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# NYCAC 2014 - Abstracts

Friday, November 14, 2014,   New York City
Room 4102 (Science Center) and room 4421

## The New York Colloquium on Algorithms and Complexity

#### Xi Chen

On the Complexity of Nash Equilibria in Anonymous Games
Anonymous games are a large and well-studied class of succinct multiplayer games, in which the payoff of each player depends only on her own strategy and number of other players playing each strategy. In a sequence of papers, Daskalakis and Papadimitriou studied the problem of finding an approximate Nash equilibrium in an anonymous game, and obtained polynomial-time approximation schemes when the number of strategies is bounded. In this talk we briefly review their results and show that the problem is complete in PPAD when the number of strategies is at least seven and the approximation parameter is exponentially small, confirming a conjecture of Daskalakis and Papadimitriou.
Joint work with David Durfee (Georgia Institute of Technology) and Anthi Orfanou (Columbia University).

#### Lane Hemaspaandra

Control in the Presence of Manipulators: Cooperative and Competitive Cases
Control and manipulation are two of the most studied types of attacks on elections. We study the complexity of control attacks on elections in which there are manipulators. We study both the case where the "chair" who is seeking to control the election is allied with the manipulators, and the case where the manipulators seek to thwart the chair. In the latter case, we see that the order of play substantially influences the complexity. We prove upper bounds, holding over every election system with a polynomial-time winner problem, for all standard control cases, and some of these bounds are at the second or third level of the polynomial hierarchy, and we provide matching lower bounds to prove these tight. Nonetheless, for important natural systems the complexity can be much lower. We prove that for approval and plurality elections, the complexity of even competitive clashes between a controller and manipulators falls far below those high bounds, even as low as polynomial time. Yet for a Borda-voting case we show that such clashes raise the complexity unless NP=coNP. This is joint work with Zack Fitzsimmons and Edith Hemaspaandra.

#### Alan Selman

Disjoint NP-pairs and Propositional Proof Systems
This talk surveys results on disjoint NP-pairs, propositional proof systems, function classes, and promise classes---including results that demonstrate close connections that bind these topics together. We illustrate important links between the questions of whether these classes have complete objects and whether optimal proof systems may exist.

#### Daniel Stefankovic

Spin models: connections between complexity, phase transition, belief propagation, and induced matrix norms
What are the typical configurations of a spin model (for example, Ising model, or Potts model) on a random regular graph ? We show that the answer to this question is characterized by tree recursions (belief propagation) and p->q operator matrix norms.
Understanding the typical configurations allows us to show hardness of approximating the partition function for certain multispin models (for example, Potts model) on d-regular graphs in the non-uniqueness regime (that is, when the Gibbs measure on infinite d-regular tree is not unique). Joint work with Andreas Galainis, and Eric Vigoda.

#### Rocco Servedio

A Probably Approximately Correct Lower Bound for Non-Adaptive Boolean Function Monotonicity Testing
Boolean function monotonicity testing is a problem that has received a lot of attention in the property testing community. In a recent work (FOCS 2014) Chen, Servedio and Tan gave an $\tilde{\Omega}(n^{1/5})$ lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown $n$-variable Boolean function is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of $\Omega(\log n)$ for this model due to Fischer et al from 2002. More recently, Chen, De, Servedio and Tan extended this lower bound to $\Omega(n^{1/2-\tau})$ for any $\tau>0$. If one believes (as is commonly conjectured) that the true nonadaptive query complexity of monotonicity testing is $\Theta(\sqrt{n})$, this new lower bound is probably approximately correct (or more precisely, probably approximately tight).
In this talk I will present the main ideas behind these recent results and talk about directions for future work on the monotonicity testing problem.

#### Moti Yung

Surveillance Adversary Protocols: past and future


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new_gaussian_control¶

qctrlopencontrols.new_gaussian_control(rabi_rotation: float, segment_count: int, duration: float, width: float, name: Optional[str] = None)qctrlopencontrols.driven_controls.driven_control.DrivenControl[source]

Generates a Gaussian driven control sequence.

Gaussian driven controls mitigate leakage out of the qubit subspace.

Parameters
• rabi_rotation (float) – Total Rabi rotation $$\theta$$ to be performed by the driven control.

• segment_count (int) – Number of segments in the control sequence.

• duration (float) – Total duration $$t_g$$ of the control sequence.

• width (float) – Width (standard deviation) $$\sigma$$ of the ideal Gaussian pulse.

• name (str, optional) – An optional string to name the control. Defaults to None.

Returns

A control sequence as an instance of DrivenControl.

Return type

DrivenControl

Notes

A Gaussian driven control 1 consists of a piecewise constant approximation to an ideal Gaussian pulse:

$\mathcal{E}_G (t) = A \exp \left[- \frac{(t - t_g/2)^2}{2\sigma^2}\right] - B$

where the two additional parameters $$A, B$$ chosen such that $$\int_{0}^{t_g} \mathcal{E}_G \,dt = \theta$$ and $$\mathcal{E}_G(0) = 0$$.

Relative values of segments are determined by sampling the ideal Gaussian at the midpoints of the segments.

References

1

Motzoi, F. et al. Physical Review Letters 103, 110501 (2009)

format: plain


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# Question about computing a Fourier transform of an integral transform related to fractional Brownian motion

I am trying to show an integral transform has a fixed point.

Let $H \in (0,1)$ and consider the following integral transform whose kernel is the density of fractional Brownian motion: $$T_H f(x) = 2 \int_{0}^{\infty} \frac{1}{\sqrt{2 \pi y^{2H}}}\exp\left(\frac{-x^2}{2y^{2H}}\right) f(y) dy.$$

I want to show $T_H$ has a fixed point given by $\phi(x) = \exp(-c|x|^{\frac{1}{2(1-H)}})$ for some $c \in \mathbb{R}$. That is $T_H \phi (x) = \phi (x)$. Although I do not know if this is true in general I know it is true for the case $H=\frac{1}{2}$.

1) What is the Fourier transform of $T_H \phi(x)$ with respect x? Is it equal to a multiple of the Fourier transform of $\phi(x)$?

The motivation for the problem is is that we know for the $H=\frac{1}{2}$ and $c=2$ then $\phi(x) = e^{-2|x|}$ is a fixed point for $T_{\frac{1}{2}}$. That is the Fourier transform of
$T_{\frac{1}{2}}e^{-2|x|} = 2 \int_{0}^{\infty} \frac{e^{\frac{-x^2}{2y}}}{\sqrt{2 \pi y}} e^{-2y} dy$ is equal to $\frac{4}{k^2+4}$ which is the same as the Fourier transform of $e^{-2|x|}$. This is from a paper titled iterated random walk by L. Turban if anyone is interested in the reference.

#### Solutions Collecting From Web of "Question about computing a Fourier transform of an integral transform related to fractional Brownian motion"

It is rather easy to verify numerically, that $\phi(x) = \exp(-c \vert x \vert^{\frac{1}{2(1-H)}} )$ does not satisfy $T_H \phi = \phi$.

To this end it is enough to observe that $(T_H \phi_c)(x) = c^{-2 (1-h)^2} (T_H \phi_1)(x c^{2 h(1-h)})$, which is obtained via changing variables $y \to c^{-2(1-h)} y$. This scaling tells you that if $T_H \phi_c = \phi_c$, then $T_H \phi_1 = c^{-2(1-h)^2} \phi_1( x c^{2 h (1-h)} )$.

Thus it suffices to show that functional forms of $T_H \phi$ and $\phi$ are different for your purported function. So we set $c=1$ and change variables into $y \to t^{2(1-h)}$. This gives

$$(T_H \phi)(x) = 2(1-h) \sqrt{\frac{2}{\pi}} \int_0^\infty t^{2(1-h)^2} \mathrm{e}^{-t} \exp(-\frac{x^2}{2} t^{-4 h (1-h)} ) \frac{\mathrm{d} t}{t}$$
The integral can be evaluate in terms of Fox H-function. Specifically, in terms of Mellin-Barnes integral:

$$(T_H \phi)(x) =\sqrt{\frac{2}{\pi}} \frac{z^{2 \rho_h}}{2h} \frac{1}{2 \pi i} \int_{\gamma – i \infty}^{\gamma+i \infty} \left( \Gamma(s) \Gamma( -\rho_h + \frac{s}{\kappa_h} ) z^{-\frac{2s}{\kappa_h }} \right)\mathrm{d}s$$
where $\gamma$ is an arbitrary $\gamma > \rho_h \kappa_h$ and $z = \frac{x}{\sqrt{2}}$ and we introduced $\kappa_h = 4 h(1-h)$ and $\rho_h = \frac{1-h}{2 h}$.

The Mellin-Barnes integral can be evaluated as the sum over poles of $\Gamma$ functions at $s= -k$ for $k \in \mathbb{Z}^+$ and at $s = \rho_h \kappa_h – k \, \kappa_h$, giving
$$\begin{eqnarray} (T_H \phi)(x) = \frac{z^{2 \rho_h} }{2 h} \sqrt{\frac{2}{\pi}} && \sum_{k=0}^\infty (- z^{\frac{2}{\kappa_h}} )^k \frac{\Gamma(-\rho_h – \kappa_h \, k )}{k!} + \\ \sqrt{\frac{4 \rho_h \kappa_h}{\pi}} && \sum_{k=0}^\infty (- z^{ 2 } )^k \frac{\Gamma(\rho_h \kappa_h – k \, \kappa_h )}{k!} \end{eqnarray}$$

When $h = \frac{1}{2}$, $\kappa_{1/2} = 1$ and $\rho_{1/2} = \frac{1}{2}$. Then the sums evaluate exactly:

$$\begin{eqnarray} T_{1/2} \phi &=& \frac{x}{\sqrt{2}} \sqrt{ \frac{2}{\pi} } \sum_{k=0}^\infty (-\frac{x^2}{2})^k \frac{\Gamma(-\frac{1}{2}-k)}{k!} + \sqrt{ \frac{2}{\pi} } \sum_{k=0}^\infty (-\frac{x^2}{2})^k \frac{\Gamma(1/2-k)}{k!} \\ &=& -\sqrt{2} \sinh( \sqrt{2} x) + \sqrt{2} \cosh( \sqrt{2} x) = \sqrt{2} \, \exp( -\sqrt{2} x) \end{eqnarray}$$

It is also possible to evaluate these sums for rational values of $h$, but expressions involve hypergeometric functions and stop being so simple.

