}\left[\Bb^{\top}\Bb \mathbf{A} \Bb^{\top}\Bb \right]\\
&\preceq c_1\mathbb{E}\left[\operatorname{tr}\left(\mathbf{A} (\bSigma-8\beta\sqrt{C}\sigma_x^2\bSigma^2+ 64\sqrt{C}\sigma_x^4\beta^2 \bSigma\operatorname{tr}(\bSigma^2))\right) (\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\bSigma(\mathbf{I}-\frac{\beta}{n_1}\mathbf{X}^{\top}\mathbf{X})\right]\\
&\preceq c_1\left(1-8\beta\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)\right)\mathbb{E}\left[\operatorname{tr}(\mathbf{A} \bSigma) \right] \mathbf{H}_{n_1,\beta}.
\end{align*}
Together with the discussions for $\beta>0$, we have $$c(\beta,\bSigma
)=c_1(1+8|\beta|\lambda_1\sqrt{C(\beta,\bSigma)}\sigma_x^2+ 64\sqrt{C(\beta,\bSigma)}\sigma_x^4\beta^2\operatorname{tr}(\bSigma^2)),$$ which completes the proof.
\end{proof}
\subsection{Bias-Variance Decomposition}
We will use the bias-variance decomposition similar to theoretical studies of classic linear regression~\cite{jain2017markov,dieuleveut2017harder,zou2021benign}. Consider the error at each iteration: $\brho_t=\bomega_t-\btheta^{*}$, where $\bomega_t$ is the SGD output at each iteration $t$. Then the update rule can be written as:
$$
\brho_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho_{t-1}+\alpha \Bb^{\top}_{t}\boldsymbol{\xi}_{t}$$
where $\Bb_t,\bxi_t$ are the meta data and noise at iteration $t$ (see \cref{ap-eq-data,ap-eq-xi}). It is helpful to consider $\brho_{t}$ as the sum of the following two random processes:
\begin{itemize}
\item If there is no meta noise, the error comes from the bias:
$$
\brho^{\text{bias}}_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho^{\text{bias}}_{t-1}\quad \brho^{\text{bias}}_{t}=\brho_{0}.$$
\item If the SGD trajectory starts from $\btheta^{*}$, the error originates from the variance:
$$
\brho^{\text{var}}_{t}:= (\Ib-\alpha\Bb^{\top}_t\Bb_t)\brho^{\text{var}}_{t-1}+\alpha \Bb^{\top}_{t}\boldsymbol{\xi}_{t}\quad \brho^{\text{var}}=\mathbf{0}
$$
and $\mathbb{E}[\brho^{\text{var}}_{t}]=0$.
\end{itemize}
With slightly abused notations, we have:
$$
\brho_{t}= \brho^{\text{bias}}_{t}+\brho^{\text{var}}_{t}.
$$
Define the averaged output of $\brho^{\text{bias}}_{t}$, $\brho^{\text{var}}_{t}$ and $\brho_t$ after $T$ iterations as:
\begin{align}\label{eq-rho}
\rhob^{\text{bias}}_T=\frac{1}{T}\sum_{t=1}^{T} \brho^{\text{bias}}_{t},\quad
\rhob^{\text{var}}_T=\frac{1}{T}\sum_{t=1}^{T} \brho^{\text{var}}_{t},\quad \rhob_T=\frac{1}{T}\sum_{t=1}^{T} \brho_{t}.
\end{align}
Similarly, we have
$$
\rhob_{T}= \rhob^{\text{bias}}_{T}+\rhob^{\text{var}}_{T}.
$$
Now we are ready to introduce the bias-variance decomposition for the excess risk.
\begin{lemma}[Bias-variance decomposition]\label{lemma-bv}
Following the notations in \cref{eq-rho}, then the excess risk can be decomposed as
\begin{align*}
R(\wl_T, \bte)\leq 2\mathcal{E}_\text{bias}+2\mathcal{E}_\text{var}
\end{align*}
where
\begin{align}
\mathcal{E}_\text{bias}=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{bias}}_T\otimes\rhob^{\text{bias}}_T] \rangle, \quad \mathcal{E}_\text{var} = \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle.
\end{align}
\end{lemma}
\begin{proof}
By \Cref{ap-lemma-excess}, we have
\begin{align*}
R(\wl_T, \bte)&=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob_T\otimes\rhob_T] \rangle \\
&=\frac{1}{2} \langle\Hte, \mathbb{E}[(\rhob^{\text{bias}}_T+\rhob^{\text{var}}_T)\otimes(\rhob^{\text{bias}}_T+\rhob^{\text{var}}_T)] \rangle\\
&\leq 2\left( \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{bias}}_T\otimes\rhob^{\text{bias}}_T] \rangle + \frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\right)
\end{align*}
where the last inequality follows because for vector-valued random variables $\ub$ and $\vb$, $\mathbb{E}\|\ub+\vb\|_{H}^{2} \leq\left(\sqrt{\mathbb{E}\|\ub\|_{H}^{2}}+\sqrt{\mathbb{E}\|\vb\|_{H}^{2}}\right)^{2}$ and from Cauchy-Schwarz inequality.
\end{proof}
For $t=0,1,\cdots,T-1$,
consider the following bias and variance iterates:
\begin{align}
\mathbf{D} _{t}=(\mathcal{I}-\alpha\mathcal{T}) \circ \Db _{t-1} \quad& \text { and } \quad \Db _{0}= (\boldsymbol{\omega}_t-\btheta^{*})(\boldsymbol{\omega}_t-\btheta^{*})^{\top}\nonumber\\ \mathbf{V} _{t}=(\mathcal{I}-\alpha\mathcal{T}) \circ \Vb _{t-1}+\alpha^{2} \Pi \quad &\text { and } \quad \Vb _{0}=\mathbf{0}\label{eq-bv}
\end{align}
where $\Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]$. One can verify that
$$
\mathbf{D}_{t}=\mathbb{E}\left[\brho_{t}^{\text {bias }} \otimes \brho_{t}^{\text {bias }}\right], \quad \mathbf{V}_{t}=\mathbb{E}\left[\brho_{t}^{\text {var }} \otimes \brho_{t}^{\text {var }}\right].
$$
With such notations, we can further bound the bias and variance terms.
\begin{lemma}\label{lemma-further-bv}
Following the notations in \cref{eq-bv}, we have
\begin{align}
\mathcal{E}_\text { bias }& \leq \frac{1}{\alpha T^{2}} \left\langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T}\right )\Htr^{-1}\Hte, \sum_{t=0}^{T-1}\Db _{t}\right\rangle,\\
\mathcal{E}_\text { var } &\leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align}
\end{lemma}
\begin{proof}
Similar calculations have appeared in the prior works~\cite{jain2017markov,zou2021benign}. However, our meta linear model contains additional terms, and hence we provide a proof here for completeness. We first have
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]& =\frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=0}^{T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]\\
& \preceq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]+\mathbb{E}[\brho^{\text{var}}_k\otimes\brho^{\text{var}}_t]
\end{align*}
where the last inequality follows because we double count the diagonal terms $t=k$.
For $t\leq k$, $\mathbb{E}[\brho^{\text{var}}_k|\brho^{\text{var}}_t]=(\mathbf{I}-\alpha\Htr)^{k-t} \brho^{\text{var}}_t$, since $\mathbb{E}[\Bb_t^{\top}\bxi_t|\brho_{t-1}]=\mathbf{0}$. From this, we have
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]
& \preceq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}+\Vb_t (\mathbf{I}-\alpha\Htr)^{k-t}.
\end{align*}
Substituting the above inequality into $\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle$, we obtain:
\begin{align*}
\mathcal{E}_\text{var} &=\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\\
&\leq \frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle \Hte, \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}\rangle + \langle \Hte,\Vb_t (\mathbf{I}-\alpha\Htr)^{k-t}\rangle\\
&=\frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Vb_t\rangle
\end{align*}
where the last inequality follows from \Cref{ass-comm} that $F$ and $\bSigma
$ commute, and hence $\Hte$ and $\mathbf{I}-\alpha\Htr$ commute.
For the bias term, similarly we have:
\begin{align}
\mathcal{E}_\text{bias}&\leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Db_t\rangle\\
&=\frac{1}{\alpha T^2} \sum_{t=0}^{T-1} \langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T-t}\right)\Htr^{-1} \Hte, \Db_t\rangle\\
&\leq \frac{1}{\alpha T^2} \langle \left(\Ib-(\mathbf{I}-\alpha\Htr)^{T}\right)\Htr^{-1} \Hte,\sum_{t=0}^{T-1} \Db_t\rangle
\end{align}
which completes the proof.
\end{proof}
\subsection{Bounding the Bias}
Now we start to bound the bias term. By \Cref{lemma-further-bv}, we focus on bounding the summation of $\Db_t$, i.e. $\sum_{t=0}^{T-1} \Db_t$. Consider $\mathbf{S}_{t}:=\sum_{k=0}^{t-1} \Db _{k}$, and the following lemma shows the properties of $\mathbf{S}_{t}$
\begin{lemma}
$\mathbf{S}_{t}$ satisfies the recursion form:
$$
\mathbf{S}_{t}=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{S}_{t-1}+\Db_{0}.
$$
Moreover, if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma
)}$, then we have:
$$
\Db_{0}=\mathbf{S}_{0} \preceq \mathbf{S}_{1} \preceq \cdots \preceq \mathbf{S}_{\infty}
$$
where $\mathbf{S}_{\infty}:=\sum_{k=0}^{\infty}(\mathcal{I}-\alpha\mathcal{T})^k \circ \Db_{0}=\alpha^{-1} \mathcal{T}^{-1} \circ \Db_{0}$.
\end{lemma}
\begin{proof}
By \cref{eq-bv}, we have
\begin{align*}
\mathbf{S}_{t}&=\sum_{k=0}^{t-1} \Db _{k}= \sum_{k=0}^{t-1}(\mathcal{I}-\alpha\mathcal{T})^{k} \circ \Db _{0}\\
&= \Db _{0}+(\mathcal{I}-\alpha\mathcal{T})\circ \left(\sum_{k=0}^{t-2}(\mathcal{I}-\alpha\mathcal{T})^{k} \circ \Db _{0}\right)\\
&= \Db _{0}+(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{S}_{t-1}.
\end{align*}
By \Cref{lemma-linearop}, $(\mathcal{I}-\alpha \mathcal{T})$ is PSD mapping, and hence $\Db_t=(\mathcal{I}-\alpha \mathcal{T})\circ \Db_{t-1}$ is a PSD matirx for every $t$, which implies $\Sb_{t-1}\preceq \Sb_{t-1}+\Db_t=\Sb_{t}$.
The form of $\Sb_{\infty}$ can be directly obtained by \Cref{lemma-linearop}.
\end{proof}
Then we can decompose $\Sb_t$ as follows:
\begin{align}
\Sb_t& = \Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha(\widetilde{\mathcal{T}}-\mathcal{T})\circ \mathbf{S}_{t-1}\nonumber\\
&=\Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha^2(\mathcal{M}-\widetilde{\mathcal{M}})\circ \mathbf{S}_{t-1}\nonumber\\
&\preceq \Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{t-1}+ \alpha^2\mathcal{M}\circ \mathbf{S}_{T}\nonumber\\
&=\sum^{t-1}_{k=0} (\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ (\Db_0+\alpha^2\mathcal{M}\circ \mathbf{S}_{T})\label{eq-st}
\end{align}
where the inequality follows because $\Sb_t\preceq \Sb_{T}$ for any $t\leq T$. Therefore, it is crucial to understand $\mathcal{M}\circ \mathbf{S}_{T}$.
\begin{lemma}\label{lemma-ms-pre}
For any symmetric matrix $\mathbf{A}$, if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma
)}$, it holds that
$$
\mathcal{M} \circ \mathcal{T}^{-1} \circ \mathbf{A} \preceq \frac{c(\btr,\bSigma)\operatorname{tr}\left(\bSigma \Htr^{-1} \mathbf{A}\right)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)} \cdot \Htr .
$$
\end{lemma}
\begin{proof}
Denote $\mathbf{C}=\mathcal{T}^{-1} \circ \mathbf{A} $. Recalling $\tilde{\mathcal{T}}=\mathcal{T}+\alpha \mathcal{M}-\alpha \widetilde{\mathcal{M}}$, we have
$$
\begin{aligned}
\widetilde{\mathcal{T}} \circ \mathbf{C}&=\mathcal{T} \circ \mathbf{C}+\alpha \mathcal{M} \circ \mathbf{C}-\alpha\widetilde{\mathcal{M}} \circ \mathbf{C}\\
& \preceq \mathbf{A}+\alpha \mathcal{M} \circ \mathbf{C}.
\end{aligned}
$$
Recalling that $\widetilde{\mathcal{T}}^{-1}$ exists and is a PSD mapping, we then have
\begin{align}
\mathcal{M} \circ \mathbf{C} &\preceq \alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathcal{M} \circ \mathbf{C}+\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A}\nonumber\\
&\preceq \sum^{\infty}_{k=0} (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A}).\label{eq-mta}
\end{align}
By \Cref{prop-4}, we have $\mathcal{M} \circ \widetilde{\mathcal{T}}^{-1} \circ \mathbf{A}\preceq \underbrace{c(\btr,\bSigma)\operatorname{tr}( \bSigma\widetilde{\mathcal{T}}^{-1} \circ \mathbf{A})}_{J_2}\Htr$. Substituting back into \cref{eq-mta}, we obtain:
\begin{align*}
\sum^{\infty}_{k=0}& (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (\mathcal{M} \circ \tilde{\mathcal{T}}^{-1} \circ \mathbf{A})\preceq \sum^{\infty}_{k=0} (\alpha \mathcal{M} \circ \tilde{\mathcal{T}}^{-1})^{k} \circ (J_2\Htr)\\
&\preceq J_2\sum^{\infty}_{k=0} (\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma))^{k} \Htr\preceq \frac{J_2}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\Htr
\end{align*}
where the second inequality follows since $\tilde{\mathcal{T}}^{-1}\circ \Htr \preceq \mathbf{I}$ (\Cref{lemma-linearop}) and $\mathcal{M} \circ \mathbf{I} \preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma) \Htr$ (\Cref{prop-4}).
Finally, we bound $J_2$ as follows:
$$
\begin{aligned}
\operatorname{tr}\left(\bSigma \widetilde{\mathcal{T}}^{-1} \circ \mathbf{A}\right) &=\alpha\operatorname{tr}\left(\sum_{k=0}^{\infty} \bSigma(\mathbf{I}-\alpha \Htr)^{k} \mathbf{A}(\mathbf{I}-\alpha \Htr)^{k}\right) \\
&=\alpha \operatorname{tr}\left(\sum_{k=0}^{\infty} \bSigma(\mathbf{I}-\alpha \Htr)^{2 k} \mathbf{A}\right) \\
&=\operatorname{tr}\left(\bSigma\left(2 \Htr-\alpha \Htr^{2}\right)^{-1} \mathbf{A}\right)\\
&\leq \operatorname{tr}\left(\bSigma \Htr^{-1} \mathbf{A}\right)
\end{aligned}
$$
where the second equality follows because $\bSigma$ and $\Htr$ commute, and the last inequality holds since $\alpha<\frac{1} { \max_{i}\{\mu_{i}(\Htr)\}}$. Putting all these results together completes the proof.
\end{proof}
\begin{lemma}[Bounding $\mathcal{M}\circ \mathbf{S}_{T}$]\label{lemma-ms}
$$
\mathcal{M} \circ \mathbf{S}_{T} \preceq \frac{c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{\alpha(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma))} \cdot \Htr.
$$
\end{lemma}
\begin{proof}
$\mathbf{S}_{T}$ can be further derived as follows:
$$
\mathbf{S}_{T}=\sum_{k=0}^{T-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Db _{0}=\alpha^{-1} \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \mathcal{T})^{T}\right]\circ \Db _{0}.
$$
Since $\tilde{\mathcal{T}}-\mathcal{T}$ is a PSD mapping by \Cref{lemma-linearop}, we have
$\mathcal{I}-\alpha \tilde{\mathcal{T}} \leq \mathcal{I}-\alpha \mathcal{T}$. Hence $\mathcal{I}-(\mathcal{I}-\alpha \mathcal{T})^{T} \preceq \mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}$. Combining with the fact that $\mathcal{T}^{-1}$ is also a PSD mapping, we have:
$$
\mathbf{S}_{T} \preceq \alpha^{-1} \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{T}\right] \circ \Db _{0}.
$$
Letting $\Ab=\left[\mathcal{I}-(\mathcal{I}- \alpha\widetilde{\mathcal{T}})^{T}\right] \circ \Db _{0}$ in \Cref{lemma-ms-pre}, we obtain:
\begin{align*}
\mathcal{M} \circ \mathbf{S}_{T}& \preceq \alpha^{-1} \mathcal{M} \circ \mathcal{T}^{-1} \circ\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0} \\
&\preceq \frac{c(\btr,\bSigma
) \cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{\alpha(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma))} \cdot \Htr .
\end{align*}
\end{proof}
Now we are ready to derive the upper bound on the bias term.
\begin{lemma}[Bounding the bias]\label{lemma-bias}
If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\mathcal{E}_\text{bias}&\leq \sum_{i}\left(\frac{1}{\alpha^{2} T^{2}}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}\\
&+ \frac{2 c(\btr,\bSigma)}{T \alpha\left(1-c(\btr,\bSigma)\alpha \operatorname{tr}(\bSigma)\right)}
\sum_{i}{\left(\frac{1}{\mu_i(\Htr)}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha \mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right) \cdot \lambda_i\omega_i^2}
\\&\times
\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2 \mu_i(\Htr)^2\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\cdot\frac{\mu_i(\Htr)}{\mu_i(\Hte)}.
\end{align*}
\end{lemma}
\begin{proof} Applying \Cref{lemma-ms} to \cref{eq-st}, we can obtain:
$$
\begin{aligned}
\mathbf{S}_{t}
&\preceq\sum_{k=0}^{t-1}(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ
\left(\frac{\alpha c(\btr,\bSigma)\cdot \operatorname{tr}\left(\bSigma \Htr^{-1}\left[\mathcal{I}-(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{T}\right] \circ \Db _{0}\right)}{1-c(\beta,\Sigma) \alpha \operatorname{tr}(\Sigma)} \cdot \Htr + \Db_{0}\right) \\
&=\sum_{k=0}^{t-1}(\mathbf{I}-\alpha \Htr)^{k}\cdot \\
&\left(\underbrace{\frac{\alpha c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma\Htr^{-1}(\Db_0-(\mathbf{I}-\alpha \Htr)^{T}\Db _{0}(\mathbf{I}-\alpha \mathbf{H}_n)^{T})\right)}{1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma)} \cdot \Htr}_{\mathbf{G}_1} +\underbrace{\Db_{0}}_{\Gb_2}\right)\\
&\cdot (\mathbf{I}-\alpha \Htr)^{k}.
\end{aligned}
$$
Letting $t=T$, and substituting the upper bound of $\mathbf{S}_{T}$ into the bias term in \Cref{lemma-further-bv}, we obtain:
$$
\begin{aligned}
\mathcal{E}_\text { bias } & \leq \frac{1}{ \alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{2 k}-(\mathbf{I}- \alpha \mathbf{H}_{n,\beta})^{T+2 k})\mathbf{H}_{n,\beta}^{-1}\Hte,\mathbf{G}_1+\Gb_2\right\rangle\\
&\leq \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_1+\Gb_2\right\rangle.
\end{aligned}
$$
We first consider
\begin{align*}
d_1= \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_1\right\rangle.
\end{align*}
Since $\Htr$, $\Hte$ and $\mathbf{I}-\alpha \Htr$ commute,
we have
\begin{align*}
d_1=& \frac{c(\btr,\bSigma) \cdot \operatorname{tr}\left(\bSigma\Htr^{-1}(\Db_0-(\mathbf{I}-\alpha \Htr)^{T}\Db _{0}(\mathbf{I}-\alpha \Htr)^{T})\right)}{(1-c(\btr,\bSigma) \alpha \operatorname{tr}(\bSigma)) T^2}\\
&\times \sum_{k=0}^{T-1}\left\langle\left( (\mathbf{I}-\alpha \Htr)^{k}-(\mathbf{I}-\alpha \Htr)^{T+k} \right),\Hte\right\rangle.
\end{align*}
For the first term, since $\bSigma$, $\Htr$ and $\mathbf{I}-\alpha \Htr$ can be diagonalized simultaneously, considering the eigen-decompositions under the basis of $\bSigma$ and recalling $\bSigma
=\Vb\bLambda\Vb^{\top}$, we have:
$$
\begin{aligned}
&\operatorname{tr}\left(\bSigma
\Htr^{-1}[\Db _{0}-(\mathbf{I}-\alpha \Htr)^{T} \Db _{0}(\mathbf{I}-\alpha\Htr)^{T}]\right)\\
&=\sum_{i}{\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{2 T}\right) \cdot\left(\left\langle\mathbf{w}_{0}-\mathbf{w}^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}}\frac{\lambda_i}{\mu_i(\Htr)}\\
&\leq 2\sum_{i}\left(\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})\geq \frac{1}{\alpha T}}+T\alpha \mu_{i}(\Hb_{n,\beta})\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T}} \right) \cdot \left(\left\langle\mathbf{w}_{0}-\mathbf{w}^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}\frac{\lambda_i}{\mu_i(\Htr)}
\end{aligned}$$
where the last inequality holds since $1-(1-\alpha x)^{2T}\leq\min\{2, 2T\alpha x\}$.
For the second term, similarly, $\Hte$ and $\mathbf{I}-\alpha \Htr$ can be diagonalized simultaneously. We then have
\begin{align*}
\sum_{k=0}^{T-1}&\left\langle\left( (\mathbf{I}-\alpha \Htr)^{k}-(\mathbf{I}-\alpha \Htr)^{T+k} \right),\Hte\right\rangle\\
\leq & \sum^{T-1}_{k=0}\sum_{i} [(1-\alpha\mu_{i}(\Htr))^{k}-(1-\alpha\mu_{i}(\Htr))^{T+k}] \mu_{i}(\Hte)\\
=&\frac{1}{\alpha} \sum_{i} [1-(1-\alpha\mu_{i}(\Htr))^{T}]^2 \frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq &\frac{1}{\alpha} \sum_i \left(\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})\geq \frac{1}{\alpha T}}+T^2\alpha^2 \lambda_{i}(\Hb_{n,\beta})\mathbf{1}_{\lambda_{i}(\Hb_{n,\beta})< \frac{1}{\alpha T}} \right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}.
\end{align*}
Now we turn to:
\begin{align*}
d_2&= \frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle ((\mathbf{I}- \alpha \Htr)^{ k}-(\mathbf{I}- \alpha \Htr)^{T+ k})\Htr^{-1}\Hte,\mathbf{G}_2\right\rangle.
\end{align*}
Considering the orthogonal decompositions of $\Hte$ and $\Htr$ under $\Vb$, $\Htr=\Vb\bLambda_1\Vb^{\top}$, $\Hte=\Vb\bLambda_2\Vb^{\top}$, where the diagonal entries of $\bLambda_1$ are $\mu_i(\Htr)$ (and $\mu_i(\Hte)$ for $\bLambda_2$). Then we have:
\begin{align*}
d_2&=\frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1}\left\langle\underbrace{\left( (\mathbf{I}- \alpha \bLambda_1)^{ k}-(\mathbf{I}- \alpha \bLambda_1)^{T+ k}\right)\bLambda_1^{-1}\bLambda_2}_{\Jb_3}, \Vb^{\top}\Db_0\Vb\right\rangle \\
&=\frac{1}{\alpha T^{2}} \sum_{k=0}^{T-1} \sum_{i}\left[\left(1-\alpha \mu_{i}(\Htr)\right)^{k}-\left(1-\alpha \mu_{i}(\Htr)\right)^{T+k}\right]
\frac{\omega_i^2\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
&=\frac{1}{\alpha^{2} T^{2}} \sum_{i}\left[1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right]^{2}\frac{\omega_i^2\mu_{i}(\Hte)}{\mu_{i}^2(\Htr)}\\
&\leq \frac{1}{\alpha^{2} T^{2}}\sum_{i}\left(\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \alpha^{2} T^{2}\mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu^2_i(\Htr)}
\end{align*}
where $\omega_i=\langle\bomega_0-\btheta^{*},\vb_i\rangle$ is the diagonal entry of $\Vb^{\top}\Db_0\Vb$ and the second equality holds since $\Jb_3$ is a diagonal matrix.
\end{proof}
\subsection{Bounding the Variance}
Note that the noisy part $\Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]$ in \cref{eq-bv} is important in the variance iterates. In order to analyze the variance term, we first understand the role of $\Pi$ by the following lemma.
\begin{lemma}[Bounding the noise]\label{lemma-noise}
\begin{align*}
& \Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]\preceq f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \Htr
\end{align*}
where $f(\beta,n,\sigma,\bSigma,\bSigma_{\btheta})=[c(\beta,\bSigma)\operatorname{tr}({\bSigma_{\btheta}\bSigma})+4c_1\sigma^2\sigma_x^2\beta^2\sqrt{C(\beta,\bSigma)}\operatorname{tr}(\bSigma^2)+{\sigma^2}/n]$.
\end{lemma}
\begin{proof} With a slight abuse of notations, we write $\btr$ as $\beta$ in this proof. By definition of meta data and noise, we have
\begin{align*}
& \Pi=\mathbb{E}[\Bb^{\top}\boldsymbol{\xi}\boldsymbol{\xi}^{\top}\Bb]\\
&=\frac{\sigma^2}{n_2}\Hb_{n_1,\beta}+\mathbb{E}[\Bb^{\top}\Bb\Sigma_{\btheta}\Bb^{\top}\Bb]+\sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}{\Xb^{\text{out}}}^{\top}\Bb].
\end{align*}
The second term can be directly bounded by \Cref{prop-4}:
$$
\mathbb{E}[\Bb^{\top}\Bb\Sigma_{\btheta}\Bb^{\top}\Bb]\preceq c(\beta,\bSigma) \operatorname{tr}({\bSigma_{\btheta}\bSigma})\Hb_{n_1,\beta}.
$$
For the third term, we utilize the technique similar to \Cref{prop-4}, and by Assumption 1, we have:
\begin{align*}
& \sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}\left[\Bb^{\top}\Xb^{\text{out}}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}{\Xb^{\text{out}}}^{\top}\Bb\right]\\
&\preceq \sigma^2c_1\cdot \frac{\beta^{2}}{n_1^2}\mathbb{E}\left[\operatorname{tr}({\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}\Sigma)(\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\bSigma (\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\right].
\end{align*}
Following the analysis for $\Jb_1$ in the proof of \Cref{prop-4}, and letting $\Ab=\mathbf{I}$, we obtain:
$$
\frac{1}{n_1^2}\mathbb{E}\left[\operatorname{tr}({\Xb^{\text{in}}}^{\top}\Xb^{\text{in}}\Sigma)(\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\bSigma (\mathbf{I}-\frac{\beta}{n_1}{\Xb^{\text{in}}}^{\top}\Xb^{\text{in}})\right] \preceq 4\sqrt{C(\beta,\bSigma)}\sigma^2_x\operatorname{tr}(\bSigma^2) \Hb_{n_1,\beta}.
$$
Putting all these results together completes the proof.
\end{proof}
\begin{lemma}[Property of $\Vb_t$]\label{lemma-v-prop}
If the stepsize satisfies $\alpha<\frac{1} {c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, it holds that
$$
\mathbf{0}=\mathbf{V}_{0} \preceq \mathbf{V}_{1} \preceq \cdots \preceq \mathbf{V}_{\infty} \preceq \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) }{1-\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma)}\mathbf{I}.
$$
\end{lemma}
\begin{proof}
Similar calculations has appeared in prior works~\cite{jain2017markov,zou2021benign}. However, our analysis of the meta linear model needs to handle the complicated meta noise, and hence we provide a proof here for completeness.
We first show that $\mathbf{V}_{t-1}\preceq\Vb_t$. By recursion:
$$
\begin{aligned}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha^{2} \Pi \\
&\overset{(a)}{=}\alpha^{2} \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi \\
&=\mathbf{V}_{t-1}+\alpha^{2}(\mathcal{I}-\alpha \mathcal{T})^{t-1} \circ \Pi \\
& \overset{(b)}{\succeq} \mathbf{V}_{t-1}
\end{aligned}
$$
where $(a)$ holds by solving the recursion and $(b)$ follows because $\mathcal{I}-\alpha \mathcal{T}$ is a PSD mapping.
The existence of $\mathbf{V}_{\infty}$ can be shown in the way similar to the proof of \Cref{lemma-linearop}. We first have
$$
\mathbf{V}_{t}=\alpha^{2} \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi \preceq \alpha^{2} \sum_{k=0}^{\infty}\underbrace{(\mathcal{I}-\alpha \mathcal{T})^{k} \circ \Pi}_{\mathbf{A}_k}.
$$
By previous analysis in \Cref{lemma-linearop} , if $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, we have
$$
\begin{aligned}
\operatorname{tr}\left(\mathbf{A}_{k}\right)
& \leq\left(1-\alpha\min_{i}\{\mu_{i}(\Htr)\}\right) \operatorname{tr}\left(\mathbf{A}_{t-1}\right).
\end{aligned}
$$
Therefore,
$$
\operatorname{tr}\left(\mathbf{V}_{t}\right) \leq \alpha^{2} \sum_{k=0}^{\infty} \operatorname{tr}\left(\mathbf{A}_{k}\right) \leq \frac{\alpha \operatorname{tr}(\Pi)}{\min_{i}\{\mu_{i}(\Htr)\}}<\infty.
$$
The trace of $\Vb_t$ is uniformly bounded from above, which indicates that $\Vb_{\infty}$ exists.
Finally, we bound $\mathbf{V}_{\infty}$. Note that $\Vb_{\infty}$ is the solution to:
$$
\mathbf{V}_{\infty} =(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{\infty}+\alpha^{2} \Pi.
$$
Then we can write $\mathbf{V}_{\infty}$ as $\Vb_{\infty} = \mathcal{T}^{-1}\circ \alpha\Pi $. Following the analysis in the proof of \Cref{lemma-ms-pre}, we have:
\begin{align*}
\tilde{\mathcal{T}} \circ \mathbf{V}_{\infty}&=\tilde{\mathcal{T}}\circ \mathcal{T}^{-1}\circ \alpha\Pi\\
&\preceq \alpha \Pi+\alpha\mathcal{M} \circ \Vb_{\infty}\\
&\preceq \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \Htr+\alpha\mathcal{M} \circ \mathbf{V}_{\infty}
\end{align*}
where the last inequality follows from \Cref{lemma-noise}.
Applying $\tilde{\mathcal{T}}^{-1}$, which exists and is a PSD mapping, to the both sides, we have
$$
\begin{aligned}
\mathbf{V}_{\infty} & \preceq \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})\cdot \tilde{\mathcal{T}}^{-1} \circ \Htr+\alpha \tilde{\mathcal{T}}^{-1} \circ \mathcal{M} \circ \mathbf{V}_{\infty} \\
& \overset{(a)}{\preceq} \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta}) \cdot \sum_{t=0}^{\infty}\left(\alpha \tilde{\mathcal{T}}^{-1} \circ \mathcal{M}\right)^{t} \circ \tilde{\mathcal{T}}^{-1} \circ \Htr\\
&\overset{(b)}{\preceq} \alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})\sum^{\infty}_{t=0} (\alpha c(\btr,\bSigma)\operatorname{tr}(\bSigma))^{t}\mathbf{I}\\
&= \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\mathbf{I}
\end{aligned}
$$
where $(a)$ holds by directly solving the recursion; $(b)$ follows from the fact that $\tilde{\mathcal{T}}^{-1} \circ \Htr\preceq \mathbf{I}$ from \Cref{lemma-linearop} and $\mathcal{M} \circ\mathbf{I}\preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma
)\Htr$ by letting $\Ab=\mathbf{I}$ in \Cref{prop-4}.
\end{proof}
Now we are ready to provide the upper bound on the variance term.
\begin{lemma}[Bounding the Variance]\label{lemma-var} If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\mathcal{E}_\text{var}\leq& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}
\\
&\times\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}.
\end{align*}
\end{lemma}
\begin{proof}
Recall
\begin{align}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \nonumber \\
&=(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2}(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \nonumber \\
& \preceq(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \mathcal{M} \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi. \label{eq-v}
\end{align}
By the uniform bound on $\mathbf{V}_t$ and $\mathcal{M}$ is a PSD mapping, we have:
\begin{align*}
\mathcal{M} \circ \mathbf{V}_{t} &\preceq \mathcal{M} \circ \mathbf{V}_{\infty}\\
&\overset{(a)}{\preceq} \mathcal{M} \circ\frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\mathbf{I}\\
&\overset{(b)}{\preceq} \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})c(\btr,\bSigma) \operatorname{tr}(\bSigma)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\cdot \Htr
\end{align*}
where $(a)$ directly follows from \Cref{lemma-v-prop}; $(b)$ holds because $\mathcal{M} \circ\mathbf{I}\preceq c(\btr,\bSigma)\operatorname{tr}(\bSigma
)\Htr$ (letting $\Ab=\mathbf{I}$ in \Cref{prop-4}). Substituting it back into \cref{eq-v}, we have:
\begin{align*}
\mathbf{V}_t&\preceq (\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha^2 \frac{\alpha
f c(\btr,\bSigma) \operatorname{tr}(\bSigma)}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\cdot \Htr+\alpha^2f \Htr\\
&=(\mathcal{I}-\alpha \tilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\frac{\alpha^2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})
}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\Htr\\
&\overset{(a)}{=} \frac{\alpha^2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}\sum^{t-1}_{k=0} (\mathbf{I}-\alpha \tilde{\mathcal{T}})^{k}\circ \Htr\\
&\overset{(b)}{\preceq} \frac{\alpha f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma)}(\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{t})
\end{align*}
where $(a)$ holds by solving the recursion and $(b)$ is due to the fact that
\begin{align*}
\sum^{t-1}_{k=0} (\mathbf{I}-\alpha \tilde{\mathcal{T}})^{k}\circ \Htr&= \sum^{t-1}_{k=0} (\mathbf{I}-\alpha\Htr)^{k}\Htr (\mathbf{I}-\alpha\Htr)^{k}\\
&\preceq \sum^{t-1}_{k=0} (\mathbf{I}-\alpha\Htr)^{k}\Htr\\
&=\frac{1}{\alpha}[\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{t}].
\end{align*}
Substituting the bound for $\Vb_t$ back into the variance term in \Cref{lemma-further-bv}, we have
$$
\begin{aligned}
\mathcal{E}_\text { var } & \leq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle \\
&=\frac{1}{\alpha T^{2}} \sum_{t=0}^{T-1}\left\langle (\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t})\Htr^{-1}\Hte, \mathbf{V}_{t}\right\rangle \\
& \leq \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2}\sum_{t=0}^{T-1}\left\langle\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{T-t},\left(\mathbf{I}-(\mathbf{I}-\alpha \mathbf{H}_{n,\beta})^{t}\right)\mathbf{H}_{n,\beta}^{-1}\mathbf{H}_{m,\eta}\right\rangle .
\end{aligned}
$$
Simultaneously diagonalizing $\Htr$ and $\Hte$ as the analysis in \Cref{lemma-bias}, we have
$$
\begin{aligned}
\mathcal{E}_\text{var}\leq&\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2}\\
&\cdot\sum_{i} \sum_{t=0}^{T-1}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T-t}\right)\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{t}\right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)} \\
\leq &\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T^2} \\
&\cdot\sum_{i} \sum_{t=0}^{T-1}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)} \\
=& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T} \sum_{i} \left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T}\right)^{2}\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq& \frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))T} \sum_{i} \left(\min \left\{1, \alpha T \mu_{i}(\Htr)\right\}\right)^2\frac{\mu_{i}(\Hte)}{\mu_{i}(\Htr)}\\
\leq &\frac{ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\\
&\cdot\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu{i}(\Hb_{n,\beta})< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)},
\end{aligned}
$$
which completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{thm-upper}}
\begin{theorem}[\Cref{thm-upper} Restated]\label{ap-thm-upper} Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$, then the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded above as follows
\[R(\wl_{T},\bte)\leq \text{Bias}+ \text{Var} \]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}+{\textstyle 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}].
\end{align*}
\end{theorem}
\begin{proof}
By \Cref{lemma-bv}, we have
\begin{align*}
R(\wl_T, \bte)\leq 2\mathcal{E}_\text{bias}+2\mathcal{E}_\text{var}.
\end{align*}
Using \Cref{lemma-bias} to bound $\mathcal{E}_\text{bias}$, and \Cref{lemma-var} to bound $\mathcal{E}_\text{var}$, we have
\begin{align*}
& R(\wl_T, \bte)\\&\leq \frac{2 f(\btr,n_2,\sigma,\bSigma,\bSigma_{\btheta})}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\\
&\times \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\\
&+\frac{4 c(\btr,\bSigma)}{T \alpha(1-c(\btr,\bSigma)\alpha \operatorname{tr}(\bSigma))}
\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2 \mu_i(\Htr)^2\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)
\\&\times
\sum_{i}{\left(\frac{1}{\mu_i(\Htr)}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+T\alpha \mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right) \cdot \lambda_i\left(\left\langle\bomega_{0}-\btheta^{*}, \mathbf{v}_{i}\right\rangle\right)^{2}}\\
&+ 2 \sum_{i}\left(\frac{1}{\alpha^{2} T^{2}}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+ \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}.
\end{align*}
Incorporating with the definition of effective meta weight
\begin{equation}
\Xi_i (\bSigma
,\alpha,T)=\begin{cases}
\mu_i(\Hb_{m,\beta^{\text{te}}})/\left(T \mu_i(\Hb_{n_1,\beta^{\text{tr}}})\right) & \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}; \\
T\alpha^2 \mu_i(\Hb_{n_1,\btr})\mu_i(\Hb_{m,\beta^{\text{te}}})& \mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T},
\end{cases}
\end{equation}
we obtain
$$
\left(\frac{1}{T}\mathbf{1}_{\mu_{i}(\Htr)\geq \frac{1}{\alpha T}}+T\alpha^2\mu_{i}^2(\Htr)\mathbf{1}_{\mu_{i}(\Htr)< \frac{1}{\alpha T} } \right)\frac{\mu_i(\Hte)}{\mu_i(\Htr)}=\Xi_i(\bSigma
,\alpha,T).
$$
Therefore,
\[ R(\wl_T, \bte)\leq \text{Bias}+\text{Var}\]
where
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Htr)} \\
\text{Var} &= \frac{2}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\left(\sum_{i}\Xi_{i} \right)
\\
\quad \times & [{f(\btr,n_2,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)}+\underbrace{ 2c(\btr,\bSigma)
\sum_{i}\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i}_{V_2}].
\end{align*}
Note that the term $V_2$ is obtained by our analysis for $\mathcal{E}_{\text{bias}}$. However, it originates from the stochasticity of SGD, and hence we treat this term as the variance in our final results.
\end{proof}
\section{Analysis for Lower Bound (Theorem~\ref{thm-lower}) }
\subsection{Fourth Moment Lower Bound for Meta Nosie}
Similarly to upper bound, we need some technical results for the fourth moment of meta data $\Bb$ and noise $\bxi$ to proceed the lower bound analysis.
\begin{lemma}\label{lemma-4l}
Suppose Assumption 1-3 hold. Given $|\btr|<\frac{1}{\lambda_1}$, for any PSD matrix $\Ab$, we have
\begin{align}\mathbb{E}[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb] &\succeq \Htr \mathbf{A }\Htr+\frac{b_1}{n_2} \operatorname{tr}(\Htr \mathbf{A}) \Htr\label{eq-lower-data}\\
\Pi&\succeq \frac{1}{n_2}g(\btr,n_1,\sigma,\bSigma_{\boldsymbol{\theta}},\bSigma)\Htr
\end{align}
where $g(\beta,n, \sigma,\bSigma, \bSigma_{\btheta}) :={\sigma^2+b_1\operatorname{tr}(\bSigma_{\btheta}\Hb_{n,\beta})+\beta^2 \mathbf{1}_{\beta\leq 0} b_1 \operatorname{tr}(\bSigma^2)/{n}}$.
\end{lemma}
\begin{proof}
With a slight abuse of notations, we write $\btr$ as $\beta$, $\mathbf{X}^{\text{in}}$ as $\mathbf{X}$ in this proof. Note that $\xb\in\mathbb{R}^{d}\sim\mathcal{P}_{\xb}$ is independent of $\Xb^{\text{in}}$. We first derive
\begin{align*}
\mathbb{E}&[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb]\\
&= \frac{1}{n_2} \mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\mathbf{x x}^{\top} (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{A}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{x} \mathbf{x}^{\top}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&+\frac{n_2-1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_2}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb) \mathbf{A}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\bSigma(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&\overset{(a)}{\succeq} \frac{b_1}{n_2} \mathbb{E}\left[\operatorname{tr}(\mathbf{A}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb) \Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb))(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Sigma(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\right]\\
&+\Hb_{n_1,\beta} \mathbf{A }\Hb_{n_1,\beta}\\
&{\succeq} \frac{b_1}{n_2} \operatorname{tr}(\Hb_{n_1,\beta} \mathbf{A}) \Hb_{n_1,\beta}+\Hb_{n_1,\beta} \mathbf{A }\Hb_{n_1,\beta}
\end{align*}
where $(a)$ is implied by Assumption 1.
Recall that $\Pi$ takes the following form:
\begin{align*}
\Pi&=\frac{\sigma^2}{n_2}\Hb_{n_1,\beta}+\mathbb{E}[\Bb^{\top}\Bb\bSigma_{\btheta}\Bb^{\top}\Bb]+\sigma^2\cdot \frac{\beta^{2}}{n_2 n_1^2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb}^{\top}\Xb{\Xb^{\text{out}}}^{\top}\Bb].
\end{align*}
The second term can be directly bounded by letting $\Ab=\bSigma_{\btheta}$ in \cref{eq-lower-data}, and we have:
$$\mathbb{E}[\Bb^{\top}\Bb\bSigma_{\btheta}\Bb^{\top}\Bb]\succeq \frac{b_1}{n_2} \operatorname{tr}(\Hb_{n_1,\beta} \bSigma_{\btheta}) \Hb_{n_1,\beta}.$$
For the third term:
\begin{align*}
& \frac{1}{n_2}\mathbb{E}[\Bb^{\top}\Xb^{\text{out}}{\Xb}^{\top}\Xb{\Xb^{\text{out}}}^{\top}\Bb]\\
&=\frac{1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\mathbf{x x}^{\top}\Xb^{\top}\Xb \mathbf{x} \mathbf{x}^{\top}(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&+\frac{n_2-1}{n_2}\mathbb{E}\left[(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\bSigma\Xb^{\top}\Xb \bSigma(\mathbf{I}-\frac{\beta}{n_1}\Xb^{\top}\Xb)\right]\\
&\succeq \frac{n_1 b_1\operatorname{tr}(\bSigma^2)}{n_2}\Hb_{n_1,\beta}\mathbf{1}_{\beta\leq 0}
\end{align*}
Putting these results together completes the proof.
\end{proof}
\subsection{Bias-Variance Decomposition}
For the lower bound analysis, we also decompose the excess risk into bias and variance terms.
\begin{lemma}[Bias-variance decomposition, lower bound]\label{lemma-bv-lower}
Following the notations in \cref{eq-bv}, the excess risk can be decomposed as follows:
$$
\begin{aligned}
R(\wl_{T},\bte)\geq \underline{\mathcal{E}_{bias}}+\underline{\mathcal{E}_{var}}
\end{aligned}
$$
where
\begin{align*}
\underline{\mathcal{E}_{bias}}= & \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{D}_{t}\right\rangle, \\
\underline{\mathcal{E}_{var}} =&\frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align*}
\end{lemma}
\begin{proof}
The proof is similar to that for \Cref{lemma-further-bv}, and the inequality sign is reversed since we only calculate the half of summation. In particular,
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]& =\frac{1}{T^{2}} \sum_{1\leq t<k\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]+\frac{1}{T^{2}} \sum_{1\leq k<t\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k]\\
& \succeq\frac{1}{T^{2}} \sum_{1\leq t<k\leq T-1} \mathbb{E}[\brho^{\text{var}}_t\otimes\brho^{\text{var}}_k].
\end{align*}
For $t\leq k$, $\mathbb{E}[\brho^{\text{var}}_k|\brho^{\text{var}}_t]=(\mathbf{I}-\alpha\Htr)^{k-t} \brho^{\text{var}}_t$, since $\mathbb{E}[\Bb_t^{\top}\bxi_t|\brho_{t-1}]=\mathbf{0}$. From this
\begin{align*}
\mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T]
& \succeq \frac{1}{T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}.
\end{align*}
Plugging this into $\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle$, we obtain:
\begin{align*}
&\frac{1}{2} \langle\Hte, \mathbb{E}[\rhob^{\text{var}}_T\otimes\rhob^{\text{var}}_T] \rangle\\
&\geq \frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t+1}^{T-1} \langle \Hte, \Vb_t(\mathbf{I}-\alpha\Htr)^{k-t}\rangle \\
&=\frac{1}{2T^{2}} \sum_{t=0}^{T-1} \sum_{k=t}^{T-1} \langle (\mathbf{I}-\alpha\Htr)^{k-t}\Hte, \Vb_t\rangle
\\&=\underline{\mathcal{E}_\text{var}}.
\end{align*}
The proof is the same for the term $\underline{\mathcal{E}_\text{bias}}$.
\end{proof}
\subsection{Bounding the Bias}
We first bound the summation of $\Db_t$, i.e. $\Sb_k= \sum^{k-1}_{t=0}\Db_t$.
\begin{lemma}[Bounding $\mathbf{S}_t$]\label{lemma-sk}
If the stepsize satisfies $\alpha <1 / (2\max_{i}\{\mu_{i}(\Htr)\})$, then for any $k \geq 2$, it holds that
\begin{align*}
\mathbf{S}_{k} &\succeq \frac{b_1}{4n_2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k / 2}\right) \mathbf{D}_{0}\right) \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k / 2}\right)\\
&+\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.
\end{align*}
\end{lemma}
\begin{proof} By \cref{eq-st}, since $\tilde{\mathcal{M}}-\mathcal{M}$ is a PSD mapping, we have
\begin{align}
\Sb_k &=\Db _{0}+(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{S}_{k-1}+ \alpha^2(\mathcal{M}-\widetilde{\mathcal{M}})\circ \mathbf{S}_{k-1}\label{eq-sk}\\
& \succeq \sum^{k-1}_{t=0} (\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{t}
\circ \Db_0\nonumber\\
&=\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.\nonumber
\end{align}
Note that for PSD $\Ab$, $$(\mathcal{M}-\widetilde{\mathcal{M}}) \circ\Ab= \mathbb{E}[\Bb^{\top}\Bb\Ab\Bb^{\top}\Bb] - \Htr \mathbf{A }\Htr
$$ By \Cref{lemma-4l}, we have
\begin{align}
(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{S}_{k} & \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\Htr \mathbf{S}_{k}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\sum_{t=0}^{k-1}(\mathbf{I}-\alpha \Htr)^{2 t} \Htr \cdot \mathbf{D}_{0}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{n_2} \operatorname{tr}\left(\sum_{t=0}^{k-1}(\mathbf{I}-2 \alpha\Htr)^{t} \Htr\cdot \mathbf{D}_{0}\right) \Htr\nonumber \\
& \succeq \frac{b_1}{2 n_2 \alpha} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k}\right) \mathbf{D}_{0}\right) \Htr.\label{eq-crude}
\end{align}
Substituting \cref{eq-crude} back into \cref{eq-sk}, and solving the recursion, we obtain
$$
\begin{aligned}
\mathbf{S}_{k} \succeq & \sum_{t=0}^{k-1}(\mathcal{I}-\alpha \tilde{\mathcal{T}})^{t} \circ\left\{\frac{b_1 \alpha}{2n_2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \mathbf{H}+\mathbf{D}_{0}\right\} \\
=& \frac{b_1 \alpha}{2n_2} \underbrace{\sum_{t=0}^{k-1} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \cdot(\mathbf{I}-\alpha\Htr)^{2 t} \Htr}_{\Jb_4} \\
&+\sum_{t=0}^{k-1}(\mathbf{I}-\alpha\Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}.
\end{aligned}
$$
The term $\Jb_4$ can be further bounded by the following:
\begin{align*}
\Jb_4&\succeq \sum_{t=0}^{k-1} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k-1-t}\right) \mathbf{D}_{0}\right) \cdot(\mathbf{I}-2\alpha\Htr)^{t}\Htr\\
&\succeq \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k/2}\right) \mathbf{D}_{0}\right) \cdot\sum_{t=0}^{k/2-1}(\mathbf{I}-2\alpha\Htr)^{t}\Htr\\
&\succeq \frac{1}{2\alpha}\operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{k/2}\right) \mathbf{D}_{0}\right) \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha\Htr)^{k/2}\right)
\end{align*}
which completes the proof.
\end{proof}
Then we can bound the bias term.
\begin{lemma}[Bounding the bias]\label{lemma-biasl}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, then we have
\begin{align*}
\underline{\mathcal{E}_{\text{bias}}}\ge& \frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} + \frac{b_1}{1000n_2(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
&\times
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i.
\end{align*}
\end{lemma}
\begin{proof}
From \Cref{lemma-bv-lower}, we have
\begin{align*}
\underline{\mathcal{E}_{bias}}&= \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{D}_{t}\right\rangle \\
&=\frac{1}{2\alpha T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \mathbf{D}_{t}\right\rangle\\
&\ge \frac{1}{2 \alpha T^{2}} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte, \sum_{t=0}^{T/2}\mathbf{D}_{t}\right\rangle\\
&\ge \frac{1}{2\alpha T^{2}} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte, \mathbf{S}_{\frac{T}{2}}\right\rangle.
\end{align*}
Applying \Cref{lemma-sk} to $\mathbf{S}_{\frac{T}{2}}$, we obtain:
\begin{align*}
\underline{\mathcal{E}_{bias}}
\ge& \underline{d_1}+\underline{d_2}
\end{align*}
where
\begin{align*}
\underline{d_1}&=\frac{b_1}{8\alpha n_2T^2} \operatorname{tr}\left(\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T / 4}\right) \mathbf{D}_{0}\right)\\ &\times\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,\right.
\left. \left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T /4}\right)\right\rangle\\
\underline{d_2}&=\frac{1}{2\alpha T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,\right.
\\&\left. \sum_{t=0}^{T/2-1}(\mathbf{I}-\alpha \Htr)^{t} \cdot \mathbf{D}_{0} \cdot(\mathbf{I}-\alpha \Htr)^{t}\right\rangle.
\end{align*}
Moreover,
\begin{align*}
\underline{d_2}&\ge\frac{1}{2\alpha T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)\Htr^{-1} \Hte,
\sum_{t=0}^{T/2-1}(\mathbf{I}-2\alpha \Htr)^{t} \mathbf{D}_{0} \right\rangle;\\
&\ge\frac{1}{4\alpha^2 T^2 }\left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T/2}\right)^2\Htr^{-2} \Hte,
\mathbf{D}_{0} \right\rangle.
\end{align*}
Using the diagonalizing technique similar to the proof for \Cref{lemma-bias}, we have
\begin{align}
\underline{d_1}&\ge \frac{b_1}{8 \alpha n_2 T^{2}}\left(\sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right) \omega_{i}^{2}\right)\\
&\times\left(\sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right)^{2} \frac{\mu_i(\Hte)}{\mu_i(\Htr)}\right)\label{b1},\\
\underline{d_2}&\geq \frac{1}{4 \alpha^{2} T^{2}} \sum_{i}\left(1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\right)^{2} \frac{\mu_i(\Hte)}{\mu_i^2(\Htr) } \omega_{i}^{2}.\label{b2}
\end{align}
We use the following fact to bound the polynomial term. For $h_1(x)=1-(1-x)^{\frac{T}{4}}$, we have
$$
h_1(x)\ge\begin{cases}
\frac{1}{5}& x\ge 1/T\\
\frac{T}{5} x& x< 1/T
\end{cases}
$$
i.e., $1-\left(1-\alpha \mu_{i}(\Htr)\right)^{T / 4}\geq \left(\frac{1}{5}\mathbf{1}_{\alpha \mu_{i}(\Htr)\ge \frac{1}{T}}+\frac{\alpha \mu_{i}(\Htr)}{5}\mathbf{1}_{\alpha \mu_{i}(\Htr)< \frac{1}{T}}\right)$. Substituting this back into \cref{b1,b2}, and using the definition of effective meta weight $\Xi_i$ complete the proof.
\end{proof}
\subsection{Bounding the Variance}
We first bound the term $\Vb_t$.
\begin{lemma}[Bounding $\mathbf{V}_{t}$]\label{lemma-vt}
If the stepsize satisfies $\alpha <1 / (\max_{i}\{\mu_{i}(\Htr)\})$, it holds that
$$
\mathbf{V}_{t} \succeq \frac{\alpha g(\btr,n_1, \bSigma, \bSigma_{\btheta})}{2} \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right).
$$
\end{lemma}
\begin{proof}
With a slight abuse of notations, we write $g(\btr,n_1, \bSigma, \bSigma_{\btheta})$ as $g$. By definition,
$$
\begin{aligned}
\mathbf{V}_{t} &=(\mathcal{I}-\alpha \mathcal{T}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \\
&=(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+(\mathcal{M}-\widetilde{\mathcal{M}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} \Pi \\
&\overset{(a)}{\succeq}(\mathcal{I}-\alpha \widetilde{\mathcal{T}}) \circ \mathbf{V}_{t-1}+\alpha ^{2} g \Htr \\
&\overset{(b)}{=}\alpha ^{2} g \cdot \sum_{k=0}^{t-1}(\mathcal{I}-\alpha \widetilde{\mathcal{T}})^{k} \circ \Htr \\
&=\alpha ^{2}g \cdot \sum_{k=0}^{t-1}(\mathbf{I}-\alpha \Htr)^{k} \Htr(\mathbf{I}-\alpha \Htr)^{k} \quad \text { (by the definition of } \mathcal{I}-\alpha \widetilde{\mathcal{T}} ) \\
&=\alpha g \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right) \cdot\left(2 \mathbf{I}-\alpha \Htr\right)^{-1} \\
& \overset{(c)}{\succeq} \frac{\alpha g}{2} \cdot\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right)
\end{aligned}
$$
where $(a)$ follows from the \Cref{lemma-4l}, $(b)$ follows by solving the recursion and $(c)$ holds since we directly replace $\left(2 \mathbf{I}-\alpha \Htr\right)^{-1} $ by $(2\Ib)^{-1}$.
\end{proof}
\begin{lemma}[Bounding the variance]\label{lemma-varl}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $\alpha<\frac{1}{c(\btr,\bSigma)\operatorname{tr}(\bSigma)}$, for sufficiently large $n_1$, s.t. $\mu_i(\Htr)>0$, $\forall i$, for $T>10$, then we have
\begin{align*}
\underline{\mathcal{E}_{\text{var}}}&\ge \frac{g(\btr,n_1, \bSigma, \bSigma_{\btheta})}{100n_2(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i}.
\end{align*}
\end{lemma}
\begin{proof}
From \Cref{lemma-bv-lower}, we have
\begin{align*}
\underline{\mathcal{E}_{var}}&= \frac{1}{2 T^{2}} \cdot \sum_{t=0}^{T-1} \sum_{k=t}^{T-1}\left\langle(\mathbf{I}-\alpha \Htr)^{k-t} \Hte, \mathbf{V}_{t}\right\rangle \\
&=\frac{1}{2\alpha T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \mathbf{V}_{t}\right\rangle.
\end{align*}
Then applying \Cref{lemma-vt}, and writting $g(\btr,n_1, \bSigma, \bSigma_{\btheta})$ as $g$, we obtain
\begin{align}
\underline{\mathcal{E}_{var}}&\ge \frac{g}{4 T^{2}} \cdot \sum_{t=0}^{T-1} \left\langle\left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{T-t}\right)\Htr^{-1} \Hte, \left(\mathbf{I}-(\mathbf{I}-\alpha \Htr)^{2 t}\right)\right\rangle\nonumber\\
&=\frac{g}{4 T^{2}} \sum_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\sum_{t=0}^{T-1}(1-(1-\alpha \mu_i(\Htr)^{T-t}))(1-(1-\alpha \mu_i(\Htr)^{2t}))\nonumber\\
&\ge \frac{g}{4 T^{2}} \sum_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)}\sum_{t=0}^{T-1}(1-(1-\alpha \mu_i(\Htr)^{T-t-1}))(1-(1-\alpha \mu_i(\Htr)^{t})) \label{eq-var}
\end{align}
where the equality holds by applying the diagonalizing technique again. Following the trick similar to that in \cite{zou2021benign} to lower bound the function $h_2(x):=\sum_{t=0}^{T-1}\left(1-(1-x)^{T-t-1}\right)\left(1-(1-x)^{t}\right)$ defined on $x\in(0,1)$, for $T>10$, we have
$$
f(x) \geq \begin{cases}\frac{T}{20}, & \frac{1}{T} \leq x<1 \\ \frac{3 T^{3}}{50} x^{2}, & 0<x<\frac{1}{T}\end{cases}
$$
Substituting this back into \cref{eq-var}, and using the definition of effective meta weight $\Xi_i$ completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{thm-lower}}
\begin{theorem}[\Cref{thm-lower} Restated]\label{ap-thm-lower}
Let $\omega_i=\left\langle\boldsymbol{\omega}_{0}-\boldsymbol{\theta}^{*}, \mathbf{v}_{i}\right\rangle$. If $|\btr|,|\bte|<1/\lambda_1$, $n_1$ is large ensuring that $\mu_i(\Hb_{n_1,\beta^{\text{tr}}})>0$, $\forall i$ and
$\alpha<1/\left(c(\btr,\bSigma) \operatorname{tr}(\bSigma)\right)$. For $T>10$, the meta excess risk $R(\overline{\boldsymbol{\omega}}_T,\bte)$ is bounded below as follows
\begin{align*}
R(\overline{\boldsymbol{\omega}}_T,\bte) \ge &\frac{1}{100\alpha^2 T} \sum_{i}\Xi_i \frac{\omega_i^2}{\mu_i(\Hb_{n_1,\beta^{\text{tr}}})} +\frac{1}{n_2}\cdot \frac{1}{(1-\alpha c(\btr,\bSigma) \operatorname{tr}(\bSigma))}\sum_{i}\Xi_{i} \\
\times & [\frac{1}{100} g(\btr,n_1, \bSigma, \bSigma_{\btheta})+\frac{b_1}{1000}
\sum_{i}\Big( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \Big) \lambda_{i}\omega^2_i].
\end{align*}
\end{theorem}
\begin{proof}
The proof can be completed by combining \Cref{lemma-biasl,lemma-varl}.
\end{proof}
\section{Proofs for Section~\ref{sec-main-task}}
\subsection{Proof of Lemma~\ref{lem-single}}
\begin{proof}[Proof of \Cref{lem-single}]
For the single task setting, we first simplify our notations in \Cref{thm-upper} as follows.
\begin{align*} c(0,\bSigma) = c_1,\quad
f(0,n_2,\sigma,\bSigma,\mathbf{0})=\sigma^2/n_2,\quad \Htr = \bSigma.
\end{align*}
By \Cref{thm-upper}, we have
\begin{align*}
\text{Bias} & = \frac{2}{\alpha^2 T} \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \frac{\omega_i^2\mu_i(\Hte)}{\lambda^2_i}\\
&\leq \frac{2}{\alpha^2 T} \sum_{i}(\alpha \lambda_i\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+\alpha\lambda_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } )\frac{\omega_i^2\mu_i(\Hte)}{\lambda^2_i}.
\end{align*}
For large $m$, we have $\mu_i(\Hte)=(1-\bte\lambda_i)^2\lambda_i+o(1)$. Therefore,
\begin{align*}
\text{Bias}
\leq \frac{2(1-\bte\lambda_d)^2}{\alpha^2 T} \sum_{i} {\omega_i^2}\leq \mathcal{O}(\frac{1}{T}).
\end{align*}
For the variance term,
\begin{align*}
\text{Var} &= \frac{2}{(1-\alpha c_1 \operatorname{tr}(\bSigma))}\underbrace{\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \frac{\mu_i(\Hte)}{\lambda_i}}_{J_5}
\\
\quad \times & [\frac{\sigma^2}{n_2}+ 2c_1
\sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha \lambda_i}+\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i].
\end{align*}
It is easy to check that
$$
\sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha \lambda_i }+\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i\leq \sum_{i}\left( \frac{\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}}{T\alpha }+\frac{1}{\alpha T}\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right) \omega^2_i \leq \mathcal{O}(1/T).
$$
Moreover,
$$
J_5\leq (1-\bte\lambda_d)^2 \sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right).
$$
The term $\sum_{i}\left(\frac{1}{T}\mathbf{1}_{\lambda_i\geq \frac{1}{\alpha T}}+T\alpha^2\lambda^2_i\mathbf{1}_{\lambda_i< \frac{1}{\alpha T} } \right)$ has the form similar to Corollary 2.3 in \cite{zou2021benign}
and we directly have
$J_5=\mathcal{O}\left(\log^{-p}(T)\right)$, which implies
\begin{align*}
\text{Var} =\mathcal{O}\left(\log^{-p}(T)\right).
\end{align*}
Thus we complete the proof.
\end{proof}
\subsection{Proof of Proposition~\ref{prop-hard}}
\begin{proof}[Proof of \Cref{prop-hard}]\label{proof-prop2}
We first consider the bias term in \Cref{thm-upper,thm-lower} (up to absolute constants):
\begin{align*}
\text{Bias}&= \frac{2}{\alpha^{2} T} \sum_{i}\left(\frac{1}{ T}\mathbf{1}_{\mu_i(\Htr)\geq \frac{1}{\alpha T}}+\alpha^{2} T \mu_i^2(\Htr)\mathbf{1}_{\mu_i(\Htr)< \frac{1}{\alpha T} } \right)\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)^2}.
\end{align*}
If $\mu_i(\Htr)\geq \frac{1}{\alpha T}$, $\frac{1}{T}\leq \alpha\mu_i(\Htr) $; and if $\mu_i(\Htr)< \frac{1}{\alpha T}$, then $\alpha^{2} T \mu_i^2(\Htr)<\alpha \mu_i(\Htr) $. Hence
\begin{align*}
\text{Bias}\le \frac{1}{\alpha^{2} T} \sum_{i}\frac{\omega_i^2\mu_i(\Hte)}{\mu_i(\Htr)}\le \frac{2}{\alpha^{2} T}\cdot\max_{i}\frac{\mu_i(\Hte)}{\mu_i(\Htr)} \|\bomega_0-\btheta^{*}\|^2=\mathcal{O}(\frac{1}{T}).
\end{align*}
Moreover,
\begin{align*}
\sum_{i}&\left( \frac{\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})\geq \frac{1}{\alpha T}}}{T\alpha \mu_i(\Hb_{n_1,\beta^{\text{tr}}})}+\mathbf{1}_{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T} } \right) \lambda_{i}\omega^2_i
\\
&\overset{(a)}{\le} \frac{1}{\alpha T}\sum_{i} \frac{\lambda_{i}}{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})}\omega^2_i\\ &\le \frac{1}{\alpha T}\max_{i} \frac{\lambda_{i}}{\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})}\|\bomega_0-\btheta^{*}\|^2=\mathcal{O}(\frac{1}{T})
\end{align*}
where $(a)$ holds since we directly upper bound $\mu_{i}(\Htr)$ by $\frac{1}{\alpha T}$ when $\mu_{i}(\Hb_{n_1,\beta^{\text{tr}}})< \frac{1}{\alpha T}$. Therefore, it is essential to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$ and $ g(\btr,n_1,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$ from variance term in the upper and lower bounds respectively.
Then we calculate some rates of interesting in \Cref{thm-upper,thm-lower} under the specific data and task distributions in \Cref{prop-hard}.
If the spectrum of $\bSigma$ satisfies $\lambda_{k}=k^{-1} \log ^{-p}(k+1)$, then it is easily verified that $\operatorname{tr}(\bSigma^{s})=O(1)$ for $s=1,\cdots,4$. By discussions on \Cref{ass:higherorder} in \Cref{sec-diss}, we have $C(\beta,\bSigma
)=\Theta(1)$ for given $\beta$. Hence,
\begin{align*} c(\btr,\bSigma) &= \Theta(1)\\
f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&=c(\btr,\bSigma)\operatorname{tr}({\bSigma_{\boldsymbol{\theta}}\bSigma})+ \Theta(1)\\
g(\btr,n_1, \sigma,\bSigma, \bSigma_{\btheta}) & =b_1\operatorname{tr}(\bSigma_{\btheta}\Htr)+\Theta(1).
\end{align*}
If $r\ge 2p-1$, then we have $g(\btr,n_1, \sigma,\bSigma, \bSigma_{\btheta})\ge \Omega\left(\log^{r-p+1}(d)\right)\ge \Omega\left(\log^{r-p+1}(T)\right)$.
Let $k^{\dagger}:= \operatorname{card}\{i: \mu_{i}(\Htr)\ge 1/\alpha T\}$. For large $n_1$, we have $\mu_i(\Htr)=(1-\btr\lambda_i)^2\lambda_i+o(1)$. If $k^{\dagger}=\mathcal{O}\left(T/\log^{p}(T+1)\right)$, then
$$\min_{1 \le i\le k^{\dagger}+1 }\mu_{i}(\Htr)=\omega\left(\frac{\log^{p}(T)}{T[\log(T)-p\log(\log(T))]^p}\right)=\omega\left(\frac{1}{T}\right)$$
which contradicts the definition of $k^{\dagger}$. Hence $k^{\dagger}=\Omega\left(T/\log^{p}(T+1)\right)$. Then
\begin{align*}
\sum_{i}\Xi_{i}\ge \Omega\left(k^{\dagger}\cdot \frac{1}{T} \right)=\Omega\left(\frac{1}{\log^{p}(T)}\right).
\end{align*}
Therefore, by \Cref{thm-lower}, $R(\wl,\bte)=\Omega\left(\log^{r-2p+1}(T)\right)$.
For $r< 2p-1$, if $d=T^l$, where $l$ can be sufficiently large ($d\gg T$) but still finite, then
\begin{itemize}
\item If $p-1<r< 2p-1$, $f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\le \mathcal{O}(\log^{r-p+1} T)$;
\item If $r\leq p-1$, $f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\le \mathcal{O}\Big(\log\left(\log(T)\right)\Big)$.
\end{itemize}
Following the analysis similar to that for Corollary 2.3 in \cite{zou2021benign}, we have $ \sum_{i}\Xi_{i}= \mathcal{O}(\frac{1}{\log^{p}(T)})$. Then by \Cref{thm-upper}
$$
R(\overline{\boldsymbol{\omega}}_T,\beta^{\text{te}})= \mathcal{O}\left(\frac{1}{\log^{p-(r-p+1)^{+}}(T)}\right).
$$
\end{proof}
\subsection{Proof of Proposition~\ref{prop-fast}}
\begin{proof}[Proof of \Cref{prop-fast}]
Following the analysis in \Cref{proof-prop2}, it is essential to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$.
If $d=T^l$, where $l$ can be sufficiently large but still finite, then $$f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})=\tilde{\Theta}(1)$$ for $\lambda_{k}=k^{q}$ $(q>1)$ or $\lambda_k=e^{-k}$.
Following the analysis similar to that for Corollary 2.3 in \cite{zou2021benign}, we have
\begin{itemize}
\item If $\lambda_{k}=k^{q}$ $(q>1)$, then $\sum_{i}\Xi_{i}=\mathcal{O}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item If $\lambda_k=e^{-k}$, then $\sum_{i}\Xi_{i}=\mathcal{O}\left(\frac{\log(T)}{T}\right)$.
\end{itemize}
Substituting these results back into \Cref{thm-upper}, we obtain
\begin{itemize}
\item If $\lambda_{k}=k^{q}$ $(q>1)$, then $ R(\overline{\boldsymbol{\omega}}_T,\bte)=\tilde{\mathcal{O}}\left(\frac{1}{T^{\frac{q-1}{q}}}\right)$;
\item If $\lambda_k=e^{-k}$, then $ R(\overline{\boldsymbol{\omega}}_T,\bte)=\tilde{\mathcal{O}}\left(\frac{1}{T}\right)$.
\end{itemize}
\end{proof}
\section{Proofs for Section~\ref{sec-main-stopping}}
\subsection{Proof of Proposition~\ref{prop-tradeoff}}\label{sec-prop4}
\begin{proof}[Proof of \Cref{prop-tradeoff}]
Following the analysis in \Cref{proof-prop2}, it is crucial to analyze $ f(\btr,n_2,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})\left(\sum_{i}\Xi_{i} \right)$.
Then we calculate the rate of interest in \Cref{thm-upper,thm-lower} under some specific data and task distributions in Proposition 4. We have $\operatorname{tr}(\bSigma^2)=\frac{1}{s}+\frac{1}{d-s}=\Theta(\frac{\log^{p}(T)}{T})$. Moreover, by discussions on Assumption 3 in \Cref{sec-diss}, $C(\beta,\bSigma)=\Theta(1)$.
Hence
\begin{align*} c(\beta,\bSigma) &:= c_1+\tilde{\mathcal{O}}(\frac{1}{T});\\
f(\beta,n,\sigma,\bSigma,\bSigma_{\boldsymbol{\theta}})&:=2c_1\mathcal{O}(1)+\frac{\sigma^2}{n}+ \tilde{\mathcal{O}}\left(\frac{1}{T}\right).
\end{align*}
By the definition of $\Xi_i$, we have
\begin{align*}
\sum_{i}\Xi_i&=\mathcal{O}\left(s\cdot \frac{\mu_1(\Hte)}{T\mu_1(\Htr)}+\frac{1}{d-s}\cdot T \frac{\mu_d(\Htr)\mu_d(\Hte)}{\lambda^2_d}\right)\\
&= \mathcal{O}\left(\frac{1}{\log^{p}(T)} \right)\frac{\mu_1(\Hte)}{\mu_1(\Htr)}+\mathcal{O}\Big(\frac{1}{\log^{q}(T)} \Big)\frac{\mu_d(\Htr)\mu_d(\Hte)}{\lambda^2_d}\\
&=\mathcal{O}\left(\frac{1}{\log^{p}(T)} \right)\frac{(1-\bte\lambda_1)^2}{(1-\btr\lambda_1)^2}+\mathcal{O}\Big(\frac{1}{\log^{q}(T)} \Big)(1-\bte\lambda_d)^2(1-\btr\lambda_d)^2
\end{align*}
where the last equality follows from the fact that for large $n$, we have $\mu_i(\Hb_{n,\beta})=(1-\beta\lambda_i)^2\lambda_i+o(1)$. Combining with the bias term which is $\mathcal{O}(\frac{1}{T})$, and applying \Cref{thm-upper} completes the proof.
\end{proof}
\subsection{Proof of Corollary~\ref{col-stop}}
\begin{proof}[Proof of \Cref{col-stop}]
For $t\in (s, K]$,
by \Cref{thm-lower}, one can verify that $t=\tilde{\Theta}(K)$ for diminishing risk. Let $t=K\log^{-l}(K)$, where $p>l>0$. Following the analysis in \Cref{sec-prop4}, we have
\begin{align}
&R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_t,\bte)\lesssim \widetilde{\mathcal{O}}(\frac{1}{K})
+(2c_1\nu^2+\frac{\sigma^2}{n_2})\\
&\times \left[\mathcal{O}\Big(\frac{1}{\log^{p-l}(K)}\Big) \frac{(1-\bte\lambda_1)^2}{(1-\btr \lambda_{1})^{2}}+\mathcal{O}\Big(\frac{1}{\log^{p+l} (K)}\Big)\Big(1-\btr \lambda_{d}\Big)^{2}\Big(1-\bte \lambda_{d}\Big)^{2}\right].
\end{align}
To clearly illustrate the trade-off in the stopping time, we let $l=0$ for convenience. If $R(\overline{\boldsymbol{\omega}}^{\beta^{\text{tr}}}_t,\bte)<\epsilon$, we have
\begin{align*}
t_{\epsilon}\leq \exp\Big(\epsilon^{-\frac{1}{p}}\Big[\frac{U_{l}}{(1-\btr\lambda_1)^2}+ U_{t} (1-\btr\lambda_d)^2\Big]^{\frac{1}{p}}\Big)
\end{align*}
where
\begin{align*}
U_{l}=\mathcal{O}\Big( (2c_1\nu^2+\frac{\sigma^2}{n_2})(1-\bte\lambda_1)^2\Big) \quad { and }\quad U_{l} =\mathcal{O}\Big( (2c_1\nu^2+\frac{\sigma^2}{n_2})(1-\bte\lambda_d)^2\Big).
\end{align*}
The arguments are similar for the lower bound, and we can obtain:
\begin{align*}
L_{l}=\mathcal{O}\Big( (2\frac{b_1\nu^2}{n_2}+\frac{\sigma^2}{n_2})(1-\bte\lambda_1)^2\Big) \quad { and }\quad L_{l} =\mathcal{O}\Big( (2\frac{b_1\nu^2}{n_2}+\frac{\sigma^2}{n_2})(1-\bte\lambda_d)^2\Big).
\end{align*}
\end{proof}
\section{Discussions on Assumptions}\label{sec-diss}
\paragraph{Discussions on \Cref{ass-comm}}
If $\mathcal{P}_{\xb}$ is Gaussian distribution, then we have
$$
F=\mathbb{E}[\xb\xb^{\top}\bSigma\xb\xb^{\top} ]= 2\bSigma^3+\bSigma\operatorname{tr}(\bSigma^2).
$$
This implies that $F$ and $\bSigma$ commute because $\bSigma^3$ and $\bSigma$ commute. Moreover, in this case
$$
\frac{\beta^2}{n}(F-\bSigma^3)=\frac{\beta^2}{n}( \bSigma^3+\bSigma\operatorname{tr}(\bSigma^2)).
$$
Therefore, if $n\gg \lambda_1(\lambda^2_1+\operatorname{tr}(\bSigma^2))$, then the eigen-space of $\Hb_{n,\beta}$ will be dominated by $(\Ib-\beta\bSigma)^2\bSigma$.
\paragraph{Discussions on \Cref{ass:higherorder}} \Cref{ass:higherorder} is an eighth moment condition for $\xb:=\bSigma^{\frac{1}{2}}\zb$, where $\zb$ is a $\sigma_x$ sub-Gaussian vector. Given $\beta$, for sufficiently large $n$ s.t. $\mu_i(\Hb_{n,b})>0$, $\forall i$, and if $\operatorname{tr}(\bSigma
^{k})$ are all $O(1)$ for $k=1,\cdots,4$, then by the quadratic form and the sub-Gaussian property, which has finite higher order moments, we can conclude that $C(\beta,\Sigma)=\Theta(1)$.
The following lemma further shows that if $\mathcal{P}_{\xb}$ is a Gaussian distribution, we can derive the analytical form for $C(\beta,\bSigma
)$.
\begin{lemma}\label{lemma-C}
Given $|\beta|<\frac{1}{\lambda_1}$, for sufficiently large $n$ s.t. $\mu_i(\Hb_{n,b})>0$, $\forall i$, and if $\mathcal{P}_{\xb}$ is a Gaussian distribution, assuming $\bSigma
$ is diagonal, we have:
$$ C(\beta, \bSigma)=210( 1+\frac{\beta^4\operatorname{tr}(\bSigma^2)^2}{(1-\beta\lambda_1)^4}).$$
\end{lemma}
\begin{proof} Let $\eb_i\in\mathbb{R}^{d}$ denote the vector that the $i$-th coordinate is $1$, and all other coordinates equal $0$. For $\xb\sim\mathcal{P}_{\xb}$, denote $\xb\xb^{\top}=[x_{ij}]_{1\leq i,j\leq d}$. Then we have:
\begin{align*}
& \mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&\leq \mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\beta\xb\xb^{\top})\bSigma (\mathbf{I}-\beta\xb\xb^{\top})\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&=\mathbb{E}\left[(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\left(\sum_{j\neq i }\beta^2\lambda_jx^2_{ij}+ \lambda_i(1-\beta x_{ii})^2\right)^2\right]
\end{align*}
For Gaussian distributions, we have $$
\mathbb{E}[x^2_{ij}x^2_{ik}]=\begin{cases}9\lambda^2_{i}\lambda^2_{j} & j=k \text{ and } \neq i\\
105\lambda_{i}^4& i=j=k\\
3\lambda^{2}_i\lambda_j\lambda_k & i\neq j\neq k
\end{cases}$$
We can further obtain:
\begin{align*}
&\mathbb{E}[\|\eb_{i}^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\eb_{i}\|^2]\\
&\leq 105(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\left(\sum_{j\neq i }\beta^2 \lambda^2_{j}+ (1-\beta \lambda_{i})^2 \right)^2\\
&\overset{(a)}{\leq} 210(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2[\beta^4\operatorname{tr}(\bSigma^2
)^2+ (1-\beta \lambda_{i})^4 ]\\
&\overset{(b)}{\leq} 210[(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2\beta^4\operatorname{tr}(\bSigma^2
)^2+ 1 ]
\end{align*}
where $(a)$ follows from the Cauchy-Schwarz inequality, and $(b)$ follows the fact that $(\eb_{i}^{\top}\Hb^{-1}_{n,\beta}\eb_{i})^2=\frac{1}{[(1-\beta\lambda_i)\lambda_i^2+\frac{\beta^2}{n}(\lambda_i^2+\operatorname{tr}(\bSigma
^2)\lambda_i)]^2}\leq 1/(1-\beta\lambda_i)^4$.
Therefore, for any unit $\vb\in\mathbb{R}^{d}$, we have
\begin{align*}
& \mathbb{E}[\|\vb^{\top}\Hb^{-\frac{1}{2}}_{n,\beta}(\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\bSigma (\mathbf{I}-\frac{\beta}{n}\Xb^{\top}\Xb)\Hb^{-\frac{1}{2}}_{n,\beta}\vb\|^2]\\
&\leq \max_{\vb} 210[(\vb^{\top}\Hb^{-1}_{n,\beta}\vb)^2\beta^4\operatorname{tr}(\bSigma^2
)^2+ 1 ]\leq 210\left( 1+\frac{\beta^4\operatorname{tr}(\bSigma^2)^2}{(1-\beta\lambda_1)^4}\right).
\end{align*}
\end{proof}
\end{document} | {"config": "arxiv", "file": "2206.09136/main.tex"} |
TITLE: Analyzing and decoding ciphertext
QUESTION [1 upvotes]: I have a worksheet which contains a dozen ciphertexts where the goal is to decrypt the encrypted English sentence(s). No information is given about what the text contains or what cipher methods are used.
I've reach the last question but am stuck on them and have spent several hours stuck on trying to figure out which cipher was used. I've tried to run frequency analysis on them and trial and error'd some different keys for things like Caesar shifts, but was unsuccessful.
So basically, I don't know how to start approaching these problems. I'm unable to identify which cipher was used and what its key is. Could you please help me here?
DGFMVXCRLCWMIDHDRLCHHDHVKLCAKYMAIHCAHEFIHZDRLHMUDRLFMVZSLIHLGGMZHLRLAHVCEEFFMVODEEMRLZYMQLFMVZDQQLKDCHLWZMSELQICAKGDAKFMVCZLZLCKFGMZUZLCHLZYXCEELAULI
Thank you in advance.
REPLY [2 votes]: The first is a substitution:
{'a': 'n', 'c': 'a', 'b': 'k', 'e': 'l', 'd': 'i', 'g': 'f', 'f': 'y', 'i': 's', 'h': 't', 'k': 'd', 'j': 'j', 'm': 'o', 'l': 'e', 'o': 'w', 'n': 'q', 'q': 'm', 'p': 'x', 's': 'b', 'r': 'v', 'u': 'g', 't': 'z', 'w': 'p', 'v': 'u', 'y': 'c', 'x': 'h', 'z': 'r'}
the cleartext is
"if you have a positive attitude and constantly strive to give your best effort eventually you will overcome your immediate problems and find you are ready for greater challenges"
I ran the ciphertext against a genetic algorithm thing I threw together until the "if you have a positive attitude" bit jumped out at me, at which point getting the rest was pretty easy by hand. If you're interested in the code see https://github.com/frrad/cipher/blob/master/tools.py (it's really hacky)
edit: updated code can usually decrypt the message by itself
edit2: If you google "cryptogram solver" you can find other tools online like http://www.blisstonia.com/software/WebDecrypto/index.php | {"set_name": "stack_exchange", "score": 1, "question_id": 712955} |
TITLE: Exercise on showing that a function with a jump discontinuity must have infinite energy
QUESTION [2 upvotes]: Let $u \in C(\bar B\setminus\{P,Q\})\cap C^1(B)$ where $B$ is the open unit ball in $\mathbb{R}^2$ centered at the origin and $P,Q$ are the points $(0,1),(0,-1)$. Suppose also that $u=1$ in the 'right' part of the boundary, i.e. $u=1$ in $\partial \bar B \cap \{(x,y) \in \mathbb{R}^2 \| \ x>0\}$ and $u=0$ in the 'left' part of the boundary, i.e. $u=0$ in $\partial \bar B \cap \{(x,y) \in \mathbb{R}^2 \| \ x<0\}$. Prove that
$$\int_B |\nabla u|^2 dxdy=+ \infty.$$
I had a similar exercise on a square and i managed to solve it, but I can't repeat the same argument for a circle. Any suggestions?
