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# Zhenghe's Blog

## January 26, 2011

### Notes 2: Kotani theory (I)-Zero Lyapunov Exponents and L^2 rotation conjugacy

This and next posts will be about Kotani theory, which is part of my syllabus. Kotani theory is first found by Shinichi Kotani when he studies spectrum theory of $Schr\ddot odinger$ operator with ergodic potentials. It gives a complete decription of  the absolutely continuous part of spectrum of  $Schr\ddot odinger$ operators in terms of Lyapunov exponents of the  corresponding of $Schr\ddot odinger$ cocycles. Thus it builds a deep and beautiful relation between operator theory and dynamical systems.

Later Artur Avila and Raphael Krikorian generalize Kotani theory to more general coycle dynamics setting and it becomes a powerful tool to study Schrodinger operators. What I am going to introduce here are main results and some preliminary stuff. Let’s start with one dimensional (1D) discrete $Schr\ddot odinger$ operators and $Schr\ddot odinger$ cocyles. From now on I set $L(E)=L(f,A^{(E-v)})$.

For simplicity, I will continue to use the triple $(X,f,\mu)$ as in last post. Assume $v\in C(X,R)$ be  a contiuous function. Then we can define a cocycle map via

$A^{(v)}(x)=\begin{pmatrix}v(x)&-1\\1&0\end{pmatrix}.$

The correspoding cocycle dynamics $(f, A^{(v)})$ is called $Schr\ddot odinger$ cocyles. It arises from the following $Schr\ddot odinger$ operator $H_{f,v,x}$ on $l^2(\mathbb Z)$

$(H_{f,v,x}u)_n=u_{n+1}+u_{n-1}+v(f^n(x))u_n,$

where $u=(u_n)_{n\in\mathbb Z}\in l^2(\mathbb Z).$ Then $u\in\mathbb C^{\mathbb Z}$ solving the eigenfunction equation $H_{f,v,x}u=Eu, E\in\mathbb R$ if and only if

$A^{(E-v)}(f^n(x))\binom{u_n}{u_{n-1}}=\binom{u_{n+1}}{u_n}.$

Thus the growth rate of $|u_n|$ with respect to $n$ characterizes both the spectrum type of the energy $E$ and the dynamics of cocycle $(f,A^{(E-v)})$, which allows one to go back and forth between spectrum theory and dynamical systems. Let $\Sigma_x$ be the spectrum of the bounded linear selfadjoint operator $H_{f,v,x}$, then the first basic fact relate operator and cocycle is for a.e. $x,$

$\Sigma_x=\{E: (f, A^{(E-v)})\notin \mathcal U\mathcal H\}.$

If furthermore $f$ is minimal, then the above relation is in fact true for all $x.$ Thus  for a.e. $x,$ the spectrum $\Sigma_x$ is independent of $x\in X,$ let’s denote it as $\Sigma$. For each $x,$ we can further decompose $\Sigma_x$ as $\Sigma_x=\Sigma_{x,pp}\cup\Sigma_{x,ac}\cup\Sigma_{x,sc}$, which correspond to pure point, absolutely continuous and singular continuous part of the spectrum of the operator $H_{f,v,x}$. These are defined by the spectral measure of the operator, which one can find in any standard functional analysis book.  It turns out that in our case, there exists sets $\Sigma_{\bullet}, \bullet\in\{pp, ac, sc\}$ such that $\Sigma_{x,\bullet}=\Sigma_{\bullet}$ for a.e. $x$ and $\bullet\in\{pp, sc,ac\}.$ (If in addition $f$ is minimal, then in fact $\Sigma_{x,ac}=\Sigma_{ac},$ for all $x,$ which is in general not true for pp and sc.)

Now we denote $\mathcal Z=\{E: L(E)=0\}.$ And for any set $\mathcal S\subset\mathbb R,$ the essential support of the set $\mathcal S$ is given by

$\overline{\mathcal S}^{ess}=\{E\in\mathbb R: Leb(\mathcal S\cap (E-\epsilon, E+\epsilon))>0$ for every $\epsilon>0\}.$

Then the next deep relation between dynamics and spectrum is the following

$\Sigma_{ac}=\overline{\mathcal Z}^{ess}.$

The relation $\Sigma_{ac}\subset\overline{\mathcal Z}^{ess}$ is relatively easy since positive Lyapunov exponents give exponential growth of the solution $u$ to the eigenfunction equation which in some sense contradicts with the absolute continuity of spectrum. The part $\overline{\mathcal Z}^{ess}\subset\Sigma_{ac}$ is a rather deep result called Kotani theory. It in fact  is a theory about that, under the assumption that Lyapunov exponents are zero, when can one in some sense conjugate the $SL(2,\mathbb R)-$valued cocycles to $SO(2,\mathbb R)-$valued cocycles. Obviously, if the cocycle map $A$ takes value in $SO(2,\mathbb R)$, then all orbits $\{A_n(x)w\}_n$ are bounded. While zero Lyapunv exponent in general just means that $\|A_n(x)w\|_n$ grows subexponentially.  What Kotani theory tells us is in fact that the following theorem

Theorem 1: For almost every $E$, if $L(E)=0,$ then there exists a map $B:X\rightarrow SL(2,\mathbb R)$ such that$B(f(x))A^{(E-v)}(x)B(x)^{-1}\in SO(2,\mathbb R)$ and $\int_X\|B(x)\|^2d\mu<\infty.$

Thus the generalized eigenfunctions for absolutely continous spectrum of the operators in question oscillate in some $L^2$ sense. Which are kind of wave like solutions and far from eigenfunctions of real eigenvalues, which decay in $l^2$ sense. As I said this is obviously stronger then zero Lyapunov exponents. Indeed, in this case we have

$\frac{1}{n}\int_X\ln\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|A_n(x)\|d\mu\leq\frac{1}{n}\int_X\|B(f^n(x))^{-1}\|\|B(x)\|d\mu$
$\leq\frac{1}{n}\int_X\|B(x)\|^2d\mu\rightarrow 0,$ as $n\rightarrow\infty$

Now let me explain how can one construct the above $B:X\rightarrow SL(2,\mathbb R).$  We first prove the following key lemma

Lemma 2: Assume $E\in\mathbb R$ satisfying $L(E+i\epsilon)=O(\epsilon),$ for $\epsilon>0$ small. Then $(f,A^{(E-v)})$ is $L^2-$conjugate to $SO(2,\mathbb R)-$valued cocycles.
Proof: First we note by the same reason that $(f,R_{i\theta}A)\in\mathcal U\mathcal H$ as in last blog, we have $(f,A^{(E+i\epsilon-v)})\in\mathcal U\mathcal H$. Thus there is invariant section $u^{\epsilon}:X\rightarrow \mathcal H$ which is the unstable direction. There are different ways to calculate Lyapunov exponents of cocylce dynamics via invaiant section of corresponding projective dynamics. We use the following: $-\frac{1}{2}$ of the contraction rate measured in $Poincar\acute e$ metric of mobius transformation at $u^{\epsilon}$. In our case we need to consider the following composition of map. Let $\mathbb H_{\epsilon}=\{E:\Im E>\epsilon\}$ with standard $Poincar\acute e$ metric, then the composition is

$\mathbb H\overset{A^{(E+i\epsilon-v)}}{\longrightarrow}\mathbb H_{\epsilon}\overset{i_{\epsilon}}{\longrightarrow}\mathbb H,$

where the first map is a isometry and the second one (which is the inclusion map) is a contraction. Thus we consider the contraction of the second map at invariant section. Then the Lyapunov exponents is given by

$L(E+i\epsilon)=-\frac{1}{2}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu.$

For simplicity, let me first assume that:

$\lim\limits_{\epsilon\rightarrow 0}u^{\epsilon}(x)$ exists for $a.e. x$ and we denote it by $u(x)$.
(then obviously, $u(x)$ is invariant for a.e. $x$, i.e. $A^{(E-v)}(x)\cdot u(x)=u(f(x))$ for a.e.$x$.)

Assuming this, we have the following straightforward estimate via Fatou’s lemma and our assumption in Lemma :

$\frac{1}{2}\int_X\frac{1}{\Im u}d\mu\leq\liminf\limits_{\epsilon\rightarrow 0}-\frac{1}{2\epsilon}\int_X\ln(1-\frac{\epsilon}{\Im u^{\epsilon}})d\mu=O(1)<\infty.$

Since $u(f(x))=E-v(x)-\frac{1}{u(x)}$, we have

$\frac{1}{2}\int_X\frac{1}{\Im u}d\mu=\frac{1}{2}\int_X\frac{1}{\Im u(f(x)}d\mu=\frac{1}{2}\int_X\frac{|u|^2}{\Im u}d\mu<\infty$.

Hence

$\int_X\frac{1+|u|^2}{\Im u}d\mu<\infty.$

Let $\phi(u)=\frac{1+|u|^2}{\Im u}$. Now we define $B:X\rightarrow SL(2,\mathbb R)$ such that $B(x)\cdot u=i,$ thus $B(f(x))A(x)B(x)^{-1}\cdot i=i$ which implies $B(f(x))A(x)B(x)^{-1}\in SO(2,\mathbb R)$. On the other hand, it’s easy to see for quite general reason $\|B(x)\|_{HS}^2=\phi(u).$ Thus $\int_X\|B(x)\|_{HS}^2d\mu<\infty.$
Here for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb R),$ $\|A\|_{HS}^2$ is the Hilbert-Schmit norm of $A$ which is just $a^2+b^2+c^2+d^2.$ It’s easy to see that if $A\cdot z=i,$ then $\|A\|_{HS}^2=\phi(z)$. Since all norms of $SL(2,R)$ are equivalent, we complete the proof of the Lemma up to the assumption.    $\square$

To get around the assumption, we need conformal barycenter, which is a Borelian function $\mathcal B:\mathcal M\rightarrow\mathbb H,$ where $\mathcal M$ is the space of probability measures on $\mathbb H.$ This function is equivariant with respect to $SL(2,\mathbb R)$ change of coordinates. i. e. for

$A\in SL(2,\mathbb R), \nu\in\mathcal M,$ we have $\mathcal B(A_*\nu)=A\cdot\mathcal B(\nu).$

Now let’s consider the probability measures $\nu_{\epsilon}=\mu\otimes\delta_{u^{\epsilon}}$ on $X\times\overline{\mathbb H}$. Then there is a subsequence converging to some $\nu=\mu\otimes\nu_x$ such that $\int_{X\times\overline{\mathbb H}}\phi d\nu<\infty$. Then apply conformal barycenter to $\nu_x$, we get a point $u(x)=\mathcal B(\nu_x)\in\mathbb H$ such that $u$ is an invariant section and

$\int_X\phi(u(x))d\mu=\int_{X\times\overline{\mathbb H}}\phi d\delta_{u(x)}d\mu\leq\int_{X\times\overline{\mathbb H}}\phi d\nu<\infty.$

Now we can construct the map $B$ as  in the proof of lemma.