Added: Below we construct the $T_H \psi$, where $\psi(x)$ is a general Fox H-function. The class of Fox H-functions is a natural place to search for solutions of $T_H \psi = \psi$ because operator $T_H$ maps a Fox H-function into another Fox H-function. Specifically, let

$$\begin{eqnarray} \psi(x) &=& H^{m,n}_{p,q}\left( \left. \begin{array}{c} (a_1, \alpha_1), \ldots, (a_p, \alpha_p) \\ (b_1, \beta_1), \ldots, (b_q, \beta_q) \end{array} \right\vert x\right) \\ &=& \frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma + i \infty} \frac{ \prod_{k=1}^m \Gamma(1-a_k – \alpha_k s) \prod_{k=1}^n \Gamma(b_k + \beta_k s) }{ \prod_{k=m+1}^p \Gamma(1-a_k – \alpha_k s) \prod_{k=n+1}^q \Gamma(b_k + \beta_k s) } x^{-s} \mathrm{d} s \end{eqnarray}$$

Then

$$(T_H \psi)(x) = \frac{1}{\sqrt{2 \pi}} \frac{1}{h} \left( \frac{x^2}{2} \right)^{\frac{(1-h)}{2h} } H^{m,n+1}_{p,q+1}\left( \left. \begin{array}{c} (a_1, \alpha_1), \ldots, (a_p, \alpha_p) \\ \left( -\frac{1-h}{2h}, \frac{1}{2h}\right), (b_1, \beta_1), \ldots, (b_q, \beta_q) \end{array} \right\vert \left(\frac{x^2}{2}\right)^{\frac{1}{2h}} \right)$$

Now let’s understand the mechanics of how it works for $h=\frac{1}{2}$. Your function $\phi(x)$ with $h=\frac{1}{2}$ is $\exp(-x) = H^{0,1}_{0,1}(x)$ with $\beta_1=1$ and $b_1 = 0$, so

$$\begin{eqnarray} \left. (T_H \phi)(x) \right\vert_{h=\frac{1}{2}} &=& \frac{2}{\sqrt{2 \pi}} \left( \frac{x^2}{2} \right)^{\frac{1}{2} } H^{0,2}_{0,2}\left( \left. \begin{array}{c} – \\ \left( -\frac{1}{2}, 1\right), (0, 1) \end{array} \right\vert \frac{x^2}{2} \right) \\ &=& \sqrt{\frac{2}{\pi}} \left( \frac{x^2}{2} \right)^{\frac{1}{2} } \frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} \Gamma\left( -\frac{1}{2} + s\right)\Gamma(s) \left( \frac{x^2}{2} \right)^{-s} \mathrm{d} s \\ &=& \sqrt{\frac{2}{\pi}} \left( \frac{x^2}{2} \right)^{\frac{1}{2} } \frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} \sqrt{\pi} \Gamma(2s-1) 4^{1-s} \left( \frac{x^2}{2} \right)^{-s} \mathrm{d} s \left. = \right|_{s \to \frac{s}{2}} \\ &=& 2 x \frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} \Gamma(s-1) \left( \sqrt{2} x \right)^{-s} \mathrm{d} s \\ &=& (2 x) \left( \frac{\exp(- \sqrt{2} x)}{\sqrt{2} x} \right) = \sqrt{2} \exp(- \sqrt{2} x) \end{eqnarray}$$

Notice that the use of tuplication identity for $\Gamma(x)$ was essential here, this essentially proves that within the family of Fox H-functions the solution to $(T_H \psi)(x) = c_1 \psi(c_2 x)$ for some constants $c_1$ and $c_2$ is only possible for $h=\frac{1}{2}$.

Now, the family of Fox H-functions is big and encompasses all hyper-geometric elementary functions, i.e. those that satisfy a linear differential equations with coefficients polynomial in $x$.

If you would like to find the eigenfunction of $T_h$ operator, I would suggest playing with
quadrature rules. Suppose $(T_H f)(x) = \sum_{i=1}^n \omega_i K(x, y_i) f(y_i)$. Solve the system of equations

$$\sum_{i=1}^n \omega_i K(x_j, y_i) f(y_i) = f(x_j)$$
numerically, which would allow you to explore the eigen-function for $h \not= \frac{1}{2}$.

As in your other problem, I will assume that the Fourier Transform is normalized as
$$\hat{f}(t)=\int_{-\infty}^\infty f(x)\;e^{-ixt}\;\mathrm{d}x$$
As shown in my answer to that question,
$$\int_{-\infty}^\infty e^{\frac{-x^2}{2y^{2H}}}e^{-ixt}dx=\sqrt{2\pi y^{2H}}\;e^{-y^{2H}/2\;t^2}$$
By linearity, the Fourier Transform of $T_Hf(x)$ is
\begin{align} &\mathcal{FT}\;\;2\int_{0}^{\infty} \frac{1}{\sqrt{2 \pi y^{2H}}}\exp\left(\frac{-x^2}{2y^{2H}}\right) f(y) dy\\ &=2\int_{0}^{\infty}\frac{1}{\sqrt{2 \pi y^{2H}}}\sqrt{2\pi y^{2H}}\;\exp(\frac{-y^{2H}}{2}t^2)f(y)\;dy\\ &=2\int_{0}^{\infty}\;\exp(\frac{-y^{2H}}{2}t^2)f(y)\;dy \end{align}
Plugging in $f(y)=\phi(y)=\exp(-cy^{\frac{1}{2(1-H)}})$, we get
\begin{align} &\mathcal{FT}\;\;T_H\phi(t)\\ &=2\int_{0}^{\infty}\;\exp(\frac{-y^{2H}}{2}t^2)\exp(-cy^{\frac{1}{2(1-H)}})\;dy \end{align}
I don’t see a way of continuing unless $H=\frac{1}{2}$, in which case,
\begin{align} &\mathcal{FT}\;\;T_{\frac{1}{2}}\phi(t)\\ &=2\int_{0}^{\infty}\;\exp(\frac{-y}{2}t^2)\exp(-cy)\;dy\\ &=\frac{4}{2c+t^2} \end{align}


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# New constructions of twistor lifts for harmonic maps

Martin Svensson, John C. Wood

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.
Original language English Manuscripta Mathematica 144 3-4 457-502 0025-2611 https://doi.org/10.1007/s00229-014-0659-9 Published - 2014

Twistors
Harmonic Maps
Symmetric Spaces
Riemann Surface
Explicit Formula
If and only if

### Keywords

• math.DG
• 53C43 (Primary) 58E20 (Secondary)

### Cite this

Svensson, Martin ; C. Wood, John. / New constructions of twistor lifts for harmonic maps. In: Manuscripta Mathematica. 2014 ; Vol. 144, No. 3-4. pp. 457-502.
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title = "New constructions of twistor lifts for harmonic maps",
abstract = "We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.",
keywords = "math.DG, 53C43 (Primary) 58E20 (Secondary)",
author = "Martin Svensson and {C. Wood}, John",
year = "2014",
doi = "10.1007/s00229-014-0659-9",
language = "English",
volume = "144",
pages = "457--502",
journal = "Manuscripta Mathematica",
issn = "0025-2611",
publisher = "Heinemann",
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}

In: Manuscripta Mathematica, Vol. 144, No. 3-4, 2014, p. 457-502.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - New constructions of twistor lifts for harmonic maps

AU - Svensson, Martin

AU - C. Wood, John

PY - 2014

Y1 - 2014

N2 - We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.

AB - We show that given a harmonic map \varphi from a Riemann surface into a classical simply connected compact inner symmetric space, there is a J_2-holomorphic twistor lift of \varphi (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.

KW - math.DG

KW - 53C43 (Primary) 58E20 (Secondary)

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DO - 10.1007/s00229-014-0659-9

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ER -


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2-1520-1.1-c1-0-19 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 1.1 $$1.0 1 0.5 1 1.34946 Elliptic curve 1520.b Modular form 1520.2.a.b Modular form 1520.2.a.b.1.1 2-1520-1.1-c1-0-2 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 1.1$$ $1.0$ $1$ $0$ $0$ $0.557668$ Modular form 1520.2.a.t.1.1
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2-1520-1.1-c1-0-4 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 1.1 $$1.0 1 0 0 0.751727 Modular form 1520.2.a.r.1.2 2-1520-1.1-c1-0-5 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 1.1$$ $1.0$ $1$ $0$ $0$ $0.773675$ Modular form 1520.2.a.p.1.2
2-1520-1.1-c1-0-6 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 1.1 $$1.0 1 0 0 0.861758 Modular form 1520.2.a.s.1.1 2-1520-1.1-c1-0-7 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 1.1$$ $1.0$ $1$ $0$ $0$ $0.904002$ Modular form 1520.2.a.t.1.3
2-1520-1.1-c1-0-8 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 1.1 $$1.0 1 0 0 0.936674 Modular form 1520.2.a.s.1.2 2-1520-1.1-c1-0-9 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 1.1$$ $1.0$ $1$ $0$ $0$ $0.938881$ Elliptic curve 1520.e Modular form 1520.2.a.e Modular form 1520.2.a.e.1.1
2-1520-19.11-c1-0-0 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 0.463 0 0.0794656 Modular form 1520.2.q.o.961.2 2-1520-19.11-c1-0-1 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $0.285$ $0$ $0.0800602$ Modular form 1520.2.q.h.961.1
2-1520-19.11-c1-0-10 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.0199 0 0.730578 Modular form 1520.2.q.h.961.2 2-1520-19.11-c1-0-11 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $-0.227$ $0$ $0.746077$ Modular form 1520.2.q.n.961.2
2-1520-19.11-c1-0-12 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.0355 0 0.771609 Modular form 1520.2.q.p.961.1 2-1520-19.11-c1-0-13 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $-0.115$ $0$ $0.778638$ Modular form 1520.2.q.i.961.2
2-1520-19.11-c1-0-14 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.0677 0 0.803641 Modular form 1520.2.q.j.961.2 2-1520-19.11-c1-0-15 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $0.00948$ $0$ $0.860797$ Modular form 1520.2.q.m.961.1
2-1520-19.11-c1-0-16 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.327 0 0.896903 Modular form 1520.2.q.n.961.1 2-1520-19.11-c1-0-17 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $-0.0627$ $0$ $0.898947$ Modular form 1520.2.q.k.961.1
2-1520-19.11-c1-0-18 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 0.234 1 1.02783 Modular form 1520.2.q.a.961.1 2-1520-19.11-c1-0-19 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $-0.00846$ $0$ $1.04711$ Modular form 1520.2.q.k.961.3
2-1520-19.11-c1-0-2 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.358 0 0.266098 Modular form 1520.2.q.m.961.3 2-1520-19.11-c1-0-20 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $0.0229$ $0$ $1.05840$ Modular form 1520.2.q.o.961.3
2-1520-19.11-c1-0-21 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 0.141 0 1.09526 Modular form 1520.2.q.l.961.3 2-1520-19.11-c1-0-22 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $-0.0859$ $0$ $1.10426$ Modular form 1520.2.q.l.961.2
2-1520-19.11-c1-0-23 $3.48$ $12.1$ $2$ $2^{4} \cdot 5 \cdot 19$ 19.11 $$1.0 1 -0.122 0 1.13822 Modular form 1520.2.q.o.961.1 2-1520-19.11-c1-0-24 3.48 12.1 2 2^{4} \cdot 5 \cdot 19 19.11$$ $1.0$ $1$ $0.157$ $0$ $1.31380$ Modular form 1520.2.q.p.961.4
Next