REPLY [2 votes]: It suffices to show that $\int_D |\nabla u|^2\, dx\, dy = +\infty$ for a small region $D$ around one of the discontinuities. By taking an appropriate smooth map, therefore, we can reformulate the problem in a more convenient way:
Reformulation: Let $H = \{(x, y) \in \mathbb{R}^2 : y \geq 0\}$ be the closed upper half-plane. Let $f: H \setminus \{(0, 0)\} \to \mathbb{R}$ be continuous, be differentiable in the interior of $H$, and satisfy $f(x, 0) = 1$ for $x < 0$ and $f(x, 0) = 0$ for $x > 0$. Let $D_\eta = \{(x, y) \in H: 0 < x^2 + y^2 < \eta^2\}$ be a half-disk of radius $\eta$ around the origin in $H$. Then $\int_{D_\eta} |\nabla f|^2\, dx\, dy = +\infty$ for any $\eta$.
Proof of the reformulation: Recast in polar coordinates: $$\int_{D_\eta} |\nabla f|^2\, dx\, dy = \int_0^\eta g(r)\, dr$$
where
$$g(r) = r \int_0^\pi |\nabla f(r \cos \theta, r \sin \theta)|^2\, d\theta = \int_{S_r} |\nabla f|^2\, ds$$
is the line integral of $|\nabla f|^2$ around a semicircular path $S_r$ of radius $r$. Note that $|\nabla f| \geq |\nabla f \cdot \hat{\theta}|$, where $\hat {\theta}$ is a counterclockwise-pointing unit vector, and that $\nabla f \cdot \hat \theta$ is the directional derivative of $f$ along the contour $S_r$, so $\int_{S_r} \nabla f \cdot \hat{\theta}\, ds = f(-r, 0) - f(r, 0) = 1$ by the Fundamental Theorem of Calculus. Therefore:
$$\begin{align*}
g(r) &= \int_{S_r} |\nabla f|^2\, ds \\
&\geq \frac{1}{\pi r} \left( \int_{S_r} |\nabla f|\,ds \right)^2 \, \tag{Cauchy-Schwarz} \\
&\geq \frac{1}{\pi r} \left( \int_{S_r} |\nabla f \cdot \hat{\theta}|\, ds \right)^2 \\
&\geq \frac{1}{\pi r} \left( \int_{S_r} (\nabla f \cdot \hat{\theta})\, ds \right)^2 \\
&= \frac{1}{\pi r}
\end{align*}$$
so $$\int_{D_\eta} |\nabla f|^2\, dx\, dy = \int_0^\eta g(r)\, dr \geq \int_0^\eta \frac{1}{\pi r}\, dr = +\infty.$$ | {"set_name": "stack_exchange", "score": 2, "question_id": 2626900} |
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\begin{abstract}
We study ``distance spheres'': the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is ``too dense'' and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be ``straightened'' into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.
\end{abstract}
\maketitle
\section{Introduction}
Let $K$ be an arbitrary set in $\RR^d$ and $r\geq 0$. The set of all points whose distance from $K$ is equal to $r$ forms a new set that we call a ``distance sphere'', and denote $S_K(r)$. (A precise definition is given below; in fact, we will focus our attention on the unit cube of $\RR^d$ rather than the whole space.)
If $K$ consists of a single point, then $S_K(r)$ is simply the sphere of radius $r$ centered on $K$. If $K$ is a general set, the distance spheres may be rather complicated objects, whose structure may change wildly as $r$ varies. Figures \ref{fig:finite} and \ref{fig:cantor} below depict some examples. These sets have been studied (under different names) by many authors, e.g., \cite{Brown, Ferry, Fu, VellisWu}.
This paper is concerned with the geometric structure of distance spheres, from a quantitative perspective. Our goal is to find large subsets of $\RR^d$ on which all the distance spheres can be simultaneously ``straightened out'' into (subsets of) parallel $(d-1)$-dimensional planes by a global mapping with controlled distortion. Moreover, we control the number of subsets and the distortion of the ``straightening map'' by constants that depend on the dimension $d$ but are otherwise independent of the set $K$.
In order to accomplish this, we must ``throw away'' some pieces of the domain on which we cannot straighten the distance spheres. These pieces come in two types: one a piece of small $d$-dimensional volume, and one the union of all locations where the set $K$ is ``too dense''. These are defined precisely below, and our main theorem is then stated as Theorem \ref{thm:main}.
The main tools in our arguments are the results of \cite{AzzamSchul} and \cite{DavidSchul} for general Lipschitz functions, combined with an analysis of the ``mapping content'' defined in \cite{AzzamSchul} in the special case of the distance function $\dist(\cdot, K)$.
\subsection{Main definitions and results}
\begin{definition}
Let $K\subseteq [0,1]^d$ be a set. For $r\geq 0$, the \textit{distance spheres} for $K$ are the sets
$$ S_K(r) = \{x\in [0,1]^d: \dist(x,K)=r\}.$$
\end{definition}
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{finite1.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{finite2.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{finite3.png}
\end{subfigure}
\caption{Examples of distance spheres $S_K(r)$ for a fixed finite set $K\subseteq [0,1]^2$ and three different values of $r$.}
\label{fig:finite}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{cantor01.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{cantor1.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{cantor2.png}
\end{subfigure}
\caption{Examples of distance spheres $S_K(r)$ for three different values of $r$ and a fixed set $K\subseteq [0,1]^2$ that is an approximation of a Cantor set.}
\label{fig:cantor}
\end{figure}
\begin{definition}
Let $K\subseteq [0,1]^d$ be a set. A set $E\subseteq [0,1]^d$ is called \textit{$K$-straightenable} if there is a bi-Lipschitz map $$g\colon\RR^d\rightarrow\RR^d$$
and an injective function
$$ \phi\colon \{r\geq 0: S_K(r)\cap E \neq \emptyset\} \rightarrow \RR$$
such that
\begin{equation}\label{eq:sc1}
g(S_K(r) \cap E) = \left(\{\phi(r)\} \times \RR^{d-1}\right) \cap g(E) \text{ for all } r\text{ such that } S_K(r) \cap E \neq \emptyset.
\end{equation}
\end{definition}
In other words, $g$ simultaneously ``straightens'' all the sets $S_K(r)\cap E$ into (subsets of) distinct vertical $(d-1)$-dimensional planes.
\begin{example}
If $K=\{\left(0,0\right)\}\subseteq [0,1]^2$, then the set
$$ E = \{ (x,y) \in [0,1]^2 : \frac{1}{2} \leq \sqrt{x^2+y^2} \leq 1 \}$$
is an example of a $K$-straightenable set. (See Figure \ref{fig:example}.) Since $K$ is a single point, the distance spheres $S_K(r)$ are simply arcs of circles. The map
$$ g(x,y) = (\sqrt{x^2+y^2}, \arctan(y/x)),$$
i.e., the map that converts rectangular to polar coordinates, straightens out the distance spheres $S_K(r) \cap E$ into distinct vertical line segments $\left(\{\phi(r)\}\times \RR\right) \cap g(E)$, where we simply take $\phi(r)=r$. One can show that $g$ is bi-Lipschitz on $E$ and extends to a bi-Lipschitz map from $\RR^2$ to $\RR^2$. Note that, while in this example $E$ is the closure of a simple open domain, we do not require this in general.
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{E1.png}
\caption{The set $E$ with a marked distance sphere (circle) $S_k(r)$ in red.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{E2.png}
\caption{The set $g(E)$ with straightened $g(S_k(r))$ in red.}
\end{subfigure}
\caption{A simple example of a straightenable set when $K$ is the one-point set $\{(0,0)\}$.}
\label{fig:example}
\end{figure}
\end{example}
\begin{definition}
Let $K\subseteq [0,1]^d$ be a set and $\epsilon>0$. We define
$$ \QQ(K,\epsilon) = \{ \text{ dyadic cubes } Q: N_{\epsilon\side(Q)}(K\cap Q) \supseteq Q\}$$
and
$$ D_\epsilon(K) := \cup_{Q\in\QQ(K,\epsilon)} Q.$$
Here $N_\eta(E)$ refers to the open $\eta$-neighborhood of a set $E$; see section \ref{sec:prelims}. In other words, $D_\epsilon(K)$ is the union of all dyadic cubes $Q$ in which $K\cap Q$ is $\epsilon\side(Q)$-dense.
\end{definition}
\begin{theorem}\label{thm:main}
Let $K\subseteq [0,1]^d$ be a set and $\epsilon>0$. Then we can write
$$ [0,1]^d = E_1 \cup \dots \cup E_M \cup D_{\epsilon}(K) \cup G,$$
where each $E_i$ is $K$-straightenable and $|G|<\epsilon$.
Moreover, the number of straightenable sets $M$ and the associated bi-Lipschitz constants depend only on $\epsilon$ and $d$. In particular, they do not depend on the set $K$.
\end{theorem}
In this result, $|G|$ refers to the $d$-dimensional volume (Lebesgue measure) of the set $G$; see section \ref{sec:prelims} for notation.
We emphasize that a large part of our interest in Theorem \ref{thm:main} lies in the fact that, in our decomposition, the number of straightenable sets and their associated constants are independent of the starting set $K$.
While Theorem \ref{thm:main} applies to arbitrary sets $K\subseteq [0,1]^d$, we also prove a stronger corollary for a specific class of sets known as \textit{porous sets}. A set $K\subseteq \RR^d$ is \textit{porous} if there is a constant $c>0$ such that, for each $r>0$ and $p\in \RR^d$, the ball $B(p,r)$ contains a ball $B(q,cr)$ that is disjoint from $K$. Many classical fractals, such as the Cantor set and Sierpi\'nski carpet, are porous. More discussion of porous sets can be found, e.g., in \cite[Ch. 5]{TysonMackay}.
If $K$ is a porous set, then we can decompose the entirety of $[0,1]^d$, outside of a set of small measure, into $K$-straightenable sets:
\begin{corollary}\label{cor:porous}
Let $K\subseteq [0,1]^d$ be a porous set with constant $c$. Let $0<\epsilon<c/2$. Then we can write
$$ [0,1]^d = E_1 \cup \dots \cup E_M \cup G,$$
where each $E_i$ is $K$-straightenable and $|G|<\epsilon$.
The number of straightenable sets $M$ and the associated bi-Lipschitz constants depend only on $\epsilon$ and $d$, and not on the set $K$.
\end{corollary}
\subsection*{Acknowledgments}
The first named author would like to thank Raanan Schul for helpful conversations at an early state of this project.
\section{Notation and preliminaries}\label{sec:prelims}
\subsection{Basics}
We use the following basic definitions. A function $f$ from a metric space $(X,d_X)$ to a metric space $(Y,d_Y)$ is called \textit{Lipschitz} (or \textit{$L$-Lipschitz} to emphasize the constant) if there is a constant $L$ such that
$$ d_Y(f(x), f(x')) \leq Ld_X(x,x') \text{ for all } x,x'\in X.$$
It is called \textit{bi-Lipschitz} (or \textit{$L$-bi-Lipschitz}) if
$$ L^{-1}d_X(x, x') \leq d_Y(f(x), f(x')) \leq Ld_X(x,x') \text{ for all } x,x'\in X.$$
We use $B(x,r)$ to denote an open ball of radius $r$ centered at $x$ in a metric space, and $\overline{B}(x,r)$ for the corresponding closed ball.
The distance from a point $p$ to a set $K$ in $\RR^d$ is defined as
$$ \dist(p, K) := \inf\{|p-q| : q\in K\}.$$
If $K$ is a set in $\RR^d$ and $\eta>0$, then $N_\eta(K)$ is the open $\eta$-neighborhood of $K$, defined as
$$ N_\eta(K) = \{p\in \RR^d : \dist(p,K)<\eta.\}$$
In $\RR^d$, we will also use the collection of \textit{dyadic cubes}. These consist of all cubes $Q$ in $\RR^d$ of the form
$$ [a_1 2^n, (a_1+1)2^n] \times \dots \times [a_d 2^n, (a_d+1)2^n], $$
where $a_1, \dots, a_d$ and $n$ are integers.
\subsection{Measure, Hausdorff content, and mapping content}
We use $|E|$ to denote the $d$-dimensional volume (Lebesgue measure) of a set in $\RR^d$.
\begin{definition}
Let $E$ be a subset of a metric space $X$, and $k\geq 0$. The \textit{$k$-dimensional Hausdorff content of $E$} is defined by
$$ \HH^k_\infty(E) = \inf_{\mathcal{B}} \sum_{B\in\mathcal{B}} \diam(B)^k,$$
where the infimum is taken over all finite or countable collections of closed balls $\mathcal{B}$ whose union contains $E$.
\end{definition}
The following definition appears first in \cite{AzzamSchul}.
\begin{definition}
Let $f\colon [0,1]^{n+m}\rightarrow Y$ be a function into a metric space, and let $A\subseteq [0,1]^{n+m}$. The \textit{$(n,m)$-mapping content of $f$ on $A$} is:
$$ \HH^{n,m}_\infty(f,A) = \inf_{\mathcal{Q}} \sum_{Q\in \mathcal{Q}} \HH^n_\infty(f(Q))\side(Q)^m,$$
where the infimum is taken over all collections of dyadic cubes $\mathcal{Q}$ in $[0,1]^{n+m}$ whose union contains $A$.
\end{definition}
\subsection{Hard Sard sets}
The following definition was first introduced in \cite{AzzamSchul}. We present the slightly altered version from \cite[Definition 1.3]{DavidSchul}.
\begin{definition}\label{def:HSpair}
Let $n,m\geq 0$. Let $E\subseteq Q_0=[0,1]^{n+m}$ and $f\colon Q_0 \rightarrow X$ a Lipschitz mapping into a metric space.
We call $E$ a \textbf{Hard Sard set for $f$} if there is a constant $C_{Lip}$ and a $C_{Lip}$-bi-Lipschitz mapping $g\colon \RR^d \rightarrow \RR^d$ such that the following conditions hold.
Write $\RR^{n+m}=\RR^n \times \RR^m$ in the standard way, and points of $\RR^{n+m}$ as $(x,y)$ with $x\in \RR^n$ and $y\in \RR^m$. Let $F = f \circ g^{-1}$.
We ask that:
\begin{enumerate}[(i)]
\item\label{HS3} If $(x,y)$ and $(x',y')$ are in $g(E)$, then $F(x,y) = F(x',y')$ if and only if $x=x'$. Equivalently,
$$ F^{-1}(F(x,y)) \cap g(E) = (\{x\} \times \RR^m) \cap g(E)$$
\item\label{HS4} The map
$$(x,y) \mapsto (F(x,y),y)$$
is $C_{Lip}$-bi-Lipschitz on the set $g(E)$.
\end{enumerate}
\end{definition}
Only condition (i) of the definition of Hard Sard set will play a role in this paper.
A slightly simplified version of the main theorem of \cite{DavidSchul} is the following:
\begin{theorem}\label{thm:davidschul}
Let $Q_0$ be the unit cube in $\RR^{n+m}$ and let $f\colon Q_0\rightarrow \RR^n$ be a $1$-Lipschitz map.
Given any $\gamma>0$, we can write
$$ Q_0 = E_1 \cup \dots \cup E_M \cup G,$$
where $E_i$ are Hard Sard sets and
$$ \HH^{n,m}_\infty(f,G) < \gamma.$$
The constant $M$ and the constants $C_{Lip}$ associated to the Hard Sard sets $E_i$ depend only on $n$, $m$, and $\gamma$.
\end{theorem}
\section{Lemmas}
\begin{lemma}\label{lem:distancefunction}
If $K$ is any set in $\RR^d$, the function
$$ f(x) = \dist(x,K)$$
is $1$-Lipschitz.
\end{lemma}
\begin{proof}
Let $x,y \in \mathbb{R}^d$, and let $K \subseteq \mathbb{R}^d$. Without loss of generality, assume $f(x) \geq f(y)$. Let $z_y$ be a point in the closure of $K$ such that $\inf\{|y-z|:z \in K\} = |y-z_y|.$ Then $f(y) = \dist(y,K) = |y-z_y|$. Then applying the triangle inequality, we have $$\dist(x,K) = \inf\{|x-z|:z \in K\} \leq |x-z_y| \leq |x-y| + |y-z_y| = |x-y| + \dist(y,K).$$ Then $$\dist(x,K) - \dist(y,K) \leq |x-y|.$$ Thus, since $f(x) \geq f(y)$, $$|f(x) - f(y)| = |\dist(x,K) - \dist(y,K)| = \dist(x,K) - \dist(y,K) \leq |x-y|,$$ and so $f(x) = \dist(x,K)$ is $1$-Lipschitz.
\end{proof}
\begin{lemma}\label{lem:intervalcontent}
If $[a,b]$ is a compact interval in $\RR$, then $\HH^1_\infty([a,b])=b-a$.
\end{lemma}
\begin{proof}
Notice that a closed ball in $\RR$ is just a closed interval $[a_{i},b_{i}]$. Then for an interval $[a,b]$, we have
$$\overline{B}\left(\frac{a+b}{2},\frac{b-a}{2}\right)=[a,b],$$
which implies $\HH^1_ \infty([a,b])\le \diam(\overline{B}(\frac{a+b}{2},\frac{b-a}{2}))=b-a$.
Now let $\{\overline{B}_{i} = [a_i,b_i]\}$ be a collection of closed balls that cover the interval $[a,b]$. Then
$$\displaystyle \sum_{i}\diam(\overline{B}_{i})=\displaystyle \sum_{i}\diam([a_{i},b_{i}])\ge b-a,$$
where the inequality is a basic fact in measure theory. Taking the infimum of both sides we get $\HH^1_ \infty([a,b])\ge b-a$. Hence, $\HH^1_ \infty([a,b])=b-a$, as desired.
\end{proof}
Now fix $K\subseteq [0,1]^d$. Let $f(x) = \dist(x,K)$.
\begin{lemma}\label{lem:segment}
Let $x\in [0,1]^d$ and $z\in \overline{K}$ such that
$$ f(x) = |z-x|.$$
If $y$ is a point on the line segment from $x$ to $z$, then
$$ |f(y)-f(x)| = |y-x|$$
\end{lemma}
\begin{proof}
By Lemma \ref{lem:distancefunction}, we know $f(x)=\dist(x,K)$ is 1-Lipschitz. Then, $|f(x)-f(y)|\le |x-y|$. However,
$$|f(x)-f(y)|=|\dist(x,K)-\dist(y,K)|=\dist(x,K)-\dist(y,K)\ge |x-z|-|y-z|,$$
as $\dist(y,K)=\inf\{|y-z|:z\in K\}$. Then,
$$\dist(x,z)-\dist(y,z)=|x-z|-|y-z|=|x-y|,$$
as $y$ is on the line segment from $x$ to $z$. Thus, $|f(x)-f(y)|=|f(y)-f(x)|=|y-x|.$
\end{proof}
\begin{lemma} \label{cubelemma}
Let $\delta>0$ and let $Q$ be a dyadic cube in $\RR^d$ such that
$$ \HH^1_\infty(f(Q)) < \delta \side(Q).$$
Then $Q\in \QQ(K,c_d\delta)$, where $c_d=\sqrt{d}+1$.
\end{lemma}
\begin{proof}
Let $\delta>0$ and let $Q$ be a dyadic cube in $\RR^d$ such that $\HH^1_\infty(f(Q)) < \delta \side(Q)$. Let $Q' \subseteq Q$ be the set of points $x$ in $Q$ such that $\dist(x,\partial Q) \geq \delta\side(Q)$, where $\partial Q$ is the set of boundary points of $Q$.
\begin{claim}\label{claim:point} Let $x \in Q'$. Then there must be a point of $K$ inside the ball $B(x, \delta \side(Q))\subseteq Q$.
\end{claim}
\begin{proof}[Proof of Claim \ref{claim:point}] Let $x \in Q'$, and let $z'$ be a point in the closure of $K$ such that $$f(x) = \dist(x,K) = |x-z'|.$$
If $z'$ is not in $Q$, then let $S$ be the line segment from $z'$ to $x$, and let $y$ be the point on the boundary of $Q$ such that $y \in S$. Then by Lemma \ref{lem:segment}, $$|f(y) - f(x)| = |y - x| \geq \delta\side(Q).$$
Now since $Q$ is closed and bounded, it is compact. Also, since $Q$ is convex, it is connected. Then since $f(x) = \dist(x,K)$ is continuous, $f(Q) \subseteq \RR$ is also compact and connected. Then $f(Q) = [a,b]$ for some $a \leq b$. Then by Lemma \ref{lem:intervalcontent}, $$\HH^1_\infty(f(Q)) = \HH^1_\infty([a,b]) = b - a.$$
Then we have $$\HH^1_\infty(f(Q)) = b - a \geq |f(y) - f(x)| \geq \delta\side(Q).$$ This contradicts the assumption that $\HH^1_\infty(f(Q)) < \delta\side(Q)$. Thus it must be that $z'$ is in $Q$. Then suppose for the sake of contradiction that $z'$ is not contained in $B(x, \delta \side(Q))$. Then $$f(x) = \dist(x,K) = |x-z'| \geq \delta\side(Q),$$ which leads us to the same contradiction as above. Thus it must be that $z'$ is contained in $B(x, \delta\side(Q))$. Since $z'$ is in the closure of $K$, $B(x,\delta\side(Q))$ must contain a point of $K$.
\end{proof}
Thus for any $x \in Q'$, there is a point $z$ of $K$ inside $B(x, \delta \side(Q))$, and so $$|x-z| < \delta\side(Q) < c_d\delta\side(Q)$$.
Now consider $x \in Q$ such that $x \not\in Q'$. Then there is some $x' \in Q'$ such that $|x-x'|\leq \sqrt{d}\delta\side(Q).$ Since $x' \in Q'$, there is some $z \in K$ such that $z \in B(x', \delta\side(Q))$. Then
$$|x-z| \leq |x-x'| + |x'-z|< \sqrt{d}\delta\side(Q) + \delta\side(Q) < c_d\delta\side(Q).$$
Thus for any $x \in Q$, there exists $z \in K \cap Q$ such that $|x-z| < c_d\delta\side(Q)$, and so $Q \in \QQ(K,c_d\delta)$.
\end{proof}
The last lemma concerns the concept of mapping content $\HH^{n,m}_\infty$ defined above.
\begin{lemma} \label{rewritelemma}
Let $f:Q_0 \rightarrow X$ be $1$-Lipschitz and $n,m\geq 1$. Let $A\subseteq Q_0$ and suppose
$$ \HH^{n,m}_\infty(f,A) < \delta$$
Then we can write
$$ A \subseteq A' \cup \bigcup_{i} Q_i,$$
where
\begin{enumerate}[(i)]
\item $|A'|< \sqrt{\delta}$,
\item $Q_i$ are dyadic cubes,
\item $\HH^n_\infty(f(Q_i)) < \sqrt{\delta}\side(Q_i)^n$ for each $i$.
\end{enumerate}
\end{lemma}
\begin{proof}
We have $$\HH^{n,m}_\infty(f,A) = \inf_{\mathcal{Q}} \sum_{Q\in \mathcal{Q}} \HH^n_\infty(f(Q))\side(Q)^m < \delta,$$ where the infimum is taken over all collections of dyadic cubes $\mathcal{Q}$ in $Q_0$ whose union contains $A$. By definition of infimum, there exists a collection of dyadic cubes $\mathcal{R} = \{R_j\}_{j\in J}$, whose union contains $A$, such that
\begin{equation}\label{eq:Rj}
\sum_{j\in J}\HH^n_\infty(f(R_j))\side(R_j)^m < \delta.
\end{equation}
We split these cubes $R_j$ into two collections:
$$\mathcal{R}^1 = \{R_j \in \mathcal{R} : \HH^n_\infty(f(R_j)) < \sqrt{\delta}\side(R_j)^n\},$$
and
$$\mathcal{R}^2 = \{R_j \in \mathcal{R} : \HH^n_\infty(f(R_j)) \geq \sqrt{\delta}\side(R_j)^n\}.$$ $\mathcal{R}^1$ will become our collection of dyadic cubes $\{Q_i\}$. The union of cubes in $\mathcal{R}^2$ will be our set $A'$, so we want to show that $\Big|\bigcup_{R_j \in R^2}R_j\Big| < \sqrt{\delta}$.
Let $J_2 := \{j \in J : R_j \in \mathcal{R}^2\}$. Then, using \eqref{eq:Rj}, we have $$\delta > \sum_{j \in J}\HH^n_\infty(f(R_j))\side(R_j)^m \geq \sum_{j \in J_2}\HH^n_\infty(f(R_j))\side(R_j)^m.$$ Then by the definition of our set $\mathcal{R}^2$, we have $$\delta > \sum_{j \in J_2}\HH^n_\infty(f(R_j))\side(R_j)^m \geq \sum_{j \in J_2}\sqrt{\delta}\side(R_j)^n \side(R_j)^m = \sqrt{\delta}\sum_{j \in J_2}|R_j| \geq \sqrt{\delta}\Big|\bigcup_{j \in J_2}R_j\Big|.$$
Thus we have $$\Big|\bigcup_{j \in J_2}R_j\Big| < \sqrt{\delta}.$$ Therefore, if we define $A'$ to be the union of the cubes in $\mathcal{R}^2$ and define $\{Q_i\}$ to be the collection of cubes in $\mathcal{R}^1$, then we can write $$A \subseteq A' \cup \bigcup_i Q_i,$$ where properties (i)-(iii) hold for $A'$ and each $Q_i$.
\end{proof}
\section{Proofs of the main results}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Take $K\subseteq[0,1]^{d}$ and $\epsilon>0$. Let $f(x)=\dist({x,K})$, which is $1$-Lipschitz by Lemma \ref{lem:distancefunction}. Applying Theorem \ref{thm:davidschul} to $f$ with $n=1$, $m=d-1$, and $\gamma=\frac{\epsilon^2}{c_d^2}$ (where $c_d=\sqrt{d}+1$ as in Lemma \ref{cubelemma}) we have that $$[0,1]^{d}=E_{1}\cup...\cup E_{M}\cup G_{0},$$
where $E_{i}$ are Hard Sard sets for $f$ and $\HH^{1,d-1}_\infty(f,G_{0})<\frac{\epsilon^2}{c_d^2}$.
The following two claims combine to complete the proof of Theorem \ref{thm:main}.
\begin{claim}\label{claim:straight}
The Hard Sard sets $E_{i}$ are $K$-straightenable sets.
\end{claim}
\begin{proof}[Proof of Claim \ref{claim:straight}]
Throughout this proof, we write points of $\RR^d$ as $(x,y)$, where $x\in \RR$ and $y\in\RR^{d-1}$. Let $E=E_i$ for some $i\in\{1, \dots, M\}$.
By Definition \ref{def:HSpair}, there is a bi-Lipschitz map $g:\RR^d \to \RR^d$ such that if $F=f\circ g^{-1}$, then for $(x,y),(x^{'},y^{'})\in g(E)$, $F(x,y)=F(x^{'},y^{'})$ if and only if $x=x^{'}$.
Now, take any $r$ such that $S_{K}(r)$ intersects $E$ and consider any point $p\in S_{K}(r)\cap g(E)$. Then $g(p)=(x,y)$, and if any other point $q\in S_{k}(r)\cap g(E)$, then $f(p)=f(q)=r$, which implies that $F(g(q))=F(g(p))$. Then, $g(q)\in (\{x\}\times\RR^{d-1})\cap g(E)$. Hence, $g(S_{K}(r))\cap g(E)\subseteq (\{x\}\times\RR^{d-1})\cap g(E)$.
Now take any point $(x,y')\in (\{x\}\times\RR^{d-1})\cap g(E)$, where $(x,y^{'})=g(p')$, for some $p'\in E$. Then $F(x,y')=F(x,y)=F(g(p))=f(p)=r=f(p')=\dist(p^{'},K)$. Thus $p'\in S_{K}(r)\cap g(E)$, and it follows that $(x,y')=g(p')\in g(S_{K}(r))\cap g(E)$. Thus $(\{x\}\times\RR^{d-1})\cap g(E)\subseteq g(S_{K}(r))\cap g(E)$, yielding the desired equality.
Lastly, define $$\phi:\{r\ge0:S_{K}(r)\cup E\neq \emptyset\}\to \RR$$
so that $\phi(r)$ is equal to the first coordinate $x$ of all points $(x,y)\in g(S_{K}(r)\cap E)$. (Note that all such points share a common first coordinate by our work above.)
By definition, $g(S_{K}(r)\cap E)=(\{\phi(r)\}\times\RR^{d-1})\cap g(E)$. Now suppose $\phi(r)=\phi(r^{'})$. Then there are points $p\in S_{k}(r)\cap E$ and $p^{'}\in S_{K}(r^{'})\cap E$ such that $g(p)=(x,y)=g(p^{'})$. Thus $F(g(p))=F(g(p^{'}))$, which implies that $f(p)=f(p^{'})$, and therefore $r=r^{'}$. Hence, $\phi$ is injective.
\end{proof}
\begin{claim}\label{claim:garbage}
The set $G_0$ is contained in $G\cup D_\epsilon(K)$, where $G$ is a subset of $[0,1]^d$ with $|G|<\epsilon$.
\end{claim}
\begin{proof}[Proof of Claim \ref{claim:garbage}]
Applying Lemma \ref{rewritelemma}, we can write $$G_0 \subseteq G \cup \bigcup_i Q_i,$$ where $|G| < \frac{\epsilon}{c_d}$ and $Q_i$ are dyadic cubes with $\HH_\infty^1(f(Q_i)) < \frac{\epsilon}{c_d} \side (Q_i)$ for each $i$. Then by Lemma \ref{cubelemma}, we have $Q_i \in \QQ(K,\epsilon)$ for every $i$, and so $Q_i \in \QQ(K,\epsilon)$ for all $i$. Then we have $$\bigcup_i Q_i \subseteq D_\epsilon(K),$$ and so $$G_0 \subseteq G \cup \bigcup_i Q_i \subseteq G \cup D_\epsilon(K),$$ where $|G| < \frac{\epsilon}{c_d} < \epsilon$
\end{proof}
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:porous}]
Let $K \subseteq [0,1]^d$ be a porous set with constant $c$, and let $0 < \epsilon < c/2$. Let $Q$ be any dyadic cube in $[0,1]^d$. Let $p$ be the point in the center of $Q$, and let $r := \frac{1}{2}\side(Q)$. Consider the ball $B(p,r) \subseteq Q$. Since $K$ is porous, there exists some point $q$ in $B(p,r)$ such that $$B(q,cr) \subseteq B(p,r)$$ and $$B(q,c r) \cap K = \emptyset.$$ Then for every $z \in K$, $$|q-z| \geq cr = \frac{c}{2}\side(Q) > \epsilon\side(Q).$$ Thus $Q \notin \QQ(K,\epsilon)$. Since this is true for every dyadic cube in $[0,1]^d$, $$\QQ(K,\epsilon) = \emptyset,$$ and so $$D_\epsilon(K) = \emptyset.$$ Then by Theorem \ref{thm:main}, we can write $$[0,1]^d = E_1 \cup ...\cup E_M \cup D_\epsilon (K) \cup G = E_1 \cup ...\cup E_M \cup G,$$ where each $E_i$ is $K$-straightenable, $|G|<\epsilon$ and the number of straightenable sets $M$ and the associated bi-Lipschitz constants depend only on $\epsilon$ and $d$, and not on the set $K$.
\end{proof}
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\bibliography{distancespherebib}
\end{document} | {"config": "arxiv", "file": "2107.09634/main.tex"} |
TITLE: A linear transformation can be decomposed as a sum of two invertible tranformations
QUESTION [1 upvotes]: Prove that if $\text{Char}(K) \neq 2$ and $V$ is finite dimensional, then $T \in \text{Hom}_{K}(V,V)$ can be expressed as the sum of two invertible linear transformations.
Proof: Choose $\lambda \neq 0 \in K$ such that $S = T - \lambda I$ is invertible. Then $T = S + \lambda I$ with both $S$ and $\lambda I$ invertible linear transformations.
P.S: I have seen the same claim made for matrices previously and the solution I saw broke into cases such as assuming first the matrix A is invertible and then considering the sum involving division by two and then the other case used writing A in a block form that would make it easier. But my one line proof above seems to work nicely though or am I making an obvious mistake?
REPLY [1 votes]: As pointed out by a comment, your proof is wrong because $\lambda$ may not exist. For instance, suppose $V=GF(2)=\{0,1\}$ and $T=I$ is the identity map. Now $T$ is the only invertible linear map on $V$, but $T+T=0\ne T$.
To prove the proposition, you may consider first the special case where $V$ is two-dimensional, $\{x_1,x_2\},\{y_1,y_2\}$ are two ordered bases of $V$ and $T$ is a non-invertible linear transformation such that
$$
\begin{cases}
T(x_1)=0,\\
T(x_2)=y_2.
\end{cases}
$$
Try to write $T$ as a sum of two invertible linear transformations in this case. Then try to generalise this construction for a general $V$ and a general $T$ with a general (and possibly zero) nullspace. | {"set_name": "stack_exchange", "score": 1, "question_id": 2048328} |
\chapter{Conclusion}
\label{chap:conclusion}
While we have gone a long way in establishing the properties of toposes of the form $\Cont(M,\tau)$ and their canonical representatives in this thesis, it is clear that there are multiple avenues for future exploration of the subject.
\section{Discrete monoids}
\label{sec:disc2}
\subsection{Further properties}
\label{ssec:further}
In Chapter \ref{chap:mpatti}, we saw many instances of how properties of the global sections morphism of a topos of the form $\Setswith{M}$ are reflected as Morita-invariant properties of $M$. Despite the rich variety that arose just from considering the global sections morphism, there are many other sources of such properties; we summarize some ideas and examples of these here:
\begin{itemize}
\item \textit{Diagonal properties}: Since the (2-)category of toposes and geometric morphisms has pullbacks, any geometric morphism $\Fcal \to \Ecal$ induces a diagonal $\Fcal \to \Fcal \times_{\Ecal} \Fcal$. We may in particular apply this to the global sections morphism, to express properties such as \textit{separatedness} of a topos (cf. \cite[Definition C3.2.12(b)]{Ele}). However, a more detailed understanding of geometric morphisms between toposes of the form $\Setswith{M}$ is needed to analyze these.
\item \textit{Relative properties}: Some properties of (Grothendieck) toposes are most succinctly expressed by the existence of geometric morphisms of a particular type to or from toposes with certain properties, as we saw in the example of \'{e}tendues at the end of Chapter \ref{chap:mpatti}.
\item \textit{Categorical properties}: There are some categorical properties of Grothendieck toposes that ostensibly aren't expressible in terms of the global sections morphism in a straightforward way, although they might be expressible in the relative sense above. This includes the property of there being a separating set of objects with a particular property, as we saw in the example of locally decidable toposes, also at the end of Chapter \ref{chap:mpatti}.
\item \textit{Internal logic properties}: As a variant of the preceding point, the internal logic of a topos is determined by the structure of subobject lattices, and so is embodied in the structure of their subobject classifiers. As we saw in Chapter \ref{chap:mpatti}, the structure of the subobject classifier in $\Setswith{M}$ corresponds to the structure of the right ideals of $M$, but we have only tackled the most basic cases in which $\Setswith{M}$ is a Boolean or de Morgan topos. There are surely further algebraic or logical properties of this lattice to investigate.
\end{itemize}
Each of these classes merits a systematic study in its own right. In the other direction, there are some notable elementary properties of monoids which we have not yet found a topos-theoretic equivalent for. The most basic is the left Ore condition, dual to Definition \ref{dfn:rOre}; of course, we could simply examine the category of \textit{left} actions of our monoid, and dualize the results presented in Chapter \ref{chap:mpatti}, but we believe it will be more informative to seek a condition intrinsic to the topos of right actions, given the variety of equivalent conditions we reached in Theorem \ref{thm:deMorgan}.
\subsection{Other presentations}
\label{ssec:other}
A related thread for future investigation is that of extending the dictionary of properties between different \textit{presentations} of toposes. We saw an instance of this realized in the comparison of toposes of the form $\Setswith{M}$ with toposes of sheaves on spaces in Chapter \ref{chap:mpatti}.
There are three types of presentation in particular that spring to mind:
\begin{itemize}
\item \textit{Syntactic presentations}: We saw how toposes can be build from geometric theories in Chapter \ref{chap:logic}. How do syntactic and model-theoretic properties translate into monoid properties?
\item \textit{Site presentations}: We know that we can use sites to present toposes, and we can further restrict to principal or finitely generated sites of Chapter \ref{chap:sgt}. How do properties of the underlying category or the Grothendieck topology translate into monoid properties?
\item \textit{Groupoid presentations}: Any Grothendieck topos can be presented as a topos of equivariant actions of a localic groupoid, and this can be chosen to be a topological monoid when the topos has enough points, cf. the work of Butz and Moerdijk, \cite{Grpoid}. Indeed, Jens Hemelaer showed in \cite{TGRM} that a topos of actions of a discrete monoid can presented as the topos of equivariant sheaves on a \textit{posetal groupoid}. Such groupoids possess both algebraic and spatial properties which might profitably be transferred to monoid properties.
\end{itemize}
A natural approach to all of these comparisons as well as those above is Caramello's `toposes as bridges' principle \cite{TST}: by finding a way to translate a property of a given presentation into an invariant of the corresponding topos, we may subsequently translate it into an invariant of each of the other presentations.
\subsection{Relativization}
\label{ssec:relativize}
Recall that in Theorem \ref{thm:2equiv0}, we demonstrated an equivalence between the 2-category of monoids, semigroup homomorphisms and conjugations, and the 2-category of their presheaf toposes, essential geometric morphisms between these and geometric transformations. This means that we can just as systematically explore how properties of semigroup or monoid homomorphisms are reflected as properties of essential geometric morphisms between toposes of the form $\Setswith{M}$. This is a direct extension, or `relativization to a different base monoid,' of the work we did in Chapter \ref{chap:mpatti}, since the unique homomorphism $M \to 1$ corresponds under this equivalence to the global sections morphism of $\Setswith{M}$. More generally, we will be able to use the biequivalence of Theorem \ref{thm:2equiv1} to compare not-necessarily-essential geometric morphisms with biactions, as we did in Scholium \ref{schl:tidiness}. Since such tensor-hom expressions exist for geometric morphisms between presheaf toposes more generally (see \cite[Section VII.2]{MLM}), toposes of monoid actions may provide a good context from which to build an algebraic analysis of geometric morphisms.
This idea has already yielded some success: in further joint work with Jens Hemelaer \cite{NonLC}, we obtained an instance of a geometric morphism induced by a monoid homomorphism which is hyperconnected, essential and local but not locally connected, providing a counterexample for an open problem posed by Thomas Streicher.
A final direction for future investigation, related to the above, is relativization in the usual topos-theory sense of considering (elementary) toposes over a base topos other than $\Set$. Amongst internal categories in arbitrary toposes (cf.\ Section \ref{ssec:proper} of Chapter \ref{chap:sgt}), monoids are naturally defined as those whose object of objects is the terminal object. Accordingly, one might be interested in examining toposes of internal right actions of internal monoids relative to a topos other than $\Set$. While many of the results we obtained in Chapter \ref{chap:mpatti} were arrived at constructively or are expressed in a way that relativizes directly, there are some which cannot be transferred directly into an arbitrary topos. For instance, our inductive construction of the submonoid right-weakly generated by $S$ in Lemma \ref{lem:construct} requires the presence of a natural number object, while the application of Proposition \ref{prop:cc2} in Theorem \ref{thm:labsorb} relies on the law of excluded middle. More significantly, the proof that condition \ref{item:rab6} implies condition \ref{item:rab7} in Theorem \ref{thm:rabsorb} explicitly relies on a form of the axiom of choice. This investigation will therefore be non-trivial, and it will be interesting to discover the relative analogues of the results presented in this thesis.
\section{Supercompactly generated toposes}
\label{sec:sgt2}
The theory-laden middle chapters contained results in various possible directions, which already hint at directions for possible future exploration. We highlight two in particular which we spent some time investigating in the course of research for those chapters, but which didn't make it into this thesis.
\subsection{Points}
\label{ssec:pts2}
A result which appeared in the original preprint version (\cite{SGT}) of Chapter \ref{chap:sgt}, but which had to be removed due to an error in its proof, was a generalization of Deligne's classical completeness theorem for coherent toposes to the class of compactly generated toposes. This result was not needed for the purposes of the thesis, since all of our toposes have enough points constructively, but the question remains whether Deligne's original proof, \cite{SGA4Coh}, can be extended to the setting of sites without pullbacks.
Determining this will be no easy task, not least because there are various results of a similar nature, such as \cite[Theorem 6.2.4]{MakkaiReyes} which demonstrates a completeness theorem for toposes of sheaves on sites which are countable in a suitable sense (albeit still with pullbacks), which are proved by totally different means.
\subsection{Reductive logic}
\label{ssec:redlogic}
We discussed the usual account of categorical logic as it applies to toposes in Chapter \ref{chap:logic}. However, this approach does not seem the ideal fit for analyzing toposes whose canonical sites do not have finite limits. We anticipate that there is scope for developing branches of logic which admit interpretations in reductive and coalescent categories.
This too will be more of a challenge than it first appears, because pullbacks and equalizers were fundamental in the interpretations of even the most basic formulae appearing in Section \ref{ssec:geomsem}. The logics for these categories will therefore not have terms in the sense that first order theories do, which already distances it from classical logic.
\subsection{Exactness Properties}
\label{ssec:effects}
In Proposition \ref{prop:effective} and Example \ref{xmpl:countable}, we explored the relationship between the conditions of effectualness on reductive and coalescent categories introduced in Definition \ref{dfn:redeff} and the more familiar notion of effectiveness recalled in Definition \ref{dfn:regeff}. More generally, one might wonder how this relates to the general notion of \textit{exactness}, which refers to the types of interaction between classes of colimits and pullbacks which occurs in Grothendieck toposes. This notion has been explored in situations admitting finite limits (including the analogous concepts for enriched categories) in \cite{GarnerLack}. It is reasonable to hope that an examination of exactness properties resembling effectualness for sites which lack finite limits might enable one to refine this notion into one which makes sense outside of the `lex' context. This may be as simple as replacing Garner and Lack's finite limit conditions with suitable flatness conditions, although the precise nature of these conditions is not clear \textit{a priori}.
\section{Topological monoids}
\label{sec:topmonoid}
There are some natural questions which arose during the developments of Chapter \ref{chap:TTMA} that we were unable to resolve. We record them here and suggest some future directions this research might proceed, independently of the content of Chapter \ref{chap:TSGT}.
\subsection{Pathological powder monoids}
\label{ssec:moremonoid}
The reader may have noticed in Chapter \ref{chap:sgt} that we did not exhibit any examples illustrating the asymmetry in the definition of powder monoids. This is because our main classes of examples, prodiscrete monoids and powder groups, are both blind to this distinction, since their definitions are stable under dualizing. Similarly, any commutative right powder monoid is also a left powder monoid. These cases make constructing examples of right powder monoids which are not left powder monoids difficult. Nonetheless, we posit that:
\begin{conj}
\label{conj:powdery}
There exists a right powder monoid which is not a left powder monoid.
\end{conj}
Scholium \ref{schl:G3'} puts some limits on Conjecture \ref{conj:powdery}, since it says that any right powder monoid is at most one step away from also being a left powder monoid. In particular, we never get an infinite nested sequence of topologies on a monoid by repeatedly computing the associated right and left action topologies. We have not demonstrated comparable results for complete monoids, but we expect them to hold:
\begin{conj}
\label{conj:complete}
The right completion of a left powder monoid or left complete monoid retains the respective property, and dually for left completions of right powder monoids or right complete monoids. However, we expect that there exists a right complete monoid which is not a left powder monoid.
\end{conj}
Another way of expressing Conjectures \ref{conj:powdery} and \ref{conj:complete} is to say that we expect the diagram of monadic full and faithful functors \eqref{eq:monadic2} to extend as follows:
\[\begin{tikzcd}
& & \ar[dl] \mathrm{CMon}_s & & \\
& \ar[dl] \mathrm{PMon}_s & &
\ar[ul] \ar[dl] \mathrm{CP'Mon}_s & \\
\mathrm{T_0Mon}_s & &
\ar[dl] \ar[ul] \mathrm{P''Mon}_s & &
\ar[ul] \ar[dl] \mathrm{C''Mon}_s,\\
& \ar[ul] \mathrm{P'Mon}_s & &
\ar[dl] \ar[ul] \mathrm{C'PMon}_s \\
& & \ar[ul] \mathrm{C'Mon}_s & &
\end{tikzcd}\]
where the notation is the intuitive extension of that employed in \eqref{eq:monadic2} and each inclusion represented is non-trivial.
\subsection{Finitely generated complete monoids}
\label{ssec:fgcompmon}
Besides these conjectures characterizing pathological examples, there is plenty of ground still to cover in understanding these classes of monoids. What does a `generic' complete monoid look like, beyond what was shown in Chapter \ref{chap:TTMA}? Is it possible to classify them?
For example, given an element $x$ in a complete monoid, we may consider the closure of the submonoid generated by $x$, which by Corollary \ref{crly:closed} is a complete monoid. One might consider this an instance of a `complete monoid generated by one element'\footnote{We include the quote marks to emphasize that this submonoid is not generated by $x$ in an algebraic sense.}. We can identify such monoids as the canonical representatives of toposes admitting a hyperconnected morphism from $\Setswith{\Nbb}$. By Corollary \ref{crly:prodisc} these are commutative prodiscrete monoids. Analogously, `finitely generated complete monoids' would correspond to complete monoids representing toposes admitting hyperconnected geometric morphisms from the toposes $\Setswith{F_{n}}$, where $F_n$ is the free (discrete) monoid on $n$ elements. By Proposition \ref{prop:prince2} they correspond to filters of right congruences on $F_n$, which we expect to have a tame classification. Is it possible to identify the `finitely presented complete monoids' amongst these? One could go on to investigate the properties of various ($2$-)categories of such monoids, taking advantage of results such as such as those in Section \ref{ssec:monads}. Future applications of the theory developed in Chapter \ref{chap:TSGT} may rely on understanding these answers to these questions.
\subsection{Invariant properties}
\label{ssec:mitopmon}
In investigating complete monoids, it will be desirable to extend the results of Chapter \ref{chap:mpatti} to the topological case. In this regard, we can already glean some positive results. Whilst we saw in Example \ref{xmpl:notpro2} that a complete monoid Morita-equivalent to a topological group need not be a group, we have the next best result.
\begin{prop}
\label{prop:densegroup}
Let $(M,\tau)$ be a topological monoid. The following are equivalent:
\begin{enumerate}
\item $\Cont(M,\tau)$ is an atomic topos;
\item The completion of $(M,\tau)$ has a dense subgroup;
\item The group of units in the completion of $(M,\tau)$ is dense;
\item For each open relation $r \in \underline{\Rcal}_{\tau}$ and $m \in M$, there exists $m' \in M$ with $(mm',1) \in r$.
\end{enumerate}
\end{prop}
\begin{proof}
($3 \Rightarrow 2 \Rightarrow 1$) If the group of units of (the completion of) $(M,\tau)$ is dense, then clearly this provides a dense subgroup. If the $(G,\tau|_G)$ is a dense subgroup of (the completion of) $(M,\tau)$, then $\Cont(M,\tau)$ admits a hyperconnected morphism from $\Setswith{G}$, whence the former is an atomic topos, by Scholium \ref{schl:descend}.
($1 \Rightarrow 3$) If $\Cont(M,\tau)$ is atomic, all of the supercompact objects are necessarily atoms. The opposite of $\underline{\Rcal}_{\tau}$ is easily verified to satisfy the amalgamation property and joint embedding property of \cite[Definition 3.3]{TGT}, and the canonical point of $(M,\tau)$ provides an $\underline{\Rcal}_{\tau}\op$-universal, $\underline{\Rcal}_{\tau}\op$-ultrahomogeneous object in $\mathrm{Ind}$-$\underline{\Rcal}_{\tau}\op$, namely the completion of $(M,\tau)$ itself. Thus by \cite[Theorem 3.5]{TGT}, there is a topological group $(G,\sigma)$ representing the topos (and having the same canonical point), and by the more detailed description of this construction in \cite[Proposition 5.7]{ATGT}, the group so constructed is precisely the group of units of $(M,\tau)$, and this group is dense in $(M,\tau)$.
($1 \Leftrightarrow 4$) Consider $\underline{\Rcal}_{\tau}$ when $\Cont(M,\tau)$ is atomic. Since all of the $M/r$ are atoms, all of the morphisms in this category are (strict) epimorphisms, which means that in particular the canonical monomorphisms $[m]: m^*(r) \to r$ are isomorphisms, providing $[m']: r \to m^*(r)$ such that $(mm',1) \in r$ and $(m'm,1) \in m^*(r)$, but the latter is implied by the former, so the former suffices. Conversely, if $4$ holds, then all of the morphisms of $\underline{\Rcal}_{\tau}$ are strict epimorphisms, whence the reductive topology coincides with the atomic topology, so $\Cont(M,\tau)$ is atomic as required.
\end{proof}
We anticipate a plethora of results of this nature, where a complete monoid generates a topos having a property $Q$ if and only if it has a dense submonoid having property $P$, where $P$ is the property corresponding to $Q$ for toposes of discrete monoid actions; the above is the case where $Q$ is the property of being atomic and $P$ is the property of being a group from Theorem \ref{thm:atomic} of Chapter \ref{chap:mpatti}. In order to attain these results, some preliminary work will be needed to accumulate the relevant factorization results for properties of geometric morphisms along hyperconnected geometric morphisms.
On the subject of geometric morphisms, we have two further conjectures. In the hope of improving Scholium \ref{schl:factors} to a more elegant result, we begin with the following:
\begin{conj}
\label{conj:characterization}
There exists an intrinsic characterization, independent of the representing monoid $M$, of those hyperconnected geometric morphisms with domain $\Setswith{M}$ identifying toposes of the form $\Cont(M,\tau)$.
\end{conj}
To be more specific, observe that Proposition \ref{prop:intrinsic} provides an intrinsic sufficient condition for a hyperconnected morphism to express its codomain topos in terms of a topology on any monoid representing its domain topos; Conjecture \ref{conj:characterization} posits that it should be possible to refine this to a necessary and sufficient condition.
We also record our expectation that the converse of Scholium \ref{schl:Morita2} fails.
\begin{conj}
\label{conj:inclusion}
There exists a complete monoid $(M',\tau')$ and an idempotent $e \in M'$ such that the semigroup inclusion $M:= eM'e \hookrightarrow M'$ is \textit{not} a Morita equivalence, but the induced geometric inclusion $\Cont(M,\tau'|_{M}) \hookrightarrow \Cont(M',\tau')$ is an equivalence.
\end{conj}
\subsection{Actions on Topological Spaces}
A proof of, or counterexample to, Conjecture \ref{conj:powdery} will establish the extent of the symmetry in the type of Morita equivalence studied in this Chapter \ref{chap:TTMA}. Whichever way this result falls, however, we have shown that the category of right actions of a topological monoid on discrete spaces is a very coarse invariant of such a monoid. Moreover, anyone interested in actions of topological monoids is likely to wish to examine their actions on more general classes of topological space. A solution to this, which is viable in any Grothendieck topos $\Ecal$, is to first consider the topos $[M\op,\Ecal]$, which can be constructed as a pullback in $\TOP$, as in the lower square here:
\begin{equation}
\label{eq:pbEM}
\begin{tikzcd}
\Ecal \ar[r] \ar[d] \ar[dr, phantom, "\lrcorner", very near start] & \Set \ar[d] \\
{[M\op{,}\Ecal]} \ar[r] \ar[d] \ar[dr, phantom, "\lrcorner", very near start] & {[M\op{,}\Set]} \ar[d] \\
\Ecal \ar[r] & \Set,
\end{tikzcd}
\end{equation}
since $M$ induces an internal monoid in $\Ecal$ by its image under the inverse image functor of the global sections morphism of $\Ecal$. Taking $\Ecal$ to be the topos of sheaves on a space $X$, we can view the objects of $[M\op,\Ecal]$ as right actions of $M$ on spaces which are discrete fibrations over $X$; taking $\Ecal$ to be a more general topos of spaces, we similarly get actions of $M$ on such spaces.
In each case, we can construct the subcategory of $[M\op,\Ecal]$ on the actions which are continuous with respect to a topology $\tau$ on $M$. In the best cases, this will produce a topos hyperconnected under $[M\op,\Ecal]$, and the analysis can proceed analogously to that of Chapter \ref{chap:TTMA}, taking advantage of the $\Ecal$-valued point constructed in the upper square of \eqref{eq:pbEM}. If this can be done with sufficient generality, one will be able to address a host of interesting Morita-equivalence problems in this way.
\subsection{Topological Categories}
\label{ssec:topcat}
Another direction to generalize is to consider topologies on small categories with more than one object. Let $\Ccal$ be a small category with set of objects $C_0$, set of morphisms $C_1$, identity map $i:C_0 \to C_1$, domain and codomain maps $d,c: C_1 \to C_0$, and composition $m:C_2\to C_1$, where $C_2$ is the pullback:
\[\begin{tikzcd}
C_2 \ar[r] \ar[d] \ar[dr, phantom, "\lrcorner", very near start] & C_1 \ar[d, "d"] \\
C_1 \ar[r, "c"'] & C_0;
\end{tikzcd}\]
this matches the presentation of internal categories recapitulated in Section \ref{ssec:proper} of Chapter \ref{chap:sgt}. As such, a presheaf on $\Ccal$ can be expressed as an object $a:F_0 \to C_0$ of $\Set/C_0$, equipped with a morphism $b:F_1 \to F_0$ where $F_1$ is the pullback,
\[\begin{tikzcd}
F_1 \ar[r, "\pi_2"] \ar[d, "\pi_1"'] \ar[dr, phantom, "\lrcorner", very near start] & F_0 \ar[d, "a"] \\
C_1 \ar[r, "c"'] & C_0,
\end{tikzcd}\]
satisfying $a \circ b = d \circ \pi_2$, $b \circ (\id_{F_0} \times_{C_0} i) = \id_{F_0}$ and $b \circ (b \times_{C_0} \id_{C_1}) = b \circ (\id_{F_0} \times_{C_0} m)$.
Equipping $C_0$ and $C_1$ with topologies such that $c$ and $d$ are continuous, we might call a presheaf as above \textit{continuous} if $b$ is continuous when $C_0$ and $F_0$ are equipped with the discrete topology and $F_1$ is equipped with the pullback topology\footnote{The map $a$ is automatically continuous when the topology on $F_0$ is discrete.}. Yet again, we can consider the full subcategory of $\Setswith{\Ccal}$ on these presheaves, and we expect it to be coreflective. In good cases, we will have the analogue of Proposition \ref{prop:hyper}, and the analysis can proceed as in Chapter \ref{chap:TTMA}, leading to a class of genuine topological categories representing these toposes.
\subsection{Localic monoids and constructiveness}
\label{ssec:localicmon}
Topos theorists tend to try to work constructively wherever possible, since doing so ensures that all results can be applied over an arbitrary topos. In this light, our frequent reliance on complementation in the underlying sets of our monoids in Chapter \ref{chap:TTMA} is quite restrictive, since \textit{a priori} it means our results are applicable only over Boolean toposes, and we have not formally demonstrated here that they apply even to this level of generality.
From a constructive perspective, more suitable objects of study than topological monoids would be \textbf{localic monoids}, which are monoids in the category of locales over a given base topos, typically $\Set$. Early on in the research for this thesis, Steve Vickers suggested that we consider pursuing this direction. However, while the category of actions of a localic monoid on sets (again viewed as discrete spaces) is easy to define, it is much harder to show that such a category is a topos. In \cite[Example B3.4.14(b)]{Ele}, we see that the more powerful results of \textit{descent theory} are required to show that categories of actions of localic groups are toposes. While descent theory is an important tool, it is far more abstract than the comonadicity theorem we used in Corollary \ref{crly:topos}, making concrete characterization results for these toposes more challenging to prove.
While we did not end up treating localic monoids in this thesis, we anticipate that the present work will be valuable in that analysis. Indeed, the functor sending a locale to its topological space of points preserves limits, so that it provides a canonical `forgetful' functor from a category of actions of a localic monoid to a category of actions of a topological monoid. We anticipate that, just as in Section \ref{sec:properties}, this functor can be used to constrain the properties of a category of actions of a localic monoid.
We should mention that another obstacle in our study of localic monoids is a lack of easily tractable examples, especially examples of localic monoids (or even localic groups) which one can show are \textit{not} Morita equivalent to topological monoids in their actions on discrete spaces. While the construction of the localic group $\mathrm{Perm}(A)$ of permutations of a locale $A$, described by Wraith in \cite{LocGrp}, is used as a basis for the Localic Galois Theory of Dubuc \cite{LGT}, the latter author provides no specific examples of instances of these. We expect that the construction of such examples will further illuminate the appropriate approach to studying categories of actions of localic monoids.
\section{Applying toposes of topological monoid actions}
\label{ssec:TSGT}
Orthogonal factorization systems abound in category theory, and we anticipate many applications of the results of Chapter \ref{chap:TSGT}; already there is much work to do in elucidating the algebraic examples sketched there. It is worth noting that in a category with pullbacks, the conditions needed for those results coincides with the concept of \textit{stable factorization system}, which is an orthogonal factorization system in which the left class is stable under pullback.
We know from Theorem \ref{thm:characterization} of Chapter \ref{chap:TTMA} that any hyperconnected morphism $\Setswith{M} \to \Ecal$ is enough to ensure the existence of a topological monoid presentation for $\Ecal$. As such, we can weaken the statement of Theorem \ref{thm:basic} and its derivatives by replacing $L$ with any monoid $M$ which acts by endomorphisms on a point of $\Ecal$. However, extra conditions along the lines of those discussed in Section \ref{ssec:factor} of the Chapter \ref{chap:TTMA} would be required to guarantee that $\Ecal$ can be identified with the actions of $M$ which are continuous with respect to some topology.
A final direction which would tie together many of the themes of the present thesis would be to complement the translation of topos-theoretic invariants into properties of topological monoid suggested in Section \ref{ssec:mitopmon} above with a translation of the same properties into site-theoretic ones described in Section \ref{ssec:other}, since this would yield concrete results about the presenting monoids without needing to explicitly calculate them.
We hope that some of these ideas will eventually lead to fruitful research uniting diverse areas of mathematics. | {"config": "arxiv", "file": "2112.10198/The_Conclusion.tex"} |
TITLE: Proving that $p \implies q$ and $\neg p \implies \neg q$ are not equivalent, without using truth tables
QUESTION [0 upvotes]: How do I prove that
$$p \implies q$$
and
$$\neg p \implies \neg q$$
are not equivalent?
I have to prove this using expression manipulation and not truth tables.
REPLY [1 votes]: Taking $q=\neg p$ we get
$$(p \Rightarrow \neg p) \Leftrightarrow (\neg p \Rightarrow \neg \neg p)$$
$$(p \Rightarrow \neg p) \Leftrightarrow (\neg p \Rightarrow p)$$
$$(\neg p \lor \neg p) \Leftrightarrow (\neg \neg p \lor p)$$
$$(\neg p \lor \neg p) \Leftrightarrow (p \lor p)$$
$$\neg p \Leftrightarrow p$$
which is clearly false.
Tautologies used: $(p \Rightarrow q) \Leftrightarrow (\neg p \lor q)$, $\neg \neg p \Leftrightarrow p$, $(p \lor p) \Leftrightarrow p$. | {"set_name": "stack_exchange", "score": 0, "question_id": 4538631} |
TITLE: Probability - trick question?
QUESTION [0 upvotes]: Let $\mu$ be the mean annual salary of Major League Baseball players for 2002. Assume that the standard deviation of the salaries of these players is $\$82,000$. What is the probability that the 2002 mean salary of a random sample of $80$ baseball players was within $\$15,000$ of the population mean, $\mu$? Assume that $n/N \leq 0.05$.
this is a trick question isn't it? the probability is one, since the mean of a sample is always the same as the mean of a population.
REPLY [2 votes]: Suppose I show you a fair coin. You take this coin and flip it 10 times, and record the number of heads and the number of tails you observed. Even though it is "fair" in the sense that each coin toss has equal probability of landing heads versus landing tails, is it guaranteed that you will observe exactly 5 heads and 5 tails? Of course not. Indeed, if you were to flip the coin an odd number of times, it isn't even possible to get an equal number of heads and tails, despite the coin being fair.
So, this illustrates that a sample mean is not the same as a population mean. The sample mean is a statistic, which is subject to sampling variation--in other words, it is the result of the observations we made, and may change each time we take a sample because there is randomness involved in the process of sampling.
Now, regarding your specific question, we are given that the distribution of salaries has mean $\mu$ and standard deviation $\sigma = 82000$. Then by the Central Limit Theorem, the sample mean $\bar x$ of the salaries of $n = 80$ randomly selected players is approximately normally distributed with mean $\mu$ and standard deviation $\sigma/\sqrt{n} = 9167.88$. The probability that $\bar x$ is within $15000$ of the population mean is then $$\Pr[|\bar x - \mu| \le 15000] = \Pr\left[\left| \frac{\bar x - \mu}{\sigma/\sqrt{n}} \right| \le \frac{15000}{9167.88} \right] = \Pr[|Z| \le 1.63615],$$ where $Z$ is a standard normal random variable (mean = 0, standard deviation = 1). Now we can look this probability up in a $z$-table, which gives us $0.898191$, or about an $89.8\%$ chance that the sample mean is within $15000$ of the population mean. | {"set_name": "stack_exchange", "score": 0, "question_id": 633131} |
\subsection{Problem setup}
We consider a nonlinear perturbed discrete-time system
\begin{subequations}
\label{eq:sys_w}
\begin{align}
\label{eq:sys_w_1}
x_{t+1}&=f_{\mathrm{w}}(x_t,u_t,w_t),\\
\label{eq:sys_w_2}
y_t&=h_{\mathrm{w}}(x_t,u_t,w_t),
\end{align}
\end{subequations}
with state $x_t\in\mathbb{X}=\mathbb{R}^n$, control input $u_t\in\mathbb{U}\subseteq\mathbb{R}^m$, disturbances/noise $w_t\in\mathbb{W}\subseteq\mathbb{R}^q$, noisy measurement $y_t\in\mathbb{Y}\subseteq\mathbb{R}^p$, time $t\in\mathbb{I}_{\geq 0}$, continuous dynamics $f_{\mathrm{w}}:\mathbb{X}\times\mathbb{U}\times\mathbb{W}\rightarrow\mathbb{X}$, and continuous output equations $h_{\mathrm{w}}:\mathbb{X}\times\mathbb{U}\times\mathbb{W}\rightarrow\mathbb{Y}$.
We assume w.l.o.g. that $0\in\mathbb{W}$ and define the nominal system equations $f(x,u):=f_{\mathrm{w}}(x,u,0)$, $h(x,u):=h_{\mathrm{w}}(x,u,0)$.
We impose point-wise in time constraints on the state and input
$(x_t,u_t)\in \mathbb{Z}$, $t\in\mathbb{I}_{\geq 0}$.
The overall control goal is to minimize some user chosen performance measure/cost $\ell$ while ensuring constraint satisfaction.
To this end, we develop an output feedback MPC scheme that uses the past measured outputs $y_j$, $j\in\mathbb{I}_{[0,t-1]}$ and some initial state estimate $\hat{x}_0$ with a known bound on the estimation error to compute a control action $u_t$ at time $t$.
In order to derive robust bounds on the estimation error and ensure robust constraint satisfaction, we assume that the disturbances are bounded.
\begin{assumption}
\label{ass:disturbance} (Bounded disturbance)
There exists a constant $\overline{w}>0$, such that $\|w_t\|\leq \overline{w}$ for all $t\in\mathbb{I}_{\geq 0}$.
\end{assumption} | {"config": "arxiv", "file": "2105.03427/Setup_1.tex"} |
TITLE: Class equation of normal subgroups
QUESTION [0 upvotes]: I am puzzled with a thought or a question regarding class equation and normality of subgroups.Consider the following situation.
Let $G$ be a finite group and $N\trianglelefteq G$.Let $G$ act on $N$ by conjugation which is allowed since $N$ is normal and let the representatives of the different equivalence classes be $n_1,n_2,...,n_k$ (excluding the equivalence class of identity).
Then, the equivalence class of $n_i$ is
$[n_i]=\{gn_ig^{-1} : g\in G\},i=1,2,..,k$
Then $N=\{e\}\displaystyle\cup_{i=1}^k[n_i]$
$\Rightarrow |N|=1+\sum_{i=1}^k |[n_i]| \quad (1)$
Now, let $N$ act on itself by conjugation.If
$cl(n_i)=\{nn_in^- : n\in N\}(i=1,2,3,..,k)$ be the equivalence of $n_i$ under this action.Then I can see that
$cl(n_i)\subseteq [n_i]$ but can it happen that
$cl(n_i)\subsetneq [n_i]$ for some $i$(i.e a proper subset) or if the question is put in other way
Will the class equation of $N$ be the same as $(1)$?
This may be a silly question but I am trying to keep my understandings clear.
I have seen a question with the same title here(Class equation of normal subgroup) but I don't think that answers my question.
Any suggestions or advice?
Thanks for your time.
REPLY [1 votes]: Let's see an example, We'll take $$G=S_3=\{e,(12),(13),(23),(123),(132)\}$$
and
$$N=A_3=\{e,(123),(132)\}$$
We look at the action of $G$ on $N$ by conjugation and we get the equivalent classes
$$\{\{e\},\{(123),(132)\} \}$$
So here the class equation is :
$$3=|N|=1+|[(123)]|=1+2$$
We look at the action of $N$ on itself we get:
$$\{\{e\},\{(123)\},\{(132)\}\}$$
So here the class equation is:
$$3=|N|=1+cl([(123)])+cl(|[(132)])=1+1+1$$
So,the $n_i$ are the representatives of the equivalent classes for the action of conjugation of $G$ on $N$.
But the there could be more equivalence classes when you take the action of N on itself.
so in the example here, we have that $[(123)]=\{(123),(132)\}$ but $cl((123))=\{(123)\}$
So when you write down the class equation for the action of $N$ on itself, you cannot just take the $cl(n_i)$, where $n_i$ are the class representative of the action of $G$ on $N$. Because you are missing more equivalence classes for in the action of $N$ on itself. | {"set_name": "stack_exchange", "score": 0, "question_id": 3715498} |
\begin{document}
\maketitle
\begin{abstract}
We study convergence rates of Gibbs measures, with density proportional to $e^{-f(x)/t}$, as $t \rightarrow 0$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ admits a unique global minimum at $x^\star$.
We focus on the case where the Hessian is not definite at $x^\star$. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of $f$ at $x^\star$, which depends on every coordinate. We give an algorithm yielding such an expansion if the polynomial order of $x^\star$ is no more than $8$, in connection with Hilbert's $17^{\text{th}}$ problem. However, we prove that the case where the order is $10$ or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.
\end{abstract}
\section{Introduction}
Gibbs measures and their convergence properties are often used in stochastic optimization to minimize a function defined on $\mathbb{R}^d$. That is, let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be a measurable function and let $x^\star \in \mathbb{R}^d$ be such that $f$ admits a global minimum at $x^\star$. It is well known \cite{hwang1980} that under standard assumptions, the associated Gibbs measure with density proportional to $e^{-f(x)/t}$ for $t >0$, converges weakly to the Dirac mass at $x^\star$, $\delta_{x^\star}$, when $t \rightarrow 0$. The Langevin equation $dX_s = -\nabla f(X_s) ds + \sigma dW_s$ consists in a gradient descent with Gaussian noise. For $\sigma = \sqrt{2t}$, its invariant measure has a density proportional to $e^{-f(x)/t}$ (see for example \cite{khasminskii2012}, Lemma 4.16), so for small $t$ we can expect it to converge to $\text{argmin}(f)$ \cite{dalalyan2016} \cite{barrera2020}.
The simulated annealing algorithm \cite{laarhoven1987} builds a Markov chain from the Gibbs measure where the parameter $t$ converges to zero over the iterations.
This idea is also used in \cite{gelfand-mitter}, giving a stochastic gradient descent algorithm where the noise is gradually decreased to zero.
Adding a small noise to the gradient descent allows to explore the space and to escape from traps such as local minima and saddle points which appear in non-convex optimization problems \cite{lazarev1992} \cite{dauphin2014}.
Such methods have been recently brought up to light again with SGLD (Stochastic Gradient Langevin Dynamics) algorithms \cite{welling2011} \cite{li2015}, especially for Machine Learning and calibration of artificial neural networks, which is a high-dimensional non-convex optimization problem.
\medskip
The rates of convergence of Gibbs measures have been studied in \cite{hwang1980}, \cite{hwang1981} and \cite{athreya2010} under differentiability assumptions on $f$. It turns out to be of order $t^{1/2}$ as soon as the Hessian matrix $\nabla^2 f(x^\star)$ is positive definite. Furthermore, in the multiple well case i.e. if the minimum of $f$ is attained at finitely many points $x_1^\star$, $\ldots$, $x_m^\star$, \cite{hwang1980} proves that the limit distribution is a sum of Dirac masses $\delta_{x_i^\star}$ with coefficients proportional to $\det(\nabla^2 f(x_i^\star))^{-1/2}$ as soon as all the Hessian matrices are positive definite. If such is not the case, we can conjecture that the limit distribution is concentrated around the $x_i^\star$ where the degeneracy is of the highest order.
\medskip
The aim of this paper is to provide a rate of convergence in this degenerate setting, i.e. when $x^\star$ is still a strict global minimum but $\nabla^2 f(x^\star)$ is no longer definite, which extends the range of applications of Gibbs measure-based algorithms where positive definiteness is generally assumed.
A general framework is given in \cite{athreya2010}, which provides rates of convergence based on dominated convergence. However a strong and rather technical assumption on $f$ is needed and checking it seems, to some extent, more demanding than proving the result. To be more precise, the assumption reads as follows: there exists a function $g : \mathbb{R}^d \rightarrow \mathbb{R}$ with $e^{-g} \in L^1(\mathbb{R}^d)$ and $\alpha_1, \ \ldots, \ \alpha_d \in (0,+\infty)$ such that
\begin{equation}
\label{eq:intro}
\forall h \in \mathbb{R}^d, \ \ \frac{1}{t} \left[ f(x^\star + \left( t^{\alpha_1} h_1,\ldots, t^{\alpha_d} h_d \right) ) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} g(h_1,\ldots,h_d).
\end{equation}
Our objective is to give conditions on $f$ such that \eqref{eq:intro} is fulfilled and then to elucidate the expression of $g$ depending on $f$ and its derivatives by studying the behaviour of $f$ at $x^\star$ in every direction. Doing so we can apply the results from \cite{athreya2010} yielding the convergence rate of the corresponding Gibbs measures. The orders $\alpha_1$, $\ldots$, $\alpha_d$ must be chosen carefully and not too big, as the function $g$ needs to depend on every of its variables $h_1$, $\ldots$, $h_d$, which is a necessary condition for $e^{-g}$ to be integrable.
We also extend our results to the multiple well case.
We generally assume $f$ to be coercive, i.e. $f(x) \rightarrow + \infty$ as $||x|| \rightarrow + \infty$, $\mathcal{C}^{2p}$ in a neighbourhood of $x^\star$ for some $p \in \mathbb{N}$ and we assume that the minimum is polynomial strict, i.e. the function $f$ is bounded below in a neighbourhood of $x^\star$ by some non-negative polynomial function, null only at $x^\star$. Thus we can apply a multi-dimensional Taylor expansion to $f$ at $x^\star$, where the successive derivatives of $f :\mathbb{R}^d \rightarrow \mathbb{R}$ are seen as symmetric tensors of $\mathbb{R}^d$. The idea is then to consider the successive subspaces where the derivatives of $f$ are null up to some order ; using that the Taylor expansion of $f(x^\star+h)-f(x^\star)$ is non-negative, some cross derivative terms are null. However a difficulty arises at orders $6$ and higher, as the set where the derivatives of $f$ are null up to some order is no longer a vector subspace in general. This difficulty is linked with Hilbert's $17^{\text{th}}$ problem \cite{hilbert1888}, stating that a non-negative multivariate polynomial cannot be written as the sum of squares of polynomials in general. We thus need to change the definition of the subspaces we consider.
Following this, we give a recursive algorithm yielding an adapted decomposition of $\mathbb{R}^d$ into vector subspaces and a function $g$ satisfying \eqref{eq:intro} up to a change of basis, giving a canonical higher order nested decomposition of $f$ at $x^\star$ in degenerate cases. An interesting fact is that the case where the polynomial order of $x^\star$ is $10$ or higher fundamentally differs from those of orders $2$, $4$, $6$ and $8$, owing to the presence of even cross terms which may be not null. The algorithm we provide works at the orders $10$ or higher only under the assumption that all such even cross terms are null. In general, it is more difficult to get a general expression of $g$ for the orders $10$ and higher.
We then apply our results to \cite{athreya2010}, where we give conditions such that the hypotheses of \cite{athreya2010}, especially \eqref{eq:intro}, are satisfied so as to infer rates of convergence of Gibbs measures in the degenerate case where $\nabla^2 f(x^\star)$ is not necessarily positive definite.
The function $g$ given by our algorithm is a non-negative polynomial function and non-constant in any of its variables, however it needs to be assumed to be coercive to be applied to \cite{athreya2010}. We study the case where $g$ is not coercive and give a method to deal with simple generic non-coercive cases, where our algorithm seems to be a first step to a more general procedure. However, we do not give a general method in this case.
Our results are applied to Gibbs measures but they can also be applied to more general contexts, as we give a canonical higher order nested expansion of $f$ at a minimum, in the case where some derivatives are degenerate.
For general properties of symmetric tensors we refer to \cite{comon2008}. In the framework of stochastic approximation, \cite{fort1999} Section 3.1 introduced the notion of strict polynomial local extremum and investigated their properties as higher order "noisy traps".
\medskip
The paper is organized as follows. In Section \ref{section:gibbs_measures}, we recall convergence properties of Gibbs measures and revisit the main theorem from \cite{athreya2010}. This theorem requires, as an hypothesis, to find an expansion of $f$ at its global minimum ; we properly state this problem in Section \ref{subsection:statement_of_problem} under the assumption of strict polynomial minimum. In Section \ref{section:main_result}, we state our main result for both single well and multiple well cases, as well as our algorithm.
In Section \ref{section:expansion}, we detail the expansion of $f$ at its minimum for each order and provide the proof. We give the general expression of the canonical higher order nested expansion at any order in Section \ref{sec:expansion_any_order}, where we distinguish the orders $10$ and higher from the lower ones. We then provide the proof for each order $2$, $4$, $6$ and $8$ in Sections \ref{section:order_2}, \ref{section:order_4}, \ref{section:order_6} and \ref{section:order_8} respectively. We need to prove that, with the exponents $\alpha_1$, $\ldots$, $\alpha_d$ we specify, the convergence in \eqref{eq:intro} holds ; we do so by proving that, using the non-negativity of the Taylor expansion, some cross derivative terms are zero. Because of Hilbert's $17^{\text{th}}$ problem, we need to distinguish the orders $6$ and $8$ from the orders $2$ and $4$, as emphasized in Section \ref{section:hilbert}. For orders $10$ and higher, such terms are not necessarily zero and must then be assumed to be zero. In Section \ref{section:order_10}, we give a counter-example if this assumption is not satisfied before proving the result.
In Section \ref{subsec:unif_non_constant}, we prove that for every order the resulting function $g$ is constant in none of its variables and that the convergence in \eqref{eq:intro} is uniform on every compact set.
In Section \ref{section:non_coercive}, we study the case where the function $g$ is not coercive and give a method to deal with the simple generic case.
In Section \ref{section:proofs_athreya}, we prove our main theorems stated in Section \ref{section:main_result} using the expansion of $f$ established in Section \ref{section:expansion}. Finally, in Section \ref{section:flat}, we deal with a "flat" example where all the derivatives in the local minimum are zero and where we cannot apply our main theorems.
\section{Definitions and notations}
We give a brief list of notations that are used throughout the paper.
We endow $\mathbb{R}^d$ with its canonical basis $(e_1,\ldots,e_d)$ and the Euclidean norm denoted by $|| \boldsymbol{\cdot} ||$. For $x \in \mathbb{R}^d$ and $r >0$ we denote by $\mathcal{B}(x,r)$ the Euclidean ball of $\mathbb{R}^d$ of center $x$ and radius $r$.
For $E$ a vector subspace of $\mathbb{R}^d$, we denote by $p_{_E} : \mathbb{R}^d \rightarrow E$ the orthogonal projection on $E$. For a decomposition of $\mathbb{R}^d$ into orthogonal subspaces, $\mathbb{R}^d = E_1 \oplus \cdots \oplus E_p$, we say that an orthogonal transformation $B \in \mathcal{O}_d(\mathbb{R})$ is adapted to this decomposition if for all $j \in \lbrace 1, \ldots, p \rbrace $,
$$ \forall i \in \lbrace \dim(E_1)+\cdots+\dim(E_{j-1})+1, \ldots, \dim(E_1)+\cdots+\dim(E_j) \rbrace, \ B \cdot e_i \in E_j .$$
For $a, \ b \in \mathbb{R}^d$, we denote by $a \ast b$ the element-wise product, i.e.
$$ \forall i \in \lbrace 1, \ldots, d \rbrace, \ (a \ast b)_i = a_i b_i .$$
For $v^1$, $\ldots$, $v^k$ vectors in $\mathbb{R}^d$ and $T$ a tensor of order $k$ of $\mathbb{R}^d$, we denote the tensor product
$$ T \cdot (v^1 \otimes \cdots \otimes v^k) = \sum_{i_1,\ldots,i_k \in \{1,\ldots,d \}} T_{i_1 \cdots i_k} v^1_{i_1} \ldots v^k_{i_k} . $$
More generally, if $j \le k$ and $v^1, \ \ldots, \ v^j$ are $j$ vectors in $\mathbb{R}^d$, then $T \cdot (v^1 \otimes \cdots \otimes v^j)$ is a tensor of order $k-j$ such that:
$$ T \cdot (v^1 \otimes \cdots \otimes v^j)_{i_{j+1}\ldots i_k} = \sum_{i_1,\ldots,i_j \in \{1,\ldots ,d \}} T_{i_1 \ldots i_k} v^1_{i_1} \ldots v^j_{i_j}. $$
For $h \in \mathbb{R}^d$, $h^{\otimes k}$ denotes the tensor of order $k$ such that
$$ h^{\otimes k} = (h_{i_1} \ldots h_{i_k} )_{i_1,\ldots,i_k \in \{1,\ldots,d \}}.$$
For a function $f \in \mathcal{C}^p\left(\mathbb{R}^d, \mathbb{R}\right)$, we denote $\nabla^k f(x)$ the differential of order $k \le p$ of $f$ at $x$, as $\nabla^k f(x)$ is the tensor of order $k$ defined by:
$$ \nabla^k f(x) = \left(\frac{\partial^k f(x)}{\partial x_{i_1} \cdots \partial x_{i_k}}\right)_{i_1,i_2,\ldots,i_k \in \{1,\ldots,d \}} .$$
By Schwarz's theorem, this tensor is symmetric, i.e. for all permutation $\sigma \in \mathfrak{S}_k$,
$$ \frac{\partial^k f(x)}{\partial x_{i_{\sigma(1)}} \cdots \partial x_{i_{\sigma(k)}}} = \frac{\partial^k f(x)}{\partial x_{i_1} \cdots \partial x_{i_k}} .$$
We recall the Taylor-Young formula in any dimension, and the Newton multinomial formula.
\begin{theorem}[Taylor-Young formula]
\label{theorem:taylor}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be $\mathcal{C}^p$ and let $x \in \mathbb{R}^d$. Then:
$$ f(x + h) \underset{h \rightarrow 0}{=} \sum_{k=0}^{p} \frac{1}{k!} \nabla^k f(x) \cdot h^{\otimes k} + ||h||^p o(1) .$$
\end{theorem}
\noindent We denote by $\binom{k}{i_1, \ldots, i_p}$ the $p$-nomial coefficient, defined as:
$$ \binom{k}{i_1,\ldots ,i_p} = \frac{k!}{i_1!\ldots i_p!} .$$
\begin{theorem}[Newton multinomial formula]
Let $h_1, \ \ldots, \ h_p \in \mathbb{R}^d$, then
\begin{equation}
\label{equation:multinomial}
(h_1 + h_2 + \cdots + h_p)^{\otimes k} = \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_p = k }} \binom{k}{i_1, \ldots, i_p} h_1^{\otimes i_1} \otimes \cdots \otimes h_p^{\otimes i_p} .