Although we introduce conformal barycenter to complete the proof of Lemma, to prove the Theorem we actually only need to apply Fubini and Fatou theorem, to pass the existence of limits from For every $x$, converges for  a.e. $E$  to  For a.e. $E$, converges for a.e. $x$. Then we get the conclusion in assumption for a.e. E.

In the next post, I will prove how can we get the condition in Lemma under the condtion in Theorem, which will complete the proof of theorem.

## January 22, 2011

### Notes of Artur Avila’s Course in Fields Institute, Toronto 1: A Simple Proof of Herman-Avila-Bochi formula

I am visiting my coadvisor Artur Avila in Fields Institute in Toronto from Jan.18 to Apr.01. He is giving a course entitled Ergodic and Spectral Theory of Quasiperiodic Cocycles ‘ and I am working on some related problems with him. This is the first time he gives such a  graduate level course.

Since his way of math always fasinates me, I plan to take careful notes of his lecture and  latex them. I will post some of them here. I will also post some topics I promised in the last blog. In this post, I am starting with a simple proof of the following nice so-called Herman-Avila-Bochi formula:

$\int_{\mathbb R/\mathbb Z}L(f,R_{\theta}A)d\theta=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.$

Here is the setting: $X$ is compact metric space with a probability measure $\mu$ and $f$ is a homeomorphism on it preserving $\mu;$ $A:X\rightarrow SL(2,\mathbb R)$ is a continuous cocyle map. Thus $(f,A):X\times \mathbb R^2\rightarrow X\times \mathbb R^2$ gives a $SL(2,\mathbb R)$-valued cocyle dynamics over base dynamics $( X,f)$, namely, $(x,w)\rightarrow (f(x), A(x)w)$ for $(x,w)\in X\times \mathbb R^2.$ $L(f, A)$ stands for the Lyapunov exponent of this dynamical systems. If I denote $(f,A)^n$ by $(f^n, A_n)$, then it is given by

$L(f,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{X} \ln\|A_n(x)\|d\mu.$

The limit exists since $\{\int_{X} \ln\|A_n(x)\|d\mu\}_{n\geq 1}$ is a subadditive sequence. $R_{\theta}$ is the rotation matrix with rotation angle $2\pi\theta$.  Thus the above formula gives the averaged Lyapunov exponent of a one-parameter family of cocycle dynamics.

I forgot to say a word about the history of this formula: It’s first proved by Michael Herman as an inequality

$\int_{R/Z}L(f,R_{\theta}A)d\theta\geq\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu$

in his famous 1983 paper in CMH  ‘Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2’. Artur Avila and Jairo Bochi made it an equality in their 2002 paper A formula with some applications to the theory of Lyapunov exponents‘. The proof I am going to post here is a simpler version which Artur did in his class.

What lies behind this formula is in fact the mean value formula for harmonic functions.   Let $z=e^{2\pi i\theta}$ and write $R_{\theta}$ as $R_z$; let $\mathcal U\mathcal H$ stands for uniformly hyperbolic systems,  then the main steps are in the following:

1. Extending z from the unit circle $S$ to unit disk $\overline{\mathbb D}.$ Then for any $z \in \mathbb D\setminus\{0\}$, $(f,R_zA(x))\in\mathcal U\mathcal H.$
2. $L(f, zR_{z}A)=\ln |z|+L(f, R_zA)$ is harmonic in $z$ for $z\in \mathbb D\setminus\{0\}$ and bounded and  well-defined for all $z\in \overline{\mathbb D}.$ It obviously agrees with $L(f,R_{z_o}A)$ for $z_0\in S$. Furthermore, $L(f, zR_{z}A)$ converges to $L(f, z_0R_{z_0}A)$ nontangentially.
3. $L(f,0R_0A)=\int_{X}\ln\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}d\mu.$
4. Mean value formula implies the formula in question.

There are quite a few notions, observations and standard results need to be explained. Let’s do them one by one. First is the defintion of uniformly hyperbolic cocycle dynamcal systems, let’s consider the more general $SL(2,\mathbb C)-$valued cocyle case, i.e. $A:X\rightarrow SL(2,\mathbb C)$ which is continuous. Let $\mathbb C\mathbb P^1$ be the Riemann surface $\mathbb C\cup\{\infty\}$ and $A$ acts on it as Mobius transforamtion, i.e. for $A=\begin{pmatrix}a& b\\c&d\end{pmatrix}$, $A\cdot z=\frac{az+b}{cz+d}$.

Defintion 1.  $(f, A)$ is said to be uniformly hyperbolic if the are two contiuous map $u, s:X\rightarrow \mathbb C\mathbb P^1$ such that
1. they are invariant in the sense that $A(x)\cdot u(x)=u(f(x))$ and $A(x)\cdot s(x)=s(f(x))$.
2. there exist constant $C>0, 0<\lambda<1$ such that for each vector $w_s\in s(x), \|A_n(x)w_s\|\leq C\lambda^n\|w_s\|$ for $n\geq 1$,  and for each $w_u\in s(x), \|A_n(x)w_u\|\leq C\lambda^{-n}\|w_u\|$ for $n\leq 0$.

One well-known equivalent condition for $\mathcal U\mathcal H$ systems is the existence of invariant cone field. More concretely, $(f, A)\in \mathcal U\mathcal H$ if and only if for each $x\in X,$ there is open disk $U(x)\subset \mathbb C\mathbb P^1$ such that there exists positive integer $N$

$\overline{A_n(x)\cdot U(x)}\subset U(f^n(x)), \forall x\in X, \forall n\geq N.$

The existence of $U(x)$  implies the existence of a corresponding $S(x).$ Using this it easy to obtain the following lemma

Lemma 2. Consider a holomorphic 1-parameter family of $\mathcal U\mathcal H$ systems $(f, A_{\lambda})$, $\lambda$ is in some open region in complex plane. Holomorphicity in the sense that for each fixed $x,$ $A_{\lambda}(x)$ is  holomorphic in $\lambda.$ Then $u(x,\lambda)$ and $s(x,\lambda)$ are  both holomorphic in $\lambda$.
Proof:  It’s easy to see by the equivalent condition above,
$u(x,\lambda)=\lim\limits_{n\rightarrow\infty}(A_{\lambda})_n(f^{-n}(x))\cdot u_{-n}$,
for any $u_{-n}\in U(f^{-n}(x))$ and the convergence is uniform in $x$ and $\lambda$. Thus the result follows . $\square$

By this lemma, we can show the following theorem,

Theorem 3. $L(f,A)$ is pluriharmonic in $\mathcal U\mathcal H$ (Pluriharmonicity means that for $(f, A_{\lambda})$ in the above Lemma , $L(f,A_{\lambda})$ is harmonic in $\lambda$).
Proof: Define a map $B_{\lambda}:X\rightarrow SL(2,\mathbb C)$ with column vectors in $u_{\lambda}(x)$ and $s_{\lambda}(x)$. Then $B_{\lambda}(f(x))A_{\lambda}(x)B_{\lambda}(x)^{-1}=diag(\gamma_{\lambda}(x), \gamma_{\lambda}(x)^{-1}),$
where all maps are holomorphical in $\lambda$. Then by Birkhoff Ergodic Theorem, we have
$L(f, A_{\lambda})=\int_X\ln|\gamma_{\lambda}(x)|d\mu.$
Hence, the result follows easily. $\square$

Now what left in the main steps are 1 and 3. Step1 is based on a key observation such that $\overline{R_{i\theta}\cdot\mathbb H_-}\subset\mathbb H_-,$ where $\theta>0$ and $\mathbb H_-$ is the low-half plane in $\mathbb C\mathbb P^1.$ An easy way to see the above fact is to Mobius transform $\mathbb H$ to $\mathbb D$ via $Q=\frac{-1}{1+i}\begin{pmatrix}1&-i\\1&i\end{pmatrix}\in \mathbb U(2).$ Then the action of $R_{\theta}$ on $\mathbb H$ is conjugated to the action of $\hat{R_{\theta}}=QR_{\theta}Q^*=diag(e^{-2\pi i\theta}, e^{2\pi i\theta})$ on $\mathbb D.$ Then it’s easy to see that $\hat{R_{i\theta }}$ contracts $\overline{\mathbb D}^c$ into $\overline{\mathbb D}^c.$ Hence, the above fact follows. Now an easy appliction of the equivalent condition for $\mathcal U\mathcal H$ systems implies the results in step1.

The proof of step3 is a straightforward computation. Indeed, it’s easy to see $P=0R_0=\frac{1}{2}\begin{pmatrix}1&-i\\i&1\end{pmatrix}$ and $P\cdot(\mathbb H^c)=-i$, thus $PA\cdot (-i)=-i$ for all $A\in SL(2, \mathbb R),$ which means that $-i$ is a constant invariant section of the projective dynamics$(f, PA)$ on $X\times CP^1.$ If $PA(x)\binom{-i}{1}=\gamma(x)\binom{-i}{1},$ then it’s easy to see $L(f,PA)=\int_X\ln|\gamma(x)|d\mu$ and $|\gamma(x)|=\frac{\|A(x)\|+\|A(x)\|^{-1}}{2}$.
Hence, step3 follows.

Finally, mean value formula for harmonic functions implies

$L(f,0R_0A)=\int_{R/Z}L(f, zR_zA)d\theta=\int_{R/Z}L(f, R_{\theta}A)d\theta,$

which establishes the formula we want to show.