text:
# Quantum fate of singularities in a dark-energy dominated universe

@article{BouhmadiLpez2009QuantumFO,
title={Quantum fate of singularities in a dark-energy dominated universe},
author={Mariam Bouhmadi-L{\'o}pez and Claus Kiefer and Barbara Sandhoefer and Paulo Vargas Moniz},
journal={Physical Review D},
year={2009},
volume={79},
pages={124035}
}
• Published 14 May 2009
• Physics
• Physical Review D
Classical models for dark energy can exhibit a variety of singularities, many of which occur for scale factors much bigger than the Planck length. We address here the issue of whether some of these singularities, the big freeze and the big demarrage, can be avoided in quantum cosmology. We use the framework of quantum geometrodynamics. We restrict our attention to a class of models whose matter content can be described by a generalized Chaplygin gas and be represented by a scalar field with an…
56 Citations

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## References

SHOWING 1-10 OF 97 REFERENCES
Quantum cosmology with a big-brake singularity
• Physics, Mathematics
• 2007
We investigate a cosmological model with a big-brake singularity in the future: while the first time derivative of the scale factor goes to zero, its second time derivative tends to minus infinity.
Properties of singularities in the (phantom) dark energy universe
• Physics, Mathematics
• 2005
The properties of future singularities are investigated in the universe dominated by dark energy including the phantom-type fluid. We classify the finite-time singularities into four classes and
Avoidance of future singularities in loop quantum cosmology
• Physics
• 2006
We consider the fate of future singularities in the effective dynamics of loop quantum cosmology. Nonperturbative quantum geometric effects which lead to ${\ensuremath{\rho}}^{2}$ modification of the
Quantum effects, soft singularities and the fate of the universe in a braneworld cosmology
• Physics
• 2006
We examine a class of braneworld models in which the expanding universe encounters a 'quiescent' future singularity. At a quiescent singularity, the energy density and pressure of the cosmic fluid as
Escaping the big rip
• Physics
• 2005
We discuss dark energy models which might describe effectively the actual acceleration of the universe. More precisely, for a four-dimensional Friedmann–Lemaitre–Robertson–Walker (FLRW) universe we
Singularities in loop quantum cosmology.
• Physics, Mathematics
Physical review letters
• 2008
While LQC avoids big bang or big rip type singularities, it is found that sudden singularities where the Hubble rate is bounded, but the Ricci curvature scalar diverges are found.
Dynamics of dark energy
• Physics
• 2006
We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent
Quantum phantom cosmology
• Physics
• 2006
We apply the formalism of quantum cosmology to models containing a phantom field. Three models are discussed explicitly: a toy model, a model with an exponential phantom potential, and a model with
Quantum cosmological Friedman models with an initial singularity
We consider the Wheeler–DeWitt equation Hψ = 0 in a suitable Hilbert space. It turns out that this equation has countably many solutions ψi which can be considered as eigenfunctions of a Hamilton


text:
# At Sunrise Corporation, direct materials are added at the beginning of the process and conversion...

At Sunrise Corporation, direct materials are added at the beginning of the process and conversions costs are uniformly applied. Other details include: What is the total cost of units completed and transferred out? $574, 164$526, 640 $584, 455$602, 873


text:
# Why don't decomposition reactions form pure elements as products?

For example, if I had a reaction like this:

$$\ce{NaHCO3 -> Na2CO3 + H2O + CO2}$$

• Why does it not break down all the way down to its elements?
• What makes it form such "intermediate" products, stopping at water, carbon dioxide, etc.?
• If you heat sodium bicarbonate strongly you can convert it to $\ce{Na2O}$. In general the reaction favors the most thermodynamically stable chemical. – MaxW Feb 21 '17 at 1:28

Using your example, you simply have not given the system enough energy to do so. Water and $\ce{CO2}$ are very stable compounds under ordinary conditions of temperature and pressure, and thus are the preferred products under standard conditions for that reaction.

If you were to heat the system further, further decomposition would occur. For example, at higher temperatures you will convert $\ce{Na2CO3}$ into $\ce{NaO}$ and more $\ce{CO2}$. Under extreme conditions, i.e. thousands of degrees K, you will ultimately have enough energy to break all of the chemical bonds and then you will be left with the elements.

• Quibble - I'd explicitly note that "extreme conditions" mean that the whole sample will be atomized to the gaseous elements. – MaxW Feb 21 '17 at 6:41
• Just one disagreement with this answer. Not elements, rather atom of those elements. – CupC_56 Feb 21 '17 at 6:55
• Or a plasma I guess. – airhuff Feb 21 '17 at 6:57
• @CupC_56, my plasma comment was a response to Max. Another good point though. And for that matter we could go nuclear with our energy source to get to subatomic particles, but I guess at that point we have to move the discussion to Physics.SE ;) – airhuff Feb 21 '17 at 7:02


text:
# Need help understanding webpage

HomogenousCow

Now everything was fine up til the point where he said:

"Now, if we want to know the total field at point Z, then we need to add together every point in the aperture. That is, we need to integrate across the aperture to get the field at point Z. To get this, we'll have to know a little more physics. Specifically, the fields from a point source can be modeled as:"

As I understand it, he is treating every point on the aperture as a point source, I do not understand why this is so.

(Btw, does anyone like the username, couldn't do spherical cow so had to think of something else physicists always use to simplify problems)


text:
# Difference between revisions of "Differentiability"

If the function $f: D \to {\mathbb R}$ is differentiable in $x_0\in D$ then there is a function $f_1: D \to {\mathbb R}$ that is continuous in $x_0$ such that
$f(x) = f(x_0) + (x-x_0) f_1(x)$


text:
## Discovering the oddity

The M1 Max is a phenomenally powerful peice of kit, but can it write almost 300GB of data to memory in one second? Let’s answer that question with some system performance and Python profiling exploration.

I was running through Julia Evans’ computers-are-fast.github.io quiz and got to the question asking how many bytes can be written to memory in just one second. Having memorized ‘napkin math’ system performance numbers I knew that sequential memory R/W throughput is on the order of 10GB/s on fast machines. Taking that into consideration I correctly guessed the biggest option, 1,000,000,000 bytes. (The exact result on the quiz author’s older laptop was 2GB/s.)

Interested in how my new Macbook M1 Max would fare, I converted the provided write_to_memory.py code to Python 3 and tested it.

from io import StringIO
CHUNK_SIZE = 1_000_000
s = "a" * CHUNK_SIZE

def f(NUMBER):
output = StringIO()
bytes_written = 0
while bytes_written < NUMBER:
num_written = output.write(s)
bytes_written += CHUNK_SIZE
import sys
f(int(sys.argv[1]))

Running this snippet to write 2GB took not one second but a remarkable 40ms. I thought, shit, just how fast is this machine? Suspiciously fast. Sequential memory write throughput is fast these days, but not that good. I wondered: how fast does this snippet run on 100GB?

And that’s how I found a giant performance wall in the code.

time python3 write_to_memory.py 99_000_000_000 ... 0.361s total
time python3 write_to_memory.py 100_000_000_000  ... 28.526s total

Writing 99GB to memory took ~350ms, impossibly fast, and writing 100GB took 30 seconds, around 80x slower. What gives?

## Initial debugging

This kind of blow up was dramatic and initially unexplainable. It wouldn’t have been in the spirit of the quiz to shrug and move on, so I stopped and began spelunking in the deeper layers of this short code snippet.

### plasma-umass / scalene

The first tool I reached for was Scalene, a new and shiny Python profiler. Profiling the snail-pace 100GB run showed a few interesting things:

• Essentially zero time was being spent executing .py code. Most execution time was in “native” code (C code) and “system” (most likely I/O)
• Scalene’s COPY (MB/s) column was calling out that the program was copying about 3.4GB of memory a second. 100GB divided by 3.4GB/s memory copy bandwidth gives approximately the program’s execution time of 28.1 seconds.
• Peak memory usage was 95.36 GiB (102.39 GB), which accords with expected program behavior.

When running Scalene on write_to_memory.py 99_000_000_000 (GUI results not pictured), I could see why it was finishing so fast. Memory usage was minimal, <2MB. This program wasn’t actually writing 99GB to memory; the first clue.

### Good ol’ print() debugging.

I was curious to know more precisely where the snippet’s data writing while loop was hitting a performance wall, so I added print(f"Written {bytes_written:_} bytes") and re-ran python3 write_to_memory.py 100_000_000_000.