\end{equation}
\end{theorem}
\noindent For $T$ a tensor of order $k$, we say that $T$ is non-negative (resp. positive) if
\begin{equation}
\label{eq:tensor_positive_def}
\forall h \in \mathbb{R}^d, \ T \cdot h^{\otimes k} \ge 0 \text{ (resp. } T \cdot h^{\otimes k} > 0 \text{)}.
\end{equation}
We denote $L^1(\mathbb{R}^d)$ the set of measurable functions $f:\mathbb{R}^d \rightarrow \mathbb{R}$ that are integrable with respect to the Lebesgue measure on $\mathbb{R}^d$. We denote by $\lambda_d$ the Lebesgue measure on $\mathbb{R}^d$. For $f : \mathbb{R}^d \rightarrow \mathbb{R}$ such that $e^{-f} \in L^1(\mathbb{R}^d)$, we define for $t > 0$, $C_t := \left( \int_{\mathbb{R}^d} e^{-f/t} \right)^{-1}$ and $\pi_t$ the Gibbs measure
$$ \pi_t(x)dx := C_t e^{-f(x)/t} dx .$$
For a family of random variables $(Y_t)_{t \in (0,1]}$ and $Y$ a random variable, we write $Y_t \underset{t \rightarrow 0}{\overset{\mathscr{L}}{\longrightarrow}} Y$ meaning that $(Y_t)$ weakly converges to $Y$.
We give the following definition of a strict polynomial local minimum of $f$:
\begin{definition}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be $\mathcal{C}^{2p}$ for $p \in \mathbb{N}$ and let $x^\star$ be a local minimum of $f$. We say that $f$ has a strict polynomial local minimum at $x^\star$ of order $2p$ if $p$ is the smallest integer such that:
\begin{equation}
\label{eq:polynomial_strict}
\exists r >0, \ \forall h \in \mathcal{B}(x^\star, r) \setminus \{0 \} , \ \sum_{k=2}^{2p} \frac{1}{k!} \nabla^k f (x^\star) \cdot h^{\otimes k} > 0 .
\end{equation}
\end{definition}
\textbf{Remarks :}
\begin{enumerate}
\item A local minimum $x^\star$ of $f$ is not necessarily strictly polynomial, for example, $f : x \mapsto e^{-||x||^{-2}}$ and $x^\star = 0$.
\item If $x^\star$ is polynomial strict, then the order is necessarily even, because if $x^\star$ is not polynomial strict of order $2l$ for some $l \in \mathbb{N}$, then we have $h_n \rightarrow 0$ such that the Taylor expansion in $h_n$ up to order $2l$ is zero ; by the minimum condition, the Taylor expansion in $h_n$ up to order $2l+1$ must be non-negative, so we also have $\nabla^{2l+1}f(x^\star) \cdot h_n^{\otimes 2l+1} = 0$.
\end{enumerate}
\medskip
For $f :\mathbb{R}^d \rightarrow \mathbb{R}$ such that $\min_{\mathbb{R}^d}(f)$ exists, we denote by $\text{argmin}(f)$ the arguments of the minima of $f$, i.e.
$$ \text{argmin}(f) = \left\lbrace x \in \mathbb{R}^d : \ f(x) = \min_{\mathbb{R}^d}(f) \right\rbrace .$$
Without ambiguity, we write "minimum" or "local minimum" to designate $f(x^\star)$ as well as $x^\star$.
Finally, we define, for $x^\star \in \mathbb{R}^d$ and $p \in \mathbb{N}$:
\begin{align*}
\mathscr{A}_p(x^\star) & := \left\lbrace f \in \mathcal{C}^{2p}(\mathbb{R}^d, \mathbb{R}) : \ f \text{ admits a local minimum at } x^\star \right\rbrace . \\
\mathscr{A}_p^\star(x^\star) & := \left\lbrace f \in \mathcal{C}^{2p}(\mathbb{R}^d, \mathbb{R}) : \ f \text{ admits a strict polynomial local minimum at } x^\star \text{ of order } 2p \right\rbrace .
\end{align*}
\section{Convergence of Gibbs measures}
\label{section:gibbs_measures}
\subsection{Properties of Gibbs measures}
Let us consider a Borel function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ with $e^{-f} \in L^1(\mathbb{R}^d)$. We study the asymptotic behaviour of the probability measures of density for $t \in (0,\infty)$:
$$ \pi_t(x) dx = C_t e^{-\frac{f(x)}{t}} dx $$
when $t \rightarrow 0$. When $t$ is small, the measure $\pi_t$ tends to the set $\argmin(f)$. The following proposition makes this statement precise.
\begin{proposition}
\label{proposition:gibbs}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be a Borel function such that
$$ f^\star := \textup{essinf}(f) = \inf \{y : \ \lambda_d\{f \le y \} >0 \} > - \infty ,$$
and $e^{-f} \in L^1(\mathbb{R}^d)$. Then
$$ \forall \varepsilon >0, \ \pi_t(\{ f \ge f^\star + \varepsilon\} ) \underset{t \rightarrow 0}{\longrightarrow} 0 .$$
\end{proposition}
\begin{proof}
As $f^\star > -\infty$, we may assume without loss of generality that $f^\star = 0$ by replacing $f$ by $f-f^\star$. Let $\varepsilon>0$. It follows from the assumptions that $f\ge 0$ $\lambda_d$-$a.e.$ and $\lambda_d \lbrace f \le \varepsilon \rbrace >0$ for every $\varepsilon>0$. As $e^-f \in L^1(\mathbb{R}^d)$, we have
$$ \lambda_d \lbrace f \le \varepsilon/3 \rbrace \le e^{\varepsilon/3} \int_{\mathbb{R}^d} e^{-f} d\lambda_d<+\infty. $$
Moreover by dominated convergence, it is clear that
$$ C_t^{-1} \downarrow \lambda_d\lbrace f=0 \rbrace<+\infty. $$
We have
$$C_t \le \left(\int_{f \le \varepsilon/3} e^{-\frac{f(x)}{t}}dx \right)^{-1} \le \left( e^{-\frac{\varepsilon}{3t}} \underbrace{\lambda_d \{ f \le \frac{\varepsilon}{3} \} }_{>0} \right)^{-1}. $$
Then
\begin{align*}
\pi_t\{f \ge \varepsilon\} = C_t \int_{f \ge \varepsilon} e^{-\frac{f(x)}{t}}dx \le \frac{e^{\varepsilon/3t} \int_{f \ge \varepsilon} e^{-f(x) / t}dx }{\lambda_d \{ f \le \frac{\varepsilon}{3} \}} \le \frac{e^{-\varepsilon/3t}C_{3t}^{-1}}{\lambda_d \{ f \le \frac{\varepsilon}{3} \}} \underset{t \rightarrow 0}{\longrightarrow} 0,
\end{align*}
because if $f(x) \ge \varepsilon$, then $e^{-\frac{f(x)}{t}} \le e^{-\frac{2\varepsilon}{3t}}e^{-\frac{f(x)}{3t} }$, and where we used that $C_{3t}^{-1}\le C_1^{-1}$ if $t\le 1/3$
\end{proof}
Now, let us assume that $f : \mathbb{R}^d \rightarrow \mathbb{R}$ is continuous, $e^{-f} \in L^1(\mathbb{R}^d)$ and $f$ admits a unique global minimum at $x^\star$ so that $\text{argmin}(f) = \{ x^\star \}$.
In \cite{athreya2010} is proved the weak convergence of $\pi_t$ to $\delta_{x^\star}$ and a rate of convergence depending on the behaviour of $f(x^\star + h)-f(x^\star)$ for small enough $h$. Let us recall this result in detail ; we may assume without loss of generality that $x^\star=0$ and $f(x^\star) = 0$.
\begin{theorem}[Athreya-Hwang, 2010]
\label{theorem:athreya:1}
Let $f : \mathbb{R}^d \rightarrow [0,\infty)$ be a Borel function such that :
\begin{enumerate}
\item $e^{-f} \in L^1(\mathbb{R}^d)$.
\item For all $\delta > 0$, $\inf \{ f(x), \ ||x|| > \delta \} > 0$.
\item There exist $\alpha_1, \ \ldots, \ \alpha_d > 0$ such that for all $(h_1,\ldots,h_d) \in \mathbb{R}^d$,
$$ \frac{1}{t} f(t^{\alpha_1} h_1,\ldots, t^{\alpha_d} h_d) \underset{t \rightarrow 0}{\longrightarrow} g(h_1,\ldots,h_d) \in \mathbb{R}. $$
\item $ \displaystyle\int_{\mathbb{R}^d} \sup_{0<t<1} e^{-\frac{f\left(t^{\alpha_1}h_1,\ldots ,t^{\alpha_d}h_d\right)}{t}} dh_1\ldots dh_d < \infty$.
\end{enumerate}
For $0<t<1$, let $X_t$ be a random vector with distribution $\pi_t$. Then $e^{-g} \in L^1(\mathbb{R}^d)$ and
\begin{equation}
\label{equation:athreya_assumption:3}
\left( \frac{(X_t)_1}{t^{\alpha_1}}, \ldots, \frac{(X_t)_d}{t^{\alpha_d}} \right) \overset{\mathscr{L}}{\longrightarrow} X \ \text{ as } t \rightarrow 0
\end{equation}
where the distribution of $X$ has a density proportional to $e^{-g(x_1,\ldots,x_d)}$.
\end{theorem}
\noindent \textbf{Remark:} Hypothesis 2. is verified as soon as $f$ is continuous, coercive (i.e. $f(x) \longrightarrow + \infty$ when $||x|| \rightarrow + \infty$) and that $\text{argmin}(f) = \lbrace 0 \rbrace$.
\medskip
To study the rate of convergence of the measure $\pi_t$ when $t \rightarrow 0$ using Theorem \ref{theorem:athreya:1}, we need to identify $\alpha_1,\ldots, \ \alpha_d$ and $g$ such that the condition \eqref{equation:athreya_assumption:3} holds, up to a possible change of basis.
Since $x^\star$ is a local minimum, the Hessian $\nabla^2 f(x^\star)$ is positive semi-definite. Moreover, if $\nabla^2 f(x^\star)$ is positive definite, then choosing $\alpha_1=\cdots=\alpha_d=\frac{1}{2}$, we have:
$$ \frac{1}{t} f(t^{1/2} h) \underset{t \rightarrow 0}{\longrightarrow} \frac{1}{2} h^T \cdot \nabla^2f(x^\star) \cdot h :=g(x) .$$
And using an orthogonal change of variable:
$$ \int_{\mathbb{R}^d} e^{-g(x)}dx = \int_{\mathbb{R}^d} e^{-\frac{1}{2} \sum_{i=1}^d \beta_i y_i^2} dy_1\ldots dy_d < \infty ,$$
where the eigenvalues $\beta_i$ are positive. However, if $\nabla^2f(x^\star)$ is not positive definite, then some of the $\beta_i$ are zero and the integral does not converge.
\subsection{Statement of the problem}
\label{subsection:statement_of_problem}
We still consider the function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and assume that $f \in \mathscr{A}_p^\star(x^\star)$ for some $x^\star \in \mathbb{R}^d$ and some integer $p \ge 1$. Then our objective is to find $\alpha_1 \ge \cdots \ge \alpha_d \in (0,+\infty)$ and an orthogonal transformation $B \in \mathcal{O}_d(\mathbb{R})$ such that:
\begin{equation}
\label{eq:alpha_developpement}
\forall h \in \mathbb{R}^d, \ \ \frac{1}{t} \left[ f(x^\star + B \cdot (t^\alpha \ast h)) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} g(h_1,\ldots,h_d),
\end{equation}
where $t^\alpha$ denotes the vector $(t^{\alpha_1},\ldots,t^{\alpha_d})$ and where $g : \mathbb{R}^d \rightarrow \mathbb{R}$ is a measurable function which is not constant in any $h_1, \ \ldots, \ h_d$, i.e. for all $i \in \lbrace 1,\ldots,d \rbrace$, there exist $h_1, \ \ldots, \ h_{i-1}, \ h_{i+1},\ \ldots, \ h_d \in \mathbb{R}^d$ such that
\begin{equation}
\label{eq:def_non_constant}
h_i \mapsto g(h_1,\ldots,h_d) \text{ is not constant.}
\end{equation}
Then we say that $\alpha_1, \ \ldots, \ \alpha_d$, $B$ and $g$ are a solution of the problem \eqref{eq:alpha_developpement}.
The hypothesis that $g$ is not constant in any of its variables is important ; otherwise, we could simply take $\alpha_1 = \cdots = \alpha_d = 1$ and obtain, by the first order condition:
$$ \frac{1}{t} \left[ f(x^\star + t(h_1,\ldots,h_d)) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} 0 .$$
\subsection{Main results : rate of convergence of Gibbs measures}
\label{section:main_result}
\begin{theorem}[Single well case]
\label{theorem:single_well}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be $\mathcal{C}^{2p}$ with $p \in \mathbb{N}$ and such that:
\begin{enumerate}
\item $f$ is coercive, i.e. $f(x) \longrightarrow + \infty$ when $||x|| \rightarrow + \infty$.
\item $\textup{argmin}(f) = {0}$.
\item $f \in \mathscr{A}_p^\star(0)$ and $f(0)=0$.
\item $e^{-f} \in L^1(\mathbb{R}^d)$.
\end{enumerate}
Let $(E_k)_k$, $(\alpha_i)_i$, $B$ and $g$ to be defined as in Algorithm \ref{algo:algorithm} stated right after, so that for all $h \in \mathbb{R}^d$,
\begin{equation*}
\frac{1}{t} \left[ f\left( x^\star + B \cdot (t^\alpha \ast h) \right) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} g(h) ,
\end{equation*}
and where $g$ is not constant in any of its variables. Moreover, assume that $g$ is coercive and the following technical hypothesis if $p \ge 5$:
\begin{align}
\label{equation:even_terms_null}
&\forall h \in \mathbb{R}^d, \ \forall (i_1,\ldots,i_p) \in \lbrace 0,2,\cdots,2p \rbrace^p, \\
& \frac{i_1}{2} + \cdots + \frac{i_p}{2p} < 1 \implies \ \nabla^{i_1+\cdots+i_p}f(x^\star) \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_{p}}}(h)^{\otimes i_{p}} = 0 . \nonumber
\end{align}
Then the conclusion of Theorem \ref{theorem:athreya:1} holds, with:
$$ \left(\frac{1}{t^{\alpha_1}}, \ldots, \frac{1}{t^{\alpha_d}}\right) \ast (B^{-1} \cdot X_t) \overset{\mathscr{L}}{\longrightarrow} X \ \text{ as } t \rightarrow 0,$$
where $X$ has a density proportional to $e^{-g(x)}$.
\end{theorem}
\begin{algorithm}
\label{algo:algorithm}
Let $f \in \mathscr{A}_p^\star(x^\star)$ for $p \in \mathbb{N}$.
\begin{enumerate}
\item Define $(F_k)_{0 \le k \le p-1}$ recursively as:
$$ \left\lbrace \begin{array}{l}
F_0 = \mathbb{R}^d \\
F_k = \lbrace h \in F_{k-1} : \ \forall h' \in F_{k-1}, \ \nabla^{2k} f(x^\star) \cdot h \otimes h'^{\otimes 2k-1} = 0 \rbrace.
\end{array} \right. $$
\item For $1 \le k \le p-1$, define the subspace $E_k$ as the orthogonal complement of $F_k$ in $F_{k-1}$. By abuse of notation, define $E_p := F_{p-1}$.
\item Define $B \in \mathcal{O}_d(\mathbb{R})$ as an orthogonal transformation adapted to the decomposition
$$ \mathbb{R}^d = E_1 \oplus \cdots \oplus E_p .$$
\item Define for $1 \le i \le d$,
\begin{equation}
\alpha_i := \frac{1}{2j} \ \text{ for } i \in \lbrace \dim(E_1)+\cdots+\dim(E_{j-1})+1, \ldots, \dim(E_1)+\cdots+\dim(E_j) \rbrace .
\end{equation}
\item Define $g : \mathbb{R}^d \rightarrow \mathbb{R}$ as
\begin{equation}
\label{eq:def_g}
g(h) = \sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}=1 }} \binom{k}{i_1,\ldots,i_p} \nabla^k f(x^\star) \cdot p_{_{E_1}}(B \cdot h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(B \cdot h)^{\otimes i_p} .
\end{equation}
\end{enumerate}
\end{algorithm}
\textbf{Remarks :}
\begin{enumerate}
\item The function $g$ is not unique, as we can choose any base $B$ adapted to the decomposition $\mathbb{R}^d = E_1 \oplus \cdots \oplus E_p$.
\item The case $p \ge 5$ is fundamentally different from the case $p \le 4$, since Algorithm \ref{algo:algorithm} may fail to provide such $(E_k)_k$, $(\alpha_i)_i$, $B$ and $g$ if the technical hypothesis \eqref{equation:even_terms_null} is not fulfilled, as explained in Section \ref{section:order_10}. This yields fewer results for the case $p \ge 5$.
\item For $p \in \lbrace 1,2,3,4 \rbrace$, the detail the expression of $g$ in \eqref{equation:order_2}, \eqref{equation:order_4}, \eqref{equation:order_6} and \eqref{equation:order_8:2} respectively.
\item The function $g$ has the following general properties : $g$ is a non-negative polynomial of order $2p$; $g(0)=0$ and $\nabla g(0) = 0$.
\item The condition on $g$ to be coercive may seem not natural. We give more details about the case where $g$ is not coercive in Section \ref{section:non_coercive} and give a way to deal with the simple generic case of non-coercivity. However dealing with the general case where $g$ is not coercive goes beyond the scope of our work.
\item The hypothesis that $g$ is coercive is a necessary condition for $e^{-g} \in L^1(\mathbb{R}^d)$. We actually prove in Proposition \ref{prop:coercive} that it is a sufficient condition.
\end{enumerate}
\medskip
Still following \cite{athreya2010}, we study the multiple well case, i.e. the global minimum is attained in a finite number of points in $\mathbb{R}^d$, say $\lbrace x_1^\star,\ldots,x_m^\star \rbrace$ for some $m \in \mathbb{N}$. In this case, the limiting measure of $\pi_t$ will have its support in $\lbrace x_1^\star,\ldots,x_m^\star \rbrace$, with different weights.
\begin{theorem}[Athreya-Hwang, 2010]
\label{theorem:athreya:2}
Let $f:\mathbb{R}^d \rightarrow [0,\infty)$ measurable such that:
\begin{enumerate}
\item $e^{-f} \in L^1(\mathbb{R}^d)$.
\item For all $\delta > 0$, $\inf \lbrace f(x), \ ||x - x_i^\star|| > \delta, \ 1 \le i \le m \rbrace > 0$.
\item There exist $(\alpha_{ij})_{\substack{1\le i \le m \\ 1 \le j \le d}}$ such that for all $i$, $j$, $\alpha_{ij} \ge 0$ and for all $i$:
$$ \frac{1}{t} f(x_i^\star + (t^{\alpha_{i1}}h_1,\ldots,t^{\alpha_{id}}h_d)) \underset{t \rightarrow 0}{\longrightarrow} g_i(h_1,\ldots,h_d) \in [0,\infty) .$$
\item For all $i \in \lbrace 1, \ldots, m \rbrace$,
$$ \int_{\mathbb{R}^d} \sup_{0<t<1} e^{-\frac{f(x_i^\star + (t^{\alpha_{i1}}h_1,\ldots,t^{\alpha_{id}}h_d))}{t}}dh_1\ldots dh_d < \infty .$$
\end{enumerate}
Then, let $\alpha := \min_{1 \le i \le m} \left\lbrace \sum_{j=1}^d \alpha_{ij} \right\rbrace $ and let $J := \left\lbrace i \in \lbrace 1,\ldots,m \rbrace : \ \sum_{j=1}^d \alpha_{ij} = \alpha \right\rbrace$. For $0 < t < 1$, let $X_t$ be a random vector with distribution $\pi_t$. Then:
$$ X_t \overset{\mathscr{L}}{\underset{t \rightarrow 0}{\longrightarrow}} \frac{1}{\sum_{j \in J} \int_{\mathbb{R}^d} e^{-g_j(x)}dx} \sum_{i \in J} \int_{\mathbb{R}^d} e^{-g_i(x)}dx \cdot \delta_{x_i^\star} .$$
\end{theorem}
\begin{theorem}[Multiple well case]
\label{theorem:multiple_well}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be $\mathcal{C}^{2p}$ for $p \in \mathbb{N}$ and such that:
\begin{enumerate}
\item $f$ is coercive i.e. $f(x) \longrightarrow + \infty$ when $||x||\rightarrow +\infty$.
\item $\argmin(f) = \lbrace x_1^\star, \ldots, x_m^\star \rbrace$ and for all $i$, $f(x_i^\star)=0$.
\item For all $i \in \lbrace 1, \ldots, m \rbrace$, $f \in \mathscr{A}^\star_{p_i}(x_i^\star)$ for some $p_i \le p$.
\item $e^{-f} \in L^1(\mathbb{R}^d)$.
\end{enumerate}
Then, for every $i \in \lbrace 1, \ldots, m \rbrace$, we consider $(E_{ik})_k$, $(\alpha_{ij})_j$, $B_i$ and $g_i$ as defined in Algorithm \ref{algo:algorithm}, where we consider $f$ to be in $\mathscr{A}_{p_i}^\star(x_i^\star)$, so that for every $h \in \mathbb{R}^d$:
$$ \frac{1}{t} f(x_i^\star + B_i \cdot (t^{\alpha_i} \ast h)) \underset{t \rightarrow 0}{\longrightarrow} g_i(h_1,\ldots,h_d) \in [0,\infty) ,$$
where $t^{\alpha_i}$ is the vector $(t^{\alpha_{i1}},\ldots, t^{\alpha_{id}})$ and where $g_i$ is not constant in any of its variables. Furthermore, we assume that for all $i$, $g_i$ is coercive and the following technical hypothesis for every $i$ such that $p_i \ge 5$:
\begin{align*}
& \forall h \in \mathbb{R}^d, \ \forall (i_1,\ldots,i_{p_i}) \in \lbrace 0,2,\ldots,2{p_i} \rbrace^{p_i}, \\
& \frac{i_1}{2} + \cdots + \frac{i_{p_i}}{2p} < 1 \implies \ \nabla^{i_1+\cdots+i_{p_i}}f(x_i^\star) \cdot p_{_{E_{i1}}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_{ip_i}}}(h)^{\otimes i_{p_i}} = 0 . \nonumber
\end{align*}
Let $\alpha := \min_{1 \le i \le m} \left\lbrace \sum_{j=1}^d \alpha_{ij} \right\rbrace $ and let $J := \left\lbrace i \in \lbrace 1,\ldots,m \rbrace : \ \sum_{j=1}^d \alpha_{ij} = \alpha \right\rbrace$. Then:
$$ X_t \underset{t \rightarrow 0}{\longrightarrow} \frac{1}{\sum_{j \in J} \int_{\mathbb{R}^d} e^{-g_j(x)}dx} \sum_{i \in J} \int_{\mathbb{R}^d} e^{-g_i(x)}dx \cdot \delta_{x_i^\star} .$$
Moreover, let $\delta > 0$ be small enough so that the balls $\mathcal{B}(x_i^\star,\delta)$ are disjoint, and define the random vector $X_{it}$ to have the law of $X_t$ conditionally to the event $||X_t - x_i^\star||< \delta$. Then:
$$\left(\frac{1}{t^{\alpha_{i1}}}, \ldots,\frac{1}{t^{\alpha_{id}}}\right) \ast (B_i^{-1} \cdot X_{it}) \overset{\mathscr{L}}{\longrightarrow} X_i \ \text{ as } t \rightarrow 0 ,$$
where $X_i$ has a density proportional to $e^{-g_i(x)}$.
\end{theorem}
\section{Expansion of $f$ at a local minimum with degenerate derivatives}
\label{section:expansion}
In this section, we aim at answering to the problem stated in \eqref{eq:alpha_developpement} in order to devise conditions to apply Theorem \ref{theorem:athreya:1}. This problem can also be considered in a more general setting, independently of the study of the convergence of Gibbs measures. It provides a non degenerate higher order nested expansion of $f$ at a local minimum when some of the derivatives of $f$ are degenerate. Note here that we only need $x^\star$ to be a local minimum instead of a global minimum, since we only give local properties.
For $k \le p$, we define the tensor of order $k$, $T_k := \nabla^k f(x^\star)$.
\subsection{Expansion of $f$ for any order $p$}
\label{sec:expansion_any_order}
In this section, we state our result in a synthetic form. The proofs of the cases $p =1,2,3,4$ are individually detailled in Sections \ref{section:order_2}, \ref{section:order_4}, \ref{section:order_6} and \ref{section:order_8} respectively.
\begin{theorem}
\label{theorem:main}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be $\mathcal{C}^{2p}$ for some $p \in \mathbb{N}$ and assume that $f \in \mathscr{A}_p^\star(x^\star)$ for some $x^\star \in \mathbb{R}^d$.
\begin{enumerate}
\item If $p \in \lbrace 1,2,3,4 \rbrace$, then there exists orthogonal subspaces of $\mathbb{R}^d$, $E_1, \ \ldots, \ E_{p}$ such that
$$ \mathbb{R}^d = E_1 \oplus \cdots \oplus E_{p},$$
and satisfying for every $h \in \mathbb{R}^d$:
\begin{align}
\label{equation:order_p:1}
& \frac{1}{t} \left[ f\left(x^\star + t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_{p}}}(h) \right) - f(x^\star) \right] \\
\label{equation:order_p:2}
\underset{t \rightarrow 0}{\longrightarrow} & \sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\cdots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}=1 }} \binom{k}{i_1,\ldots,i_{p}} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_{p}}}(h)^{\otimes i_{p}}.
\end{align}
The convergence is uniform with respect to $h$ on every compact set. Moreover, let $B \in \mathcal{O}_d(\mathbb{R})$ be an orthogonal transformation adapted to the decomposition $E_1 \oplus \cdots \oplus E_{p}$, then
\begin{equation}
\label{equation:order_p:3}
\frac{1}{t} \left[ f\left( x^\star + B \cdot (t^\alpha \ast h) \right) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} g(h) ,
\end{equation}
where
\begin{equation}
\label{eq:def_g:2}
g(h) = \sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}=1 }} \binom{k}{i_1,\ldots,i_{p}} T_k \cdot p_{_{E_1}}(B \cdot h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_{p}}}(B \cdot h)^{\otimes i_p}
\end{equation}
is not constant in any of its variables $h_1, \ \ldots, \ h_d$ and
\begin{align}
\label{eq:def_alpha:2}
\alpha_i & := \frac{1}{2j} \ \text{ for } i \in \lbrace \dim(E_1)+\cdots+\dim(E_{j-1})+1, \ldots, \dim(E_1)+\cdots+\dim(E_j) \rbrace .
\end{align}
\item If $p \ge 5$ and if there exist orthogonal subspaces of $\mathbb{R}^d$, $E_1, \ \ldots, \ E_{p}$ such that
$$ \mathbb{R}^d = E_1 \oplus \cdots \oplus E_{p}$$
and satisfying the following additional assumption
\begin{align}
\label{equation:even_terms_null:2}
& \forall h \in \mathbb{R}^d, \ \forall (i_1,\ldots,i_p) \in \lbrace 0,2,\ldots,2p \rbrace^{p}, \\
& \frac{i_1}{2} + \cdots + \frac{i_p}{2p} < 1 \implies \ T_{i_1+\cdots+i_p} \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_{p}}}(h)^{\otimes i_{p}} = 0 , \nonumber
\end{align}
then \eqref{equation:order_p:2} stills holds true, as well as the uniform convergence on every compact set. Moreover, if $B \in \mathcal{O}_d(\mathbb{R})$ is an orthogonal transformation adapted to the previous decomposition, then \eqref{equation:order_p:3} still hold true. However, depending on the function $f$, such subspaces do not necessarily exist.
\end{enumerate}
\end{theorem}
\medskip
\textbf{Remarks:}
\begin{enumerate}
\item The limit \eqref{equation:order_p:2} can be rewritten as:
$$\sum_{k=2}^{2p} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\cdots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}=1 }} T_k \cdot \frac{p_{_{E_1}}(h)^{\otimes i_1}}{i_1!} \otimes \cdots \otimes \frac{p_{_{E_{p}}}(h)^{\otimes i_{p}}}{i_p!}. $$
\item For $p \in \lbrace 1,2,3,4 \rbrace$, we explicitly give the expression of the sum \eqref{equation:order_p:2} and the $p$-tuples $(i_1,\ldots,i_p)$ such that $\frac{i_1}{2}+\cdots+\frac{i_p}{2p}=1$, in \eqref{equation:order_2}, \eqref{equation:order_4}, \eqref{equation:order_6} and \eqref{equation:order_8:2} respectively.
\item For $p \in \lbrace 1,2,3,4 \rbrace$, we give in Algorithm \ref{algo:algorithm} an explicit construction of the orthogonal subspaces $E_1, \ \ldots, \ E_{p}$ as complementaries of annulation sets of some derivatives of $f$.
\item The case $p \ge 5$ is fundamentally different from the case $p \in \lbrace 1,2,3,4 \rbrace$. The strategy of proof developed for $p \in \lbrace 1,2,3,4 \rbrace$ fails if the assumption \eqref{equation:even_terms_null:2} is not satisfied. In \ref{section:order_10} a counter-example is detailed. The case $p \ge 5$ yields fewer results than for $p \le 4$, as the assumption \eqref{equation:even_terms_null:2} is strong.
\item For $p \ge 5$, such subspaces $E_1$, $\ldots$, $E_p$ may also be obtained from Algorithm \ref{algo:algorithm}, however \eqref{equation:even_terms_null:2} is not necessarily true in this case.
\end{enumerate}
\medskip
The proof of Theorem \ref{theorem:main} is given first individually for each $p \in \lbrace 1,2,3,4 \rbrace$, in Sections \ref{section:order_2}, \ref{section:order_4}, \ref{section:order_6}, \ref{section:order_8} respectively. The proof for $p \ge 5$ is given in Section \ref{section:order_10}. The proof of the uniform convergence and of the fact that $g$ is not constant is given in Section \ref{subsec:unif_non_constant}.
\subsection{Review of the one dimensional case}
We review the case $d=1$, as it guides us for the proof in the case $d \ge 2$. The strategy is to find the first derivative $f^{(m)}(x^\star)$ which is non zero and then to choose $\alpha_1 = 1/m$.
\begin{proposition}
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be $\mathcal{C}^p$ for some $p \in \mathbb{N}$ and let $x^\star$ be a strict polynomial local minimum of $f$. Then :
\begin{enumerate}
\item The order of the local minimum $m$ is an even number and $f^{(m)}(x^\star) > 0$.
\item $ f(x^\star + h) \underset{h \rightarrow 0}{=} f(x^\star) + \frac{f^{(m)}(x^\star)}{m!}h^p + o(h^m) $
\end{enumerate}
\end{proposition}
Then $\alpha_1 := 1/m$ is the solution of \eqref{eq:alpha_developpement} and
$$\frac{1}{t} (f(x^\star + t^{1/m}h) - f(x^\star)) \underset{t \rightarrow 0}{\longrightarrow} \frac{f^{(m)}(x^\star)}{m!} h^m $$
which is a non-constant function of $h$, since $f^{(m)}(x^\star) \ne 0$.
The direct proof using the Taylor formula is left to the reader.
\subsection{Proof of Theorem \ref{theorem:main} for $p=1$}
\label{section:order_2}
Let $f \in \mathscr{A}^\star_1(x^\star)$. The assumption that $x^\star$ is a strict polynomial local minimum at order $2$ implies that $\nabla^2 f(x^\star)$ is positive definite. Let us denote $(\beta_i)_{1 \le i \le d}$ its positive eigenvalues. By the spectral theorem, let us write $\nabla^2 f(x^\star ) = B {\rm Diag}(\beta_{1:d})B^T$ for some $B \in \mathcal{O}_d(\mathbb{R})$. Then:
\begin{equation}
\label{equation:order_2}
\frac{1}{t} (f(x^\star + t^{1/2}B \cdot h) - f(x^\star)) \underset{t \rightarrow 0}{\longrightarrow} \frac{1}{2}\sum_{i=1}^{d} \beta_i h_i^2.
\end{equation}
Thus, a solution of \eqref{eq:alpha_developpement} is $\alpha_1=\cdots=\alpha_d=\frac{1}{2}$, $B$, and $g(h_1,\ldots,h_d)=\frac{1}{2} \sum_{i=1}^d \beta_i h_i^2$, which is a non-constant function of every $h_1, \ \ldots, \ h_d$, since for all $i$, $\beta_i$ is positive.
In the following, our objective is to establish a similar result when $\nabla^2 f(x^\star)$ is not necessarily positive definite.
\subsection{Proof of Theorem \ref{theorem:main} for $p=2$}
\label{section:order_4}
\begin{theorem}
\label{theorem:order_4}
Let $f \in \mathscr{A}_2(x^\star)$. Then there exist orthogonal subspaces $E$ and $F$ such that $\mathbb{R}^d = E \oplus F$, and that for all $h \in \mathbb{R}^d$:
\begin{align}
\frac{1}{t} & \left[ f(x^\star + t^{1/2}p_{_{E}}(h) + t^{1/4}p_{_F}(h)) - f(x^\star)\right] \nonumber \\
\label{equation:order_4}
\underset{t \rightarrow 0}{\longrightarrow} & \ \frac{1}{2} \nabla^2 f(x^\star) \cdot p_{_E}(h)^{\otimes 2} + \frac{1}{2} \nabla^3 f(x^\star) \cdot p_{_E}(h) \otimes p_{_F}(h)^{\otimes 2} + \frac{1}{4!} \nabla^4 f(x^\star)\cdot p_{_F}(h)^{\otimes 4} .
\end{align}
Moreover, if $f \in \mathscr{A}_2^\star (x^\star)$, then this is a solution to the problem \eqref{eq:alpha_developpement}, with $E_1 = E$, $E_2=F$, $\alpha$ defined in \eqref{eq:def_alpha:2}, $B$ adapted to the previous decomposition and $g$ defined in \eqref{eq:def_g:2}.
\end{theorem}
\noindent \textbf{Remark:} The set of $2$-tuples $(i_1,i_2)$ such that $\frac{i_1}{2} + \frac{i_2}{4} = 1$, are $(2,0)$, $(1,2)$ and $(0,4)$, which gives the terms appearing in the sum in \eqref{equation:order_p:2}.
\begin{proof}
Let $F := \{ h \in \mathbb{R}^d : \ \nabla^2 f(x^\star) \cdot h^{\otimes 2} = 0 \}$. By the spectral theorem and since $\nabla^2 f(x^\star)$ is positive semi-definite, $F = \{ h \in \mathbb{R}^d : \ \nabla^2 f(x^\star) \cdot h = 0^{\otimes 1} \}$ is a vector subspace of $\mathbb{R}^d$. Let $E$ be the orthogonal complement of $F$ in $\mathbb{R}^d$.
For $h \in \mathbb{R}^d$ we expand the left term of \eqref{equation:order_4} using the Taylor formula up to order $4$ and the multinomial formula \eqref{equation:multinomial}, giving
$$ \sum_{k=2}^4 \frac{1}{k!} \sum_{\substack{i_1,i_2 \in \lbrace 0,\ldots,k \rbrace \\ i_1+i_2=k}} \binom{k}{i_1,i_2} t^{\frac{i_1}{2} + \frac{i_2}{4}-1} T_k \cdot p_{_E}(h)^{\otimes i_1} \otimes p_{_F}(h)^{\otimes i_2} + o(1).$$
The terms with coefficient $t^a$, $a>0$, are $o(1)$ as $t \rightarrow 0$.
By definition of $F$ we have $\nabla^2 f(x^\star) \cdot p_{_F}(h) = 0^{\otimes 1}$, so we also have
$$\nabla ^3 f(x^\star) \cdot p_{_F}(h)^{\otimes 3} = 0$$ by the local minimum condition. This yields the convergence stated in \eqref{equation:order_4}.
Moreover, if $x^\star$ is a local minimum of polynomial order 4, then by the local minimum condition, $\nabla^4 f(x^\star) > 0$ on $F$ in the sense of \eqref{eq:tensor_positive_def}. Moreover, since $\nabla^2 f(x^\star) > 0$ on $E$, then the limit is not constant in any $h_1, \ \ldots, \ h_d$.
\end{proof}
\noindent \textbf{Remark:} The cross odd term is not necessarily null. For example, consider
$$ \begin{array}{rrl}
f : & \mathbb{R}^2 & \longrightarrow \mathbb{R} \\
& (x,y) & \longmapsto x^2 + y^4 + xy^2.
\end{array} $$
Then $f$ admits a global minimum at $x^\star=0$ since $|xy^2| \le \frac{1}{2}(x^2+y^4)$. We have $E_1 = \mathbb{R}(1,0)$, $E_2 = \mathbb{R}(0,1)$ and for all $(x,y) \in \mathbb{R}^2$, $T_3 \cdot (xe_1) \otimes (ye_2)^{\otimes 2} = 2xy^2$ is not identically null.
\subsection{Difficulties beyond the 4th order and Hilbert's $17^{\text{th}}$ problem}
\label{section:hilbert}
If we do not assume as in the previous section that $\nabla^4 f(x^\star)$ is not positive on $F$, then we carry on the development of $f(x^\star + h)$ up to higher orders.
A first idea is to consider $F_2 := \{ h \in F: \ \nabla^4 f(x^\star) \cdot h^{\otimes 4}=0 \} \subseteq F$ and $E_2$ a complement subspace of $F_2$ in $F$, and to continue this process by induction as in Section \ref{section:order_4}. However, $F_2$ is not necessarily a subspace of $F$.
Indeed, let $T$ be a symmetric tensor defined on $\mathbb{R}^{d'}$ of order $2k$ with $k \in \mathbb{N}$. As $T$ is symmetric, there exist vectors $v^1, \ \ldots, \ v^q \in \mathbb{R}^{d'}$, and scalars $\lambda_1, \ \ldots, \ \lambda_q \in \mathbb{R}$ such that $T = \sum_i \lambda_i (v^i)^{\otimes 2k}$ (see \cite{comon2008}, Lemma 4.2.), so
$$ \forall h \in \mathbb{R}^{d'}, \ T \cdot h^{\otimes 2k} = \sum_{i=1}^q \lambda_i (v^i)^{\otimes 2k} \cdot h^{\otimes 2k} = \sum_{i=1}^q \lambda_i \langle v^i , h \rangle^{2k} .$$
For $k = 2$ and $T = \nabla^{2k} f(x^\star)_{|_F}$, since $x^\star$ is a local minimum, we have, identifying $F$ and $\mathbb{R}^{d'}$,
$$ \forall h \in \mathbb{R}^{d'}, \ T \cdot h^{\otimes 2k} \ge 0 $$
Then, we could think it implies that for all $i$, $\lambda_i \ge 0$, and then
$$ T \cdot h^{\otimes 2k} = 0 \ \implies \forall i, \ \langle v^i, h \rangle = 0$$
which would give a linear caracterization of $\{h \in \mathbb{R}^{d'} : \ T \cdot h^{\otimes 2k} = 0 \}$ and in this case, $F_2$ would be a subspace of $F$.
However this reasoning is not correct in general as we do not have necessarily that for all $i$, $\lambda_i \ge 0$.
We can build counter-examples as follows. Since $T$ is a non-negative symmetric tensor, $T$ can be seen as a non-negative homogeneous polynomial of degree $2k$ with $d'$ variables. A counter-example at order $2k=4$ is $T(X,Y,Z) = ((X-Y)(X-Z))^2$, which is a non-negative polynomial of order 4, but $\{T=0\} = \{ X=Y \text{ or } X=Z \}$, which is not a vector space.
Another counterexample given in \cite{motzkin1967} at order $2k = 6$ is the following. We define
$$ T(X,Y,Z) = Z^6 + X^4 Y^2 + X^2 Y^4 - 3 X^2 Y^2 Z^2 $$
By the arithmetic-geometric mean inequality and its equality case, $T$ is non-negative and $T(x,y,z) = 0$ if and only if $z^6 = x^4 y^2 = x^2 y^4 $, so that
$$\{ T = 0 \} =
\mathbb{R} \begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix} \
\cup \ \mathbb{R} \begin{pmatrix}
- 1 \\
1 \\
1
\end{pmatrix} \
\cup \ \mathbb{R} \begin{pmatrix}
1 \\
- 1 \\
1
\end{pmatrix} \
\cup \ \mathbb{R} \begin{pmatrix}
1 \\
1 \\
- 1
\end{pmatrix}. $$
Hence, $\{ T = 0 \}$ is not a subspace of $\mathbb{R}^{3}$. In particular $T$ cannot be written as $\sum_i \lambda_i (v^i)^{\otimes 2k}$ with $\lambda_i \ge 0$.
In fact, this problem is linked with the Hilbert's seventeenth problem that we recall below.
\begin{problem}[Hilbert's seventeeth problem]
Let $P$ be a non-negative polynomial with $d'$ variables, homogeneous of even degree $2k$. Find polynomials $P_1, \ \ldots, \ P_r$ with $d'$ variables, homogeneous of degree $k$, such that $P = \sum_{i=1}^r P_i^2$
\end{problem}
Hilbert proved in 1888 \cite{hilbert1888} that there does not always exist a solution. In general $\{T=0\}$ is not even a submanifold of $\mathbb{R}^{d'}$. Indeed, taking $T :h \mapsto \nabla^{2k} f(x^\star) \cdot h^{\otimes 2k}$, we have $\partial_h T\cdot h = 2k \nabla^{2k} f(x^\star) \cdot h^{\otimes 2k-1} $ is not surjective in $h=0$, so the surjectivity condition for $\lbrace T=0 \rbrace$ to be a submanifold is not fulfilled.
\subsection{Proof of Theorem \ref{theorem:main} for $p=3$}
\label{section:order_6}
We slightly change our strategy of proof developed in Section \ref{section:order_4}. For $k \ge 2$, we define $F_k$ recursively as
\begin{equation}
\label{eq:F_k_def}
F_k := \lbrace h \in F_{k-1} : \ \forall h' \in F_{k-1}, \ \nabla^{2k} f(x^\star) \cdot h \otimes h'^{\otimes 2k-1} = 0 \rbrace,
\end{equation}
instead of $\lbrace h \in F_{k-1} : \ \nabla^{2k} f(x^\star) \cdot h^{\otimes 2k} = 0 \rbrace$. Then, by construction, $F_k$ is a vector subspace of $\mathbb{R}^d$.
\begin{theorem}
\label{theorem:order_6}
Let $f \in \mathscr{A}_3(x^\star)$. Then there exist orthogonal subspaces of $\mathbb{R}^d$, $E_1$, $E_2$ and $F_2$, such that
$$ \mathbb{R}^d = E_1 \oplus E_2 \oplus F_2 ,$$
and such that for all $h \in \mathbb{R}^d$,
\begin{align}
\label{equation:order_6}
\frac{1}{t} & \left[ f(x^\star + t^{1/2}p_{_{E_1}}(h) + t^{1/4}p_{_{E_2}}(h) + t^{1/6}p_{_{F_2}}(h)) - f(x^\star) \right] \\
\underset{t \rightarrow 0}{\longrightarrow} & \ \frac{1}{2} \nabla^2 f(x^\star) \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{1}{2} \nabla^3 f(x^\star) \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h)^{\otimes 2} + \frac{1}{4!} \nabla^4 f(x^\star)\cdot p_{_{E_2}}(h)^{\otimes 4} \nonumber \\
& + \frac{4}{4!}\nabla^4 f(x^\star)\cdot p_{_{E_1}}(h)\otimes p_{_{F_2}}(h)^{\otimes 3} + \frac{10}{5!}\nabla^5 f(x^\star) \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_2}}(h)^{\otimes 3} + \frac{1}{6!}\nabla^6 f(x^\star) \cdot p_{_{F_2}}(h)^{\otimes 6}. \nonumber
\end{align}
Moreover, if $f \in \mathscr{A}_3^\star(x^\star)$, then this is a solution to the problem \eqref{eq:alpha_developpement}, with $E_3 = F_2$, $\alpha$ defined in \eqref{eq:def_alpha:2}, $B$ adapted to the previous decomposition and $g$ defined in \eqref{eq:def_g:2}.
\end{theorem}
\noindent \textbf{Remark:} The set of $3$-tuples $(i_1,i_2,i_3)$ such that $\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6} = 1$, are $(2,0,0)$, $(1,2,0)$, $(0,4,0)$, $(1,0,3)$, $(0,2,3)$, $(0,0,6)$, which gives the terms appearing in \eqref{equation:order_p:2}.
\begin{proof}
We consider the subspace
$$ F_1 := \lbrace h \in \mathbb{R}^d : \ T_2 \cdot h^{\otimes 2} = 0 \rbrace = \lbrace h \in \mathbb{R}^d : \ T_2 \cdot h = 0^{\otimes 1} \rbrace, $$
since $T_2 \ge 0$.
Then, let $E_1$ be the orthogonal complement of $F_1$ in $\mathbb{R}^d$ and consider the vector subspace of $F_1$ defined by
$$ F_2 = \lbrace h \in F_1 : \ \forall h' \in F_1, \ T_4 \cdot h \otimes h'^{\otimes 3} = 0 \rbrace .$$
Let $E_2$ be the orthogonal complement of $F_2$ in $F_1$. Then we have
$$ \mathbb{R}^d = E_1 \oplus F_1 = E_1 \oplus E_2 \oplus F_2 .$$
For $h \in \mathbb{R}^d$ we expand the left term of \eqref{equation:order_6} using the Taylor formula up to order $6$ and the multinomial formula \eqref{equation:multinomial}, giving
$$ \sum_{k=2}^6 \frac{1}{k!} \sum_{\substack{i_1,i_2,i_3 \in \lbrace 0,\ldots,k \rbrace \\ i_1+i_2+i_3=k}} \binom{k}{i_1,i_2,i_3} t^{\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6}-1} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes p_{_{E_2}}(h)^{\otimes i_2} \otimes p_{_{F_2}}(h)^{\otimes i_3} + o(1),$$
and we prove the convergence stated in \eqref{equation:order_6}.
\medskip
All the terms with coefficient $t^a$ where $a>0$ are $o(1)$ as $t \rightarrow 0$.
\textbf{Order 2:} we have $T_2 \cdot p_{_{E_2}}(h) = 0^{\otimes 1}$ and $T_2 \cdot p_{_{F_2}}(h) = 0^{\otimes 1}$ so the only term for $k=2$ is $\frac{1}{2} T_2 \cdot p_{_{E_1}} (h)^{\otimes 2}$.
\medskip
\textbf{Order 3:} $\triangleright$ Since $x^\star$ is a local minimum and $T_2 \cdot p_{_{F_1}}(h)^{\otimes 2} = 0$, we have $T_3 \cdot p_{_{F_1}}(h)^{\otimes 3} = 0$. Then, using property Proposition \ref{proposition:null_tensor:1}, if the factor $p_{_{E_1}}(h)$ does not appear as an argument in $T_3$, then the corresponding term is zero.
$\triangleright$ Let us prove that
\begin{equation*}
T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 2} = 0 .
\end{equation*}
Using Theorem \ref{theorem:order_4} with $E = E_1$, $F = E_2 \oplus F_2$, we have in particular that for all $h \in \mathbb{R}^d$,
\begin{equation}
\label{equation:order_4_positive}
\frac{1}{2} T_2 \cdot p_{_E}(h)^{\otimes 2} + \frac{1}{2} T_3 \cdot p_{_E}(h) \otimes p_{_F}(h)^{\otimes 2} + \frac{1}{4!} T_4 \cdot p_{_F}(h)^{\otimes 4} \ge 0 .
\end{equation}
Then taking $h \in E_1 \oplus F_2$ so that $h= p_{_{E_1}}(h)+p_{_{F_2}}(h)$ and with
\begin{equation}
\label{eq:proof:T4_null}
\left[T_4\cdot p_{_{F_2}}(h)\right]_{|F_1} \equiv 0^{\otimes 3},
\end{equation}
we may rewrite \eqref{equation:order_4_positive} as
$$ \frac{1}{2} T_2 \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{1}{2} T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 2} \ge 0 .$$
Now, considering $h' = \lambda h$, we have that for all $\lambda \in \mathbb{R}$,
$$ \lambda^2 \left(\frac{1}{2} T_2 \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{\lambda}{2} T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 2} \right) \ge 0 ,$$
so that necessarily $T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 2} = 0$.
$\triangleright$ Let us prove that
$$ T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h) \otimes p_{_{F_2}}(h) = 0 .$$
We use again \eqref{equation:order_4_positive}, with $p_{_F}(h) = p_{_{E_2}}(h) + p_{_{F_2}}(h)$, so that
$$ \frac{1}{2} T_2 \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{1}{2} T_3 \cdot p_{_{E_1}}(h) \otimes \left(p_{_{E_2}}(h) + p_{_{F_2}}(h)\right)^{\otimes 2} + \frac{1}{4!} T_4 \cdot \left(p_{_{E_2}}(h) + p_{_{F_2}}(h)\right)^{\otimes 4} \ge 0 . $$
But using \eqref{eq:proof:T4_null} and that $T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 2} = 0$, we obtain
$$ \frac{1}{2} T_2 \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{1}{2} T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h)^{\otimes 2} + T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h) \otimes p_{_{F_2}}(h) + \frac{1}{4!} T_4 \cdot p_{_{E_2}}(h)^{\otimes 4} \ge 0 . $$
Now, considering $h' = p_{_{E_1}}(h) + p_{_{E_2}}(h) + \lambda p_{_{F_2}}(h)$, we have that for all $\lambda \in \mathbb{R}$,
$$ \frac{1}{2} T_2 \cdot p_{_{E_1}}(h)^{\otimes 2} + \frac{1}{2} T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h)^{\otimes 2} + \lambda T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h) \otimes p_{_{F_2}}(h) + \frac{1}{4!} T_4 \cdot p_{_{E_2}}(h)^{\otimes 4} \ge 0 ,$$
so necessarily $T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h) \otimes p_{_{F_2}}(h) = 0$.
$\triangleright$ The last remaining term for $k=3$ is $\frac{1}{2} T_3 \cdot p_{_{E_1}}(h) \otimes p_{_{E_2}}(h)^{\otimes 2}$.
\medskip
\textbf{Order 4:} If the factor $p_{_{E_1}}(h)$ does not appear and if the factor $p_{_{F_2}}(h)$ appears at least once, then using \eqref{eq:proof:T4_null} the corresponding term is zero. If $p_{_{E_1}}(h)$ appears, the only term with a non-positive exponent of $t$ is $\frac{4}{4!} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{F_2}}(h)^{\otimes 3}$. So the only terms for $k=4$ are $\frac{1}{4!}T_4 \cdot p_{_{E_2}}(h)^{\otimes 4}$ and $\frac{4}{4!} T_4 \cdot p_{_{E_1}}(h)\otimes p_{_{F_2}}(h)^{\otimes 3}$.
\medskip
\textbf{Order 5:} $\triangleright$ The terms where $p_{_{E_1}}(h)$ appears at least once have a coefficient $t^a$ with $a>0$ so are $o(1)$ when $t \rightarrow 0$.
$\triangleright$ We have $T_2 \cdot p_{_{F_2}}(h)^{\otimes 2} = 0$, $T_3 \cdot p_{_{F_2}}(h)^{\otimes 3} = 0$, $T_4 \cdot p_{_{F_2}}(h)^{\otimes 4} = 0$ and since $x^\star$ is a local minimum, we have
$$ T_5 \cdot p_{_{F_2}}(h)^{\otimes 5} = 0 .$$
$\triangleright$ Let us prove that
$$ T_5 \cdot p_{_{E_2}}(h) \otimes p_{_{F_2}}(h)^{\otimes 4} = 0 .$$
Let $h \in \mathbb{R}^d$. We have
$$ \frac{1}{t^{11/12}} \left[ f(x^\star + t^{1/4}p_{_{E_2}}(h) + t^{1/6}p_{_{F_2}}(h)) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} \frac{1}{4!} T_5 \cdot p_{_{E_2}}(h) \otimes p_{_{F_2}}(h)^{\otimes 4} \ge 0 .$$
Hence, considering $h'=\lambda h$, we have for every $\lambda \in \mathbb{R}$,
$$ \lambda^5 T_5 \cdot p_{_{E_2}}(h) \otimes p_{_{F_2}}(h)^{\otimes 4} \ge 0 ,$$
which yields the desired result.
$\triangleright$ The only remaining term for $p=5$ is
$$\frac{10}{5!} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_2}}(h)^{\otimes 3} .$$
\medskip
\textbf{Order 6:} The only term for $k=6$ is $\frac{1}{6!} T_6 \cdot p_{_{F_2}}(h)^{\otimes 6}$ ; the other terms have a coefficient $t^a$ with $a > 0$, so are $o(1)$ when $t \rightarrow 0$.
\end{proof}
\textbf{Remark :} As in Theorem \ref{theorem:order_4} and the remark that follows, the remaining odd cross-terms cannot be proved to be zero using the same method of proof, and may be actually not zero. For example, consider:
$$ \begin{array}{rrl}
f : & \mathbb{R}^2 & \longrightarrow \mathbb{R} \\
& (x,y) & \longmapsto x^4 + y^6 + x^2y^3,
\end{array} $$
which satisfies $h \mapsto \nabla^5 f(x^\star) \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_2}}(h)^{\otimes 3} \not\equiv 0$.
\subsection{Proof of Theorem \ref{theorem:main} for $p=4$}
\label{section:order_8}
\begin{theorem}
\label{theorem:order_8}
Let $f \in \mathscr{A}_4(x^\star)$. Then there exist orthogonal subspaces of $\mathbb{R}^d$, $E_1$, $E_2$, $E_3$ and $F_3$ such that
$$ \mathbb{R}^d = E_1 \oplus E_2 \oplus E_3 \oplus F_3 ,$$
and for all $h \in \mathbb{R}^d$,
\begin{align}
& \frac{1}{t} \left[ f(x^\star + t^{1/2}p_{_{E_1}}(h) + t^{1/4}p_{_{E_2}}(h) + t^{1/6}p_{_{E_3}}(h) + t^{1/8}p_{_{F_3}}(h)) - f(x^\star) \right] \nonumber \\
\label{equation:order_8:2}
\underset{t \rightarrow 0}{\longrightarrow} \ & \sum_{k=2}^{8} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_{4} \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{4} = k }} \binom{k}{i_1,\ldots,i_{4}} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes p_{_{E_2}}(h)^{\otimes i_2} \otimes p_{_{E_3}}(h)^{\otimes i_3} \otimes p_{_{F_3}}(h)^{\otimes i_4}.
\end{align}
These terms are summarized as tuples $(i_1,\ldots,i_4)$ in Table \ref{figure:terms_exponent_1}. Moreover, if $f \in \mathscr{A}_4^\star(x^\star)$, then this is a solution to \eqref{eq:alpha_developpement}, with $E_4 = F_3$, $\alpha$ defined in \eqref{eq:def_alpha:2}, $B$ adapted to the previous decomposition and $g$ defined in \eqref{eq:def_g:2}.
\end{theorem}
\begin{table}
\centering
\begin{tabular}{|c|c|}
\hline
\rule[-1ex]{0pt}{2.5ex} Order $2$ & $(2,0,0,0)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $3$ & $(2,1,0,0)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $4$ & $(0,4,0,0), \ (1,1,0,2), \ (1,0,3,0)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $5$ & $(1,0,0,4), \ (0,2,3,0), \ (0,3,0,2)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $6$ & $(0,1,3,2), \ (0,2,0,4), \ (0,0,6,0)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $7$ & $(0,1,0,6), \ (0,0,3,4)$ \\
\hline
\rule[-1ex]{0pt}{2.5ex} Order $8$ & $(0,0,0,8)$ \\
\hline
\end{tabular}
\caption{Terms expressed as $4$-tuples in the development \eqref{equation:order_8:2}}
\label{figure:terms_exponent_1}
\end{table}
\begin{proof}
As before, we define the subspaces $F_0 := \mathbb{R}^d$ and by induction:
$$ F_k = \left\lbrace h \in F_{k-1} : \ \forall h' \in F_{k-1}, \ T_{2k} \cdot h \otimes h'^{\otimes 3} = 0 \right\rbrace $$
for $k=1,2,3$. We define $E_k$ as the orthogonal complement of $F_k$ in $F_{k-1}$ for $k=1,2,3$, so that
$$ \mathbb{R}^d = E_1 \oplus E_2 \oplus E_3 \oplus F_3 .$$
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$E_1$ & \multicolumn{3}{c|}{$F_1$} \\
\hline
\multirow{5}{*}{$T_2 \ge 0$} & \multicolumn{3}{c|}{$T_2=0$} \\
\hhline{~---}
& $E_2$ & \multicolumn{2}{c|}{$F_2$} \\
\hhline{~---}
& \multirow{3}{*}{$T_4 \ge 0$} & \multicolumn{2}{c|}{$T_4=0$} \\
\hhline{~~--}
& & $E_3$ & $F_3$ \\
\hhline{~~--}
& & $T_6 \ge 0$ & $T_6=0$ \\
\hline
\end{tabular}
\caption{Illustration of the subspaces}
\label{figure:subspaces}
\end{table}
\noindent Then we apply a Taylor expansion up to order $8$ to the left side of \eqref{equation:order_8:2} and the multinomial formula \eqref{equation:multinomial}, which reads
$$ \sum_{k=2}^{8} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_{4} \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{4} = k }} \binom{k}{i_1,\ldots,i_{4}} t^{\frac{i_1}{2} + \cdots + \frac{i_4}{8} -1} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes p_{_{E_2}}(h)^{\otimes i_2} \otimes p_{_{E_3}}(h)^{\otimes i_3} \otimes p_{_{F_3}}(h)^{\otimes i_4} + o(1) .$$
$\triangleright$ If $\frac{i_1}{2} + \cdots + \frac{i_4}{8} > 1$ then the corresponding term is in $o(1)$ when $t \rightarrow 0$.
\noindent $\triangleright$ If $\frac{i_1}{2} + \cdots + \frac{i_4}{8} < 1$ then the corresponding term diverges when $t \rightarrow 0$, so we need to prove that actually
\begin{equation}
\label{eq:proof:order_8_null_terms}
T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes p_{_{E_2}}(h)^{\otimes i_2} \otimes p_{_{E_3}}(h)^{\otimes i_3} \otimes p_{_{F_3}}(h)^{\otimes i_4} = 0 .
\end{equation}
-- If $\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6} + \frac{i_4}{8} < 1$ but if we also have $\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6} + \frac{i_4}{6} < 1$, then by applying the property at the order $6$ (Theorem \ref{theorem:order_6}) with the $3$-tuple $(i_1,i_2,i_3+i_4)$, we get \eqref{eq:proof:order_8_null_terms}.
-- So we only need to consider $4$-tuples such that $\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6} + \frac{i_4}{8} < 1$ and $\frac{i_1}{2} + \frac{i_2}{4} + \frac{i_3}{6} + \frac{i_4}{6} \ge 1$. We can remove all the terms which are null by the definitions of the subspaces $E_1, \ E_2, \ E_3, \ F_3$. The remaining terms are:
For $k=4$: $\frac{t^{21/24}}{6} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{F_3}}(h)^{\otimes 3}$, $\frac{t^{11/12}}{2} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{E_3}}(h) \otimes p_{_{F_3}}(h)^{\otimes 2}$, $\frac{t^{23/24}}{2} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{E_3}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)$.
For $k=5$ : $\frac{t^{21/24}}{12} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)^{\otimes 3}$, $\frac{t^{11/12}}{4} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{E_3}}(h) \otimes p_{_{F_3}}(h)^{\otimes 2}$, $\frac{t^{23/24}}{4} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{E_3}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)$.
For $k=6$ : $\frac{t^{21/24}}{5!} T_6 \cdot p_{_{E_2}}(h) \otimes p_{_{F_3}}(h)^{\otimes 5}$, $\frac{t^{11/12}}{4!} T_6 \cdot p_{_{E_2}}(h) \otimes p_{_{E_3}}(h) \otimes p_{_{F_3}}(h)^{\otimes 4}$, $\frac{t^{23/24}}{12} T_6 \cdot p_{_{E_2}}(h) \otimes p_{_{E_3}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)^{\otimes 3}$.
First, we note that
\begin{align*}
& \frac{1}{t^{21/24}} \left[ f(x^\star + t^{1/2}p_{_{E_1}}(h) + t^{1/4}p_{_{E_2}}(h) + t^{1/6}p_{_{E_3}}(h) + t^{1/8}p_{_{F_3}}(h)) - f(x^\star) \right] \\
\underset{t \rightarrow 0}{\longrightarrow} \ & \frac{1}{6} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{F_3}}(h)^{\otimes 3} + \frac{1}{12} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)^{\otimes 3} + \frac{1}{5!} T_6 \cdot p_{_{E_2}}(h) \otimes p_{_{F_3}}(h)^{\otimes 5} \ge 0.
\end{align*}
Then, considering $h' = \lambda p_{_{E_1}}(h) + p_{_{E_2}}(h) + p_{_{E_3}}(h) + p_{_{F_3}}(h)$, we have that for all $\lambda \in \mathbb{R}$,
\begin{equation*}
\frac{\lambda}{6} T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{F_3}}(h)^{\otimes 3} + \frac{1}{12} T_5 \cdot p_{_{E_2}}(h)^{\otimes 2} \otimes p_{_{F_3}}(h)^{\otimes 3} + \frac{1}{5!} T_6 \cdot p_{_{E_2}}(h) \otimes p_{_{F_3}}(h)^{\otimes 5} \ge 0 ,
\end{equation*}
so necessarily $$T_4 \cdot p_{_{E_1}}(h) \otimes p_{_{F_3}}(h)^{\otimes 3} = 0.$$
Then, considering $h' = p_{_{E_2}}(h) + \lambda p_{_{F_3}}(h)$ for $\lambda \in \mathbb{R}$, we get successively that the two other terms are null.
\medskip
Likewise, we prove successively that the terms in $t^{11/12}$ are null, and then that the terms in $t^{23/24}$ are null. This yields the convergence stated in \eqref{equation:order_8:2}.
\end{proof}
\subsection{Counter-example and proof of Theorem \ref{theorem:main} with $p\ge 5$ under the hypothesis \eqref{equation:even_terms_null:2}}
\label{section:order_10}
Algorithm \ref{algo:algorithm} may fail to yield such expansion of $f$ for orders no lower than $10$ if the hypothesis \eqref{equation:even_terms_null:2} is not fulfilled. Indeed for $p \ge 5$, there exist $p$-tuples $(i_1,\ldots,i_p)$ such that $\frac{i_1}{2}+\cdots+\frac{i_p}{2p} < 1$ and $i_1$, $\ldots$, $i_p$ are all even. Such tuples do not appear at orders $8$ and lower, but they do appear at orders $10$ and higher, for example $(0,2,0,0,4)$ for $k=6$.
In such a case, we cannot use the positiveness argument to prove that the corresponding term $T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p}$ is zero, and in fact, it may be not zero.
Let us give a counter example. Consider
$$ \begin{array}{rrl}
f : & \mathbb{R}^2 & \longrightarrow \mathbb{R} \\
& (x,y) & \longmapsto x^4 + y^{10} + x^2 y^4.
\end{array} $$
Then $f \in \mathscr{A}_5^\star(0)$ and we have $E_1 = \lbrace 0 \rbrace$, $E_2 = \mathbb{R}\cdot(1,0)$, $E_3 = \lbrace 0 \rbrace$, $E_4 = \lbrace 0 \rbrace$, $F_4 = \mathbb{R}\cdot(0,1)$. But
$$ \frac{1}{t} f(t^{1/4}, t^{1/10}) = \frac{1}{t} \left(t + t + t^{9/10} \right) $$
goes to $+\infty$ when $t \rightarrow 0$.
\medskip
\textbf{Now, let us give the proof of Theorem \ref{theorem:main} for $p \ge 5$.} In this proof, we assume that the subspaces $E_1, \ \ldots, \ E_p$ given in Algorithm \ref{algo:algorithm} satisfy the hypothesis \eqref{equation:even_terms_null:2}.
\begin{proof}
We develop \eqref{equation:order_p:1}, which reads:
$$ \sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_{p} \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k }} \binom{k}{i_1,\ldots,i_{p}} t^{\frac{i_1}{2} + \cdots + \frac{i_p}{2p} -1} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} + o(1) =: S .$$
The terms such that $\frac{i_1}{2} + \cdots + \frac{i_p}{2p} < 1$ may diverge when $t \rightarrow 0$, so let us prove that they are in fact null.
Let
$$ \alpha := \inf \left\lbrace \frac{i_1}{2}+\cdots+\frac{i_p}{2p} : \ h \longmapsto\sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha }} \binom{k}{i_1,\ldots,i_{p}} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \not\equiv 0 \right\rbrace ,$$
and assume by contradiction that $\alpha < 1$. Then we have for all $h \in \mathbb{R}^d$:
$$ t^{1-\alpha} S \underset{t \rightarrow 0}{\longrightarrow} \left( \sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha }} \binom{k}{i_1,\ldots,i_{p}} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \right) \ge 0,$$
by the local minimum property. Then, considering $h' = \lambda_1 p_{_{E_1}}(h) + \cdots + \lambda_p p_{_{E_p}}(h)$, we have, for all $h \in \mathbb{R}^d$ and $\lambda_1, \ \ldots, \ \lambda_d \in \mathbb{R}$,
\begin{equation}
\label{eq:order_5_polynomial}
\sum_{k=2}^{2p} \frac{1}{k!} \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k \rbrace \\ i_1 + \cdots + i_{p} = k \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha }} \lambda_1^{i_1}\ldots\lambda_p^{i_p} \binom{k}{i_1,\ldots,i_{p}} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \ge 0 .
\end{equation}
Now, we fix $h \in \mathbb{R}^d$ such that the polynomial in \eqref{eq:order_5_polynomial} in the variables $\lambda_1, \ \ldots, \ \lambda_p$ is not identically zero, and we consider $k_{\max}$ its highest homogeneous degree, so that we have
$$ \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k_{\max} \rbrace \\ i_1 + \cdots + i_{p} = k_{\max} \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha }} \lambda_1^{i_1}\ldots\lambda_p^{i_p} \binom{k_{\max}}{i_1,\ldots,i_{p}} T_{k_{\max}} \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \ge 0 .$$
If $k_{\max}$ is odd, this yields a contradiction, taking $\lambda_1 = \cdots = \lambda_p =: \lambda \rightarrow \pm \infty$. If $k_{\max}$ is even, we consider the index $l_1$ such that $i_{l_1} =: a_1$ is maximal and the coefficients in the above sum with $i_{l_1} = a_1$ are not all zero.
Then fixing all the $\lambda_l$ for $l \ne l_1$ and taking $\lambda_{l_1} \rightarrow \infty$, we have
$$ \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k_{\max} \rbrace \\ i_1 + \cdots + i_{p} = k_{\max} \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha \\ i_{l_1} = a_1 }} \lambda_1^{i_1}\ldots\lambda_p^{i_p} \binom{k_{\max}}{i_1,\cdots,i_{p}} T_{k_{\max}} \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \ge 0 .$$
Thus, if $a_1$ is odd, this yields a contradiction. If $a_1$ is even, we carry on this process by induction : knowing $l_1, \ \ldots, \ l_r$, we choose the index $l_{r+1}$ such that $l_{r+1} \notin \lbrace l_1,\ldots,l_r \rbrace$, the corresponding term
$$ \sum_{\substack{i_1,\ldots,i_p \in \lbrace 0,\ldots,k_{\max} \rbrace \\ i_1 + \cdots + i_{p} = k_{\max} \\ \frac{i_1}{2} + \cdots + \frac{i_p}{2p}= \alpha \\ i_{l_1}=a_1,\ldots,i_{l_{r+1}}=a_{r+1} }} \lambda_1^{i_1}\ldots\lambda_p^{i_p} \binom{k_{\max}}{i_1,\ldots,i_{p}} T_{k_{\max}} \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} $$
is not identically null and such that $i_{l_{r+1}} =: a_{r+1}$ is maximal. Necessarily, $a_{r+1}$ is even.
In the end we will find a non-zero term whose exponents $i_{\ell}$ are all even which contradicts assumption \eqref{equation:even_terms_null:2}.
\end{proof}
\subsection{Proofs of the uniform convergence and of the non-constant property}
\label{subsec:unif_non_constant}
In this section we prove the additional properties claimed in Theorem \ref{theorem:main} : the uniform convergence with respect to $h$ on every compact set and the fact that the function $g$ is not constant in any of its variables $h_1, \ \ldots, \ h_d$.
\begin{proof}
First, let us prove that the convergence is uniform with respect to $h$ on every compact set. Let $\varepsilon >0$ and let $R>0$. By the Taylor formula at order $2p$, there exists $\delta>0$ such that for $||h|| < \delta$,
$$ \left| f(x^\star + h) - f(x^\star) - \sum_{k=2}^{2p} \frac{1}{k!} \sum_{i_1+\cdots+i_p=k} \binom{k}{i_1,\ldots,i_p} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \right| \le \varepsilon ||h||^{2p} .$$
Now, let us consider $t \rightarrow 0$ and $h \in \mathbb{R}^d$ with $||h|| \le R$. Then we have:
$$ \forall t \le \max\left(1, \left(\frac{\delta}{R}\right)^{1/(2p)}\right), \ ||t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_p}}(h) || \le \delta ,$$
so that
\begin{align*}
\left|\frac{1}{t} \left[ \right.\right. & \left. \left. f(x^\star + t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_p}}(h)) - f(x^\star)\right] - \sum_{k=2}^{2p} \frac{1}{k!} \sum_{i_1+\cdots+i_p=k} \binom{k}{i_1,\ldots,i_p} \right. \\
& \left. \cdot t^{\frac{i_1}{2}+\cdots+\frac{i_p}{2p}-1} T_k \cdot p_{_{E_1}}(h)^{\otimes i_1} \otimes \cdots \otimes p_{_{E_p}}(h)^{\otimes i_p} \right| \le \frac{\varepsilon}{t}||t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_p}}(h)||^{2p}.
\end{align*}
We proved or assumed that the terms such that $\frac{i_1}{2}+\cdots+\frac{i_p}{2p} < 1$ are zero. We denote by $g_1(h)$ the sum in the last equation with the terms such that $\frac{i_1}{2}+\cdots+\frac{i_p}{2p}=1$ and by $g_2(h)$ the sum with the terms such that $\frac{i_1}{2}+\cdots+\frac{i_p}{2p} > 1$. We also define $a$ as the smallest exponent of $t$ appearing in $g_2(h)$:
$$ a := \min \left\lbrace \frac{i_1}{2}+\cdots+\frac{i_p}{2p} \ : \ i_1,\ldots,i_p \in \lbrace 0,\ldots,2p\rbrace, \ i_1+\cdots+i_p \le 2p, \ \frac{i_1}{2}+\cdots+\frac{i_p}{2p} > 1 \right\rbrace > 1 .$$
So that:
\begin{align}
\label{eq:uniform_proof:1}
& \left| \frac{1}{t} \left[ f(x^\star + t^{1/2}p_{E_1}(h) +\cdots + t^{1/(2p)}p_{_{E_p}}(h)) - f(x^\star) \right] - g_1(h) \right| \\
& \le t^{a-1}|g_2(h)| + \frac{\varepsilon}{t}||t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_p}}(h)||^{2p}. \nonumber
\end{align}
We remark that $h \mapsto g_2(h)$ is a polynomial function so is bounded on every compact set. We also have:
$$ \frac{\varepsilon}{t}||t^{1/2}p_{_{E_1}}(h) + \cdots + t^{1/(2p)}p_{_{E_p}}(h)||^{2p} \le \frac{\varepsilon (t^{1/(2p)})^{2p}}{t} ||h||^{2p} = \varepsilon ||h||^{2p} .$$
So \eqref{eq:uniform_proof:1} converges to $0$ as $t \rightarrow 0$, uniformly with respect to $h$ on every compact set.
\bigskip
Now let us assume that $f \in \mathscr{A}_p^\star(x^\star)$ ; we prove that the function $g$ defined in \eqref{eq:def_g} is not constant in any of its variables in the sense of \eqref{eq:def_non_constant}. Let $B \in \mathcal{O}_d(\mathbb{R})$ adapted to the decomposition $\mathbb{R}^d = E_1 \oplus \cdots \oplus E_p$. We have:
$$ \frac{1}{t} \left[ f\left( x^\star + B \cdot (t^\alpha \ast h)\right) - f(x^\star) \right] \underset{t \rightarrow 0}{\longrightarrow} g(h) .$$
Let $i \in \lbrace 1, \ldots, p \rbrace$ and $k$ such that $v_i := B \cdot e_i \in E_k$. Let us assume by contradiction that $g$ does not depend on the $i^{\text{th}}$ coordinate. Considering the expression of $g$ in \eqref{eq:def_g} and setting all the variables outside $E_k$ to $0$, we have:
$$ \forall h \in E_k, \ \lambda \in \mathbb{R} \mapsto T_{2k} \cdot (h + \lambda v_i)^{\otimes 2k} $$
is constant. Then applying \eqref{equation:multinomial}, we have:
$$ \forall h \in E_k, \ T_{2k} \cdot v_i \otimes h^{\otimes 2k-1} = 0 .$$
Moreover, for $h \in F_{k-1}$, let us write $h = h' + h''$ where $h' \in E_k$ and $h'' \in F_k$, so that
$$T_{2k} \cdot v_i \otimes h^{\otimes 2k-1} = T_{2k} \cdot v_i \otimes h'^{\otimes 2k-1} = 0,$$
where we used that $$ \forall h^{(3)} \in F_{k-1}, \ T_{2k} \cdot h'' \otimes \left(h^{(3)}\right)^{\otimes 2k-1} = 0$$
following \eqref{eq:F_k_def}, and Proposition \ref{proposition:null_tensor:1}.
Considering the definition of $E_k$ as the orthogonal complement of $F_k$, which is defined in \eqref{eq:F_k_def}, the last equation contradicts that $v_i \in E_k$.
\end{proof}
\subsection{Non coercive case}
\label{section:non_coercive}
The function $g$ we obtain in Algorithm \ref{algo:algorithm} is a non-negative polynomial function which is constant in none of its variables. However, this does not always guarantee that $e^{-g} \in L^1(\mathbb{R}^d)$, or even that $g$ is coercive. Indeed, $g$ can be null on an unbounded continuous polynomial curve, while the polynomial degree of the minimum $x^\star$ of $f$ is higher than the degree of $g$ in these variables. For example, let us consider
\begin{align}
\label{eq:non_coercive_f}
f \colon \ \mathbb{R}^2 &\to \mathbb{R}\\
(x,y) &\mapsto (x-y^2)^2 + x^6. \nonumber
\end{align}
Then $f \in \mathscr{A}_3^\star(0)$ and using Algorithm \ref{algo:algorithm}, we get
$$ g(x,y) = (x-y^2)^2 ,$$
which does not satisfy $e^{-g} \in L^1(\mathbb{R}^d)$.
In fact this case is highly degenerate, as, with
\begin{equation*}
f_\varepsilon(x,y) := f(x,y) + \varepsilon xy^2 = x^2 + y^4 - (2-\varepsilon)xy^2 + x^6 ,
\end{equation*}
we have that $g_\varepsilon(x,y) = x^2 + y^4 - (2-\varepsilon)xy^2$ satisfies $e^{-g_\varepsilon} \in L^1(\mathbb{R}^d)$ for every $\varepsilon \in (0,4)$ and that $x^\star$ is not the global minimum of $f_\varepsilon$ for every $\varepsilon \in (-\infty, 0) \cup (4, \infty)$.
We now prove that instead of assuming $e^{-g} \in L^1(\mathbb{R}^d)$, we can only assume that $g$ is coercive, which is justified in the following proposition. More specific conditions for $g$ to be coercive can be found in \cite{bajbar2015} and \cite{bajbar2019}.
\begin{proposition}
\label{prop:coercive}
Let $g : \mathbb{R}^d \rightarrow \mathbb{R}$ be the polynomial function obtained from Algorithm \ref{algo:algorithm}. If $g$ is coercive, then $e^{-g} \in L^1(\mathbb{R}^d)$.
\end{proposition}
\begin{proof}
Let
$$ A_k := \text{Span}\left(e_i: \ i \in \lbrace \dim(E_1) + \cdots + \dim(E_{k-1}) +1, \ldots, \dim(E_1)+\cdots+\dim(E_k) \rbrace \right) $$
for $k \in \lbrace 1,\ldots, p \rbrace$.
By construction of $g$, note that for all $t \in [0,+\infty)$,
$$ g\left(\sum_{k=1}^p t^{1/2k}p_{_{A_k}}(h)\right) = tg(h) .$$
Since $g$ is coercive, there exists $R \ge 1$ such that for every $h$ with $||h|| \ge R$, $g(h) \ge 1$.
Then, for every $h \in \mathbb{R}^d$, we have:
\begin{align*}
g(h) & = g\left( \sum_{k=1}^p p_{_{A_k}}(h) \right) = g\left( \sum_{k=1}^p \frac{||h||^{1/2k}}{R^{1/2k}}p_{_{A_k}}\left(R^{1/2k}\frac{h}{||h||^{1/2k}}\right) \right) \\
& = \frac{||h||}{R} g\left( \sum_{k=1}^p p_{_{A_k}}\left(R^{1/2k}\frac{h}{||h||^{1/2k}}\right) \right).
\end{align*}
Then, for $||h|| \ge R$,
\begin{align*}
\left|\left| \sum_{k=1}^p p_{_{A_k}}\left(R^{1/2k}\frac{h}{||h||^{1/2k}}\right) \right| \right|^2 = \sum_{k=1}^p \frac{R^{1/k}}{||h||^{1/k}} ||p_{_{A_k}}(h)||^2 \ge \frac{R}{||h||} ||h||^2 = R ||h|| \ge R^2 \ge R,
\end{align*}
so that $g(h) \ge \frac{||h||}{R}$ which in turn implies $e^{-g} \in L^1(\mathbb{R}^d)$.
\end{proof}
We now deal with the simplest configuration where the function $g$ is not coercive, as described in \eqref{eq:non_coercive_sum}, by dealing with the case where $f$ is given by \eqref{eq:non_coercive_f}, which is an archetype of such configuration. However, dealing with the general case is more complicated and to give a general formula for the rate of convergence of the measure $\pi_t$ in this case is not our current objective.
\begin{proposition}
\label{prop:x_y2_convergence}
Let the function $f$ be given by \eqref{eq:non_coercive_f}. Then, if $(X_t,Y_t) \sim C_t e^{-f(x,y)/t} dx dy$, we have:
\begin{equation*}
\left( \frac{X_t}{t^{1/6}}, \frac{Y_t^2-X_t}{t^{1/2}} \right) \underset{t \rightarrow 0}{\longrightarrow} C\frac{e^{-x^6}}{\sqrt{x}} \frac{e^{-y^2}}{\sqrt{\pi}} \mathds{1}_{x \ge 0} dx dy ,
\end{equation*}
where $C = \left( \int_0^\infty \frac{e^{-x^6}}{\sqrt{x}}dx \right)^{-1}$.
\end{proposition}
\begin{proof}
First, let us consider the normalizing constant $C_t$. We have :
\begin{align*}
C_t^{-1} & = \int_{\mathbb{R}^2} e^{-\frac{(x-y^2)^2+x^6}{t}} dx dy = 2t^{3/4} \int_{-\infty}^{\infty} e^{-t^2 x^6} \int_0^\infty e^{-(y^2-x)^2}dy \ dx \\
& = t^{3/4} \int_{-\infty}^{\infty} e^{-t^2 x^6} \int_{-x}^\infty \frac{e^{-u^2}}{\sqrt{u+x}}dy \ dx
= t^{7/12} \int_{-\infty}^{\infty} e^{-x^6} \int_{-t^{-1/3}x}^{\infty} \frac{e^{-u^2}}{\sqrt{t^{1/3}u+x}} du \ dx \\
& \underset{t \rightarrow 0}{\sim} t^{7/12} \int_{0}^{\infty} \frac{e^{-x^6}}{\sqrt{x}} \int_{-\infty}^{\infty} e^{-u^2} du \ dx,
\end{align*}
where the convergence is obtained by dominated convergence and where we performed the change of variables $x' = t^{-1/6}x$ and $u = t^{-1/2}(y^2-x)$. Then we consider, for $a_1 < b_1$ and $a_2 < b_2$,
$$ \mathbb{P}\left( \frac{X_t}{t^{1/6}} \in [a_1,b_1], \ \frac{Y^2-X}{t^{1/2}} \in [a_2,b_2] \right) .$$
Performing the same changes of variables and using the above equivalent of $C_t$ completes the proof.