I’ve already given a self-contained and complete proof of Herman-Avila-Bochi formula in this post. I like this proof very much. Because it’s very conceptul and few computation is involved. One can also see how does mean value formula  for harmonic functions gets into dynamical systems.

## April 21, 2010

### Syllabus of my qualifying exam: Topics in 1D discrete quasiperiodic Schrodinger cocycles

Filed under: Schrodinger Cocycles — Zhenghe @ 9:35 pm
Tags: , ,

Here is the my qualifying exam syllabus. I like these topics, because they  lie in the intersection of dynamical systems and operator theory,  both of which are my favorite areas.

Since operators are quasiperiodic, the corresponding cocycle dynamics are over irrational rotation on unit circle, consequently, the spectrum of Schrodinger operators depend in a very subtle way on the arithematic property of irrational rotation numbers, which I introduced in the last blog.

The syllabus is based on some papers I’ve read, some of which I know in detail and some I just went through roughly. I am going to introduce some of the them in future blogs.

1. Spectral measure of self-adjoint operators.
1.1. Continuous functional calculus of normal operators.
1.2. Spectral measure of self-adjoint operators.

2. Irrational rotation on unit circle.
2.1. Strict ergodicity.
2.2  Continued fraction expansion
2.2. Diophantine, Brjuno and Liouville numbers.

3.Dynamics of quasiperiodic SL(2,R)-cocyles.
3.1. Lyapunov exponents and Oseledec’s theorem.
3.2. Hyperbolicity and reducibility of SL(2,R)-cocycles.
3.3. Some global results.
3.3.1. Density of uniformly hyperbolic systems in $C^0$-topology.
3.3.2. Genericity of zero Lyapunov exponents in $C^0$-topology.
3.3.3. Continuity of Lyapunov exponents in analytic topology.
3.3.4. Ubiquity of nonuniformly hyperbolic systems.

4. 1D discrete quasiperiodic Schrodinger operators and Schrodinger cocycles.
4.1. Correspondence between uniform hyperbolicity and resolvent set.
4.2. Cantor spectrum and density of uniform hyperbolicity
4.3. Kotani theory: reducibility and absolutely continuous spectrum.
4.4. Anderson localization and nonuniform hyperbolicity.
4.5. Almost Mathieu Operator.

## February 25, 2010

### Why golden mean is optimal: an introduction to Diophantine, Brjuno and Liouville number

Irrational rotations on unit circle $S^1=R/Z$ is one of most important examples in dynamical systems, which I am going to denote it by: $\alpha: S^1\rightarrow S^1, x\mapsto x+\alpha$, where $\alpha\in [0, 1]\setminus Q$.

Then we have the following facts:

Theorem 1: Dynamical systems defined above are minimal and uniquely ergodic.
(Th0se who are not familiar with these concepts can google them or take a look at Peter Walters’ “An introduction to ergodic theory”.)

The subtlety of these dynamical systems lie in the way the orbit Orb(x) of a point x recurrent to itself, in particular, we can fix x=0. It turns out the recurrence of 0 is closely related to the arithmetic property of the rotation angle $\alpha$, which lead to interesting classification of irrational numbers. WLOG, I am going to restrict myself to I=[0,1].

Definition 2: $\alpha$ satisfies a Diophantine condition $DC(\gamma, \tau), \gamma>0, \tau>0$, if $|q\alpha-p|>\gamma |q|^{-\tau}, (p, q)\in Z^2, q\neq 0$.

Let $DC=\cup_{\gamma>0, \tau>0}DC(\gamma, \tau)$ be the so-called Diophantine number, then it is well known that $DC_{\tau}=\cup_{\gamma>0}DC(\gamma, \tau)$ has full Lebesgue measure if $\tau>1$. In some sense, Diophantine condition means that $\alpha$ cannot be approximated by rational numbers too fast, which in dynamics language means 0 cannot be recurrent to itself too fast under rotation $\alpha$

There is actually another equivalent way to define Diophantine number in terms of continued fraction expansion of $\alpha$.
Let $\alpha=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}}$ be the continued fraction expansion of $\alpha$, and $\frac{p_n}{q_n}=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots+\frac{1}{a_n}}}}}$ be the n’th step approximant, then we have the following properties:

(1)$\frac{1}{q_n}>|q_{n-1}\alpha-p_{n-1}|>\frac{1}{q_n+q_{n-1}}$,
(2)$\parallel k\alpha\parallel_{R/Z}>|q_{n-1}\alpha-p_{n-1}|, \forall q_{n-1}+1\leq k\leq q_n-1$.
where $\parallel\cdot\parallel_{R/Z}$ denote the distance to the nearest integer.

Thus, to define Diophantine condition, we need only care about the continued fraction approximants, where then Diophantine conditions are actually given by the growth rate of sequence $(q_n)_{n\in Z^+}$. Use above inequalities and definition 2, it’s not difficult to see that:

$\alpha\in DC_{\tau}$ iff $q_{n+1}, where c is some constant.

But what about the growth rate of $(q_n)_{n\in Z^+}$ in general? In fact, we have the following nice induction formula for $(q_n)_{n\in Z}$:

$q_n=a_nq_{n-1}+q_{n-2}$.

Thus, the slowest growth is when $a_n=1$ for all n, where the corresponding $\alpha$ is the golden mean and $(q_n)_{n\in Z}$ is the Fibonacci sequence if we set $q_0=1, q_{-1}=0$! So in some sense, golden mean is the most irrational irrational number, because it is approximated by rational numbers in the slowest rate.

For Fibonacci sequence $b_n$, we can find the exact expression by generating function, which is

$b_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1})$.

Hence, sequence $(q_n)_{n\in Z^+}$ grows at least exponentially fast.

Definition 3: $\alpha$ is called a Brjuno number if the associated sequence $(q_n)_{n\in Z^+}$ satisfies $\sum_n \frac{log q_{n+1}}{q_n}<\infty$.

By the equivalent condition of Diophantine number and the slowest growth of $(q_n)_{n\in Z^+}$, Brjuno number contains all Diophantine number so it’s a full Lebesgue measure set. We can define another number which decribe the growth rate of $(q_n)_{n\in Z^+}$ outside a set contains Brjuno numbers.

$\beta(\alpha)=\limsup\limits_{n\rightarrow\infty}\frac{log q_{n+1}}{q_n}$.

Then obviously we have:

$DC\subset$ Brjuno Numbers $\subset\{\alpha: \beta(\alpha)=0\}$.

Again $\{\alpha: \beta(\alpha)=0\}$ is a full Lebesgue measure set. Thus,  it seems that $\{\alpha: \beta(\alpha)>0\}$ is a rather small set which may even be empty. But that’s not true:

Definition 4: $\alpha\in${$\alpha: \beta(\alpha)=\infty$} are said to be Liouville number.
Remark: Some also define Liouville numbers to be the complement of Diophantine numbers in irrational numbers.

Obviously, the set of all Liouville numbers has zero Lebesgue measure and they are approximated by rational numbers really very fast. It is an interesting fact that the set of Liouville numbers is generic though it has zero Lebesgue measure. These arithmetic properties in many ways govern dynamical behaviors in many dynamical systems.

## February 21, 2010

### Dynamics of quadratic family: from attracting fixed point to absolutely continuous invariant probability (ACIP)

This is one of my previous notes, I post it  for testing:)

In study of Smooth Ergodic Theory, we always follow the way that geometrical properties of derivative implies statistical properties of dynamical systems.

Typical statistical behavior of dynamical systems is hyperbolicity, which includes uniform hyperbolicity, partial hyperbolicity and nonuniform hyperbolicity. Famous examples of uniformly hyperbolic systems are linear automorphisms on 2-torus and geodesic flows, while partially hyperbolic systems are given by time-1 map of Anosov flows and Frame flows. Both uniform hyperbolicity and partial hyperbolicity are open conditions.

Then how about nonuniform hyperbolicity? First of all, it is not an open condition. And there is few good example. Furthermore, we can ask ‘are typical systems nonuniformly hyperbolic?’. Here I am going to introduce an famous example related to nonuniformly hyperbolic theory, which gave rise to nice generalization.

First, let us introduce the so-called Henon map, which is given by $f_{a,b}(x,y)=(x^2-a+y, bx)$. It was given by Henon in 1976 and he found some strange attractor when a=1.4, b=0.3. Then followed by Benedicks, Carleson 1991 and Benedicks, Young 1993’s work, we have the following theorem:

Theorem: Let $f_{a,b}(x,y)=(x^{2}-a+y,bx)$, then $\forall b\neq 0$ sufficiently small, there exist$\Omega^{*}(b)$satisfies

(a)$\forall a\in\Omega^*(b)$, $f_{a,b}$ is nonuniformly hyperbolic.
(b)$m(\Omega^*(b))>0$, and there is no interval in $\Omega^{*}(b)$

Concretely, here nonuniform hyperbolicity means for every pair of a, b above, there exists an attractor $\Lambda_{a,b}$ with ergodic SRB measure with nonzero Lypapunov exponents and $\Lambda_{a,b}$ contains tagencies.

Henon map is the two dimensional case, its study was originated from the one dimensional case, namely, b=0 and $f_a(x)=x^2-a$ the quadratic family, where we have the following theorem, which is known as Jacobson theorem

Theorem(Jacobson, 1981):
There exists $\Omega^*$, s.t. i) $\forall a\in\Omega^*$, $f_a$ is nonuniformly hyperbolic
(which means it has ergodic SRB measure which is absolutely continuous w.r.t. Lebesgue measure, and positive Lyapunov exponent); ii) $m(\Omega^*)>0$ and contains no intervals.

Now we are going to focus our study on the quadratic family $f_a(x)=x^2-a$. First, we will study the change of dynamical behavior of $f_a$ when parameter $a$ varies.