The program hung at 100_000_000_000 - CHUNK_SIZE == 99_999_000_000 bytes.

Interestingly, if I reduced CHUNK_SIZE to 100KB from 1MB, the program hung 10 times. This seemed to suggest that the performance degradation was related not to amount of memory written but the number of writes issued. If this was correct, then progressively reducing chunk size by a factor of ten should increase .write(s) operations and number of stalls by a factor of ten.

This is what happened, and stalls at smaller chunk sizes were also shorter. I had my second clue.

## Instruments 🔬

scalene was helpful but it wouldn’t be able to show me what exactly was happening in the CPython interpreter’s C code, so I reached for another profiling tool. Being on a Macbook M1, I fired up the OSX Instruments app’s Time Profiler, which “performs low-overhead time-based sampling of processes running on the system’s CPUs”.

I started recording within Instruments and ran the snail-pace python3 write_to_memory.py 100_000_000_000 command. After the process finished, in around 30 secs, I stopped the recording and created a new recording run for python3 write_to_memory.py 99_000_000_000.

It did not take long to spot the issue. Here’s an image of the slow run, showing a very slow function call named flush_accumulator that was completely absent from any fast run I profiled:

Here’s a rundown of the above callstack data:

• At the top we can see functions for running the write_to_memory.py file. eg. pyrun_file
• _PyEval_EvalCode is more or less the start of the Python interpreter loop.
• write_str is the internal C function dispatched to when the .write() method is called on the StringIO object.
• flush_accumulator was not present in profiles of fast runs, in particular the run that ‘writes’ 99GB to memory. In the image we can see this function is where the code snippet spends almost all its time.
• _platform_memmove$VARIANT$Rosetta is the ‘bottom’ of our investigation. It’s memmove, and doing memmove on 100GB of data is why the run is so slow.

It makes total sense that doing memmove on 100GB is slow. The scalene profiler was showing us in the COPY (MB/s) column that about 3.4GB of memory was being copied per second.

But that memory copying was only happening on the 100,000th .write(). When asking the snippet to write 99GB, which triggers 99,999 .write() operations, no significant memory copying happened at all. Why?

Looking up the flush_accumulator function in the CPython source code gives the answer.

## The recently deceased accu.c module

The flush_accumulator function and its related code actually just got replaced 15 days ago (mid-July 2022), but until recently StringIO objects did ‘lazy writing’ of Unicode strings to their internal buffer with something called an accumulator. This accumulator functionality was in Objects/accu.c, and it contains the core function, _PyAccu_Accumulate:

int _PyAccu_Accumulate(_PyAccu *acc, PyObject *unicode)
{
Py_ssize_t nsmall;
assert(PyUnicode_Check(unicode));

if (PyList_Append(acc->small, unicode))
return -1;
nsmall = PyList_GET_SIZE(acc->small);
/* Each item in a list of unicode objects has an overhead (in 64-bit
* builds) of:
*   - 8 bytes for the list slot
*   - 56 bytes for the header of the unicode object
* that is, 64 bytes.  100000 such objects waste more than 6 MiB
* compared to a single concatenated string.
*/
if (nsmall < 100000)
return 0;
return flush_accumulator(acc);
}

Hard-coded right there is the magic 100000 number controlling the observed gaps between stalls! In order to avoid wasting memory building up many intermediate objects during creation of a StringIO’s buffer, StringIO’s .write() is lazy. It only accumulates references to the written object until it’s either forced to ‘realize’ the buffer – which can be triggered manually with .getvalue() – or if this magic number of “small” accumulations is reached.

// in cpython/Modules/stringio.c
if (self->state == STATE_ACCUMULATING) {
if (self->string_size == self->pos) {
if (_PyAccu_Accumulate(&self->accu, decoded))
goto fail;
goto success;
}
if (realize(self))
goto fail;
}

We can quickly confirm this by modifying the snippet to spit out the reference count of the s string used to build up the 100GB of data, both before and after the buffer realization trigger. Before I show the modified snippet, here’s the confirming output:

\$ python3 write_to_memory.py 100_000_000_000
's' var's reference count just before StringIO is realized: 100000
's' var's reference count after StringIO is realized: 1

And here’s the snippet:

import ctypes
import sys
from io import StringIO

CHUNK_SIZE = 1_000_000
s = "a" * CHUNK_SIZE

class CStructStruct(ctypes.Structure):
_fields_ = [
("ob_refcnt", ctypes.c_long),
("ob_type", ctypes.c_void_p),
("ob_ptr", ctypes.c_long),
]

def f(NUMBER):
output = StringIO()
bytes_written = 0
writes = 0
while bytes_written < NUMBER:
if writes + 1 == 100_000:
print(
f"'s' var's reference count just before StringIO is realized: {cstruct_s.ob_refcnt}"
)
output.write(s)
bytes_written += CHUNK_SIZE
writes += 1
if writes > 100_000:
print(f"'s' var's reference count after StringIO is realized: {cstruct_s.ob_refcnt}")

f(int(sys.argv[1]))

## Fixing the code snippet

To fix the code snippet we can just call the output.getvalue() function after exiting the while loop, which forces the internal buffer to be realized if it hasn’t been already. Though I’d almost bet a house it won’t be merged, I’ve submitted that fix to website’s repo: #20. Avoid laziness of StringIO.write spoiling benchmark correctness.

This memory filling code snippet’s odd performance behavior wasn’t obvious initially, but with persistent digging computers can be understood.


text:
View Single Post

Recognitions:
Gold Member
 Quote by RonL Thanks Om, I just did a quick calculation, and a 10 mile dia. ring (single row) would require 22,440 boxes, but then this is a big disaster. I guess the big concerne would be can the confinement area be in a location that would not interfere with the operations of killing the well or bringing it back into control ? Do you or anyone have thoughts about the tube tunnel, using air bubbles ? RonL
Did you not get my PM?

And what's this about air bubbles?

I don't think a surface containment device the size of which you are talking about is necessary. The opinion on leak rate ranges from 500,000 to 4.2 million gallons of oil per day.(ref) So worst case is around 3000 gallons per minute that would have to be pumped out. So with existing technologies, a containment area 100 feet across and maybe 20 feet deep would be all that is needed. IMHO of course.

ps. The non-water permeable ripstop material I mentioned is only $6.95 a yard. So a 60" wide pair of the materials, sewn into a tube, at 1 mile length, would only cost ~$25,000. Must less costly than the nearly \$1 billion dollars that spent so far.

And it comes in Royal, so I'm sure the queen would approve.


text:
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Contribute

### Topics

InfoQ Homepage News MSMQ and .NET Core

# MSMQ and .NET Core

Microsoft Message Queuing (MSMQ) is currently not available for .NET Core. While other message queuing systems are generally preferred, many enterprise applications were based on MSMQ and this creates a problem for teams looking to migrate from .NET Framework to .NET Core or the upcoming .NET 5. But a recent pull request for Reference Source may change the situation.

MSMQ was created in 1997 and has been available as an optional OS component since Windows 95 and Windows NT 4.0. (The current version of Windows is derived from Windows NT.) This ubiquity made it the default choice for many companies, especially in the early years where there were few competitors outside of Java’s proprietary JMS technology.

Though, accusing JMS of being proprietary is a bit misleading. While it was only available for Java, it could run on any operating system. Conversely, MSMQ was available for any COM-compatible programming language, but could only run on Windows. And for any programming language to be viable on Windows, it needed to be able to access COM libraries.

For .NET programmers, there were multiple options for working with MSMQ. The most difficult route is to use the Win32 or COM-based libraries directly. This puts the COM/.NET interop issues in the forefront. As anyone who has dealt with COM from C# can tell you, it is hard to avoid memory leaks when mixing a reference-counted library and .NET’s mark & sweep style GC.

The more popular option was a library known as System.Messaging. This was part of the .NET Framework since the beginning. However, it hasn’t been carried over to .NET Core.

The final option was to use Windows Communication Foundation (WCF). In theory this allows messages that were previously being sent to MSMQ to be rerouted to any other message queueing system merely by changing a configuration setting. But the necessary plugins for non-Microsoft technologies were not widely available, so WCF never fulfilled that promise. And the .NET Core version of WCF is neither production ready nor does it have MSMQ on the current roadmap.

Why does this matter? In his article titled MSMQ is dead, David Boike cites two major advantages of MSMQ that need to be considered when trying to switch to a different message queueing system:

MSMQ is a distributed or federated messaging system where every server hosts the queue infrastructure, and outgoing messages are sent locally before being delivered to their ultimate address. In contrast, most other messaging systems are centralized or broker-based, meaning there is only one logical queue system, which usually exists in a cluster to provide for high availability. This makes scale-out quite a bit easier as multiple physical endpoint instances can both compete for messages on the same queue using the competing consumers pattern.

MSMQ was fairly unique in that it supported distributed transactions, meaning that database operations in SQL Server and messaging operations through the queues could all be combined in one atomic transaction so that either everything succeeded, or everything failed. Most other message transports do not support distributed transactions, but if you’re using NServiceBus, we have a component called the Outbox that simulates the reliability of a distributed transaction by piggybacking on a local database transaction. See our blog post What does idempotent mean? for more details.

Recently Karel Zikmund of Microsoft posted the original .NET Framework code for System.Messaging to Reference Source. This puts it under an MIT license and makes it possible to start working on a .NET Core implementation. If this happens, presumably it would be a community effort much like Core WCF.

According to Bill Loytty, he was able to get System.Messaging working under .NET Core. But this was before System.Messaging was open sourced, so at the time he couldn’t publish his results.

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• ##### MSMQ and .NET Core

by Andrew Watts,

• ##### Re: MSMQ and .NET Core

by Jonathan Allen,

• ##### MSMQ and .NET Core

by Andrew Watts,

Your message is awaiting moderation. Thank you for participating in the discussion.

We currently uses MSMQ in our specific small system. We'd really like to see a .NET Core version. 2 Key things that stand out for MSMQ is the UX to create triggers / rules that launch an EXE by simply passing the parameters in the queue. This small touch within the UX provides great simplicity. The other aspect is the queue to restore itself on server reboot. Unfortunately despite providing those parameters, our data is being lost. We have somehow enabled a edge case bug

• ##### Re: MSMQ and .NET Core

Your message is awaiting moderation. Thank you for participating in the discussion.