\end{proof}
More generally, if the function $g$ is not coercive and if we can write, up to a change of basis,
\begin{equation}
\label{eq:non_coercive_sum}
g(h_1,\ldots,h_d) = Q_1(h_1,h_2)^2 + Q_2(h_3,h_4)^2 + \cdots + Q_r(h_{2r-1},h_{2r})^2 + \widetilde{g}(h_{2r+1},\ldots,h_d) ,
\end{equation}
where the $Q_i$ are polynomials with two variables null on an unbounded curve (for example, $Q_i(x,y) = (x-y^2)$, $Q_i(x,y) = (x^2-y^3)$, $Q_i(x,y)=x^2y^2$), and where $\widetilde{g}$ is a non-negative coercive polynomial, then
\begin{align*}
& \left( a_1\left((X_t)_{1}, (X_t)_{2}, t \right), \ldots, a_r\left((X_t)_{2r-1}, (X_t)_{2r}, t \right), \left(\frac{1}{t^{\alpha_{2r+1}}},\ldots,\frac{1}{t^{\alpha_d}}\right) \ast \left(\widetilde{B} \cdot ((X_t)_{2r+1},\ldots,(X_t)_d)\right) \right) \\
& \underset{t \rightarrow 0}{\longrightarrow} b_1(x_1,x_2) \ldots b_r(x_{2r-1},x_{2r}) Ce^{-\widetilde{g}(x_{2r+1},\ldots,x_d)} dx_1 \ldots dx_{2r} dx_{2r+1}\cdots dx_d ,
\end{align*}
where $C$ is a normalization constant, $\widetilde{B} \in \mathcal{O}_{d-2r-1}(\mathbb{R})$ is an orthogonal transformation and for all $k=1,\ldots,r$, $a_k : \mathbb{R}^2 \times (0,+\infty) \rightarrow \mathbb{R}^2$ and $b_k$ is a density on $\mathbb{R}^2$. Such $a_k$ and $b_k$ can be obtained by applying the same method as in Proposition \ref{prop:x_y2_convergence}.
Algorithm \ref{algo:algorithm} yields the first change of variable for this method, given by the exponents $(\alpha_i)$ (in the proof of Proposition \ref{prop:x_y2_convergence}, the first change of variable is $t^{-1/2}x$ and $t^{-1/4}y$) and thus seems to be the first step of a more general procedure in this case.
However, we do not give a general formula as the general case is cumbersome. Moreover, we do not give a method where the non coercive polynomials $Q_i$ depend on more than two variables, like
$$ Q(x,y,z) = (x-y^2)^2 + (x-z^2)^2 .$$
The method sketched in Proposition \ref{prop:x_y2_convergence} cannot be direclty applied to this case.
\section{Proofs of Theorem \ref{theorem:single_well} and Theorem \ref{theorem:multiple_well} using Theorem \ref{theorem:main}}
\label{section:proofs_athreya}
\subsection{Single well case}
\label{subsec:single_well_proof}
We now prove Theorem \ref{theorem:single_well}.
\begin{proof}
Using Theorem \ref{theorem:main}, we have for all $h \in \mathbb{R}^d$:
$$ \frac{1}{t} f(B \cdot (t^\alpha \ast h)) \underset{t \rightarrow 0}{\longrightarrow} g(h) .$$
To simplify the notations, assume that there is no need of a change of basis i.e. $B=I_d$. We want to apply Theorem \ref{theorem:athreya:1} to the function $f$. However the condition
$$ \int_{\mathbb{R}^d} \sup_{0<t<1} e^{-\frac{f \left(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d\right)}{t}} dh_1\ldots dh_d < \infty $$
is not necessarily true. Instead, let $\varepsilon > 0$ and we apply Theorem \ref{theorem:athreya:1} to $\widetilde{f}$, where $\widetilde{f}$ is defined as:
$$ \widetilde{f}(h) = \left\lbrace \begin{array}{ll}
f(h) & \text{ if } h \in \mathcal{B}(0,\delta) \\
||h||^2 & \text{ else} ,
\end{array} \right. $$
and where $\delta > 0$ will be fixed later. Then $\widetilde{f}$ satisfies the hypotheses of Theorem \ref{theorem:athreya:1}. The only difficult point to prove is the last condition of Theorem \ref{theorem:athreya:1}.
If $t \in (0,1]$ and $h \in \mathbb{R}^d$ are such that $(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d) \notin \mathcal{B}(0,\delta)$, then
$$ \frac{\widetilde{f} (t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d)}{t} = \frac{||(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d) ||^2}{t} \ge ||h||^2, $$
because for all $i$, $\alpha_i \le \frac{1}{2}$. If $t$ and $h$ are such that $(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d) \in \mathcal{B}(0,\delta)$, then choosing $\delta$ such that for all $(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d) \in \mathcal{B}(0,\delta)$,
$$ \left| \frac{f (t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d)}{t} - g(h) \right| \le \varepsilon, $$
which is possible because of the uniform convergence on every compact set (see Section \ref{subsec:unif_non_constant}), we derive that
$$ \frac{f (t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d)}{t} \ge g(h) - \varepsilon .$$
Hence
$$ \int_{\mathbb{R}^d} \sup_{0<t<1} e^{-\frac{\widetilde{f} \left(t^{\alpha_1}h_1,\ldots,t^{\alpha_d}h_d\right)}{t}} dh_1\ldots dh_d \le \int_{\mathbb{R}^d} e^{-||h||^2} dh + e^{\varepsilon} \int_{\mathbb{R}^d} e^{-g(h)} dh .$$
Since $g$ is coercive, using Proposition \ref{prop:coercive} we have $e^{-g} \in L^1(\mathbb{R}^d)$ and it follows from Theorem \ref{theorem:athreya:1} that if $\widetilde{X}_t$ has density $\widetilde{\pi}_t(x) := \widetilde{C}_t e^{-\widetilde{f}(x)/t}$, then
$$ \left( \frac{(\widetilde{X}_t)_1}{t^{\alpha_1}}, \ldots, \frac{(\widetilde{X}_t)_d}{t^{\alpha_d}} \right) \overset{\mathscr{L}}{\longrightarrow} X \ \text{ as } t \rightarrow 0 ,$$
where $X$ has density proportional to $e^{-g(x)}$.
Now, let us prove that if $X_t$ has density proportional to $e^{-f(x)/t}$, then we also have
\begin{equation}
\label{equation:Y_law_convergence}
\left( \frac{(X_t)_1}{t^{\alpha_1}}, \ldots, \frac{(X_t)_d}{t^{\alpha_d}} \right) \overset{\mathscr{L}}{\longrightarrow} X \ \text{ as } t \rightarrow 0 .
\end{equation}
Let $\varphi : \mathbb{R}^d \rightarrow \mathbb{R}$ be continuous with compact support. Then
\begin{align*}
& \mathbb{E}\left[ \varphi\left( \frac{(X_t)_1}{t^{\alpha_1}}, \ldots, \frac{(X_t)_d}{t^{\alpha_d}} \right) - \varphi\left( \frac{(\widetilde{X}_t)_1}{t^{\alpha_1}}, \ldots, \frac{(\widetilde{X}_t)_d}{t^{\alpha_d}} \right) \right] \\
& = \int_{\mathbb{R}^d} \varphi \left(\frac{x_1}{t^{\alpha_1}},\ldots,\frac{x_d}{t^{\alpha_d}}\right) \left(C_t e^{-\frac{f(x_1,\ldots,x_d)}{t}} - \widetilde{C}_t e^{-\frac{\widetilde{f}(x_1,\ldots,x_d)}{t}} \right) dx_1\ldots dx_d =: I_1 + I_2,
\end{align*}
where $I_1$ is the integral on the set $\mathcal{B}(0,\delta)$ and $I_2$ on $\mathcal{B}(0,\delta)^c$. We have then:
$$ |I_2| \le || \varphi ||_\infty ( \pi_t(\mathcal{B}(0,\delta)^c) + \widetilde{\pi}_t(\mathcal{B}(0,\delta)^c) ) \underset{t \rightarrow 0}{\longrightarrow} 0 ,$$
where we used Proposition \ref{proposition:gibbs}. On the other hand, we have $f=\widetilde{f}$ on $\mathcal{B}(0,\delta)$, so that
$$ |I_1| \le ||\varphi||_\infty |C_t - \widetilde{C}_t| \int_{\mathcal{B}(0,\delta)} e^{-\frac{f(x)}{t}}dx \le ||\varphi||_\infty \left|1 - \frac{\widetilde{C}_t}{C_t}\right| .$$
And we have:
$$ \frac{\widetilde{C}_t}{C_t} = \frac{\int e^{-\frac{f(x)}{t}}dx}{\int e^{-\frac{\widetilde{f}(x)}{t}}dx} = \frac{\int_{\mathcal{B}(0,\delta)} e^{-\frac{f(x)}{t}}dx + \int_{\mathcal{B}(0,\delta)^c} e^{-\frac{f(x)}{t}}dx}{\int_{\mathcal{B}(0,\delta)} e^{-\frac{f(x)}{t}}dx + \int_{\mathcal{B}(0,\delta)^c} e^{-\frac{\widetilde{f}(x)}{t}}dx} .$$
By Proposition \ref{proposition:gibbs}, we have when $t \rightarrow 0$
\begin{align*}
\int_{\mathcal{B}(0,\delta)^c} e^{-\frac{\widetilde{f}(x)}{t}}dx & = o\left( \int_{\mathcal{B}(0,\delta)} e^{-\frac{\widetilde{f}(x)}{t}}dx \right) \\
\int_{\mathcal{B}(0,\delta)^c} e^{-\frac{f(x)}{t}}dx & = o\left( \int_{\mathcal{B}(0,\delta)} e^{-\frac{f(x)}{t}}dx \right),
\end{align*}
so that $\widetilde{C}_t/C_t \rightarrow 1$, so $I_1 \rightarrow 0$, which then implies \eqref{equation:Y_law_convergence}.
\end{proof}
\subsection{Multiple well case}
We now prove Theorem \ref{theorem:multiple_well}.
\begin{proof}
The first point is a direct application of Theorem \ref{theorem:athreya:2}.
For the second point, we remark that $X_{it}$ has a density proportional to $e^{-f_i(x)/t}$, where
$$ f_i(x) := \left\lbrace\begin{array}{l}
f(x) \text{ if } x \in \mathcal{B}(x_i, \delta) \\
+ \infty \text{ else}.
\end{array} \right. $$
We then consider $\widetilde{f}_i$ as in Section \ref{subsec:single_well_proof}:
$$ \widetilde{f}_i(x) = \left\lbrace \begin{array}{ll}
f_i(x) & \text{ if } x \in \mathcal{B}(x_i,\delta) \\
||x||^2 & \text{ else} .
\end{array} \right. $$
and still as in Section \ref{subsec:single_well_proof}, we apply Theorem \ref{theorem:athreya:1} to $\widetilde{f}_i$ and then prove that random variables with densities proportional to $e^{-\widetilde{f}_i(x)/t}$ and $e^{-f_i(x)/t}$ respectively have the same limit in law.
\end{proof}
\section{Infinitely flat minimum}
\label{section:flat}
In this section, we deal with an example of infinitely flat global minimum, where we cannot use a Taylor expansion.
\begin{proposition}
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ such that
$$ \forall x \in \mathcal{B}(0,1), \ f(x) = e^{-\frac{1}{||x||^2}} $$
and
$$ \forall x \notin \mathcal{B}(0,1), \ f(x) > a $$
for some $a > 0$. Furthermore, assume that $f$ is coercive and $e^{-f} \in L^1(\mathbb{R}^d)$. Then, if $X_t$ has density $\pi_t$,
$$ \log^{1/2}\left(\frac{1}{t}\right) \cdot X_t \overset{\mathscr{L}}{\longrightarrow} X \ \text{ as } t \rightarrow 0 ,$$
where $X \sim \mathcal{U}(\mathcal{B}(0,1))$.
\end{proposition}
\begin{proof}
Noting that $\int_{||x|| >1}e^{-f(x)/t}dx \rightarrow 0$ as $t \rightarrow 0$ by dominated convergence, we have
$$ C_t \underset{t \rightarrow 0}{\sim} \left(\int_{\mathcal{B}(0,1)} e^{-e^{-\frac{1}{||x||^2}}/t} dx \right)^{-1} = \log^{d/2}\left(\frac{1}{t}\right) \left( \underbrace{\int_{\mathcal{B}(0,\sqrt{\log(1/t)})} e^{-t^{\frac{1}{||x||^2}-1}} dx}_{\underset{t \rightarrow 0}{\rightarrow} \text{Vol}(\mathcal{B}(0,1)) } \right)^{-1}, $$
where the convergence of the integral is obtained by dominated convergence. Then we have, for $-1<a_i<b_i<1$ and $\sum_i a_i^2 < 1$, $\sum_i b_i^2 < 1$:
\begin{align*}
\mathbb{P}\left(\log^{1/2}\left(\frac{1}{t}\right) \cdot X_t \in \prod_{i=1}^d [a_i,b_i] \right) = \frac{C_t}{\log^{d/2}\left(\frac{1}{t}\right)}\int_{(a_i)}^{(b_i)} e^{-t^{\frac{1}{|x|^2}-1}} dx \underset{t \rightarrow 0}{\longrightarrow} \frac{\prod_{i=1}^d (b_i-a_i)}{\text{Vol}(\mathcal{B}(0,1))}.
\end{align*}
\end{proof}
\section*{Acknowledgements}
\noindent I would like to thank Gilles Pag\`es for insightful discussions.
\input{convergence_rates_gibbs_measures.bbl}
\appendix
\section{Properties of tensors}
\begin{proposition}
\label{proposition:null_tensor:1}
Let $T_k$ be a symmetric tensor of order $k$ in $\mathbb{R}^d$. Let $E$ be a subspace of $\mathbb{R}^d$. Assume that
$$ \forall h \in E, \ T_k \cdot h^{\otimes k} = 0 .$$
Then we have
$$ \forall h_1,\ldots,h_k \in E, \ T_k \cdot h_1 \otimes \cdots \otimes h_k = 0 .$$
\end{proposition}
\begin{proof}
Using \eqref{equation:multinomial}, we have for $h_1, \ \ldots, \ h_k \in E$ and $\lambda_1, \ \ldots, \ \lambda_k \in \mathbb{R}$,
$$ T_k \cdot (\lambda_1 h_1+\cdots+\lambda_k h_k)^{\otimes k} = \sum_{i_1+\cdots+i_k=k} \binom{k}{i_1,\ldots,i_k} \lambda_1^{i_1}\ldots\lambda_k^{i_k} T_k \cdot h_1^{\otimes i_1} \otimes \cdots \otimes h_k^{\otimes i_k} = 0 ,$$
which is an identically null polynomial in the variables $\lambda_1, \ \ldots, \ \lambda_k$, so every coefficient is null, in particular
$$ \forall h_1,\ldots,h_k \in E, \ T_k \cdot h_1 \otimes \cdots \otimes h_k = 0 .$$
\end{proof}
\end{document} | {"config": "arxiv", "file": "2101.11557/convergence_rates_gibbs_measures.tex"} |
\begin{document}
\title{Networked MIMO with Fractional Joint Transmission in Energy Harvesting Systems}
\author{Jie~Gong~\IEEEmembership{Member,~IEEE}, Sheng~Zhou~\IEEEmembership{Member,~IEEE}, Zhenyu~Zhou~\IEEEmembership{Member,~IEEE}
\thanks{Jie Gong is with School of Data and Computer Science, Sun Yat-sen University, Guangdong 510006, China. Email: [email protected]}
\thanks{Sheng Zhou is with Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. Email: [email protected].}
\thanks{Zhenyu Zhou is with State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, School of Electrical and Electronic Engineering, North China Electric Power University, Beijing 102206, China. Email: zhenyu\[email protected]}
}
\maketitle
\begin{abstract}
This paper considers two base stations (BSs) powered by renewable energy serving two users cooperatively. With different BS energy arrival rates, a \emph{fractional joint transmission} (JT) strategy is proposed, which divides each transmission frame into two subframes. In the first subframe, one BS keeps silent to store energy while the other transmits data, and then they perform zero-forcing JT (ZF-JT) in the second subframe. We consider the average sum-rate maximization problem by optimizing the energy allocation and the time fraction of ZF-JT in two steps. Firstly, the sum-rate maximization for given energy budget in each frame is analyzed. We prove that the optimal transmit power can be derived in closed-form, and the optimal time fraction can be found via bi-section search. Secondly, \emph{approximate dynamic programming} (DP) algorithm is introduced to determine the energy allocation among frames. We adopt a linear approximation with the features associated with system states, and determine the weights of features by simulation. We also operate the approximation several times with random initial policy, named as \emph{policy exploration}, to broaden the policy search range. Numerical results show that the proposed fractional JT greatly improves the performance. Also, appropriate policy exploration is shown to perform close to the optimal.
\end{abstract}
\IEEEpeerreviewmaketitle
\section{Introduction}
Wireless communication with energy harvesting technology, which exploits renewable energy to power wireless devices, is expected as one of the promising trends to meet the target of green communications in the future. The advantages of energy harvesting include the sustainability with renewable energy source, the flexibility of network deployment without power line to reduce network planning cost, and etc. Recently, wireless cellular networks with renewable energy are rapidly developing. For instance, China Mobile has built about 12,000 renewable energy powered base stations (BSs) by 2014 \cite{ChinaMobile}. However, due to the randomness of the arrival process of the renewable energy and the limitation on the battery capacity, energy shortage or waste will occur when the energy arrival mismatches with the network traffic requirement. How to efficiently use the harvested energy is a big challenge.
In the literature, a lot of research work has focused on the energy harvesting based communications. For single-link case, the optimal power allocation structure, \emph{directional water-filling}, is found in both single-antenna transceiver system \cite{ozel2011transmission, jie2013optimal} and multiple-input multiple-output (MIMO) channel \cite{hu2015optimal}. The research efforts have been further extended to the network case, and the power allocation policies are proposed for broadcast channel \cite{yang2012broadcasting}, multiple access channel \cite{yang2012optimal}, interference channel \cite{tutuncuoglu2012sum}, as well as cooperative relay networks \cite{chuan2013threshold, minasian2014energy}. Nevertheless, there lacks research effort on the effect of energy harvesting on the multi-node cooperation, i.e., network MIMO.
The network MIMO technology, which shares the user data and channel state information among multiple BSs, and coordinates the data transmission and reception by transforming the inter-cell interference into useful signals, has been extensively studied in the literature \cite{karakayali2006network, zhang2009networked, huang2009increasing}. And it has already been standardized in 3GPP as Coordinated Multi-Point (CoMP) \cite{3GPP2011TR}. By applying joint precoding schemes such as zero-forcing (ZF) \cite{boccardi2006zero, kaviani2011optimal} among BSs for joint transmission (JT), the system sum-rate can be greatly increased. However, how the dynamic energy arrival influences the performance of network MIMO requires further study. Specifically, as the JT is constrained by the per-BS power budget, the performance of the network MIMO is limited if the power budgets are severely asymmetric among BSs. For example, if a solar-powered BS in a windless sunny day cooperates with a wind-powered BS, the latter will become the performance bottleneck of cooperation, while the harvested energy of the former is not efficiently utilized. To deal with this problem, people have introduced the concept of energy cooperation \cite{gurakan2013energy, chia2014energy}, where BSs can exchange energy via either wired or wireless link with some loss of energy transfer. In this case, the JT problem with energy harvesting becomes a power allocation problem with weighted sum power constraint as shown in \cite{jie2015comp}. However, the feasibility and efficiency of cooperation in energy domain strongly depends on the existence and the efficiency of energy transfer link.
In this paper, we consider how to improve the utilization of harvested energy with cooperation between the wireless radio links. Intuitively, if the energy cannot be transferred between BSs, {the BS with higher energy arrival rate should use more energy in data transmission to avoid energy waste. While to use the energy more effective, BS cooperation strategy should be carefully designed under the asymmetric energy constraints.} Based on this, we propose a \emph{fractional JT} strategy, where the network MIMO is only applied in a fraction of a transmission frame. Specifically, we consider two BSs cooperatively serving two users, and divide each transmission frame into two subframes. In the first subframe, one of the BSs serves one user while the other stores energy. In the second subframe, the two BSs perform JT to cooperatively serve the two users. With the stored energy, the power gap between two BSs in the second subframe is filled, and hence, JT can achieve higher sum-rate. Such a strategy avoids the potential energy waste in the BS with higher energy arrival rate, and hence can improve the energy utilization. The objective is to maximize the average sum-rate for given energy arrival rates, and the optimization parameters include the fraction of time for JT and the power allocation policy in each frame. Our preliminary work \cite{gong2014downlink} has studied the greedy policy that tries to use all the available energy in each frame. In this paper, we further consider the optimal policy as well as the low-complexity policy. The contributions of this paper are as follows.
\begin{itemize}
\item We propose the fractional JT strategy, and formulate the long-term average sum-rate maximization problem using Markov decision process (MDP) \cite{bertsekas2005dynamic}. The problem is divided into two sub-problems, i.e., energy management among frames, and power allocation problem for fractional JT in each frame.
\item We prove that to solve the average sum-rate maximization problem, in each frame, we only need to solve the power allocation problem with equality power constraints, which has closed-form expressions. Then the JT time fraction decision problem is proved to be a convex optimization problem, and a bi-section search algorithm is proposed to find the optimal JT time fraction.
\item We adopt the \emph{approximate dynamic programming} (DP) \cite{bertsekas2005dynamic} algorithm to reduce the computational complexity of determining the energy allocation among frames. The algorithm runs iteratively with two steps: \emph{policy evaluation} and \emph{policy improvement}. In the policy evaluation, the relative utility function in the Bellman's equation is approximated as a weighted summation of a set of features associated with system states. The weights are estimated by simulation. In the policy improvement, random initial policies are periodically selected to rerun the iteration to broaden the search range. Numerical simulations show the remarkable performance gain compared with the conventional network MIMO.
\end{itemize}
The rest of the paper is organized as follows. Section \ref{sec:model} describes the system model and Section \ref{sec:problem} describes the MDP problem formulation. In Section \ref{sec:perstage}, the per-frame optimization problem is analyzed. Then the approximate DP algorithm is proposed in Section \ref{sec:ADP}. Simulation study is presented in Section \ref{sec:sim}. Finally, Section \ref{sec:concl} concludes the paper.
\emph{Notations:} Bold upper case and lower case letters denote matrices and vectors, respectively. $|\cdot|$ denotes the absolute value of a scalar, and $[x]^+ = \max\{x, 0\}$. $(\cdot)^T$ and $(\cdot)^H$ denote the transpose and transpose conjugate of a matrix, respectively. ${\cal R}^{+}$ is the non-negative real number field. $\mathbb{E}$ represents the expectation operation.
\section{System Model} \label{sec:model}
We consider a wireless communication network consisting of two BSs powered by renewable energy (e.g., solar energy, wind energy, etc.) and two users as shown in Fig.~\ref{fig:system}. Assume the BSs are able to store the harvested energy in their battery for future usage. All the BSs and the users are equipped with a single antenna. The BSs are interconnected via an error-free backhaul link sharing all the data and the channel state information, so that they can perform JT to eliminate the interference. { However, the energy cannot be transferred between the BSs as we consider the off-grid scenario.} We consider the typical scenario for applying network MIMO, in which the two users are located at the cell boundary. In this case, the average channel gains are comparable, and hence cooperative transmission can achieve significant performance gain. The wireless channel is assumed block fading, i.e., the channel state is constant during each fading block, but changes from block to block. We define the transmission frame as a channel fading block with frame length $T_f$. The perfect channel state information is assumed known to the BSs at the beginning of each frame. { If the backhaul capacity is limited, the two BSs can exchange quantized data and channel state information, and cooperate in the same way using the imperfect information.}
\begin{figure}
\centering
\includegraphics[width=4.2in]{JTscenario.eps}
\caption{System model for joint transmission with 2 BSs and 2 users.} \label{fig:system}
\end{figure}
In the $t$-th frame, if the JT technique is utilized, the received signals $\mathbf{y}_t = [y_{t,1}, y_{t,2}]^T$ at the users are
\begin{equation}
\mathbf{y}_t = \mathbf{H}_t\mathbf{W}_t\mathbf{x}_t + \mathbf{n}_t,
\end{equation}
where $\mathbf{H}_t$ is the channel matrix with components $H_{t,ik} = l_{ik}\tilde{H}_{t,ik}, 1\le i,k \le 2$ indicating the channel coefficient from BS $k$ to user $i$ with large-scale fading factor $l_{ik}$ and i.i.d. small-scale fading factor $\tilde{H}_{t,ik}$, $\mathbf{W}_t$ is the corresponding precoding matrix with components $w_{t,ki}$, $\mathbf{x}_t = [x_{i,1}, x_{t,2}]^T$ is the intended signals for the users with $\mathbb{E}(\mathbf{x}_t\mathbf{x}_t^H) = \mathrm{diag}(p_{t,1}, p_{t,2})$, where $p_{t,i}, i = 1, 2$ is the power allocated to user $i$, and $\mathbf{n}_t$ is the additive white Gaussian noise with zero mean and variance $\mathbb{E}(\mathbf{n}_t\mathbf{n}_t^H) = \sigma_n^2\mathbf{I}$, where $\mathbf{I}$ is a $2\times 2$ unit matrix.
{In this paper, the widely used ZF precoding scheme \cite{boccardi2006zero} is adopted to completely eliminate the interference by channel inverse. Thus, the decoding process at the users can be simplified. And its performance can be guaranteed, especially when the interference dominates the noise. In addition, ZF precoding is a representative precoding scheme. Hence, the following analysis can be easily extended to other schemes.} For ZF precoding scheme, we have
\begin{equation}
\mathbf{W}_t = \mathbf{H}_t^{-1}.
\end{equation}
Hence, the data rate is
\begin{equation}
R_{t,i} = \log_2(1+\frac{p_{t,i}}{\sigma^2_n})\label{eq:Rcomp}
\end{equation}
with per-BS power constraint
\begin{equation}
\sum_{i=1}^2 |w_{t,ki}|^2 p_{t,i} \le P_{t,k}, \quad k = 1, 2. \label{eq:powerctr}
\end{equation}
where $P_{t,k}$ is the maximum available transmit power of BS $k$ in frame $t$. {Notice that if the BSs and the users are equipped with multiple antennas, ZF precoding scheme should be replaced by the multi-cell block diagonalization (BD) \cite{zhang2009networked} scheme which also nulls the inter-BS interference. As the multi-cell BD scheme is a generalization of ZF precoding scheme from single antenna case to multi-antenna case, it has similar mathematical properties with the latter. Hence, the following results can be extened to multi-antenna case.}
As the BSs are powered by the renewable energy, $P_{t,k}$ is determined by the amount of harvested energy as well as the available energy in the battery. It is pointed out in \cite{chuan2013threshold, chuan2014optimal} that in real systems, the energy harvesting rate changes in a much slower speed than the channel fading. Specifically, a fading block in current wireless communication systems is usually measured in the time scale of milliseconds, while the renewable energy such as solar power may keep constant for seconds or even minutes. Hence, the energy arrival rate (energy harvesting power) is assumed constant over a sufficient number of transmission frames, denoted by $E_k, k = 1, 2$. In this case, the key factor of the energy harvesting is the energy arrival causality constraint, i.e., the energy that has not arrived yet cannot be used in advance. In this paper, we mainly study the influence of the energy causality on the network MIMO.
{Notice that in practice, the optimization over multiple energy coherence blocks is required as the energy arrival rate varies over time. If the future energy arrival information is unknown (i.e., purely random and unpredictable), we can monitor the energy harvesting rate and once it changes, we recalculate the optimal policy under the new energy constraint, and then apply the new policy. The policy optimization problem is considered in this paper. While if the energy arrival rate is predictable, the optimization should jointly consider multiple blocks in the prediction window, which is beyond the scope the this paper.}
\subsection{Fractional Joint Transmission Strategy}
Notice that the energy arrival rates of different BSs may be different due to either utilizing various energy harvesting equipments (e.g., one with solar panel, the other with wind turbine) or encountering different environment conditions (e.g., partly cloudy). In this case, the conventional network MIMO may not be sum-rate optimal as the harvested energy is not efficiently utilized. Specifically, if the channel conditions of the two users are similar, applying network MIMO with on average the same energy usage can achieve the optimal cooperation efficiency. As a result, in the asymmetric energy arrival case, the energy of the BS with higher energy arrival rate may be not efficiently used. Hence, the performance of network MIMO may be greatly degraded. {Notice that the above fact does not only hold for ZF precoding, but also holds for other approaches (such as the approach based on dirty paper coding \cite{caire2003achievable, karakayali2007network}) as it is caused by the asymmetric per-BS power constraints, rather than the precoding scheme itself.}
To improve the utilization of the harvested energy, we propose a fractional JT strategy to adapt to the asymmetric energy arrival rates. Thanks to the energy storage ability, the BS can turn to sleep mode to store energy for a while, and then cooperatively transmits data with the other BS. In this way, it can provide higher transmit power when applying network MIMO. The strategy is detailed as follows. We divide the whole transmission frame into two subframes as shown in Fig.~\ref{fig:frame}. In the first subframe, named as \emph{single-BS transmission phase}, one of the BSs $k_t \in \{1,2\}$ is selected to serve a user, while the other one, denoted by $\bar{k}_t \neq k_t$, turns to sleep mode to store energy. In the second subframe, named as \emph{ZF-JT phase}, the two BSs jointly transmit to the two users with ZF precoding scheme as explained earlier in this section. Denote by $\alpha_tT_f$ the length of the single-BS transmission phase, where $0 \le \alpha_t \le 1$, and hence, the length of the ZF-JT phase is $(1-\alpha_t)T_f$. To get the optimal fractional JT transmission strategy, we need to choose $k_t$ and $\alpha_t$ carefully.
In the single-BS transmission phase, to be consistent with the objective of maximizing sum-rate, the active BS serves one of the users with higher instantaneous data rate. Specifically, the user $\tilde{i}$ is scheduled when satisfying $\tilde{i} = \arg \max_{1\le i \le 2} \log_2(1+ \frac{\bar{P}|H_{t,ik_t}|^2}{\sigma^2_n}),$ i.e., the user with the maximum expected data rate with transmit power $\bar{P}= E_{k_t}$. {In practice, the proposed fractional JT transmission strategy can be supported by the CoMP \cite{3GPP2011TR}, in which all the data is shared by the two BSs in both subframes. Notice that as only one BS is active in the first subframe, the data transferred to the inactive BS via the backhaul is useless, and such a backhaul data sharing protocol is inefficient.}
{However, when the backhaul capacity is limited, the proposed fractional JT strategy can make use of the backhaul capacity in the first subframe to enhance the performance. Since the shared data is required only in the second subframe, the two BSs in the first subframe can proactively exchange the data to be jointly transmitted later. Thus, the quantization noise of the shared data can be reduced and the cooperation gain can be enhanced.}
\begin{figure}
\centering
\includegraphics[width=3.2in]{framestr.eps}
\caption{Frame structure of fractional JT. The frame length is $T_f$.} \label{fig:frame}
\end{figure}
\subsection{Sum-rate Maximization Problem}
Our objective is to optimize the sum-rate under the proposed fractional JT strategy. The power constraints in each frame are detailed as follows. The available energy in the battery of the active BS $k_t$ at the beginning of each frame $t$ is denoted by $B_{t,k_t}$. Then the power in the first subframe satisfies
\begin{equation}
\tilde{p}_{t} \le \frac{B_{t,k_t}}{\alpha_t T_f} + E_{k_t}. \label{eq:power1}
\end{equation}
At the beginning of the second subframe, the amounts of available battery energy in the two BSs become $B_{t,k_t}+\alpha_t T_fE_{k_t} -\alpha_t T_f\tilde{p}_{t}$ and $B_{i,\bar{k}_t}+\alpha_t T_fE_{\bar{k}_t}$, respectively. As a result, the power constraints (\ref{eq:powerctr}) for ZF-JT become
\begin{align}
& \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i} \le \frac{B_{t,k_t}+\alpha_t T_f(E_{k_t} -\tilde{p}_{t})} {(1-\alpha_t)T_f} + E_{k_t}, \label{eq:power2}\\
& \sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i} \le \frac{B_{t,\bar{k}_t}+\alpha_t T_fE_{\bar{k}_t}}{(1-\alpha_t)T_f} + E_{\bar{k}_t}. \label{eq:power3}
\end{align}
The battery energy states are updated according to
\begin{align}
& B_{t\!+\!1\!,k_{t+1}} \!=\! B_{t,k_t} \!+\! T_f(E_{k_t} \!-\! \alpha_t\tilde{p}_{t} \!-\! (1\!-\!\alpha_t) \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i}), \label{eq:battery1}\\
& B_{t\!+\!1\!,\bar{k}_{t+1}} \!=\! B_{t,\bar{k}_t} \!+\! T_f(E_{\bar{k}_t} \!-\! (1\!-\!\alpha_t)\sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i}), \label{eq:battery2}
\end{align}
with initial state $B_{1,1} = B_{1,2} = 0$. In (\ref{eq:power1}), (\ref{eq:power2}), and (\ref{eq:power3}), we have $0<\alpha_t<1$ as the denominator cannot be zero. In fact, by multiplying $\alpha_t$ on both sides of (\ref{eq:power1}) and $1-\alpha_t$ on both sides of (\ref{eq:power2}) and (\ref{eq:power3}), the special case that $\alpha_t = 0 \textrm{~or~} 1$ can be included in a unified formulation. Denote by $\mathbf{{k}} = \{k_1, k_2, \cdots, k_N\}$, $\mathbf{{\alpha}} = \{\alpha_1, \alpha_2, \cdots, \alpha_N\}$, $\mathbf{{\tilde{p}}} = \{\tilde{p}_1, \tilde{p}_2, \cdots, \tilde{p}_N\}$, $\mathbf{{p}} = \{\mathbf{\emph{p}}_1, \mathbf{\emph{p}}_2, \cdots, \mathbf{\emph{p}}_N\}$, where $\mathbf{\emph{p}}_t = (p_{t,1}, p_{t,2})^T$, and $N$ is the number of transmission frames. Our optimization problem can be formulated as
\begin{align}
\max_{\mathbf{k}, \mathbf{\alpha}, \mathbf{\tilde{p}}, \mathbf{p}} \;& \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N \left(\alpha_t\tilde{R}_{t,\tilde{i}} \!+\! (1\!-\!\alpha_t)\sum_{i=1}^{2}R_{t,i}\right)\!\right] \label{eq:problem}\\
\mathrm{s.t.~}&\alpha_t\tilde{p}_t \le \frac{B_{t,k_t}}{T_f} + \alpha_tE_{k_t}, \label{eq:power1o}\\
& (1\!-\!\alpha_t) \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i} \!+\! \alpha_t\tilde{p}_t \le \frac{B_{t,k_t}}{T_f}\!+\!E_{k_t}, \label{eq:power2o}\\
& (1-\alpha_t)\sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i} \le \frac{B_{t,\bar{k}_t}}{T_f} + E_{\bar{k}_t}, \label{eq:power3o}\\
{} & \tilde{p}_t, p_{t, 1}, p_{t, 2} \in {\cal R}^{+}, \qquad \forall t = 1, 2, \cdots, N. \label{eq:power4o}\\
{} & 0 \le \alpha_t \le 1, \label{eq:power5o}
\end{align}
where $\tilde{R}_{t,\tilde{i}} = \log_2(1+\tilde{p}_t|H_{t,\tilde{i}k_t}|^2/\sigma^2_n)$, and $R_{t,i}$ is expressed as (\ref{eq:Rcomp}). The optimization parameters include the transmit power $\tilde{p}_t, p_{t,k}, k = 1, 2$, the frame division parameter $\alpha_t$, and the selection of BSs $k_t$ for single-BS transmission phase. Notice that if $\alpha_t=0$, the problem reduces to the conventional power allocation problem for network MIMO; if $\alpha_t=1$, the problem becomes user selection and rate maximization problem for single-BS transmission. To find the optimal solution, we need to calculate the integration of the channel distribution over all the frames and exhaustively search all the possible power allocation and frame division policies, which is computationally overwhelming. In the work, we aim to design a low-complex algorithm to achieve close-to-optimal performance.
\section{MDP Modeling and Optimization} \label{sec:problem}
In this section, we reformulate the stochastic optimization problem (\ref{eq:problem}) based on the MDP framework \cite{bertsekas2005dynamic}. Specifically, in each channel fading block, we need to decide which BS should turn to sleep to store energy in the first subframe, how long it should sleep, and how much power should be allocated. The decision in each frame will influence the decisions in the future, as it changes the remained energy in the battery. MDP is an effective mathematical framework to model such a time-correlated decision making problem. The formulation is detailed as follows.
\subsection{MDP Problem Reformulation}
A standard MDP problem contains the following elements: state, action, per-stage utility function and state transition. In our problem, the stage refers to the frame. In each stage, the system state includes the battery states of two BSs at the beginning of the frame and the channel states, i.e., $s_t = (B_{t,1}, B_{t, 2}, \mathbf{H}_t)$. Denote the state space by $\mathcal{S}$. We model the action as the power budget of each frame, i.e., $a_t(s_t) = (A_{t,1}, A_{t,2})$ which satisfies $0 \le A_{t,1} \le \frac{B_{t,1}}{T_f} + E_1$ and $0 \le A_{t,2} \le \frac{B_{t,2}}{T_f} + E_2.$
We denote the state-dependent action space by $\mathcal{A}(s_t) = \{(A_{t,1}, A_{t,2})|0 \le A_{t,1} \le \frac{B_{t,1}}{T_f} + E_1, 0 \le A_{t,2} \le \frac{B_{t,2}}{T_f} + E_2 \}$.
The per-stage sum-rate function can be expressed as
\begin{equation}
g(s_t, a_t) = \max_{k_t\!, \alpha_t\!, \tilde{p}_t\!, \mathbf{\emph{p}}_t} \alpha_t\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}_t|H_{t,\tilde{i}k_t}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha_t\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{t,i}}{\sigma^2_n}\!\Big), \label{eq:frameprobg}
\end{equation}
where the maximization is taken under the constraints (\ref{eq:power1o}), (\ref{eq:power4o}), (\ref{eq:power5o}) and
\begin{align}
& (1-\alpha_t) \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i} + \alpha_t\tilde{p}_t \le A_{t,k_t}, \label{eq:power2g}\\
& (1-\alpha_t)\sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i} \le A_{t, \bar{k}_t}, \label{eq:power3g}
\end{align}
The state transition of the battery energy is deterministic according to (\ref{eq:battery1}) and (\ref{eq:battery2}). The channel state of the next stage is obtained according to the channel transition $\mathrm{Pr}(\mathbf{H}_{t+1}|\mathbf{H}_t)$, which is independent with the battery energy state.
Consequently, the original problem (\ref{eq:problem}) can be reformulated as
\begin{align}
\max_{\bm{a}} \;& \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N g(s_t, a_t(s_t))\!\right]. \label{eq:problemMDP}
\end{align}
The optimization is taken over all the possible policies $\bm{a} = \{a_1, a_2, \ldots\}$. It is obvious that for any two states, there is a stationary policy $\bm{a}$ so that one state can be accessed from the other with finite steps \cite[Sec~4.2]{bertsekas2005dynamic}. Consequently, the optimal value is independent of the initial state and there exists an optimal stationary policy $\bm{a}^* = \{a^*(s)|s \in \mathcal{S}\}$ .
\subsection{Value Iteration Algorithm}
According to \cite[Prop.~4.2.1]{bertsekas2005dynamic}, there exists a scalar $\Lambda^*$ together with some vector $\bm{h}^* = \{h^*(s)|s \in \mathcal{S}\}$ satisfies the Bellman's equation
\begin{equation}
\Lambda^* + h^*(s) = \max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \sum_{s' \in \mathcal{S}}p_{s\rightarrow s' | a}h^*(s')\right], \label{eq:bellman}
\end{equation}
where $\Lambda^*$ is the optimal average utility, and $h^*(s)$ is viewed as \emph{relative or differential utility}\footnote{In the textbook \cite{bertsekas2005dynamic}, $h^*(s)$ is defined as \emph{relative cost} instead since the objective there is to minimize the average cost}. It represents the maximum difference between the expected utility to reach a given state $s_0$ from state $s$ for the first time and the utility that would be gained if the utility per stage was the average $\Lambda^*$. Furthermore, if $a^*(s)$ attains the maximum value of (\ref{eq:bellman}) for each $s$, the stationary policy $\bm{a}^*$ is optimal. Based on the Bellman's equation, instead of the long term average sum-rate maximization, we only need to deal with (\ref{eq:bellman}) which only relates with per-stage sum-rate $g(a, s)$ and state transition $p_{s\rightarrow s' | a}$. The value iteration algorithm \cite[Sec.~4.4]{bertsekas2005dynamic} can effectively solve the problem.
Specifically, we firstly initialize $h^{(0)}(s) = 0, \forall s \in \mathcal{S}$, and set a parameter $0 < \tau < 1$, which is used to guarantee the convergence of value iteration while obtaining the same optimal solution \cite[Prop. 4.3.4]{bertsekas2005dynamic}. Then we choose a state to calculate the relative utility. We choose a fixed state $s_0 = (0, 0, \mathbf{H}_0)$, and denote the output of the $n$-th iteration as $\bm{h}^{(n)}= \{h^{(n)}(s)|s \in \mathcal{S}\}$. For the $(n\!+\!1)$-th iteration, we first calculate
\begin{equation}
\Lambda^{(n+1)}(s_0) = \max_{a\in \mathcal{A}(s_0)}\left[ g(s_0, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H_0})h^{(n)}(s')\right], \label{eq:lambdanplus1}
\end{equation}
where $s' = (B_1', B_2', \mathbf{H}')$, and $B_1', B_2'$ are calculated according to (\ref{eq:battery1}) and (\ref{eq:battery2}), respectively. Then we calculate the relative utilities as
\begin{equation}
h^{(n+1)}(s) = (1-\tau)h^{(n)}(s) + \max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right] - \Lambda^{(n+1)}(s_0). \label{eq:hnplus1}
\end{equation}
Recall that the parameter $\tau$ is used to guarantee the convergence of the relative value iteration. It can be viewed as replacing the relative utility $h(s)$ by $\tau h(s)$, which is proved not to change the optimal value. As the optimal average utility is irrelative with the initial state, $\Lambda^{(n+1)}(s_0)$ converges to $\Lambda^*$.
Notice that the states and the actions are all in the continuous space. By discretizing the state space and the action space, the MDP framework can be applied to solve the problem. However, to make the solution accurate, the granularity of the discretization needs to be sufficiently small, which results in a tremendous number of states, especially for the $2\times2$ MIMO channels (4 elements, each with two scalars: real part and imaginary part). As a consequence, we need to not only calculate the per-stage sum-rate function $g(s, a)$ that includes maximization operation for all states, but also iteratively update all the relative utilities $h(s)$. In this sense, solving the MDP problem encounters unaffordable high computational complexity, which is termed as the \emph{curse of dimensionality} \cite{bertsekas2005dynamic}. To reduce the computational complexity, on the one hand, the maximization problem in the per-stage sum-rate function should be solved efficiently. On the other hand, the complexity of the iteration algorithm should be reduced via some approximation. In the next two sections, we will discuss these two aspects in detail.
\section{Per-Frame Sum-Rate Maximization} \label{sec:perstage}
In this section, we firstly consider the per-stage sum-rate function $g(s_t, a_t)$, i.e., the sum-rate maximization problem in each frame for the current state $s_t = (B_{t,1}, B_{t, 2}, \mathbf{H}_t)$ and the given action $a_t = (A_{t,1}, A_{t,2})$. We ignore the time index $t$ for simplicity. The per-frame optimization problem can be formulated as
\begin{align}
\max_{k\!, \alpha\!, \tilde{p}\!, p_1\!, p_2\!} \quad& \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big) \label{eq:frameprob}\\
\mathrm{s.t.~}\quad&\alpha\tilde{p} \le \frac{B_{k}}{T_f} + \alpha E_{k}, \label{eq:power1p}\\
& (1-\alpha) \sum_{i=1}^2 |w_{ki}|^2 p_{i} + \alpha\tilde{p} \le A_{k}, \label{eq:power2p}\\
& (1-\alpha)\sum_{i=1}^2 |w_{\bar{k}i}|^2 p_{i} \le A_{\bar{k}}, \label{eq:power3p}\\
{} & \tilde{p}, p_{1}, p_{2} \in {\cal R}^{+}. \label{eq:power4p}\\
{} & 0 \le \alpha \le 1. \label{eq:power5p}
\end{align}
As $k \in \{1, 2\}$, the optimization over $k$ can be done by solving the problem for all $k$, and selecting the one with larger sum-rate. Thus, we only need to consider the problem for a given $k$.
Then the optimization problem can be rewritten as
\begin{align}
\max_{\alpha\!, \tilde{p}\!, p_1\!, p_2\!} \quad& \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big) \label{eq:frameprobk}
\end{align}
The problem (\ref{eq:frameprobk}) with constraints (\ref{eq:power1p})-(\ref{eq:power5p}) is not convex in general. However, as shown later, given $\alpha$, the power allocation problem is a convex optimization, and the optimization over $\alpha$ given the optimal power allocation is also convex. According to these properties, we study the optimization of power allocation and subframe division separately.
\subsection{Power Allocation Optimization}
If we fix the variables $k$ and $\alpha$ in (\ref{eq:frameprobk}), we obtain a power allocation optimization problem, which has the following property.
\begin{theorem}\label{prop:convex}
For given $k$ and $\alpha$, the problem
\begin{align}
\max_{\tilde{p}\!, p_1\!, p_2\!} \quad& \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big) \label{prob:poweralloc}
\end{align}
with constraints (\ref{eq:power1p}) - (\ref{eq:power4p}) is a convex optimization problem.
\end{theorem}
\begin{IEEEproof}
Once $\alpha$ is fixed, the objective function is the maximization of a summation of concave functions, and all the constraints are linear. As a result, the problem is convex.
\end{IEEEproof}
Theorem \ref{prop:convex} tells us that for a given $k$ and $\alpha$, the optimal solution can be found by solving a convex optimization problem for power allocation. According to the convex optimization theory \cite{boyd2004convex}, we have the following observation.
\begin{proposition} \label{prop:greedy}
For a given $k$, when the optimal solution for the problem (\ref{eq:frameprobk}) with constraints (\ref{eq:power1p})-(\ref{eq:power5p}) is achieved, either (\ref{eq:power2p}) or (\ref{eq:power3p}) is satisfied with equality.
\end{proposition}
\begin{IEEEproof}
See Appendix \ref{proof:greedy}.
\end{IEEEproof}
However, Proposition \ref{prop:greedy} cannot guarantee the equality holds in both (\ref{eq:power2p}) and (\ref{eq:power3p}). If both are satisfied with equality, the problem can be simplified and the solution can be given in closed-form. As a matter of fact, an equivalent problem can be formulated which only needs to solve the power allocation problem with equality held in (\ref{eq:power2p}) and (\ref{eq:power3p}). To get the result, we firstly provide a useful lemma as follows.
\begin{lemma} \label{lemma:hincre}
The relative utility $h^*(s) = h^*(B_1, B_2, \mathbf{H})$ is nondecreasing w.r.t. $B_1$(or $B_2$) for given $B_2$(or $B_1$) and $\mathbf{H}$.
\end{lemma}
\begin{IEEEproof}
See Appendix \ref{proof:hincre}.
\end{IEEEproof}
Intuitively, more energy in the battery can support higher data rate. Hence, the utility increases with the increase of the battery energy. Based on Lemma \ref{lemma:hincre}, we have the following conclusion.
\begin{theorem} \label{prop:gbar}
Define $\bar{g}(s, a) = g(s, a)$ where the optimization is under the constraints (\ref{eq:power1p}), (\ref{eq:power4p}), (\ref{eq:power5p}) and the equality constraints
\begin{align}
& (1-\alpha) \sum_{i=1}^2 |w_{ki}|^2 p_{i} + \alpha\tilde{p} = A_{k}, \label{eq:power2gb}\\
& (1-\alpha)\sum_{i=1}^2 |w_{\bar{k}i}|^2 p_{i} = A_{\bar{k}}, \label{eq:power3gb}
\end{align}
we have
\begin{align}
\Lambda^* = \max \; \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N g(s_t, a_t(s_t))\!\right] = \max \; \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N \bar{g}(s_t, a_t(s_t))\!\right] \nonumber
\end{align}
\end{theorem}
\begin{IEEEproof}
See Appendix \ref{proof:gbar}.
\end{IEEEproof}
Based on Theorem \ref{prop:gbar}, we only need to solve the maximization problem under the equality constraints (\ref{eq:power2gb}) and (\ref{eq:power3gb}). The optimal power allocation solution as follows.
\begin{proposition} \label{prop:ptilde}
For a given $k$ and $0 < \alpha < 1$, we denote
\begin{align}
\tilde{p}_{\mathrm{min}} &= \max\Big\{0, \frac{C_{2}}{\alpha|w_{\bar{k}1}|^2}\Big\}, \label{ptilde:min}\\
\tilde{p}_{\mathrm{max}} &= \min\Big\{\frac{B_k}{\alpha T_f} + E_k, \frac{C_{1}}{\alpha|w_{\bar{k}2}|^2}\Big\}, \label{ptilde:max}
\end{align}
define the set ${\cal P}_{k,\alpha} = \left\{\tilde{p} \Big| \tilde{p}_{\mathrm{min}} \le \tilde{p} \le \right. \left.\tilde{p}_{\mathrm{max}} \right\}$, and denote $\tilde{p}_0$ as the nonnegative root of
\begin{align}
\frac{|H_{\tilde{i}k}|^2}{\sigma^2_n + \tilde{p}|H_{\tilde{i}k}|^2} - \frac{(1-\alpha) |w_{\bar{k}2}|^2}{\sigma^2_nC_0 + C_1 - \alpha |w_{\bar{k}2}|^2\tilde{p}} + \frac{(1-\alpha) |w_{\bar{k}1}|^2}{\sigma^2_nC_0 + C_2 + \alpha |w_{\bar{k}1}|^2\tilde{p}} = 0,
\label{eq:quadratic}
\end{align}
where $C_0 = (1-\alpha)(|w_{k1}|^2|w_{\bar{k}2}|^2 - |w_{k2}|^2|w_{\bar{k}1}|^2),
C_{1} = A_k|w_{\bar{k}2}|^2 - A_{\bar{k}}|w_{k2}|^2,
C_{2} = A_{k}|w_{\bar{k}1}|^2 - A_{\bar{k}}|w_{k1}|^2.$ Then the solution for the problem (\ref{prob:poweralloc}) with constraints (\ref{eq:power1p}), (\ref{eq:power4p}), (\ref{eq:power2gb}) and (\ref{eq:power3gb}) is
\begin{itemize}
\item If ${\cal P}_{k,\alpha} = \emptyset$, the problem is infeasible.
\item Otherwise, we have
(1) if $\tilde{p}_0 \in {\cal P}_{k,\alpha}$, $\tilde{p}^* = \tilde{p}_0$ is the optimal power for the single-BS transmission phase;
(2) if $\tilde{p}_0 > \tilde{p}_{\mathrm{max}}$, $\tilde{p}^* = \tilde{p}_{\mathrm{max}}$ is optimal;
(3) if $\tilde{p}_0 < \tilde{p}_{\mathrm{min}}$, $\tilde{p}^* = \tilde{p}_{\mathrm{min}}$ is optimal;
and the optimal $p_i^*, i = 1, 2$ can be obtained via
\begin{align}
p_1^* = & \frac{C_{1} - \alpha|w_{\bar{k}2}|^2\tilde{p}^*}{C_0}, \label{eq:p1star}\\
p_2^* = & \frac{\alpha|w_{\bar{k}1}|^2\tilde{p}^* - C_{2}}{C_0}. \label{eq:p2star}
\end{align}
\end{itemize}
\end{proposition}
\begin{IEEEproof}
See Appendix \ref{proof:ptilde}.
\end{IEEEproof}
Notice the solutions for $\alpha = 0$ and $\alpha = 1$ are not included in the proposition as they are trivial. For $\alpha = 0$, ZF-JT is applied in the whole frame. Then $\tilde{p} = 0$ and $p_i, i = 1, 2$ are obtained by solving (\ref{eq:power2gb}) and (\ref{eq:power3gb}). For $\alpha = 1$, the problem is feasible only when $A_{\bar{k}} = 0$, then $p_i = 0, i = 1, 2$ and $\tilde{p}$ can be obtained by solving (\ref{eq:power2gb}). According to Proposition \ref{prop:ptilde}, for $0 < \alpha < 1$, the power allocation problem (\ref{prob:poweralloc}) for the fixed $k$ and $\alpha$ with equality constraints (\ref{eq:power2gb}) and (\ref{eq:power3gb}) can be solved by calculating and comparing the values of $\tilde{p}_{\mathrm{min}}, \tilde{p}_{\mathrm{max}},$ and $\tilde{p}_0$. As they can be expressed in closed-form, the calculation is straightforward and simple. On the contrary, solving the original power allocation problem with inequality constraints (\ref{eq:power2p}) and (\ref{eq:power3p}) requires searching over the feasible set via iterations such as interior-point method \cite[Chap. 11]{boyd2004convex}.
\subsection{Optimization Over $\alpha$}
Besides the power allocation policy, we need to further determine optimal time ratio $\alpha$. As a matter of fact, the following theorem tells us that the optimization over $\alpha$ is also convex.
\begin{theorem} \label{prop:alphaconcave}
For a given $k$, define a function
\begin{equation}
F_k(\alpha) = \max_{\tilde{p}\!, p_1\!, p_2\!} \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big), \label{func:alpha}
\end{equation}
where $0 \le \alpha \le 1$ and the maximization is constrained by (\ref{eq:power1p})-(\ref{eq:power4p}). $F_k(\alpha)$ is a concave function.
\end{theorem}
\begin{IEEEproof}
See Appendix \ref{proof:alphaconcave}.
\end{IEEEproof}
\begin{corollary}
The function $\bar{F}_k(\alpha) = F_k(\alpha)$, where the maximization is under constraints (\ref{eq:power1p}), (\ref{eq:power4p}), (\ref{eq:power2gb}), and (\ref{eq:power3gb}), is a concave function.
\end{corollary}
\begin{IEEEproof}
The proof simply follows the lines of Appendix \ref{proof:alphaconcave}.
\end{IEEEproof}
Since $\bar{F}_k(\alpha)$ is a concave function, the optimal $\alpha$ either satisfies $\bar{F}_k'(\alpha) = 0$ or takes the boundary values $\alpha_{\mathrm{min}}$ or $1$, where $\alpha_{\mathrm{min}} \le 1$ is presented in (\ref{eq:alphamin}) in Appendix \ref{proof:ptilde}. However, the closed-form solution for $\bar{F}_k'(\alpha) = 0$ is not easy to be obtained as the expression of $\bar{F}_k$ with respect to $\alpha$ is complex. Giving the condition that the value of $\bar{F}_k(\alpha)$ itself is easy to be computed, we can adopt the bi-section search algorithm and in each iteration check the monotonicity of $\bar{F}_k(\alpha)$ in a small neighborhood of $\alpha$. The bi-section search algorithm is detailed in Algorithm \ref{alg:bisection}.
\begin{algorithm}[th]
\caption{Bi-section search algorithm to find the maximum $\bar{F}_k(\alpha)$} \label{alg:bisection}
\begin{algorithmic}[1]
\STATE Initialize $\delta \alpha > 0, \uline{\alpha} = \alpha_{\mathrm{min}}, \bar{\alpha} = 1, I = 0$.
\WHILE {$I = 0$}
\STATE Set $\hat{\alpha} = \frac{1}{2}(\uline{\alpha} + \bar{\alpha})$.
\IF {$\bar{F}_k(\hat{\alpha}) \ge \bar{F}_k(\hat{\alpha} - \delta \alpha)$ and $\bar{F}_k(\hat{\alpha}) \ge \bar{F}_k(\hat{\alpha} + \delta \alpha)$}
\STATE Set $I = 1$.
\ELSE
\IF {$\bar{F}_k(\hat{\alpha} - \delta \alpha) \le \bar{F}_k(\hat{\alpha}) \le \bar{F}_k(\hat{\alpha} + \delta \alpha)$}
\STATE Set $\uline{\alpha} = \hat{\alpha}$.
\ELSE
\STATE Set $\bar{\alpha} = \hat{\alpha}$.
\ENDIF
\ENDIF
\ENDWHILE
\STATE The optimal solution is $\bar{F}_k(\hat{\alpha})$.
\end{algorithmic}
\end{algorithm}
In Algorithm \ref{alg:bisection}, $\delta \alpha$ should be carefully selected to balance the accuracy of the optimal solution $\hat{\alpha}$ and the convergence speed of the iteration. Before running the bi-section algorithm, we need to firstly check if the optimal is obtained at the boundary points. Altogether, the algorithm for calculating $\bar{g}(s, a)$ is summarized in Algorithm \ref{alg:perframeopt}.
\begin{algorithm}[th]
\caption{Per-stage Utility Calculation Algorithm} \label{alg:perframeopt}
\begin{algorithmic}[1]
\STATE Initialize $\bar{g}(s, a) = 0$ and $\delta \alpha > 0$.
\FORALL{$k = 1$ to $2$}
\IF {${\cal P}_{k,\alpha_{\mathrm{min}}} \neq \emptyset$, and $\bar{F}_k(\alpha_{\mathrm{min}}) > \bar{F}_k(\alpha_{\mathrm{min}} + \delta \alpha)$}
\STATE Update $\bar{g}(s, a) \leftarrow \max\{\bar{g}(s, a), \bar{F}_k(\alpha_{\mathrm{min}})\}$.
\ELSIF {${\cal P}_{k,1} \neq \emptyset$, and $\bar{F}_k(1) > \bar{F}_k(1-\delta \alpha)$}
\STATE Update $\bar{g}(s, a) \leftarrow \max\{\bar{g}(s, a), \bar{F}_k(1)\}$.
\ELSE
\STATE Run Algorithm \ref{alg:bisection}, and then update $\bar{g}(s, a) \leftarrow \max\{\bar{g}(s, a), \bar{F}_k(\hat{\alpha})\}$.
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Approximate Dynamic Programming} \label{sec:ADP}
In this section, we adopt the approximate DP \cite[Chap.~6]{bertsekas2005dynamic} to solve the policy optimization problem and deal with the complexity issue due to the large number of system states. The basic idea of the approximate DP is to estimate the relative utility $h(s)$ via a set of parameters $\bm{c} = (c_1, c_2, \cdots, c_M)^T$ rather than to calculate the exact value. In this way, we only need to train the parameter vector $\bm{c}$ based on a small set of simulation samples. Specifically, we apply \emph{approximate policy iteration} algorithm as the convergence property can be guaranteed. Firstly, we briefly introduce the \emph{policy iteration} algorithm and its approximation version. Then we will implement the algorithm to solve our problem.
\subsection{Policy Iteration Algorithm}
The policy iteration algorithm includes two steps in each iteration: \emph{policy evaluation} and \emph{policy improvement}. It starts with any feasible stationary policy, and improves the objective step by step. Suppose in the $n$-th iteration, we have a stationary policy denoted by $\bm{a}^{(n)} = \{a^{(n)}(s)|s\in \mathcal{S}\}$. Based on this policy, we perform policy evaluation step, i.e., we solve the following linear equations
\begin{equation}
\lambda^{(n)} + h^{(n)}(s) = g(s, a^{(n)}(s)) + \sum_{s' \in \mathcal{S}}p_{s\rightarrow s' | a^{(n)}(s)}h^{(n)}(s') \label{eq:linear}
\end{equation}
for $\forall s \in \mathcal{S}$ to get the average cost $\lambda^{(n)}$ and the relative utility vector $\bm{h}^{(n)}$. Notice that the number of unknown parameters $(\lambda^{(n)}, \bm{h}^{(n)})$ is one more than the number of equations. Hence, more than one solutions exist, which are different with each other by a constant value for all $h^{(n)}(s)$. Without loss of generality, we can select a fixed state $s_0$ so that $h^{(n)}(s_0)=0$, then the solution for (\ref{eq:linear}) is unique.
The second step is to execute the policy improvement to find a stationary policy $\bm{a}^{(n+1)}$ which minimizes the right hand side of Bellman's equation
\begin{equation}
a^{(n+1)}(s) \!=\! \arg\!\max_{a\in \mathcal{A}(s)} \!\left[g(s, a) \!+\! \sum_{s' \in \mathcal{S}}\!p_{s\rightarrow s' | a}h^{(n)}(s')\right]. \label{eq:ukplus1}
\end{equation}
If $\bm{a^{(n+1)}} = \bm{a^{(n)}}$, the algorithm terminates, and the optimal policy is obtained $\bm{a^*} = \bm{a^{(n)}}$. Otherwise, repeat the procedure by replacing $\bm{a^{(n)}}$ with $\bm{a^{(n+1)}}$. It is proved that the policy \emph{does} improve the performance, i.e., $\lambda^{(n)} \le \lambda^{(n+1)}$ \cite[Prop.~4.4.2]{bertsekas2005dynamic}\footnote{For the average cost minimization problem discussed in Bertsekas's book, the direction of the inequality reverses.}, and the policy iteration algorithm terminates in finite number of iterations \cite[Prop.~4.4.1]{bertsekas2005dynamic}.
\subsection{Approximate Policy Evaluation}
For the policy evaluation step, the approximation DP tries to approximate the relative utility $h^{(n)}(s)$ by
\begin{equation}
\tilde{h}^{(n)}(s, \bm{c}^{(n)}) = \bm{\phi}(s)^T\bm{c}^{(n)}, \label{eq:hest}
\end{equation}
where $\bm{\phi}(s) = (\phi_1(s), \phi_2(s), \cdots, \phi_M(s))^T$ is an $M\times 1$ vector representing the features associated with state $s$, and $\bm{c}^{(n)} = (c_1^{(n)}, c_2^{(n)}, \cdots, c_M^{(n)})^T$ is an $M\times 1$ parameter vector. Instead of calculating all the relative utilities, we can train the parameter vector $\bm{c}^{(n)}$ using a relative small number of utility values and then estimate the others by (\ref{eq:hest}). Based on the estimated relative utility, the approximation of parameter vector for the next iteration is obtained by minimizing the least square error based on a weighted Euclidean norm, i.e.,
\begin{equation}
\bm{c}^{(n+1)} = \arg\min_{\bm{c} \in \mathcal{R}^M} ||\bm{\hat{h}}^{(n+1)} - \Phi\bm{c}||_{\bm{\xi}}^2,
\end{equation}
where $||\bm{J}||_{\bm{\xi}} = \sqrt{\sum_{s\in\mathcal{S}}\xi(s)J^2(s)}$ with a vector of positive weights $\xi(s), \forall s \in \mathcal{S}, \sum_s \xi(s) = 1$, $\mathcal{R}^M$ represents the $M$-dimensional real space, $\Phi$ is a matrix that has all the feature vectors $\phi(s)^T$, $\forall s \in \mathcal{S}$ as rows, and $\bm{\hat{h}}^{(n+1)} = F(\Phi\bm{c}^{(n)})$, where $F(\Phi\bm{c}^{(n)}) = (F(\bm{\phi}(s_1)^T\bm{c}^{(n)}), F(\bm{\phi}(s_2)^T\bm{c}^{(n)}), \cdots)^T$ and for each state $s$,
\begin{equation}
F(\bm{\phi}(s)^T\bm{c}^{(n)}) = g(s, a^{(n)}(s)) - \lambda^{(n)} + \sum_{s' \in \mathcal{S}}p_{s\rightarrow s' | a^{(n)}(s)}\phi(s')^T\bm{c}^{(n)}, \quad \forall s \in \mathcal{S}.
\end{equation}
For simplicity, the mapping $F$ can be written in matrix form as in \cite[Sec 6.6]{bertsekas2005dynamic}, i.e., $F(\bm{h}) = \bm{g} - \lambda \bm{e} + \bm{P h}$, where $\lambda$ is the average utility, $\bm{P}$ is the transition probability matrix and $\bm{e}$ is the unit vector. Further more, the mapping $F$ can be replaced by a parameterized mapping $F^{(\beta)} = (1-\beta)\sum_{i = 0}^{+\infty}\beta^iF^{i+1}$, where $\beta \in [0, 1)$, and $F^{i+1}(\bm{h}) = F^{i}(F(\bm{h}))$. The algorithm is called \emph{least square policy evaluation with parameter $\beta$ (LSPE($\beta$))} \cite[Chap. 6]{bertsekas2005dynamic}. The benefit of introducing the parameter $\beta$ is as follows. On the one hand, a higher convergence rate and smaller error bound can be obtained by setting larger $\beta$. On the other hand, when simulation is applied for approximation, larger $\beta$ results in more pronounced simulation noise. Hence, tuning the parameter $\beta$ helps to balance these factors. If $\beta = 0$, the mapping reduces to $F$.
Actually, we do not need to calculate samples of ${\hat{h}}(s)$ to estimate $\bm{c}$. Instead, the calculation can be done by simulation. Specifically, we generate a long simulated trajectory $s_0, s_1, \cdots$ based on the given action $\bm{a}^{(n)}$, and update $\bm{c}$ for each simulation realization according to the least square error metric. The advantage of simulation is that we only need a simulated trajectory rather than the state transition probability for a given policy. In reality, it means that we can use the simulated samples or the historical samples to directly calculate the estimated relative utility, instead of firstly estimate the transition probability and then estimate the utility. In the simulation-based LSPE($\beta$) algorithm, $\bm{c}$ is updated iteratively according to each simulation sample. It can be expressed in matrix form \cite[Sec 6.6]{bertsekas2005dynamic} as for the $i$-th sample,
\begin{equation}
\bm{c}_{i+1} = \bm{c}_i + \bm{B}_i^{-1}(\bm{A}_i\bm{c}_i+\bm{b}_i), \label{eq:cLSPE}
\end{equation}
where
\begin{align}
\bm{A}_i &= \frac{i}{i+1}\bm{A}_{i-1} + \frac{1}{i+1}\bm{z}_i(\bm{\phi}(s_{i+1})^T-\bm{\phi}(s_{i})^T),\nonumber\\
\bm{B}_i &= \frac{i}{i+1}\bm{B}_{i-1} + \frac{1}{i+1}\bm{\phi}(s_{i})\bm{\phi}(s_{i})^T,\nonumber\\
\bm{b}_i &= \frac{i}{i+1}\bm{b}_{i-1} + \frac{1}{i+1}\bm{z}_i(g(s_i, a^{(n)}(s_i))-\lambda_i),\nonumber\\
\bm{z}_i &= \beta \bm{z}_{i-1} + \bm{\phi}(s_{i}), \nonumber\\
\lambda_i &= \frac{1}{i+1}\sum_{j=0}^ig(s_j, a^{(n)}(s_j)),\nonumber
\end{align}
for all $i \ge 0$ and the boundary values $\bm{A}_{-1} = 0, \bm{B}_{-1} = 0, \bm{b}_{-1} = 0, \bm{z}_{-1} = 0.$ Note that there are two iterations in the approximate DP. The outer iteration runs policy evaluation and policy improvement to update the policy, the inner iteration runs the LSPE($\beta$) algorithm to update the parameter vector $\bm{c}$. In the $n$-th policy evaluation, the policy $\bm{a}^{(n)}$ is viewed as an input to generate the simulation trajectory and calculate $\bm{c}_i$ according to (\ref{eq:cLSPE}) in the inner iteration. When the difference between $\bm{c}_{i+1}$ and $\bm{c}_i$ is small enough, the policy evaluation process terminates and we get $\bm{c}^{(n)} = \bm{c}_i$. Then the policy is updated using $\bm{c}^{(n)}$, i.e.,
\begin{equation}
a^{(n+1)}(s) \!=\! \arg\!\max_{a\in \mathcal{A}(s)} \!\left[g(s, a) \!+\! \sum_{s' \in \mathcal{S}}\!p_{s\rightarrow s' | a}\bm{\phi}(s')^T\bm{c}^{(n)}\right]. \nonumber
\end{equation}
Generally, the length of the simulation trajectory is small than the number system state. Hence, the computational complexity of policy evaluation step can be reduced, especially when the number of states is large. Notice that the policy improvement step still needs to go through all the states due to the existence of the maximization operation.
\subsection{Implementation Issues}
To get an efficient approximate DP algorithm, the features of each state $\bm{\phi}(s)$ needs to be carefully selected. In our problem, we consider the following features.
\begin{itemize}
\item Energy-related features to indicate the influence of available energy on the utility. As the utility is represented in terms of data rate, the energy-related features are defined as $\log_2(1+\frac{B_k/T_f+E_k}{\sigma^2_n}), k = 1, 2.$
\item Channel-related features to indicate the influence of channel gain. Similarly, they are defined as $\log_2(1+|H_{ik}|^2), i = 1, 2, k = 1, 2.$
\item Cooperation features to indicate the influence of JT. As a MIMO system, the eigenvalues are the key indicator of the MIMO link performance. Hence, we define this type of feature as $\log_2(1+\rho_i), i = 1, 2,$ where $\rho_i, i = 1, 2$ are the eigenvalues of matrix $\mathbf{H}\mathbf{H}^H$.
\item The 2nd-order features. As the actual data rate is calculated by the product of power and channel gain, we further consider the following features: $\log_2(1+\frac{(B_k/T_f+E_k)|H_{ik}|^2}{\sigma^2_n}), i = 1, 2, k = 1, 2$ and $\log_2(1+\frac{(B_k/T_f+E_k)\rho_k}{\sigma^2_n}), k = 1, 2$.
\end{itemize}
The second issue concerning the approximate DP is that as the estimated relative utility is calculated based on the simulation samples generated for a given policy. Thus, some states that are unlikely to occur under this policy are under-represented. As a result, the relative utility estimation of these states may be highly inaccurate, causing potentially serious errors in the policy improvement process. This problem is known as \emph{inadequate exploration} \cite[Sec. 6.2]{bertsekas2005dynamic} of the system dynamics. One possible way for guaranteeing adequate exploration of the state space is to frequently restart the simulation from a random state under a random policy. We call it as \emph{policy exploration}. We will show later in the next section the influence of policy exploration on the performance.
\section{Simulation Study} \label{sec:sim}
We study the performance of the proposed algorithms by simulations. We adopt the outdoor pico-cell physical channel model from 3GPP standard \cite{3GPP2010TR}. The pathloss is $\mathrm{PL} = 140.7 + 36.7\log_{10}d$ (dB), where the distance $d$ is measured in km. The distance between pico BSs is 100m. The shadowing fading follows log-normal distribution with variance 10dB. The small-scale fading follows Rayleigh distribution with zero mean and unit variance. The average SNR at the cell edge (50m to the pico BS) with transmit power 30dBm is set to 10dB. We set the two users are placed in the cell edge of the two pico BSs depicted in Fig.~\ref{fig:system}. Hence, they experience the same large-scale fading. The BSs are equipped with energy harvesting devices (e.g.~solar panels). {The transmit power of pico BSs is around hundreds of mW, and we set the energy harvesting rate accordingly.}
\begin{figure}
\centering
\includegraphics[width=4.5in]{LSPEcmp.eps}
\caption{The influence of number of iterations and number of policy explorations on the sum-rate performance of approximate DP. The energy arrival rate of BS1 is 0.1W.} \label{fig:LSPE}
\end{figure}
Firstly, we evaluate the influence of number of iterations in the approximate DP on the performance. We fix the energy arrival rate of BS1 as 0.1W and change that of BS2. Denote the number of iterations for policy improvement by $N_I$, and the number of policy explorations which restarts the policy iteration by $N_E$. We set different values of $N_I$ and $N_E$ to run the approximate DP algorithm and compare the achievable sum-rate. The result is shown in Fig.~\ref{fig:LSPE}. From this figure, we can see that if policy exploration is not considered, i.e., $N_E = 1$, the approximate DP reveals some random fluctuation. Solely increasing the number of policy iterations is not guaranteed to improve the performance. On the other hand, by increasing the number of policy explorations, the fluctuation can be efficiently reduced and the performance can be greatly improved, even with relatively small number of policy iterations. This validates the claim that the simulation-based policy iteration may be inaccurate, and it is quite important to adopt policy exploration in the approximate DP algorithm design.
\begin{figure}
\centering
\includegraphics[width=4.5in]{CoMPcmp.eps}
\caption{Average sum-rate comparison of different algorithms. The energy arrival rate of BS1 is 0.1W.} \label{fig:compcmp}
\end{figure}
Then we show the performance of approximate DP compared with the optimal policy obtained via DP optimal algorithm. And the following baselines are also considered for comparison. In the conventional network MIMO, the whole frame applies ZF-JT without sub-frame spitting. In the greedy policy, we do not optimize the energy allocation among frames, but greedily use all the available energy for sum-rate maximization in each frame. Mathematically, we solve the problem (\ref{eq:frameprob}) under constraints (\ref{eq:power1p})-(\ref{eq:power5p}) with $A_k = \frac{B_k}{T_f}+E_k, k = 1,2.$ Hence, instead of finding the policy for each state before the system runs, we can get the online solution based on current system state. According to Theorem \ref{prop:convex} and Theorem \ref{prop:alphaconcave}, the problem can be solved by firstly applying bi-section search over $\alpha$ and then for each $\alpha$ calculating optimal power allocation via convex optimization. {Besides, we consider always selecting the BS with higher energy arrival rate to transmit in the single-BS transmission subframe. Finally, we also consider a more general fractional JT scheme that divides each frame into three subframes: Each BS transmits individually in the first and second subframe, and then they jointly transmit in the third subframe. We also solve the sum-rate maximization problem via DP.}
{By fixing the energy arrival rate of BS1 as 0.1W and changing that of BS2, the results are shown in Fig.~\ref{fig:compcmp}. It can be seen that the generalized fractional JT scheme with three subframes provides little performance gain compared with the scheme with two subframes, even with symmetric energy arrival rates. Intuitively, the fractional JT with three subframes may perform better in symmetric case. However, the performance depends not only on the energy arrival rates of two BSs, but also on the channel states. When the energy arrival rates are asymmetric, dividing each frame into two subframes and letting the BS with higher energy arrival rate to transmit in the first subframe is sufficient. When the energy arrival rates are symmetric, the channel states become the key factor. In fact, the case with asymmetric channel gains is analogous to the case with asymmetric energy profiles. Hence, letting the BS with higher channel gain to transmit in the first subframe is sufficient. The scheme with three subframes may be better in symmetric case, which is however of low probability as it requires the energy arrival rates and the channel states are jointly symmetric. In addition, the optimization for three subframes is much more complex than that for two subframes. Therefore, the fractional JT with two subframes is preferred.}
It can be also seen in Fig.~\ref{fig:compcmp} that the proposed approximate DP algorithm with $N_I = 10, N_E = 10$ performs very close to the optimal one. In addition, the greedy policy show a noticeable gap to the optimal policy, which illustrates the necessity of inter-frame energy allocation optimization. {Always choosing BS2 to transmit in the first subframe degrades the performance compared with the proposed algorithm, while the gap diminishes as the energy asymmetry becomes stronger. This is also due to the dependence of performance on both the energy profiles and the channel states. When the channel state of the BS with more energy is much worse than the other, it would be preferred to sleep to wait for a better channel.} Also, the proposed fractional JT algorithm dramatically outperforms the conventional network MIMO algorithm, especially when the asymmetry of energy arrival rate between two BSs becomes severe. {Notice that the performance gain is remarkable even for the symmetric case (energy arrival rate of BS2 is also 0.1W). As mentioned before, the gain comes from the asymmetry of channel states, which is analogous to the asymmetry of energy arrival rates.} With the increase of energy arrival rate in BS2, the sum-rate of conventional algorithm saturates to around 2.5bps/Hz. {The reason is that according to the power constraint (\ref{eq:powerctr}), the power constraint of BS2 associated with sufficiently large budget $P_{t,2}$ is usually satisfied with strict inequality. Then, increasing $P_{t,2}$ does not affect the optimization result. That is, the sum-rate does not increase as the higher energy arrival rate of BS2 does not contribute.} On the other hand, the sum-rate of the fractional JT increases in the speed of $\log$ function. It also shows the importance of applying fractional JT in energy harvesting system.
\begin{figure}
\centering
\includegraphics[width=4.5in]{CoMPcmpPmax.eps}
\caption{Average sum-rate comparison of different algorithms. The energy arrival rate of BS2 is 1.2W.} \label{fig:compPmax}
\end{figure}
{We further simulate the case that the energy arrival rate is sufficient for transmission. We set the maximum transmit power per frame as 1.2W. The energy arrival rate of BS2 is equal to the maximum power per frame, and we vary the rate of BS1 to obtain the curves in Fig.~\ref{fig:compPmax}. It can be seen that the performance gain of the proposed fractional JT strategy compared with the conventional network MIMO decreases as the energy arrival rate of BS1 becomes closer to that of BS2. And all the curves tend to be flat when the maximum transmit power can be satisfied by energy harvesting. Besides, always choosing BS2 to transmit in the first subframe approaches optimal then the energy asymmetry is strong. But it performs even worth than the greedy policy in symmetric case when the maximum transmit power is achieved in both BSs.}
\begin{figure}
\centering
\includegraphics[width=4.5in]{alpha.eps}
\caption{Average time ratio $\alpha$ for single-transmission phase of different algorithms. The energy arrival rate of BS1 is 0.1W.} \label{fig:alpha}
\end{figure}
Fig.~\ref{fig:alpha} shows the average time ratio $\alpha$ for single-transmission phase versus the energy arrival rate of BS2. It can be seen that average $\alpha$ increases as the asymmetry of energy arrival rates increases. Furthermore, the average $\alpha$ of DP optimal algorithm increases at the lowest speed, and the approximate DP algorithm performs very close to it. The greedy policy can only increase the time ratio for single-transmission to better utilize the higher energy arrival rate, and hence $\alpha$ increases at a higher speed w.r.t. the increase of energy arrival rate of BS2. On the contrary, by averaging the available energy over the transmission frames in the DP optimal and approximate DP algorithms, relatively more time ratio can be used to apply network MIMO to enhance the sum-rate.
\begin{figure}
\centering
\includegraphics[width=4.5in]{CDF.eps}
\caption{Cumulative distribution function of user data rate with different algorithms. The energy arrival rate of BS1 is 0.1W, and that of BS2 is 0.8W.} \label{fig:CDF}
\end{figure}
Finally, the cumulative distribution function (CDF) of user data rate is depicted in Fig.~\ref{fig:CDF} with energy arrival rates of the two BSs as 0.1W and 0.8W, respectively. It shows that the proposed fractional JT algorithm greatly enhances the user data rate compared with the conventional network MIMO, and the proposed approximate DP algorithm achieves close-to-optimal performance. Since the energy arrival rate of BS2 is much larger than BS1, simply choosing BS2 to transmit in the first subframe also performs close to the optimal. Notice that the greedy policy reduces the percentage of zero data rate since it transmits with all the available energy in each frame, with the sacrifice of channel fading diversity for opportunistic inter-frame scheduling. As a result, the ratio of low data rate is much higher than the DP-based algorithms. For instance, about 43\% of users' data rate is lower than 1bps/Hz. With DP-based algorithms, the ratio reduces by about 8\%.
\section{Conclusion}\label{sec:concl}
In this paper, we have proposed a fractional JT scheme for BS cooperation that divides a transmission frame to firstly apply single-BS transmission and then adopt ZF-JT transmission to enhance the average sum-rate. The MDP-based problem is formulated and solved by firstly allocating energy among frames and then optimizing per-frame sum-rate. By analyzing the convexity of per-frame sum-rate optimization problem, and applying approximate DP algorithm, the computational complexity is greatly reduced. The proposed fractional JT scheme has been shown to achieve much higher sum-rate compared with the conventional ZF-JT only scheme. As the energy arrival asymmetry increases, the achievable rate of ZF-JT saturates (2.5bps/Hz in our settings), while the proposed scheme reveals a logarithmic increase. The proposed approximate DP algorithm can approach the DP optimal algorithm with sufficient number of policy explorations.