Let $f_{a}=x^{2}-a$, $\varepsilon>0$ is a sufficiently small number, $p_{1}(a)\leq p_{0}(a)$ are two roots of equation $x^{2}-a=x$ and $I_{a}=[-p_{0}(a), p_{0}(a)]$, then then we have the following description:

Let $a_{0}$ be  the number that $x^{2}-a_{0}=x$ has a unique solution, then

If $a, then $\forall x\in R$, $f_{a}^{n}(x)\rightarrow\infty$ when $n\rightarrow\infty$

If $a=a_{0}$ then:
$f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}$
$f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}$

If $a=a_{0}+\varepsilon$ then:
$f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}$
$f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}$

Let $a_{1}$ be the number that $f'_{a_{1}}(p_{1}(a_{1}))=-1$then:

If $a=a_{1}-\varepsilon$ then:
$f_{a}^{n}(x)\rightarrow p_{1}(a), \forall x\in I_{a}$
$f_{a}^{n}(x)\rightarrow\infty, \forall x\notin I_{a}$

If $a=a_{1}+\varepsilon$, then $f'_{a_{1}}(p_{1}(a))<-1$ so there is no fix point. But the graph of $y=f^{2}_{a}(x)$ and y=x will intersect  in four points and one of them is an attracting fix point. So $f_{a}$ will have an attracting 2-periodic orbit such that $\forall x\in I, f^{n}_{a}(x)$ will approach this orbit.

Then following the  idea of renormalization, we can take part of the graph of $f^{2}_{a}$ and analyze it just as those of $f_{a}$ for $a>a_{1}$. Then we will find that the attracting 2-periodic orbit will preserve until a reaches an $a_{2}$. For $a>a_{2}$, we consider $f^{4}_{a}$, then we will find a fix point which is an attracting 4-periodic orbit of $f_{a}$.

Keep going this way, we will find a sequence $\{a_{i}\}_{i\geq0}$ such that for every $a_{i}$, there will be a doubling of period of attracting orbit. In the end, $\{a_{i}\}_{i\geq0}$ will converge to a number $a_{\infty}<2$.

$a=a_{\infty}$ is the so-called infinitely renormalizable case,   it was dicovered by Feigenbanm in 1970s. In this case, $f_{a}$ has an invariant interval $I$, for almost all $x\in I, f^{n}_{a}(x)\rightarrow C$, where C is an invariant Cantor set and the dynamical behavior on C just likes the irrational rotation of unit circle.

For $a=2$,Ulam-Von Neumann found the behavior of $f_{a}(x)=x^{2}-a$ on $[-2, 2]$ is conjugate with the tent map T on $[0,1]$ via map $h(z)=2\cos(\pi z)$, where T is given by

$T(x)=-2x+1$ for $0\leq x\leq \frac{1}{2}$
$T(x)=2x-1$ for $\frac{1}{2}< x\leq1.$

That is, $f_{2}=h\circ T\circ h^{-1}$. But we know that the Lebesgue measure $m$ of $[0, 1]$ is the ergodic and invariant measure of tent map $T$. So the smooth map $h$ will push $m$ forward to be a ergodic acip (absolutely continuous invariant probability) $\mu$ of $f_{2}$ on $[-2, 2]$. But for one dimensional map with single critical point, existence of acip implies positive Lyapunov exponent. So $f_{2}$ is a nonuniformly hyperbolic system on $[-2, 2]$ and $\mu$ is its SRB measure (in one dimensional case SRB coinsides with acip).

When $a>2$, there will be no invariant interval of $f_{a}$, but there does exist an invariant Cantor set on $I_{a}$ such that the dynamical behavior on it is topologically conjugate with the left shift map on $\Sigma_{2}=\{0, 1\}^{N}$.

Now we are going to study the parameter interval $(a_{\infty}, 2)$, which has rich phenomena and deep results. For example, there are lots of parameters with attracting periodic orbits which give rise to many cascades of doubling bifurcation which just like the process from $a_0$ to $a_{\infty}$. Actually, Simo and Tatjer numerically found about 30 million “periodic windows” with period $\leq 24$, and the total length of them is about $10\%$ of $(a_{\infty}, 2)$.

If we let $\Omega= (a_{\infty}, 2)$, $\Omega^-= \{a: f_a$ admits attracting periodic orbits$\}$ and $\Omega^+=\{a: f_a$ admits acip$\}$, then we have the following serial deep results:

Theorem(Jacobson, 1981):
$m(\Omega^+)>0$.

Theorem(Luzzatto, Takahashi, 2006):
$m(\Omega^+)>10^{-5000}$.

Theorem(Craczjk, Smotek, Lyubich, 1997):
$\Omega^-$ is open and dense in $\Omega$.

Theorem(Lyubich, 2002):
$\Omega^+\bigcup\Omega^-$ has full measure of $\Omega$.

Remark:
1) $(a_{\infty}, 2)\setminus (\Omega^{+}\bigcup\Omega^{-})$ is not empty, for example, there are infinitely renormalizable systems and systems whose physical measure support on repelling fixed point.

2) Generalization: first,  $m(\Omega^+)>0$ for the following one dimensional maps: Multimodel map(Tsujii, 1993), Contracting Lorenz map(Rovella, 1993), Lorenz like map with critical point(Luzzatto and Tacker 2000) and infinite critical points map(Pacifer, Parclla and Viena, 2000); second, $\Omega^-$ is open and dense in space of$C^r$maps$(r\geq1)$(Kozlosky, Shen and Von Stier, 2005).

## February 17, 2010

### This should be my math blog:)

Filed under: Uncategorized — Zhenghe @ 1:28 am

Hahaha

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Blog at WordPress.com.


text:
Draw the graphs of the lines x – y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis.
Question:

Draw the graphs of the lines x – y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis. Also, find this area.

Solution:

$x-y=1$

$\Rightarrow y=x-1$

When $x=0, y=0-1=-1$

When $x=1, y=1-1=0$

When $x=2, y=2-1=1$

Thus, the points on the line x – y = 1 are as given in the following table:

Plotting the points (0, –1), (1, 0) and (21) and drawing a line passing through these points, we obtain the graph of of the line x – y = 1.

$2 x+y=8$

$\Rightarrow y=-2 x+8$

When $x=1, y=-2 \times 1+8=-2+8=6$

When $x=2, y=-2 \times 2+8=-4+8=4$

When $x=3, y=-2 \times 3+8=-6+8=2$

Thus, the points on the line  2x + y = 8 are as given in the following table:

Plotting the points (1, 6), (2, 4) and (32) and drawing a line passing through these points, we obtain the graph of of the line  2x + y = 8.

The shaded region represents the area bounded by the lines x – y = 1, 2x + y = 8 and the y-axis. This represents a triangle.

It can be seen that the lines intersect at the point C(3, 2). Draw CD perpendicular from C on the y-axis.

Height = CD = 3 units

Base = AB = 9 units

$\therefore$ Area of the shaded region $=$ Area of $\Delta \mathrm{ABC}=\frac{1}{2} \times \mathrm{AB} \times \mathrm{CD}=\frac{1}{2} \times 9 \times 3=\frac{27}{2}$ square units

format: plain


text:
Proof : Let G be the centroid of ΔABC i. e., the point of intersection of AD, BE and CF.In triangles BEC and BFC, we have ∠B = ∠C = 60. Trapezoid is a quadrilateral with two parallel sides and centroid of a trapezoid lies between two bases. Ask Question Asked 9 years, 1 month ago. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of … Subscribe. Square. Matplotlib: Centroid of Scatter Plot Circles. x = a / 2 (2a) y = b / 2 (2b) Area. C++ Code Mat canny_output; vector > contours; … A lens is a convex-convex area bounded by two circular arcs. [citation needed] It is used as a substitute for the older terms "center of gravity," and "center of mass", when the purely geometrical aspects of that point are to be emphasized.The term is peculiar to the English language. Find the Centroid of the Semi-Circle Below: 4 r 3 π = 4 (2) 3 (3.14) = 0.849 Centroid is 0.8 in. Centroid is defined as the centre mass of a geometric object which has uniform density. Well then, we will have to use findContours to find the number of contours in the Image and find the center of each of them. If we sum the moment of all these strips, normalised over the area of the entire half-circle, we can find the centroid. 1. A = a b (2c) Circle. by M. Bourne. Construction : Draw medians, AD, BE and CF. To answer your first question - why the centroids of a half-disk and half-arc are different, imagine their physical manifestations: both are objects having the same mass. From the definition also, it is also apparent that the moment of inertia should always have a positive … Feel … Centroid of semicircle is at a distance of 4R/3π from the base of semicircle. The centroid is the location that we can model the shape as if the entire weight acts through this point. I want to calculate the weighted centroid of each circle with OpenCV in Python. In tilt-slab construction, we have a concrete wall (with doors and windows cut out) which we need to raise into position.We don't want the wall to crack as we raise it, so we need to know the center of mass of the wall. I have a Matplotlib scatter plot, plotting 3 different circles and I am trying to find the axes x, y coordinates of the center point of each of these circles, i.e. Mark as New; Bookmark; Subscribe; Mute; Subscribe to RSS Feed; Permalink; Print; Email to a Friend; Report Inappropriate Content; My company just switched us from MapInfo to Arcmap. Often though, one may use the term "moment of inertia of circle", missing to specify an axis. 7.3 CENTROID OF A QUADRANT OF A CIRCLE. Centroid of a circle . Half circle is known as semi-circle. Centroid of an Area by Integration. Q4 Explain parallel axes theorem for a rectangle area of width B and height H. Q5 Explain the method to determine moment of inertia … Distance. Results for symmetrical objects like square, circle, cylinder, rectangle, ring etc are omitted. The area of the element is approximated as the area of the triangle given by Fig. 7.3 Centroid of a quadrant of a circle. Centroids and Centers of Gravity | Engineering Mechanics Review centroid and length of semicircular arc: pin. Centroid. Answer to: Use polar coordinates to find the centroid of a quarter circle of radius 3. I'm using this piece of code: im2, contours, hierarchy = cv2.findContours(thresh, cv2.RETR_TREE, cv2.CHAIN_APPROX_SIMPLE) for c in contours: # calculate moments for each contour M = cv2.moments(c) # calculate x,y coordinate of center if M["m00"] != 0: cX = int(M["m10"] / … History. Class 11th Physics | System of Particles | Centre of Mass of a Class 11th Physics | System of … For very simple shapes like rectangles or circles, it's easy to find the centroid because it's … Let us see how it works! x = readWKT (paste ("GEOMETRYCOLLECTION(POLYGON((0 0,10 0,10 10,0 10,0 0)),", "POLYGON((15 0,25 15,35 0,15 0)))")) # Centroids of both the square and circle independently c1 = gCentroid (x, byid = TRUE) # Centroid of square and circle together c2 = gCentroid (x) plot (x) plot (c1, col = 'red', add = TRUE) plot (c2, col = 'blue', add = TRUE) Contents. Yi for your instruction could be the end result of the centroid direction of Y for your shape. 2” Centroid Practice Problems 6. 69 Statics by Dr. Ibrahim Find the centroid of a quarter circle by double integration in rectangular coordinates Procedure for locatingcenter of gravity of a body or the centroid of a The latter is also a half-disk, but it has all of its mass concentrated at the edge. Please see the image for the full question. As shown in the figure below: Centroid of Semi-circle Formula: $$\bar{Y}= \frac{4R}{3π }$$ Example. For the left semicircle the centroid is at x=-4r/3pi, but I'm not sure how to tackle the right part to ultimately find the total centroid. Setting the circle center at the origin (0,0), y-bar is the y coordinate of the centroid when its x coordinate is zero (as the figure in the link is drawn). In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Area under the plotted curve represents volume of whole body and … Jump to solution. How do we find the center of mass for such an uneven shape? New Contributor II 10-31-2013 11:30 AM. Active 9 years ago. Find the centroid of the complex shape below: (0,0) Rectangle Semi Circle Area = 24 Area = 1. Centroid of a circle Drag the vertices of the triangle to create different triangles (acute, right, and obtuse) to see how the centroid location changes. by MikeMellin. This is true if the figure is rotated/translated by any amounts. I think in England we would tend to interpret 'circle' as 'area of circle' in this context. a number within the bbox between 0 and 1. Developed by Roger … From symmetry, we can see that the centroid lies along the y-axis, which acts like a mirror to the quarter circles either side. 1248. In other words, y-bar is the distance from the circle's center to the centroid. Finding the center of only one blob is quite easy, but what if there are multiple blobs in the Image? Center of Mass and Centroids: Composite Bodies and Figures Integration vs Appx Summation: Irregular Volume Reduce the problem to one of locating the centroid of area Appx Summation may be used instead of integration Divide the area into several strips Volume of each strip = A∆x Plot all such A against x. Learn more about centroid of a trapezoid formula at vedantu.com and also download free pdf format of Textbook Solutions, Revision Notes and Board Questions Papers. I got graphs of my homework. Students can measure segments BG and GF and noticing the relationship between the two parts of each median formed. CENTROID OF SOME STANDARD GEOMETRIC FIGURES Following results are obtained by integration which will be explained later. Viewed 2k times 2. We take more care when referring to arcs, sectors and segments in this context in mechanics. I know that you should divide the circle into different areas. Distance. Centroid of semi-circle is at a distance of 4R/3π from the base of semi-circle. 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text:
# Definition:Sine Integral Function