Honestly, I have serious doubts about the longevity of MSMQ in general. I would recommend moving to a different queueing platform as soon as reasonable.

Almost everyone I talked to while researching this recommended RabbitMQ as the default choice.

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# TinyGAN: Distilling BigGAN for Conditional Image Generation

29 Sep 2020  ·  , ·

Generative Adversarial Networks (GANs) have become a powerful approach for generative image modeling. However, GANs are notorious for their training instability, especially on large-scale, complex datasets. While the recent work of BigGAN has significantly improved the quality of image generation on ImageNet, it requires a huge model, making it hard to deploy on resource-constrained devices. To reduce the model size, we propose a black-box knowledge distillation framework for compressing GANs, which highlights a stable and efficient training process. Given BigGAN as the teacher network, we manage to train a much smaller student network to mimic its functionality, achieving competitive performance on Inception and FID scores with the generator having $16\times$ fewer parameters.

PDF Abstract


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# Alternate parameterization of a line

There are several ways to specify a line. In Mathematica's graphics, it is done by specifying two points and then connecting the points with the Line command. But it is sometimes desirable to parameterize a line, and the most familiar way is the form $y = m x + b$ where $m$ is the slope and $b$ is the offset. This parameterization is particularly awkward for near-vertical lines (where the slope becomes near infinite and numerically sensitive). An alternative is the $(\rho, \theta)$ parameterization which specifies the line as an angle $\theta$ and a distance $\rho$ from the origin, as shown in the theory section of this wikipedia article. The gist is that the line consists of all points $x$ and $y$ for which $\rho = x \cos(\theta) + y \sin(\theta)$, essentially a conversion to polar coordinates.

Here is a function which takes a pair of points {p1, p2} as input in the form {{x1,y1},{x2,y2}} and outputs the corresponding $\rho$ and $\theta$:

linePolar[{p1_, p2_}] := Module[{},
{rx, ry} = Rationalize[p1]; {sx, sy} = Rationalize[p2];
sol = Solve[{rx Cos[t] + ry Sin[t] == rho,
sx Cos[t] + sy Sin[t] == rho}, {rho, t}] // N;
First[Select[{rho, t} //. sol //. C[1] -> 0, #[[1]] >= 0 &]]];

For example:

linePolar[{{0, -1}, {-1, 0}}]
{0.707107, -2.35619}

which shows that the line connecting the two points is Sqrt[2]/2 away from the origin at an angle of -135 degrees. The almost vertical line

linePolar[{{1, -1}, {1.1, 1}}]
{1.04869, -0.0499584}

has slope near zero, and the answer appears to be nicely behaved numerically as the line crosses vertical.

I have two questions. First, it bothers me that the function is essentially re-solving the system of equations each time it is called, and this must be inefficient. Is there a nice way to make it only do the solution once? Second, I would like to have an "inverse" for this function -- something that would take a pair of $(\rho, \theta)$ values and allow it to be plotted. This would be useful because many subsequent calculations need to occur in the $(\rho, \theta)$ space, and it would be good to be able to visualize this by plotting.

-
To be clear are we talking about (infinite) lines, or line segments here? –  Mr.Wizard Aug 29 at 1:52
The question is really about (full) lines, though of course we can only ever plot portions of them. This is why I enclosed inverse in scary quotes -- you can't really get back to the particular specified endpoints, only back to the line itself. –  bill s Aug 29 at 1:53
No time to get into this today but it seems like a variation of a relatively common rectangular to polar conversion. I presume you've already looked at such things; wherein lies the difference and difficulty? –  Mr.Wizard Aug 29 at 1:58
Here's a decent reference I believe: mathforum.org/dr.math/faq/formulas/faq.polar.html –  Mr.Wizard Aug 29 at 2:01
What is the actual problem you want to solve? It's not clear why you would prefer the polar parametrization over e.g. p1 + t*(p2-p1) where p1 and p2 are the given points in R^2 and t is the parameter. Is it that you have a particular need for the distance from the origin? –  Daniel Lichtblau Aug 29 at 4:20

The Cross product is very useful here. It can be used with 3D vectors to get the perpendicular distance of a line to the origin, but it also works with a 2D vector as an easy way to specify its perpendicular:

This function returns the radial distance $\rho$ and the angle $\theta$ as in the question, but doesn't require resolving the coordinates of the given points explicitly:

lineToPolar[{p1_, p2_}] :=
{
Abs[#], ArcTan @@ Cross[Sign[#] (p2 - p1)]} &@
Last[Apply[Cross, PadRight[#, 3] & /@ {p1, Normalize[p1 - p2]}]]

lineToPolar[{{0, -1}, {-1, 0}}]

(* ==> {1/Sqrt[2], -((3 Pi)/4)} *)

Simplify[
lineToPolar[{{x, y}, {u, v}}], {x, y, u, v} ∈ Reals]

$$\begin{array}{c}\{\frac{\left| v x-u y\right|}{\sqrt{(u-x)^2+(v-y)^2}},\\\tan^{-1}\left(\frac{(v-y) \text{sgn}(v x-u y)}{\sqrt{\text{sgn}\left((u-x)^2+(v-y)^2\right)}},\frac{(u-x) \text{sgn}(u y-v x)}{\sqrt{\text{sgn}\left((u-x)^2+(v-y)^2\right)}}\right)\}\end{array}$$

This is the general formula for the polar representation. You could use this directly to define the conversion function if desired.

To optimize the conversion function some more, the component form of the given vectors could be used together with this form, where the Sign function has been turned into a case distinction:

FullSimplify[
lineToPolar[{{x, y}, {u, v}}], {x, y, u, v} ∈ Reals] /.
Indeterminate -> 0

$$\{\frac{\left| v x-u y\right| }{\sqrt{(u-x)^2+(v-y)^2}}, \begin{cases} 0 & v x=u y \\ \tan ^{-1}(v-y,x-u) & u y<v x \\ \tan ^{-1}(y-v,u-x) & \text{True} \\ \end{cases} \}$$

Edit:

In an earlier version, the parameter $\rho$ was allowed to be negative if the order of p1 and p2 was reversed. That means $\rho$ is the 2D impact parameter of the line, but for a real polar coordinate you may want to require $\rho>0$. In the above code, I make sure that when the impact parameter is negative we exchange the order of p1 and p2 to get back to a positive $\rho$.

End edit

The inverse is even simpler:

Clear[ρ, θ];
lineFromPolar[ρ_, θ_] =
Function[{t}, {t, ρ}.#] &[RotationMatrix[Pi/2 - θ]]

With[{ρ = 1/2, θ = 3 Pi/4},
ParametricPlot[lineFromPolar[ρ, θ][t], {t, -10, 10}]
]

Here I used the fact that the offset from the origin is specified by a vector perpendicular to the line, of length $\rho$. This required orthogonality is provided automatically by the rows of RotationMatrix.

-
I have never checked that Cross works that way too and that ArcTan has this handy extension. And I should have! :) +1 –  Kuba Aug 29 at 6:49
A fine answer, however your lineFromPolar is not written efficiently. You should evaluate the body of the function: lineFromPolar2[ρ_, θ_] := Evaluate[{#, ρ}.RotationMatrix[-θ]] &. Timing for your method: lineFromPolar[1.4, 0.3] /@ RandomReal[9, 150000] // Timing // First is 0.936 seconds; and for mine is 0.01436 seconds. –  Mr.Wizard Aug 29 at 8:02
@Mr.Wizard You're right, I didn't think about efficiency in the inverse. The version in my edit is even faster than yours now. –  Jens Aug 29 at 15:40
@Jens -- thanks for looking at this. I think there is still a Pi (or maybe Pi/2) confusion somewhere. If we take lineToPolar[{{1, -1}, {1, 1}}] (note that this is a vertical line that connects these two points) then you get {-1, Pi}. But this is a horizontal line, as you can see by putting it into lineFromPolar. –  bill s Aug 29 at 16:20
@bills Oh sorry, I used different definitions of $\theta$ for the two functions. What do you prefer: is $\theta$ the angle of the actual line with the horizontal? Or is it the angle of the normal to the line with the horizontal, as in the article you linked here? I have to change one or the other, but I'll wait until you let me know which you like better... –  Jens Aug 29 at 16:54

Evaluating lineToPolar[{{1,-1},{1.1,1}}] gives a negative rho, and an angle t differing by $\pi$ from the OP example. Incorporating a simple Abs[rho] and keeping the same angle t gives a different line than I think is required. I used

lineToPolarAlt[{{x1_, y1_}, {x2_, y2_}}] :=
With[{numerator = -y1 x2 + x1 y2},
{Abs[(numerator)/Norm[{x1, y1} - {x2, y2}]],
If[numerator >= 0, Arg[ I x1 - y1 - I x2 + y2],
Arg[-I x1 + y1 + I x2 - y2]]}]

to check the two different cases.

-


text:
# Any classical inequalities to use?

Algebra Level 3

$\large (X-1)(X-3)(X-5)(X-7)\ldots (X-97) < 0$

How many positive integers $$X$$ satisfy the inequality above?

×


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# a tangent vector which belongs to intersection of a manifold and a subspace is tangent to their intersection?

I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace $P\subsetneq R^n$.

There are also a point $p\in P\cap M$ and a tangent vector point $v\in P\cap T_pM$. (Since $M$ is a submanifold of $R^n$ I am using the standard identification and considering $T_pM\subsetneq R^n$ so $P\cap T_pM$ makes sense).

It is also given that the intersection $P\cap M$ is a submanifold of $M$. My question is: does $v\in T_p(P\cap M)$?

My ideas so far are: $v \in P∩T_pM$ implies we can build two curves: $\alpha :I \mapsto P, \beta:I \mapsto M$ such that $\alpha (0)=\beta(0)=p, \alpha' (0)=\beta'(0)=v$. But I am not sure it is possible to find a single curve which passes throug $p$ with velocity $v$ that lies inside $P\cap M$.