{ In this paper, fractional JT with two subframes is considered since we only consider the transmit power consumption. If the non-ideal circuit power is considered, more general frame structure is required to further save energy. Specifically, the BSs may turn to idle mode to reduce the circuit power consumption. This would be an interesting research direction for future work.}
\appendices
\section{Proof of Proposition \ref{prop:greedy}} \label{proof:greedy}
For any given $\alpha$, the power allocation solution satisfies the Karush-Kuhn-Tucker (KKT) conditions \cite{boyd2004convex}. Define the Lagrangian function for any multipliers $\lambda\ge 0, \mu\ge 0, \eta\ge 0$ as
\begin{align}
\mathcal{L} = &-\Bigg(\alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2} {\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big)\Bigg) +\lambda\Big(\alpha\tilde{p} - \frac{B_{k}}{T_f} - \alpha E_k\Big)\nonumber\\
&+\mu\Big((1-\alpha) \sum_{i=1}^2 |w_{ki}|^2 p_{i} + \alpha\tilde{p} - A_k\Big) +\eta\Big((1-\alpha)\sum_{i=1}^2 |w_{\bar{k}i}|^2 p_{i} - A_{\bar{k}}\Big) \label{eq:lagrangian}
\end{align}
with additional complementary slackness conditions
\begin{align}
\lambda\Big(\alpha\tilde{p} - \frac{B_{k}}{T_f} - \alpha E_k\Big) &= 0, \nonumber\\
\mu\Big((1-\alpha) \sum_{i=1}^2 |w_{ki}|^2 p_{i} + \alpha\tilde{p} - A_k\Big) &=0,\nonumber\\
\eta\Big((1-\alpha)\sum_{i=1}^2 |w_{\bar{k}i}|^2 p_{i} - A_{\bar{k}}\Big) &= 0.\nonumber
\end{align}
Here, we ignore the non-negative power constraints in the above formulation to simplify the expression. It can be directly added to the result. We apply the KKT optimality conditions to the Lagrangian function (\ref{eq:lagrangian}). By setting $\partial\mathcal{L}/\partial \tilde{p} = \partial\mathcal{L}/\partial p_{i} =0$, we obtain
\begin{eqnarray}
\tilde{p}^* & = & \left[\frac{1}{\lambda+\mu}-\frac{\sigma^2_n}{|H_{\tilde{i}k}|^2}\right]^+, \label{opt:ptilde}\\
p_i^* & = & \left[\frac{1}{\mu|w_{ki}|^2+|w_{\bar{k}i}|^2\eta}-\sigma^2_n\right]^+, i = 1, 2.
\label{opt:pj}
\end{eqnarray}
Notice that to guarantee the validity of (\ref{opt:pj}), either $\mu$ or $\eta$ should be non-zero, which means that at least one of (\ref{eq:power2p}) and (\ref{eq:power3p}) is satisfied with equality.
\section{Proof of Lemma \ref{lemma:hincre}} \label{proof:hincre}
Since $h^*(s) = \lim\limits_{n\rightarrow+\infty}h^{(n)}(s)$, we prove the monotonicity property by induction. In addiction, we only need to prove the monotonicity for $B_1$. The proof for $B_2$ follows the same procedure.
Obviously, it is true for $n=0$ as $h^{(0)}(s) = 0, \forall s \in \mathcal{S}$. Assume that $h^{(n)}(B_1, B_2, \mathbf{H})$ is nondecreasing w.r.t $B_1$, and the optimal action for state $s = (B_1, B_2, \mathbf{H})$ is $a^* = (A_1^*, A_2^*)$, i.e.,
\begin{align}
\max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right]
= g(s, A_1^*, A_2^*) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1', B_2', \mathbf{H}'). \nonumber
\end{align}
Then consider the state $s'' = (B_1+\delta B, B_2, \mathbf{H})$, where $\delta B > 0$. We have
\begin{align}
&h^{(n+1)}(s'')\nonumber\\
= &(1-\tau)h^{(n)}(s'') + \max_{a\in \mathcal{A}(s'')}\left[ g(s'', a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right] - \Lambda^{(n+1)}(s_0) \nonumber\\
\buildrel (a) \over \ge &(1-\tau)h^{(n)}(s'') + g(s'', A_1^*, A_2^*) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1'+\delta B, B_2', \mathbf{H}') - \Lambda^{(n+1)}(s_0) \nonumber\\
\buildrel (b) \over \ge &(1-\tau)h^{(n)}(s) + g(s, A_1^*, A_2^*) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1', B_2', \mathbf{H}') - \Lambda^{(n+1)}(s_0)
= h^{(n+1)}(s), \nonumber
\end{align}
where the inequality (a) holds as the action $(A_1^*, A_2^*) \in \mathcal{A}(s'')$, and (b) holds due to the following two reasons. Firstly, $g(s'', A_1^*, A_2^*) \ge g(s, A_1^*, A_2^*)$ as the constraint (\ref{eq:power1p}) for the latter is not looser than the former. Secondly, $h^{(n)}(B_1'+\delta B, B_2', \mathbf{H}') \ge h^{(n)}(B_1', B_2', \mathbf{H}')$ due to the monotonicity of $h^{(n)}(B_1, B_2, \mathbf{H})$ w.r.t. $B_1$. As a result, we prove that $h^{(n+1)}(B_1, B_2, \mathbf{H})$ is also nondecreasing w.r.t. $B_1$.
In summary, $h^{(n)}(B_1, B_2, \mathbf{H})$ is nondecreasing w.r.t. $B_1$ for all $n = 0, 1, 2, \cdots$. Hence, we also have that $h^*(B_1, B_2, \mathbf{H})$ is nondecreasing w.r.t. $B_1$. The same holds for $B_2$.
\section{Proof of Theorem \ref{prop:gbar}} \label{proof:gbar}
Regarding the per-stage utility $\bar{g}$, the Bellman's equation also holds for a scalar $\bar{\Lambda}^*$ and some vector $\bm{\bar{h}}^* = \{\bar{h}^*(s)|s \in \mathcal{S}\}$, and the value iteration algorithm works in the same way. Hence, we only need to prove by induction that $\Lambda^{(n)}(s_0) = \bar{\Lambda}^{(n)}(s_0)$ and $h^{(n)}(s) = \bar{h}^{(n)}(s)$.
We initialize that $\Lambda^{(0)}(s_0) = \bar{\Lambda}^{(0)}(s_0) = 0$ and $h^{(0)}(s) = \bar{h}^{(0)}(s) = 0, \forall s \in \mathcal{S}$. Suppose that $\Lambda^{(n)}(s_0) = \bar{\Lambda}^{(n)}(s_0), h^{(n)}(s) = \bar{h}^{(n)}(s), \forall s \in \mathcal{S}$. For the $(n+1)$-th iteration and $\forall s = (B_1, B_2, \mathbf{H}), a = (A_1, A_2)$, we have
\begin{align}
\bar{g}(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1', B_2', \mathbf{H}')
\buildrel (c) \over \le & {g}(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1', B_2', \mathbf{H}') \nonumber\\
\buildrel (d) \over \le & {g}(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(B_1'', B_2'', \mathbf{H}')\nonumber
\end{align}
where $B_k' = B_k + T_fE_k - A_k, \forall k = 1, 2,$ while $B_k'', k = 1, 2$ are calculated via (\ref{eq:battery1}) and (\ref{eq:battery2}), respectively. Hence we have $B_k'' \ge B_k', \forall k = 1, 2$. Inequality (c) holds as the maximization of $g$ has larger feasible region than that of $\bar{g}$, while (d) holds due to the monotonicity of the relative utility $h(s)$. As a result, we have
\begin{align}
\max_{a \in \mathcal{A}(s)}\left[\bar{g}(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})\bar{h}^{(n)}(s')\right] \le \max_{a \in \mathcal{A}(s)}\left[g(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right] \label{proof:le}
\end{align}
On the other hand, there exists an action $(A_1^*, A_2^*)$ such that
\begin{align}
\max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right]
= & g(s, A_1^*, A_2^*) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})h^{(n)} (B_1^*, B_2^*, \mathbf{H}'),\nonumber\\
\buildrel (e) \over = & \bar{g}(s, A_1^*, A_2^*) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})\bar{h}^{(n)} (B_1^*, B_2^*, \mathbf{H}'),\nonumber\\
\buildrel (f) \over \le & \max_{a \in \mathcal{A}(s)}\left[\bar{g}(s, a) + \tau\sum_{ \mathbf{H}'} \mathrm{Pr} (\mathbf{H}'|\mathbf{H})\bar{h}^{(n)}(s')\right],\label{proof:ge}
\end{align}
where $B_k^* = B_k + T_fE_k - A_k^*, \forall k = 1, 2$, and hence, equality (e) holds. Inequality (f) holds as $(A_1^*, A_2^*) \in \mathcal{A}(s)$. It can be seen by (\ref{proof:le}), (\ref{proof:ge}) jointly with (\ref{eq:lambdanplus1}) and (\ref{eq:hnplus1}) that $\Lambda^{(n+1)}(s_0) = \bar{\Lambda}^{(n+1)}(s_0)$ and $h^{(n+1)}(s) = \bar{h}^{(n+1)}(s)$.
In summary, we have $\Lambda^{(n)}(s_0) = \bar{\Lambda}^{(n)}(s_0), h^{(n)}(s) = \bar{h}^{(n)}(s)$ for all $n = 0, 1, 2, \cdots$. Hence, we have $\Lambda^* = \max \; \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N \bar{g}(s_t, a_t(s_t))\!\right] = \bar{\Lambda}^*$.
\section{Proof of Proposition \ref{prop:ptilde}} \label{proof:ptilde}
According to the equality constraints (\ref{eq:power2gb}) and (\ref{eq:power3gb}), $p_i, i = 1, 2$ can be represented as functions of $\tilde{p}$, i.e., $p_1 = \frac{C_{1} - \alpha|w_{\bar{k}2}|^2\tilde{p}}{C_0}, p_2 = \frac{\alpha|w_{\bar{k}1}|^2\tilde{p} - C_{2}}{C_0}$, where $C_0, C_{1}, C_{2}$ are presented in the proposition.
As the elements of $\mathbf{H}$ are i.i.d., we have $C_0 \neq 0$. Hence, the per-stage sum rate function can be written as a function of $\tilde{p}$:
\begin{align}
f_{k,\alpha}(\tilde{p}) = \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2} {\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\left[\log_2\Big(\!1\!+\!\frac{C_{1} - \alpha|w_{\bar{k}2}|^2\tilde{p}}{\sigma^2_n C_0}\!\Big) + \log_2\Big(\!1\!+\!\frac{ \alpha|w_{\bar{k}1}|^2\tilde{p} - C_{2}}{\sigma^2_n C_0}\!\Big)\right]. \nonumber
\end{align}
The constraints can be written as the feasible set of $\tilde{p}$. Without loss of generality, we assume $C_0 > 0$. The feasible set for $C_0 < 0$ can be derived in the similar way. With the non-negative constraints $p_i \ge 0, i = 1, 2$, we have $\frac{C_{2}}{\alpha|w_{\bar{k}1}|^2} \le \tilde{p} \le \frac{C_{1}}{\alpha|w_{\bar{k}2}|^2}$. Jointly with (\ref{eq:power1p}) and $\tilde{p} \ge 0$, the feasible set can be expressed as ${\cal P}_{k,\alpha} = \left\{\tilde{p} \Big| \tilde{p}_{\mathrm{min}} \le \tilde{p} \le \right. \left.\tilde{p}_{\mathrm{max}} \right\}$, where $\tilde{p}_{\mathrm{min}}$ and $\tilde{p}_{\mathrm{max}}$ are expressed as (\ref{ptilde:min}) and (\ref{ptilde:max}), respectively. To guarantee that ${\cal P}_{k,\alpha} \neq \emptyset$, we have $\tilde{p}_{\mathrm{min}} \le \tilde{p}_{\mathrm{max}}$, which results in $\alpha \ge \frac{1}{E_k}\big( \frac{C_{2}}{|w_{\bar{k}1}|^2} - \frac{B_k}{T_f}\big)$. We set
\begin{equation}
\alpha_{\mathrm{min}} = \max\left\{0, \frac{1}{E_k}\Big( \frac{C_{2}}{|w_{\bar{k}1}|^2} - \frac{B_k}{T_f}\Big)\right\}. \label{eq:alphamin}
\end{equation}
Hence, there are two cases so that ${\cal P}_{k,\alpha} = \emptyset$. The first is $\alpha_{\mathrm{min}} > 1$, and the second is that $0 < \alpha_{\mathrm{min}} \le 1$ and $0 \le \alpha < \alpha_{\mathrm{min}}$. Otherwise, the per-frame optimization problem can be reformulated as
\begin{equation}
\max_{\tilde{p} \in {\cal P}_{k,\alpha}} f_{k,\alpha}(\tilde{p}), \label{eq:maxptilde}
\end{equation}
whose convexity still holds according to the following lemma.
\begin{lemma}
The problem (\ref{eq:maxptilde}) is a convex optimization problem.
\end{lemma}
\begin{IEEEproof}
As the $\log$ function is concave and the functions inside the $\log$ operation are linear function of $\tilde{p}$, the composition of a linear function with a concave function is still concave. Hence, $f_{k,\alpha}(\tilde{p})$ is a concave function. On the other hand, the feasible set ${\cal P}_{k,\alpha}$ is convex. Therefore, the considered problem is a convex optimization problem.
\end{IEEEproof}
Due to the concavity of the function $f_{k,\alpha}(\tilde{p})$, the optimal solution can be found by solving $f_{k,\alpha}'(\tilde{p}) = 0$, which is expressed as (\ref{eq:quadratic}). It can be transformed into a quadratic equation, and hence, the nonnegative root can be easily solved. Denote the solution for $f_{k,\alpha}'(\tilde{p}) = 0$ by $\tilde{p}_0$. Then according to the concavity of the function $f_{k,\alpha}$, the optimal solution for the problem $\max\limits_{\tilde{p} \in {\cal P}_{k,\alpha}} f_{k,\alpha}(\tilde{p})$ is either $\tilde{p}_0$ or the boundary points of the feasible set ${\cal P}_{k,\alpha}$ depending on whether $\tilde{p}_0 \in {\cal P}_{k,\alpha}$ or not.
\section{Proof of Theorem \ref{prop:alphaconcave}} \label{proof:alphaconcave}
For any $\alpha^{(1)}, \alpha^{(2)} \in [0, 1]$, we assume that
\begin{equation}
F_k(\alpha^{(j)}) = \alpha^{(j)}\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}^{(j)}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha^{(j)}\!)\sum_{i=1}^{2}\!\log_2\!\Big(\!1\!+\!\frac{p_{i}^{(j)}}{\sigma^2_n}\!\Big), \nonumber
\end{equation}
for $j = 1, 2$, i.e., $\tilde{p}^{(j)}, p_{i}^{(j)}, i = 1, 2$ achieve the maximum sum-rate. For any $0 < \gamma < 1$, we have
\begin{align}
{}\gamma F_k(\alpha^{(1)}) + (1-\gamma) F_k(\alpha^{(2)})
\le\alpha'\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}'|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha'\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}'}{\sigma^2_n}\!\Big) \label{eq:concave1}
\end{align}
where
\begin{align}
\alpha' =& \gamma\alpha^{(1)} + (1-\gamma)\alpha^{(1)}, \label{eq:alphap}\\
\tilde{p}' =& \frac{\gamma\alpha^{(1)}}{{\alpha'}}\tilde{p}^{(1)} +\frac{(1 - \gamma) \alpha^{(2)}}{{\alpha'}}\tilde{p}^{(2)}, \nonumber\\
p_i' = & \frac{\gamma(1-\alpha^{(1)})}{{1-\alpha'}}{p}_i^{(1)} +\frac{(1 - \gamma) (1-\alpha^{(2)})}{{1-\alpha'}}{p}_i^{(2)}, \quad i = 1, 2, \nonumber
\end{align}
and the inequality in (\ref{eq:concave1}) is due to the concavity of $\log$ function. In addition,
\begin{align}
\alpha'\tilde{p}' =& {\gamma\alpha^{(1)}}\tilde{p}^{(1)} +{(1 - \gamma) \alpha^{(2)}}\tilde{p}^{(2)} \nonumber\\
{}\le& \gamma\left( \frac{B_k}{T_f}+\alpha^{(1)}E_k \right) +(1 - \gamma) \left( \frac{B_k}{T_f}+\alpha^{(2)}E_k \right)
= \frac{B_k}{T_f} + \alpha'E_k, \nonumber
\end{align}
i.e., $\tilde{p}'$ satisfies the constraint (\ref{eq:power1p}). Similarly, $\tilde{p}'$ and $p_i', i = 1, 2$ also satisfy the constraints (\ref{eq:power2p}) and (\ref{eq:power3p}). Hence, $\tilde{p}', p_i', i = 1, 2$ is a feasible power allocation solution. As $F_k(\alpha)$ is maximal over all power allocation policies, we have
\begin{equation}
\alpha'\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}'|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha'\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}'}{\sigma^2_n}\!\Big) \le F_k(\alpha'). \label{eq:concave2}
\end{equation}
Combining (\ref{eq:concave1}), (\ref{eq:alphap}) and (\ref{eq:concave2}), we have
\begin{equation}
\gamma F_k(\alpha^{(1)}) + (1-\gamma) F_k(\alpha^{(2)}) \le F_k(\gamma\alpha^{(1)} + (1-\gamma)\alpha^{(1)}). \nonumber
\end{equation}
As a consequence, $F_k$ is a concave function. | {"config": "arxiv", "file": "1607.00575/EHbasedCoMP.tex"} |
TITLE: Generalizations of Inverse Function Theorem
QUESTION [2 upvotes]: A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem:
Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a
compact submanifold of $M$ such that $\left. f\right|_Z$ is injective,
and $f_*$ is an isomorphism at each point of $Z$. Then $f$ maps an
open neighborhood of $Z$ diffeomorphically onto an open neighborhood
of $f(Z)$.
At the risk of asking a slippery question, is this the "most general" version of the IFT, or is there one more general yet?
REPLY [1 votes]: Eric, even in Guillemin & Pollack you'll find a more general version. Look at Exercise 14 on p. 56. It removes the compactness hypothesis on $Z$.
There are also interesting questions to ask along the lines of this: If $f\colon\Bbb R^n\to\Bbb R^n$ (replace with manifolds if you wish) is a local diffeomorphism at each point, what condition(s) are sufficient to guarantee that $f$ is a global diffeomorphism? | {"set_name": "stack_exchange", "score": 2, "question_id": 1405743} |
TITLE: Lebesgue Integration by Riesz Method textbook
QUESTION [0 upvotes]: As an undergraduate student I am recommended to use Soo Bong Chae's Lebesgue Integration as a textbook for a course of Lebesgue Integral. The book is far from satisfying my personal needs as it gets quite complicated in proofs and quite a lot of arguments in proofs are not explained (simply left to reader). But this book approaches Lebesgue Integration by Riesz Method, meaning that Lebesgue integration is approximated by sequences of step functions and also Lebesgue measure is considered to be the consequence of the integration theory. Nonetheless, in my search through Google, I did not come across a textbook which consists of mainly Lebesgue Integration and Measure by Riesz approach and written for undergraduates. So, I need your help to suggest me another book on the subject.
REPLY [0 votes]: This answer is obviously a little bit dated, but for anybody looking in the future: The Lebesgue Integral for Undergraduates by William presents Lebesgue Integration using the Riesz approach. In the book, he calls it the "Daniell-Riesz" approach (after Percy J. Daniell). I read the book with a professor through an independent study course.
It's not a full treatise on the subject (the author is upfront about its limitations in the introduction). The book definitely helped me when I later encountered measure theory in a real analysis course.
If anyone is interested, you can find PDFs of the Preface and Table of Contents on the MAA's website:
Preface: https://www.maa.org/sites/default/files/pdf/ebooks/TLI_Preface.pdf
Table of Contents: https://bookstore.ams.org/cdn-1646297439663/text-27/~~FreeAttachments/text-27-toc.pdf
** I'm not affiliated with the author/MAA/etc | {"set_name": "stack_exchange", "score": 0, "question_id": 2051694} |
TITLE: Cobordism monopole Floer homology
QUESTION [6 upvotes]: From the famous book: Monopole and three manifold, Kronheimer and Mrowka(https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf). It is known that:
Let $Y$ be a closed oriented $3$ manifold, choosing a spinc structure $\mathfrak s$ and metric $g$ and a generic perturbation $p$, one can construct the monopole Floer homology groups:
$$\check{HM}_*(Y,\mathfrak s, g,p),~\hat{HM}_*(Y,\mathfrak s, g,p),~\overline{HM}_*(Y,\mathfrak s, g,p).$$
The groups are graded over a set $\mathbb J_s$ admitting a $\mathbb Z$ action.( details are given in Section 20-22). We define the negative completions(Definition 23.1.3 of the book) by
$$\check{HM}_\bullet(Y,\mathfrak s, g,p),~\hat{HM}_\bullet(Y,\mathfrak s, g,p),~\overline{HM}_\bullet(Y,\mathfrak s, g,p).$$
If we want to consider all spinc
structures at the same time, we need to consider the completed monopole Floer homology
$$\check{HM}_\bullet(M,F;\mathbb F)=\bigoplus_\mathfrak s\check{HM}_\bullet(M,F,\mathfrak;\mathbb F).$$
To show that these homology groups are independent of the metric and the perturbation, the authors gave a property: a cobordism between 3-manifolds gives rise to homomorphisms between
their monopole Floer homologies(see Section 23-26). They construct a homomorphism from $\check{HM}_\bullet(Y,g_1,p_1)$ to $\check{HM}_\bullet(Y',g_2,p_2)$, where there is a cobordism from $Y$ to $Y'$.
Q I do not understand the two points below:
Why the authors use the negative completion, where we need it?
If we just want to show that the monopole Floer homology $\check{HM}_*(Y,\mathfrak s)$ is independent of the metric and perturbation, can we just using the trivial cobordism $[0,1]\times Y$ to show a homomorphism $\check{HM}_*(Y,\mathfrak s,g_1,p_1) \to \check{HM}_*(Y,\mathfrak s, g_2,p_2)$? The homomorphism is given by counting the number of solutions of the zero-dim moduli space $M([a_1],W^*,[b_2])$, where $W^*=(-\infty,0]\times Y\cup I\times Y\cup[1,\infty)\times Y$, and $[a_1]$ and $[b_2]$ are the critical points of $(Y,\mathfrak s,g_1,p_1)$ and $(Y,\mathfrak s,g_2,p_2)$ respectively. I think the arguments of Section23-25 also work before taking the negative completion .
PS Let $G_*$ be an abelian group graded by the set $\mathbb J$ equipped with a $\mathbb Z$-action. Let $O_a(a\in A)$ be the set of free $\mathbb Z$-orbits in $\mathbb J$ and fix an element $j_a\in O_a$ for each $a$. Consider the subgroups
$$G_*[n]=\bigoplus_a\bigoplus_{m\geq n} G_{j_a-m},$$
which form a decreasing filtration of $G_*$. We define the negative completion of $G_*$ as the topological group $G_\bullet\supset G_*$ obtained by completing with respect to this filtration.
REPLY [3 votes]: The first bullet is definitely explained in the book! Surely around where it was introduced, it has to do with summing over all spin-c structures. We need to pass to the completion because the 4-manifold can have infinitely many spin-c structures that would need to be used.
The second bullet, yes. In general we should not expect results concerning completions of a graded group to also hold for the uncompleted group. But here we consider the trivial cobordism, and spin-c structures on $[0,1]\times Y$ are the same as spin-c structure on $Y$, so no completion is needed in this situation. | {"set_name": "stack_exchange", "score": 6, "question_id": 361462} |
TITLE: Uniform Convergence of the series : $\sum\frac{1}{1+x^n}, x\ge0$
QUESTION [0 upvotes]: It is clear that the series converges uniformly on the interval $[a,\infty)$ , for $a>1$ as follows on using the Weirstrass M-Test as
$$\frac{1}{1+x^n} < \frac{1}{a^n}$$
Kindly suggest what about rest of the domain. My book says discuss the convergence and uniform convergence. I am confused how to discuss the two concepts in this example. Please help.
REPLY [4 votes]: Let's denote $u_n(x)=\frac{1}{1+x^n}$ then it's clear that if $0\leq x\leq 1$ then $u_n(x)\not \to_{n\to\infty}0$ then $\sum_n u_n(x)$ is divergent and if $x>1$ then $u_n(x)\sim x^{-n}$ so $\sum_n u_n(x)$ is convergent by comparaison with the geometric series and hence the series $\sum_n u_n(x)$ is pointwise convergent on the interval $(1,+\infty)$.
Now if $x\geq a$ for $a>1$ we have
$$||u_n||_\infty\leq a^{-n}$$
hence the series $\sum_n ||u_n||_\infty$ is convergent then the series $\sum_n u_n(x)$ is normal convergent and then uniformly convergent on the interval $[a,+\infty)$.
Recall We have: the normal convergence implies the uniform convergence implies the pointwise convergence but the converse isn't true. | {"set_name": "stack_exchange", "score": 0, "question_id": 353323} |
\section{Additional Simulations}\label{sec:sim_appendix}
\subsection{Table of results for the AR$(1)$ and MA$(1)$ for a Gaussian time series}\label{sec:AR-MA-Gaussian}
\begin{landscape}
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{c|rrrrr|rrrrr}
\multirow{2}{*}{\textit{Likelihoods}} & \multicolumn{10}{c}{$\theta$} \\
\cline{2-11}
& 0.1 & 0.3 & 0.5 & 0.7 & 0.9 & 0.1 & 0.3 & 0.5 & 0.7 & 0.9 \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=20$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=20$} \\ \hline
Gaussian & \color{blue}{-$0.012$}{\scriptsize (0.22)} & \color{blue}{-$0.028$}{\scriptsize (0.21)} & \color{red}{-$0.043$}{\scriptsize (0.19)} & \color{red}{-$0.066$}{\scriptsize (0.18)} & \color{red}{-$0.072$}{\scriptsize (0.14)}
& \color{red}{$0.010$}{\scriptsize (0.28)} & $0.016${\scriptsize (0.28)} & $0.025${\scriptsize (0.24)} & \color{red}{$0.012$}{\scriptsize (0.21)} & \color{red}{$0.029$}{\scriptsize (0.17)} \\
Whittle & \color{red}{-$0.015$}{\scriptsize (0.21)} & \color{red}{-$0.041$}{\scriptsize (0.20)} & \color{blue}{-$0.063$}{\scriptsize (0.19)} & -$0.095${\scriptsize (0.18)} & -$0.124${\scriptsize (0.15)} &
\color{blue}{$0.005$}{\scriptsize (0.29)} & \color{blue}{$0.002$}{\scriptsize (0.28)} & \color{red}{-$0.004$}{\scriptsize (0.24)} & -$0.052${\scriptsize (0.23)} & -$0.152${\scriptsize (0.21)} \\
\color{blue}{Boundary} & -$0.015${\scriptsize (0.22)} & -$0.037${\scriptsize (0.21)} & -$0.054${\scriptsize (0.19)} & -$0.079${\scriptsize (0.18)} & -$0.103${\scriptsize (0.14)}
& $0.007${\scriptsize (0.30)} & $0.009${\scriptsize (0.29)} & $0.009${\scriptsize (0.24)} &-$0.022${\scriptsize (0.24)} & -$0.111${\scriptsize (0.20)} \\
\color{blue}{Hybrid} & -$0.012${\scriptsize (0.22)} & -$0.030${\scriptsize (0.21)} & -$0.049${\scriptsize (0.19)} & \color{blue}{-$0.072$}{\scriptsize (0.18)} & \color{blue}{-$0.095$}{\scriptsize (0.14)}
& $0.011${\scriptsize (0.30)} & $0.021${\scriptsize (0.29)} & $0.026${\scriptsize (0.25)} & -$0.007${\scriptsize (0.22)} & \color{blue}{-$0.074$}{\scriptsize (0.17)} \\
Tapered & -$0.014${\scriptsize (0.22)} & -$0.036${\scriptsize (0.21)} & -$0.063${\scriptsize (0.19)} & -$0.090${\scriptsize (0.18)} & -$0.117${\scriptsize (0.14)}
& $0.004${\scriptsize (0.29)} & \color{red}{$0.004$}{\scriptsize (0.28)} & \color{blue}{-$0.006$}{\scriptsize (0.24)} & \color{blue}{-$0.043$}{\scriptsize (0.21)} & -$0.122${\scriptsize (0.18)} \\
Debiased & -$0.013${\scriptsize (0.22)} & -$0.033${\scriptsize (0.21)} & -$0.049${\scriptsize (0.19)} & -$0.069${\scriptsize (0.19)} & -$0.085${\scriptsize (0.16)}
& $0.005${\scriptsize (0.29)} & $0.013${\scriptsize (0.28)} & $0.021${\scriptsize (0.25)} & -$0.005${\scriptsize (0.24)} & -$0.088${\scriptsize (0.21)} \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=50$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=50$} \\ \hline
Gaussian & \color{blue}{-$0.006$}{\scriptsize (0.14)} &\color{red}{-$0.011$}{\scriptsize (0.14)} & \color{red}{-$0.013$}{\scriptsize (0.12)} & \color{red}{-$0.033$}{\scriptsize (0.11)} & \color{red}{-$0.030 $}{\scriptsize (0.07)}
& -$0.002${\scriptsize (0.16)} & \color{blue}{$0.008$}{\scriptsize (0.15)} & \color{blue}{$0.017$}{\scriptsize (0.14)} & $0.018${\scriptsize (0.12)} & \color{blue}{$0.014$}{\scriptsize (0.08)} \\
Whittle & \color{red}{-$0.008$}{\scriptsize (0.14)} & \color{blue}{-$0.016$}{\scriptsize(0.14)} & \color{blue}{-$0.023$}{\scriptsize (0.12)} & -$0.045${\scriptsize (0.11)} & -$0.049${\scriptsize (0.08)}
& \color{red}{-$0.004$}{\scriptsize (0.15)} & \color{red}{$0.001$}{\scriptsize (0.15)} & \color{red}{$0.001$}{\scriptsize (0.14)} & -$0.020${\scriptsize (0.13)} & -$0.067${\scriptsize (0.11)} \\
\color{blue}{Boundary} & \color{blue}{-$0.007$}{\scriptsize (0.14)} &-$0.012${\scriptsize (0.14)} & -$0.015${\scriptsize (0.12)} & \color{blue}{-$0.034$}{\scriptsize (0.11)} & \color{blue}{-$0.036$}{\scriptsize (0.07)}
& -$0.003${\scriptsize (0.16)} & $0.006${\scriptsize (0.16)} & $0.013${\scriptsize (0.14)} & $0.005${\scriptsize (0.13)} & -$0.026${\scriptsize (0.09)} \\
\color{blue}{Hybrid} & -$0.005${\scriptsize (0.14)} & -$0.011${\scriptsize (0.14)} & -$0.015${\scriptsize (0.13)} & -$0.033${\scriptsize(0.11)} & -$0.035${\scriptsize (0.07)}
&-$0.001${\scriptsize (0.16)} & $0.010${\scriptsize (0.16)} & $0.015${\scriptsize (0.14)} & \color{blue}{$0.014$}{\scriptsize (0.12)} & \color{red}{-$0.010$}{\scriptsize (0.07)} \\
Tapered & -$0.005${\scriptsize (0.14)} & -$0.013${\scriptsize(0.14)} &-$0.018${\scriptsize (0.13)} & -$0.038${\scriptsize (0.11)} & -$0.039${\scriptsize (0.08)}
& $0${\scriptsize (0.16)} & $0.008${\scriptsize (0.16)} &$0.010${\scriptsize (0.14)} & \color{red}{$0.003$}{\scriptsize (0.12)} & -$0.023${\scriptsize (0.08)} \\
Debiased & -$0.006${\scriptsize (0.14)} &-$0.011${\scriptsize (0.14)} & -$0.015${\scriptsize (0.12)} & -$0.035${\scriptsize (0.11)} & -$0.032${\scriptsize (0.08)}
& \color{blue}{-$0.002$}{\scriptsize (0.16)} & $0.009${\scriptsize (0.16)} & $0.019${\scriptsize (0.15)} & $0.017${\scriptsize (0.15)} & -$0.011${\scriptsize (0.11)} \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=300$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim \mathcal{N}(0,1)$, $n=300$} \\ \hline
Gaussian & \color{blue}{$0$}{\scriptsize (0.06)} &\color{red}{-$0.002$}{\scriptsize (0.06)} & \color{blue}{-$0.001$}{\scriptsize (0.05)} & \color{red}{-$0.004$}{\scriptsize (0.04)} & \color{red}{-$0.005$}{\scriptsize (0.03)}
& $0.002${\scriptsize (0.06)} & \color{red}{$0$}{\scriptsize (0.06)} & \color{blue}{$0.003$}{\scriptsize (0.05)} & \color{red}{$0$}{\scriptsize (0.04)} & \color{blue}{$0.004$}{\scriptsize (0.03)} \\
Whittle & \color{red}{$0$}{\scriptsize (0.06)} & \color{blue}{-$0.003$}{\scriptsize(0.06)} & \color{red}{-$0.003$}{\scriptsize (0.05)} & -$0.007${\scriptsize (0.04)} & -$0.008${\scriptsize (0.03)}
& \color{red}{$0.001$}{\scriptsize (0.06)} & \color{blue}{-$0.001$}{\scriptsize (0.06)} & \color{red}{$0$}{\scriptsize (0.05)} & -$0.007${\scriptsize (0.04)} & -$0.020${\scriptsize (0.04)} \\
\color{blue}{Boundary} & \color{blue}{$0$}{\scriptsize (0.06)} &-$0.002${\scriptsize (0.06)} & \color{blue}{-$0.001$}{\scriptsize (0.05)} & \color{blue}{-$0.004$}{\scriptsize (0.04)} & \color{blue}{-$0.006$}{\scriptsize (0.03)}
& \color{blue}{$0.002$}{\scriptsize (0.06)} & $0${\scriptsize (0.06)} & $0.003${\scriptsize (0.05)} & \color{blue}{$0$}{\scriptsize (0.04)} & -$0.002${\scriptsize (0.03)} \\
\color{blue}{Hybrid} & $0${\scriptsize (0.06)} & -$0.002${\scriptsize (0.06)} & -$0.001${\scriptsize (0.05)} & -$0.005${\scriptsize(0.04)} & -$0.006${\scriptsize (0.03)}
&$0.002${\scriptsize (0.06)} & $0${\scriptsize (0.06)} & $0.004${\scriptsize (0.05)} & $0.001${\scriptsize (0.05)} & \color{red}{$0.003$}{\scriptsize (0.03)} \\
Tapered & $0${\scriptsize (0.06)} & -$0.002${\scriptsize (0.06)} & -$0.001${\scriptsize (0.05)} & -$0.005${\scriptsize(0.05)} & -$0.006${\scriptsize (0.03)}
&$0.002${\scriptsize (0.06)} & $0${\scriptsize (0.06)} & $0.004${\scriptsize (0.05)} & $0.001${\scriptsize (0.05)} & $0.003${\scriptsize (0.03)} \\
Debiased & \color{blue}{$0$}{\scriptsize (0.06)} & -$0.002${\scriptsize (0.06)} & \color{blue}{-$0.001$}{\scriptsize (0.05)} & -$0.004${\scriptsize(0.04)} & -$0.006${\scriptsize (0.03)}
&\color{blue}{$0.002$}{\scriptsize (0.06)} & $0${\scriptsize (0.06)} & $0.003${\scriptsize (0.05)} & $0${\scriptsize (0.05)} & $0.009${\scriptsize (0.05)} \\
\end{tabular}
\caption{\textit{Bias and the standard deviation (in the parentheses) of six different quasi-likelihoods for an AR(1) (left) and MA(1) (right) model for
the standard normal innovations. Length of the time series $n=20, 50$, and $300$. We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.}}
\label{tab:AR}
\end{table}
\end{landscape}
\subsection{Figures and Table of results for the AR$(1)$ and MA$(1)$ for a non-Gaussian time series}\label{sec:AR-MA-chi}
In this section, we provide figures and table of the results in Section \ref{sec:specified}
when the innovations follow a standardized chi-squared distribution
two degrees of freedom, i.e.
$\varepsilon_t \sim (\chi^2(2)-2)/2$
(this time the asymptotic bias will contain the fourth order cumulant term).
The results are very similar to the Gaussian innovations.
\begin{figure}[ht]
\begin{center}
\textbf{AR$(1)$ model}
\vspace{1em}
\includegraphics[scale=0.35,page=1]{plot/AR_chisq2.pdf}
\includegraphics[scale=0.35,page=3]{plot/AR_chisq2.pdf}
\includegraphics[scale=0.35,page=5]{plot/AR_chisq2.pdf}
\includegraphics[scale=0.35,page=2]{plot/AR_chisq2.pdf}
\includegraphics[scale=0.35,page=4]{plot/AR_chisq2.pdf}
\includegraphics[scale=0.35,page=6]{plot/AR_chisq2.pdf}
\vspace{1em}
\textbf{MA$(1)$ model}
\vspace{1em}
\includegraphics[scale=0.35,page=1]{plot/MA_chisq2.pdf}
\includegraphics[scale=0.35,page=3]{plot/MA_chisq2.pdf}
\includegraphics[scale=0.35,page=5]{plot/MA_chisq2.pdf}
\includegraphics[scale=0.35,page=2]{plot/MA_chisq2.pdf}
\includegraphics[scale=0.35,page=4]{plot/MA_chisq2.pdf}
\includegraphics[scale=0.35,page=6]{plot/MA_chisq2.pdf}
\caption{\textit{Bias (first row) and the RMSE (second row) of the parameter estimates for the AR(1) and MA(1) models
where the innovations follow the standardized chi-squared distribution with 2 degrees of freedom. Length of the time series $n=20$(left), $50$(middle), and $300$(right). } }
\label{fig:ar1ma1.chisq}
\end{center}
\end{figure}
\begin{landscape}
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{c|rrrrr|rrrrr}
\multirow{2}{*}{\textit{Likelihoods}} & \multicolumn{10}{c}{$\theta$} \\
\cline{2-11}
& 0.1 & 0.3 & 0.5 & 0.7 & 0.9 & 0.1 & 0.3 & 0.5 & 0.7 & 0.9 \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=20$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=20$} \\ \hline
Gaussian & \color{blue}{-$0.007$}{\scriptsize (0.21)} & \color{blue}{-$0.007$}{\scriptsize (0.20)} & \color{red}{-$0.029$}{\scriptsize (0.19)} & \color{red}{-$0.053$}{\scriptsize (0.17)} & \color{red}{-$0.069$}{\scriptsize (0.13)}
& -$0.001${\scriptsize (0.28)} & \color{blue}{$0.030$}{\scriptsize (0.25)} & \color{red}{$0.020$}{\scriptsize (0.23)} & \color{red}{$0.004$}{\scriptsize (0.20)} & \color{red}{$0.056$}{\scriptsize (0.17)} \\
Whittle & -$0.009${\scriptsize (0.21)} & -$0.016${\scriptsize (0.20)} & -$0.043${\scriptsize (0.20)} & -$0.086${\scriptsize (0.18)} & -$0.119${\scriptsize (0.14)} &
\color{blue}{-$0.005$}{\scriptsize (0.27)} & $0.018${\scriptsize (0.26)} & $0${\scriptsize (0.24)} & -$0.061${\scriptsize (0.22)} & -$0.153${\scriptsize (0.21)} \\
\color{blue}{Boundary} & -$0.007${\scriptsize (0.22)} & -$0.013${\scriptsize (0.20)} & -$0.035${\scriptsize (0.20)} & -$0.068${\scriptsize (0.18)} & -$0.097${\scriptsize (0.13)}
& -$0.002${\scriptsize (0.28)} & $0.024${\scriptsize (0.26)} & $0.009${\scriptsize (0.25)} &-$0.030${\scriptsize (0.23)} & -$0.113${\scriptsize (0.20)} \\
\color{blue}{Hybrid} & -$0.002${\scriptsize (0.22)} & -$0.005${\scriptsize (0.20)} & -$0.026${\scriptsize (0.20)} & \color{blue}{-$0.058$}{\scriptsize (0.18)} & \color{blue}{-$0.088$}{\scriptsize (0.13)}
& $0.005${\scriptsize (0.29)} & $0.035${\scriptsize (0.26)} & $0.021${\scriptsize (0.24)} & \color{blue}{$0.004$}{\scriptsize (0.20)} & \color{blue}{-$0.074$}{\scriptsize (0.17)} \\
Tapered & -$0.003${\scriptsize (0.21)} & -$0.011${\scriptsize (0.20)} & -$0.037${\scriptsize (0.20)} & -$0.077${\scriptsize (0.18)} & -$0.109${\scriptsize (0.13)}
& $0.002${\scriptsize (0.28)} & $0.023${\scriptsize (0.25)} & \color{blue}{$0.002$}{\scriptsize (0.23)} & -$0.032${\scriptsize (0.21)} & -$0.112${\scriptsize (0.18)} \\
Debiased & \color{red}{-$0.011$}{\scriptsize (0.21)} & \color{red}{-$0.018$}{\scriptsize (0.19)} & \color{blue}{-$0.040$}{\scriptsize (0.20)} & -$0.070${\scriptsize (0.19)} & -$0.090${\scriptsize (0.15)}
& \color{red}{-$0.007$}{\scriptsize (0.27)} & \color{red}{$0.021$}{\scriptsize (0.25)} & $0.010${\scriptsize (0.24)} & -$0.039${\scriptsize (0.24)} & -$0.140${\scriptsize (0.23)} \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=50$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=50$} \\ \hline
Gaussian & $0.004${\scriptsize (0.13)} &-$0.011${\scriptsize (0.13)} & \color{red}{-$0.012$}{\scriptsize (0.11)} & \color{red}{-$0.031$}{\scriptsize (0.10)} & \color{red}{-$0.029$}{\scriptsize (0.07)}
& $0.009${\scriptsize (0.15)} & \color{blue}{$0.003$}{\scriptsize (0.15)} & \color{red}{$0.017$}{\scriptsize (0.13)} & $0.014${\scriptsize (0.12)} & \color{red}{$0.010$}{\scriptsize (0.08)} \\
Whittle & \color{red}{$0.001$}{\scriptsize (0.13)} & \color{blue}{-$0.016$}{\scriptsize(0.13)} & -$0.019${\scriptsize (0.12)} & -$0.044${\scriptsize (0.10)} & -$0.049${\scriptsize (0.07)}
& \color{red}{$0.005$}{\scriptsize (0.14)} & \color{red}{-$0.004$}{\scriptsize (0.14)} & $0.004${\scriptsize (0.14)} & -$0.020${\scriptsize (0.13)} & -$0.065${\scriptsize (0.12)} \\
\color{blue}{Boundary} & $0.001${\scriptsize (0.13)} &-$0.013${\scriptsize (0.13)} & -$0.012${\scriptsize (0.12)} & \color{blue}{-$0.033$}{\scriptsize (0.10)} & -$0.036${\scriptsize (0.07)}
& $0.006${\scriptsize (0.15)} & $0.001${\scriptsize (0.15)} & $0.015${\scriptsize (0.14)} & $0.001${\scriptsize (0.12)} & -$0.030${\scriptsize (0.10)} \\
\color{blue}{Hybrid} & $0.003${\scriptsize (0.13)} & -$0.009${\scriptsize (0.14)} & -$0.010${\scriptsize (0.12)} & \color{blue}{-$0.032$}{\scriptsize(0.11)} & \color{blue}{-$0.034$}{\scriptsize (0.07)}
&$0.008${\scriptsize (0.15)} & $0.005${\scriptsize (0.15)} & $0.018${\scriptsize (0.13)} & \color{blue}{$0.010$}{\scriptsize (0.12)} & \color{blue}{-$0.014$}{\scriptsize (0.09)} \\
Tapered & $0.003${\scriptsize (0.13)} & -$0.011${\scriptsize(0.14)} &-$0.013${\scriptsize (0.12)} & -$0.036${\scriptsize (0.11)} & -$0.038${\scriptsize (0.07)}
& $0.007${\scriptsize (0.15)} & $0.004${\scriptsize (0.15)} & \color{blue}{$0.014$}{\scriptsize (0.13)} & \color{red}{$0$}{\scriptsize (0.11)} & -$0.026${\scriptsize (0.08)} \\
Debiased & \color{blue}{$0.002$}{\scriptsize (0.13)} & \color{red}{-$0.013$}{\scriptsize (0.13)} & \color{blue}{-$0.014$}{\scriptsize (0.11)} & -$0.034${\scriptsize (0.11)} & -$0.030${\scriptsize (0.08)}
& \color{blue}{$0.007$}{\scriptsize (0.15)} & \color{blue}{$0.001$}{\scriptsize (0.15)} & $0.017${\scriptsize (0.14)} & $0.015${\scriptsize (0.14)} & -$0.027${\scriptsize (0.13)} \\ \hline \hline
& \multicolumn{5}{c}{ \textbf{AR(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=300$}
& \multicolumn{5}{c}{ \textbf{MA(1)}, $\{e_{t}\}\sim (\chi^2(2)-2)/2$, $n=300$} \\ \hline
Gaussian & $0${\scriptsize (0.06)} &\color{red}{-$0.005$}{\scriptsize (0.05)} & \color{red}{-$0.004$}{\scriptsize (0.05)} & \color{red}{-$0.004$}{\scriptsize (0.04)} & \color{red}{-$0.006$}{\scriptsize (0.03)}
& $0${\scriptsize (0.06)} & \color{red}{-$0.002$}{\scriptsize (0.05)} & \color{red}{$0$}{\scriptsize (0.05)} & \color{red}{$0.003$}{\scriptsize (0.04)} & \color{blue}{$0.003$}{\scriptsize (0.03)} \\
Whittle & \color{red}{-$0.001$}{\scriptsize (0.06)} & \color{blue}{-$0.006$}{\scriptsize(0.05)} & -$0.005${\scriptsize (0.05)} & -$0.006${\scriptsize (0.04)} & -$0.009${\scriptsize (0.03)}
& \color{red}{$0$}{\scriptsize (0.06)} & \color{blue}{-$0.003$}{\scriptsize (0.05)} & -$0.003${\scriptsize (0.05)} & -$0.004${\scriptsize (0.04)} & -$0.018${\scriptsize (0.04)} \\
\color{blue}{Boundary} & \color{blue}{$0$}{\scriptsize (0.06)} &-$0.005${\scriptsize (0.05)} & \color{blue}{-$0.004$}{\scriptsize (0.05)} & \color{blue}{-$0.004$}{\scriptsize (0.04)} & \color{blue}{-$0.007$}{\scriptsize (0.03)}
& \color{blue}{$0$}{\scriptsize (0.06)} & -$0.002${\scriptsize (0.05)} & \color{blue}{$0$}{\scriptsize (0.05)} & \color{blue}{$0.002$}{\scriptsize (0.04)} & -$0.002${\scriptsize (0.03)} \\
\color{blue}{Hybrid} & $0${\scriptsize (0.06)} & -$0.006${\scriptsize (0.06)} & -$0.004${\scriptsize (0.05)} & -$0.004${\scriptsize(0.04)} & -$0.007${\scriptsize (0.03)}
&$0.001${\scriptsize (0.06)} & -$0.002${\scriptsize (0.06)} & $0${\scriptsize (0.05)} & \color{blue}{$0.003$}{\scriptsize (0.04)} & \color{red}{$0.002$}{\scriptsize (0.03)} \\
Tapered & $0${\scriptsize (0.06)} & -$0.006${\scriptsize (0.06)} & -$0.005${\scriptsize (0.05)} & -$0.004${\scriptsize(0.04)} & -$0.007${\scriptsize (0.03)}
&$0.001${\scriptsize (0.06)} & -$0.002${\scriptsize (0.06)} & $0${\scriptsize (0.05)} & $0.003${\scriptsize (0.04)} & $0.001${\scriptsize (0.03)} \\
Debiased & $0${\scriptsize (0.06)} & -$0.005${\scriptsize (0.05)} & -$0.004${\scriptsize (0.05)} & -$0.004${\scriptsize(0.04)} & -$0.006${\scriptsize (0.03)}
&\color{blue}{$0$}{\scriptsize (0.06)} & -$0.002${\scriptsize (0.05)} & $0${\scriptsize (0.05)} & $0.003${\scriptsize (0.05)} & $0.013${\scriptsize (0.05)} \\
\end{tabular}
\caption{\textit{Bias and the standard deviation (in the parentheses) of six different quasi-likelihoods for an AR(1) (left) and MA(1) (right) model for
the standardized chi-squared innovations. Length of the time series $n=20, 50$, and $300$. We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.}}
\label{tab:AR.chisq}
\end{table}
\end{landscape}
\subsection{Misspecified model for a non-Gaussian time series}
In this section, we provide figures and table of the results in Section \ref{sec:misspecifiedmodel}
when the innovations follow a standardized chi-squared distribution
two degrees of freedom, i.e.
$\varepsilon_t \sim (\chi^2(2)-2)/2$.
The results are given in Tables \ref{tab:arma11.chisq} and \ref{tab:ar2.chisq}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{cc|ccccccc}
$n$ & Parameter & Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased \\ \hline \hline
\multirow{3}{*}{20} & $\phi$ & {\color{blue} $0.029(0.1)$} & -$0.102(0.16)$ & -$0.032(0.12)$
&{\color{red} -$0.001(0.1)$} & -$0.088(0.13)$ & $0.170(0.12)$ \\
& $\psi$ & {\color{blue} $0.066(0.08)$} & -$0.184(0.20)$ & -$0.039(0.15)$
& {\color{red}$0.030(0.09)$} & -$0.064(0.12)$ & $0.086(0.09)$ \\
& $I_{n}(f;f_\theta)$ & $1.573(0.82)$ & $1.377(3.11)$ & $0.952(0.91)$
& {\color{blue} $1.006(0.84)$} & {\color{red}$0.675(0.63)$} & $2.618(0.84)$\\ \hline
\multirow{3}{*}{50} & $\phi$ & {\color{red}$0.014(0.07)$} & -$0.051(0.10)$ & -$0.004(0.07)$
& $0.007(0.07)$ & {\color{blue}-$0.003(0.07)$} & $0.143(0.11)$ \\
& $\psi$ & $0.027(0.06)$ & -$0.118(0.13)$ & -$0.013(0.09)$
& {\color{blue} $0.008(0.07)$} & {\color{red} $0.009(0.06)$} & $0.090(0.03)$ \\
& $I_{n}(f;f_\theta)$ & $0.342(0.34)$ & $0.478(0.53)$ & $0.298(0.32)$
& {\color{blue}$0.230(0.27)$} & {\color{red} $0.222(0.27)$} & $1.158(0.37)$ \\ \hline
\multirow{3}{*}{300} & $\phi$ & {\color{red} $0.001(0.03)$} & -$0.015(0.03)$ & {\color{blue} -$0.002(0.03)$}
& $0(0.03)$ & -$0.001(0.03)$ & $0.090(0.08)$ \\
& $\psi$ & {\color{red}$0.006(0.03)$} & -$0.033(0.05)$ & $0.002(0.03)$
& {\color{red}$0.003(0.03)$} & $0.003(0.03)$ & $0.091(0.02)$ \\
& $I_{n}(f;f_\theta)$ & $0.029(0.05)$ & $0.067(0.10)$ & $0.034(0.06)$
& {\color{red}$0.027(0.04)$} & {\color{blue}$0.028(0.04)$} & $0.747(0.23)$ \\ \hline
\multicolumn{8}{l}{Best fitting ARMA$(1,1)$ coefficients $\theta = (\phi, \psi)$ and spectral divergence:} \\
\multicolumn{8}{l}{~~$-$ $\theta_{20}=(0.693, 0.845)$, $\theta_{50}=(0.694,0.857)$, $\theta_{300}=(0.696,0.857)$. } \\
\multicolumn{8}{l}{~~$-$ $I_{20}(f; f_{\theta}) = 3.773$, $I_{50}(f; f_{\theta}) = 3.415$, $I_{300}(f; f_{\theta}) = 3.388$.} \\
\end{tabular}
\caption{\textit{Best fitting (bottom lines) and the bias of estimated
coefficients for six different methods for the ARMA$(3,2)$
misspecified case fitting ARMA$(1,1)$ model for the standardized chi-squared innovations. Standard deviations are in the parentheses. We
use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.}
}
\label{tab:arma11.chisq}
\end{table}
\begin{table}[ht]
\centering
\footnotesize
\begin{tabular}{cc|ccccccc}
$n$ & Parameter & Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased \\ \hline \hline
\multirow{3}{*}{20} & $\phi_1$ & {\color{blue}$0.017(0.13)$} & -$0.178(0.23)$ & -$0.047(0.17)$
&{\color{red} -$0.006(0.14)$} & -$0.134(0.15)$ & $0.044(0.14)$ \\
& $\phi_2$ & {\color{red} $0.002(0.09)$} & $0.176(0.2)$ & $0.057(0.16)$
& $0.023(0.12)$ & $0.135(0.13)$ & {\color{blue}-$0.019(0.13)$} \\
& $I_{n}(f;f_\theta)$ & {\color{red}$0.652(0.72)$} & $1.3073(1.46)$ & $0.788(0.85)$
& {\color{blue}$0.671(0.8)$} & $0.887(0.97)$ & $0.658(0.81)$\\ \hline
\multirow{3}{*}{50} & $\phi_1$ & {\color{blue}$0.018(0.09)$} & -$0.079(0.12)$ & -$0.010(0.09)$
& {\color{red}$0.002(0.09)$} & {\color{blue}-$0.018(0.09)$} & $0.140(0.15)$ \\
& $\phi_2$ & -$0.018(0.06)$ & $0.072(0.11)$ & $0.012(0.07)$
& {\color{red} $0.001(0.06)$} & {\color{blue}$0.016(0.06)$} & -$0.1(0.09)$ \\
& $I_{n}(f;f_\theta)$ & {\color{red}$0.287(0.36)$} & $0.406(0.52)$ & $0.302(0.39)$
& $0.298(0.39)$ & {\color{blue}$0.293(0.38)$} & $0.631(0.7)$ \\ \hline
\multirow{3}{*}{300} & $\phi_1$ & {\color{red} $0.002(0.04)$} & -$0.015(0.04)$ & {\color{blue}-$0.002(0.04)$}
& $0(0.04)$ & -$0.001(0.04)$ & $0.012(0.04)$ \\
& $\phi_2$ & -$0.005(0.02)$ & $0.011(0.03)$ & {\color{red}-$0.001(0.02)$}
& {\color{blue}-$0.001(0.02)$} & -$0.001(0.02)$ & -$0.016(0.04)$ \\
& $I_{n}(f;f_\theta)$ & {\color{red}$0.050(0.07)$} & $0.056(0.07)$ & {\color{blue} $0.051(0.07)$}
& $0.052(0.07)$ & $0.054(0.08)$ & $0.061(0.08)$ \\ \hline
\multicolumn{8}{l}{Best fitting AR$(1)$ coefficients $\theta = (\phi_1, \phi_2)$ and spectral divergence:} \\
\multicolumn{8}{l}{~~$-$ $\theta_{20}=(1.367, -0.841)$, $\theta_{50}=(1.364,-0.803)$, $\theta_{300}=(1.365,-0.802)$. } \\
\multicolumn{8}{l}{~~$-$ $I_{20}(f; f_{\theta}) = 2.902$, $I_{50}(f; f_{\theta}) = 2.937$, $I_{300}(f; f_{\theta}) = 2.916$.} \\
\end{tabular}
\caption{\textit{Same as in Table \ref{tab:ar2.chisq} but fitting an AR(2) model.
}}
\label{tab:ar2.chisq}
\end{table}
\subsection{Comparing the the new likelihoods constructed with the
predictive DFT with AR$(1)$ coefficients and AIC order selected AR$(p)$ coefficients} \label{sec:fixedP}
In this section we compare the performance of new likelihoods where the order of the AR
model used in the predictive DFT is determined using the AIC with a fixed choice
of order with the AR model (set to $p=1$). We use ARMA$(3,2)$ model considered in
Section \ref{sec:misspecifiedmodel} and fit the the ARMA$(1,1)$ and
AR$(2)$ to the data. We compare the new likelihoods with the Gaussian
likelihood and the Whittle likelihood. The results are given in Tables
\ref{tab:arma11P} and \ref{tab:arma20P}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{c|l|ccc}
\multicolumn{2}{c}{} & $\phi$ & $\psi$ & $I_{n}(f;f_\theta)$ \\ \hline \hline
\multicolumn{2}{c}{Best} & $0.694$ & $0.857$ & $3.415$ \\ \hline
\multirow{6}{*}{Bias}
& Gaussian & $0.012$\scriptsize{(0.07)} & $0.029$\scriptsize{(0.06)} & $0.354$\scriptsize{(0.34)} \\
& Whittle & -$0.054$\scriptsize{(0.09)} & -$0.116$\scriptsize{(0.12)} & $0.457$\scriptsize{(0.46)} \\ \cline{2-5}
& Boundary(AIC) & -$0.006$\scriptsize{(0.07)} & -$0.008$\scriptsize{(0.08)} & $0.292$\scriptsize{(0.3)} \\
& Boundary($p$=1) & -$0.020$\scriptsize{(0.08)} & -$0.045$\scriptsize{(0.09)} & $0.299$\scriptsize{(0.29)} \\ \cline{2-5}
& Hybrid(AIC) & $0.004$\scriptsize{(0.07)} & $0.009$\scriptsize{(0.07)} & $0.235$\scriptsize{(0.28)} \\
& Hybrid($p$=1) & $0.003$\scriptsize{(0.07)} & $0.010$\scriptsize{(0.07)} & $0.261$\scriptsize{(0.3)} \\
\end{tabular}
\caption{\textit{Best fitting (top row) and the bias of estimated
coefficients for six different methods for the Gaussian ARMA$(3,2)$
misspecified case fitting ARMA$(1,1)$ model. Length of the time series $n$=50. Standard deviations are in the parentheses.
(AIC): an order $p$ is chosen using AIC; ($p$=1): an order $p$ is set to 1.
} }
\label{tab:arma11P}
\end{table}
\begin{table}[ht]
\centering
\small
\begin{tabular}{c|l|ccc}
\multicolumn{2}{c}{} & $\phi_1$ & $\phi_2$ & $I_{n}(f;f_\theta)$ \\ \hline \hline
\multicolumn{2}{c}{Best} & $1.364$ & -$0.803$ & $2.937$ \\ \hline
\multirow{6}{*}{Bias}
& Gaussian & $0.019$\scriptsize{(0.09)} & -$0.024$\scriptsize{(0.06)} & $0.275$\scriptsize{(0.33)} \\
& Whittle & -$0.077$\scriptsize{(0.12)} & $0.066$\scriptsize{(0.1)} & $0.382$\scriptsize{(0.45)} \\ \cline{2-5}
& Boundary(AIC) & -$0.009$\scriptsize{(0.09)} & $0.006$\scriptsize{(0.07)} & $0.283$\scriptsize{(0.37)} \\
& Boundary($p$=1) & -$0.030$\scriptsize{(0.1)} & $0.032$\scriptsize{(0.07)} & $0.295$\scriptsize{(0.35)} \\ \cline{2-5}
& Hybrid(AIC) & $0.003$\scriptsize{(0.09)} & -$0.006$\scriptsize{(0.07)} & $0.283$\scriptsize{(0.37)} \\
& Hybrid($p$=1) & -$0.003$\scriptsize{(0.09)} & $0.003$\scriptsize{(0.06)} & $0.276$\scriptsize{(0.35)} \\
\end{tabular}
\caption{\textit{Same as in Table \ref{tab:arma11P}, but fitting an AR$(2)$.
} }
\label{tab:arma20P}
\end{table}
\section{Simulations: Estimation for long memory time series}\label{sec:sim-long}
\subsection{Parametric estimation for long memory Gaussian time series}\label{sec:longGaussian}
We conduct some simulations for time series whose spectral
density, $f$, does not satisfies Assumption \ref{assum:A}. We focus on
the ARFIMA$(0,d,0)$ model where
\begin{eqnarray*}
(1-B)^{d}W_t = \varepsilon_t,
\end{eqnarray*}
$B$ is the backshift operator, $-1/2<d<1/2$ is a fractional
differencing parameter, and $\{\varepsilon_t\}$ is an i.i.d. standard
normal random variable. Let $\Gamma(x)$ denote the gamma function.
The spectral density and autocovariance of the ARFIMA$(0,d,0)$ model (where the variance of
the innovations is set to $\sigma^2=1$) is
\begin{eqnarray}
\label{eq:arfima_formula}
f_W(\omega) = (1-e^{-i\omega})^{-2d} = (2\sin(\omega/2))^{-2d}
\quad \text{and} \quad
c_W(k) = \frac{\Gamma(k+d)\Gamma(1-2d)}{\Gamma(k-d+1) \Gamma(1-d) \Gamma(d)}
\end{eqnarray}
respectively (see \cite{b:gir-12}, Chapter 7.2). Observe that for
$-1/2<d<0$, the $f_{W}(0)=0$, this is called antipersistence. On the other
hand, if $0< d <1/2$, then $f_{W}(0)=\infty$ and $W_{t}$ has long memory.
We generate ARFIMA$(0,d,0)$ models with $d = -0.4,-0.2,0.2$ and $0.4$ and
Gaussian innovations. We fit both the ARFIMA$(0,d,0)$
model, with $d$ unknown (specified case) and the AR$(2)$ model (with
unknown parameters $\theta = (\phi_1,\phi_2)$)
(misspecified case) to the data. To do so,
we first demean the time series. We evaluate the (plug-in)
Gaussian likelihood using the autocovariance function in (\ref{eq:arfima_formula})
and the autocovariance function of AR$(2)$ model.
For the other 5 frequency domain likelihoods, we evaluate the likelihoods
at all the fundamental frequencies with the exception of the zero frequency
$\omega_{n,n}=0$. We fit using the spectral density in (\ref{eq:arfima_formula})
or the spectral density $f_{\theta}(\omega) = |1-\phi_{1}e^{-i\omega} - \phi_{2}e^{-2i \omega}|^{-2}$ where
$\theta=(\phi_{1}, \phi_{2})$ (depending on whether the model is
specified or misspecified).
For each simulation, we calculate six different parameter estimators. For the misspecified case,
we also calculate the spectral divergence
\begin{eqnarray*}
\widetilde{I}_n(f;f_\theta) = \frac{1}{n-1} \sum_{k=1}^{n-1}\left( \frac{f(\omega_{k,n})}{f_\theta(\omega_{k,n})} +
\log f_\theta(\omega_{k,n}) \right),
\end{eqnarray*}
where we omit the zero frequency.
The best fitting AR(2) model is $\theta^{Best} = \arg \min_{\theta
\in \Theta} \widetilde{I}_n(f;f_\theta)$. In Tables \ref{tab:FI} and
\ref{tab:misFI} we give a bias and standard deviation
for the parameter estimators for the correctly specified and
misspecified model (in the misspecified case we also give the spectral
divergence).
\vspace{1em}
\noindent \underline{Correctly specified model} From Table \ref{tab:FI} we observe that the bias of both new
likelihood estimators is consistently the smallest over all sample sizes and all $d$ except for $d=-0.2$.
The new likelihoods have the
smallest or second smallest RMSE for $n$=300, but not for the small sample sizes (e.g. $n$=20 and 50). This is probably due to increased variation in the new likelihood
estimators caused by the estimation of the AR parameter for the predictive
DFT. Since the time series has a long memory, the AIC is likely to choose a
large order autoregressive order $p$, which will increase the variance
in the estimator (recall that the second order error of the boundary corrected Whittle is $O(p^{3}n^{-3/2})$).
The (plug-in) Gaussian likelihood has a relatively large bias for all $d$, which
matches the observations in \cite{p:lib-05}, Table 1. However, it has
the smallest variance and this results in the smallest RMSE for almost all $n$ when $d$ is negative.
The debiased Whittle also has a larger bias
than most of the other estimators.
However, it has a smaller variance and thus, having the
smallest RMSE for almost all $n$ and positive $d$.
\begin{table}[ht]
\centering
\small
\begin{tabular}{c|cccc}
\multirow{2}{*}{\textit{Likelihoods}} & \multicolumn{4}{c}{$d$} \\
\cline{2-5}
& -0.4 & -0.2 & 0.2 & 0.4 \\ \hline \hline
& \multicolumn{4}{c}{$n=20$}\\ \hline
Gaussian & {\color{red}-$0.097$}{\scriptsize (0.23)} & {\color{red}-$0.148$}{\scriptsize (0.22)} & -$0.240${\scriptsize (0.23)} & -$0.289${\scriptsize (0.22)} \\
Whittle & $0.027${\scriptsize (0.3)} & $0.006${\scriptsize (0.28)} & {\color{blue}-$0.008$}{\scriptsize (0.29)} & {\color{blue}-$0.016$}{\scriptsize (0.29)} \\
\color{blue}{Boundary} & $0.014${\scriptsize (0.31)} & $0${\scriptsize (0.29)} & -$0.005${\scriptsize (0.30)} & -$0.007${\scriptsize (0.30)} \\
\color{blue}{Hybrid} & $0.009${\scriptsize (0.31)} & -$0.007${\scriptsize (0.3)} & $0.005${\scriptsize (0.30)} & -$0.001${\scriptsize (0.30)} \\
Tapered & $0.026${\scriptsize (0.3)} & $0${\scriptsize (0.3)} & $0.006${\scriptsize (0.30)} & $0.003${\scriptsize (0.29)} \\
Debiased & {\color{blue}$0.015$}{\scriptsize (0.29)} & {\color{blue}-$0.003$}{\scriptsize (0.27)} & {\color{red}-$0.029$}{\scriptsize (0.26)} & {\color{red}-$0.044$}{\scriptsize (0.27)} \\ \hline \hline
& \multicolumn{4}{c}{$n=50$}\\ \hline
Gaussian & {\color{red}-$0.042$}{\scriptsize (0.13)} & -$0.073${\scriptsize (0.14)} & -$0.097${\scriptsize (0.14)} & -$0.123${\scriptsize (0.12)} \\
Whittle & {\color{blue}$0.006$}{\scriptsize (0.15)} & {\color{red}-$0.016$}{\scriptsize (0.15)} & {\color{blue}-$0.013$}{\scriptsize (0.15)} & {\color{blue}-$0.005$}{\scriptsize (0.15)} \\
\color{blue}{Boundary} & -$0.005${\scriptsize (0.15)} & -$0.020${\scriptsize (0.16)} & -$0.011${\scriptsize (0.16)} & $0.001${\scriptsize (0.16)} \\
\color{blue}{Hybrid} & -$0.011${\scriptsize (0.15)} & -$0.021${\scriptsize (0.15)} & -$0.012${\scriptsize (0.16)} & $0${\scriptsize (0.16)} \\
Tapered & -$0.007${\scriptsize (0.15)} & -$0.019${\scriptsize (0.16)} & -$0.011${\scriptsize (0.16)} & $0.010${\scriptsize (0.16)} \\
Debiased& -$0.008${\scriptsize (0.16)} & {\color{blue}-$0.020$}{\scriptsize (0.16)} & {\color{red}-$0.019$}{\scriptsize (0.15)} & {\color{red}-$0.021$}{\scriptsize (0.14)} \\ \hline \hline
& \multicolumn{4}{c}{$n=300$}\\ \hline
Gaussian& {\color{red}-$0.006$}{\scriptsize (0.05)} & -$0.013${\scriptsize (0.05)} & -$0.020${\scriptsize (0.05)} & -$0.083${\scriptsize (0.02)} \\
Whittle& $0.006${\scriptsize (0.05)} & {\color{red}-$0.001$}{\scriptsize (0.05)} & {\color{blue}-$0.004$}{\scriptsize (0.05)} & {\color{blue}$0.002$}{\scriptsize (0.05)} \\
\color{blue}{Boundary}& {\color{blue}$0.002$}{\scriptsize (0.05)} & {\color{blue}-$0.003$}{\scriptsize (0.05)} & {\color{blue}-$0.003$}{\scriptsize (0.05)} & {\color{red}$0.002$}{\scriptsize (0.05)} \\
\color{blue}{Hybrid}& $0${\scriptsize (0.05)} & -$0.004${\scriptsize (0.05)} & -$0.004${\scriptsize (0.05)} & $0.001${\scriptsize (0.05)} \\
Tapered& $0${\scriptsize (0.05)} & -$0.004${\scriptsize (0.05)} & -$0.004${\scriptsize (0.05)} & $0.003${\scriptsize (0.05)} \\
Debiased& -$0.001${\scriptsize (0.05)} & -$0.003${\scriptsize (0.05)} & {\color{red}-$0.007$}{\scriptsize (0.05)} & -$0.060${\scriptsize (0.02)} \\ \hline \hline
\end{tabular}
\caption{\textit{Bias and the standard deviation (in the parentheses) of six different quasi-likelihoods for ARFIMA$(0,d,0)$ model for
the standard normal innovations. Length of the time series $n=20, 50$, and $300$. We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.
}}
\label{tab:FI}
\end{table}
\vspace{1em}
\noindent \underline{Misspecified model} We now compare the estimator when we fit the misspecified AR$(2)$ model to the data. From Table \ref{tab:misFI}
we observe that the Gaussian likelihood performs uniformly well for all $d$ and $n$, it usually has the smallest bias and RMSE for
the positive $d$. The tapered Whittle also performs uniformly well for all $d$ and $n$, especially for the negative $d$. In comparison, the new
likelihood estimators do not perform that well as compared with Gaussian and
Whittle likelihood. As mentioned above, this may be due to the
increased variation caused by estimating many AR parameters. However,
it is interesting to note that when $d=0.4$ and $n=300$, the estimated spectral divergence outperforms the Gaussian
likelihood. We leave the theoretical development of the sampling
properties of the new likelihoods and long memory time series for future research.
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{ccc|c|cccccc}
\multirow{2}{*}{$d$} & \multirow{2}{*}{$n$} & \multirow{2}{*}{Par.} & \multirow{2}{*}{Best} &
\multicolumn{6}{c}{Bias} \\ \cline{5-10}
&&&&
Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased \\ \hline \hline
\multirow{9}{*}{-0.4} &
\multirow{3}{*}{20} & $\phi_1$ & -$0.300$ & -$0.028(0.22)$ & {\color{blue}-$0.015(0.22)$} & -$0.022(0.23)$
&-$0.026(0.23)$ & {\color{red}-$0.010(0.21)$} & -$0.026(0.22)$ \\
& & $\phi_2$ & -$0.134$ & -$0.067(0.19)$ & {\color{blue}-$0.058(0.19)$} & -$0.064(0.2)$
& -$0.068(0.2)$ & {\color{red}-$0.062(0.18)$} & -$0.067(0.19)$ \\
& & $I_{n}(f;f_\theta)$ & $1.141$ & $0.103(0.1)$ & $0.099(0.1)$ & $0.110(0.12)$
& $0.108(0.11)$ & {\color{red}$0.095(0.1)$} & {\color{blue}$0.106(0.1)$} \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & -$0.319$ & {\color{blue}-$0.006(0.14)$} & {\color{red}$0.02(0.14)$} & -$0.004(0.15)$
& -$0.004(0.15)$ & $0.003(0.15)$ & -$0.006(0.15)$ \\
& & $\phi_2$ & -$0.152$ & -$0.028(0.13)$ & {\color{blue}-$0.022(0.13)$} & -$0.027(0.13)$
& -$0.029(0.13)$ & {\color{red}-$0.022(0.13)$} & -$0.029(0.13)$ \\
& & $I_{n}(f;f_\theta)$ & $1.092$ & {\color{blue}$0.043(0.05)$} & {\color{red}$0.043(0.05)$} & $0.045(0.05)$
& $0.046(0.05)$ & $0.043(0.05)$ & $0.045(0.05)$ \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & -$0.331$ & {\color{blue}-$0.003(0.06)$} & {\color{red}-$0.002(0.06)$} & -$0.003(0.06)$
& -$0.002(0.06)$ & -$0.001(0.06)$ & {\color{blue}-$0.003(0.06)$} \\
& & $\phi_2$ & -$0.164$ & {\color{blue}-$0.005(0.06)$} & {\color{red}-$0.004(0.05)$} & {\color{blue}-$0.005(0.06)$}
& -$0.005(0.06)$ & -$0.004(0.06)$ & -$0.005(0.06)$ \\
& & $I_{n}(f;f_\theta)$ & $1.062$ & {\color{red}$0.008(0.01)$} & {\color{red}$0.008(0.01)$} & $0.008(0.01)$
& $0.008(0.01)$ & $0.008(0.01)$ & $0.008(0.01)$ \\ \hline \hline
\multirow{9}{*}{-0.2} &
\multirow{3}{*}{20} & $\phi_1$ & -$0.157$ & -$0.027(0.23)$ & {\color{blue}-$0.024(0.23)$} & -$0.028(0.24)$
&-$0.029(0.23)$ & {\color{red}-$0.020(0.22)$} & -$0.028(0.23)$ \\
& & $\phi_2$ & -$0.066$ & -$0.084(0.21)$ & {\color{blue}-$0.085(0.2)$} & -$0.088(0.21)$
& -$0.089(0.21)$ & {\color{red}-$0.084(0.19)$} & -$0.089(0.21)$ \\
& & $I_{n}(f;f_\theta)$ & $1.066$ & $0.107(0.1)$ & $0.104(0.1)$ & $0.111(0.11)$
& $0.110(0.11)$ & {\color{red}$0.096(0.1)$} & {\color{blue}$0.108(0.1)$} \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & -$0.170$ & {\color{blue}-$0.020(0.15)$} & {\color{red}-$0.015(0.15)$} & -$0.018(0.15)$
& -$0.020(0.15)$ & -$0.016(0.15)$ & -$0.019(0.15)$ \\
& & $\phi_2$ & -$0.079$ & -$0.035(0.14)$ & {\color{blue}-$0.032(0.14)$} & -$0.034(0.14)$
& -$0.035(0.14)$ & {\color{red}-$0.031(0.13)$} & -$0.036(0.14)$ \\
& & $I_{n}(f;f_\theta)$ & $1.038$ & $0.043(0.04)$ & {\color{blue}$0.043(0.04)$} & $0.045(0.04)$
& $0.045(0.04)$ & {\color{red}$0.043(0.04)$} & $0.045(0.04)$ \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & -$0.001$ & {\color{blue} $0(0.06)$} & {\color{red}-$0.001(0.06)$} & $0(0.06)$
& $0.001(0.06)$ & $0.001(0.06)$ & -$0.006(0.06)$ \\
& & $\phi_2$ & -$0.088$ & {\color{blue}-$0.007(0.05)$} & {\color{red}-$0.007(0.05)$} & -$0.007(0.05)$
& -$0.007(0.06)$ & -$0.007(0.06)$ & -$0.007(0.05)$ \\
& & $I_{n}(f;f_\theta)$ & $1.019$ & {\color{red}$0.007(0.01)$} & {\color{red}$0.007(0.01)$} & $0.007(0.01)$
& $0.008(0.01)$ & $0.008(0.01)$ & $0.007(0.01)$ \\ \hline \hline
\multirow{9}{*}{0.2} &
\multirow{3}{*}{20} & $\phi_1$ & $0.167$ & -$0.066(0.24)$ & -$0.072(0.24)$ & -$0.071(0.25)$
& -$0.068(0.25)$ & {\color{red}-$0.080(0.23)$} & {\color{blue}-$0.070(0.24)$} \\
& & $\phi_2$ & $0.057$ & -$0.098(0.2)$ & {\color{blue}-$0.106(0.19)$} & -$0.107(0.19)$
& -$0.108(0.2)$ & {\color{red}-$0.111(0.18)$} & -$0.105(0.19)$ \\
& & $I_{n}(f;f_\theta)$ & $0.938$ & {\color{blue}$0.1(0.11)$} & $0.1(0.11)$ & $0.103(0.12)$
& $0.105(0.12)$ & $0.098(0.11)$ & {\color{red}$0.1(0.1)$} \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & $0.186$ & {\color{blue}-$0.025(0.15)$} & {\color{red}-$0.027(0.15)$} & -$0.026(0.15)$
& -$0.027(0.15)$ & -$0.034(0.15)$ & -$0.025(0.15)$ \\
& & $\phi_2$ & $0.075$ & -$0.040(0.15)$ & {\color{red}-$0.043(0.15)$} & {\color{blue}-$0.043(0.15)$}
& -$0.042(0.15)$ & -$0.047(0.15)$ & -$0.042(0.15)$ \\
& & $I_{n}(f;f_\theta)$ & $0.971$ & $0.046(0.05)$ & {\color{red}$0.044(0.05)$} & $0.046(0.05)$
& $0.048(0.05)$ & $0.047(0.05)$ & {\color{blue}$0.046(0.05)$} \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & $0.208$ & {\color{red}-$0.007(0.06)$} & {\color{blue}-$0.007(0.06)$} & -$0.007(0.06)$ & -$0.007(0.06)$ & -$0.008(0.06)$ & -$0.007(0.06)$ \\
& & $\phi_2$ & $0.097$ & -$0.006(0.06)$ & {\color{red}-$0.007(0.06)$} & {\color{blue}-$0.006(0.06)$} & -$0.007(0.07)$ & -$0.008(0.07)$ & -$0.006(0.06)$ \\
& & $I_{n}(f;f_\theta)$ & $1.002$ & {\color{red}$0.008(0.01)$} & {\color{blue}$0.008(0.01)$} & {\color{blue}$0.008(0.01)$} & $0.008(0.01)$ & $0.009(0.01)$ & {\color{blue}$0.008(0.01)$ } \\ \hline \hline
\multirow{9}{*}{0.4} &
\multirow{3}{*}{20} & $\phi_1$ & $0.341$ & -$0.072(0.25)$ & {\color{red}-$0.082(0.25)$} & -$0.075(0.26)$ & -$0.074(0.26)$ & -$0.103(0.24)$ & {\color{blue}-$0.077(0.25)$} \\
& & $\phi_2$ & $0.094$ & {\color{blue}-$0.116(0.21)$} & -$0.134(0.2)$ & -$0.135(0.2)$ & {\color{blue}-$0.133(0.2)$} & -$0.133(0.19)$ & -$0.130(0.2)$ \\
& & $I_{n}(f;f_\theta)$ & $0.877$ & $0.111(0.12)$ & {\color{blue}$0.113(0.12)$} & $0.116(0.12)$
& $0.116(0.13)$ & $0.114(0.12)$ & {\color{red}$0.113(0.11)$} \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & $0.378$ & {\color{red}-$0.024(0.15)$} & {\color{blue} -$0.030(0.15)$} & -$0.027(0.15)$ & -$0.029(0.15)$ & -$0.039(0.15)$ & -$0.027(0.15)$ \\
& & $\phi_2$ & $0.129$ & -$0.058(0.15)$ & {\color{red}-$0.069(0.14)$} & {\color{blue}-$0.067(0.15)$} & -$0.064(0.15)$ & -$0.072(0.15)$ & -$0.066(0.15)$ \\
& & $I_{n}(f;f_\theta)$ & $0.944$ & {\color{red}$0.051(0.06)$} & $0.053(0.06)$ & $0.053(0.06)$ & {\color{blue}$0.054(0.06)$} & $0.055(0.06)$ & $0.054(0.06)$ \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & $0.428$ & {\color{red}-$0.004(0.06)$} & -$0.004(0.06)$ & {\color{blue}-$0.004(0.06)$} & -$0.007(0.06)$ & -$0.009(0.06)$ & -$0.004(0.06)$ \\
& & $\phi_2$ & $0.178$ & -$0.003(0.06)$ & {\color{red}-$0.005(0.06)$} & -$0.004(0.06)$ & -$0.004(0.06)$ & -$0.006(0.06)$ & {\color{blue}-$0.004(0.06)$} \\
& & $I_{n}(f;f_\theta)$ & $1.010$ & {\color{red}$0.009(0.01)$} & $0.009(0.01)$ & {\color{red}$0.009(0.01)$} & $0.010(0.01)$ & $0.010(0.01)$ & $0.009(0.01)$ \\
\end{tabular}
\caption{\textit{ Best fitting and the bias of estimated coefficients using the six different methods
for misspecified Gaussian ARFIMA$(0,d,0)$ case fitting AR$(2)$ model. Standard deviations are in the parentheses.
We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.
}}
\label{tab:misFI}
\end{table}
\subsection{Parametric estimation for long memory non-Gaussian time series}
We fit the parametric models described in Appendix \ref{sec:longGaussian}. However, the underlying time series is non-Gaussian and
generated from the ARFIMA$(0,d,0)$
\begin{eqnarray*}
(1-B)^{d}W_t = \varepsilon_t,
\end{eqnarray*}
where $\{\varepsilon_t\}$ are i.i.d. standardized chi-square random variables with two-degrees of freedom i.e.
