Wallis-Ramanujan-Schur-Feynman

This article by Tewodros Amdeberhan, Olivier R. Espinosa, Victor H. Moll and Armin Straub has been published in American Mathematical Monthly (August/September 2010).

You can also find this article on the arXiv as 1004.2453v1 [math.CA].

Abstract

One of the earliest examples of analytic representations for π is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula

$$ \frac{2}{\pi} \int_0^\infty \frac{dx}{(x^2+1)^{n+1}} = \frac{1}{2^{2n}} \binom{2n}{n}. $$
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ws.pdf

Closed-form evaluation of integrals appearing in positronium decay

This article by Tewodros Amdeberhan, Victor H. Moll and Armin Straub has been published in Journal of Mathematical Physics (Volume 50, Issue 10, October 2009, 6 Pages) and is available at doi:10.1063/1.3246615.

Abstract

A theoretical prediction for the total width of the positronium decay in QED has been given by B. Kniehl et al. in the form of an expansion in Sommerfeld's fine-structure constant. The coefficients of this expansion are given in the form of two-dimensional definite integrals, with an integrand involving the polylogarithm function. We provide here an analytic expression for the one-loop contribution to this problem.

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A fast numerical algorithm for the integration of rational functions

This article by Dante Manna, Luis Medina, Victor H. Moll and Armin Straub has been published in Numerische Mathematik (Volume 115, Number 2, April 2010, Pages 289-307) and is available at doi:10.1007/s00211-009-0284-9.

Abstract

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numlanden.pdf
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Landen.m
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examples.nb

The local recognition of reflection graphs of spherical Coxeter groups

This article by Ralf Gramlich, Jonathan I. Hall and Armin Straub has been published in Journal of Algebraic Combinatorics (published online Sep 2009) and is available at doi:10.1007/s10801-009-0201-4.

You can also find this article on the arXiv as 0808.2173v2 [math.GR].

Abstract

Based on the third author's thesis in this article we complete the local recognition of commuting reflection graphs of spherical Coxeter groups arising from irreducible crystallographic root systems.

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The p-adic valuation of k-central binomial coefficients

This article by Tewodros Amdeberhan, Victor H. Moll and Armin Straub has been published in Acta Arithmetica (Volume 140, 2009, Pages 31-42) and is available at doi:10.4064/aa140-1-2.

You can also find this article on the arXiv as 0811.2028v1 [math.NT].

Abstract

The coefficients c(n,k) defined by

$$(1-k^{2}x)^{-1/k} = \sum_{n \geq 0} c(n,k)x^n$$

reduce to the central binomial coefficients $ \binom{2n}{n} $ for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.

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cenbin.pdf
Date: Jul 8, 2010

The "What Is...?" column

I very much enjoy reading the "What Is…?" column in the Notices of the AMS. Unfortunately, there seemed to be no index to this column. I have therefore created this one in the hope that it'll be helpful to others as well.

  1. What is…an amoeba? — Oleg Viro, September 2002
  2. What is…the monster? — Richard Borcherds, October 2002
read more »
in
Date: Mar 5, 2010

Computer proved monotonicity of some coefficients

Ming-Hua Lin from the University of Regina sent me the following problem:

Problem: Let p ≥ 2 be an integer, and define

$$ f (t) = \left( \frac{1}{1 - t \left( \frac{1}{p} + \frac{p - 1}{2 p^2} t<br /> \right)} \right)^p = \sum_{n \geqslant 0} c_n t^n . $$

Show that $ c_2 > c_3 > c_4 > \cdots $. read more »

in
Date: Aug 18, 2009

CARMA workshop in Newcastle, 2009

On August 18, 2009, the University of Newcastle hosted a CARMA workshop on Multidimensional Numerical Integration and Special Function Evaluation. Besides enjoying very interesting talks it's been my pleasure to present, together with James Wan, on progress of our joint work with Jon Borwein, Peter Donovan and Dirk Nuyens on expectations of random walks. read more »

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in
Date: Jun 12, 2009

Apollonian circle packings in Mathematica

Today, I have been playing a little bit with Apollonian circle packings. Here is the code I wrote in Mathematica to visualize such packings (see below for an example). read more »

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in
Date: Jun 10, 2009

Fast q-binomials in Mathematica

Recently, I have been doing experiments involving q-binomial coefficients in Mathematica. Starting with version 7, Mathematica is prepared for some q-business; in particular, there exists a function named QBinomial giving the q-analog of Binomial. However, this implementation turned out to not be fast enough for my needs. Here is an alternative approach which is not only way faster but provides a full factorization. read more »

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