## Definition

The sine integral function is the real function defined as:

$\Si: \R \to \R$:
$\Si \paren x = \begin{cases} \displaystyle \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t & : x \ne 0 \\ \vphantom x \\ 0 & : x = 0 \\ \end{cases}$

## Also see

• Results about the sine integral function can be found here.


text:
A Neighborhood of Infinity

Saturday, February 11, 2012

Using Lawvere theories to combine effects

> {-# LANGUAGE MultiParamTypeClasses, ExplicitForAll, RankNTypes, FlexibleInstances, FlexibleContexts, TypeSynonymInstances #-}

> import Data.Monoid
> import Data.Functor.Identity

In an earlier post I talked about how monads arise from free algebras. Let me recap a bit.

In Part 1 I described algebras. They're sets with operations on them satisfying some laws. We can build new elements of an algebra from old ones by using its operations. Eg. if x and y are in an algebra then x mappend y must be in it too. Starting with a bunch of symbols, thought of as leaves, we can consider the set of all expressions trees we can build from them. If we consider pairs of trees to be equivalent if the laws say the corresponding expressions are equal, then the set of trees itself forms an algebra known as a free algebra (for the given theory).

Let's start with some code. This type class says that the type b has leaves of type a:

> class Free a b where
>   leaf :: a -> b

Effects from monoids
Now we can make the type of all trees built from Monoid operations and including all leaves of type a:

> data FreeMonoid a = FreeMonoid (forall b. (Monoid b, Free a b) => b)

And we have:

> instance Monoid (FreeMonoid a) where
>   mempty = FreeMonoid mempty
>   FreeMonoid a mappend FreeMonoid b = FreeMonoid (a mappend b)

Unfortunately elements like e1 and e2 two ought to be equal but Haskell doesn't know this:

> e1, e2 :: FreeMonoid Char
> e1 = FreeMonoid (leaf 'a' mappend (leaf 'b' mappend leaf 'c'))
> e2 = FreeMonoid ((leaf 'a' mappend leaf 'b') mappend leaf 'c')

Instead we can manually construct a type that does respect equality in monoids. Elements of FreeMonoid are binary trees with a mappend at each node. Associativity means that we can always replace a tree with an equivalent one where the left branch is a leaf. We can also use the laws to eliminate any occurrence of mempty. So every element of FreeMonoid a is equivalent to one of the form:
Leaf x1 mappend (Leaf x2 mappend (... mempty))

In other words, free monoids are lists. We can make this explicit. The standard prelude already makes [] an instance of Monoid so we just need:

> instance Free a [a] where
>       leaf x = [x]

Here's the isomorphism (modulo tree equivalence):

> iso1 :: FreeMonoid a -> [a]
> iso1 (FreeMonoid x) = x

> iso1' :: [a] -> FreeMonoid a
> iso1' [] = FreeMonoid mempty
> iso1' (a : as) = let FreeMonoid r = iso1' as
>                  in FreeMonoid (leaf a mappend r)

As I talked about in that earlier article, free algebras give monads and the trees representing expressions in the algebra can be thought of as abstract syntax trees for domain specific languages. In this case it's the usual list monad. So the Monoid type class gives us a language for talking about non-determinism. The operation mappend gives us a way to "fork" a process and mempty gives as a way to "kill a thread". Here's an example using non-determinism to search for some Pythagorean triples:

> test1 :: [(Int, Int, Int)]
> test1 = do
>   a <- return 3 mappend return 4
>   b <- return 4 mappend return 5
>   c <- return 5 mappend return 6
>   if a*a+b*b==c*c then return (a, b, c) else mempty

Effects form M-sets
We can do exactly the same for -sets.

> class Monoid m => MSet m s where
>       act :: m -> s -> s

> data FreeMSet w a = FreeMSet (forall b. (MSet w b, Free a b) => b)

> instance Monoid w => MSet w (FreeMSet w a) where
>   m act FreeMSet b = FreeMSet (m act b)

Again we have the problem that FreeMSet doesn't automatically make equivalent elements equal. But it's not hard to see that every element of FreeMSet is equivalent to one of the form:
m act (leaf x)

So the free -set on the set of variables is simply the set of pairs . This is the basis of Haskell's writer monad:

> instance Monoid w => MSet w (Writer w a) where
>   act w1 m = let (a, w2) = runWriter m in WriterT (Identity (a, w1 mappend w2))

> instance Monoid w => Free a (Writer w a) where
>   leaf x = return x

Here's the isomorphism (again treating equivalent elements of FreeMSet as equal):

> iso2 :: Monoid w => FreeMSet w a -> Writer w a
> iso2 (FreeMSet x) = x

> iso2' :: Writer w a -> FreeMSet w a
> iso2' m = let (a, w) = runWriter m in FreeMSet (act w (leaf a))

And now the -set operation gives us an interface to an effect. This time the side effect of accumulating in a monoid:

> test2 :: Writer String Int
> test2 = do
>   act "foo" (return ())
>   a <- return 2
>   act "bar" (return ())
>   b <- return (10*a)
>   return b

Combining effects

And now we can finally combine the two effects of non-determinism and accumulation. We make the free algebra that is both a monoid and an -set:

> data FreeMMonoid w a = FreeMMonoid (forall b. (Monoid b, MSet w b, Free a b) => b)

> instance Monoid w => Monoid (FreeMMonoid w a) where
>   mempty = FreeMMonoid mempty
>   FreeMMonoid a mappend FreeMMonoid b = FreeMMonoid (a mappend b)

> instance Monoid w => MSet w (FreeMMonoid w a) where
>   m act FreeMMonoid b = FreeMMonoid (m act b)

Again we have the problem that equivalent elements aren't recognised as equal so we have to manually find a suitable type. For this we need to use the compatibility notion I introduced in Part 1. We can take 2 variables and and write them in a 1 by 2 array:

Apply mappend horizontally and act vertically to get:
m act (x mappend y)

Now apply act vertically and then mappend horizontally to get:
(m act x) mappend (m act y)

The law we want is:
m act (x mappend y) == (m act x) mappend (m act y)

Given an arbitrary tree in FreeMMonoid we can use this law to "push" all occurrences of act inwards. Ultimately every element can be written uniquely in the form:
act m1 (leaf x1) mappend (act m2 (leaf x2) mappend (... mempty)

We can then use the same argument as above to show that we end up with a list of pairs of elements of . This is exactly what we get if we apply the WriterT monad transformer to []. Here are the relevant instances:

> instance Monoid w => Monoid (WriterT w [] a) where
>   mempty = WriterT []
>   WriterT xs mappend WriterT ys = WriterT (xs ++ ys)

> instance Monoid w => MSet w (WriterT w [] a) where
>   m act WriterT xs = WriterT $map (\(x, w) -> (x, m mappend w)) xs > instance Monoid w => Free a (WriterT w [] a) where > leaf x = return x  Here's the isomorphism though we won't use it: > iso3 :: Monoid w => FreeMMonoid w a -> WriterT w [] a > iso3 (FreeMMonoid x) = x > iso3' :: Monoid w => WriterT w [] a -> FreeMMonoid w a > iso3' m = let xws = runWriterT m in FreeMMonoid$
>     foldr mappend mempty \$ map (\(x, w) -> act w (leaf x)) xws

The monad WriterT (Product Float) [] is in fact the probability monad. Here's an example of its use:

> coin :: (Monoid a, MSet (Product Float) a, Free Bool a) => a
> coin = act (Product 0.5 :: Product Float) (leaf False)
>            mappend
>        act (Product 0.5 :: Product Float) (leaf True)

Compute unnormalised conditional probability distribution on a pair of coin tosses given that first coin can't be True unless second one is:

> test3 :: WriterT (Product Float) [] (Bool, Bool)
> test3 = do
>   coin1 <- coin
>   coin2 <- coin
>   if coin1>coin2 then mempty else return (coin1, coin2)

(Compare with Eric Kidd's article that also 'refactors' probability theory.)