$\varepsilon_t \sim (\chi^2(2)-2)/2$. The results in the specified setting are given in Table
\ref{tab:FI.chisq} and from the non-specified setting in Table \ref{tab:misFI.chisq}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{c|cccc}
\multirow{2}{*}{\textit{Likelihoods}} & \multicolumn{4}{c}{$d$} \\
\cline{2-5}
& -0.4 & -0.2 & 0.2 & 0.4 \\ \hline \hline
& \multicolumn{4}{c}{$n=20$}\\ \hline
Gaussian& {\color{red}-$0.077$}{\scriptsize (0.22)} & {\color{red}-$0.130$}{\scriptsize (0.23)} & {\color{blue}-$0.219$}{\scriptsize (0.21)} & -$0.277${\scriptsize (0.20)} \\
Whittle& $0.089${\scriptsize (0.34)} & $0.092${\scriptsize (0.36)} & $0.077${\scriptsize (0.32)} & {\color{blue}$0.058$}{\scriptsize (0.31)} \\
\color{blue}{Boundary}& $0.077${\scriptsize (0.34)} & $0.086${\scriptsize (0.37)} & $0.079${\scriptsize (0.33)} & $0.070${\scriptsize (0.32)} \\
\color{blue}{Hybrid}& $0.077${\scriptsize (0.35)} & $0.084${\scriptsize (0.37)} & $0.087${\scriptsize (0.33)} & $0.086${\scriptsize (0.32)} \\
Tapered& $0.092${\scriptsize (0.35)} & $0.096${\scriptsize (0.37)} & $0.089${\scriptsize (0.33)} & $0.078${\scriptsize (0.31)} \\
Debiased & {\color{blue}$0.057$}{\scriptsize (0.33)} & {\color{blue}$0.051$}{\scriptsize (0.31)} & {\color{red}$0.009$}{\scriptsize (0.25)} & {\color{red}-$0.021$}{\scriptsize (0.26)} \\ \hline \hline
& \multicolumn{4}{c}{$n=50$}\\ \hline
Gaussian & {\color{red}-$0.047$}{\scriptsize (0.13)} & {\color{red}-$0.065$}{\scriptsize (0.14)} & -$0.097${\scriptsize (0.13)} & -$0.130${\scriptsize (0.12)} \\
Whittle& {\color{blue}$0.008$}{\scriptsize (0.15)} & $0.004${\scriptsize (0.16)} & {\color{blue}-$0.001$}{\scriptsize (0.15)} & {\color{blue}$0.001$}{\scriptsize (0.16)} \\
\color{blue}{Boundary} & -$0.002${\scriptsize (0.16)} & $0${\scriptsize (0.16)} & $0.002${\scriptsize (0.15)} & $0.006${\scriptsize (0.16)} \\
\color{blue}{Hybrid} & -$0.005${\scriptsize (0.15)} & -$0.001${\scriptsize (0.16)} & $0.004${\scriptsize (0.16)} & $0.006${\scriptsize (0.16)} \\
Tapered & $0.001${\scriptsize (0.15)} & $0.002${\scriptsize (0.16)} & $0.006${\scriptsize (0.16)} & $0.016${\scriptsize (0.17)} \\
Debiased & -$0.004${\scriptsize (0.16)} & {\color{blue}-$0.001$}{\scriptsize (0.16)} & {\color{red}-$0.013$}{\scriptsize (0.14)} & {\color{red}-$0.025$}{\scriptsize (0.14)} \\ \hline \hline
& \multicolumn{4}{c}{$n=300$}\\ \hline
Gaussian & {\color{red}-$0.011$}{\scriptsize (0.05)} & -$0.012${\scriptsize (0.05)} & -$0.017${\scriptsize (0.05)} & -$0.085${\scriptsize (0.02)} \\
Whittle & $0.002${\scriptsize (0.05)} & {\color{red}$0$}{\scriptsize (0.05)} & {\color{blue}-$0.001$}{\scriptsize (0.05)} & {\color{blue}-$0.001$}{\scriptsize (0.05)} \\
\color{blue}{Boundary} & {\color{blue}-$0.003$}{\scriptsize (0.05)} & {\color{blue}-$0.001$}{\scriptsize (0.05)} & {\color{blue}$0$}{\scriptsize (0.05)} & {\color{red}$0.001$}{\scriptsize (0.05)} \\
\color{blue}{Hybrid} & -$0.006${\scriptsize (0.05)} & -$0.003${\scriptsize (0.05)} & -$0.001${\scriptsize (0.05)} & -$0.003${\scriptsize (0.05)} \\
Tapered & -$0.006${\scriptsize (0.05)} & -$0.003${\scriptsize (0.05)} & -$0.001${\scriptsize (0.05)} & -$0.001${\scriptsize (0.05)} \\
Debiased & -$0.006${\scriptsize (0.05)} & -$0.002${\scriptsize (0.05)} & {\color{red}-$0.004$}{\scriptsize (0.05)} & -$0.061${\scriptsize (0.03)} \\ \hline \hline
\end{tabular}
\caption{\textit{Same as in Table \ref{tab:FI} but for the chi-squared innovations.
}}
\label{tab:FI.chisq}
\end{table}
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{ccc|c|cccccc}
\multirow{2}{*}{$d$} & \multirow{2}{*}{$n$} & \multirow{2}{*}{Par.} & \multirow{2}{*}{Best} &
\multicolumn{6}{c}{Bias} \\ \cline{5-10}
&&&&
Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased \\ \hline \hline
\multirow{9}{*}{-0.4} &
\multirow{3}{*}{20} & $\phi_1$ & -$0.300$ & {\color{blue}-$0.017(0.21)$} & -$0.004(0.21)$ & -$0.012(0.22)$
& -$0.018(0.22)$ & {\color{red}$0.001(0.21)$} & -$0.009(0.22)$ \\
& & $\phi_2$ & -$0.134$ & -$0.073(0.18)$ & {\color{blue}-$0.071(0.18)$} & -$0.076(0.18)$
& -$0.077(0.18)$ & {\color{red}-$0.064(0.16)$} & -$0.074(0.18)$ \\
& & $I_{n}(f;f_\theta)$ & $1.141$ & $0.096(0.11)$ & {\color{red}$0.097(0.11)$} & $0.105(0.11)$
& $0.102(0.11)$ & {\color{blue}$0.088(0.11)$} & $0.100(0.11)$ \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & -$0.319$ & {\color{blue}-$0.008(0.14)$} & {\color{red}$0(0.14)$} & -$0.005(0.14)$
& -$0.007(0.15)$ &$0.001(0.14)$ & -$0.006(0.14)$ \\
& & $\phi_2$ & -$0.152$ & -$0.035(0.12)$ & {\color{blue}-$0.030(0.12)$} & -$0.035(0.12)$
& -$0.035(0.12)$ & {\color{red}-$0.026(0.12)$} & -$0.035(0.12)$ \\
& & $I_{n}(f;f_\theta)$ & $1.092$ & $0.042(0.05)$ & {\color{red}$0.040(0.04)$} & $0.043(0.05)$
& $0.043(0.05)$ & {\color{blue}$0.041(0.04)$} & {\color{blue}$0.043(0.05)$} \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & -$0.331$ & {\color{red}-$0.004(0.06)$} & {\color{blue}-$0.002(0.06)$} & -$0.004(0.06)$ & -$0.004(0.06)$ & -$0.003(0.06)$ & -$0.004(0.06)$ \\
& & $\phi_2$ & -$0.164$ & {\color{red}-$0.007(0.05)$} & {\color{red}-$0.006(0.05)$} & -$0.007(0.05)$ & -$0.008(0.05)$ & -$0.007(0.05)$ & -$0.007(0.05)$ \\
& & $I_{n}(f;f_\theta)$ & $1.062$ & {\color{red}$0.007(0.01)$} & {\color{blue}$0.007(0.01)$} & {\color{blue}$0.007(0.01)$} & {\color{blue}$0.008(0.01)$} & $0.008(0.01)$ & $0.007(0.01)$ \\ \hline \hline
\multirow{9}{*}{-0.2} &
\multirow{3}{*}{20} & $\phi_1$ & -$0.157$ & -$0.036(0.22)$ & -$0.034(0.22)$ & -$0.037(0.23)$
& -$0.039(0.23)$ & {\color{red}-$0.028(0.21)$} & {\color{blue}-$0.034(0.22)$} \\
& & $\phi_2$ & -$0.066$ & -$0.083(0.19)$ & {\color{blue}-$0.080(0.18)$} & -$0.082(0.19)$
& -$0.083(0.19)$ & {\color{red}-$0.076(0.18)$} & -$0.082(0.19)$ \\
& & $I_{n}(f;f_\theta)$ & $1.066$ & $0.099(0.11)$ & {\color{blue}$0.095(0.11)$} & $0.100(0.11)$
& $0.101(0.11)$ & {\color{red}$0.086(0.1)$} & $0.097(0.11)$ \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & -$0.170$ & {\color{blue}-$0.018(0.15)$} & {\color{red}-$0.016(0.14)$} & -$0.019(0.15)$
& -$0.017(0.15)$ & -$0.012(0.15)$ & -$0.019(0.15)$ \\
& & $\phi_2$ & -$0.079$ & -$0.034(0.13)$ & {\color{blue}-$0.033(0.13)$} & -$0.034(0.13)$
& -$0.035(0.13)$ & {\color{red}-$0.031(0.13)$} & -$0.034(0.13)$ \\
& & $I_{n}(f;f_\theta)$ & $1.038$ & $0.042(0.05)$ & {\color{blue}$0.041(0.05)$} & $0.043(0.05)$
& $0.043(0.05)$ & {\color{red}$0.041(0.04)$} & $0.043(0.05)$ \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & -$0.179$ & {\color{red}-$0.001(0.06)$} & {\color{red}$0(0.06)$} & -$0.001(0.06)$ & $0(0.06)$ & $0(0.06)$ & -$0.001(0.06)$ \\
& & $\phi_2$ & -$0.088$ & {\color{red}-$0.008(0.05)$} & {\color{red}-$0.007(0.05)$} & -$0.008(0.06)$ & -$0.008(0.06)$ & -$0.007(0.06)$ & -$0.008(0.06)$ \\
& & $I_{n}(f;f_\theta)$ & $1.019$ & {\color{red}$0.007(0.01)$} & {\color{red}$0.007(0.01)$} & {\color{red}$0.007(0.01)$} & $0.007(0.01)$ & $0.007(0.01)$ & {\color{red}$0.007(0.01)$} \\ \hline \hline
\multirow{9}{*}{0.2} &
\multirow{3}{*}{20} & $\phi_1$ & $0.167$ & -$0.053(0.23)$ & -$0.064(0.22)$ & -$0.062(0.23)$
& -$0.055(0.23)$ & {\color{blue}-$0.071(0.21)$} & {\color{red}-$0.066(0.21)$} \\
& & $\phi_2$ & $0.057$ & -$0.091(0.2)$ & -$0.100(0.2)$ & -$0.100(0.2)$
& -$0.101(0.2)$ & {\color{red}-$0.099(0.19)$} & {\color{blue}-$0.095(0.19)$} \\
& & $I_{n}(f;f_\theta)$ & $0.938$ & $0.094(0.1)$ & $0.091(0.09)$ & $0.094(0.1)$
& $0.097(0.1)$ & {\color{blue}$0.086(0.09)$} & {\color{red}$0.086(0.08)$} \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & $0.186$ & {\color{red}-$0.023(0.15)$} & {\color{blue}-$0.026(0.15)$} & -$0.025(0.15)$
& -$0.024(0.15)$ & -$0.030(0.15)$ & -$0.024(0.15)$ \\
& & $\phi_2$ & $0.075$ & -$0.044(0.14)$ & {\color{blue}-$0.051(0.14)$} & -$0.051(0.14)$
& -$0.047(0.14)$ & {\color{red}-$0.051(0.13)$} & -$0.050(0.14)$ \\
& & $I_{n}(f;f_\theta)$ & $0.971$ & {\color{red}$0.043(0.04)$} & {\color{red}$0.042(0.04)$} & $0.043(0.04)$
& $0.044(0.04)$ & {\color{blue}$0.042(0.04)$} & $0.043(0.04)$ \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & $0.208$ & {\color{red}-$0.003(0.06)$} & {\color{blue}-$0.004(0.06)$} & -$0.003(0.06)$ & -$0.003(0.06)$ & -$0.005(0.06)$ & -$0.003(0.06)$ \\
& & $\phi_2$ & $0.097$ & {\color{blue}-$0.008(0.06)$} & {\color{red}-$0.009(0.06)$} & -$0.008(0.06)$ & -$0.010(0.07)$ & -$0.011(0.06)$ & -$0.008(0.06)$ \\
& & $I_{n}(f;f_\theta)$ & $1.002$ & {\color{red}$0.007(0.01)$} & {\color{red}$0.007(0.01)$} & {\color{red}$0.007(0.01)$} & $0.007(0.01)$ & $0.008(0.01)$ & {\color{red}$0.007(0.01)$ } \\ \hline \hline
\multirow{9}{*}{0.4} &
\multirow{3}{*}{20} & $\phi_1$ & $0.341$ & -$0.073(0.24)$ & {\color{blue}-$0.080(0.23)$} & -$0.074(0.24)$ & -$0.070(0.25)$ & -$0.103(0.23)$ & {\color{red}-$0.085(0.23)$} \\
& & $\phi_2$ & $0.094$ & -$0.106(0.21)$ & -$0.129(0.19)$ & -$0.128(0.2)$ & -$0.126(0.2)$ & {\color{red}-$0.122(0.19)$} & {\color{blue}-$0.123(0.19)$} \\
& & $I_{n}(f;f_\theta)$ & $0.877$ & {\color{red}$0.103(0.11)$} & {\color{blue}$0.103(0.11)$} & $0.107(0.12)$
& $0.108(0.12)$ & $0.103(0.11)$ & $0.101(0.11)$ \\ \cline{2-10}
& \multirow{3}{*}{50} & $\phi_1$ & $0.378$ & {\color{red}-$0.021(0.14)$} & {\color{red} -$0.027(0.14)$} & -$0.023(0.15)$ & -$0.026(0.15)$ & -$0.037(0.15)$ & -$0.026(0.14)$ \\
& & $\phi_2$ & $0.129$ & {\color{red}-$0.044(0.14)$} & {\color{blue}-$0.054(0.14)$} & -$0.051(0.14)$ & -$0.049(0.14)$ & -$0.059(0.14)$ & -$0.051(0.14)$ \\
& & $I_{n}(f;f_\theta)$ & $0.944$ & $0.044(0.05)$ & {\color{blue}$0.045(0.05)$} & $0.046(0.05)$ & $0.046(0.05)$ & $0.047(0.05)$ & {\color{red}$0.045(0.05)$} \\ \cline{2-10}
& \multirow{3}{*}{300} & $\phi_1$ & $0.428$ & {\color{blue}-$0.003(0.06)$} & {\color{blue}-$0.003(0.06)$} & {\color{red}-$0.003(0.06)$} & -$0.004(0.06)$ & -$0.006(0.06)$ & -$0.002(0.06)$ \\
& & $\phi_2$ & $0.178$ & {\color{blue}-$0.004(0.06)$} & {\color{red}-$0.006(0.06)$} & {\color{blue}-$0.004(0.06)$} & -$0.006(0.06)$ & -$0.008(0.06)$ & {\color{blue}-$0.005(0.06)$} \\
& & $I_{n}(f;f_\theta)$ & $1.010$ & $0.010(0.01)$ & {\color{red}$0.009(0.01)$} & {\color{red}$0.009(0.01)$} & $0.010(0.01)$ & $0.010(0.01)$ & {\color{red}$0.009(0.01)$} \\
\end{tabular}
\caption{\textit{Best fitting and the bias of estimated coefficients using the six different methods
for misspecified ARFIMA$(0,d,0)$ case fitting AR$(2)$ model for the chi-squared innovations.
Standard deviations are in the parentheses.
We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.
}}
\label{tab:misFI.chisq}
\end{table}
\subsection{Semi-parametric estimation for Gaussian time series}\label{sec:LWGaussian}
Suppose the time series $\{X_t\}$ has a spectral density $f(\cdot)$ with $\lim_{\omega \rightarrow 0+} f(\omega) \sim C\omega^{-2d}$ for some
$d \in (-1/2, 1/2)$. The local Whittle (LW) estimator is an estimation method for estimating $d$ without using assuming any parametric structure on $d$. It was first proposed in
\cite{p:kun-87,p:rob-95,p:che-03}, see also \cite{b:gir-12}, Chapter 8). The LW estimator is defined as
$\widehat{d} = \arg\min R(d)$ where
\begin{eqnarray} \label{eq:LW}
R(d) = \log \left( M^{-1} \sum_{k=1}^{M} \frac{|J_n(\omega_{k,n})|^2}{\omega_{k,n}^{-2d}} \right)
- \frac{2d}{M} \sum_{k=1}^{M} \log \omega_{k,n},
\end{eqnarray}
and $M=M(n)$ is an integer such that $M^{-1} + M/n \rightarrow 0$ as $n \rightarrow \infty$.
The objective function can be viewed as ``locally'' fitting a spectral density of form $f_{\theta}(\omega) = C\omega^{-2d}$ where $\theta = (C,d)$ using
the Whittle likelihood.
Since $\widetilde{J}_n (\omega_{k,n} ; f) \overline{J_n(\omega_{k,n})}$ is an unbiased estimator of
true spectral density $f(\omega_{k,n})$, it is possible that replacing the periodogram with
the (feasible) complete periodogram my lead to a better estimator of $d$. Based on this
we define the (feasible) hybrid LW criterion,
\begin{eqnarray*}
Q(d) = \log \left( M^{-1} \sum_{k=1}^{M} \frac{\widetilde{J}_n (\omega_{k,n} ; \widehat{f}_p) \overline{J_{n,\underline{h}_n}(\omega_{k,n})}}{\omega_{k,n}^{-2d}} \right) - \frac{2d}{M} \sum_{k=1}^{M} \log \omega_{k,n}.
\end{eqnarray*}
In a special case that the data taper $h_{t,n} \equiv 1$, we call it the boundary corrected LW criterion.
To empirically assess the validity of the above estimation scheme,
we generate a Gaussian ARFIMA$(0,d,0)$ model from Section \ref{sec:longGaussian} for $d=-0.4, -0.2, 0.2$ and $0.4$ and evaulate the LW, tapered LW (using tapered DFT in (\ref{eq:LW})), boundary corrected LW, and hybrid LW. We set $M \approx n^{0.65}$ where $n$ is a length of the time series and we use Tukey taper with 10\% of the taper on each end of the time series. For each simulation, we obtain four different LW estimators.
Table \ref{tab:LW} summarizes the bias and standard deviation (in the parentheses) of LW estimators. We observe that the bondary corrected LW has a smaller bias than the regular Local Whittle likelihood except when $d=-0.2$ and $n=50, 300$. However, the standard error tends to be larger (this is probably because of the additional error caused by estimating the AR$(p)$ parameters in the new likelihoods). Despite the larger standard error, in terms of RMSE, the
boundary corrected LW (or hybrid) tends to have overall at least the second smallest RSME for most $d$ and $n$.
\begin{table}[h]
\centering
\small
\begin{tabular}{c|cccc}
\multirow{2}{*}{\textit{Local likelihoods}} & \multicolumn{4}{c}{$d$} \\
\cline{2-5}
& -0.4 & -0.2 & 0.2 & 0.4 \\ \hline \hline
& \multicolumn{4}{c}{$n=20$}\\ \hline
Whittle& {\color{red}$0.283$}{\scriptsize (0.46)} & {\color{red}$0.111$}{\scriptsize (0.5)} & {\color{red}-$0.075$}{\scriptsize (0.48)} & -$0.226${\scriptsize (0.43)} \\
\color{blue}{Boundary}& $0.282${\scriptsize (0.46)} & {\color{blue}$0.109$}{\scriptsize (0.51)} & {\color{blue}-$0.075$}{\scriptsize (0.48)} & -$0.220${\scriptsize (0.44)} \\
\color{blue}{Hybrid}& $0.275${\scriptsize (0.46)} & $0.115${\scriptsize (0.52)} & -$0.067${\scriptsize (0.49)} & {\color{blue}-$0.209$}{\scriptsize (0.44)} \\
Tapered& {\color{blue}$0.279$}{\scriptsize (0.46)} & $0.121${\scriptsize (0.52)} & -$0.068${\scriptsize (0.49)} & {\color{red}-$0.204$}{\scriptsize (0.43)} \\ \hline \hline
& \multicolumn{4}{c}{$n=50$}\\ \hline
Whittle& $0.060${\scriptsize (0.26)} & -$0.056${\scriptsize (0.33)} & {\color{red}-$0.089$}{\scriptsize (0.37)} & -$0.109${\scriptsize (0.32)} \\
\color{blue}{Boundary} & $0.045${\scriptsize (0.26)} & {\color{blue}-$0.063$}{\scriptsize (0.34)} & {\color{blue}-$0.088$}{\scriptsize (0.38)} & {\color{blue}-$0.106$}{\scriptsize (0.32)} \\
\color{blue}{Hybrid} & {\color{blue}$0.033$}{\scriptsize (0.25)} & -$0.069${\scriptsize (0.34)} & -$0.090${\scriptsize (0.38)} & -$0.110${\scriptsize (0.32)} \\
Tapered & {\color{red}$0.035$}{\scriptsize (0.25)} & {\color{red}-$0.068$}{\scriptsize (0.34)} & -$0.085${\scriptsize (0.38)} & {\color{red}-$0.085$}{\scriptsize (0.31)} \\ \hline \hline
& \multicolumn{4}{c}{$n=300$}\\ \hline
Whittle & {\color{red}$0.056$}{\scriptsize (0.12)} & {\color{red}-$0.014$}{\scriptsize (0.11)} & {\color{red}-$0.010$}{\scriptsize (0.12)} & {\color{blue}$0.004$}{\scriptsize (0.11)} \\
\color{blue}{Boundary} & {\color{blue}$0.052$}{\scriptsize (0.12)} & {\color{blue}-$0.017$}{\scriptsize (0.11)} & {\color{blue}-$0.009$}{\scriptsize (0.12)} & {\color{red}$0.003$}{\scriptsize (0.11)} \\
\color{blue}{Hybrid} & $0.054${\scriptsize (0.12)} & -$0.018${\scriptsize (0.12)} & -$0.011${\scriptsize (0.12)} & -$0.002${\scriptsize (0.11)} \\
Tapered & $0.057${\scriptsize (0.12)} & -$0.018${\scriptsize (0.12)} & -$0.011${\scriptsize (0.12)} & $0.003${\scriptsize (0.12)} \\ \hline \hline
\end{tabular}
\caption{\textit{Bias and the standard deviation (in the parentheses) of four different Local Whittle estimators for ARFIMA$(0,d,0)$ model for
the standard normal innovations. Length of the time series $n=20, 50$, and $300$. We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.
}}
\label{tab:LW}
\end{table}
\subsection{Semi-parametric estimation for long memory non-Gaussian time series}
Once again we consider the semi-parametric Local Whittle estimator described in Appendix \ref{sec:LWGaussian}. However, this time we assess the estimation scheme for non-Gaussian time series. We generate the ARFIMA$(0,d,0)$
\begin{eqnarray*}
(1-B)^{d}W_t = \varepsilon_t,
\end{eqnarray*}
where $\{\varepsilon_t\}$ are i.i.d. standardarized chi-square random variables with
two-degrees of freedom i.e. $\varepsilon_t \sim (\chi^2(2)-2)/2$.
The results are summarized in Table \ref{tab:LW.chisq}.
\begin{table}[ht]
\centering
\small
\begin{tabular}{c|cccc}
\multirow{2}{*}{\textit{Local likelihoods}} & \multicolumn{4}{c}{$d$} \\
\cline{2-5}
& -0.4 & -0.2 & 0.2 & 0.4 \\ \hline \hline
& \multicolumn{4}{c}{$n=20$}\\ \hline
Whittle& $0.354${\scriptsize (0.52)} & {\color{blue}$0.235$}{\scriptsize (0.54)} & {\color{red}$0.002$}{\scriptsize (0.51)} & -$0.154${\scriptsize (0.41)} \\
\color{blue}{Boundary}& {\color{blue}$0.349$}{\scriptsize (0.52)} & {\color{red}$0.231$}{\scriptsize (0.54)} & $0.003${\scriptsize (0.51)} & -$0.146${\scriptsize (0.41)} \\
\color{blue}{Hybrid}& $0.347${\scriptsize (0.53)} & $0.229${\scriptsize (0.55)} & $0.016${\scriptsize (0.52)} & {\color{red}-$0.133$}{\scriptsize (0.4)} \\
Tapered& {\color{red}$0.351$}{\scriptsize (0.52)} & $0.229${\scriptsize (0.54)} & {\color{blue}$0.023$}{\scriptsize (0.51)} & {\color{blue}-$0.144$}{\scriptsize (0.41)} \\ \hline \hline
& \multicolumn{4}{c}{$n=50$}\\ \hline
Whittle& $0.019${\scriptsize (0.25)} & {\color{red}-$0.080$}{\scriptsize (0.34)} & {\color{red}-$0.125$}{\scriptsize (0.38)} & -$0.149${\scriptsize (0.33)} \\
\color{blue}{Boundary} & $0.007${\scriptsize (0.25)} & {\color{blue}-$0.086$}{\scriptsize (0.35)} & {\color{blue}-$0.123$}{\scriptsize (0.39)} & {\color{blue}-$0.146$}{\scriptsize (0.33)} \\
\color{blue}{Hybrid} & {\color{red}-$0.003$}{\scriptsize (0.24)} & -$0.098${\scriptsize (0.35)} & -$0.128${\scriptsize (0.39)} & -$0.153${\scriptsize (0.34)} \\
Tapered & {\color{blue}$0.001$}{\scriptsize (0.24)} & -$0.100${\scriptsize (0.35)} & -$0.125${\scriptsize (0.39)} & {\color{red}-$0.133$}{\scriptsize (0.33)} \\ \hline \hline
& \multicolumn{4}{c}{$n=300$}\\ \hline
Whittle & {\color{red}$0.104$}{\scriptsize (0.14)} & {\color{red}-$0.006$}{\scriptsize (0.14)} & {\color{red}-$0.015$}{\scriptsize (0.14)} & {\color{blue}-$0.011$}{\scriptsize (0.13)} \\
\color{blue}{Boundary} & $0.101${\scriptsize (0.15)} & {\color{blue}-$0.008$}{\scriptsize (0.14)} & {\color{blue}-$0.014$}{\scriptsize (0.14)} & {\color{red}-$0.012$}{\scriptsize (0.13)} \\
\color{blue}{Hybrid} & {\color{blue}$0.100$}{\scriptsize (0.15)} & -$0.011${\scriptsize (0.15)} & -$0.017${\scriptsize (0.14)} & -$0.020${\scriptsize (0.13)} \\
Tapered & $0.104${\scriptsize (0.15)} & -$0.012${\scriptsize (0.15)} & -$0.017${\scriptsize (0.15)} & -$0.016${\scriptsize (0.13)} \\ \hline \hline
\end{tabular}
\caption{\textit{Same as in Table \ref{tab:LW} but for the chi-square innovations.
}}
\label{tab:LW.chisq}
\end{table}
\section{Simulations: Alternative methods for estimating the
predictive DFT} \label{sec:alternative}
As pointed out by the referees, using the Yule-Walker estimator to
estimate the prediction coefficients in the predictive DFT may in
certain situations be problematic. We discuss the issues and potential
solutions below.
The first issue is that Yule-Walker
estimator suffers a finite sample bias, especially when the spectral density has a root close to the
unit circle (see, e.g., \cite{p:tjo-83}). One remedy to reduce the
bias is via data tapering (\cite{p:dah-88} and \cite{p:zha-92}).
Therefore, we define the \textbf{b}oundary \textbf{c}orrected Whittle likelihood using \textbf{t}apered \textbf{Y}ule-\textbf{W}alker (BC-tYW) replace
$\widehat{f}_p$ with $\widetilde{f}_p$ in (\ref{eq:Winf})
where
$\widetilde{f}_p$ is a spectral density of AR$(p)$ process where the AR coefficients are estimated using
Yule-Walker with tapered time series. In the simulations we use the Tukey taper with
$d=n/10$ and select the order $p$ using the AIC.
The second issue is if the underlying time series is complicated in
the sense that the underlying AR representation has multiple roots. Then fitting a large order
AR$(p)$ model may result in a loss of efficiency. As an alternative,
we consider a fully nonparametric estimator of $\widehat{J}_n(\omega;f)$ based on the estimated spectral density function.
To do so,
we recall from Section \ref{sec:firstorder} the first order approximation of $\widehat{J}_n(\omega;f)$ is $\widehat{J}_{\infty,n}(\omega;f)$ where
\begin{eqnarray*}
\widehat{J}_{\infty,n}(\omega_{};f)
&=&
\frac{n^{-1/2}}{\phi(\omega;f)} \sum_{t=1}^{n}X_{t}\phi_{t}^{\infty}(\omega;f)
+e^{i(n+1)\omega}
\frac{n^{-1/2}}{ \overline{\phi(\omega;f)}} \sum_{t=1}^{n}X_{n+1-t}\overline{\phi_{t}^{\infty}(\omega;f)} \\
&=&
\frac{\psi(\omega;f)}{\sqrt{n}} \sum_{t=1}^{n}X_{t} \sum_{s=0}^{\infty} \phi_{s+t}(f)e^{-is\omega}
+e^{i(n+1)\omega}
\frac{\overline{\psi(\omega;f)}}{\sqrt{n}} \sum_{t=1}^{n}X_{n+1-t} \sum_{s=0}^{\infty} \phi_{s+t}(f)e^{is\omega}
,
\end{eqnarray*}
where
$\psi(\omega;f) = \sum_{j=0}^{\infty} \psi_j(f) e^{-ij \omega}$ be an MA transfer function.
Our goal is to estimate $\psi(\omega;f)$ and $\{\phi_j(f)\}$ based on the observed time series.
We use the method proposed in Section 2.2. of \cite{p:kra-18}.
We first start from the well known Szeg{\"o}'s identity
\begin{eqnarray*}
\log f(\cdot) = \log \sigma^2
|\psi(\cdot;f)|^2 = \log \sigma^2 + \log \psi(\cdot;f) +
\log \overline{\psi(\cdot;f)}.
\end{eqnarray*}
Next, let $\alpha_{k}(f)$ be the $k$-th Fourier coefficient of $\log f$,
i.e., $\alpha_{k}(f) = (2\pi)^{-1} \int_{-\pi}^{\pi} \log f(\lambda) e^{-ik \lambda} d\lambda$. Then,
since $\log f$ is real, $\alpha_{-k}(f) = \overline{\alpha_k(f)}$.
Plug in the expansion of $\log f$ to the above identity gives
\begin{eqnarray*}
\log \psi(\omega;f) =
\sum_{j=1}^{\infty} \alpha_j(f) e^{-ij\omega}.
\end{eqnarray*}
Using above identity, we estimator $\psi(\cdot;f)$. let $\widehat{f}$
be a spectral density estimator and let $\widehat{\alpha}_k$ be the estimated
$k$-th Fourier coefficient of $\log \widehat{f}$. Then define
\begin{eqnarray*}
\widehat{\psi}(\omega;\widehat{f}) = \exp \left( \sum_{j=1}^{M} \widehat{\alpha}_j e^{-ij\omega} \right)
\end{eqnarray*}
for some large enough $M$.
To estimate the AR$(\infty)$ coefficients
we use the recursive formula in equation (2.7) in \cite{p:kra-18},
\begin{eqnarray*}
\widehat{\phi}_{k+1} = -\sum_{j=0}^{k} \left( 1-\frac{j}{k+1} \right) \widehat{\alpha}_{k+1-j} \widehat{\phi}_{j}
\qquad k=0,1,...,M-1
\end{eqnarray*}
where $\widehat{\phi}_0=-1$.
Based on this a nonparametric estimator of $\widehat{J}_{n}(\omega_{};f)$ is
\begin{eqnarray*}
\widehat{J}_n(\omega;\widehat{f})
=
\frac{\widehat{\psi}(\omega;\widehat{f})}{\sqrt{n}} \sum_{t=1}^{n \wedge M}X_{t} \sum_{s=0}^{M-t} \widehat{\phi}_{s+t}e^{-is\omega}
+e^{i(n+1)\omega}
\frac{\overline{\widehat{\psi}(\omega;\widehat{f})}}{\sqrt{n}} \sum_{t=1}^{n \wedge M}X_{n+1-t} \sum_{s=0}^{M -t} \widehat{\phi}_{s+t}e^{is\omega}
\end{eqnarray*}
where $n \wedge M = \min(n,M)$.
In the simulations we estimate $\widehat{f}$ using \texttt{iospecden} function in R (smoothing with infinite order Flat-top kernel)
and set $M$=30.
By replacing $\widehat{J}_n(\omega;f)$
with its nonparametric estimator $\widehat{J}_n(\omega;\widehat{f})$
in (\ref{eq:Winf}) leads us to define a new feasible criterion which we call the
\textbf{b}oundary \textbf{c}orrected Whittle likelihood using
\textbf{N}on\textbf{p}arametric estimation (BC-NP).
\subsection{Alternative methods for estimating the predictive DFT results for a Gaussian time series}
To access the performance of all the different likelihoods (with different
estimates of the predictive DFT), we generate the AR$(8)$ model
\begin{eqnarray*}
U_{t} = \phi_{U}(B)\varepsilon_{t}
\end{eqnarray*}
where $\{\varepsilon_{t}\}$ are i.i.d. normal random variables,
\begin{eqnarray}
\label{eq:phiU}
\phi_U(z) = \prod_{j=1}^{4} (1-r_j e^{i \lambda_j}z) (1-r_j e^{-i \lambda_j}z) = 1-\sum_{j=1}^{8}\phi_j z^{j}
\end{eqnarray}
$\underline{r} = (r_1,r_2,r_3,r_4) = (0.95,0.95,0.95,0.95)$ and
$\underline{\lambda} = (\lambda_1, \lambda_2, \lambda_3,\lambda_4) = (0.5,1,2,2.5)$.
We observe that corresponding spectral density $f_U(\omega) = |\phi_U(e^{-i\omega})|^{-2}$
has pronounced peaks at $\omega=0.5,1,1.5$ and $2$. For all the simulations below we use $n=100$.
For each simulation, we fit AR$(8)$ model, evaluate six likelihoods from the previous sections plus two likelihoods (BC-tYW and BC-NP),
and calculate the parameter estimators. Table \ref{tab:AR8} summarizes the bias and standard derivation of the estimators and the last row is
an average $\ell_2$-distance between the true and estimator scaled with $n$. The Gaussian likelihood has the smallest bias and the smallest RMSE.
As mentioned in Section \ref{sec:specified}, our methods still need to estimate AR coefficients which has an additional error of order $O(p^3n^{-3/2})$
and it could potentially increase the bias compared to the Gaussian likelihood.
The boundary corrected Whittle and hybrid Whittle have smaller bias than the Whittle, tapered, and debiased Whittle. Especially, the hybrid Whittle usually has the second smallest RMSE.
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{c|cccccccc}
\multirow{2}{*}{Par.} &
\multicolumn{8}{c}{Bias} \\ \cline{2-9}
&
Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased
& {\color{blue} BC-tYW} & {\color{blue} BC-NP} \\ \hline \hline
$\phi_1 (0.381)$ & {\color{red}-$0.008(0.08)$} & -$0.025(0.09)$ & -$0.009(0.08)$ & -$0.006(0.09)$ & -$0.012(0.09)$ & -$0.008(0.09)$ & {\color{blue}-$0.008(0.08)$} & -$0.005(0.12)$ \\
$\phi_2 (\text{-}0.294)$ & {\color{red}$0.002(0.09)$} & $0.024(0.1)$ & $0.005(0.09)$ & {\color{blue}$0.002(0.09)$} & $0.010(0.09)$ & $0.003(0.1)$ & $0.003(0.09)$ & $0.002(0.13)$ \\
$\phi_3 (0.315)$ & {\color{red}-$0.009(0.08)$} & -$0.038(0.09)$ & -$0.011(0.09)$ & {\color{blue}-$0.009(0.09)$} & -$0.023(0.09)$ & -$0.010(0.09)$ & -$0.009(0.09)$ & -$0.010(0.12)$ \\
$\phi_4 (\text{-}0.963)$ & {\color{red}$0.031(0.09)$} & $0.108(0.1)$ & $0.042(0.09)$ & {\color{blue}$0.034(0.09)$} & $0.075(0.09)$ & $0.043(0.1)$ & $0.037(0.09)$ & $0.076(0.12)$ \\
$\phi_5 (0.285)$ & {\color{red}-$0.015(0.08)$} & -$0.049(0.09)$ & -$0.020(0.09)$ & -$0.016(0.08)$ & {\color{blue}-$0.029(0.08)$} & -$0.017(0.1)$ & -$0.018(0.09)$ & -$0.022(0.12)$ \\
$\phi_6 (\text{-}0.240)$ & {\color{red}$0.010(0.08)$} & $0.040(0.09)$ & $0.014(0.09)$ & $0.010(0.09)$ & {\color{blue}$0.024(0.08)$} & $0.012(0.1)$ & $0.011(0.09)$ & $0.022(0.11)$ \\
$\phi_7 (0.280)$ & {\color{red}-$0.017(0.08)$} & -$0.053(0.09)$ & -$0.021(0.09)$ & -{\color{blue}$0.020(0.09)$} & -$0.039(0.08)$ & -$0.022(0.09)$ & -$0.020(0.09)$ & -$0.027(0.1)$ \\
$\phi_8 (\text{-}0.663)$ & {\color{red}$0.049(0.08)$} & $0.116(0.08)$ & $0.059(0.08)$ & {\color{blue}$0.055(0.08)$} & $0.096(0.08)$ & $0.061(0.09)$ & $0.056(0.08)$ &$0.101(0.1)$ \\
$n\|\underline{\phi}-\widehat{\underline{\phi}}\|_2$ & {\color{red}$6.466$} & $18.607$ & $8.029$ & {\color{blue}$7.085$} & $13.611$ & $8.164$ & $7.470$ & $13.280$ \\
\end{tabular}
\caption{\textit{ Bias and the standard deviation (in the parenthesis) of eight different quasi-likelihoods for the Gaussian AR$(8)$ model. Length of time series $n$=100.
True AR coefficients are in the parenthesis of the first column.
}}
\label{tab:AR8}
\end{table}
Bear in mind that neither of the two new criteria uses a hybrid method (tapering on the actual DFT), the
BC-tYW significantly reduces the bias than the boundary corrected Whittle and it is comparable with the hybrid Whittle.
This gives some credence to the referee's claim that the bias due to the Yule-Walker estimation can be alleviated using tapered Yule-Walker estimation.
Whereas, BC-NP reduces the bias for the first few coefficients but overall, has a larger bias than the boundary corrected Whittle. Also, the standard deviation of BC-NP is quite large than other methods. We suspect that the nonparametric estimator $\widehat{J}(\omega;\widehat{f})$ is
sensitive to the choice of the tuning parameters (e.g. bandwidth, kernel function, etc). Moreover, since the true model follows a finite autoregressive process, other methods (boundary corrected Whittle, BC-tYW, and hybrid Whittle) have an advantage over the nonparametric method. Therefore, by choosing appropriate tuning parameters under certain underlying process (e.g., seasonal ARMA model) can improve the estimators, and this will be
investigated in future research.
\subsection{Alternative methods for estimating the predictive DFT results for a non-Gaussian time series}
This time we assess the different estimation schemes for non-Gaussian time series. We generate the same AR$(8)$ model as above with
\begin{eqnarray*}
V_{t} = \phi_{U}(B)\varepsilon_{t}
\end{eqnarray*}
where $\{\varepsilon_t\}$ are i.i.d. standardarized chi-square random variables with
two-degrees of freedom i.e. $\varepsilon_t \sim (\chi^2(2)-2)/2$ and
$\phi_{U}(z)$ is defined as in (\ref{eq:phiU}).
For each simulation, we fit AR$(8)$ model, evaluate six likelihoods from the previous sections plus two likelihoods (BC-tYW and BC-NP), and calculate the parameter estimators.
The results are summarized in Table \ref{tab:AR8.chisq}.
\begin{table}[ht]
\centering
\scriptsize
\begin{tabular}{c|cccccccc}
\multirow{2}{*}{Par.} &
\multicolumn{8}{c}{Bias} \\ \cline{2-9}
&
Gaussian & Whittle & {\color{blue}Boundary} & {\color{blue}Hybrid} & Tapered & Debiased
& {\color{blue} BC-tYW} & {\color{blue} BC-NP} \\ \hline \hline
$\phi_1 (0.381)$ & {\color{red}$0.001(0.08)$}& -$0.013(0.09)$ & -$0.002(0.09)$ & $0.001(0.09)$ & -$0.003(0.09)$ & $0.004(0.09)$ & {\color{blue}$0(0.09)$} & $0.001(0.12)$ \\
$\phi_2 (\text{-}0.294)$ & {\color{red}-$0.001(0.09)$} & $0.014(0.1)$ & -$0.001(0.09)$ & {\color{blue} -$0.002(0.09)$} & $0.006(0.09)$ & -$0.008(0.11)$ & -$0.002(0.09)$ & -$0.010(0.13)$ \\
$\phi_3 (0.315)$ & {\color{red}-$0.004(0.09)$} & -$0.027(0.1)$ & -$0.005(0.09)$ & {\color{blue} -$0.003(0.09)$} & -$0.015(0.09)$ & $0(0.1)$ & -$0.003(0.09)$ & -$0.005(0.12)$ \\
$\phi_4 (\text{-}0.963)$ & {\color{red}$0.034(0.09)$} & $0.097(0.09)$ & $0.040(0.09)$ & {\color{blue}$0.034(0.09)$} & $0.073(0.09)$ & $0.038(0.11)$ & $0.036(0.09)$ & $0.068(0.12)$ \\
$\phi_5 (0.285)$ & {\color{red}-$0.007(0.09)$} & -$0.032(0.09)$ & -$0.009(0.09)$ & -$0.005(0.09)$ & {\color{blue}-$0.018(0.09)$} & -$0.004(0.1)$ & -$0.007(0.09)$ & -$0.005(0.12)$ \\
$\phi_6 (\text{-}0.240)$ & {\color{red}$0.007(0.09)$} & $0.029(0.09)$ & $0.009(0.09)$ & $0.006(0.09)$ & {\color{blue}$0.018(0.09)$} & $0.003(0.1)$ & $0.007(0.09)$ & $0.006(0.12)$ \\
$\phi_7 (0.280)$ & {\color{red}-$0.019(0.08)$} & -$0.047(0.09)$ & -$0.021(0.09)$ & {\color{blue}-$0.018(0.09)$} & -$0.034(0.09)$ & -$0.020(0.1)$ & -$0.019(0.09)$ & -$0.026(0.11)$ \\
$\phi_8 (\text{-}0.663)$ & {\color{red}$0.058(0.08)$} & $0.114(0.08)$ & $0.062(0.09)$ & {\color{blue}$0.059(0.09)$} & $0.098(0.08)$ & $0.065(0.1)$ & $0.060(0.08)$ &$0.107(0.1)$ \\
$n\|\underline{\phi}-\widehat{\underline{\phi}}\|_2$ & {\color{red}$7.006$} & $16.607$ & $7.728$ & {\color{blue}$7.107$} & $13.054$ & $7.889$ & $7.319$ & $13.001$ \\
\end{tabular}
\caption{\textit{Bias and the standard deviation (in the parenthesis) of eight different quasi-likelihoods for the AR$(8)$ model for the standardized chi-squared innovations. Length of time series $n$=100.
True AR coefficients are in the parenthesis of the first column. We use {\color{red}red} to denote the smallest RMSE and
{\color{blue}blue} to denote the second smallest RMSE.
}}
\label{tab:AR8.chisq}
\end{table} | {"config": "arxiv", "file": "2001.06966/appendix_simulations.tex"} |
TITLE: Prove the relation $\mathcal{L}u - L_h u = O(h^2)$ for $h \to 0$
QUESTION [1 upvotes]: Exercise: Prove the relation $$\mathcal{L}u - L_h u = O(h^2)\,\, \text{for} \,\,h \to 0,$$ for sufficiently smooth functions $u$ for the standard five-point discretization
$$-\Delta_h = \frac{1}{h^2}\begin{bmatrix}
& -1 & \\
-1 & 4 & -1 \\
& -1 &\\
\end{bmatrix}_h$$
of the operator $\mathcal{L} = - \Delta$ using Taylor expansion.
Question: how do I solve/handle this?
I have no idea what to do, thanks in advance!
REPLY [0 votes]: On the one hand, $-\mathcal{L}$ is the Laplace operator, i.e. $-\mathcal{L}u = u_{xx} + u_{yy}$. On the other hand, $-L_h$ is the finite-difference approximation of $-\mathcal{L}$, obtained by centered differencing of the second derivatives $u_{xx} \simeq (u_{i+1,j} - 2u_{i,j} + u_{i-1,j})/h^2$ and $u_{yy} \simeq (u_{i,j+1} - 2u_{i,j} + u_{i,j-1})/h^2$, where $u_{i,j}\simeq u(ih,jh)$. For each approximation, we know --or we obtain by Taylor series expansion as $h\to 0$, see e.g. this post-- that the finite difference formula is second-order accurate, which ends the proof. | {"set_name": "stack_exchange", "score": 1, "question_id": 2510467} |
\begin{document}
\begin{abstract}
We study automorphisms of order four on K3 surfaces. The symplectic ones have been first
studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique.
In this paper we give a classification of the purely non-symplectic automorphisms by relating the
structure of their fixed locus to their action on cohomology, in the following cases: the fixed locus contains a curve of genus $g>0$; the fixed locus contains at least a curve and all the curves fixed by the square of the automorphism are rational. We give partial results in the other cases.
Finally, we classify non-symplectic automorphisms of order four
with symplectic square.
\end{abstract}
\maketitle
\section*{Introduction}
Let $X$ be a K3 surface over $\mathbb C$ with an order four automorphism.
Such automorphism acts on the one-dimensional vector space $H^{2,0}(X)$ of holomorphic two-forms of $X$ either as the identity,
minus the identity or as the multiplication by $\pm i$.
Accordingly, the automorphism is called symplectic, with symplectic square or purely non-symplectic.
Symplectic automorphisms of finite order have been investigated by several authors, their fixed locus contains only isolated points (six if the order is four) and their action on the K3 lattice $H^2(X,\IZ)$ is known to be independent on the surface (cf.\cite{Nikulin1, Alice1, Alice2}).
Non-symplectic automorphisms have been classified in \cite{Nikulin1, MO} (see also \cite{zhang} for a survey on the topic) and the fixed locus has been identified if the order is prime in \cite{AS3,takiauto, ast}.
In \cite{Taki} Taki classified order four non-symplectic automorphisms acting as the identity on the Picard lattice of the surface. Moreover, Sch\"u{tt} \cite{Schuett} studied the special case when the transcendental lattice of the surface has rank four.
This paper deals mainly with purely non-symplectic automorphisms of order four under the assumption that their square is the identity on the Picard lattice.
By the Torelli type theorem, this holds for the generic element of the family of K3 surfaces having an order four non-symplectic automorphism with a given action on the K3 lattice.
We give a complete classification of the fixed locus of such automorphisms when it either contains a curve of genus $g>0$ or it contains a curve of genus $0$ and all the curves fixed by the square of the automorphism are rational. Moreover, we provide partial results for the remaining cases and we give several examples.
We also consider the case when the automorphism has symplectic square: we prove that its fixed locus is empty and its invariant lattice has rank $6$.
The study of such automorphisms and their fixed locus is
interesting also in relation with the Borcea-Voisin construction of Calabi-Yau varieties
and the investigation of Mirror-Simmetry (cf. \cite{Borcea, Voisin}). In fact Borcea and Voisin consider the product between a K3 surface with a non-symplectic automorphism of order 2,3,4 or 6, and an elliptic curve with an automorphism of the same order. A resolution of the quotient variety
is then a Calabi-Yau threefold. In \cite{Alice0} Garbagnati used some non-symplectic automorphism of order four to give examples of Calabi-Yau threefolds using this construction.
We now give a short description of the paper's sections. Let $\sigma$ be a non-symplectic automorphism of order four on a K3 surface $X$.
In section 1 we give a general description of the fixed locus of $\sigma$. In case it is purely non-symplectic, it is the disjoint union of $n$ points, $k$ smooth rational curves and possibly a smooth curve of genus $g$. By means of Lefschetz's formulas we provide two relations, between the invariants $n,k,g$ and the ranks of the eigenspaces of $\sigma^*$ on the lattice $H^2(X,\IZ)$.
If $\sigma$ has symplectic square, we prove that the fixed locus is empty and such ranks are uniquely determined.
In section 2 we study elliptic fibrations $\pi:X\map \IP^1$ such that $\sigma$ preserves each fiber of $\pi$. In Corollary \ref{cor} the configuration of the singular fibers, which are of Kodaira type $III, I_0^*$ or $III^*$, is related to the structure of the fixed locus of $\sigma$.
In section 3 we assume that $\sigma$ fixes pointwisely an elliptic curve $E$. In Theorem \ref{g1} we describe the singular fibers of the elliptic fibration with fiber $E$ and the corresponding structure of the fixed locus of $\sigma$.
In section 4 and 5 we classify the case when $\sigma$ contains a curve of genus $g>1$ in its fixed locus or a rational curve and $\sigma^2$ fixes only rational curves.
In section 6 we assume that $\sigma$ is the identity on the Picard lattice and
we give an independent proof of \cite[Proposition 4.3]{Taki}.
In section 7 we consider the case when $\sigma$ only fixes isolated points and we provide families of examples.
In section 8, we study the case when $\sigma$ only fixes isolated points and rational curves, and $\sigma^2$ fixes a curve of genus $g\geq 1$. \\
{\em Acknowledgements:} We warmly thank Bert van Geemen and Alice Garbagnati for useful discussions.
\section{The fixed locus}
Let $X$ be a K3 surface with a {\it non-symplectic} automorphism $\sigma$ of order four,
i.e. such that the action of $\sigma^*$ on the vector space $H^{2,0}(X)=\IC \omega_X$ of holomorphic two-forms is not trivial. We will call the automorphism {\it purely non-symplectic} if $\sigma^*\omega_X=\pm i \omega_X$. Otherwise $\sigma^*\omega_X=- \omega_X$ and $\sigma^2$ is a symplectic involution.
We will denote by $r,l,m$ the rank of the eigenspace of $\sigma^*$ in $H^2(X,\IZ)$ relative to the eigenvalues $1,-1$ and $i$ respectively. Moreover, let
$$
S(\sigma)=\{x\in H^2(X,\IZ)\,|\, \sigma^*(x)=x\},
$$
$$
S(\sigma^2)=\{x\in H^2(X,\IZ)\,|\, ({\sigma^2})^*(x)=x\},\quad T(\sigma^2)=S(\sigma^2)^{\perp}\cap H^2(X,\IZ).
$$
Observe that $r=\rank S(\sigma)$, $r+l=\rank S(\sigma^2)$ and $2m=\rank T(\sigma^2)$.
\begin{pro}\label{rel4}
Let $\sigma$ be a purely non-symplectic automorphism of order four on a K3 surface $X$.
Then:
\begin{itemize}
\item $\Fix(\sigma^2)$ is either the disjoint union of two elliptic curves or the disjoint union of a smooth curve $C$ of genus $g\geq 0$ and
$j$ smooth rational curves;
\item $\Fix(\sigma)\subset \Fix(\sigma^2)$ is the disjoint union of smooth curves and $n$ isolated points.
\end{itemize}
Moreover, the following relations hold:
$$
n=2\alpha+4,\quad \alpha=\frac{r-l-2}{4}=\frac{10-l-m}{2},
$$
where $\alpha=\sum_{C_i\subset \Fix(\sigma)} (1-g(C_i))$.
\end{pro}
\begin{proof} Since $\sigma$ is purely non-symplectic, then $\sigma^2$ is a non-symplectic involution.
By \cite[Theorem 4.2.2]{nikulinfactor} or \cite[Theorem 4.1]{ast} the fixed locus of $\sigma^2$ is either empty, the disjoint union of two elliptic curves or the disjoint union of a curve of genus $g\geq 0$ and smooth rational curves.
The action of $\sigma$ at a point in $\Fix(\sigma)$ can be locally diagonalized as follows (see \cite[\S 5]{Nikulin1}):
$$
A_{4,0}= \left(\begin{array}{cc}
i &0\\
0& 1
\end{array}
\right),\ \
A_{4,1}= \left(\begin{array}{cc}
-i &0\\
0& -1
\end{array}
\right).
$$
In the first case the point belongs to a smooth fixed curve, while in the second case it is an isolated fixed point.
We will apply holomorphic and topological Lefschetz's formulas to obtain the last two relations in the statement.
The Lefschetz number of $\sigma$ is
$$
L(\sigma)=\sum_{j=0}^{2}(-1)^j \tr(\sigma^*|H^j(X,\calO_X))=1-i,
$$
since $\sigma^*$ acts as multiplication by $i$ on $H^{2,0}(X)$.
By \cite[p. 567]{atiyahsinger} one obtains:
$$
L(\sigma)=\frac{n}{\det(I-\sigma^*|T_P)}+\frac{1+i}{(1-i)^2}\sum_i (1-g(C_i))
=\frac{n}{\det(I-A_{4,1})}+\alpha \frac{1+i}{(1-i)^2},
$$
where $n$ is the number of isolated fixed points, $P$ is an isolated fixed point, $T_P$ denotes the tangent space at $P$ and $C_i$ are the curves in the fixed locus.
Comparing the two formulas for $L(\sigma)$
we obtain the relation $n=2\alpha+4$. In particular this implies that the fixed locus of $\sigma$ (and thus that of $\sigma^2$) is not empty.
We consider now the topological Lefschetz fixed point formula
\begin{eqnarray*}
\chi(\Fix(\sigma))=\sum_{j=0}^{4}(-1)^j\tr(\sigma^*|H^j(X,\IR))
=2+\tr (\sigma^*|H^2(X,\IR)).
\end{eqnarray*}
Since $\tr (\sigma^*|H^2(X,\IR))=r-l$, then:
$$
\chi(\Fix(\sigma))=n+2\alpha=2+r-l
.$$
Using the relation $n=2\alpha+4$, this gives the two expressions for $\alpha$ in the statement.
\end{proof}
We now provide a similar result in case $\sigma^2$ is symplectic.
\begin{pro}
Let $\sigma$ be a non-symplectic automorphism of order four on a K3 surface $X$ such that $\sigma^2$ is symplectic.
Then $\Fix(\sigma)$ is empty and $r=6, l=8, m=4$.
\end{pro}
\begin{proof}
Since $\sigma$ has symplectic square, then the local action of $\sigma$ at a fixed point is of the following type:
$$A_{4,2}= \left(\begin{array}{cc}
i &0\\
0& i
\end{array}
\right).
$$
The holomorphic Leftschetz formula immediately gives that $n=0$ since $\sigma^*=-id$ on $H^{2,0}(X)$.
Moreover, the topological Lefschetz formula gives that $l-r=2$.
Since the invariant lattice of a symplectic involution has rank $14$, \cite{Nikulin1}, then $l+r=14$, so that $l=8, r=6$ and $2m=22-14=8$.
\end{proof}
\begin{example}
Consider the following family of quartics surfaces in $\IP^3$:
$$a_1x_0^4+x_0^2(a_2x_1^2+a_3x_2x_3)+x_0x_1(a_4x_2^2+a_5x_3^2)+$$
$$x_1^2(a_6x_1^2+a_7x_2x_3)+x_2^2(a_{8}x_2^2+a_{9}x_3^2)+a_{10}x_3^4=0.
$$
The generic element $X_a$ of the family is a smooth quartic surface, hence a K3 surface, and carries the order four automorphism:
$$\sigma(x_0,x_1,x_2,x_3)=(x_0,-x_1,ix_2,-ix_3),$$
which has no fixed points and whose square fixes the eight intersection points between $X_a$ and the lines $x_0=x_1=0$, $x_2=x_3=0$.
Since the space of matrices in $\rm GL_3(\IC)$ commuting with $\sigma$ has dimension $4$, then the family has $10-4=6$ moduli.
\end{example}
\begin{pro}\label{lat}
Let $\sigma$ be a non-symplectic automorphism of order four on a K3 surface $X$. Then $S(\sigma)$ is a hyperbolic sublattice of $\Pic(X)$.
If $\sigma$ is purely non-symplectic, then $S(\sigma^2)\subset \Pic(X)$ and it is a $2$-elementary lattice with determinant $2^d$, such that $\rank S(\sigma^2)=r+l=10, d=8$ if $\sigma^2$ fixes two elliptic curves and otherwise
$$2g=22-r-l-d,\quad 2j=r+l-d.$$
\end{pro}
\begin{proof}
If $x\in S(\sigma)$, then $(x,\omega_X)=(\sigma^*(x),\sigma^*(\omega_X))=(x,\alpha\omega_X)$, with $\alpha\not=1$ since $\sigma$ is non-symplectic. Thus $x\in \Pic(X)=\omega_X^{\perp}\cap H^2(X,\IZ)$.
A similar argument shows that $S(\sigma^2)\subset \Pic(X)$ if $\sigma$ is purely non-symplectic.
Observe that, by \cite[Theorem 3.1]{Nikulin1}, the surface $X$ is algebraic.
Moreover, it is easy to construct a $\sigma$-invariant class with positive self-intersection. This implies that $S(\sigma)$ is a hyperbolic lattice, since $\Pic(X)$ is hyperbolic by Hodge index theorem.
The proof that $S(\sigma^2)$ is $2$-elementary and the relations in the statement are given in \cite[Theorem 4.2.2]{nikulinfactor} or \cite[Theorem 4.1]{ast}.
\end{proof}
\begin{remark} The moduli space of K3 surfaces carrying a purely non-symplectic automorphism of order four with a given action on the K3 lattice is known to be a complex ball quotient of dimension $m-1$, see \cite[\S 11]{DK}. The generic element of such space is a K3 surface such that $\omega_X$ is the generic element of an eigenspace of $\sigma^*$ in $T(\sigma^2)\otimes \IC$, so that $\Pic(X)=S(\sigma^2)$ .
On the other hand, if the automorphism has symplectic square, then the period belongs to the eigenspace where $\sigma^*=-\id$, so that $\Pic(X)$ contains $S(\sigma)\oplus T(\sigma^2)$, $\rk\Pic(X)\geq 14$ and, given the action on the K3 lattice, the dimension of the moduli space is equal to $6$. \end{remark}
The following result will be useful later.
\begin{lemma}\label{even1}
If $x\in S(\sigma^2)$, then $x\cdot \sigma(x)$ is even.
\end{lemma}
\begin{proof}
If $\sigma(x)=x$, then the statement is obvious since $H^2(X,\IZ)$ is an even lattice.
Otherwise, since $x$ belongs to $S(\sigma^2)$, it is of the form $x=\frac{a+b}{n}$ for some positive integer $n$ where $a\in S(\sigma)$ and $b$ belongs to its orthogonal complement in $S(\sigma^2)$, where $\sigma^*=-\id$.
Thus $x\cdot \sigma(x)=\frac{a^2-b^2}{n^2}=\frac{2a^2}{n^2}-x^2$, which is even.
\end{proof}
\section{Elliptic fibrations}
In this section we will study elliptic fibrations on K3 surfaces carrying a purely non-symplectic automorphism of order four.
The following result is proved with an argument contained in the proof of \cite[Proposition 2.9]{DKe}.
\begin{pro}\label{ellinv}
Let $\sigma$ be an automorphism of a K3 surface $X$.
If the rank of the invariant lattice of $\sigma^*$ in $H^2(X,\IZ)$ is $\geq 5$, then there is a $\sigma$-invariant elliptic fibration $\pi:X\map \IP^1$.
\end{pro}
\begin{proof} We will denote by $S(\sigma)$ the invariant lattice of $\sigma^*$ in $H^2(X,\IZ)$.
By \cite[Corollary 2, pag. 43]{Se}, since $\rk(S(\sigma))\geq 5$, there exists a primitive isotropic vector $x\in S(\sigma)$.
After applying a finite number of reflections with respect to $(-2)$-curves, we obtain a nef class $x'$ which is uniquely determined by $x$ (see \cite[\S 6, Theorem 1]{PSS}). Observe that $x'$ is primitive and $x'^2=0$.
It is easy to see that $\sigma^*$ acts
on the orbit of $x$ with respect to the reflection group. Since $x'$ is the unique
nef member in the orbit and any automorphism preserves nefness, then
$\sigma^*(x') = x'$. The morphism associated to $x'$ is a $\sigma$-invariant elliptic fibration on $X$.
\end{proof}
Let $\pi:X\map \IP^1$ be a $\sigma$-invariant elliptic fibration such that any of its fibers is invariant for $\sigma$ and contains at least a fixed point. The last assumption is not necessary if the fibration is jacobian: indeed if $\sigma$ is fixed points free on the generic fiber, then it acts as a translation on it and it can be easily proved, by writing explicitly a holomorphic 2-form, that the automorphism would be symplectic.
If $\sigma$ has order four, then the generic fiber of $\pi$ contains two fixed points for $\sigma$ and four fixed points of $\sigma^2$.
Thus $\pi$ has two bisections (not necessarily irreducible): a curve $E_{\sigma}\subset \Fix(\sigma)$ and a curve $E_{\sigma^2}\subset \Fix(\sigma^2)$.
We now describe the singular fibers of the elliptic fibration and the action of $\sigma$ on them.
\begin{pro}\label{ell} Let $X$ be a K3 surface with a non-symplectic order four automorphism $\sigma$ and $\pi:X\map \IP^1$ be an elliptic fibration such that $\sigma$ preserves each fiber of $\pi$ and has a fixed point on it.
Then the singular fibers of $\pi$ are of the following Kodaira types:
\begin{enumerate}[$\bullet$]
\item $III$: $R_1\cup R_2$, where either\\
a) the $R_i$'s are exchanged by $\sigma$, $E_{\sigma^2}$ intersects each $R_i$ at one point and $E_{\sigma}$ intersects in $R_1\cap R_2$ or \\
b) the $R_i$'s are $\sigma$-invariant, $E_{\sigma}$ intersects each $R_i$ at one point and $E_{\sigma^2}$ intersects in $R_1\cap R_2$.
\item $I_0^*$: $2R_1+R_2+R_3+R_4+R_5$, where either\\
a) $R_2, R_3$ are $\sigma$-invariant (intersected by $E_{\sigma}$) and $R_4, R_5$ are exchanged by $\sigma$ (intersected by $E_{\sigma^2}$) or\\
b) $R_2,\dots, R_5$ are permuted by $\sigma$, $E_{\sigma}$ and $E_{\sigma^2}$ intersect $R_1$.
\item $III^*$: $R_1+2R_2+3R_3+4R_4+2R_5+3R_6+2R_7+R_8$, where either\\
a) $\sigma$ preserves each irreducible component of the fiber, $R_2, R_4, R_7\subset \Fix(\sigma)$, $E_{\sigma}$ intersects $R_1, R_8$ and $E_{\sigma^2}$ intersects $R_5$ or\\
b) $\sigma$ preserves each irreducible component of the fiber, $R_4\subset \Fix(\sigma)$, $R_2, R_7$ contain two isolated fixed points, $E_{\sigma}$ intersects $R_1, R_8$, $E_{\sigma^2}$ intersects $R_5$ or\\
c) $\sigma$ exchanges the two branches of the fiber, $E_{\sigma^2}$ intersects $R_1, R_8$ and $E_{\sigma}$ intersects $R_5$.\\
\end{enumerate}
\end{pro}
\begin{proof}
By the previous argument, the restriction of $\sigma$ to the generic fiber of $\pi$ has order four and two fixed points. Thus any smooth fiber of $\pi$ has $j$-invariant equal to $1$. By the Kodaira classification it follows that the singular fibers of $\pi$ are either of type $I_0^*$, $III$, or $III^*$. We now analyze the possible actions of $\sigma$ on these fibers.
If $F$ is a reducible fiber of type $I_0^*$, then the component $R_1$ is clearly $\sigma$-invariant.
Observe that $R_1$ is not fixed by $\sigma$, since otherwise each $R_i$, $i=2,\dots,5$, should contain a fixed point for $\sigma$ in the intersection with either $E_{\sigma^2}$ or $E_{\sigma}$. This is absurd because $\sigma$ exchanges the two (distinct) points in $F\cap E_{\sigma^2}$.
Thus $\sigma$ has either order two or four on $R_1$.
If $\sigma^2=\id$ on $R_1$, then each $R_i$, $i=2,\dots,5$ contains a fixed point of $\sigma^2$ in the intersection with either $E_{\sigma^2}$ or $E_{\sigma}$, thus we are in case $I_0^* a)$.
If $\sigma$ has order $4$ on $R_1$, then $\sigma$ permutes the curves $R_i$, $i=2,\dots,5$ and $R_1$ contains two fixed points for $\sigma$ in the intersection with $E_{\sigma}$ and $E_{\sigma^2}$, giving case $I_0^* b)$.
If $F$ is a fiber of type $III^*$, then $R_4$ and $R_5$ are clearly $\sigma$-invariant.
If $\sigma$ preserves each irreducible component of $F$, then $R_4\subset \Fix(\sigma)$ (since it contains $3$ fixed points) and, since $E_{\sigma^2}$ contains at most a fixed point, $E_{\sigma}$ intersects $R_1, R_8$ and $E_{\sigma^2}$ intersects $R_5$. The curves $R_2$ and $R_7$ are either contained in $\Fix(\sigma)$ or contain each two isolated fixed points. These give the cases $III^*a) $ and $b)$ respectively.
Otherwise, if $\sigma$ exchanges the two branches of the fiber, then $\sigma^2=\id$ on $R_4$, $E_{\sigma^2}$ intersects $R_1$ and $R_8$ in two points exchanged by $\sigma$ and $E_{\sigma}$ intersects $R_5$.
The case of a fiber of type $III$ can be discussed in a similar way.
\end{proof}
We will denote by $g_{\sigma}$ and $g_{\sigma^2}$ the genus of $E_{\sigma}$ and $E_{\sigma^2}$ respectively, by $n$ the number of isolated points in $\Fix(\sigma)$, by $k$ the number of smooth rational curves in $\Fix(\sigma)$ and by $a$ the number of smooth rational curves in $\Fix(\sigma^2)$ exchanged by $\sigma$.
\begin{cor}\label{cor} Under the hypotheses of Proposition \ref{ell}, we have the following possibilities for the invariants defined above.
\begin{itemize}
\item If $E_{\sigma}$ is irreducible and $E_{\sigma^2}$ is reducible (hence it is the union of two smooth rational curves exchanged by $\sigma$):
$$
\begin{array}{cccc|l}
g_{\sigma} & n & k & a & \mbox{reducible fibers} \\
\hline
3 & 0 & 0 & 1 & 8IIIa)\\
2 & 2 & 0 & 1 & 6IIIa)+I_0^*a)\\
& 2 & 0 & 2 & 5IIIa)+III^*c)\\
1 & 4 & 0 & 1 & 4IIIa)+2I_0^*a)\\
& 4 & 0 & 2 & 3IIIa)+I_0^*a)+III^*c)\\
& 4 & 0 & 3 & 2IIIa)+2III^*c)\\
0& 6& 1 & 1 & 2IIIa)+3I_0^*a)\\
& 6& 1 & 2 & IIIa)+2I_0^*a)+III^*c)\\
& 6& 1 & 3 & I_0^*a)+2III^*c)\\
\end{array}
$$
\item If $E_{\sigma}$ is reducible and $E_{\sigma^2}$ is irreducible: $a=0$ and $$
\begin{array}{ccc|l}
g_{\sigma^2} & n & k & \mbox{reducible fibers} \\
\hline
3 & 8 & 2 & 8IIIb)\\
2 & 8 & 2 & 6IIIb)+I_0^*a)\\
& 10&3 & 5IIIb)+III^*b)\\
1 & 8 & 2 & 4IIIb)+2I_0^*a) \\
& 10 & 3 & 3IIIb)+I_0^*a)+III^*b)\\
&12 & 4 & 2IIIb)+2III^*b)\\
0 & 8 & 2 & 2IIIb)+3I_0^*a)\\
& 10 & 3 & IIIb)+2I_0^*a)+III^*b)\\
& 12 & 4 & I_0^*a)+2III^*b)
\end{array}
$$
\end{itemize}
\begin{itemize}
\item If $E_{\sigma}$ and $E_{\sigma^2}$ are both irreducible:
$$
\begin{array}{ccccc|l}
g_{\sigma} & g_{\sigma^2} & n & k & a & \mbox{reducible fibers} \\
\hline
2 & 0 & 2 & 0 & 0 & 6IIIa)+2IIIb)\\
1 & 1 & 4 & 0 & 0 & 4IIIa)+4IIIb) \\
& & 4 & 0 & 0 & 4I_0^*b)\\
1 & 0 & 6 & 1 &0 & 3IIIa)+I_0^*b)+III^*b)\\
& & 6 & 1 & 0 & 4IIIa)+IIIb)+III^*b)\\
& & 4 & 0 & 0 & 4IIIa)+2IIIb)+I_0^*a)\\
& & 4 & 0 & 1 & 3IIIa)+2IIIb)+III^*c)\\
0& 2 & 6 & 1 & 0 & 2IIIa)+6IIIb)\\
& & 6 & 1 & 0 & 4IIIb)+2I_0^*b)\\
0 & 1 & 8 & 2 & 0 & 2IIIa)+3IIIb)+III^*b)\\
& & 6 & 1 & 0 & 2IIIa)+4IIIb)+I_0^*a)\\
& & 6 & 1 & 1 & IIIa)+4IIIb)+III^*c) \\
0 & 0 & 10 & 3 & 0 & 2IIIa)+2III^*b)\\
& & 8 & 2 & 0 & 2IIIa)+IIIb)+I_0^*a)+III^*b)\\
& & 8 & 2 & 1 & IIIa)+IIIb)+III^*b)+III^*c)\\
&& 6 & 1 & 0 & 2IIIa)+2IIIb)+2I_0^*a)\\
& & 6 & 1 & 1 & IIIa)+2IIIb)+I_0^*a)+III^*c)\\
& & 6 & 1 & 2 & 2IIIb)+2III^*c)
\end{array}$$
\item If both $E_{\sigma}$ and $E_{\sigma^2}$ are reducible: $n=8, k=2, a=1$ and the reducible fibers are of type $4I_0^*a)$.
\end{itemize}
\end{cor}
\begin{proof}
Observe that the restrictions of $\pi$ to $E_{\sigma}$ and to $E_{\sigma^2}$ are double covers of $\IP^1$.
If $E_{\sigma}$ (or $E_{\sigma^2}$) is irreducible, then it contains $2g_{\sigma}+2$ (or $2g_{\sigma^2}+2$) ramification points.
Since a smooth fiber of $\pi$ contains exactly $4$ fixed points for $\sigma^2$, then such ramification points belong to singular fibers of $\pi$, which are classified in Proposition \ref{ell}.
The ramification points of $E_{\sigma}$ belong either to a fiber of type $III\, a)$, $I_0^*\,b)$ or $III^*\,c)$.
On the other hand, the ramification points of $E_{\sigma^2}$ belong either to a fiber of type $III\, b)$, $I_0^*\, b)$, $III^*\,a)$ or $III^*\,b)$.
This implies that $g_{\sigma}\leq 3$ (or $g_{\sigma^2}\leq 3$) since otherwise the Euler-Poincar\'e characteristic of the singular fibers
would give at least $e(III)(2g_{\sigma}+2)=3(2g_{\sigma}+2)>24=e(X)$ (similarly for $g_{\sigma^2}$).
If $E_{\sigma}$ and $E_{\sigma^2}$ are both irreducible, this implies that $g_{\sigma}, g_{\sigma^2}\leq 2$. We obtain the tables in the statement by enumerating all cases which are compatible with Proposition \ref{ell} and Proposition \ref{rel4}.
\end{proof}
The following result allows, in some cases, to prove that a given elliptic fibration is $\sigma$-invariant.
\begin{pro}\label{ineq}
Let $X$ be a K3 surface with an automorphism $\sigma$ and $\pi:X\map \IP^1$ be an elliptic fibration whose general fiber has class $f$.
If $\sigma^*$ fixes a class $x\in \Pic(X)$ with $x^2>0$, then
$$(f\cdot\sigma^*(f))x^2\leq 2(x\cdot f)^2.$$
Moreover, if in addition $\pi$ is jacobian and there is a section of $\pi$ not intersecting $x$, the following holds
$$x^2\leq \frac{2(x\cdot f)^2}{f\cdot \sigma^*(f)+1}.$$
\end{pro}
\begin{proof}
Let $M$ be the sublattice of $\Pic(X)$ generated by $x$ and $f+\sigma^*(f)$. Its intersection matrix has negative determinant
$\det(M)=2(x^2(f\cdot \sigma^*(f))-2(x\cdot f)^2)\leq 0$ by Hodge index theorem. This gives the first inequality.
The second inequality follows from a similar argument with the lattice generated by $x, f+\sigma^*(f)$ and the class of the section not intersecting $x$.
\end{proof}
\begin{theorem}\label{jac}
Let $\sigma$ be a purely non-symplectic automorphism of order $4$ on a K3 surface $X$ such that $\Pic(X)=S(\sigma^2)\cong U\oplus R$, where $R$ is a direct sum of root lattices. Then $X$ carries a jacobian elliptic fibration $\pi:X\map \IP^1$ which is $\sigma^2$-invariant, has reducible fibers described by $R$ and a unique section $E$.
The involution $\sigma^2$ acts as an involution on the simple components of the reducible fibers of $\pi$ and on the fibers of types $I^*_{N}, III^*, II^*$ as in Figure \ref{action}, where $\sigma^2$ acts identically on dotted components and as an involution on the other ones.
\begin{figure}[ht!]
\begin{center}
\includegraphics[scale=0.29]{fibers}
\end{center}
\caption{Action of $\sigma^2$ on reducible fibres of types $I^*_{N}, III^*, II^*$}
\label{action}
\end{figure}
\noindent Moreover, if $\Fix(\sigma^2)$ contains a curve $C$ of genus $g>1$, then:
\begin{enumerate}[a)]
\item $\sigma^2$ preserves each fiber of $\pi$, $C$ intersects the generic fiber at three points and $E\subset \Fix(\sigma^2)$;
\item $\pi$ is $\sigma$-invariant if $g> 4$;
\item the genus of a curve in $\Fix(\sigma)$ is $\leq 2$.
\end{enumerate}
\end{theorem}
\begin{proof}
The first half of the statement follows from \cite[Lemma 2.1, 2.2]{kondoell}.
If $\sigma^2$ fixes a curve $C$ of genus $g>1$, then this curve is transversal to the fibers of $\pi$. This implies that $\sigma^2$ preserves each fiber of $\pi$ and has $4$ fixed points on it: one on $E$ and three on $C$ (because $C$ intersects each fiber in at least two points and there are no other sections). This proves $a)$.
Let $x$ be the class of $C$ and $f$ be the class of a fiber of $\pi$.
If $f\not=\sigma^*(f)$, then $f\cdot \sigma^*(f)\geq 2$. It follows from Proposition \ref{ineq} that
$2g-2=x^2\leq \frac{2(x\cdot f)^2}{f\cdot \sigma^*(f)+1}\leq 6$, proving $b)$.
Observe that, if $\sigma$ fixes a curve $C$ of genus $g>1$, then $f\not=\sigma^*(f)$ since otherwise the generic
fiber would contain at most $2$ fixed points by $\sigma$. Moreover in this case $f\cdot \sigma^*(f)\geq 4$ since each fiber contains at least $3$ fixed points and the intersection $f\cdot \sigma^*(f)$ is even by Lemma \ref{even1}. This implies $g\leq 2$ by Proposition \ref{ineq} and proves $c)$.
\end{proof}
\section{$\Fix(\sigma)$ contains an elliptic curve}
We now assume that $\sigma$ fixes an elliptic curve $C$. In this case the K3 surface $X$ has an elliptic fibration $\pi_C:X\map \IP^1$ having $C$ as a smooth fiber. Observe that all curves fixed by $\sigma^2$, since they are disjoint from $C$, are contained in the fibers of $\pi_C$. In particular the genus of a fixed curve is $\leq 1$ so that $\alpha\geq 0$.
We will now classify the reducible fibers of $\pi_C$. For the following result, see also \cite[\S 4.2]{nikulinfactor}.
\begin{theorem}\label{g1}
Let $\sigma$ be a purely non-symplectic order four automorphism on a K3 surface $X$ with
$\Pic(X)=S(\sigma^2)$ and $\pi_C:X\map \IP^1$ be an elliptic fibration with a smooth fiber $C\subset \Fix(\sigma)$.
Then $\sigma$ preserves $\pi_C$ and acts on its basis as an order four automorphism with two fixed points corresponding to the fiber $C$ and a fiber $C'$ which is either smooth, of Kodaira type $I_{4M}$ or $IV^*$. The corresponding invariants of $\sigma$ are given in Table \ref{g=1}.
\begin{table}[ht]
$$
\begin{array}{ccc|ccc|c}
m & r & l & n &k & a & \mbox{type of } C' \\
\hline
5 & 7 & 5 & 4 & 0 & 0 & I_0\ or\ I_4 \\
\hline
4 & 10 & 4 & 6 & 1 & 0 & I_8 \ or\ IV^* \\
& 8 & 6 & 4&0 & 1 & I_8 \ or\ IV^* \\
\hline
3 & 9 & 7 & 4& 0 & 2 & I_{12} \\
\hline
2 & 10 & 8 & 4 & 0 & 3 & I_{16} \\
\end{array}$$
\vspace{0.2cm}
\caption{The case $g=1$}\label{g=1}
\end{table}
\end{theorem}
\begin{proof} We first observe that $\sigma^2$ is not the identity on the basis of $\pi_C$, since otherwise
it would act as the identity on the tangent space at a point of $C$, contradicting the fact that $\sigma^2$ is non-symplectic. Hence $\sigma$ has order four on $\IP^1$ and has two fixed points, corresponding to $C$ and another fiber $C'$.
If $C'$ is irreducible, then $\alpha=0$ and $n=4$ by Proposition \ref{rel4}, which implies that $C'$ is smooth elliptic and $\sigma$ has order two on it.
We now assume that $C'$ is reducible. Since $n\geq 4$ by Proposition \ref{rel4}, then $C'$ contains at least two (disjoint) smooth rational curves fixed by $\sigma^2$. This immediately excludes the Kodaira types $I_2, I_3, III, IV$ for $C'$.
Observe that if a component of $C'$ is preserved by $\sigma^2$ and it is ``external'', i.e. it only intersects one other component, then it is fixed by $\sigma^2$ since otherwise it should contain a fixed point outside of any curve fixed by $\sigma^2$.
If $C'$ is of type $I_N^*$, then either $\sigma^2$ preserves each component of $C'$ or $\sigma$ has order four on them.
In the first case the four external components are fixed by $\sigma^2$ by the previous remark and the same should hold for the multiplicity two components intersecting them, since they contain four fixed points, giving a contradiction.
In the second case $\sigma^2$ exchanges two components of $C'$ of multiplicity one. If they intersect the same component of multiplicity two, then this is invariant for $\sigma^2$ but contains at most a fixed point of $\sigma^2$, a contradiction; if they intersect distinct components of multiplicity two, then it can be easily seen that $n=0$, a contradiction again.
If $C'$ is either of type $II^*$ or $ III^*$, then $\sigma^2$ preserves each component, since the set of components has no order four automorphism. As before, this implies that both the central component (of multiplicity $6$ and $4$ respectively) and the external component (of multiplicity $3$ and $2$ respectively) intersecting it are fixed by $\sigma^2$, giving a contradiction.
If $C'$ is of type $IV^*$, then the central component of multiplicity $3$ is clearly invariant for $\sigma$.
If the central component is fixed by $\sigma$, then $k=1$, $a=0$ and $n=6$ by Proposition \ref{rel4}.
Otherwise two branches of the fiber are exchanged, and the same Proposition gives $k=0$, $n=4$, $a=1$.
Finally we assume that $C'$ is of type $I_N$, $N\geq 4$. By the previous remark $C'$ contains at least two components fixed by $\sigma^2$, this implies that all components are preserved by $\sigma^2$ and a component which is not fixed intersects two fixed ones. Moreover, it follows from Proposition \ref{rel4} that the number of components of $C'$ in $\Fix(\sigma^2)$ is even, since it equals
$k+ n/2+2a=\alpha+n/2+2a=2+2\alpha+2a.$ Thus $N$ is a multiple of four, i.e. $C'$ is of type $I_{4M}$ for some positive integer $M$.
If $a=0$, then $n=4M-2k$. Since $n=2k+4$ by Proposition \ref{rel4}, this gives $k=M-1$ and $n=2M+2$.
We now prove that the cases $M=3,4$ do not exist if $a=0$.
If $M=4$, then $r=16$ by Proposition \ref{rel4} and \ref{lat}. The classes of irreducible components of the fiber $I_{16}$ generate a parabolic sublattice of finite index in $S(\sigma)$. This gives a contradiction since $S(\sigma)$ is a hyperbolic lattice by Proposition \ref{lat}, thus this case does not appear.
If $M=3$, then $r=13$ by Proposition \ref{rel4} and \ref{lat}. By the latter proposition and by the classification theorem of $2$-elementary lattices \cite[Theorem 4.3.1]{nikulinfactor} we have that $S(\sigma^2)\cong U\oplus E_8\oplus D_4\oplus A_1^2$.
By Theorem \ref{jac} the surface has a $\sigma^2$-invariant elliptic fibration $\pi$ with a section $E$ fixed by $\sigma^2$, a reducible fiber of type $II^*$, one of type $I_0^*$ and two of type $\tilde{A}_1$.