• Of course you are right, the (local) uniqueness of solutions ensures that my two curves coincide on a small enough interval. So we have a curve $\alpha :I \mapsto P\cap M$ which is smooth when considered as a curve in $M$, and as a curve in $P$ separately. but how do we know it is smooth as a curve into the intersection manifold $P\cap M$? – Asaf Shachar Apr 1 '15 at 10:09
• I don't think anything like uniqueness of solutions applies here; you have a tangent vector, not a tangent vector field. It's certainly not the case that your curves need to coincide on a small enough interval. Even two curves just in $M$ through a given tangent vector $v$ need not coincide on a small interval. – mollyerin Apr 1 '15 at 11:41

No, this isn't enough to guarantee that $v \in T_p(P \cap M)$. As a (degenerate-looking) counterexample, take $M$ to be the graph of $y = x^2$ in $\mathbb{R}^2$, $p$ the origin, and $P$ the subspace corresponding to the $x$-axis. Then $(1,0)$ is in both $P$ and $T_pM$, but it's not in $T_p(P \cap M)$ because the latter is too small.
On the other hand, I believe the following general fact is true: If $M$ and $N$ are embedded submanifolds of $\mathbb{R}^n$ which intersect transversely at $p$ (which means $T_p M + T_p N = \mathbb{R}^n$), then $M \cap N$ is a submanifold in a neighborhood of $p$ and $T_p(M \cap N) = T_pM \cap T_pN$. Your result follows by letting $N = P$.
Sketch of proof of this fact: write $M = f^{-1}(0)$ and $N = g^{-1}(0)$, on some neighborhood $U$ of $p$, where $f: U \to \mathbb{R}^{n - k}$ and $g : U \to \mathbb{R}^{n-l}$, where $k$ and $l$ are the dimensions of $M$ and $N$, respectively, and $0$ is a regular value of both $f$ and $g$. The point then is that the transversality assumption guarantees that $0$ is also a regular value of $(f, g) : U \to \mathbb{R}^{2n-k-l}$, which guarantees that $M \cap N \cap U$ is a submanifold and that the desired equality on tangent spaces holds.


text:
Seungho Kuk, Hyogon Kim and Yongtae Park, "Detecting False Position Attack in Vehicular Communications Using Angular Check," Proceedings of 23rd ACM International Conference on Mobile Computing and Networking (MobiCom 2017), 2nd ACM International Workshop on Smart, Autonomous, and Connected Vehicular Systems and Services (CarSys 2017), Snowbird, UT, October 2017, pp. 25-29.

Quick access

Original Version (at publishers web site)
BibTeX

Seungho Kuk
Hyogon Kim
Yongtae Park

BibTeX reference

@inproceedings{kuk2017detecting,     author = {Kuk, Seungho and Kim, Hyogon and Park, Yongtae},     title = {{Detecting False Position Attack in Vehicular Communications Using Angular Check}},     booktitle = {23rd ACM International Conference on Mobile Computing and Networking (MobiCom 2017), 2nd ACM International Workshop on Smart, Autonomous, and Connected Vehicular Systems and Services (CarSys 2017)},     pages = {25-29},     year = {2017},     month = {October},     address = {Snowbird, UT},     publisher = {ACM},     doi = {10.1145/3131944.3131949},    }

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text:
# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeOct 25th 2009

Todd started Schur functor. I added internal links and wrote linear category to be the target of one of them.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeOct 25th 2009

Thanks, but I should say that I was in the middle of an edit when you did all that, and my submission erased your links, which I tried to reinsert. Please check: the link to "change of base" was supposed to point to an extant page, but I'm not sure which.

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeOct 25th 2009

Ah, your edit took longer than 30 minutes, so the lock timed out! And here I thought that you had cancelled it.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 25th 2009

Ah, I had no idea that locks time out! That's annoying, but at least now I know. I was busy trying to get the kids in bed and so on when it happened.

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeOct 25th 2009

I believe that they time out so that people can't lock a page forever if they just forget about it or their Internet connection crashes or something.

Sometimes I'll save a draft every 15 minutes or so to prevent it.

• CommentRowNumber6.
• CommentAuthorGuest
• CommentTimeOct 26th 2009
To get around the spam filter in the minimum time, I'll drop my intended comment for Schur functor here:

is the divided powers functor Ab --> Ab a Schur functor? The preprint arXiv:0910.2817, section 2.1 has details.
How about the Lie functor Ab --> Ab, which is just the tensor algebra with the obvious Z-Lie structure? Same paper for details.

David Roberts
• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeOct 26th 2009

Thanks, David. For the time being I'm restricting to categories enriched in rational vector spaces, so for the time being let me apply the query to $Vect_{fd}$ (finite-dimensional spaces) instead of $Ab$. Then yes, it looks like their paper gives formulas for the homogeneous components of these constructions as direct sums of certain classical $S_\lambda$'s.

• CommentRowNumber8.
• CommentAuthorbwebster
• CommentTimeOct 27th 2009
• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeOct 27th 2009

Thanks, Ben. I added a small and maybe slightly cryptic comment, to be followed up on later.

• CommentRowNumber10.
• CommentAuthorbwebster
• CommentTimeOct 28th 2009

I also created a stub on Specht modules, which should be key for defining analogues of Schur functors for all abelian monoidal categories.


text:
How to check an algebraic number for membership in a list

I need to check an algebraic number for membership in a list of algebraic numbers. The numbers can be expressed in different forms (combinations of radicals, Root objects, trig functions), but the check should be exact. Now I use the following code:

ContainsAlgebraicQ[list_, a_] :=
MatchQ[Intersection[ list, {a}, SameTest -> (MinimalPolynomial[#1 - #2] === (#&) &)], {_}]

but in some cases evaluation of MinimalPolynomial takes significant time, although I only want to check the difference for zero. Is there a better approach for my task?

-
PossibleZeroQ might be of some help. – Spawn1701D Jun 5 '13 at 19:46
Consider MemberQ[{Sqrt[2], Sqrt[3], Sin[\[Pi]/5], GoldenRatio, Root[#^3 - 2 &, 1]}, x_ /; RootReduce[x - 2^(1/3)] == 0]... – J. M. Jun 5 '13 at 19:47
I'd do this with PossibleZeroQ[ # - a, "ExactAlgebraics"]@list, or just MemberQ[PossibleZeroQ[ #, Method -> "ExactAlgebraics"]& @ list, True]. This post is closely related: Most efficient way to determine conclusively whether an algebraic number is zero – Artes Jun 5 '13 at 21:42

PossibleZeroQ[ #, Method -> "ExactAlgebraics"] provides the most efficient and exact test whether two algebraics are equal, (see e.g. Most efficient way to determine conclusively whether an algebraic number is zero for some benchmarks). It is Listable so we can define the following function:

ContainsAlgebraicQ[list_, a_] :=
MemberQ[PossibleZeroQ[list - a, Method -> "ExactAlgebraics"], True]

E.g. check if 1 + Sqrt[3] is an element of

list = { Root[4 + 2 #1^4 + #1^8 &, 8],
(Sqrt[2] + Sqrt[3] + Sqrt[6] + 3)/Sqrt[5 + 2 Sqrt[6]],
((7 - 2 I)/(1 + I Sqrt[2]) + (4 + 14 I)/(Sqrt[2] + 2 I) - 8 + 2 I)^(1/4) };

ContainsAlgebraicQ[ list, 1 + Sqrt[3]]

True

but

ContainsAlgebraicQ[ list, Root[4 + 2 #1^4 + #1^8 &, 7]]

False

and of course

ContainsAlgebraicQ[ list, Root[4 + 2 #1^4 + #1^8 &, 8]]

True

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Thanks! Based on my observations of how this code works, I assume that trig functions of rational multiples of π are not considered explicit algebraics by Mathematica. So I added a step that invokes RootReduce on expressions containing any trig functions before passing them to PossibleZeroQ. Please let me know if you are aware of a better approach. – Vladimir Reshetnikov Jun 6 '13 at 1:32
I think you need not to use RootReduce before PossibleZeroQ. Though Daniel Lichtblau has been surprised that PossibleZeroQ[ #, Method -> "ExactAlgebraics"]& worked for non-explicit algebraics. But for a more reliable approach it might be safer. – Artes Jun 6 '13 at 9:54

If list has sufficiently many repeated subexpressions, it becomes efficient to convert the elements to explicit algebraic numbers in a common number field.

list=ToNumberField[list,All]

Performance testing suggests that ToNumberField memoizes evaluations of subexpressions. Evaluation is fast when the input is an algebraic combination of previously evaluated subexpressions.

(* all elements of list must be explicit algebraic numbers in a common number field *)
ContainsAlgebraicQ[{},a_,extension_]= False
ContainsAlgebraicQ[list_,a_,extension_]:=
Quiet[MemberQ[list,ToNumberField[a,AlgebraicNumberPolynomial[list[[1]]]]]]

Here, Quiet suppresses a message that occurs when a is not in the number field generated by extension. I'm not sure what performance is like in this case; your application may allow better options for choice of extension.

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text:
distributed-closure-0.3.5: Serializable closures for distributed programming.

Data.Functor.Static

Description

Closure is not a functor, since we cannot map arbitrary functions over it. But it sure looks like one, and an applicative one at that. What we can do is map static pointers to arbitrary functions over it (or in general, closures). Closure is not just an applicative functor, it's also a monad, as well as a comonad, if again we limit the function space to those functions that can be statically pointed to.

In fact an entire hierarchy of classes mirroring the standard classes can be defined, where nearly the only difference lies in the fact that higher-order arguments must be a proof of static-ness (i.e. a Closure). The other difference is that composing static values requires a proof of typeability, so we carry those around (Typeable constraints).

This module and others define just such a class hierarchy in the category of static functions (aka values of type Closure (a -> b)).

Synopsis

# Documentation

class Typeable f => StaticFunctor f where Source #

Instances of StaticFunctor should satisfy the following laws:

staticMap (static id) = id
staticMap (static (.) cap f cap g) = staticMap f . staticMap g

Minimal complete definition

staticMap

Methods

staticMap :: (Typeable a, Typeable b) => Closure (a -> b) -> f a -> f b Source #

Instances

 Source # MethodsstaticMap :: (Typeable * a, Typeable * b) => Closure (a -> b) -> Closure a -> Closure b Source #


text:
# Can the minimum kinetic energy of a simple harmonic oscillator be non zero?