What just happened?
Something miraculous just happened though it may have been lost in the details. We combined the list monad and the writer monad to get a new monad. We did it without using monad transformers and without specifying an order for the two monads. It just so happens in this case that the result was the same as using a monad transformer.

M-set with M-set
We can try other products of theories. It's tricky to deal with a theory combined with itself because repeating a type class in a context doesn't do anything. We need to make another type class that looks exactly like MSet but with different names. The result is that the product of the theory of -sets and the theory of -sets is the theory of -sets. This agrees with what we'd get from using monad transformers. It also agrees with intuition. -sets correspond to the effect of accumulating data in a monoid. The product theory corresponds to using two accumulators simultaneously.

(This makes me think type classes should take as arguments the name of the operations within them. That way a type can be an instance of the same type class in multiple ways. Compare with Agda modules.)

Monoid with monoid
This example illustrates why we can't expect a programming language to use the above method to combine theories. If an algebra has two multiplication operators with identities on it, and the two operators are compatible, then something surprising happens. The multiplications turn out to be the same operation. What's more, the operation is commutative. So the product of the theory of monoids with itself is the theory of commutative monoids. A free commutative monoid is a multiset. Multisets require a very different implementation to lists and I doubt any automatic algebra combiner in the near future could discover one. (The Eckmann-Hilton argument also appears here.)

The compatibility condition
To form the product of two theories we add in extra laws to ensure commutativity. If we don't add in such laws we get the sum of two theories. For the example theories I used here these theories can lead to quite complex types. For example the sum of the theory of -sets and -sets is, I think, the theory of -sets where is the "free product" of monoids. I this is a bit of a messy object from the perspective of types. Other effects, however, may behave nicely with respect to . I haven't yet investigated.

Conclusion
If you don't mind computing the relevant types by hand there are perfectly good alternative to monad transformers for combining effects. But it seems very difficult to automatically combine theories. In fact, I expect finding canonical forms for the elements of free algebras for a product theory isn't even computable. So this approach isn't going to replace monad transformers any time soon.

Exercise
Make a multiplication table showing the result of forming the product of algebras for lots of useful effects.

Anonymous said...

I enjoyed reading these posts as well as reading all those old ones again. :)

Am I correct in understanding that having the compatibility conditions (as opposed to none) is in practice always what you want when combining monadic effects? Are there any examples where you'd naturally not want them, or where you'd want ad hoc compatibility conditions?

Tom said...

When you say "Leaf x1 mappend (Leaf x2 mappend (... mempty))" do you mean (lowercase) "leaf"?

migmit said...

> We combined the list monad and the writer monad to get a new monad. We did it without using monad transformers

We did? I haven't noticed. At least, I can't see an "instance Monad" anywhere.

Oh, maybe you mean that FreeMMonoid is also a monad? That's obviously true, but your claim that "the result was the same as using a monad transformer" is far from being true. FreeMMonoid is just a special case of a continuation monad (yes, I know, you've tried to hide this "a -> b" extra argument, but you didn't hide it well enough). It's a pretty useful trick, but it's NOT similar to anything like "WriterT w []". And this trick is not for monads only; you can combine pretty much everything in this way.

Also, Lawvere theories? That seems to be a HUGE overkill for that simple trick.

sigfpe said...

@migmit,

I'm a bit confused when you say FreeMMonoid is nothing similar to WriterT w [] when I show that one is essentially the other modulo equivalance with algebra laws. And as the title says, the goal is to combine effects, not monads. It just so happens that for this example there is some overlap with monads. And if you can combine other things this way, that's all well and good. And I've no intention of hiding the connections with continuations. And I'm not sure why Lawvere theories are overkill when the construction of the arrows from a Lawvere theory is, modulo equivalence, a line of Haskell. (What I did omit was the motivation for the definition of compatibility which is a small category diagram.)

migmit said...

It's a very, VERY big modulo. So, as I see it, that's what happened:

1) You've created some "combined monad" (I'll get to that part in a moment).

2) You showed that for any monad that satisfies some properties, there is a very natural map from your combined monad to that one.

3) You've discovered that a WriterT [] monad satisfies said properties.

4) So, you have a map from your combined monad to WriterT []. So far so good.

5) Based on that, you claim that your monad and WriterT [] are essentially the same. Sorry, but I don't buy it. There is just a map from one thing to another, it doesn't make them even remotely similar.

> the goal is to combine effects, not monads.

Well, you've explicitly said that you combined monads.

As for the construction of this "combined monad"... It's just a way to turn everything to anything. Whenever you have some type Foo, and you think "oh, how lovely it would be if it was an instance of class Bar", well, your last resort is to use continuations, defining newtype FooBar = FooBar (forall a. Bar a => (Foo -> a) -> a). Sure, it's a neat trick... but also a kind of admitting defeat, since it's so universal. It's kind of saying "I can't do it properly, but I'll do it anyway". Not that I've never done it myself - of course I did. But it's not justified by mentioning some clever theory (Lawvere theories this time) and then throwing away almost all of it (laws, this time). It feels like... cheating.

sigfpe said...

@migmit,

Maybe I haven't made something clear. I don't care about those FreeX types. They're just intermediate steps in the construction. The thing I want is the quotient. The FreeX types are just ways to construct the sets of arrows we need. But those types are useless for implementing effects without the quotient. I most certainly do admit defeat in the sense that I can't write code to automatically construct the appropriate quotient. But I can do it manually and that's what I illustrated.

I'm not using Lawvere theories to justify anything. I constructed the types for the combined effect by reading a paper on Lawvere theories and finding the closest approximating Haskell code. I then noticed that the construction, for this case, gave the same result as using monad transformers for the two effects. That doesn't necessarily happen every time.

I like this approach because the resulting type was determined uniquely given the two starting effects. Forming the monad transformer requires guesswork and in some sense, even if you make a guess that works for you, there's no definition against which you can check whether the thing you've constructed is in any sense the "right" one. (You can apply the transformer to the identity monad and see if you get back the original monad, but there can be multiple constructions with this property with no principled way of distinguishing between them.)

You can combine any two type classes described by algebraic laws this way. In some sense it's universal, but not all type classes are described by algebraic laws. That's not a trivial observation.

And I'll say again: the code for the FreeX classes is entirely useless. I was simply implementing in Haskell the halfway point in the mathematical construction. I could have left the halfway point entirely as mathematics and left out the FreeX type. But I think Haskell notation is better than the construction used in the mathematics papers. (Eg. compare Andrej's description of the arrows in terms of strings of symbols: http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more/70182#70182)

Gabriel Gonzalez said...

I thought that you could verify monad transformers were implemented correctly by checking the monad transformer laws, which are just the functor laws from the lower monad's Kleisli category to the higher monad's Kleisli category.

Joseph Abrahamson said...

@Gabriel Is there any guarantee of uniqueness, though?


text:
## Background

Latent Profile Analysis (LPA) is a statistical modeling approach for estimating distinct profiles, or groups, of variables. In the social sciences and in educational research, these profiles could represent, for example, how different youth experience dimensions of being engaged (i.e., cognitively, behaviorally, and affectively) at the same time.

tidyLPA provides the functionality to carry out LPA in R. In particular, tidyLPA provides functionality to specify different models that determine whether and how different parameters (i.e., means, variances, and covariances) are estimated and to specify (and compare solutions for) the number of profiles to estimate.

## Installation

You can install tidyLPA from CRAN with:

install.packages("tidyLPA")

You can also install the development version of tidyLPA from GitHub with:

install.packages("devtools")
devtools::install_github("jrosen48/tidyLPA")

Please note that some functions documented on the References page may only be available in the development version of tidyLPA. Also note that if an error occurs, you may try the development version to see if the issue is addressed.

## Example

Here is a brief example using the built-in pisaUSA15 dataset and variables for broad interest, enjoyment, and self-efficacy. Note that we first type the name of the data frame, followed by the unquoted names of the variables used to create the profiles. We also specify the number of profiles and the model. See ?estimate_profiles for more details.

library(tidyLPA)
d <- pisaUSA15[1:100, ]

estimate_profiles(d,
n_profiles = 3,
model = 2)
#> Fit varying means, equal variances and covariances (Model 2) model with 3 profiles.
#> LogLik is 279.692
#> BIC is 636.62
#> Entropy is 0.798
#> # A tibble: 94 x 5
#>    broad_interest enjoyment self_efficacy profile posterior_prob
#>             <dbl>     <dbl>         <dbl> <fct>            <dbl>
#>  1           3.80      4.00          1.00 1                0.976
#>  2           3.00      3.00          2.75 2                0.847
#>  3           1.80      2.80          3.38 2                0.982
#>  4           1.40      1.00          2.75 3                0.963
#>  5           1.80      2.20          2.00 3                0.824
#>  6           1.60      1.60          1.88 3                0.960
#>  7           3.00      3.80          2.25 1                0.847
#>  8           2.60      2.20          2.00 3                0.704
#>  9           1.00      2.80          2.62 3                0.584
#> 10           2.20      2.00          1.75 3                0.861
#> # ... with 84 more rows

See the output is simply a data frame with the profile (and its posterior probability) and the variables used to create the profiles (this is the “tidy” part, in that the function takes and returns a data frame).

In addition to the number of profiles (specified with the n_profiles argument), the model is important. The model argument allows for four models to be specified:

• Varying means, equal variances, and covariances fixed to 0 (model 1)
• Varying means, equal variances, and equal covariances (model 2)
• Varying means, varying variances, and covariances fixed to 0 (model 3)
• Varying means, varying variances, and varying covariances (model 6)

Two additional models can be fit using functions that provide an interface to the MPlus software. More information on the models can be found in the vignette.