The section $E$ and a subset of the irreducible components of the first two reducible fibers of $\pi$
give a chain of $11$ smooth rational curves which are contained in the fiber of type $I_{12}$ of $\pi_C$.
Moreover, the remaining components of the two fibers of $\pi$ give sections of $\pi_C$.
Let $A$ and $B$ be two simple components of the fiber of type $I_0^*$ and $\tilde{A}_1$ respectively not intersecting the section $E$. Let $M$ be the sublattice of $\Pic(X)$ generated by the classes of the components of the fiber of type $I_{12}$ and by the classes of $A\cup \sigma(A)$ and $B\cup \sigma(B)$. Observe that $M$ is a sublattice of $S(\sigma)$ since $a=0$.
An easy computation shows that the intersection matrix of $M$ has determinant equal to $3(a+b-4c)+20$, where $a$ and $b$ are the self-intersections of the classes of $A\cup \sigma(A)$ and $B\cup \sigma(B)$ respectively and $c$ is the intersection between $A$ and $\sigma(B)$.
Since such determinant is obviously not zero for any $a,b,c$ ($20\not\equiv 0 \mod 3$), then $M$ is a rank $14$ sublattice of $S(\sigma)$, giving a contradiction.
If $a=M-1$, $M\geq 2$, then $\sigma$ acts as an order two symmetry on the graph of $C'$, so that $k=0$ and $n=4$.
Observe that $\sigma$ can not act as a rotation of $I_{4M}$ because otherwise $n=0$, contradicting the fact that $n\geq 4$.
\end{proof}
\begin{example}\label{wei}
We now assume that the fibration $\pi_C:X\map \IP^1$ in Theorem \ref{g1} has a $\sigma$-invariant section. Then a Weierstrass equation for the fibration is the following:
$$
y^2=x^3+a(t)x+b(t),
$$
where $a(t)=ft^8+at^4+b$, $b(t)=gt^{12}+ct^8+dt^4+e$ and
$$\sigma(x,y,t)=(x,y,it).$$
The fibers preserved by $\sigma$ are over $0, \infty$ and the action at infinity is
$$(x/t^4,y/t^6,1/t)\mapsto (x/t^4,-y/t^6,-i/t).$$
The discriminant polynomial of $\pi_C$ is:
$$\Delta(t):=4a(t)^3+27 b(t)^2=g_1t^{24}+g_2t^{20}+g_3t^{16}+g_4t^{12}+O(t^8),$$
where
$$
g_1=4f^3+27g^2,\quad g_2=12f^2a+54gc,\quad g_3=12f^2b+54gd+12fa^2+27c^2,
$$
$$
g_4=24fab+4a^3+54ge+54cd
.$$
For a generic choice of the coefficients of $a(t)$ and $b(t)$ the fibration has 24 fibers of type $I_1$ over the zeros of $\Delta(t)$, $\sigma$ fixes pointwisely the fiber over $0$ and it acts as an involution on the fiber over $\infty$ (both fibers are smooth).
If $g_1=0$ the fibration acquires a fiber of type $I_4$ by a generic choice of the parameters,
if $g_1=g_2=g_3=0$ we generically get a fiber of type $I_{12}$ and for $g_1=g_2=g_3=g_4=0$ we get a fiber of type $I_{16}$.
If $g_1=g_2=0$ one gets two possible solutions: if $f=g=0$ the fibration acquires a fiber of type
$IV^*$, otherwise it gets a fiber of type $I_8$.
\end{example}
More examples for the case $g=1$ will be given in Examples \ref{quartic} and \ref{hyp}.
\section{$\Fix(\sigma)$ contains a curve of genus $>1$}
We now assume that $\Fix(\sigma)$ contains a curve $C$ of genus $g>1$.
By Proposition \ref{rel4} we have
$$\Fix(\sigma^2)=C\cup (E_1\cup\cdots\cup E_k)\cup (F_1\cup F'_1 \cup\cdots \cup F_a\cup F'_a)\cup (G_1\cup \cdots\cup G_{n/2}),$$
$$\Fix(\sigma)=C\cup E_1\cup\cdots\cup E_k\cup\{p_1,\dots,p_n\},$$
where $E_i, F_i, G_i$ are smooth rational curves such that
$\sigma(F_i)=F'_i,\ \sigma(G_i)=G_i$ and each $G_i$ contains exactly two isolated fixed points of $\sigma$.
\begin{lemma}\label{rel4a}
$k\leq r+m-8$,\ $l-m=2a$.
\end{lemma}
\begin{proof}
The curves $C,\ E_i,\ F_i\cup F'_i,\ G_i$ are $\sigma$-invariant and are orthogonal to each other, thus their classes in $\Pic(X)$ give independent elements in $S(\sigma)$.
Then $r=\rank (S(\sigma))\geq 1+k+a+n/2$ and this gives the inequality.
An easy computation shows that
$$\mathcal X(\Fix(\sigma^2))-\mathcal X(\Fix(\sigma))=4a.$$
On the other hand, by Proposition \ref{rel4} and topological Lefschetz fixed point formula applied to $\sigma^2$:
$$\mathcal X(\Fix(\sigma))=24-2m-2l,\ \mathcal X(\Fix(\sigma^2))=24-4m.$$
Comparing these equalities, we obtain the statement.
\end{proof}
\begin{theorem}\label{propocurve}
Let $X$ be a K3 surface and $\sigma$ be a purely non-symplectic automorphism of order four on it such that $\Pic(X)=S(\sigma^2)$.
If $\Fix(\sigma)$ contains a curve of genus $g> 1$ then the invariants associated to $\sigma$ are as in Table \ref{g>1}.
\begin{table}[ht]
$$
\begin{array}{ccc|ccccccl}
m & r & l & n &k & a & g \\
\hline
7 & 1 & 7 & 0 &0 & 0 & 3 \\
\hline
6 & 4 & 6 & 2& 0& 0 & 2 \\
& 2 & 8 & 0& 0 & 1 & 3 \\
\hline
5 & 5 & 7 & 2& 0 & 1 & 2 \\
\hline
4 & 6 & 8 & 2& 0& 2 & 2 \\
\end{array}
$$
\vspace{0.3cm}
\caption{The case $g>1$}\label{g>1}
\end{table}
\end{theorem}
\begin{proof}
If $r=\rk(S(\sigma))\geq 5$ then, by Proposition \ref{ellinv}, $X$ carries a $\sigma$-invariant elliptic fibration $\pi$.
If $C\subset \Fix(\sigma)$ has genus $g>1$, then $C$ is transversal to the fibers of $\pi$, so that any fiber of $\pi$ is preserved by $\sigma$ and we are in the first two cases of Corollary \ref{cor}.
Thus, if $r\geq 5$ and $g>1$, then either $g=n=2$, $k=0$ and $a\in\{0,1,2\}$ or $g=3, n=k=0$ and $a=1$.
Thus we now assume that $r<5$. By Proposition \ref{rel4} and Lemma \ref{rel4a}
we are left for $(r,k,g,a)$ with the cases in Table \ref{g>1} and the cases $(4,0,3,3), (3,0,3,2), (4,1,3,0), (4,2,4,0)$.
In any case we can compute $S(\sigma^2)$ (up to isomorphism) by the classification theorem of $2$-elementary lattices \cite[Theorem 4.3.1]{nikulinfactor}:
\begin{eqnarray*}
\begin{array}{c|c}
(r,k,g,a)&S(\sigma^2)\\
\hline
(4,0,3,3)&U\oplus E_8\oplus D_4\\
\hline
(3,0,3,2)& U\oplus D_8\oplus A_1^{\oplus 2}\\
\hline
(4,1,3,0)&U\oplus D_4\oplus A_1^{\oplus 4};\,U(2)\oplus D_4^{\oplus 2}\\
\hline
(4,2,4,0)&U\oplus D_4^{\oplus 2};\, U\oplus D_6\oplus A_1^{\oplus 2}.
\end{array}
\end{eqnarray*}
In the cases when $S(\sigma^2)\cong U\oplus R$, Theorem \ref{jac} implies that $g\leq 2$, giving a contradiction.
We are left with the case $(4,1,3,0)$ and $S(\sigma^2)\cong U(2)\oplus D_4^{\oplus 2}$. By \cite[Lemma 2.1, 2.2]{kondoell} $X$ carries a $\sigma^2$- invariant elliptic fibration $\pi$ with no sections and two reducible fibers of type $I_0^*$:
$2R+R_1+R_2+R_3+R_4$ and $2R'+R_1'+R_2'+R_3'+R_4'$.
Since $\sigma^2$ fixes the curve $C$ of genus $3$, then any fiber of $\pi$ is preserved by $\sigma^2$.
Moreover, since $\sigma^2=\id$ on $\Pic(X)$, then each smooth rational curve is $\sigma^2$-invariant. This implies that $\sigma^2$ fixes $R$ and $R'$ since each of them contains $4$ fixed points.
Since $k=1$, one of these two curves is also fixed by $\sigma$, we can assume it to be $R$.
The curve $C$ meets the generic fiber in $4$ points and it intersects all the $R_i$'s and the $R_i'$'s.
This implies that the fibration is not $\sigma$-invariant, since otherwise the generic fiber should contain only $2$ fixed points for $\sigma$.
Thus we can assume that $\sigma(R_1)\not=R_1$ and we have that $\beta:=R_1\cdot\sigma(R_1)\geq 2$. The sublattice of $S(\sigma)$ generated by the classes of $C,R$ and $R_1\cup \sigma(R_1)$ has the following intersection matrix
\begin{eqnarray*}
M:=\left(\begin{array}{ccc}
4&0&2\\
0&-2&2\\
2&2&-4+2\beta
\end{array}
\right).
\end{eqnarray*}
Since $\det(M)=-16\beta+24<0$, we get a contradiction with the fact that $S(\sigma)$ has hyperbolic signature, thus this case does not appear.
\end{proof}
\begin{example}[\textbf{plane quartics}]\label{quartic} This construction is due to Kond\=o \cite{K1}. Let $C$ be a smooth plane quartic, defined as the zero set of a homogeneous polynomial $f_4\in \IC[x_0,x_1,x_2]$ of degree four. The fourfold cover of $\IP^2$ branched along $C$ is a K3 surface with equation $$t^4=f_4(x_0,x_1,x_2).$$
The covering automorphism
$$\sigma(x_0,x_1,x_2,t)= (x_0,x_1,x_2, it)$$ is a non-symplectic automorphism of order four whose fixed locus is the plane section $t=0$, which is isomorphic to the curve $C$. Thus we have $a=n=k=0$ and $g=3$.
In case $C$ has ordinary double points (i.e. nodes) or cusps,
then the fourfold cover $X$ of $\IP^2$ branched along $C$ has rational double points of type $A_3$ and $E_6$ at the inverse images of a node and of a cusp of $C$ respectively.
The minimal resolution $\tilde X$ of $X$ is a K3 surface and the covering automorphism of $X$ lifts to a non-symplectic automorphism of $\tilde X$.
If $C$ has a node, then the central component of the exceptional divisor of type $A_3$ is fixed by $\sigma^2$ and contains two fixed points for $\sigma$.
If $C$ has a cusp, then the exceptional curve is of type $E_6$, $\sigma^2$ fixes its two simple components and $\sigma$ exchanges them.
Thus, if $C$ is irreducible with $x$ nodes and $y$ cusps, then the invariants of $\sigma$ are $g=3-x-y$, $a=y$, $n=2x$ and $k=0$. Taking $(x,y)=(0,0), (1,0)$ and $(0,1)$ we obtain examples for the first, second and fourth case in Table \ref{g>1}.
If $(x,y)=(2,0), (1,1)$ or $(0,2)$ we obtain examples with $g=1$ corresponding to the cases in Table \ref{g=1} with a fiber $C'$ of type $I_4, I_8$ (with $(k,n,a)=(0,4,1)$) and $I_{12}$ respectively.
If $C$ is the union of a cubic and a line we obtain the case in Table \ref{g=1} with a fiber $C'$ of type $IV^*$ and $(k,n,a)=(1,6,0)$.
In \cite[Proposition 1.7]{a} the lattice $S(\sigma^2)$ has been computed in case $C$ is irreducible and generic with $x$ nodes and $y$ cusps.
\end{example}
\begin{example}[\textbf{hyperelliptic genus three curves}]\label{hyp} Let $\mathbb F_4$ be a Hirzebruch surface and $e,f \in \Pic(\mathbb F_4)$ be the classes of the rational curve $E$ with $E^2=-4$ and the class of a fiber respectively.
A smooth curve $C$ with class $2e+8f$ is a hyperelliptic genus three curve.
Let $Y$ be the double cover of $\mathbb F_4$ branched along $C$ and $X$ be the double cover of $Y$ branched along $C\cup R_1\cup R_2$, where $R_1\cup R_2$ is the inverse image of the curve $E$.
The surface $X$ is a K3 surface (see \cite[\S 3]{a}) with a non-symplectic automorphism $\sigma$ of order $4$ whose fixed locus is the inverse image of the curve $C$ and exchanges the curves $R_i$. Observe that $\sigma^2$ fixes $C\cup R_1\cup R_2$.
An alternative construction which associates the K3 surface $X$ to the curve $C$ has been given by Kond\=o \cite{K2}. In this case we have $g=3$, $n=k=0$ and $a=1$.
As in the previous example, if $C$ has at most nodes and cusps, then the minimal resolution $\tilde X$ of $X$ is again a K3 surface with a non-symplectic automorphism of order $4$. If $C$ is irreducible with $x$ nodes and $y$ cusps, then the invariants of $\sigma$ are $g=3-x-y$, $a=y+1$, $n=2x$, $k=0$.
Taking $C$ with a node and a cusp we obtain examples for the third and the last case respectively in Table \ref{g>1}.
If $(x,y)=(2,0), (1,1)$ or $(0,2)$ we obtain examples with $g=1$ corresponding to the cases in Table \ref{g=1} with a fiber $C'$ of type $I_8$ (with $(k,n,a)=(0,4,1)$), $I_{12}$ and $I_{16}$ respectively.
In \cite[\S4.9]{K2} the lattice $S(\sigma^2)$ has been computed in case $C$ is irreducible and generic with $x$ nodes and $y$ cusps.
\end{example}
\section{$\Fix(\sigma^2)$ only contains rational curves}
In this section we assume that the curves fixed by $\sigma^2$ are rational and that at least one of them is fixed by $\sigma$.
\begin{theorem}\label{rational}
Let $X$ be a K3 surface and $\sigma$ be a purely non-symplectic automorphism of order four on it.
If $\Fix(\sigma)$ contains a smooth rational curve and all curves fixed by $\sigma^2$ are rational,
then the invariants associated to $\sigma$ are as in Table \ref{g=0} (where $k+1$ is the number of curves in $\Fix(\sigma)$).
\end{theorem}
\begin{table}[ht]
$$
\begin{array}{ccc|ccccccl}
m & r & l & n &k+1 & a \\
\hline
4 & 10 & 4 & 6 & 1 & 0 \\
\hline
3 & 13 & 3 & 8 &2 & 0 \\
& 11 & 5 & 6& 1 & 1 \\
\hline
2 & 16 & 2 & 10 &3 & 0 \\
& 14 & 4 & 8 &2 & 1 \\
& 12 & 6 & 6& 1 & 2 \\
\hline
1 & 19 & 1 & 12 & 4 & 0 \\
& 13 & 7 & 6 & 1 & 3 \\
\end{array}
$$
\vspace{0.2cm}
\caption{The case $g=0$}\label{g=0}
\end{table}
\begin{proof}
By Proposition \ref{rel4} and Lemma \ref{rel4a}
we have the cases in Table \ref{g=0} and the cases $(r,k,a)=(17, 2,1)$ and $(15, 1,2)$.
In both cases the surface $X$ has a $\sigma$-invariant jacobian elliptic fibration by \cite[Lemma (1.5), (2)]{MO}. By Example \ref{vinberg} these two cases do not appear since $a\in \{0,3\}$.
\end{proof}
\begin{example}[\textbf{Vinberg's K3 surface}]\label{vinberg}
If $m=1$, then $S(\sigma^2)=\Pic(X)$ has maximal rank and $X$ is isomorphic to the unique K3 surface with
$$T(X)=T(\sigma^2)\cong \left( \begin{array}{cc} 2 & 0 \\
0 & 2\end{array}\right),$$
since it can be easily proved that, up to isometry, this is the only rank two even positive definite, $2$-elementary lattice and it has moreover an order four isometry without fixed vectors.
The automorphism group of this K3 surface is known to be infinite and has been computed by Vinberg in \cite[\S 2.4]{V}.
In particular, it is known that $X$ is birationally isomorphic to the following quartics in $\IP^3$:
$$x_0^4=-x_1x_2x_3(x_1+x_2+x_3),$$
$$x_0^4=x_2^2x_3^2+x_3^2x_1^2+x_1^2x_2^2-2x_1x_2x_3(x_1+x_2+x_3),$$
which are degree four covers of $\IP^2$ branched along the union of four lines in general position and an irreducible quartic with three cusps respectively.
The two covering automorphisms $x_0\mapsto ix_0$ induce non-conjugate order four non-symplectic automorphisms on $X$:
the first one has $a=0$, $n=12$ and fixes $4$ smooth rational curves (the proper transforms of the lines), the second one has $a=3$ (coming from the cusps), $n=6$ and fixes one smooth rational curve (the proper transform of the quartic). These give the last two examples in Table \ref{g=0}.
All elliptic fibrations $\pi:X\map \IP^1$ are known to be jacobian by \cite[Theorem 2.3]{Ke} and have been classified by Nishiyama in \cite[Theorem 3.1]{Nishiyama}. We recall the classification here, where $R$ is the lattice generated by components of reducible fibers not intersecting the zero section and $MW$ is the Mordell-Weil group of $\pi$.
\begin{table}[ht]
$$\begin{array}{cccc}
\mbox{No.} & R & MW & a\\
\hline
\vspace{0.1cm}
1) & E_8^{\oplus 2}\oplus A_1^{\oplus 2} & 0 & 0,4\\
2)& E_8\oplus D_{10} & 0 & 0\\
3)& D_{16}\oplus A_1^{\oplus 2}& \IZ_2 & 0,3 \\
4)& E_7^{\oplus 2}\oplus D_4 & \IZ_2 & 0,3,4 \\
5)& E_7\oplus D_{10}\oplus A_1 & \IZ_2 & --\\
6)& A_{17}\oplus A_1 & \IZ_3 & 0\\
7)& D_{18} & 0 & 0\\
8)& D_{12}\oplus D_6 & \IZ_2 & 0,3\\
9)& D_8^{\oplus 2}\oplus A_1^{\oplus 2}& (\IZ_2)^2 & 0,3,4 \\
10)& A_{15}\oplus A_3 & \IZ_4 & 0,3,4\\
11)& E_6\oplus A_{11} & \IZ\oplus \IZ_3& 0,3\\
12)& D_6^{\oplus 3} & (\IZ_2)^2& 3\\
13)& A_9^{\oplus 2} & \IZ_5& 0
\end{array}$$
\vspace{0.2cm}
\caption{Elliptic fibrations of Vinberg's K3 surface}
\end{table}
In each case we will compute the possible values taken by $a$.
We will apply the height formula \cite[Theorem 8.6]{shioda} and the notation therein. Moreover, we will apply Theorem \ref{jac} to determine the number of fixed curves of $\sigma^2$ inside the reducible fibers.\\
1)
All curves fixed by $\sigma^2$ are contained in the reducible fibers.
The automorphism $\sigma$ either preserves the $E_8$ fibers or it exchanges them, giving
$a=0$ or $a=4$ respectively.\\
2)
The reducible fibers contain exactly $8$ curves fixed by $\sigma^2$ and the two remaining fixed curves of $\sigma^2$ are transversal to the fibers, one of them is the unique section, the other is a $3$-section. This implies that both are $\sigma$-invariant and $\sigma$ preserves all components of $D_{10}$, giving $a=0$.\\
3)
The reducible fibers contain exactly $7$ curves fixed by $\sigma^2$. The remaining fixed curves of $\sigma^2$ are transversal to the fibers and give two sections $s_0, s_1$ (assume $s_0$ to be the zero section) and a $2$-section.
The translation by the order two element in the Mordell-Weil group gives rise to a symplectic automorphism of order two. Since such automorphism has $8$ fixed points by \cite{Nikulin1}, then it acts on the fiber $D_{16}$ as a reflection with respect to the central component, i.e. $s_0$ and $s_1$ intersect the fiber in simple components not meeting the same component.
This implies that $\sigma$ acts on the components of $D_{16}$ either as the identity or as a reflection with respect to the central component, giving $a=0$ and $a=3$ respectively.\\
4)
As before, the reducible fibers only contain $7$ curves fixed by $\sigma^2$ and the fibration has two sections $s_0, s_1$ and a $2$-section fixed by $\sigma^2$.
If $\sigma$ preserves each fiber, then either $s_0, s_1$ are fixed by $\sigma$ or they are exchanged giving $a=0$ and $a=3$ respectively by Corollary \ref{cor}.
Otherwise, if $\sigma$ has order two on the basis of $\pi$, then it exchanges the two fibers of type $E_7$, so that either $a=3$ or $a=4$. \\
5) Since the fibration has three reducible fibers of distinct types, then $\sigma$ acts as the identity on $\IP^1$. This is not possible by Proposition \ref{ell}, thus this fibration can not be $\sigma$-invariant.\\
6) The fiber of type $A_{17}$ contains $9$ curves fixed by $\sigma^2$ and is clearly $\sigma$-invariant.
Since $18\not\equiv 0\ (\!\!\!\mod 4)$, then $a=0$.\\
7) The fiber of type $D_{18}$ contains $8$ curves fixed by $\sigma^2$, the remaining two curves fixed by $\sigma^2$ give a section and a $3$-section. Thus $a=0$.\\
8) The reducible fibers contain $7$ curves fixed by $\sigma^2$, the remaining fixed curves of $\sigma^2$ give two sections $s_0,s_1$.
By the height formula, $s_0$ and $s_1$ meet simple components intersecting distinct components in the $D_{12}$ fiber and intersecting the same component in the $D_6$ fiber. Thus either $a=0$ or $a=3$.\\
9) The reducible fibers contain $6$ curves fixed by $\sigma^2$ and the fibration has four sections $s_0,s_1, s_2, s_3$ fixed by $\sigma^2$.
Observe that the elliptic fibration has two fibers of type $D_8$, two of type $I_2$ and no other singular fibers (since the Euler characteristic of $X$ is $24$).
By the height formula, we can assume that $s_1$ intersects $\Theta_1^1, \Theta_2^2, \Theta_1^3, \Theta_1^4$, $s_2$ intersects $\Theta_2^1, \Theta_1^2, \Theta_1^3, \Theta_1^4$ and $s_3$ intersects $\Theta_3^1, \Theta_3^2, \Theta_0^3, \Theta_0^4$.
Observe that $\sigma$ either preserves the fibers $D_8$ and exchanges the fibers $I_2$, or it exchanges both pairs of reducible fibers, or it exchanges the fibers $D_8$ and it preserves the fibers $I_2$.
In the first case $a=0,3$ or $4$, according to the action of $\sigma$ on the sections $s_i$'s.
In the second case $a=4$ (observe that the $4$ fixed points by $\sigma$ are contained in two smooth fibers).
The last case does not appear since otherwise $\sigma$ should preserve all components of the fibers $I_2$, which gives a contradiction since the components intersecting $s_1, s_2$ contain no fixed points for $\sigma$.\\
10) In this case all fixed curves by $\sigma^2$ are contained in the two reducible fibers, thus $\sigma$ has order $4$ on $\IP^1$. Observe that the fibration has four sections $s_i, i=0,1,2,3$, preserved by $\sigma^2$, so that each of them intersects the curves fixed by $\sigma^2$.
This remark and the height formula imply that we can assume that $s_1$ intersects $\Theta_8^1, \Theta_0^2$, $s_2$ intersects $\Theta_4^1, \Theta_2^2$ and $s_3$ intersects $\Theta_{12}^1, \Theta_2^2$.
This implies that $\sigma$ either preserves all components of the reducible fibers ($a=0$), or it acts
on the sections as a permutation $(s_2 s_3)$ or $(s_0 s_1)$ ($a=3$), or as a permutation of type $(s_0 s_1)(s_2 s_3)$ ($a=4$).\\
11) In this case all fixed curves by $\sigma^2$ are contained in the two reducible fibers, thus $\sigma$ has order $4$ on $\IP^1$.
By the height formula we can assume that $s_1$ intersects $\Theta_1^1, \Theta_4^2$ and $s_2$ intersects $\Theta_2^1, \Theta_8^2$.
This implies that $\sigma$ either preserves all components of reducible fibers ($a=0$), or it acts
on the sections as a transposition ($a=3$).\\
12) The reducible fibers contain $6$ curves fixed by $\sigma^2$, the remaining fixed curves give four sections $s_i, i=0,1,2,3$. Observe that $\sigma$ has order two on $\IP^1$, it exchanges two fibers of type $D_6$ and it preserves the third one.
By the height formula we can assume that $s_1$ intersects $\Theta_1^1, \Theta_3^2, \Theta_3^3$, $s_2$ intersects $\Theta_3^1, \Theta_3^2, \Theta_1^3$ and $s_3$ intersects $\Theta_3^1, \Theta_1^2, \Theta_2^3$. This implies that $\sigma$ acts on the sections as a transposition, so that $a=3$.\\
13) All fixed curves by $\sigma^2$ are contained in the two reducible fibers and
$\sigma$ preserves the two fibers of type $I_{10}$ (since otherwise $a=5$, which is not possible), so that $a=0$.
\end{example}
\begin{example}
Let $\tilde X$ and $C$ as in Example \ref{quartic}.
Taking $C$ with 3 nodes or with 2 nodes and a cusp we obtain examples for the first and third case in Table \ref{g=0} respectively. If $C$ is the union of two conics (or the union of a line and a nodal cubic), the union of a conic and two lines or the union of a line and a cuspidal cubic we obtain examples for the second, fourth and fifth case in the table respectively.
Finally, as mentioned in the previous example, if $C$ is the union of four lines or has 3 cusps, we obtain the last two cases in the table.
Similarly, if $\tilde X$ and $C$ are as in Example \ref{hyp} and $C$ has 3 nodes, 2 nodes and a cusp or 1 node and two cusps, we obtain examples for the third, sixth and the last case in Table \ref{g=0} respectively.
\end{example}
\section{The case $l=0$}\label{trivpic}
In this section we will assume that $l=0$, i.e. $\sigma$ acts as the identity on $S(\sigma^2)$.
\begin{pro}\label{ell0}
Let $\sigma$ be a non-symplectic automorphism on a K3 surface $X$ such that $l=0$.
Then $\Fix(\sigma)$ is the disjoint union of smooth rational curves and points, $r=2\,(\!\!\!\!\mod 4)$ and $a=0$.
\end{pro}
\begin{proof}
Assume that $\sigma$ fixes a curve of genus $g\geq 1$, then we get a contradiction by Theorem \ref{g1} and by Lemma \ref{rel4a} (since it says that $-m=2a$).
Observe that $r=2\,(\!\!\!\!\mod 4)$ by Proposition \ref{rel4}.
If $a$ is not zero, then there are two rational curves $F_1, F'_1$ fixed by $\sigma^2$ such that $\sigma(F_1)=F'_1$.
If $f_1,f'_1$ are their classes in $\Pic(X)$, then $\sigma^*(f_1-f'_1)=f'_1-f_1$ and $f_1-f'_1$ is not zero, contradicting $l=0$.
\end{proof}
\begin{lemma}\label{iso}
Let $L$ be a lattice which is the direct sum of lattices isomorphic to $U\oplus U$, $U\oplus U(2)$, $E_8$ or $D_{4k}$, $k\geq 1$. Then $L$ has an isometry $\tau$ with $\tau^2=-\id$ acting trivially on $A_L=L^{\vee}/L$.
\end{lemma}
\begin{proof}
It is known that the Weyl group of a lattice isometric to either $E_8$ or $D_{4k}$, $k\geq 1$, contains an isometry $\tau$ with $\tau^2=-\id$ acting trivially on $A_L$, \cite{cc}.
An isometry $\tau$ of $U\oplus U$ as in the statement can be defined as follows:
$$\tau:\ e_1\mapsto e_2,\ e_2\mapsto -e_1,\ e_3\mapsto e_4,\ e_4\mapsto -e_3,$$
where $e_1,e_2$ and $e_3,e_4$ are the natural generators of the first and the second copy of $U$.
Such an action can be defined similarly on $U\oplus U(2)$.
\end{proof}
By Proposition \ref{ell0} the fixed locus of $\sigma$ and of its square are as follows:
$$\Fix(\sigma^2)=C\cup (E_1\cup\cdots\cup E_k)\cup (G_1\cup \cdots\cup G_{n_1/2}),$$
$$\Fix(\sigma)=E_1\cup\cdots\cup E_k\cup\{p_1,\dots,p_n\},$$
where $C$ is a curve of genus $g$, $E_i, G_i$ are smooth rational curves such that
$\sigma(G_i)=G_i$ and each $G_i$ contains exactly two isolated fixed points of $\sigma$.
We will denote by $n_2=n-n_1$ the number of isolated fixed points of $\sigma$ contained in $C$.
\begin{theorem}
Let $\sigma$ be a non-symplectic automorphism on a K3 surface $X$ such that $S(\sigma^2)=S(\sigma)=\Pic(X)$. Then the invariants of the fixed locus of $\sigma$, the lattices $S(\sigma^2)$ and $T(\sigma^2)$ (up to isomorphism) appear in the following table.
\begin{table}[ht]
$$
\begin{array}{cc|cccc|ll}
m & r & n_1 &n_2 & k & g & S(\sigma^2) & T(\sigma^2) \\
\hline
10 & 2 & 2 & 2 & 0 & 10 & U & U\oplus U\oplus E_8^{\oplus 2}\\
& 2 & 0 & 4 & 0 & 9 & U(2) & U\oplus U(2)\oplus E_8^{\oplus 2}\\
\hline
8 & 6 & 2 & 4 & 1 & 7 & U\oplus D_4 & U \oplus U\oplus E_8\oplus D_4\\
& 6 & 0 & 6 & 1 & 6 & U(2)\oplus D_4 & U\oplus U(2)\oplus E_8\oplus D_4\\
\hline
6 & 10 & 6 & 2& 2& 6 & U\oplus E_8 & U \oplus U \oplus E_8\\
& 10 & 4 & 4 & 2 & 5 & U(2)\oplus E_8 & U\oplus U(2)\oplus E_8\\
& 10 & 2 & 6 & 2& 4 & U\oplus D_4^{\oplus 2} & U\oplus U\oplus D_4^{\oplus 2}\\
& 10 & 0 & 8 & 2 & 3 & U(2)\oplus D_4^{\oplus 2} & U\oplus U(2)\oplus D_4^{\oplus 2}\\
\hline
4 & 14 & 6 & 4 & 3 & 3 & U\oplus E_8\oplus D_4 & U \oplus U \oplus D_4\\
& 14 & 4 & 6 & 3 & 2 & U(2)\oplus E_8\oplus D_4 & U\oplus U(2)\oplus D_4\\
\hline
2 & 18 & 10 & 2 & 4 & 2 & U\oplus E_8^{\oplus 2} & U\oplus U\\
& 18 & 8 & 4 & 4 & 1 & U(2)\oplus E_8^{\oplus 2} & U\oplus U(2)\\
\end{array}
$$
\ \\
\ \\
\caption{The case $l=0$}\label{l=0}
\end{table}
\end{theorem}
\begin{proof}
By Proposition \ref{rel4} we have that
$$\mathcal X(\Fix(\sigma^2))-\mathcal X(\Fix(\sigma))=2-2g-n_2=-2m.$$
Moreover, since $\sigma$ acts on $C$ as an order two automorphism with $n_2$ fixed points, then $2g-2-n_2\geq -4$, i.e. $g\geq \frac{n_2}{2}-1$ by Riemann-Hurwitz formula.
These remarks, together with Proposition \ref{rel4} and the classification theorem of $2$-elementary even lattices \cite[Theorem 4.3.1]{nikulinfactor} give a list of possible cases, the ones appearing in the table and some more with $S(\sigma^2)$ isomorphic to one of the following lattices:
$$U\oplus A_1^{\oplus 4},\ U\oplus D_6\oplus A_1^{\oplus 2},\ U\oplus D_4\oplus A_1^{\oplus 4},\ U\oplus E_8\oplus A_1^{\oplus 4},\ U\oplus E_8\oplus E_7\oplus A_1,$$
$$U\oplus E_7\oplus A_1,\ (2)\oplus A_1.$$
In the first five cases $X$ has a $\sigma$-invariant jacobian elliptic fibration $\pi:X\map \IP^1$ with more than two reducible fibers by Theorem \ref{jac}. Since $\sigma$ fixes $>2$ points in the basis of $\pi$, then it preserves each fiber of $\pi$.
This gives a contradiction since $\sigma$ should have two fixed points in each fiber while the fibration has a unique section fixed by $\sigma$.
We now show that the case $S(\sigma^2)\cong(2)\oplus A_1$ does not appear. Let $e,f$ be the generators of $S(\sigma^2)$ with $e^2=2, f^2=-2, e\cdot f=0$. The class $e$ is nef and the associated morphism is a degree two map $\pi:X\map \IP^2$ which is the minimal resolution of the double cover branched along an irreducible plane sextic $S$ with a node at the image of the curve with class $\pm f$.
Since $\sigma^*(e)=e$, then $\sigma$ induces a projectivity $\bar \sigma$ of $\IP^2$ with $\bar\sigma^2=\id$ (since it fixes $\pi(C)=S$).
This implies that, up to a choice of coordinates, a birational model of $X$ and $\sigma$ are given as follows:
$$X:\ w^2=f_6(x_0,x_1,x_2),\quad \sigma(x_0,x_1,x_2,w)=(-x_0,x_1,x_2, iw),$$
where $f_6$ is a homogeneous degree six polynomial with $\sigma(f_6)=-f_6$ and singular at one point.
Observe that such polynomial $f_6$ contains $x_0=0$ as a component, giving a contradiction.
If $S(\sigma)\cong U\oplus E_7\oplus A_1\cong (2)\oplus A_1\oplus E_8$,
then $X$ is the minimal resolution of the double cover of $\IP^2$ branched along an irreducible sextic with a node and a triple point of type $E_8$. We can exclude this case by an argument similar to the previous one.
If $T(\sigma^2)$ is any lattice appearing in Table \ref{l=0}, then it carries an isometry $\tau$ with $\tau^2=-\id$ and acting trivially on the discriminant group by Lemma \ref{iso}.
It follows that the isometries $\id_{S(\sigma^2)}$ and $\tau$ glue to give an order four isometry $\rho$ of $L_{K3}$.
By the Torelli-type Theorem \cite[Theorem 3.10]{namikawa} there exists a K3 surface $X$ with a non-symplectic automorphism $\sigma$ of order four such that $\sigma^*=\rho$ up to conjugacy.
Thus all cases in the table do exist.
\end{proof}
\begin{remark}
In all cases of Table \ref{l=0} the lattices $S(\sigma^2)$ and $T(\sigma^2)$ are $2$-elementary even lattices with $x^2\in\IZ$ for all $x\in L^{\vee}$.
A lattice theoretical proof of this fact and an alternative proof of Theorem \ref{l=0} are given by Taki in\cite[Proposition 2.4]{Taki}.
\end{remark}
\begin{example}\label{hirz}
Consider the elliptic fibration $\pi:X\map \IP^1$ in Weierstrass form given by
$$y^2=x^3+a(t)x+b(t),$$
where $a$ is an even polynomial and $b$ is an odd polynomial in $t$.
Observe that it carries the order four automorphism
$$(x,y,t)\mapsto (-x, iy,-t).$$
For generic coefficients $X$ is a K3 surface, the fixed locus of $\sigma^2$ is the union of the curve of genus $10$ defined by $y=0$ and the section $x=z=0$ and $\sigma$ fixes four points on them (in the fibers over $t=0, \infty$).
It follows by propositions \ref{rel4}, \ref{lat} and \cite[Theorem 4.3.1]{nikulinfactor} that $S(\sigma^2)\cong U$, $r=2$, $l=0$ and $m=10$.
A geometric construction of this family of K3 surfaces can be given as follows.
Let $Y$ be the Hirzebruch surface $\mathbb F_4$ and $e,f\in \Pic(Y)$ be the classes of the $(-4)$-curve and of a fiber respectively.
Observe that a section of $-2K_Y=4e+12f$ is the disjoint union of the $(-4)$-curve $E$ and a curve $C$ with class $3e+12f$ ($e$ is in the base locus of $4e+12f$). The generic such $C$ is smooth
and the double cover $p:X\map Y$ branched along $C\cup E$ is a K3 surface. We denote by $\tilde C$ and $\tilde E$ the pull-backs of $C$ and $E$ by $p$, observe that $g(\tilde C)=10$ and $g(\tilde E)=0$.
We will consider the affine coordinates $t=u_1/u_2$ and $x=v_1/v_2$, where $u_1, u_2$ give a basis of $H^0(Y,f)$ and $v_1\in H^0(Y,e+4f)$, $v_2\in H^0(Y,e)$ are non-zero sections.
Let $\iota\in {\rm Aut}(Y)$ be the involution on $Y$ given by $(t, x)\mapsto (-t,-x)$ and let $f=0$ be the equation of $\tilde C$ in local coordinates.
If $f(-t,-x)=-f(t,x)$, then $f(t,x)=x^3+a(t)x+b(t)$ where $a$ is an even, degree $8$ and $b$ is an odd, degree $11$ polynomial in $t$.
In this case $\iota$ lifts to an order $4$ automorphism $\sigma$ on $X$, in local coordinates:
$$X:\ y^2=f(t,x),\qquad \sigma(t,x,y)=(-t,-x,iy).$$
The ruling of $Y$ induces a jacobian elliptic fibration on $X$ having $\tilde C$ as a trisection and $\tilde E$ as a section, such that its Weierstrass equation and the action of $\sigma$ on it are clearly the same as the ones given for the fibration $\pi$ at the beginning of this example.
\end{example}
\begin{example}\label{quadric}
Consider the involution $\iota: ((x_0,y_0), (x_1,y_1))\mapsto ((x_0,-x_1), (y_0,-y_1))$ of $\IP^1\times \IP^1$ and let $f$ be a bihomogeneous polynomial of degree $(4,4)$ such that $\iota(f)=-f$.
If $C=\{f=0\}$ is a smooth curve, then the double cover of $\IP^1\times \IP^1$ branched along $C$
$$X:\ w^2=f(x_0,x_1,y_0,y_1)$$ is a K3 surface and carries the order four automorphism:
$$\sigma:\ (((x_0,x_1), (y_0,y_1)), w)\mapsto (((x_0,-x_1), (y_0,-y_1)), iw).$$
The fixed locus of $\sigma^2$ is the genus $9$ curve defined by $w=0$ and $\sigma$ fixes $4$ points on it.
It follows by propositions \ref{rel4}, \ref{lat} and \cite[Theorem 4.3.1]{nikulinfactor} that $S(\sigma^2)\cong U(2)$ (equals the pull-back of $\Pic(\IP^1\times \IP^1)$), $r=2$, $l=0$ and $m=10$.
\end{example}
\section{$\Fix(\sigma)$ only contains isolated points}
Let $\sigma$ be a purely non-symplectic automorphism of order four on a K3 surface $X$ having only isolated fixed points. It follows from Proposition \ref{rel4}
that $\Fix(\sigma)$ contains exactly four points $p_1,\dots,p_4$. Moreover, the fixed locus of $\sigma^2$ is as follows:
$$\Fix(\sigma^2)=C\cup (F_1\cup F'_1)\cup \cdots\cup (F_a\cup F'_a)\cup G_1\cup\cdots\cup G_{n_1/2},$$
where each $G_i$ is a smooth rational curve which contains $2$ fixed points of $\sigma$ and $C$ is a smooth genus $g$ curve which contains the remaining $4-n_1$ fixed points.
\begin{theorem}
Let $\sigma$ be a purely non-symplectic automorphism of order $4$ having only isolated fixed points on a K3 surface $X$. Then $\sigma$ fixes exactly $4$ points. Moreover, if $\Pic(X)=S(\sigma^2)$ and $l>0$, then the invariants of $\Fix(\sigma^2)$ and the lattice $S(\sigma^2)$ appear in Table \ref{alpha=0}.
\begin{table}[ht]
$$
\begin{array}{ccc|cccc|c}
m & r &l & n_1 &n_2 & g &a & S(\sigma^2) \\
\hline
9 & 3&1 & 2 & 2& 8 & 0 &U\oplus A_1^{\oplus 2}\\
& & & 0 & 4 & 7 & &U(2)\oplus A_1^{\oplus 2} \\
\hline
8 & & & 2 & 2 & 6 & 0 &U\oplus A_1^{\oplus 4} \mbox{ o } \boxed{U(2)\oplus D_4} \\
& & & 0 & 4 & 5 & & U(2)\oplus A_1^{\oplus 4}\\
\hline
7 & & & 2 & 2 & 4 & &U\oplus A_1^{\oplus 6}\\
& & & 0 & 4 & 3 & &U(2)\oplus A_1^{\oplus 6}\\
\hline
6 & 6 & 4 & 4 & 0& 3 & 0 & \boxed{U(2)\oplus D_4^{\oplus 2}} \\
& & & 2 & 2& 2 & &U\oplus A_1^{\oplus 8} \\
& & & 0 & 4& 1 & &U(2)\oplus A_1^{\oplus 8} \\
& & & 0 & 4& 3 & 1&\boxed{ U\oplus D_4\oplus A_1^{\oplus 4}} \mbox{ o } U(2)\oplus D_4^{\oplus 2} \\
& & & 2 & 2& 4 & &U\oplus D_4^{\oplus 2} \mbox{ o } \boxed{ U\oplus D_6\oplus A_1^{\oplus 2}}\\
\hline
5 & 7 & 5 & 4 & 0 & 1 & 0& \boxed{U(2)\oplus D_4^{\oplus 2}\oplus A_1^{\oplus 2} } \\
& & & 2 & 2 & 0 & &U\oplus A_1^{\oplus 10} \\
& & & 0 & 4 & 1 & 1 & U(2)\oplus D_4^{\oplus 2}\oplus A_1^{\oplus 2} \\
& & & 2 & 2 & 2 & & U\oplus D_4^{\oplus 2}\oplus A_1^{\oplus 2} \\
& & & 0 & 4 & 3 & 2& \boxed{U\oplus E_7\oplus A_1^{\oplus 3}} \\
& & & 2 & 2 & 4 & &\boxed{U\oplus E_8\oplus A_1^{\oplus 2}}\\
\hline
4 & 8 & 6 & 2 & 2 & 0 & 1& U\oplus D_4^{\oplus 2}\oplus A_1^{\oplus 4} \\
& & & 4 & 0 & 1 & & \boxed{U(2)\oplus D_6^{\oplus 2}}\mbox{ o } \boxed{U\oplus D_4^{\oplus 3}}\\
& & & 0 & 4 & 1 & 2&U(2)\oplus D_6^{\oplus 2} \mbox{ o } \boxed{U\oplus D_4^{\oplus 3}}\\
& & & 2 & 2 & 2 & & \boxed{U\oplus E_8\oplus A_1^{\oplus 4}} \mbox{ o } \boxed{U(2)\oplus D_4\oplus E_8}\\
& & &0 & 4 & 3 & 3 & \boxed{U\oplus E_8\oplus D_4 }\\
\hline
3 & 9 & 7 & 2 & 2 & 0 & 2& U\oplus D_6^{\oplus 2}\oplus A_1^{\oplus 2} \\
& & & 4 & 0 & 1 & & \boxed{U(2)\oplus E_7^{\oplus 2}}\\
& & & 0 & 4 & 1 & 3& U(2)\oplus E_7^{\oplus 2} \\
& & & 2 & 2 & 2 & & U\oplus E_7^{\oplus 2} \\
\hline
2 & 10 & 8 & 2 & 2 & 0 & 3 &U\oplus E_7^{\oplus 2}\oplus A_1^{\oplus 2} \mbox{ o } U\oplus D_8^{\oplus 2} \\
& & & 4 & 0 & 1 & & \boxed{U\oplus E_8\oplus E_7\oplus A_1} \mbox{ o } U(2)\oplus E_8^{\oplus 2} \\
& & & 0 & 4 & 1 & 4& \boxed{U\oplus E_8\oplus E_7\oplus A_1} \mbox{ o } U(2)\oplus E_8^{\oplus 2} \\
\hline
1 & 11 & 9 & 2 & 2 & 0 & 4&U\oplus E_8^{\oplus 2}\oplus A_1^{\oplus 2} \\
\end{array}
$$
\ \\
\ \\
\caption{The case $\alpha=k=0,\ l>0$}\label{alpha=0}
\end{table}
\end{theorem}
\begin{proof} Since $\alpha=k=0$, it follows from Proposition \ref{rel4} that $n=4$, $r=l+2$ and $m=10-l$. The fixed locus of $\sigma^2$ contains $2a+n_1/2$ smooth rational curves and a curve of genus $g$. Thus by Proposition \ref{lat} we obtain $r+l=11-g+2a+n_1/2$.
Observe that the case $l=1, n_1=4$ does not exist since in this case, by Proposition \ref{lat} and \cite[Theorem 4.3.1]{nikulinfactor} (see Figure 1 in \cite{ast}), a curve fixed by $\sigma^2$ has genus $\leq 8$.
If $S(\sigma^2)\cong U\oplus R$, where $R$ is a direct sum of root lattices, and $g>4$, then by Theorem \ref{jac} the surface $X$ carries a $\sigma$-invariant jacobian elliptic fibration $\pi$ with reducible fibers of type $R$.
The automorphism $\sigma$ acts as an involution on $\IP^1$ and preserves two fibers $F,F'$ of $
\pi$. The curve $C\subset \Fix(\sigma^2)$ of genus $g$ intersects each fiber at $3$ points and $C\cap F$, $C\cap F'$ are $\sigma$-invariant. This implies that $C$ contains at least two fixed points, i.e. $n_2\geq 2$.
Observe that $\pi$ has a section fixed by $\sigma^2$ and $\sigma$-invariant, thus $n_1\geq 2$. Moreover, $a=0$ if the rational curves fixed by $\sigma^2$ are at most $2$, i.e. if $n_1/2+2a\leq 2$.
Since $C$ intersects the generic fiber of $\pi$ in $3$ points, then it is trigonal.
If $g\geq 3$, this implies that $C$ is not hyperelliptic, hence the canonical morphism of $C$ is an embedding in $\IP^{g-1}$.
The involution $\sigma$ on $C$ is thus induced by an automorphism of the projective space.
If $g=3$, this implies that $C$ is isomorphic to a plane quartic and, since an involution of $\IP^2$ fixes a line, that $n_2>0$.
If $(m,r,g,a)=(7,5,6,1)$, then $S(\sigma^2)\cong U\oplus D_6$. Observe that $\sigma^2$ fixes the section of $\pi$ and two irreducible components of the fiber of type $D_6$. Since $a=1$ and the fibration is $\sigma$-invariant, then $\sigma$ should act as a reflection on the fiber $D_6$, but this is not possible since the fibration has a unique section.
The cases in the table are then obtained by taking all possible values for $l$, using the previous equations and remarks and the fact that $n_1\in \{0,2,4\}$. The lattices can be computed by means of propositions \ref{rel4}, \ref{lat} and the classification theorem of $2$-elementary lattices \cite[Theorem 4.3.1]{nikulinfactor}.
\end{proof}
\begin{example}
Let $C, E\subset \mathbb F_4$ as in Example \ref{hirz}. If $C$ has rational double points, then the minimal resolution $X$ of the double cover of $\mathbb F_4$ branched along $C\cup E$ is still a K3 surface.
If $C$ has two nodes exchanged by $\iota$, then $\iota$ lifts to an order four automorphism $\sigma$ of $X$ such that $\sigma^2$ fixes a curve of genus $8$ (the pull-back of the proper transform of $C$) and a smooth rational curve $\tilde E$ (the pull-back of the $(-4)$-curve $E$).
Observe that here $l>0$ since the exceptional divisors over the two nodes are exchanged by $\sigma$.
The lattice $S(\sigma^2)$ in this case is isomorphic to $U\oplus A_1^{\oplus 2}$.
If $C$ has two triple points exchanged by $\iota$, then $\sigma^2$ fixes the pull-back of the proper transform of the curve $C$, $\tilde E$ and the central components of the resolution trees over the two singular points, which are of type $\tilde D_4$. In this case $a=1$ since such components are exchanged by $\sigma$. In this case the lattice $S(\sigma^2)$ is isomorphic to $U\oplus D_4^{\oplus 2}$.
Similarly, we get examples if the triple points of $C$ are simple singularities of type $D_n (n\geq 6), E_7, E_8$.
Considering $C$ with ordinary nodes (up to $10$) and triple points exchanged by $\iota$ we obtain several examples for the cases in the table with $S(\sigma^2)\cong U\oplus R$.
Similarly, we can construct examples for the cases of type $U(2)\oplus R$ by generalizing Example \ref{quadric} to the case when the curve $C$ in $\mathbb P^1\times \mathbb P^1$ has simple singularities.
In this way, we obtain examples for the cases in Table \ref{alpha=0}, excepted the boxed ones.
\end{example}
\begin{example} Consider the jacobian elliptic fibration $\pi:X\map \IP^1$ defined as follows:
$$y^2=x(x^2+a(t^4)x+b(t^4)),$$
where $a, b$ are polynomials of degree $1$ and $2$ respectively.
Observe that $\pi$ has a $2$-torsion section $t\mapsto (0,0,t)$. The translation by this section gives a symplectic involution $\tau$ on $X$.
Moreover, $X$ has the order four non-symplectic automorphisms $\sigma: (x,y,z,t)\mapsto (x,y,z,it)$ and $\sigma':=\sigma\circ \tau$.
For generic $a,b$, $\pi$ has $8$ fibers of type $I_2$ and $8$ fibers of type $I_1$.
The automorphisms $\sigma$ and $\sigma'$ act with order four on $\IP^1$, preserve the two smooth fibers over $t=0$ and $t=\infty$ and act as an involution over $t=\infty$. Moreover, $\sigma$ fixes pointwisely the fiber over $t=0$, while $\sigma'$ acts as an order two translation on it.
For special choices of $a$ and $b$ we can obtain examples with reducible fibers of type $I_{4M}$ over $t=0$ or $t=\infty$.
For example, if $a(t^4)=t^4$ and $b(t^4)=1$, then $\pi$ has a smooth fiber over $t=0$ and a fiber of type $I_{16}$ over $t=\infty$.
The automorphism $\sigma$ fixes pointwisely the fiber over $t=0$ and acts on the fiber over $t=\infty$ as a reflection which leaves invariant the components $\Theta_0, \Theta_8$ (see the notation in Example \ref{vinberg}), giving
$k=0, a=3, n=4$.
The symplectic involution $\tau$ acts over $t=\infty$ as a rotation sending $\Theta_0$ to $\Theta_8$.
Finally, the automorphism $\sigma'$ acts over $t=0$ as a translation and over $t=\infty$ as a reflection which leaves invariant the components $\Theta_4,\Theta_{12}$, giving $k=0, a=3, n_2=0, n_1=4$ (see Table \ref{alpha=0}).
For more details on this example se also \cite[Proposition 4.7]{vGS}.
\end{example}
\section{The other cases}\label{other}
At this point of the classification the cases left out are those with
$$
\Fix(\sigma)=E_1\cup\cdots\cup E_k\cup\{p_1,\ldots,p_n\}
$$
$$
\Fix(\sigma^2)=C\cup (E_1\cup\cdots\cup E_k)\cup(F_1\cup F_1'\cup\cdots\cup F_a\cup F_a')\cup (G_1\cup\cdots \cup G_{\frac{n_1}{2}})
$$
where $C$ is a curve of genus $g>0$, $n_2=n-n_1$ is the number of fixed points on $C$ and we can assume that $k>0$ and $l>0$ (recall that we have also $m>0$). Observe that in this case $\alpha=k$ so $n=2k+4$
and $2k=10-l-m$ by Proposition \ref{rel4}. In particular, observe that:
$$
m+l\equiv 0\ (\!\!\!\!\!\!\mod 2)\,\,\,\,\mbox{and}\,\,\,\,m+l\leq 8.
$$
On the other hand, by computing the
difference $\chi(\Fix(\sigma^2))-\chi(\Fix(\sigma))$ topologically and using the Lefschetz's formula, one gets
the relation
\begin{eqnarray*}
2-2g-n_2+4a=2l-2m
\end{eqnarray*}
so that
\begin{eqnarray}\label{punti}
g-2a=m-l+1-\frac{n_2}{2}.
\end{eqnarray}
Using Hurwitz formula on $C$ we obtain
\begin{eqnarray}\label{hurwitz}
n_2\leq 2g+2.
\end{eqnarray}
By Proposition \ref{lat} we also have:
\begin{eqnarray}\label{punti2}
g+j=g+2a+k+\frac{n_1}{2}\leq 11
\end{eqnarray}
Combining \eqref{punti} and \eqref{punti2} we get
\begin{eqnarray}\label{genus}
g\leq 5-k+\frac{m-l}{2}=m
\end{eqnarray}
\begin{eqnarray}\label{a}
a\leq 2-\frac{k}{2}+\frac{n_2}{4}+\frac{(l-m)}{4}\leq 3+\frac{l-m}{4}
\end{eqnarray}
where we obtain the last inequality using $n_2\leq 2k+4$. Finally observing that $|m-l|\leq 6$ and $k\geq 1$
we get $g\leq 7$ and $a\leq 4$.
\begin{lemma}\label{curvefissate}
Assume that $g=g(C)>1$. Then $g\leq m$ and we are in one of the following cases:
$$
\begin{array}{c|c|c|c}
m+l&k&g\leq &a\leq \\
\hline
4&3&3&2\\
6&2&5&3\\
8&1&7&4
\end{array}
$$
\end{lemma}
\bprf By \eqref{genus} we have that $g\leq m$, in particular $m\geq 2$.
Moreover, by the previous conditions we have that $m+l=4,6,8$.
This gives the inequalities for $g$ in the table.
If $m+l=4,6$ then $|m-l|\leq 2$ and $|m-l|\leq 4$ respectively, so by \eqref{a} we get $a\leq 3$ and $a\leq 4$ respectively.
Similarly, if $m+l=8$ then $a\leq 4$.\\
We show that the case $m+l=4$, $k=3$ and $a=3$ is not possible. By \eqref{a} we get $l\geq m$ and by \eqref{punti}
with $n_2\leq 2k+4=10$ we get $g\leq 2+(m-l)$. If $l>m$ then $g<2$, which is not possible. If $l=m$ then we have
$g=2$, hence again $n_2=10$ by \eqref{punti}, contradicting \eqref{hurwitz}. The case
$m+l=6$, $a=4$ can be excluded similarly.\\
\qed
\begin{example}\label{wei2}
Consider the following elliptic K3 surface $\pi:X\map \IP^1$ in Weierstrass form:
$$
y^2=x^3-a(t)x,\ \ \deg a(t)=8.
$$
with the order four automorphism
$$\sigma(x,y,t)=(-x,iy,t).$$
The automorphism $\sigma$ fixes the two sections $x=y=0$, $x=z=0$ (hence $k\geq 2$), while $\sigma^2$ also fixes
the curve $C:\ y=x^2-a(t)=0$.
The discriminant of the fibration is $\Delta(t)=4a^3(t)$, hence for a generic polynomial $a(t)$ the fibration has $8$ fibers of type $III$ (more precisely these are of type $III b)$).
Moreover, the curve $C$ has genus $3$ and $\sigma$ has $8$ fixed points on it, so that: $k=2$, $n_2=8$, $a=0$. One can compute also that $l=0$, so that this case appears in Table \ref{l=0}.
We now study how $a(t)$ can split:
\begin{enumerate}[i)]
\item If $a(t)=a_1(t)^2a_6(t)$, then $\pi$ has a fiber $I_0^*$ and $6$ fibers $III b)$. In this case the ramification points on $C$ are $6$, so that $g(C)=2$. Here $k=2$, $n_1=2$, $n_2=6$, $a=0$.
Thus we have an example in Lemma \ref{curvefissate}.
\item If $a(t)=a_1(t)^2b_1(t)^2a_4(t)$ then $\pi$ two fibers $I_0^*$ and four fibers $III b)$. Here $k=2$, $n_1=4$, $n_2=4$, $a=0$ and $g=1$. This case appears in Proposition \ref{othergenus1}.
\item If $a(t)=a_1(t)^2b_1(t)^2c_1(t)^2a_2(t)$ then we have $3$ fibers $I_0^*$ and two $III b)$, so that $k=2$, $n_1=6$, $n_2=2$, $a=0$ and $g=0$. This case appears in Table \ref{g=0}.
\item If $a(t)=a_1(t)^2b_1(t)^2c_1(t)^2d_1(t)^2$ then we have four fibers $I_0^*$. In this case $C$ splits into the union of two rational curves. In this case we have $k=2$, $n=8$, $a=1$. We are again in one case of Table \ref{g=0}.
\end{enumerate}
For generic $a(t)$ the Mordell-Weil group of $X$ is isomorphic to $\IZ_2$. The translation by a generator of the group is a symplectic involution $\iota$ on $X$ and
the composition $\sigma'=\iota \circ \sigma$ is again a non-symplectic automorphism of order $4$ on $X$ which fixes $C$ and exchanges the two sections of $\pi$.
\end{example}
\begin{example} In Lemma \ref{curvefissate} an easy computation with MAPLE
finds 63 possible cases.
One can produce some examples with the fixed locus described there
starting from Example \ref{hirz} (resp. Example \ref{quadric}) and imposing simple
singularities on the curve $C$ of genus 10 (resp. genus 9) such that at least one singular point
is invariant for $\iota$.
For example, one can assume that the curve $C$ in Example \ref{hirz} has an ordinary triple point at an invariant point of $\iota$ and two nodes exchanged by $\iota$. The Dynkin diagram of the resolution over the triple point, which is of type $\tilde D_4$, is $\sigma$-invariant: its simple components are preserved and the double component is fixed by $\sigma$. Thus $k=1$, $n_2=4$, $n_1=2$, $a=0$ and $g=5$.
Observe that, if $C$ has just an ordinary triple point at an invariant point, then we are in the case $g=7, k=1, n_1=2, n_2=4$ of Table \ref{l=0}.
Similar examples can be constructed from Example \ref{quadric}.
\end{example}
We now consider the case when $\sigma^2$ fixes an elliptic curve.
\begin{pro}\label{othergenus1}
With the previous notation, if $g(C)=1$ then we are in one of the cases appearing in Table \ref{others}.
\begin{table}[ht]
$$
\begin{array}{ccc|cccc|c}
m & r & l & n_1& n_2 & k&a & \mbox{type of } C'\\
\hline
5&9&3&2&4&1&0 & I_4\\
\hline
4&12&2&4&4&2&0 & I_8\\
&10&4&6&0&1&0 & \\
\hline
3&15&1&6&4&3&0& I_{12}\\
&13&3&8&0&2&0\\
\hline
4&12&2&4&4&2&0& IV^*\\
&10&4&2&4&1&1\\
&10&4&6&0&1&0\\
\end{array}
$$
\caption{The case $g=1, k>0, l>0$.}\label{others}
\end{table}
\end{pro}
\bprf
Using the relations at the beginning of the paragraph one can find the values in the table, we now show that these are the only possibilities.\\
Since $\sigma$ preserves $C$, then there is a $\sigma$-invariant elliptic fibration $\pi_C:X\map \IP^1$ with fiber $C$. Observe that $\sigma$ has order four on the basis of $\pi_C$, since otherwise $\sigma^2$ would act as the identity on the tangent space at a point of $C$.
Thus $\sigma$ has two fixed points on $\IP^1$, corresponding to the fiber $C$ and a fiber $C'$ of $\pi_C$.
This implies that all rational curves fixed by $\sigma$ are contained in $C'$, so that $C'$ is reducible (since $k>0$).
Observe that $\sigma$ acts on $C$ either as an involution with four fixed points or as a translation.
By Proposition \ref{rel4} we have $n\geq 6$, so that $C'$ contains at least two fixed points of $\sigma$.
This excludes the Kodaira types $I_2,\,I_3,\,III,\,IV$ for $C'$.
By similar arguments as in the proof of Theorem \ref{g1} also the types $I_N^*$, $II^*$ and $III^*$ can be excluded. If $C'$ has Kodaira type $IV^*$ then $\sigma$ either preserves each component or it exchanges two branches. In the first case $a=0$ and either $n=n_1=6$ or $n_1=n_2=4$. In the second case $a=1$, $k=1$, $n_1=2$ and $n_2=4$.
We now consider the case when $C'$ is of type $I_N$, $N\geq 4$.
Since $C'$ contains at least a fixed curve for $\sigma$, then all
components of $I_N$ are preserved by $\sigma$, hence $a=0$. By the previous remarks, since $n_2=4$ or $0$, then either $n_1=2k$ or $n_1=2k+4$ respectively.
If $n_2=4$, then $N=2k+n_1=4k$. For $N>12$ we get $k>3$, which gives $m<3$ by Proposition \ref{rel4}. By the equality \eqref{punti} we get $m-l=2$ so $m=2$ and $l=0$, contradicting the assumption $l>0$.
Hence we only have the cases in the table.
On the other hand, if $n_2=0$ we get $N=2k+(2k+4)=4k+4$. By equation \eqref{punti} we get $l=m$ and $k+m=5$ by Proposition \ref{rel4}. If $N\geq 16$ then $k\geq 3$, which gives $m<3$ as before.
If $m=l=1$, then $k=4$, $n_1=12$ and $C'$ is of type $I_{20}$. Since $a=0$ the lattice $S(\sigma)$ contains the classes of the $20$ components of $C'$, contradicting $r=19$.
If $m=l=2$, then $k=3$, $n_1=10$ and $C'$ is of type $I_{16}$á Since $a=0$ the lattice $S(\sigma)$ contains the classes of the $16$ components of $C'$. This gives a contradiction since $r=16$ and $S(\sigma)$ is hyperbolic by Proposition \ref{lat}. \qed
\begin{example} Observe that the elliptic K3 surface $\pi_C:X\map \IP^1$ in Example \ref{wei} also carries the non-symplectic order four automorphism
$$\sigma'(x,y,t)=(x,-y,it),$$
obtained by composing the automorphism $\sigma$ defined there with the non-symplectic involution $y\mapsto -y$.
Generically, the automorphism $\sigma'$ acts on the elliptic curve over $t=0$ as an involution with $4$ isolated fixed points and as the identity on the fiber over $t=\infty$.
If the fiber over $t=\infty$ is reducible, then $\sigma$ fixes at most rational curves, in particular $k=\alpha$.
Observe that $\sigma'$ preserves the section at infinity $t\mapsto (0:1:0;t)$ and has two fixed points on it, one on the fiber over $t=0$ the other over $t=\infty$. Using this condition and the equality $n=2\alpha+4$ given by Proposition \ref{rel4}, one sees that $n=6$, $k=1$ if $\pi_C$ has a fiber of type $I_4$, $(k,n,a)=(2,8,0)$ for a fiber
$I_8$, $(3,10,0)$ for a fiber $I_{12}$ and $(k,n,a)=(4,12,0)$ for a fiber $I_{16}$.
For a fiber of type $IV^*$ there are two possibilities: either $\sigma'$ fixes two curves, and so we have $(k,n,a)=(2,8,0)$, or it acts as a reflection fixing one curve, giving $(k,n,a)=(1,6,0)$.
The automorphism $\sigma^2=(\sigma')^2$ is an involution fixing the smooth elliptic curve over $t=0$ and some rational curves in the singular fiber over $t=\infty$.
The cases with the fibers $I_4$, $I_8$, $I_{12}$ and $IV^*$ give examples with $l\not=0$ that appear in Table \ref{others}.
In case there is a fiber of type $I_{16}$
we have instead $l=0$ (see the last line of Table \ref{l=0}).
\end{example}
\bibliographystyle{amsplain}
\bibliography{Biblio}
\end{document} | {"config": "arxiv", "file": "1102.4436/Ordine4nonsympl14marzo2011.tex"} |
TITLE: Complex number systems of equations fx-115es plus calc?
QUESTION [1 upvotes]: Anyone here know how to use the EQN mode on the casio fx-115es plus to find solutions to a system of equations involving complex numbers? Also, if that's not possible, what about entering complex numbers to a matrix on this calculator? I haven't been able to figure it out, and searching the web hasn't returned any relevant results.
I'm guessing its possible as this calculator is allowed on the engineering FE exams. So far I've only been able to solve systems using real numbers in EQN mode, and have only been able to perform complex calculations in CMPLX mode.
Thanks.
REPLY [3 votes]: Complex numbers are not supported—on simultaneous equations or on matrices—on the fx-115es plus scientific calculator.
Indeed, no scientific calculator (neither T.I., Sharp or Casio) supports these two kinds of problem solving in CPLX mode.
For that, you will need to choose a graphing calculator with a CAS (Computer Algebra System).
/Silicon Valley Regards | {"set_name": "stack_exchange", "score": 1, "question_id": 887308} |
TITLE: Integral containing Associated Legendre Polynomials
QUESTION [0 upvotes]: I need to evaluated the following integral:
$\int_0^\pi \sin(x) \cos(x) P_l^m(\cos x) P_k^m(\cos x) \mathrm{d}x$
and I thought since a solution is known to a similar thing
$\int_0^\pi \sin(x) P_l^m(\cos x) P_k^m(\cos x) \mathrm{d}x = \frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l,k}$
maybe this is the case with an additional $\cos x$ as well.
REPLY [1 votes]: The previous answer about using Gaunt's theorem won't hold unless some condition regarding the degrees are satisfied. | {"set_name": "stack_exchange", "score": 0, "question_id": 1116464} |
TITLE: Convergence of a sequence in metric space
QUESTION [1 upvotes]: Can someone please help me with this problem? Thanks!
Check if the sequence $x_n= (1+1/n)^n$ is convergent in $ (X,d)$ where $d(x,y)=$ $\frac {2|x-y|}{3+2|x-y|}$, and if it is convergent, then find its limit.
REPLY [0 votes]: let $g:[0,\infty) \to [0,\infty)$ be $g(t) = {2 t \over 3+2t}$. Note that $g(t) \le {2 \over 3} t$ and for $t \le 1$, we have $g(t) \ge {2 \over 5} t$.
Note that $d(x,y) = g(|x-y|)$. In particular, $x_n$ is convergent with respect to $g$ iff it is convergent with respect to $|\cdot|$.
REPLY [0 votes]: Since that distance is topologically equivalent to the usual distance and since, with respect to the usual distance, that sequence converges to $e$, then the sequence converge to $e$ in $(\mathbb R,d)$ too. | {"set_name": "stack_exchange", "score": 1, "question_id": 3255658} |
TITLE: Classical mechanics kinetic energy string
QUESTION [0 upvotes]: A particle of mass m on a smooth horizontal table is attached to a string of length l passing through a small hole in the table and carries a particle of equal mass hanging vertically.
The position of the particle on the table is given in terms of its distance r from the hole and of the angle θ the string makes with some fixed line in the table.
The position of the other particle is given in terms of its vertical distance from the table.
Question: Derive Lagrange's equation for the system in terms of the generalized coordinates r and and $\theta$.
I was able to deduce that the potential energy is $U=mgr$. But how do I construct the kinetic energy in terms of $r$ and $\theta$?
REPLY [0 votes]: Hint.
We know
$$
p = \rho(\cos\theta,\sin\theta)\Rightarrow \dot p = \dot\rho(\cos\theta,\sin\theta)+\rho(-\sin\theta,\cos\theta)\dot\theta\Rightarrow \|\dot p\|^2=\dot \rho^2+\rho^2\dot\theta^2
$$
now falling vertically
$$
\cases{
T_1 = \frac 12 m_1 \dot\rho_1^2\\
U_1 = -\rho_1 m _1 g
}
$$
and on the table
$$
\cases{
T_2 = \frac 12m_2(\dot \rho_2^2+\rho_2^2\dot\theta^2)\\
U_2 = 0
}
$$
Note that
$$
\rho_1 + \rho_2 = \rho_0
$$ | {"set_name": "stack_exchange", "score": 0, "question_id": 4087742} |
\begin{document}
\title{Dynamical Analysis of a Networked Control System}
\author{Guofeng Zhang\thanks{School of Automation, Hangzhou Dianzi University,
Hangzhou, Zhejiang 310038, P. R. China} \and Tongwen Chen\thanks{Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, Alberta, Canada T6G 2V4} \and Guanrong Chen\thanks{Department of Electronic Engineering,
City University of Hong Kong, Hong Kong, P. R. China} \and Maria D'Amico\thanks{Depto. de Ingenier«õa El«ectrica y de Computadoras,
Universidad Nacional del Sur, Avda Alem 1253,
B8000CPB Bah«õa Blanca, Argentina} }
\maketitle
\begin{abstract}
A new network data transmission strategy was proposed in Zhang \&
Chen [2005] (arXiv:1405.2404), where the resulting nonlinear system was analyzed and
the effectiveness of the transmission strategy was demonstrated via
simulations. In this paper, we further generalize the results of
Zhang \& Chen [2005] in the
following ways: 1) Construct first-return maps of the nonlinear systems formulated in
Zhang \& Chen [2005] and derive several existence conditions of
periodic orbits and study their properties. 2) Formulate the new
system as a hybrid system, which will ease the succeeding analysis.
3) Prove that this type of hybrid systems is not structurally stable
based on phase transition which can be applied to higher-dimensional
cases effortlessly. 4) Simulate a higher-dimensional model with
emphasis on their rich dynamics. 5) Study a class of continuous-time
hybrid systems as the counterparts of the discrete-time systems
discussed above. 6) Propose new controller design methods based on
this network data transmission strategy to improve the performance
of each individual system and the whole network. We hope that this
research and the problems posed here will rouse interests of
researchers in such fields as control, dynamical systems and
numerical analysis.
\noindent \textbf{Keywords}: bifurcation, computational complexity,
first-return map, hybrid system, networked control system,
stability.
\end{abstract}
\newpage
\listoffigures
\newpage
\addtolength{\baselineskip}{0.4\baselineskip}
\section{Introduction}
Consider the networked control system shown in Fig.~\ref{Figure1}.
\begin{figure}[tbh]
\epsfxsize=6in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure1.eps}} \caption{A standard networked
control system} \label{Figure1}
\end{figure}
It is obvious that connecting various system components via
communication media can reduce wiring, ease installation and improve
maintenance, among others. As far as distributed control systems are
concerned, communication among individual controllers provides each
of them with more information so that better control performance can
be achieved [Ishii \& Francis, 2002]. These advantages endorse the
network control technology a promising future in systems engineering
and applications.
Unfortunately, since the encoded system output, controller output
and other information are transmitted via communication networks
shared by many users, data traffic congestion is always unavoidable.
This usually give rise to time delays, packet loss and other
undesirable behaviors to the control systems. These problem have
become a major subject of research in this and several closely
related fields. Many network protocols and control strategies have
already been proposed to tackle the problems. Loosely speaking,
these considerations fall into three categories, which are further
discussed as follows.
The first category simply models a networked control system as a
control system with bounded time delays. In a series of recent
papers [Walsh {\it et al.}, 1999, 2001, 2002, 2002b], [Walsh \& Ye,
2001], the try-once-discard (TOD) protocol is proposed and studied
intensively, where an upper bound of sensor-to-controller time
delays induced by the network is derived, for which exponential
stability of the closed-loop system is guaranteed. This idea is
further generalized in Nesic \& Teel [2004] to derive a set of
Lyapunov UGES (Uniformly Globally Exponentially Stable) protocols in
the $L^{p}$ framework. In Yue {\it et al.} [2005], assuming bounded
time delays and packet dropouts, a robust $H_{\infty}$ control
problem is studied for networked control systems. In general this
approach is quite conservative, as has been widely acknowledged.
The second category models network time delays and packet dropouts
as random processes, typically Markov chains. In this way, some
specific features of these random processes can be utilized to
design controllers that guarantee desired system performance. In
Krtolica {\it et al.} [1994], a random model of time delays based on
Markov chain is established via augmentation. Necessary and
sufficient conditions for zero-state mean-square exponential
stability have been derived for this system. In Nilsson {\it et al.}
[1998], both sensor-to-controller and controller-to-actuator time
delays are modeled as independent white-noise with zero mean and
unit variance and consequently a (sub)optimal stochastic control
problem is studied. Two Matlab toolboxes, Jitterbug and TrueTime,
are introduced in Cervin {\it et al.} [2003], based on the principle
that networked control systems can be viewed as delayed sampled-data
systems with quantization effects. These two toolboxes can be used
as experiment platforms for research on real-time dynamical control
systems. They can be easily employed to quickly determine how
sensitive a control system is to delays, jitters, lost samples, etc.
These two categories of methods deal with network effects passively,
i.e., they solely consider the effects of network traffic on the
control systems separately, leaving aside the interactions of the
control systems and communication network. This latter consideration
is very important, which leads to the third category of
methodologies. This approach takes into account the tradeoff between
data rate and control performance. In order to minimize bandwidth
utilization, Goodwin {\it et al.} [2004] proposed some methods of
using quantization to reduce the size of the transmitted data and
solved the problem via a moving horizon technique. In Wong \&
Brockett [1999], the effect of quantization error, quantization, and
propagation time on the containability, a weaker stability concept,
of networked control systems is studied. In Takikonda \& Mitter
[2004], the tradeoff of data rate and desirable control objectives
is considered with emphasis on observability and stabilizability
under communication constraints. A necessary condition is
established on the rate for asymptotic observability and
stabilizability of a linear discrete-time system. More specifically,
the rate must be bigger than the summation of the logarithms of
modules of the unstable system poles. Then, these results are
further generalized to the study of control over noisy channels in
Takikonda \& Mitter [2004b]. The problem of asymptotic stabilization
is considered in Brockett \& Liberzon [2000], where time-varying
quantizers are designed to achieve the stabilization of an unstable
system. For the LQG optimal control of an unstable scalar system
over an additive white Gaussian noise (AWGN) channel, it is reported
[Elia, 2004] that the achievable transmission rate is given by the
Bode sensitivity integral formula, thereby establishing the
equivalence between feedback stabilization through an analog
communication channel and a communication scheme based on feedback
which unifies the design of control systems and communication
channels.
In this paper, we continue the study of the data transmission
strategy proposed in our earlier paper [Zhang \& Chen, 2005], where
a new network data transmission strategy was proposed to reduce
network traffic congestion. By
adding constant deadbands to both the controller and the plant shown in Fig.~
\ref{Figure1}, signals will be sent only when it is necessary. By
adjusting the deadbands, a tradeoff between control performance and
reduction of network data transmission rate can be achieved. The
data transmission strategy proposed is suitable for fitting a
control network into an integrated communication network composed of
control and data networks, to fulfill the need for a new breed
geared toward total networking (see [Raji, 1994]). This problem is
of course very appealing as depicted by Raji [1994; and at the same
time it is fundamentally important so is listed in Murray {\it et
al.} [2003] as a future direction in control research in an
information-rich world: ``Current control systems are almost
universally based on synchronous, clocked systems, so they require
communication networks that guarantee delivery of sensor, actuator,
and other signals with a known, fixed delay. Although current
control systems are robust to variations that are included in the
design process (such as a variation in some aerodynamic coefficient,
motor constant, or moment of inertia), they are not at all tolerant
of (unmodeled) communication delays or dropped or lost sensor or
actuator packets. Current control system technology is based on a
simple communication architecture: all signals travel over
synchronous dedicated links, with known (or worst-case bounded)
delays and no packet loss. Small dedicated communication networks
can be configured to meet these demanding specifications for control
systems, but a very interesting question is whether we can develop a
theory and practice for control systems that operate in a
distributed, asynchronous, packet-based environment.''
Essentially speaking, under the network data transmission strategy
proposed here, in an integrated network composed of data and control
networks, it is asked that the network should provide sufficient
communication bandwidth upon request of control systems. As a
payoff, control systems will save network resources by deliberately
dropping packets without degrading system performance severely. This
is a crucial tradeoff. On the one hand, control signals are normally
time critical, hence the priority should be given to them whenever
requested; on the other hand, due to one characteristic of control
networks, namely, small packet size but frequent packet flows, it is
somewhat troublesome to manage because it demands frequent
transmissions. Our scheme aims to relieve this burden for the whole
communication network.
As we proceed, readers will find that the simple network transmission data
strategy analyzed here gives rise to many unexpected and interesting
dynamical phenomena and mathematical problems, which have innocent
appearance but are hard to deal with. More specifically, we investigate the
following issues: the stability of the control systems under the proposed
scheme; existence of periodic orbits by means of first-return maps;
bifurcation and phase transition phenomena; computational complexity. Since
this research project is oriented toward the study of control problems in a
network setting, the effectiveness of the scheme and the corresponding
controller design problem will also be addressed after system analysis.
The layout of this paper is as follows: the proposed network
protocol is presented in Sec.~2, where its advantages are discussed.
The resulting closed-loop system under this network protocol is
analyzed in Secs.~3-7. More precisely, Sec.~3 contains a study of a
closed-loop system consisting of a scalar plant controlled by a
proportional controller with a constant gain, as the simplest case
under this framework to provide a detailed analysis. The structural
stability of the system is studied in Sec.~4. Sec.~5 studies the
rich dynamics of a higher-dimensional model. The continuous-time
counterpart of this type of discrete-time systems is discussed in
Sec.~5. The controller design problem is then addressed in Sec.~7.
Some concluding remarks, open problems and future research issues
are finally posed and discussed in Sec.~8.
\section{The Proposed Network Protocol}
In Zhang \& Chen [2005], a new data transmission strategy was
proposed, which is briefly reviewed here. Consider the feedback system shown in Fig.~\ref
{Figure2},
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure2.eps}} \caption{A typical feedback
system} \label{Figure2}
\end{figure}
where $G$ is a discrete-time system of the form:
\begin{eqnarray}
x(k+1) &=&Ax(k)+Bu(k), \label{sysG} \\
y(k) &=&Cx(k), \notag
\end{eqnarray}
with the state $x\in \mathbb{R}^{n}$, the input $u\in \mathbb{R}^{m}$, the
output $y\in \mathbb{R}^{p}$, and the reference input $r\in \mathbb{R}^{p}$,
respectively; $C$ is a stabilizing controller:
\begin{eqnarray}
x_{d}(k+1) &=&A_{d}x_{d}(k)+B_{d}e(k), \label{conC} \\
u(k) &=&C_{d}x_{d}(k)+D_{d}e(k), \notag \\
e\left( k\right) &=&r\left( k\right) -y\left( k\right) , \notag
\end{eqnarray}
with its state $x_{d}\in \mathbb{R}^{n_{c}}$. Let $\xi =\left[
\begin{array}{c}
x \\
x_{d}
\end{array}
\right] $. Then, the closed-loop system from $r$ to $e$ is described by
\begin{eqnarray}
\xi \left( k+1\right) &=&\left[
\begin{array}{cc}
A-BD_{d}C & BC_{d} \\
-B_{d}C & A_{d}
\end{array}
\right] \xi \left( k\right) +\left[
\begin{array}{c}
BD_{d} \\
B_{d}
\end{array}
\right] r(k), \label{clsys1} \\
e(k) &=&\left[
\begin{array}{cc}
-C & 0
\end{array}
\right] \xi \left( k\right) +r(k). \notag
\end{eqnarray}
Now, we add some nonlinear constraints on both $u$ and $y$.
Specifically, consider the system shown in Fig.~\ref{Figure3}.
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure3.eps}} \caption{A constrained feedback
system} \label{Figure3}
\end{figure}
The nonlinear constraint $H_{1}$ is defined as follows: for a given $\delta
_{1}>0$, let $v(-1)=0$; and for $k\geq0$, let
\begin{equation}
v(k)=H_{1}\left( u_{c}\left( k\right) ,v(k-1)\right) =\left\{
\begin{array}{ll}
u_{c}(k), & \mbox{if~}\left\| u_{c}\left( k\right) -v\left( k-1\right)
\right\| _{\infty }>\delta _{1}, \\
v(k-1), & \mbox{otherwise.}
\end{array}
\right. \label{constraint1}
\end{equation}
Similarly, $H_{2}$ is defined as follows: for a given $\delta _{2}>0$, let $
z(-1)=0$; for $k\geq0$, let
\begin{equation}
z(k)=H_{2}\left( y_{c}\left( k\right) ,z(k-1)\right) =\left\{
\begin{array}{ll}
y_{c}(k), & \mbox{if~}\left\| y_{c}\left( k\right) -z\left( k-1\right)
\right\| _{\infty }>\delta _{2}, \\
z(k-1), & \mbox{otherwise.}
\end{array}
\right. \label{constraint2}
\end{equation}
It can be shown that $\left\| H_{1}\right\| $, the induced norm of $H_{1}$,
equals $2$, and so is $\left\| H_{2}\right\| $.