I was practicing physics problems when I faced one which said that minimum kinetic energy of a simple harmonic oscillator (in the question it was a spring block system) can be non zero.

However, it occurred to me that it cannot be possible since in such a motion a time will come when the spring will be stretched to its maximum and the velocity of the block will change direction. So, at a time between the change in direction of the velocity there will be an instant of zero velocity where the kinetic energy will be zero(the instant of minimum Kinetic Energy).

So who is right here, and if the minimum kinetic energy of a simple harmonic oscillator can be non zero then cite an example.

Edit: Sorry the question is actually about "minimum Kinetic Energy"

• The kinetic energy is $\frac12 m\dot{x}^2.$ Plug in $x(t)$ for the standard harmonic oscillator and you'll see it's non-zero for certain $t$, as expected... Jan 22 '17 at 19:51
• There are times when it is zero and times when it is not. The total energy, however, is always non-zero and positive. Jan 22 '17 at 22:54
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:03

So who is right here, and if the kinetic energy of a simple harmonic oscillator can be non zero then cite an example.

A simple harmonic oscillator has to have a non-zero velocity at some point by definition.

The point of a simple harmonic oscillator is that it oscillates, or moves back and forth. If there were no kinetic energy (velocity), it wouldn't move.

Can you site any examples of a harmonic oscillator that has zero velocity?

• Well, a classical oscillator with zero total energy would be permanently at rest at the equilibrium point. But even that special case is not possible in QM. Jan 23 '17 at 0:55
• Actually the question is about minimum Kinetic Energy. Sorry for the inconvenience caused. Jan 23 '17 at 5:04
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:04

There are times where there is zero kinetic energy in a harmonic motion. However, there are times where it is non-zero. And the question is just whether it can be non-zero. And it certainly can be non-zero.

Update to reflect the updated question:

The minimum kinetic energy can be non-zero if the oscillator is two dimensional. Then for every trajectory that is circular or elliptical, the pendulum would never completely stop.

• Please refer to the recent edit. Jan 22 '17 at 19:33
• What's to refer to? His answer remains correct. Keep in mind that while total energy is conserved the kinetic and potential energies can (and do) both change as a function of time: it is only their sum which remains constant. Jan 22 '17 at 21:49
• Actually the question is about minimum Kinetic Energy. Sorry for the inconvenience caused. Jan 23 '17 at 5:03
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:04


text:
# Definition:Jacobi's Equation of Functional/Dependent on N Functions

## Definition

Let

$\displaystyle\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

be a functional, where $\map {\mathbf y} a=A$ and $\map {\mathbf y} b=B$.

Let

$\displaystyle\int_a^b\paren{\mathbf h'\mathbf P\mathbf h'+\mathbf h\mathbf Q\mathbf h}\rd x$

be a quadratic functional, where $\displaystyle P_{ij}=\frac 1 2 F_{y_i'y_j'}$ and $\displaystyle Q_{ij}=\frac 1 2 \paren{F_{y_iy_j}-\frac \d {\d x} F_{y_iy_j'} }$

Then the Euler's equation of the latter functional:

$-\dfrac \d {\d x} \paren{\mathbf P\mathbf h'}+\mathbf Q\mathbf h=\mathbf 0$

is called Jacobi's Equation of the former functional.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.


text:
• # question_answer 20) Which of the following is an improper fraction? A)  B)  C)  D)

(b) The fraction represented in the figure B is $\frac{4}{4}$ = 1. It is an improper fraction.


text:
# A stronger Flow Decomposition Theorem?

In the classic Network Flows: Theory, Algorithms, and Applications book (pages 80/81) the flow decomposition theorem is stated as follows:

Every nonnegative arc flow x can be represented as a path and cycle flow (though not necessarily uniquely) with the following two properties:
(a) Every directed path with positive flow connects a deficit node to an excess node
(b) At most n+m paths and cycles have nonzero flow; out of these at most m cycles have nonzero flow with m the number of edges and n the number of nodes in the graph

My question is, if the theorem can be formulated more strongly modifying point b as follows:

(b) At most m paths and cycles have nonzero flow

My question arises as going through the lecture nodes of a more modern optimization course at Stanford, I find the flow decomposition theorem formulated exactly with this "stronger" assertion (page 58 with proof here or a similar theorem here).

Going through the proofs in the lecture nodes and the book, I find the stronger formulation justified. Especially in Network Flows: Theory, Algorithms, and Applications the following argument is given for their m+n bound on paths:

Now observe that each time we identify a directed path, we reduce the excess/deficit of some node to zero or the flow on some arc to zero;

As I understand it, reducing the excess/deficit of some node to zero implies setting the flow on some arc to zero, as the excess/deficit flow of a node has to go/come over an arc whose flow is now set to zero. This however would imply bounding the number of paths with m instead of n+m, as each construction of a flow-path would remove one edge and not one edge or node from the graph.

Have I misunderstood the proof and are the lecture nodes from Stanford wrong? Do you know other books which one could possible cite for the stronger version of the flow decomposition theorem?

Thank you!

Note: In this question with the very promising title, a different question has been asked. It took the theorem from one of the aforementioned Stanford notes, and in the answers the correctness of it was discussed albeit for different reasons.

• The proof in Plotkin's notes you link to is actually not quite correct. When we find an (s,t) path we are not allowed to subtract the path-flow with the minimum of the flow on the path. We would also need to take the minimum with the balance of s and the negative balance of t. (Trevisan's notes have the same issue). Feb 21, 2021 at 10:38
• The quoted argument of Ahuja, Magnanti, and Orlin is the correct way to argue. Feb 21, 2021 at 10:40
• Reducing excess/deficit of some node to zero implies that the node becomes balanced. Feb 21, 2021 at 10:44
• Typically one works with s-t flows at the start with no explicit demand/supply. It is easy to reduce the general case with multiple sources and sinks with demands/supplies to s-t flow by adding new source, sink. Adding new source/sink creates at most n new edges so that is another way to think about the problem. Feb 21, 2021 at 22:37
• And one may go all the way to a circulation by adding an arc from t to s. Feb 22, 2021 at 9:46

The statement of Ahuja, Magnanti, and Orlin applies to general flows, not only $$(s,t)$$-flows. During decomposition, when removing a path flow, either one of the two endpoints become balanced or one the arcs will get flow 0. This gives the bound $$m+n$$.
In the case of $$(s,t)$$-flows, we may note that during decomposition the nodes $$s$$ and $$t$$ becomes balanced at the same time. This gives the already the bound $$m+1$$ on the number of paths and cycles. If we look at the flow decomposition, when finding the final path flow (this is where the balance of $$s$$ and $$t$$ becomes balanced), if no flow on some arc is made 0, it means that we will still find a cycle flow. Now the final cycle flow will make at least two arcs have flow 0 at the same time. This sharpens the bound to $$m$$ in case of $$(s,t)$$-flows.


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# Three points

Three points K (-3; 2), L (-1; 4), M (3, -4) are given. Find out:
(a) whether the triangle KLM is right
b) calculate the length of the line to the k side
c) write the coordinates of the vector LM
d) write the directional form of the KM side
e) write the directional form of the axis of the KM side

Result

p =  1
t =  4.4721
x =  4
y =  -8
s = (Correct answer is: s=y=-x-1)

#### Solution:

$x=mx-lx=3-\left(-1\right)=4$
$y=my-ly=\left(-4\right)-4=-8$

We would be very happy if you find an error in the example, spelling mistakes, or inaccuracies, and please send it to us. We thank you!

Tips to related online calculators
For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.
Two vectors given by its magnitudes and by included angle can be added by our vector sum calculator.
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Pythagorean theorem is the base for the right triangle calculator.

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text:
# Finding the Mean and Variance of this distribution

Given the following:

$$Y|\Lambda=\mathrm{Poisson(\Lambda)}$$ and $$\Lambda=\mathrm{Gamma}(\alpha, \beta)$$

Find the distribution, mean and variance of $$Y$$.

After a lot of work, I was able to find the distribution of $$Y$$ which is:

$$f_Y(y)={{\Gamma(y+\alpha)e^{(-y/\beta)}}\over{y!\Gamma(\alpha)\beta^\alpha}}$$

Now using this distribution, I wanted to find the mean and variance. I tried integrating the above results but I couldn't find a closed form for the product $$\Pi_{i=1}^n{(y+\alpha-i)}$$ and from their the integration breaks down.

I would appreciate if someone could provide me a way to calculate the mean and varaiance of Y. Thanks

• $Y$ should have a negative-binomial distribution (see en.wikipedia.org/wiki/…), from which the moments follow. Jan 19, 2020 at 16:21
• Yes the question tells us to show that it is a negative binomial distribution but could you please show me how you came to this conclusion through the distribution of Y? As my distribution doesn't consist of probability but the exp function Jan 19, 2020 at 16:34
• Did you see the derivation on the wiki page? You made a mistake somewhere. Jan 19, 2020 at 16:38
• Ah yes. I see my mistake. Thank you very much Jan 19, 2020 at 17:27
• You can get displayed equations by enclosing them in double instead of single dollar signs. This is particularly helpful when you're mixing fractions and exponents, which tend to become hard to read the way you typeset them. Jan 19, 2020 at 17:44

I solved this question by setting $$\beta = (p/(1-p))$$. I then followed the derivation of the Negative Binomial distribution as shown in this link. Since I wanted an answer in terms of $$\beta$$, I simply solved the above expression for $$p$$ and replaced $$\beta$$ with that expression, giving me Y~NB($$\alpha$$, $$\beta/(1+\beta)$$). From there the mean and variance can easily be found.


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# Landau pole

In physics, the Landau pole or the Moscow zero is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues.[1] The fact that coupling constants depend on the momentum (or length) scale is one of the basic ideas behind the renormalization group.

Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or φ 4 theory—a scalar field with a quartic interaction—such as may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling constant becomes infinite at a finite energy scale. In a theory intended to be complete, this could be considered a mathematical inconsistency. A possible solution is that the renormalized charge could go to zero as the cut-off is removed, meaning that the charge is completely screened by quantum fluctuations (vacuum polarization). This is a case of quantum triviality, which means that quantum corrections completely suppress the interactions in the absence of a cut-off.