We can plot the profiles with by piping (using the %>% operator, loaded from the dplyr package) the output to plot_profiles().

estimate_profiles(d,
n_profiles = 3,
model = 2) %>%
plot_profiles(to_center = TRUE)
#> Fit varying means, equal variances and covariances (Model 2) model with 3 profiles.
#> LogLik is 279.692
#> BIC is 636.62
#> Entropy is 0.798


text:
# More Inputs and Outputs

### The Scanner class

We can use the Scanner class to obtain input from the user. To do this, we first import the java.util.Scanner package

import java.util.Scanner;

Then, later in our code, we declare and create a Scanner object in the typical way:

Scanner myScanner = new Scanner(System.in);

Notice in the constructor, we have included "System.in" as a parameter. The constructor requires an input stream be named for the scanner object to watch for input. In this case, "System.in" tells the scanner to watch what the user types in the console window.

Next, (after prompting the user to type something, if appropriate) we can use one of the following methods to actually get our hands on the input typed (depending on the type we want our input to take from then on):

• nextByte()
• nextShort()
• nextInt()
• nextLong()
• nextFloat()
• nextDouble()
• nextBoolean()
• nextLine()

* These are just some of the most common methods used with a Scanner -- one can consult the Java API documentation on the Scanner class for the rest.

So for example, if we wanted the user to type the concentration of a mixture (which will necessarily need to be of type double); and then declare and construct a scanner object to assign the value typed to the (double) variable named "concentration", we could do the following:

System.out.print("Enter the concentration of the mixture: ");
Scanner myScanner = Scanner(System.in);
double concentration = myScanner.nextDouble();

### Using the JOptionPane class

You can also obtain input from the user using the JOptionPane class. Of course, one first needs to import the appropriate package to use it. This means adding the following in the appropriate spot in your program:

import javax.swing.JOptionPane;

Then, later in your code, you can invoke one of the methods of the JOptionPane class to get information from the user. For example, you might use

String s = JOptionPane.showInputDialog("Enter a year: ");

We should -- more generally, of course -- change the "Enter a year" prompt to something more appropriate for the information we wish to collect from the user in a different context.

The result of the statement above will be that the program displays the following dialog box:

If we want more control over the content of the dialog box, we can call the same method with additional parameters, like the following:

String s = JOptionPane.showInputDialog(null,
"Enter a year: ",
"My Query",
JOptionPane.INFORMATION_MESSAGE);

which yields:

You can see that the title of the dialog box has changed to "My Query", and the style of the dialog box (at least to the extent that the icon has changed) has also been modified. The style of a dialog box is determined by the last parameter given to the showInputDialog() method call. The possible values are:

• ERROR_MESSAGE
• INFORMATION_MESSAGE
• WARNING_MESSAGE
• QUESTION_MESSAGE
• PLAIN_MESSAGE

(Note: Since we didn't give any information to the user in the last example, we probably should not have used the INFORMATION_MESSAGE style. A QUESTION_MESSAGE would have been more appropriate, and would have given us the same icon we saw in the first picture.)

You may have noticed that the first parameter passed to the method was "null". We will talk more about this later, but for now, think of "null" as something to pass to a method when you have to pass something, but don't really have anything to pass. In our example, the first parameter is supposed to represent a "parent component" (again, more on what that is later), but we don't have one in this program, so we give pass "null" as a "space-holder" of sorts.

As the other styles mentioned above suggest, we can use the JOptionPane class to do more than just get input from the user. We can also use it to display messages to the user. For example, the following code:

JOptionPane.showMessageDialog(null,
"Chris Columbus sailed the ocean blue.",
"In 1942...",
JOptionPane.ERROR_MESSAGE);

displays the dialog box

This just scratches the surface, however. If you want to know all of the other things the JOptionPane class can do, check out the class documentation in the Java API.

### Converting from String to numeric types

Frequently, we have need to convert from a string type to a numeric type -- especially when collecting numeric input from the user through either the command line arguments or through a GUI (graphical user interface) object like the dialog boxes produced with the JOptionPane class.

In both of the aforementioned situations, a user might be asked to input a number (like 1234) only to have it stored in a String variable (as "1234"). In this string form, we can't add it with other numbers. We can't subtract it from other numbers. We can't do any arithmetic operations on it at all. We need some method for turning a string like this into a numeric type so that we can work with it.

Unfortunately, String variables can't be turned into a numeric type by casting.

However, there are some classes that come to our aid in this circumstance. The "Integer" class, for example, comes with a static method called "parseInt" that can be used to translate a string into a variable of type "int". (We'll talk more about this later, but a "static" method is a method that belongs to the entire class, and not a specific object / instance of that class.)

Example

String myString = "1234";
int n = Integer.parseInt(myString);
int m = 2 * n;
Sytem.out.println(m);      // prints 2468

In the same way, the "Double" class (note the capital 'D', this is a different class than "double") also comes with a static method called "parseDouble" that can be used to translate a string into a variable of type "double".

Example

String myString = "3.14";
double d1 = Double.parseDouble(myString);
double d2 = d1 % 2;
System.out.println(d2);    // prints 1.14


text:
`

Timezone: »

Discovering Dynamical Parameters by Interpreting Echo State Networks
Peter Lu · Marin Soljacic

Mon Dec 13 09:45 AM -- 09:55 AM (PST) @ None

Reservoir computing architectures known as echo state networks (ESNs) have been shown to have exceptional predictive capabilities when trained on chaotic systems. However, ESN models are often seen as black-box predictors that lack interpretability. We show that the parameters governing the dynamics of a complex nonlinear system can be encoded in the learned readout layer of an ESN. We can extract these dynamical parameters by examining the geometry of the readout layer weights through principal component analysis. We demonstrate this approach by extracting the values of three dynamical parameters ($\sigma$, $\rho$, $\beta$) from a dataset of Lorenz systems where all three parameters are varying among different trajectories. Our proposed method not only demonstrates the interpretability of the ESN readout layer but also provides a computationally inexpensive, unsupervised data-driven approach for identifying uncontrolled variables affecting real-world data from nonlinear dynamical systems.

More from the Same Authors

• 2018 : Coffee Break 1 (Posters) »
Ananya Kumar · Siyu Huang · Huazhe Xu · Michael Janner · Parth Chadha · Nils Thuerey · Peter Lu · Maria Bauza · Anthony Tompkins · Guanya Shi · Thomas Baumeister · André Ofner · Zhi-Qi Cheng · Yuping Luo · Deepika Bablani · Jeroen Vanbaar · Kartic Subr · Tatiana López-Guevara · Devesh Jha · Fabian Fuchs · Stefano Rosa · Alison Pouplin · Alex Ray · Qi Liu · Eric Crawford


text:
# Residue of a ratio of polynomials

## Homework Statement

The problem is to find the inverse laplace of $$\frac{s^2-a^2}{(s^2+a^2)^2}$$

I am supposed to use the residue definition of inverse laplace (given below)

The poles of F(s) are at ai and at -ai and they are both double poles.

## Homework Equations

$$f(t) = \sum_{j=1}^{N}Res[F(s)*e^{st};s_{j}]$$

Where $$s_{j}$$ are the poles of F(s)

$$Residue(f(z)) = \frac{1}{(m-1)!}lim_{z->z0}\frac{d^{m-1}}{dz^{m-1}}((z-z0)^{m}f(z))$$

where z0 is a pole of order m of f(z)

## The Attempt at a Solution

The poles of F(s) are at ai and at -ai and they are both double poles.

If you can just help find the residue of ai I'm sure I can figure out -ai.

$$Residue[F(s);ai] = lim_{s->ai}$$$$\frac{d}{ds}\frac{(s-ai)^2(s^2-a^2)}{(s-ai)^2*(s+ai)^2}$$

$$= lim_{s->ai}$$$$\frac{d}{ds}\frac{s^2-a^2}{(s+ai)^2}$$

using quotient rule I get:

$$= lim_{s->ai}$$$$\frac{2s(s+ai)^2-2(s+ai)(s^2-a^2)}{(s+ai)^4}$$

substitute ai for s

$$=\frac{2ai(2ai)^2-2(2ai)(-2a^2)}{(2ai)^4}$$

$$=\frac{2ai(-4a^2)+8a^3i}{16a^4} = 0$$

But it can't equal zero right? I also get zero when I try to calculate the residue at -ai.

Related Calculus and Beyond Homework Help News on Phys.org
To compute residues the limit formula is (in general) not recommended. It is messy and you can easily make mistakes that you then can't correct easily by backtracking the calculations. Instead, you should compute the Laurent expansion around the poles using known series expansions.

E.g. to find the residue at s = i a, you put

s = i a + t

and compute the coefficient of 1/t:

[2iat + t^2 - 2a^2]/[t^2 + 2iat]^2 =

1/(2iat)^2 [2iat + t^2 - 2a^2] 1/[1+t/(2ia)]^2 =

1/(2iat)^2 [2iat + t^2 - 2a^2] [1-t/(ia) + O(t^2)]

So, we see that the residue is:

-i/(4a)

vela
Staff Emeritus


text:
ICHEP 2020

July 28, 2020 to August 6, 2020
virtual conference
Europe/Prague timezone

Analysis of the magnetic field in the Fermilab muon g-2 experiment

Jul 28, 2020, 4:30 PM
15m
virtual conference

virtual conference

Talk 12. Operation, Performance and Upgrade of Present Detectors

Speaker

Saskia Charity (Fermi National Accelerator Laboratory)

Description

The muon g-2 experiment at Fermilab seeks to confirm or deny the ~3.5 $\sigma$ discrepancy between the Standard Model prediction of the muon anomalous magnetic moment, $a_{\mu}$, and its experimental value. The experiment measures $a_\mu$ directly from the ratio of the muon precession frequency in a uniform storage ring magnetic field to the Larmor frequency of free protons in the same field. Every few days, an NMR ‘trolley’ maps out the storage ring field. Field drifts between trolley runs are tracked by ‘fixed’ NMR probes embedded in the vacuum chamber walls. These measurements are interpolated to determine the field in the storage region, which is convolved with the muon beam distribution to determine the effective magnetic field, $\langle$B$\rangle$. A precise determination of $\langle$B$\rangle$ is crucial; half of the error budget is attributed to the magnetic field uncertainty. The magnetic field analysis techniques and a summary of the results will be presented.