In Octanez {\it et al.} [2002] \textit{adjustable} deadbands are
proposed to reduce network traffics, where the closed-loop system
with deadbands is modeled as a perturbed system, with exponential
stability followed from that of the original system [Khalil, 1996].
The constraints, $\delta _{1}$ and $\delta _{2}$, proposed here are
fixed. We have observed [Zhang \& Chen, 2005] that the stability of
the system shown in Fig.~\ref{Figure3} is fairly complicated and
only local stability can be obtained. However, the main advantage of
fixed deadbands is that it will reduce network traffics more
effectively. Furthermore, the stability region can be scaled as
large as desired.
For the ``constrained'' system shown in Fig.~\ref{Figure2}, let $p$
denote the state of the system $G$ and $p_{d}$ denote the state of
the controller $C$. Then
\begin{eqnarray*}
p(k+1) &=&Ap(k)+Bv(k), \\
y_{c}(k) &=&Cp(k),
\end{eqnarray*}
and
\begin{eqnarray*}
p_{d}(k+1) &=&A_{d}p_{d}(k)+B_{d}e_{c}(k), \\
u_{c}(k) &=&C_{d}p_{d}(k)+D_{d}e_{c}(k), \\
e_{c}(k) &=&r(k)-z(k).
\end{eqnarray*}
Let $\eta =\left[
\begin{array}{c}
p \\
p_{d}
\end{array}
\right] $. Then, the closed-loop system from $r$ to $e$ is
\begin{eqnarray}
\eta (k+1) &=&\left[
\begin{array}{cc}
A & 0 \\
0 & A_{d}
\end{array}
\right] \eta (k)+\left[
\begin{array}{cc}
B & 0 \\
0 & B_{d}
\end{array}
\right] \left[
\begin{array}{c}
v(k) \\
-z(k)
\end{array}
\right] +\left[
\begin{array}{c}
0 \\
B_{d}
\end{array}
\right] r(k), \label{clsys2} \\
e_{c}(k) &=&\left[
\begin{array}{cc}
-C & 0
\end{array}
\right] \eta \left( k\right) +r(k), \notag
\end{eqnarray}
where $v$ and $z$ are given in Eqs. (\ref{constraint1})-(\ref{constraint2}).
At this point, one can see that in the framework of the
communication network containing both data and control networks,
this proposed data transmission strategy will provide sufficient
communication bandwidth upon request of control networks used by the
control systems. As a payoff, the control systems will save some
network resources by deliberately dropping packets. This
consideration is well tailored to the requirement of control
networks in general. On the one hand, control signals are normally
time critical, hence the priority should be given to them whenever
requested. On the other hand, due to the characteristics of control
networks, namely, small packet size but frequent packets flows, it
is somewhat troublesome to manage because it demands frequent
transmissions. Our scheme aims to relieve this burden for the whole
communication network.
\section{First-Return Maps}
To simplify the following discussions, suppose that the system $G$ in Fig.~
\ref{Figure2} is a scalar system, the controller $C$ is simply -1,
there is no $H_{2}$ involved, and $r=0$. That is, in this section,
we consider the following simplified system:
\begin{equation}
x(k+1)=ax(k)+bv(k), \label{original system}
\end{equation}
with $v(-1)\in \mathbb{R}$, and for $k\geq 0$,
\begin{eqnarray}
v\left( k\right) &=&H\left( x\left( k\right) ,v\left( k-1\right) \right)
\notag \\
&:=&\left\{
\begin{array}{ll}
x\left( k\right), & \mbox{if~}\left| x\left( k\right) -v\left( k-1\right)
\right| >\delta , \\
v\left( k-1\right), & \mbox{otherwise,}
\end{array}
\right. \label{original_constraint}
\end{eqnarray}
where $|a+b|<1$ and $\delta $ is a positive number.
In Zhang \& Chen [2005], the system composed of Eqs. (\ref{original system})-(\ref
{original_constraint}) was studied in great detail, where a
necessary condition for the existence of periodic orbits was
derived. We now generalize it and provide a necessary and sufficient
condition and give a characterization of the state space based on
it. This analysis is important: To achieve good control, a nonlinear
system may be desired to work near an equilibrium point or a limit
cycle. In the case that a limit cycle is preferred, this result will
reveal under what condition limit cycles may exist and starting from
where a trajectory may converge to the desired limit cycle. In the
case that an equilibrium is desirable, this result will provide the
designer with some information as how to design controllers to
prevent trajectories from being stuck into a limit cycle. Hence,
this analysis will provide useful insights into the design of
control systems under the proposed data transmission strategy. We
now investigate this important problem case by case.
\subsection{Case 1: $0<a<1$, $b<0$}
For convenience, we present Fig.~\ref{Figure4},
\begin{figure}[tbp]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure4.eps}} \caption{Diagram for the case
with $0<a<1$ and $b<0$} \label{Figure4}
\end{figure}
where
\begin{eqnarray*}
L_{o} &:=&\left\{ \left( v_{-},x\right) \in I_{a}:x=\frac{b}{1-a}
v_{-}\right\} , \\
L_{\delta +} &:=&\left\{ \left( v_{-},x\right) \in I_{b}:x-v_{-}=\delta
\right\} , \\
L_{\delta -} &:=&\left\{ \left( v_{-},x\right) \in I_{b}:x-v_{-}=-\delta
\right\} \\
L_{1\delta +} &:=&\left\{ \left( v_{-},x\right) \in I_{b}:x=\left(
a+b\right) v_{-} ,v_{-}>\frac{1-\left| a\right| }{1-\left( a+b\right) }
\delta \right\} , \\
L_{1\delta -} &:=&\left\{ \left( v_{-},x\right) \in I_{b}:x=\left(
a+b\right) v_{-} ,v_{-}<-\frac{1-\left| a\right| }{1-\left( a+b\right) }
\delta \right\}, \\
L_{Hb+} &:=&\left\{ \left( v_{-},\frac{-b}{1-\left| a+b\right| }\delta
\right) :\left| v_{-}\right| \leq \frac{-b}{1-\left| a+b\right| }\delta
\right\} , \\
L_{Hb-} &:=&\left\{ \left( v_{-},\frac{b}{1-\left| a+b\right| }\delta
\right) :\left| v_{-}\right| \leq \frac{-b}{1-\left| a+b\right| }\delta
\right\} , \\
L_{Vb+} &:=&\left\{ \left( \frac{-b}{1-\left| a+b\right| }\delta ,x\right)
:\left| x\right| \leq \frac{-b}{1-\left| a+b\right| }\delta \right\} , \\
L_{Vb-} &:=&\left\{ \left( \frac{b}{1-\left| a+b\right| }\delta ,x\right)
:\left| x\right| \leq \frac{-b}{1-\left| a+b\right| }\delta \right\}, \\
L_{Va+} &:=&\left\{ \left( \frac{1-\left| a\right| }{1-\left( a+b\right) }
\delta ,x\right) ,:\left| x\right| \leq \frac{-b}{1-\left( a+b\right) }
\delta \right\} , \\
L_{Va-} &:=&\left\{ \left( -\frac{1-\left| a\right| }{1-\left( a+b\right) }
\delta ,x\right) ,:\left| x\right| \leq \frac{-b}{1-\left( a+b\right) }
\delta \right\}.
\end{eqnarray*}
Assume that an initial condition $(v_{0},x_{0})$ is located on the line $
L_{1\delta +}$ satisfying
\begin{equation*}
x_{0}=(a+b)v_{0}.
\end{equation*}
Moreover, suppose that the trajectory starting from it does not
converge to a fixed point (see [Zhang \& Chen, 2005] for details).
Hence, the successive iterations are given by
\begin{eqnarray*}
x_{1} &=&ax_{0}+bv_{0}=\left( a^{2}+ab+b\right) v_{0}=\left(
a^{2}+\sum_{i=0}^{1}a^{i}b\right) v_{0}, \\
v_{1} &=&v_{0},
\end{eqnarray*}
\begin{eqnarray*}
x_{2} &=&ax_{1}+bv_{0}=\left( a^{3}+\sum_{i=0}^{2}a^{i}b\right) v_{0}, \\
v_{2} &=&v_{0},
\end{eqnarray*}
\begin{equation*}
\vdots
\end{equation*}
\begin{eqnarray*}
x_{m} &=&ax_{m-1}+bv_{0}=\left( a^{m+1}+\sum_{i=0}^{m}a^{i}b\right) v_{0}, \\
v_{m} &=&v_{0}.
\end{eqnarray*}
Accordingly,
\begin{equation*}
v_{m}-x_{m}=\left( 1-a^{m+1}-\sum_{i=0}^{m}a^{i}b\right) v_{0}.
\end{equation*}
As indicated in Fig.~\ref{Figure4}, the orbit moves downward
following the line $v = v_0$ on the right part of the region. In
this way, there will
exist a value of $m$ such that the trajectory crosses the line segment $
L_{\delta-}$, i.e.,
\begin{equation}
\left| x_{m}-v_{m}\right| >\delta , \label{chaos-1-switch}
\end{equation}
and note that such an $m$ always exists. Thus
\begin{eqnarray*}
\left( 1-a^{m+1}-\sum_{i=0}^{m}a^{i}b\right) v_{0} >\delta & \Leftrightarrow
& \frac{\left( 1-a\right) \delta }{(1-(a+b))v_{0}}<1-a^{m+1} \\
&\Leftrightarrow &a^{m+1}<1-\frac{\left( 1-a\right) \delta }{(1-(a+b))v_{0}}
\\
&\Leftrightarrow &m>\frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{
(1-(a+b))v_{0}}\right) }{\ln a}-1.
\end{eqnarray*}
Hence, the smallest $m$ is given by
\begin{equation}
m=\left\lceil \frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{
(1-(a+b))v_{0}}\right) }{\ln a}\right\rceil -1, \label{m}
\end{equation}
where $\lceil r \rceil$ is the least integer bigger than $r$. Note that
\begin{eqnarray*}
x_{m+1} &=&\left( a+b\right) x_{m}, \\
v_{m+1} &=&x_{m},
\end{eqnarray*}
Since $x_{m}<0$, this point is located on the left part of Fig.~\ref
{Figure4}, and also
\begin{equation*}
\left| x_{m+1}-v_{m+1}\right| <\delta .
\end{equation*}
Hence
\begin{eqnarray*}
x_{m+2} &=&ax_{m+1}+bv_{m}=\left( a^{2}+\sum_{i=0}^{1}a^{i}b\right) x_{m}, \\
v_{m+2} &=&x_{m},
\end{eqnarray*}
\begin{equation*}
\vdots
\end{equation*}
\begin{eqnarray*}
x_{m+n} &=&\left( a^{n}+\sum_{i=0}^{n-1}a^{i}b\right) x_{m}, \\
v_{m+n} &=&x_{m},
\end{eqnarray*}
\begin{equation*}
x_{m+n}-v_{m+n}=\left( a^{n}+\sum_{i=0}^{n-1}a^{i}b-1\right) x_{m}.
\end{equation*}
The orbit now moves upward along the line $v=x_{m}$ (see Fig.~\ref{Figure4}
). Then, there will exist a value of $n$ such that the trajectory crosses $
L_{\delta +}$, i.e.,
\begin{equation*}
\left| x_{m+n}-v_{m+n}\right| >\delta.
\end{equation*}
Note that
\begin{equation*}
v_{m+n}=x_{m}<0.
\end{equation*}
Then
\begin{eqnarray*}
\left( a^{n}+\sum_{i=0}^{n-1}a^{i}b-1\right) x_{m} &=&\left(
a^{n}+\sum_{i=0}^{n-1}a^{i}b-1\right) \left(
a^{m+1}+\sum_{i=0}^{m}a^{i}b\right) v_{0}>\delta \\
&\Leftrightarrow &1-a^{n}>\frac{\left( 1-a\right) \delta }{(a+b-1)\left(
\left( 1-a^{m+1}\right) \frac{a+b-1}{1-a}+1\right) v_{0}} \\
&\Leftrightarrow &n>\frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{
(a+b-1)\left( \left( 1-a^{m+1}\right) \frac{a+b-1}{1-a}+1\right) v_{0}}
\right) }{\ln a}.
\end{eqnarray*}
Hence, the smallest $n$ is
\begin{equation}
n=\left\lceil \frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{
(a+b-1)\left( \left( 1-a^{m+1}\right) \frac{a+b-1}{1-a}+1\right) v_{0}}
\right) }{\ln a}\right\rceil . \label{n}
\end{equation}
The switching law (Eq. \ref{original_constraint}) provokes that the new
iteration point
\begin{eqnarray*}
x_{m+n+1} &=&\left( a+b\right) x_{m+n}, \\
v_{m+n+1} &=&x_{m+n},
\end{eqnarray*}
returns to the zone where the trajectory was originated. In particular, if
\begin{equation*}
\left( v_{m+n+1},x_{m+n+1}\right) =\left( v_{0},x_{0}\right)
\end{equation*}
then one will get a closed orbit. This motivates us to define the first-return map
\begin{eqnarray}
\varphi &:&L_{1}\rightarrow L_{1} \notag \\
v &\mapsto &\left( a^{n}+\sum_{i=0}^{n-1}a^{i}b\right) \left(
a^{m+1}+\sum_{i=0}^{m}a^{i}b\right) v, \label{poincare_map}
\end{eqnarray}
where $L_{1}$ is the projection of $L_{1\delta +}$ onto the $v$
axis, and $m$ and $n$ satisfy Eqs. (\ref{m})-(\ref{n}),
respectively.
\begin{definition}
\label{chaos-1-firstreturn-def} (Type 1 periodic orbits) A periodic orbit
starting from $\left( v_{0},x_{0}\right)\in L_{1\delta +}$ is said to be of
type 1 if
\begin{equation}
\varphi \left( v_{0}\right) =v_{0}, \label{criterion_periodic}
\end{equation}
where $\varphi $ is defined by Eq. (\ref{poincare_map}), and $m$ and
$n$ satisfy Eqs. (\ref{m})-(\ref{n}), respectively. In this case,
the period of this orbit starting from $\left( v_{0},x_{0}\right) $
is $m+n+1$.
\end{definition}
\begin{remark}
\label{chaos-1-firstreturn-rema-type1} {\rm A periodic orbit is of
type 1 if it forms a closed loop right after the first return. There
are possibly other periodic orbits that become closed loops after
several returns. These orbits can be studied in a similar way, but
it is more computationally involved.}
\end{remark}
The following result follows immediately from the foregoing discussions.
\begin{theorem}
\label{chaos-1-firstreturn-thm-type1} The trajectory starting from $\left(
v_0,x_0\right) $ is periodic of type 1 if and only if Eq. (\ref
{criterion_periodic}) holds.
\end{theorem}
Actually, we can find all periodic orbits of type 1: If Eq. (\ref
{criterion_periodic}) holds, i.e.,
\begin{equation*}
\varphi \left( v_0\right) =v_0,
\end{equation*}
then
\begin{equation*}
\left( a^{n}+\sum_{i=0}^{n-1}a^{i}b\right) \left(
a^{m+1}+\sum_{i=0}^{m}a^{i}b\right) =1,
\end{equation*}
i.e.,
\begin{equation}
\left( \left( 1-a^{n}\right) \frac{a+b-1}{1-a}+1\right) \left( \left(
1-a^{m+1}\right) \frac{a+b-1}{1-a}+1\right) =1 \label{periodic_orbits}
\end{equation}
for some $m,n>0$. Given that $0<a<1$, $b<0$ and $\left| a+b\right|
<1$, $m$ and $n$ satisfying (\ref{periodic_orbits}) are both finite.
Hence, all periodic orbits of type 1 can be found.
\begin{remark}
\label{chaos-1-firstreturn-dense-periodic} {\rm If $\left(
v,x\right) $
leads to a periodic orbit of type 1, according to Eqs (\ref{m}) and (\ref{n}
), there exists a neighborhood of $\left( v,x\right) $ on
$L_{1\delta +}$ such that each point of which will lead to a
periodic orbit of type 1, so all such orbits are together dense.}
\end{remark}
\subsection{Case 2: $a=1$}
Assume that an initial condition $\left( v,x\right) $ satisfies
\begin{equation*}
x=\left( 1+b\right) v, v>0,
\end{equation*}
and also suppose that the orbit starting from it is within the oscillating
region. Then
\begin{eqnarray*}
x_{1} &=&x+bv=\left( 1+2b\right) v, \\
v_{1} &=&v,
\end{eqnarray*}
\begin{equation*}
\vdots
\end{equation*}
\begin{eqnarray*}
x_{m} &=&x_{1}+bv_{1}=\left( 1+\left( m+1\right) b\right) v, \\
v_{m} &=&v.
\end{eqnarray*}
Suppose that
\begin{equation*}
v_{m}-x_{m}=-\left( m+1\right) bv>\delta .
\end{equation*}
Then
\begin{equation*}
m+1>\frac{\delta }{\left( -b\right) v}.
\end{equation*}
Hence, the least $m$ is given by
\begin{equation*}
m=\left\lceil \frac{\delta }{\left( -b\right) v}\right\rceil -1.
\end{equation*}
Moreover,
\begin{eqnarray*}
x_{m+1} &=&\left( 1+b\right) x_{m}, \\
v_{m+1} &=&x_{m}<0,
\end{eqnarray*}
\begin{equation*}
\vdots
\end{equation*}
\begin{eqnarray*}
x_{m+n} &=&\left( 1+nb\right) x_{m}, \\
v_{m+n} &=&x_{m}.
\end{eqnarray*}
Suppose that
\begin{equation*}
x_{n+m}-v_{n+m}=nbx_{m}>\delta .
\end{equation*}
Then
\begin{equation*}
n>\frac{\delta }{bx_{m}}=\frac{\delta }{b\left( 1+\left( m+1\right) b\right)
v}.
\end{equation*}
Hence, the smallest $n$ is given by
\begin{equation*}
n=\left\lceil \frac{\delta }{b\left( 1+\left( m+1\right) b\right) v}
\right\rceil .
\end{equation*}
Define
\begin{eqnarray}
\varphi &:&L_{1}\rightarrow L_{1} \notag \\
v &\mapsto &\left( 1+nb\right) \left( 1+\left( m+1\right) b\right) v,
\label{Poinc_map_2}
\end{eqnarray}
If
\begin{equation*}
\varphi \left( v\right) =v,
\end{equation*}
then
\begin{equation}
\frac{1}{m+1}+\frac{1}{n}=-b. \label{peirodic}
\end{equation}
\begin{theorem}
\label{chaos-1-firstreturn-a=1-thm} The trajectory starting from $\left(
v,(1+b)v\right) $ is periodic of type 1 if and only if $v$ is a fixed point
of the first-return map defined in Eq. (\ref{Poinc_map_2}).
\end{theorem}
\begin{remark}
\label{chaos-1-firstreturn-a=1-rem} {\rm This result is a
generalization of Theorem 3 in Zhang \& Chen [2005], where the
condition is only necessary. For example, given $a=1$ and $b=-1/2$,
the origin is the unique invariant set.
It is obvious that $p=q=4$ is a solution to
\begin{equation*}
\frac{1}{p}+\frac{1}{q}=-b.
\end{equation*}
However, there are no periodic orbits. This indicates that the
necessary condition given by Theorem 3 in Zhang \& Chen [2005] is
not sufficient.}
\end{remark}
Next we find all periodic orbits of type 1 for the case of $a=1$.
For convenience, here we use $m$ instead of $m+1$ in Eq. (\ref{peirodic}).
Suppose
\begin{equation*}
b=-\frac{q}{p},
\end{equation*}
where $p>0$, $q>0$, $\gcd \left( p,q\right) =1$. According to Eq. (\ref
{peirodic}),
\begin{equation*}
\frac{1}{m}=-b-\frac{1}{n}=\frac{pn-q}{qn},
\end{equation*}
i.e.,
\begin{equation*}
m=\frac{qn}{pn-q}.
\end{equation*}
Obviously,
\begin{equation*}
m>\frac{q}{p}.
\end{equation*}
Furthermore, $m$ is a decreasing function of $n$. By symmetry, let
\begin{equation*}
n_{0}=\left\lceil \frac{q}{p}\right\rceil .
\end{equation*}
Then
\begin{equation*}
\left\lceil \frac{q}{p}\right\rceil \leq m\leq \left\lceil \frac{qn_{0}}{pn_{0}-q}
\right\rceil .
\end{equation*}
Similarly,
\begin{equation*}
\left\lceil \frac{q}{p}\right\rceil \leq n\leq \left\lceil \frac{qm_{0}}{
pm_{0}-q}\right\rceil ,
\end{equation*}
where
\begin{equation*}
m_{0}=\left\lceil \frac{q}{p}\right\rceil .
\end{equation*}
Based on this analysis and Theorem \ref{chaos-1-firstreturn-a=1-thm}, all
periodic orbits of type 1 can be determined.
\subsection{Case 3 $a=-1$ and $\left| a+b\right| <1$}
For this case, each trajectory is an eventually periodic orbit of period 2.
\subsection{Case 4 $a>1$}
This is similar to the case of $0<a<1$. The only difference is
\begin{equation*}
m<\frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{(1-(a+b))v}\right) }{
\ln a}-1,
\end{equation*}
due to $\ln a>0$. The least $m$ is
\begin{equation*}
m=\left\lceil \frac{\ln \left( 1-\frac{\left( 1-a\right) \delta }{(1-(a+b))v}
\right) }{\ln a}\right\rceil -1.
\end{equation*}
The complex dynamics exhibited in this system is due to its
nonlinearity induced by switching. This is different from that of a
quantized system. The complicated behavior of an unstable quantized
scalar system has been extensively studied in Delchamps [1988, 1989,
1990] and Fagnani \& Zampieri [2003], and the MIMO case is addressed
in Fagnani \& Zampieri [2004]. In Delchamps [1990], it is mentioned
that if the system parameter $a$ is stable, a quantized system may
have many fixed points as well as periodic orbits, which are all
asymptotically stable. However, for the constrained systems here,
almost all trajectories are not periodic orbits. For systems with
$a=1$, periodic orbits are locally stable, which is not the case for
a quantized system [Delchamps, 1990]. Given that $a$ is unstable,
the ergodicity of the quantized system is investigated in Delchamps
[1990]. In essence, related results there depend heavily on the
affine representation of the system by which the system is piecewise
expanding, i.e., the absolute value of the derivative of the
piecewise affine map in each interval partitioned naturally by it is
greater than $1$. Based on this crucial property, the main theorem
(Theorem 1) in Lasota \& Yorke [1973] and then that in Li \& Yorke
[1978] are employed to show that there exists a unique invariant
measure under the affine map on which the map is also ergodic.
Therefore, ergodicity has been established for scalar unstable
quantized systems. However, this is not the case for the system
studied here. Though the system is still piecewise linear, it is
\textbf{singular} with respect to the Lebesgue measure and,
furthermore, the derivative of the system map in a certain
region is $\left( a+b\right) $, whose absolute value is strictly less than $
1 $. Hence, the results in Lasota \& Yorke [1973] and Li \& Yorke
[1978] are not applicable here. However, by extensive experiments,
we strongly believe that the system indeed has the property of
ergodicity. This will be left as a conjecture for future research to
verify theoretically.
\section{Structural Stability}
Loosely speaking, a nonlinear system is \textit{structurally stable} if a
slight perturbation of its system parameters will not change its phase
portrait qualitatively. In Zhang \& Chen [2005], we proved that given $a=1$ in system (\ref
{original system}), if there are periodic orbits, then the system is not
structurally stable. In this section, we will show that the condition of $
a=1 $ is actually unnecessary.
\begin{figure}[tbh]
\epsfxsize=3.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure5.eps}} \caption{A global attracting
region of a type-1 generic system} \label{Figure5}
\end{figure}
\begin{figure}[tbh]
\epsfxsize=3.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure6.eps}} \caption{A global attracting
region of the generic system with $a=0.3$ and $b=-0.9$}
\label{Figure6}
\end{figure}
We begin with the simplest case, namely, a generic system, whose
unique attractor is the line segment of fixed points. Even such a
simple case can still be classified into at least two categories.
We proved in Zhang \& Chen [2005] that if the parameters of the
system defined by Eqs. (\ref{original
system})-(\ref{original_constraint}) satisfy
\begin{equation}
\frac{1-|a|}{1-(a+b)}>\frac{|b|}{1-|a+b|} \label{unique condition}
\end{equation}
with $|a+b|<1$ and $|a|<1$, then it is generic. One of its global
attracting regions is shown in Fig.~\ref{Figure5}, where $I_{o}$ is
the line segment of fixed points, which runs from the point
$(x_{2},x_{1})=\left(-\frac{(1-|a|)\delta}{1-(a+b)},
\frac{b}{1-a}\frac{(1-|a|)\delta}{1-(a+b)}\right)$ on the left to
the point $(x_{2},x_{1})=\left(\frac{(1-|a|)\delta}{1-(a+b)},
-\frac{b}{1-a}\frac{(1-|a|)\delta}{1-(a+b)}\right)$ on the right. A
trajectory will converge {\it vertically} to a certain point on
$I_o$. On the other hand, we proved that the system with parameters
$a=\frac{3}{10}$ and $b=-\frac{9}{10}$, which do not satisfy Eq.
(\ref{unique condition}), is also generic whose typical trajectories
are like that shown in Fig.~\ref{Figure6} (the trajectory starting
from $(0.005,0.005)$ around converges to a fixed point close to
$(0.0004,0.004)$ after several oscillations). Till now, we have not
found a third type of generic systems. Apparently the first generic
system is simpler than the second one. Hence, we first investigate
the structural stability of the first type of systems. For
convenience, we call such systems type-1 generic systems or generic
systems of type 1. Observing that each type-1 generic system has a
global attracting region as shown in Fig.~\ref{Figure5}, thereby we
focus on its behavior in this region.
The following result asserts that two generic systems of type 1 are
`identical' in the sense of topology:
\begin{proposition}\label{paper-prop1}
Two type-1 generic systems are homeomorphic, i.e., there exists
a bijective map form one to the other which has a continuous
inverse.
\end{proposition}
\noindent \textbf{Proof.}~~ For convenience, define
\begin{equation*}
x_{2}\left( k\right) :=v\left( k-1\right) ,~~k\geq 0.
\end{equation*}
Then, the original system defined in Eqs. (\ref{original system})-(\ref
{original_constraint}) is equivalent to
\begin{equation}
\left[
\begin{array}{c}
x_{1}\left( k+1\right) \\
x_{2}\left( k+1\right)
\end{array}
\right] =\left\{
\begin{array}{ll}
\left[
\begin{array}{cc}
a+b & 0 \\
1 & 0
\end{array}
\right] \left[
\begin{array}{c}
x_{1}\left( k\right) \\
x_{2}\left( k\right)
\end{array}
\right] , & \mbox{if~}\left| x_{1}\left( k\right) -x_{2}\left( k\right)
\right| >\delta , \\
\left[
\begin{array}{ll}
a & b \\
0 & 1
\end{array}
\right] \left[
\begin{array}{c}
x_{1}\left( k\right) \\
x_{2}\left( k\right)
\end{array}
\right] , & \mbox{otherwise.~~~}
\end{array}
\right. \label{modified system}
\end{equation}
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure7.eps}} \caption{Global attracting
regions of systems $\Sigma _{1}$ and $\Sigma _{2}$ } \label{Figure7}
\end{figure}
Consider the following two type-1 generic systems whose global
attracting regions are shown in Fig.~\ref{Figure7}:
\begin{equation*}
\Sigma _{1}:~~ x\left( k+1\right) =\left\{
\begin{array}{ll}
B_{1}x\left( k\right) , & \mbox{if~}\left| x_{1}\left( k\right) -x_{2}\left(
k\right) \right| >\delta , \\
A_{1}x\left( k\right) , & \mbox{otherwise,~~~}
\end{array}
\right.
\end{equation*}
\begin{equation*}
\Sigma _{2}:~~ y\left( k+1\right) =\left\{
\begin{array}{ll}
B_{2}x\left( k\right) , & \mbox{if~}\left| y_{1}\left( k\right) -y_{2}\left(
k\right) \right| >\delta , \\
A_{2}x\left( k\right) , & \mbox{otherwise,~~~}
\end{array}
\right.
\end{equation*}
in which
\begin{equation*}
x=\left[
\begin{array}{c}
x_{1} \\
x_{2}
\end{array}
\right] , ~ y=\left[
\begin{array}{c}
y_{1} \\
y_{2}
\end{array}
\right] , ~ A_{i}=\left[
\begin{array}{cc}
a_{i} & b_{i} \\
0 & 1
\end{array}
\right] ,~B_{i}=\left[
\begin{array}{cc}
a_{i}+b_{i} & 0 \\
1 & 0
\end{array}
\right] ,~~i=1,2.
\end{equation*}
Next, we define a map $h$ from $I_{x}$ to $I_{y}$ by
\begin{eqnarray}
h &:&I_{x}\longrightarrow I_{y}, \label{projection-fixed points} \\
&&\left( \frac{b_{1}}{1-a_{1}}x_{2},x_{2}\right) \mapsto \left( \frac{b_{2}
}{1-a_{2}}\frac{\frac{1-\left| a_{2}\right| }{1-(a_{2}+b_{2})}}{\frac{
1-\left| a_{1}\right| }{1-(a_{1}+b_{1})}}x_{2},\frac{\frac{1-\left|
a_{2}\right| }{1-(a_{2}+b_{2})}}{\frac{1-\left| a_{1}\right| }{
1-(a_{1}+b_{1})}}x_{2}\right) . \nonumber
\end{eqnarray}
Clearly, $h$ is one-to-one, onto and has a continuous inverse.
Next, define
\begin{eqnarray}
\tilde{h} &:&\Sigma _{1}\longrightarrow \Sigma _{2}, \label{new map} \\
&&\left( x_{1},x_{2}\right) \mapsto \left( \frac{\frac{b_{2}}{1-a_{2}}}{
\frac{b_{1}}{1-a_{1}}}\frac{\frac{1-\left| a_{2}\right| }{1-(a_{2}+b_{2})}}{
\frac{1-\left| a_{1}\right| }{1-(a_{1}+b_{1})}}x_{1},\frac{\frac{1-\left|
a_{2}\right| }{1-(a_{2}+b_{2})}}{\frac{1-\left| a_{1}\right| }{
1-(a_{1}+b_{1})}}x_{2}\right) . \nonumber
\end{eqnarray}
It is easy to see that the projection of $\tilde{h}$ on $I_{x}$ is
exactly $h $, and furthermore $\tilde{h}$ is a homeomorphism.
Actually system $\Sigma _{2}$ can be obtained by stretching (or
contracting) system $\Sigma _{1}$, therefore they are topologically
equivalent. $\hfill$ $\blacksquare$
Two remarks are in order.
\begin{remark}{\rm
It is hard to apply the foregoing method to other types of generic
systems because they may not have such simple global attracting
regions.}
\end{remark}
\begin{remark}{\rm
As can be conjectured, this method is probably not applicable
to non-generic systems that have some complex attractors besides the
line segment of fixed points (see [Zhang \& Chen, 2005] for
details).}
\end{remark}
Before investigating the structural stability of generic systems, we
first discuss their $\omega $-stability. A dynamical system is
$\omega $-stable if there exists a homeomorphism from its
\textit{non-wandering} set (here it is $I_{o}$) to that of the
system obtained by perturbing it slightly [Smale, 1967].
Hence, a structurally stable dynamical system is necessarily $\omega $
-stable, but the converse may not be true. Because there always
exists a homeomorphism between two given line segments, it seems
plausible to infer that a generic system is $\omega $-stable.
Unfortunately, it is not true. Observe that Proposition
\ref{paper-prop1} holds upon the assumption that two given systems
are generic, some other ``unusual'' types of perturbations may lead
to a system that is not generic, thus destroying the $\omega
$-stability of generic systems. To that end, a new point of view is
required.
Define a family of systems:
\begin{equation}
\left[
\begin{array}{c}
x_{1}\left( k+1\right) \\
x_{2}\left( k+1\right)
\end{array}
\right] =\left\{
\begin{array}{ll}
A_{1}\left[
\begin{array}{c}
x_{1}\left( k\right) \\
x_{2}\left( k\right)
\end{array}
\right], & \mbox{if~}\left| x_{1}\left( k\right) -x_{2}\left( k\right)
\right| >\delta , \\
A_{2}\left[
\begin{array}{c}
x_{1}\left( k\right) \\
x_{2}\left( k\right)
\end{array}
\right], & \mbox{otherwise,~~~}
\end{array}
\right. \label{family}
\end{equation}
where
\begin{equation}
A_{1}=\left[
\begin{array}{cc}
a+b & 0 \\
1 & 0
\end{array}
\right] , ~~ A_{2}=\left[
\begin{array}{cc}
a+\lambda b & \left( 1-\lambda \right) b \\
\lambda & \left( 1-\lambda \right)
\end{array}
\right] . \label{system matx.}
\end{equation}
Note that when $\lambda =1$, $A_{2}=A_{1}$, and that this system is a stable
linear system. When $\lambda =0$, $A_{2}=\left[
\begin{array}{cc}
a & b \\
0 & 1
\end{array}
\right] $, giving the system defined by Eq. (\ref{modified system}). Hence,
by introducing $\lambda \in \left[ 0,1\right] $, one gets a family of
systems.
It is easy to verify the following result:
\begin{theorem}
\label{ss_thm_2} For each $\lambda \in (0,1]$, system (\ref{family}) has a
unique fixed point $\left( 0,0\right) $.
\end{theorem}
Consider a perturbation of a system in the form of (\ref{modified system})
by choosing a $\lambda$ sufficiently close to but not equal to zero. Theorem
\ref{ss_thm_2} tells us that the new system has a unique fixed
point. Clearly there are no homeomorphisms between these two systems
since there exist no one-to-one maps from a line segment to a single
point. Moreover, a non-generic system also has a line segment of
fixed points. So, we have the following conclusion:
\begin{theorem}
\label{ss_thm_omega_stability} System (\ref{modified system}) is not $\omega$
-stable.
\end{theorem}
The following is an immediate consequence.
\begin{corollary}
\label{ss_coro_1} System (\ref{modified system}) is not structurally stable.
\end{corollary}
\begin{remark}{\rm
The above investigation tells us that the system with $\lambda
=0$ is a rather ill-conditioned one. Will a system with $\lambda \neq 0$ be $
\omega $-stable (or even structurally stable): We are convinced that
this generally holds, but till now we have not found a proof.}
\end{remark}
\begin{remark}{\rm
For a generic system,
no matter it is of type 1 or not, its non-wandering set is just a
line segment of fixed points, therefore its $\omega $-stability is
preserved if a perturbation is on $a$ and $b$, while {\it not}
destroying the structure shown in Eq. (\ref{modified system}).
Consider the discussion in Sec.~2, the system composed of
(\ref{original system})-(\ref{original_constraint}) is proposed for
a new data transmission strategy, hence though the perturbation of
$a$ and $b$ is reasonable, the perturbation of the form
(\ref{family})-(\ref{system matx.}) induced by $\lambda$ does not
make sense physically. Based on this, we can say that $\omega
$-stability is ``robust'' with respect to uncertainty which is
meaningful (The same argument is proposed in Robbin [1972] for
structural stability). However, it is pretty fragile with respect to
such rare uncertainty as that in Eqs. (\ref{family})-(\ref{system
matx.}). In other words, it is robust yet fragile. It is argued in
Doyle [2004] that `robust yet fragile' is the most important
property of complex systems.}
\end{remark}
\begin{remark}
{\rm It follows form the above results that there is a transition
process in the family of systems defined by Eqs.
(\ref{family})-(\ref{system matx.}) as $\lambda $ moves from 1 to 0,
which has been discussed in our another paper [Zhang {\it et al.},
2005].}
\end{remark}
\section{Higher-Order Systems}
In this section, we briefly discuss the high-dimensional cases.
Consider the following two-dimensional system:
\begin{eqnarray*}
x_{1}\left( k+1\right) &=&a_{1}x_{1}\left( k\right) +b_{1}x_{2}\left(
k\right) , \\
x_{2}\left( k+1\right) &=&a_{2}x_{2}\left( k\right) +b_{2}v\left( k\right),
\end{eqnarray*}
where
\begin{equation*}
v\left( k\right) =\left\{
\begin{array}{ll}
x_{1}\left( k\right), & \mbox{if} \left| x_{1}\left( k\right) -v\left(
k-1\right) \right| >\delta, \\
v\left( k-1\right), & \mbox{otherwise.}
\end{array}
\right.
\end{equation*}
Introduce a new variable,
\begin{equation*}
x_{3}\left( k\right) =v\left( k-1\right) ,
\end{equation*}
and define
\begin{equation*}
x=\left[
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}
\right] , ~~ A_{1}=\left[
\begin{array}{ccc}
a_{1} & b_{1} & 0 \\
b_{2} & a_{2} & 0 \\
1 & 0 & 0
\end{array}
\right] , ~~ A_{2}=\left[
\begin{array}{ccc}
a_{1} & b_{1} & 0 \\
0 & a_{2} & b_{2} \\
0 & 0 & 1
\end{array}
\right] .
\end{equation*}
Then
\begin{equation}
x\left( k+1\right) =\left\{
\begin{array}{ll}
A_{1}x\left( k\right), & \mbox{if} \left| x_{1}\left( k\right) -x_{3}\left(
k\right) \right| >\delta, \\
A_{2}x\left( k\right), & \mbox{otherwise.}
\end{array}
\right. \label{2-d system}
\end{equation}
\subsection{Fixed points and switching surfaces}
Suppose that $\left( \bar{x}_{1},\bar{x}_{2},\bar{x}_{3}\right) $ is a fixed
point of system (\ref{2-d system}). Then
\begin{eqnarray*}
\bar{x}_{1} &=&a_{1}\bar{x}_{1}+b_{1}\bar{x}_{2}, \\
\bar{x}_{2} &=&a_{2}\bar{x}_{2}+b_{2}\bar{x}_{3}, \\
\bar{x}_{3} &=&\bar{x}_{3}.
\end{eqnarray*}
If $a_{1}\neq 1$ and $a_{2}\neq 1$, then
\begin{equation}
\bar{x}_{2}=\frac{b_{2}}{1-a_{2}}\bar{x}_{3}, ~~ \bar{x}_{1}=\frac{b_{1}}{
1-a_{1}}\frac{b_{2}}{1-a_{2}}\bar{x}_{3}, \label{case1a}
\end{equation}
where
\begin{equation}
\left| \bar{x}_{3}\right| \leq \frac{\delta }{\left| \frac{b_{1}b_{2}}{
\left( 1-a_{1}\right) \left( 1-a_{2}\right) }-1\right| }. \label{case1b}
\end{equation}
If $a_{1}\neq 1$ and $a_{2}=1$, then
\begin{equation}
\bar{x}_{3}=0, ~~ \bar{x}_{1}=\frac{b_{1}}{1-a_{1}}\bar{x}_{2}
\label{case2a}
\end{equation}
and
\begin{equation}
\left| \bar{x}_{2}\right| \leq \left| \frac{\left( 1-a_{1}\right) \delta }{
b_{1}}\right| . \label{case2b}
\end{equation}
If $a_{1}=1$ and $a_{2}\neq 1$, then
\begin{equation}
\bar{x}_{2}=0, ~~ \bar{x}_{3}=0 \label{case3a}
\end{equation}
and
\begin{equation}
\left| \bar{x}_{1}\right| \leq \delta . \label{case3b}
\end{equation}
In all the three cases, fixed points constitute a line segment in $\mathbb{R}
^{3} $. Note that the case of $a_{1}=1$ and $a_{2}=1$ is contained
in the case defined by Eqs. (\ref{case3a})-(\ref{case3b}).
Next, we consider the first case. Obviously the switching surfaces of this
system are
\begin{equation*}
x_{1}-x_{3}=\pm \delta ,
\end{equation*}
hence the two end points of the line of fixed points are
\begin{equation*}
\pm \left( \frac{k_{1}}{k_{1}-1}\delta , \ \frac{b_{2}}{1-a_{2}}\frac{k_{1}}{
k_{1}-1}\delta , \ \frac{1}{k_{1}-1}\delta \right) ,
\end{equation*}
where $k_{1}=\frac{b_{1}}{1-a_{1}}\frac{b_{2}}{1-a_{2}}$. They are symmetric
with respect to the origin.
\subsection{An example}
Based on the analysis in Sec.~4, one can see that system (\ref{2-d
system}) is not structurally stable. In this section, we illustrate
the complex behavior of this system.
Consider the following system:
\begin{equation}
x\left( k+1\right) =\left\{
\begin{array}{ll}
A_{1}x\left( k\right), & \mbox{if~}\ \left| x_{1}\left( k\right)
-x_{3}\left( k\right) \right| >1, \\
A_{2}x\left( k\right), & \mbox{otherwise,}
\end{array}
\right. \label{paper2_2-d system}
\end{equation}
where
\begin{equation*}
A_{1}=\left[
\begin{array}{ccc}
1-\epsilon & 1 & 0 \\
-\epsilon/2 & 1 & 0 \\
1 & 0 & 0
\end{array}
\right] , ~ A_{2}=\left[
\begin{array}{ccc}
1-\epsilon & 1 & 0 \\
0 & 1 & -\epsilon/2 \\
0 & 0 & 1
\end{array}
\right].
\end{equation*}
The variations of the trajectory, starting from $(1, ~ 1/10^5, ~ -1)
$ as $\epsilon$ varies, are plotted in
Figs.~\ref{Figure8}-\ref{Figure9}. One can see the phase transition
process vividly from these figures.
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure8.eps}} \caption{Attractors in 3-d:I }
\label{Figure8}
\end{figure}
\begin{figure}[tbh]
\epsfxsize=4.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure9.eps}} \caption{Attractors in 3-d:II }
\label{Figure9}
\end{figure}
Figs.~\ref{Figure8}-\ref{Figure9} reveal the rich dynamics of a 3-d
system governed by the switching law, which will be our future
research topic.
\section{The Continuous-Time Case}
In this section, we study the continuous-time counterpart of the
discrete-time system (\ref{2-d system}). The first motivation is to
check the data transmission strategy for analog channels; the second
is that discrete-time systems can be regarded as continuous ones if
their base frequencies are much bigger than the network data
transmission rate [Walsh {\it et al.}, 2002]; the third is that a
system governed by this transmission strategy possesses very rich
dynamics, thus it is also interesting in its own right.
\subsection{System setting}
Consider the following system:
\begin{eqnarray}
\dot{x}_{1}(t) &=&a_{1}x_{1}(t)+b_{1}x_{2}(t), \label{chap8_main_system} \\
\dot{x}_{2}(t) &=&a_{2}x_{2}(t)+b_{2}v(t), \notag
\end{eqnarray}
where the matrix $A=\left(
\begin{array}{cc}
a_{1} & b_{1} \\
b_{2} & a_{2}
\end{array}
\right)$ is stable, and the switching law is given by
\begin{equation} \label{chap8_main_switching}
v(t)=\left\{
\begin{array}{ll}
x_{1}(t), & \mbox{if~} \left| x_{1}(t)-v(t_{-})\right| >\delta , \\
v(t_{-}), & \mbox{otherwise,}
\end{array}
\right.
\end{equation}
in which $\delta$ is a positive scalar.
As can be observed, the system governed by Eqs. (\ref{chap8_main_system})-(
\ref{chap8_main_switching}) is the continuous-time counterpart of system (
\ref{2-d system}). Note that the system consists of two first-order
ordinary differential equations. Furthermore, if we fix $v(0_{-})$
to be $0$, then the above system is autonomous. Moreover, due to the
switching nature of the system, the vector field of this system may
not be continuous for some set of parameters, not to mention
differentiability. This suggests that the well-known
Poincar$\acute{e}$-Bendixson theorem might not be applicable [Hale
\& Kocak, 1991], i.e., besides equilibria and periodic orbits,
the $\omega$-limit set of this system may contain other attractors.
This turns out to be true as shown by the following simulations.
However, before doing that, let us first state a
general result regarding the system composed of Eqs. (\ref{chap8_main_system})-
(\ref{chap8_main_switching}).
\begin{theorem}
\label{chap8_bdd} The trajectories of the system given by Eqs. (\ref
{chap8_main_system})-(\ref{chap8_main_switching}) are bounded.
Moreover, they converge to the origin as $\delta$ tends to zero.
\end{theorem}
Its proof is omitted due to space limitation.
\subsection{Simulations}
\begin{figure}[t]
\epsfxsize=3.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure10.eps}} \caption{A continuous-time
switching system} \label{Figure10}
\end{figure}
Consider the simulink model shown in Fig.~\ref{Figure10}. The system
$G$ and the controller $C$ are modeled by the first and the second
equations in Eq. (\ref{chap8_main_system}), respectively. The function $
\mathrm{FCN}$, modelling the switching function (\ref{chap8_main_switching}
), is defined by
\begin{equation}
f(u):=u_1+u_2-u_2\ast (\left| u_1-u_2\right| >\delta)-u_1\ast
(\left| u_1-u_2\right| \leq\delta)~. \label{Chap8_n}
\end{equation}
Thus, by letting
\begin{equation*}
u=\left[
\begin{array}{l}
u_1 \\
u_2
\end{array}
\right] ,
\end{equation*}
one has
\begin{equation}
v:=f\left( u\right) =\left\{
\begin{array}{ll}
u_1, & \text{if }\left| u_1-u_2\right| >\delta, \\
u_2, & \text{otherwise.}
\end{array}
\right. \label{chap8_constraint}
\end{equation}
Is the block \textrm{FCN}, the function $f$, well-defined? It
suffices to verify the case at time $0$. Firstly, choose $\left|
x_{1}\left( 0\right) \right| <\delta$. Then $u_1=x_{1}\left(
0\right) $. By simulation, it is found that $v=0$. Secondly, choose
$\left| x_{1}\left( 0\right) \right|
>\delta$. Then $u_1=x_{1}\left( 0\right) $. Simulation shows that $
v=x_{1}\left( 0\right) $. To simplify the notation, denote $u_2$ at time $0$
by $v\left( 0_{-}\right) $. Summarizing the above, one has
\begin{eqnarray*}
v\left( 0_{-}\right) &=&0, \\
v\left( 0\right) &=&\left\{
\begin{array}{ll}
x_{1}\left( 0\right) , & \text{if }\left| x_{1}\left( 0\right) -v\left(
0_{-}\right) \right| >\delta, \\
v\left( 0_{-}\right) , & \text{otherwise.}
\end{array}
\right.
\end{eqnarray*}
Similarly,
\begin{equation*}
v\left( t\right) =\left\{
\begin{array}{ll}
x_{1}\left( t\right) , & \text{if }\left| x_{1}\left( 0\right) -v\left(
0_{-}\right) \right| >\delta, \\
v\left( t_{-}\right) , & \text{otherwise,}
\end{array}
~~\mbox{~for~ }t>0,\right.
\end{equation*}
where $v\left( t_{-}\right) $ is either some previous value of the state $
x_{1}$, say $x_{1}\left( t-t_{0}\right) $ for some $t_{0}$ satisfying $
0<t_{0}\leq t$, or $v\left( 0_{-}\right) =0$. Hence, the block
\textrm{FCN} is well-defined.
Next, we find the equilibria of the system. As expected, the
equilibria of the system constitute a line segment, just as in the
discrete-time case. The equilibria are given by
\begin{equation} \label{Chap8_fixed_points}
\Lambda =\left\{ \left( x_{1}=\frac{b_{1}b_{2}}{a_{1}a_{2}}v, ~ x_{2}=-\frac{
b_{2}}{a_{2}}v, ~v\right) :\left| v\right| \leq \frac{\delta }{\left| 1-
\frac{b_{1}b_{2}}{a_{1}a_{2}}\right| }\right\}.
\end{equation}
For the system composed of Eqs. (\ref{chap8_main_system})-(\ref
{chap8_main_switching}), an interesting question is: Given an initial
condition $x(0)$, will $x(t)$ settle to a certain equilibrium or converge to
a periodic orbit or have more complex behavior? There are two ways of
tracking a trajectory $x(t)$: one is to solve Eqs. (\ref{chap8_main_system}
)-(\ref{chap8_main_switching}) directly, and the other is by means of
numerical methods. To get an analytic solution, one has to detect the
discontinuous points of the right-hand side of Eq. (\ref{chap8_main_system}
). We first show that the number of the discontinuous points within any
given time interval is finite.
Start at some time $t_{0}\geq 0$ and assume that $\left( x_{1}\left(
t_{0}\right), x_{2}\left( t_{0}\right) \right) $ and $v\left( t_{0-}\right)
=x_{1}\left( t_{0}\right) $ are given, without loss of generality. Suppose
that the first jump of $v$ is at instant $t_{0}+T$. To be specific in the
following calculation, let $t_{0}=0$. Then
\begin{eqnarray*}
x_{1}\left( T\right) &=&e^{a_{1}T}x_{1}\left( 0\right)
+\int_{0}^{T}e^{a_{1}\left( T-\tau \right) }b_{1}x_{2}\left( \tau \right)
d\tau , \\
x_{2}\left( T\right) &=&e^{a_{2}T}x_{2}\left( 0\right)
+\int_{0}^{T}e^{a_{2}\left( T-\tau \right) }b_{2}x_{1}\left( 0\right) d\tau
\\
&=&e^{a_{2}T}x_{2}\left( 0\right) +\int_{0}^{T}e^{a_{2}u}dub_{2}x_{1}\left(
0\right) .
\end{eqnarray*}
Consequently,
\begin{equation} \label{large}
x_{1}\left( T\right) -x_{1}\left( 0\right) =\left(
e^{a_{1}T}-1\right)x_{1}(0)\left (1-\frac{b_{1}b_{2}}{a_{1}a_{2}}
\right)+\left( e^{a_{2}T}-e^{a_{1}T}\right) b_{1}\frac{x_{2}\left( 0\right) +
\frac{b_{2}}{a_{2}}x_{1}\left( 0\right) }{a_{2}-a_{1}}.
\end{equation}
As $T\rightarrow 0$, $e^{a_{1}T}-1\rightarrow 0$, and $
e^{a_{2}T}-e^{a_{1}T}\rightarrow 0$. Moreover, we have already shown
the boundedness of solutions, so there exists a $T^{\ast }>0$ such
that
\begin{equation}
\left| x_{1}\left( T\right) -v\left( 0\right) \right|=\left| x_{1}\left(
T\right) -x_{1}\left( 0\right) \right| <\delta \label{chap8_terminal}
\end{equation}
for all $T<T^{\ast }$. Thus, the finiteness of the number of the
discontinuous points within any given time interval is established.
Based on this result, theoretically one can find the analytic solution of
the system. However, it is difficult since the condition in Eq. (\ref
{chap8_terminal}) has to be checked all the time to determine the
switching time $T$. Moreover, this process depends on the initial
point, $\left( x_{1}\left( t_{0}\right), x_{2}\left( t_{0}\right)
\right) $, which is hard
due to the impossibility of finding the exact $T$ satisfying $
\left| x_{1}\left( T\right) -v\left( 0\right) \right|=\delta$. This
problem will be addressed in more details in Sec.~6.7.
Another route to study this type of systems is by means of numerical
solutions. In the following, some simulations will be shown to
analyze the complexity of the system depicted in
Fig.~\ref{Figure10}. In all the following trajectory figures, the
horizontal axes stands for $x_1$ and the vertical one is $x_2$.
\subsection{Converging to some fixed point}
\begin{figure}[th]
\epsfxsize=3.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure11.eps}} \caption{Converging to an
equilibrium other than the origin} \label{Figure11}
\end{figure}
Fix those parameters shown in Fig.~\ref{Figure10} to be:
\begin{equation*}
a_{1}=-1,~~ b_{1}=2,~~ a_{2}=-2,~~ b_{2}=-2,
\end{equation*}
and choose an initial condition $(10, -10)$. Then, we get simulation
results shown in Fig.~\ref{Figure11}. One can see that this
trajectory converges to a point specified by Eq.
(\ref{Chap8_fixed_points}), which is close, but not equal, to the
origin.
\subsection{Sensitive dependence on initial conditions}
First, fix system parameters as
\begin{equation} \label{Chap8_unstable_parmi}
a_{1}=1,~b_{1}=2,~a_{2}=-2,~b_{2}=-2,~\delta=1,
\end{equation}
and note that there is an \textit{unstable} pole in the system $G$.
Suppose
\begin{equation*}
x_{1}\left( 0\right) =2,\quad x_{2}\left( 0\right) =1,
\end{equation*}
and
\begin{equation*}
x_{1}\left( 0\right) =2-10^{-10},\quad x_{2}\left( 0\right) =1.
\end{equation*}
\begin{figure}[th]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure12.eps}} \caption{Sensitive dependence
on initial conditions (the horizontal axes is $x_1$ and the vertical
one is for $x_2$)} \label{Figure12}
\end{figure}
Then, we get the simulation result shown in Fig.~\ref{Figure12},
where the first two are trajectories from those two sets of initial
conditions given above and the third one is their difference.
Clearly, one can see the sensitive dependence on initial conditions.
\subsection{Coexisting attractors}
\begin{figure}[th]
\epsfxsize=5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure13.eps}} \caption{Several coexisting
attractors (the horizontal axes is $x_1$ and the vertical stands for
$x_2$)} \label{Figure13}
\end{figure}
Adopt the system parameters as in Eq. (\ref{Chap8_unstable_parmi}).
Then, we have simulation results shown in Fig.~\ref
{Figure13}, where the largest initial differences of any two such
trajectories is $10^{-3}$. These attractors are all alike; however,
they are located in different positions; that is, they are
coexisting attractors.
\subsection{A periodic orbit}
Now choose
\begin{equation} \label{Chap8_unstable_parmi2}
a_{1}=-11,~b_{1}=1/4,~a_{2}=10,~b_{2}=1/4-(a_{1}-a_{2})^2,~\delta=1.
\end{equation}
Simulations show that most trajectories behave like the one shown in Fig.~\ref
{Figure14}, which is periodic.
\begin{figure}[th]
\epsfxsize=6in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure14.eps}} \caption{A periodic orbit (the
horizontal axes is $x_1$ and the vertical one is for $x_2$)}
\label{Figure14}
\end{figure}
Having observed various complex dynamics possessed by the system
shown in Fig.~\ref{Figure10}, one may ask the following question:
\begin{center}
\textit{Is the complexity exhibited by the system due to numerical errors or
is the system truly chaotic? }
\end{center}
We received the following warning during our simulations using
Simulink: \textit{Block diagram ``A continuous-time switching
system'' contains 1 algebraic loop(s).} This warning is due to the
fact that one of the output of the function block FCN is its own
input. Certainly, this may lead to numerical errors. So, a transport
delay is added to rule out this
possibility. This consideration leads to the following scheme (Fig.~\ref
{Figure15}):
\begin{figure}[th!]
\epsfxsize=6in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure15.eps}} \caption{A modified
continuous-time switching system} \label{Figure15}
\end{figure}
To correctly implement Eq. (\ref{chap8_main_switching}), the
transport delay T must be small enough. Here, it is fixed to be
$T=5*10^{-2}$. Suppose that system parameters are given by Eq.
(\ref{Chap8_unstable_parmi}) and choose two sets of initial
conditions, $(2,1)$ and $(2-10^{-6},1)$. Then, we get
Fig.~\ref{Figure16}.
\begin{figure}[th!]
\epsfxsize=6in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure16.eps}} \caption{Sensitive dependence
on initial conditions (the horizontal axes is $x_1$ and the vertical
one is for $x_2$)} \label{Figure16}
\end{figure}
According to the upper part of this plot, two trajectories almost coincide;
however, the lower plot clearly reveals sensitive dependence on initial
conditions.
For a sufficiently small transport delay $T$, many simulations show
that the complex attractor is unique, but sensitive dependence on
initial conditions still persists. Apparently, this phenomenon needs
further investigations.
\subsection{Computational Complexity}
We have visualized some complex behaviors of system (\ref
{chap8_main_system})-(\ref{chap8_main_switching}), but we have not answered
the question posed above. In this section, we study this problem in some
details.
Consider the following system:
\begin{eqnarray}
\dot{x}_{1}\left( t\right) &=&a_{1}x_{1}\left( t\right) +b_{1}x_{2}\left(
t\right) , \label{systems} \\
\dot{x}_{2}\left( t\right) &=&a_{2}x_{2}\left( t\right) +b_{2}p, \notag
\end{eqnarray}
where
\begin{equation*}
a_{1}=1,~b_{1}=2,~a_{2}=-2,~b_{2}=-2, ~\delta=1,
\end{equation*}
and $p$ is a scalar. At $t=0$, let
\begin{equation*}
x_{1}\left( 0\right) =p,~x_{2}\left( 0\right) =q.
\end{equation*}
Then, a direct calculation gives
\begin{eqnarray}
x_{1}\left( t\right) &=&2p-\frac{1}{3}e^{t}\left( p-2q\right) -\frac{2}{3}
e^{-2t}\left( p+q\right) , \label{solutions} \\
x_{2}\left( t\right) &=&-p+e^{-2t}\left( p+q\right) . \notag
\end{eqnarray}
Suppose $p\neq 2q$. Then, there exists an instant $t_{0}>0$ such that
\begin{equation}
\left| p-x_{1}\left( t_{0}\right) \right| =1. \label{Chap8_Sec3_switching}
\end{equation}
Set
\begin{equation*}
p=x_{1}\left( t_{0}\right) .
\end{equation*}
And then solve equations (\ref{systems}) starting from $\left(
x_{1}\left( t_{0}\right) ,~ x_{2}\left( t_{0}\right) \right) $ at time $
t_{0} $. Repeat this procedure (update $p$ whenever Eq. (\ref
{Chap8_Sec3_switching}) is satisfied) to get an analytic solution of the
system starting from $\left( x_{1}\left( 0\right), x_{2}\left(
0\right)\right)=(p,q) $.
Now, it is easy to realize that the complexity may probably be due
to the following reasons:
\begin{itemize}
\item There is an unstable mode in $x_{1}(t)$ in Eq. (\ref{solutions}).
\item It is hard to find the exact switching time, e.g., $t_{0}$ in Eq. (\ref
{Chap8_Sec3_switching}), even numerically. Because of this,
numerical errors will accumulate and be exaggerated from time to
time by the unstable mode. Further research is required to study the
effect of the accumulated errors on the dynamics of the system.
\item Sec.~6.6 tells us the first item alone can not guarantee complex
behavior.
\end{itemize}
\section{Control Based on the Network Protocol}
Because this research originates from network-based control, in this section
we discuss some control problems under this transmission strategy.
In Zhang \& Chen [2005], concentrated on a scalar case, chaotic
control is investigated. Here, we consider a tracking problem:
suppose the controller $C$ has been designed for the system $G$ in
shown Fig.~\ref{Figure2}, so that the output $y$ tracks the
reference signal $r$. How does the nonlinear constraints $H_{1}$ and
$H_{2}$ affect this tracking problem? We begin with a simple
example.
\begin{example}
{\rm Consider the following discrete-time system $G$:
\[
\frac{0.005z^{-1}+0.005z^{-2}}{1-2z^{-1}+z^{-2}}.
\]
Note that this system is {\it unstable}. Suppose we have already
designed a controller $K$ of the form
\[
\frac{37.33-33.78z^{-1}}{1-0.1111z^{-1}},
\]
which achieves step tracking. First, we check the data transmission
strategy by simulating the system shown in Fig.~\ref{Figure3}, with
$r\equiv 1$ but without $H_{2}$ involved therein. Choose $\delta
_{1}=1$. Then, the tracking error is plotted (the $*$ line in
Fig.~\ref{Figure17}).
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure17.eps}} \caption{Tracking error with
$\delta_{1}=1$} \label{Figure17}
\end{figure}
After that, we
modify the control law as follows:
\[
v(k)=H_{1}\left( u_{c}\left( k\right) ,v(k-1)\right) =\left\{
\begin{array}{ll}
u_{c}(k), & \mbox{if~}\left| u_{c}\left( k\right) -v\left(
k-1\right)
\right| >1, \\
\varepsilon _{1}v(k-1)+\varepsilon _{2}x\left( k\right) , & \mbox{otherwise,}
\end{array}
\right.
\]
where $\varepsilon _{1}\in \mathbb{R},$ $\varepsilon _{2}\in
\mathbb{R}^{1\times 2}$ are to be determined. Here, we assume that
the state $x$ of the system $G$ is available. When there is no
transmission from $C$ to $G$, instead of simply using the previously
stored control value $v(k-1)$, $\varepsilon _{1}v(k-1)+\varepsilon
_{2}x\left( k\right) $ is used. The reason is that by adjusting
$\varepsilon _{1}$ and $\varepsilon _{2}$ sensibly we may achieve
better control performance. By selecting
\[
\varepsilon _{1}=0.86, ~ \varepsilon _{2}=\left[ -0.21 ~ 0.21\right]
,
\]
the tracking error is plotted (the dotted line in Fig.~\ref{Figure17}
).
\begin{figure}[tbh]
\epsfxsize=4in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure18.eps}} \caption{Tracking error with
$\delta_{1}=1/2$} \label{Figure18}
\end{figure}
In this simulation, iteration time is 350, so the dropping rate can
also be obtained: the former is $85.14\%$ and the latter is
$96.57\%$. Hence, the modified control law is more effective in
reducing data traffics. The steady-state error shown in
Fig.~\ref{Figure17} under the modified control law is around $0.0774
$. Next, we choose $\delta _{1}=1/2$, and get the result shown in Fig.~\ref
{Figure18} following the same procedure. The dropping rates are
$83.14\%$ for the original and $96.29\%$ for the modified. In this
case the steady-state tracking error for the modified system is
around $0.0079$. So, by modifying the control law, we increased the
dropping rate therefore reduced the data traffics, and at the same
time improved the performance of the control system.}
\end{example}
In the above example, it is shown that for simple control systems it
is possible to improve the performance of both the network and the
control system by modifying the underlying control law. Clearly, it
is more mathematically involved when one confronts a more complex
system. In the following we transform this problem into an
optimization problem.
Define $\varsigma =\eta -\xi $, and $e_{r}:=e_{c}-e$. By subtracting the
system in (\ref{clsys1}) from that in (\ref{clsys2}), we get
\begin{eqnarray}
\varsigma (k+1) &=&\check{A}\varsigma (k)+\left[
\begin{array}{cc}
B & BD_{d} \\
0 & B_{d}
\end{array}
\right] \left( \left[
\begin{array}{c}
H_{1}\left( u_{c}\left( k\right) ,v(k-1)\right) \\
H_{2}\left( y_{c}\left( k\right) ,z(k-1)\right)
\end{array}
\right] -\left[
\begin{array}{c}
u_{c}\left( k\right) \\
y_{c}\left( k\right)
\end{array}
\right] \right) , \notag \\
e_{r}(k) &=&\left[
\begin{array}{cc}
-C & 0
\end{array}
\right] \varsigma \left( k\right) +z\left( k\right) -y_{c}\left( k\right) .
\label{tracking}
\end{eqnarray}
Then, the tracking error $e_{c}$ of system (\ref{clsys2}) can be obtained
via tracking error $e$ of system (\ref{clsys1}), which is a standard
feedback system.
To study the tracking error $e_{r}$, we employ the $l^{\infty
}$-norm of the system signals [Bamieh, 2003]. According to
(\ref{tracking}), we have
\begin{equation}
\left\| \varsigma \right\| _{\infty }\leq \left\| \left( z^{-1}I-\left[
\begin{array}{cc}
A-BD_{d}C & BC_{d} \\
-B_{d}C & A_{d}
\end{array}
\right] \right) ^{-1}\left[
\begin{array}{cc}
B & BD_{d} \\
0 & B_{d}
\end{array}
\right] \right\| _{1}\bar{\delta}, \label{statebound}
\end{equation}
and
\begin{eqnarray}
\left\| e_{r}\right\| _{\infty } &\leq &\left\| \left[
\begin{array}{cc}
-C & 0
\end{array}
\right] \left( z^{-1}I-\left[
\begin{array}{cc}
A-BD_{d}C & BC_{d} \\
-B_{d}C & A_{d}
\end{array}
\right] \right) ^{-1}\left[
\begin{array}{cc}
B & BD_{d} \\
0 & B_{d}
\end{array}
\right] \right\| _{1} \notag \\
&&~~~\times\left\| \left[
\begin{array}{c}
H_{1}\left( u_{c}\left( k\right) ,v(k-1)\right) \\
H_{2}\left( y_{c}\left( k\right) ,z(k-1)\right)
\end{array}
\right] -\left[
\begin{array}{c}
u_{c}\left( k\right) \\
y_{c}\left( k\right)
\end{array}
\right] \right\| _{\infty }+\delta _{2} \label{trackingbound} \\
&\leq &\bar{\delta}\left\| \left[
\begin{array}{cc}
-C & 0
\end{array}
\right] \left( z^{-1}I-\left[
\begin{array}{cc}
A-BD_{d}C & BC_{d} \\
-B_{d}C & A_{d}
\end{array}
\right] \right) ^{-1}\left[
\begin{array}{cc}
B & BD_{d} \\
0 & B_{d}
\end{array}
\right] \right\| _{1}+\delta _{2}. \notag
\end{eqnarray}
\begin{figure}[tbh]
\epsfxsize=3.5in
\par
\epsfclipon
\par
\centerline{\epsffile{Figure19.eps}} \caption{An $\ell_{1}$
minimization problem} \label{Figure19}
\end{figure}
\begin{remark}
{\rm Eq. (\ref{trackingbound}) gives an upper bound of the
difference of
the tracking error for the systems shown in Figs. \ref{Figure2}-\ref
{Figure3}. In light of (\ref{statebound}) and (\ref{trackingbound}),
with $\delta _{1}$ and $\delta _{2}$ fixed, minimizing the size of
an attractor and the tracking error can be converted to the problem
of
designing a controller $K$ that achieves step tracking in Fig.~\ref
{Figure2} and minimizes $\left\| z_{1}\right\| _{\infty }$ in Fig.~\ref
{Figure19} simultaneously, with
\begin{equation*}
G_{2}=\left[
\begin{array}{c|cc}
A & B & B \\ \hline
-C & 0 & 0 \\
-C & 0 & 0
\end{array}
\right] , \ K=\left[
\begin{array}{c|c}
A_{d} & B_{d} \\ \hline
C_{d} & D_{d}
\end{array}
\right] , \ \left\| w_{1}\right\| _{\infty }\leq 1, \ \left\| w_{2}\right\|
_{\infty }\leq 1.
\end{equation*}
For this multiple-objective control problem, LMI techniques can be
applied. More specifically, by parameterizing all stabilizing
controllers, the step tracking problem has
an equality constraint for the $l^{1}$control problem shown in Fig.~\ref
{Figure19}, which can be modified as an LMI minimization problem
(see [Chen \& Francis, 1995] and [Chen \& Wen, 1995] for more
details).}
\end{remark}
\section{Conclusions}
In this paper, we have generalized the results of Zhang \& Chen
[2005] in the following ways: 1) We have constructed first-return
maps of the nonlinear systems in Zhang \& Chen [2005] and derived
existence conditions for periodic orbits and studied their
properties. 3) We have formulated the involving systems as hybrid
systems, and proved that this type of hybrid systems is not
structurally stable. 4) We have examined higher-dimensional models
with detailed studies of the existence of periodic orbits. 5) We
have investigated a class of continuous-time hybrid systems as the
counterparts of the discrete-time systems. 6) We have proposed new
controller design methods based on this network transmission
strategy for improving control performance of individual systems as
well as the whole network.
Interestingly, one application of this network data transmission
strategy is the so-called limited communication control in control
and coordination of multiple subsystems. One example is: A single
decision maker controls many subsystems over a communication channel
of a finite capacity, where the decision maker can control only one
subsystem at a time. Let us consider the following situation:
Suppose there are several systems sharing a common communication
channel, where at each transmission time only one system can send a
signal. Is it possible that each subsystem adopts the transmission
strategy proposed here so that the whole system can achieve some
desired system performance? Note that under the proposed
transmission strategy, each system just sends ``necessary'' signals,
leaving communication resources to the others to use. So, if we
design the \textit{transmission sequence} carefully, the whole
system might perform well. A similar but essentially different
problem was discussed in Hristu \& Morgansen [1999], which is an
extension of the work of Brockett [1995]. The problem studied
therein is: Given a set of control systems controlled by a single
decision maker, which can communicate with only one system at a
time, design a communication sequence so that the whole network is
asymptotically stable. Using augmentation, this problem can be
converted to a mathematical programming problem for which some
algorithms are currently available. Here, under the proposed
transmission strategy, the communication strategy depends severely
on the control systems. Hence the communication sequence depends
explicitly on all subsystems, adding more constraints to the design
of the communication sequence. This important yet challenging
problem will be our future research topic.
\section{Acknowledgement}
This work was partially supported by NSERC. G. Zhang is grateful to
the discussions with Dr. Michael Y. Li and Dr. Y. Lin. M.B. D'Amico
appreciates the financial support of SGCyT at the Universidad
Nacional del Sur, CONICET, ANPCyT (PICT -11- 12524) and the City
University of Hong Kong (CERG CityU 1114/05E).
\noindent {\bf References}
Bamieh, B. [2003] ``Intersample and finite wordlength effects in
sampled-data problems,'' {\it IEEE Trans. Automat. Contr.} {\bf
48}(4), 639-643.
Brockett, R. \& Liberzon, D. [2000] ``Quantized feedback
stabilization of linear systems,'' {\it IEEE Trans. Automat. Contr.}
{\bf 45}(7), 1279-1289.
Brockett, R. [1995] ``Stabilization of motor networks,'' in {\it
Proc. of IEEE Conf. Decision and Control}, pp. 1484-1488.
Cervin, A., Henriksson, D., Lincoln, B., Eker, J., \& Arz$
\acute{e}$n, K. [2003] ``How does control timing affect
performance?'' {\it IEEE Control Systems Magazine} {\bf 23}(1),
16-30.
Chen, T. \& Francis, B. [1995] {\it Optimal Sampled-Data Control
Systems}(Springer:London).
Chen, X. \& Wen, J. [1995] ``A linear matrix inequality approach to
the discrete-time mixed $l_{1}/\mathcal{H}_{\infty }$ control
problem,'' in {\it Proc. of IEEE Conf. Decision and Control}, pp.
3670-3675.
Delchamps, D. [1988] ``The stabilization of linear systems with
quantized feedback,'' in {\it Proc. of IEEE Conf. Decision and
Control}, pp. 405-410.
Delchamps, D. [1989] ``Controlling the flow of information in
feedback systems with measurement quantization,'' in {\it Proc. IEEE
Conf. Decision and Control}, pp. 2355-2360.
Delchamps, D. [1990] ``Stabilizing a linear system with quantized
state feedback,'' {\it IEEE Trans. Automat. Contr.} {\bf 35}(8),
916-924.
Doyle, J. [2004] ``Complexity,'' Presented at Georgia Institute of
Techology.
Elia, N. [2004] ``When Bode meets Shannon: control-oriented feedback
communication schemes,'' {\it IEEE Trans. Automat. Contr.} {\bf
49}(9), 1477-1488.
Fagnani, F. \& Zampieri, S. [2003] ``Stability analysis and
synthesis for scalar linear systems with a quantized feedback,''
{\it IEEE Trans. Automat. Contr.} {\bf 48}(8), 1569-1583.
Fagnani, F. \& Zampieri, S. [2004] ``Quantized stabilization of
linear systems: complexity versus performance,'' {\it IEEE Trans.
Automat. Contr.} {\bf 49}(9), 1534-1548.
Goodwin, G., Haimovich, H., Quevedo, D. \& Welsh, J. [2004] ``A
moving horizon approach to networked control system design,'' {\it
IEEE Trans. Automat. Contr.} {\bf 49}(9), 1427-1445.
Hale, J. \& Kocak, H. [1991] {\it Dynamics and Bifurcations}
(Springer-Verlag).
Hristu, D. \& Morgansen, K. [1999] ``Limited communication
control,'' {\it Systems \& Control Letters} {\bf 37}(4), 193-205.
Ishii, H. \& Francis, B. [2002] ``Stabilization with control
networks,'' {\it Automatica} {\bf 38}(10), 1745-1751.
Khalil, H. [1996] {\it Nonlinear Systems}(Prentice Hall).
Krtolica, R., \"{O}zg\"{u}ner, \"{U}., Chan, D., G\"{o}
ktas, G., Winkelman, J. \& Liubakka, M. [1994] ``Stability of linear
feedback systems with random communication delays,'' {\it Int. J.
Control} {\bf 59}, 925-953.
Lasota, A. \& Yorke, J. [1973] ``On the existence of invariant
measures for piecewise monotonic transformations,'' {\it Trans.
Amer. Math. Soc.} (186), 481-488.
Li, T. \& Yorke, J. [1978] ``Ergodic transformations from an
interval to itself,'' {\it Trans. Amer. Math. Soc.} (235), 183-192.
Montestruque, L. \& Antsaklis, P. [2004] ``Stability of model-based
networked control systems with time-varying transmission times,''
{\it IEEE Trans. Automat. Contr.} {\bf 49}(9), 1562-1572.
Murray, R., ${\AA}$str$\ddot{o}$m, K., Boyd, S., Brockett, R. \&
Stein, G. [2003] ``Future directions in control in an
information-rich world,'' {\it IEEE Control Systems Magazine} {\bf
23}(2), 20-33.
Nilsson, J., Bernhardssont, B. \& Witenmark, B. [1998] ``Stochastic
analysis and control of real-time systems with random time delays,''
{\it Automatica} {\bf 34}(1), 57-64.
Nesic, D \& Teel, A. R. [2004] ``Input output stability properties
of networked control systems,'', {\it IEEE Trans. Automat. Contr.}
{\bf 49}(10), 1650-1667.
Octanez, P., Monyne, J., \& Tilbury, D. [2002] ``Using deadbands to
reduce communication in networked control systems,'' in {\it Proc.
of American Control Conference}, pp. 3015-2020.
Raji, R. [1994] ``Smart networks for control,''{\it IEEE Spectrum},
{\bf 31}, 49-53
Robbin, J. W. [1972] ``Topological conjugacy and structural
stability for discrete dynamical systems,'' {\it Bell. Amer.Math.
Soc.} (78), 923-952.
Smale, S. [1967] ``Differentible dynamical systems,'' {\it Bull.
Amer. Math. Soc.} (73), 747-817.
Tatikonda, S. \& Mitter, S. [2004] ``Control under communication
constraints,'' {\it IEEE Trans. Auto. Contr.} {\bf 49}(9),
1056-1068.
Tatikonda, S. \& Mitter, S. [2004b] ``Control over noisy channels,''
{\it IEEE Trans. Auto. Contr.} {\bf 49}(9), 1196-1201.
Wong, W. \& Brockett, R. [1999] ``Systems with finite communication
bandwidth constraints, part II: stabilization with limited
information feedback,'' {\it IEEE Trans. Auto. Contr.} {\bf 44}(5),
1049-1053.
Walsh, G., Beldiman, O., \& Bushnell, L. [1999] ``Error encoding
algorithms for networked control systems,'' in Proc. of IEEE Conf.
Decision and Control, pp. 4933-4938, 1999.
Walsh, G., Beldiman, O., \& Bushnell, L. [2001] ``Asymptotic
behavior of nonlinear networked control systems,'' {\it IEEE, Trans.
Automat. Contr.} {\bf 46}(6), 1093 -1097.
Walsh, G., Beldiman, O., \& Bushnell, L. [2002] ``Stability analysis
of networked control systems,'' {\it IEEE Trans. Contr. Syst.
Techno.} {\bf 10}(3), 438-446.
Walsh, G., Hong, Y., \& Bushnell, L. [2002b] ``Error encoding
algorithms for networked control systems,'' {\it Automatica} {\bf
38}(2), 261-267.
Walsh, G. \& Ye, H. [2001] ``Scheduling of networked control
systems,'' {\it IEEE Control Systems Magazine} {\bf 21}(1), 57-65.
Yue, D., Han, Q., \& Lam, J. [2005] ``Network-based robust $
H_{\infty}$ control of systems with uncertainty,'' {\it Automatica}
{\bf 41}(6), 999-1007.
Zhang, G. \& Chen, T. [2005] ``Networked control systems: a
perspective from chaos,'' {\it Int. J. Bifurcation and Chaos}, {\bf
15}(10), 3075-3101.
Zhang, G., Chen, G., Chen, T. \& Lin, Y. [2005] ``Analysis of a type
of nonsmooth dynamical systems,'' {\it Chaos, Solitons \& Fractals}
{\bf 30}, 1153-1164.
\end{document} | {"config": "arxiv", "file": "1405.4520/IJBC_ZCCD.tex"} |
TITLE: Expectation of a mixed random variable given only the CDF
QUESTION [2 upvotes]: I'm interested in the following question:
Given only the cumulative distribution function $F(x)$ of a mixed random variable $X$, how does one proceed to calculate the expectation $E(X)$?
By mixed I mean a variable which is neither continuous nor discrete. For example, the cdf could be:$$F(x)=\begin{cases}0&,x\in(-\infty,-1)\\
\frac13+\frac x3&,x\in [-1,0)\\
\frac12+\frac x3&,x\in [0,1)\\
1&,x\in [1,+\infty) \end{cases},$$
though it could be more complicated. Note that it isn't piecewise constant, nor continuous (there's a jump at $x=0$ for example).
If $X$ was absolutely continuous, I guess the simplest approach would be to take the derivative of $F$ to get the density and then integrate for the expectation.
If it was discrete, one could easily find the distribution law from the cdf itself, by seeing the size and location of jumps and then take the weighted sum for expectation.
However, I don't have an idea how to go about calculating the expectation of a mixed variable.
I should note that I'm not looking for the solution for the above example specifically, but a general method for solving the question at the top of the post.
REPLY [10 votes]: Here's a careful derivation of the formula in Gautam Shenoy's answer:
If $X$ is a non-negative random variable, this well-known result:
$$
\mathrm E(X)=\int_0^{+\infty}\mathrm P(X\gt t)\,\mathrm dt=\int_0^{+\infty}\mathrm P(X\geqslant t)\,\mathrm dt\tag1
$$
expresses the expectation of $X$ in terms of its CDF:
$$
\mathrm E(X)=\int_0^{+\infty}[1 - F(t)]\,\mathrm dt\tag2
$$
To extend (2) to the general case where $X$ may take negative values, we can write
$$E(X)=E(X^+)-E(X^-)\tag3$$
where the positive part and negative part of $X$ are defined by
$$
X^+:=\begin{cases}
X& \text{if $X>0$}\\
0&\text{otherwise}\\
\end{cases}\tag4
$$
and
$$
X^-:=\begin{cases}
-X& \text{if $X<0$}\\
0&\text{otherwise}\\
\end{cases}.\tag5
$$
Since both $X^+$ and $X^-$ are nonnegative, we can apply (1). Observe that for every $t>0$
$$
P(X^+>t)=P(X>t)=1-F(t)\tag6
$$
and
$$P(X^-\ge t)=P(X\le -t)=F(-t).\tag7$$
Plugging these into (1) and using (3) gives
$$
E(X)=\int_0^\infty[1-F(t)]dt-\int_0^\infty F(-t)dt.\tag8
$$
After a change of variable in the second integral we obtain the equivalent
$$
E(X)=\int_0^\infty[1-F(t)]dt-\int_{-\infty}^0 F(t)dt.\tag9
$$ | {"set_name": "stack_exchange", "score": 2, "question_id": 2136984} |
TITLE: Working with exponent on series
QUESTION [4 upvotes]: Hi have this sequence:
$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}$$
I understand that this is a Geometric series so this is what I've made to get the sum.
$$\sum\limits_{n=1}^\infty (-1)^n\frac{3^{n}\cdot 3^{-2}}{4^n}$$
$$\sum\limits_{n=1}^\infty (-1)^n\cdot 3^{-2}{(\frac{3}{4})}^n$$
So $a= (-1)^n\cdot 3^{-2}$ and $r=\frac{3}{4}$ and the sum is given by
$$(-1)^n\cdot 3^{-2}\cdot \frac{1}{1-\frac{3}{4}}$$
Solving this I'm getting the result as $\frac{4}{9}$ witch I know Is incorrect because WolframAlpha is giving me another result.
So were am I making the mistake?
REPLY [4 votes]: The objective here is to transform your sum into a sum of the form:
$$\sum_{n=1}^\infty ar^{n-1}$$
$$\text{Transformation: }\quad\quad\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n} = \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\frac{(-3)^{n-1}}{4^{n-1}} = \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\left(\frac{-3}{4}\right)^{n-1}$$
Hence $a = -\dfrac{1}{12}$ and $r = -\dfrac{3}{4}.\quad$ Now use the fact that
$$\sum_{n=1}^\infty ar^{n-1} = \dfrac{a}{1 - r} = -\left(\frac{1}{12}\right)\cdot \left(\frac{1}{1 - (-\frac{3}{4})}\right)$$
Simpilfy, and then you are done! | {"set_name": "stack_exchange", "score": 4, "question_id": 287405} |