Since the Landau pole is normally calculated using perturbative one-loop or two-loop calculations, it is possible that the pole is merely a sign that the perturbative approximation breaks down at strong coupling. Lattice field theory provides a means to address questions in quantum field theory beyond the realm of perturbation theory, and thus has been used to attempt to resolve this question. Numerical computations performed in this framework seems to confirm Landau's conclusion that QED charge is completely screened for an infinite cutoff.[2][3][4]

## Brief history

According to Landau, Abrikosov, Khalatnikov,[5] the relation of the observable charge gobs with the “bare” charge g0 for renormalizable field theories is given by expression

$g_{obs}=\frac{g_0}{1+\beta_2 g_0 \ln \Lambda/m} \,, \qquad \qquad\qquad (1)$

where m is the mass of the particle, and Λ is the momentum cut-off. For finite g0 and Λ → ∞ the observed charge gobs tends to zero and the theory looks trivial. In fact, the proper interpretation of Eq.1 consists in its inversion, so that g0 (related to the length scale $\Lambda^{-1})$ is chosen to give a correct value of gobs:

$g_0=\frac{g_{obs}}{1-\beta_2 g_{obs} \ln \Lambda/m}\,. \qquad\qquad\qquad (2)$

As Λ grows, the bare charge g0 = g(Λ) increases, to diverge at the renormalization point

$\Lambda_{Landau} = m \exp\left\{ \frac{1}{\beta_2 g_{obs}} \right\} \, . \qquad\qquad\qquad (3)$

This singularity is the Landau pole with a negative residue, g(Λ) ≈ −ΛLandau /(β2(Λ−ΛLandau)) .

In fact, however, the growth of g0 invalidates Eqs.1,2 in the region g0≈1, since these were obtained for g0≪ 1, so that the exact reality of the Landau pole becomes doubtful.

The actual behavior of the charge $g(\mu)$ as a function of the momentum scale $\mu$ is determined by the Gell-MannLow equation[6]

$\frac{dg}{d \ln \mu} =\beta(g)=\beta_2 g^2+\beta_3 g^3+\ldots \qquad\qquad\qquad (4)$

which gives Eqs.1,2 if it is integrated under conditions $g(\mu)=g_{obs}$ for $\mu=m$ and $g(\mu)=g_0$ for $\mu=\Lambda$, when only the term with $\beta_2$ is retained in the right hand side. The general behavior of $g(\mu)$ depends on the appearance of the function $\beta(g)$ .

According to the standard classification,[7] there are three qualitatively different cases:

(a) if $\beta(g)$ has a zero at the finite value $g*$, then growth of $g$ is saturated, i.e. $g(\mu)\to g*$ for $\mu\to\infty$ ;

(b) if $\beta(g)$ is non-alternating and behaves as $\beta(g) \propto g^\alpha$ with $\alpha\le 1$ for large $g$, then the growth of $g(\mu)$ continues to infinity;

(c) if $\beta(g) \propto g^\alpha$ with $\alpha>1$ for large $g$, then $g(\mu)$ is divergent at finite value $\mu_0$ and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of $g(\mu)$ for $\mu>\mu_0$ .

Landau and Pomeranchuk [8] tried to justify the possibility (c) in the case of QED and $\phi^4$ theory. They have noted that the growth of $g_0$ in Eq.1 drives the observable charge $g_{obs}$ to the constant limit, which does not depend on $g_0$ . The same behavior can be obtained from the functional integrals, omitting the quadratic terms in the action. If neglecting the quadratic terms is valid already for $g_0\ll 1$, it is all the more valid for $g_0$ of the order or greater than unity : it gives a reason to consider Eq.1 to be valid for arbitrary $g_0$ . Validity of these considerations on the quantitative level is excluded by non-quadratic form of the $\beta$-function. Nevertheless, they can be correct qualitatively. Indeed, the result $g_{obs} =const(g_0)$ can be obtained from the functional integrals only for $g_0\gg 1$, while its validity for $g_0\ll 1$, based on Eq.1, may be related with other reasons; for $g_0\approx 1$ this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition. The Monte Carlo results [9] seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, though a different interpretation is also possible.

The case (c) in the Bogoliubov and Shirkov classification corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum. Indeed, if $g_{obs}$ is finite, the theory is internally inconsistent. The only way to avoid it, is to tend $\mu_0$ to infinity, which is possible only for $g_{obs}\to 0$ . It is a widespread belief that both QED and φ 4 theory are trivial in the continuum limit. In fact, available information confirms only “Wilson triviality”, which just amounts to positivity of β(g) for g≠0 and can be considered as firmly established. Indications of “true” quantum triviality are not numerous and allow different interpretations.

## Phenomenological aspects

In a theory intended to represent a physical interaction where the coupling constant is known to be non-zero, Landau poles or triviality may be viewed as a sign of incompleteness in the theory. For example, QED is usually not believed to be a complete theory on its own, and the Landau pole could be a sign of new physics entering via its embedding into a Grand Unified Theory. The grand unified scale would provide a natural cutoff well below the Landau scale, preventing the pole from having observable physical consequences.

The problem of the Landau pole in QED is of pure academic interest. The role of $g_{obs}$ in Eqs.1,2 is played by the fine structure constant α ≈ 1/137 and the Landau scale for QED is estimated as 10283keV/c2, which is far beyond any energy scale relevant to observable physics. For comparison, the maximum energies accessible at the Large Hadron Collider are of order 1013 eV, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is only 1028 eV.

The Higgs boson in the Standard Model of particle physics is described by $\phi^4$ theory. If the latter has a Landau pole, then this fact is used in setting a "triviality bound" on the Higgs mass.[10] The bound depends on the scale at which new physics is assumed to enter and the maximum value of the quartic coupling permitted (its physical value is unknown). For large couplings, non-perturbative methods are required. Lattice calculations have also been useful in this context.[11]

## Recent developments

Solution of the Landau pole problem requires calculation of the Gell-Mann–Low function $\beta(g)$ at arbitrary $g$ and, in particular, its asymptotic behavior for $g\to\infty$ . This problem is very difficult and was considered as hopeless for many years: by diagrammatic calculations one can obtain only few expansion coefficients $\beta_2, \beta_3,\ldots$, which do not allow to investigate the $\beta$ function in the whole. The progress became possible after development of the Lipatov method for calculation of large orders of perturbation theory:[12] now one can try to interpolate the known coefficients $\beta_2, \beta_3,\ldots$ with their large order behavior and to sum the perturbation series. The first attempts of reconstruction of the $\beta$ function witnessed on triviality of $\phi^4$ theory. Application of more advanced summation methods gave the exponent $\alpha$ in the asymptotic behavior $\beta(g) \propto g^\alpha$ a value close to unity. The hypothesis for the asymptotics $\beta(g) \propto g$ was recently confirmed analytically for $\phi^4$ theory and QED [13] .[14] Together with positiveness of $\beta(g)$, obtained by summation of series, it gives the case (b) of the Bogoliubov and Shirkov classification, and hence the Landau pole is absent in these theories. Possibility of omitting the quadratic terms in the action suggested by Landau and Pomeranchuk is not confirmed.

## References

1. ^ Lev Landau, in Wolfgang Pauli, ed. (1955). Niels Bohr and the Development of Physics. London: Pergamon Press.
2. ^ Göckeler, M.; R. Horsley, V. Linke, P. Rakow, G. Schierholz, and H. Stüben (1998). "Is There a Landau Pole Problem in QED?". Physical Review Letters 80 (19): 4119–4122. arXiv:hep-th/9712244. Bibcode:1998PhRvL..80.4119G. doi:10.1103/PhysRevLett.80.4119.
3. ^ Kim, S.; John B. Kogut and Lombardo Maria Paola (2002-01-31). "Gauged Nambu–Jona-Lasinio studies of the triviality of quantum electrodynamics". Physical Review D 65 (5): 054015. arXiv:hep-lat/0112009. Bibcode:2002PhRvD..65e4015K. doi:10.1103/PhysRevD.65.054015.
4. ^ Gies, Holger; Jaeckel, Joerg (2004-09-09). "Renormalization Flow of QED". Physical Review Letters 93 (11): 110405. arXiv:hep-ph/0405183. Bibcode:2004PhRvL..93k0405G. doi:10.1103/PhysRevLett.93.110405.
5. ^ L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 95, 497, 773, 1177 (1954).
6. ^ Gell-Mann, M.; Low, F. E. (1954). "Quantum Electrodynamics at Small Distances". Physical Review 95 (5): 1300–1320. Bibcode:1954PhRv...95.1300G. doi:10.1103/PhysRev.95.1300.
7. ^ N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed. (Nauka, Moscow, 1976; Wiley, New York, 1980).
8. ^ L.D.Landau, I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 102, 489 (1955); I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 103, 1005 (1955).
9. ^ B.Freedman, P.Smolensky, D.Weingarten, Phys. Lett. B 113, 481 (1982).
10. ^ Gunion, J.; H. E. Haber, G. L. Kane, and S. Dawson (1990). The Higgs Hunters Guide. Addison-Wesley.
11. ^ For example, Heller, Urs; Markus Klomfass, Herbert Neuberger, and Pavols Vranas (1993-09-20). "Numerical analysis of the Higgs mass triviality bound". Nuclear Physics B 405 (2–3): 555–573. arXiv:hep-ph/9303215. Bibcode:1993NuPhB.405..555H. doi:10.1016/0550-3213(93)90559-8., which suggests MH < 710 GeV.
12. ^ L.N.Lipatov, Zh.Eksp.Teor.Fiz. 72, 411 (1977) [Sov.Phys. JETP 45, 216 (1977)].
13. ^ I. M. Suslov, JETP 107, 413 (2008); JETP 111, 450 (2010); http://arxiv.org/abs/1010.4081, http://arxiv.org/abs/1010.4317.
14. ^ I. M. Suslov, JETP 108, 980 (2009), http://arxiv.org/abs/0804.2650.