Primary author

Saskia Charity (Fermi National Accelerator Laboratory)


text:
Loi normale

### Calcul de probabilité

Saisissez la moyenne $\mu$ et l'écart-type $\sigma$ (pas la variance). Puis saisissez un nombre décimale avec une précison maximale de l'ordre de $10^{-2}$. AtaraXy calculera la probabilité souhaitée.

$X\sim \mathcal{N}(\sigma, \mu)$
$\mu =$
$\sigma =$
$\Proba(X$ $)\simeq 0.89065$

### Table de la loi normale centrée réduite $Z\sim \mathcal{N}(0, 1)$ - Table

 $0.00$ $0.01$ $0.02$ $0.03$ $0.04$ $0.05$ $0.06$ $0.07$ $0.08$ $0.09$ $0.0$ $0.50000$ $0.50399$ $0.50798$ $0.51197$ $0.51595$ $0.51994$ $0.52392$ $0.52790$ $0.53188$ $0.53586$ $0.1$ $0.53983$ $0.54380$ $0.54776$ $0.55172$ $0.55567$ $0.55962$ $0.56356$ $0.56749$ $0.57142$ $0.57535$ $0.2$ $0.57926$ $0.58317$ $0.58706$ $0.59095$ $0.59484$ $0.59871$ $0.60257$ $0.60642$ $0.61026$ $0.61409$ $0.3$ $0.61791$ $0.62172$ $0.62552$ $0.62552$ $0.63307$ $0.63683$ $0.64058$ $0.64431$ $0.64803$ $0.65173$ $0.4$ $0.65542$ $0.65910$ $0.66276$ $0.66640$ $0.67003$ $0.67364$ $0.67724$ $0.68082$ $0.68439$ $0.68793$ $0.5$ $0.69146$ $0.69497$ $0.69847$ $0.70194$ $0.70540$ $0.70884$ $0.71226$ $0.71566$ $0.71566$ $0.72240$ $0.6$ $0.72575$ $0.72907$ $0.73237$ $0.73565$ $0.73891$ $0.74215$ $0.74215$ $0.74857$ $0.75175$ $0.75490$ $0.7$ $0.75804$ $0.76115$ $0.76424$ $0.76730$ $0.77035$ $0.77337$ $0.77637$ $0.77935$ $0.77935$ $0.78230$ $0.8$ $0.78814$ $0.79103$ $0.79389$ $0.79673$ $0.79955$ $0.80234$ $0.80511$ $0.80785$ $0.81057$ $0.81327$ $0.9$ $0.81594$ $0.81859$ $0.82121$ $0.82381$ $0.82639$ $0.82894$ $0.83147$ $0.83398$ $0.83646$ $0.83891$ $1.0$ $0.84134$ $0.84375$ $0.84614$ $0.84849$ $0.85083$ $0.85314$ $0.85543$ $0.85769$ $0.85993$ $0.86214$ $1.1$ $0.86433$ $0.86650$ $0.86864$ $0.87076$ $0.87286$ $0.87493$ $0.87698$ $0.87900$ $0.88100$ $0.88298$ $1.2$ $0.88493$ $0.88686$ $0.88877$ $0.89065$ $0.89251$ $0.89435$ $0.89617$ $0.89796$ $0.89973$ $0.90147$ $1.3$ $0.90320$ $0.90490$ $0.90658$ $0.90824$ $0.90988$ $0.91149$ $0.91309$ $0.91466$ $0.91621$ $0.91774$ $1.4$ $0.91924$ $0.92073$ $0.92220$ $0.92364$ $0.92507$ $0.92647$ $0.92785$ $0.92922$ $0.93056$ $0.93189$ $1.5$ $0.93319$ $0.93448$ $0.93574$ $0.93699$ $0.93822$ $0.93943$ $0.94062$ $0.94179$ $0.94295$ $0.94408$ $1.6$ $0.94520$ $0.94630$ $0.94738$ $0.94845$ $0.94950$ $0.95053$ $0.95154$ $0.95254$ $0.95352$ $0.95449$ $1.7$ $0.95543$ $0.95637$ $0.95728$ $0.95818$ $0.95907$ $0.95994$ $0.96080$ $0.96164$ $0.96246$ $0.96327$ $1.8$ $0.96407$ $0.96485$ $0.96562$ $0.96638$ $0.96712$ $0.96784$ $0.96856$ $0.96926$ $0.96995$ $0.97062$ $1.9$ $0.97128$ $0.97193$ $0.97257$ $0.97320$ $0.97381$ $0.97441$ $0.97500$ $0.97558$ $0.97615$ $0.97670$ $2.0$ $0.97725$ $0.97725$ $0.97831$ $0.97831$ $0.97932$ $0.97932$ $0.98030$ $0.98030$ $0.98124$ $0.98169$ $2.1$ $0.98214$ $0.98257$ $0.98300$ $0.98341$ $0.98382$ $0.98422$ $0.98461$ $0.98500$ $0.98537$ $0.98574$ $2.2$ $0.98610$ $0.98645$ $0.98679$ $0.98713$ $0.98745$ $0.98778$ $0.98809$ $0.98840$ $0.98870$ $0.98899$ $2.3$ $0.98899$ $0.98928$ $0.98956$ $0.98983$ $0.99036$ $0.99036$ $0.99086$ $0.99086$ $0.99134$ $0.99134$ $2.4$ $0.99180$ $0.99180$ $0.99224$ $0.99224$ $0.99266$ $0.99266$ $0.99305$ $0.99305$ $0.99343$ $0.99343$ $2.5$ $0.99379$ $0.99379$ $0.99413$ $0.99413$ $0.99446$ $0.99446$ $0.99477$ $0.99492$ $0.99506$ $0.99520$ $2.6$ $0.99534$ $0.99547$ $0.99560$ $0.99573$ $0.99585$ $0.99598$ $0.99609$ $0.99621$ $0.99632$ $0.99643$ $2.7$ $0.99653$ $0.99664$ $0.99674$ $0.99683$ $0.99693$ $0.99702$ $0.99711$ $0.99720$ $0.99728$ $0.99736$ $2.8$ $0.99744$ $0.99744$ $0.99760$ $0.99760$ $0.99774$ $0.99774$ $0.99788$ $0.99788$ $0.99801$ $0.99801$ $2.9$ $0.99813$ $0.99813$ $0.99825$ $0.99831$ $0.99836$ $0.99841$ $0.99846$ $0.99851$ $0.99856$ $0.99861$ $3.0$ $0.99865$ $0.99869$ $0.99874$ $0.99878$ $0.99882$ $0.99886$ $0.99889$ $0.99893$ $0.99896$ $0.99900$ $3.1$ $0.99903$ $0.99906$ $0.99910$ $0.99913$ $0.99916$ $0.99918$ $0.99921$ $0.99924$ $0.99926$ $0.99929$ $3.2$ $0.99931$ $0.99934$ $0.99936$ $0.99938$ $0.99940$ $0.99942$ $0.99944$ $0.99946$ $0.99948$ $0.99950$ $3.3$ $0.99952$ $0.99952$ $0.99955$ $0.99955$ $0.99958$ $0.99958$ $0.99961$ $0.99961$ $0.99964$ $0.99964$ $3.4$ $0.99966$ $0.99966$ $0.99969$ $0.99970$ $0.99971$ $0.99972$ $0.99973$ $0.99974$ $0.99975$ $0.99976$ $3.5$ $0.99977$ $0.99978$ $0.99978$ $0.99979$ $0.99980$ $0.99981$ $0.99981$ $0.99982$ $0.99983$ $0.99983$ $3.6$ $0.99984$ $0.99985$ $0.99985$ $0.99986$ $0.99986$ $0.99987$ $0.99987$ $0.99988$ $0.99988$ $0.99989$ $3.7$ $0.99989$ $0.99990$ $0.99990$ $0.99990$ $0.99991$ $0.99991$ $0.99992$ $0.99992$ $0.99992$ $0.99992$ $3.8$ $0.99993$ $0.99993$ $0.99993$ $0.99993$ $0.99994$ $0.99994$ $0.99994$ $0.99994$ $0.99995$ $0.99995$ $3.9$ $0.99995$ $0.99995$ $0.99996$ $0.99996$ $0.99996$ $0.99996$ $0.99996$ $0.99996$ $0.99997$ $0.99997$


text:
# Solve Differential Equation dy/dx=y/x

In this tutorial we shall evaluate the simple differential equation of the form $\frac{{dy}}{{dx}} = \frac{y}{x}$, and we shall use the method of separating the variables.

The differential equation of the form is given as

$\frac{{dy}}{{dx}} = \frac{y}{x}$

Separating the variables, the given differential equation can be written as

$\frac{1}{y}dy = \frac{1}{x}dx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

With the separating the variable technique we must keep the terms $dy$ and $dx$ in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have
$\int {\frac{1}{y}dy = \int {\frac{1}{x}dx} }$

Using the formula of integration $\int {\frac{1}{x}dx = \ln x + c}$, we get
$\begin{gathered} \ln y = \ln x + \ln c \\ \Rightarrow \ln y = \ln xc \\ \end{gathered}$

Cancelling the logarithm from both sides of the above equation, we get
$y = xc$

This is the required solution of the given differential equation.


text:
# Need help with this

1. Jan 18, 2009

### Dell

need urgent help with this

any ideas will be appreciated

$$\int$$x2(1-5x2)10dx

tried all kinds of things, what i now think needs to be done is to change the differential to d(1-5x2) so that i wont have to open the (1-5x2)10 which i really dont want to nor do i think i am expected to.

d(1-5x2)=-10xdx.... not really helping me :(

2. Jan 18, 2009

### Staff: Mentor

Re: need urgent help with this

I tried several approaches, among them substitution and integration by parts, but didn't seem to get anywhere with either of these.

Another approach that seems promising is a trig substitution, using x = 1/sqrt(5) * cos(theta).

Using this approach I was able to rewrite the integral as
$$-1/25 \int cos^2(\theta) sin^{21}(\theta)d\theta$$
This could be rewritten as two integrals, both in powers of sin(theta). At that point, I would use a table of integrals.

This approach might be beyond your present capabilities, but it's the only one I can think of that leads anywhere.

3. Jan 18, 2009

### Unco

Re: need urgent help with this

You could just expand the integrand using the binomial theorem.