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53
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and t ..."
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Cited by 36 (5 self)
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Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Highprecision floatingpoint arithmetic in scientific computation
 Computing in Science and Engineering, May–June
, 2005
"... At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required: some of these applications require roughly twice ..."
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Cited by 17 (1 self)
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At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required: some of these applications require roughly twice this level; others require four times; while still others require hundreds or more digits to obtain numerically meaningful results. Such calculations have been facilitated by new highprecision software packages that include highlevel language translation modules to minimize the conversion effort. These activities have yielded a number of interesting new scientific results in fields as diverse as quantum theory, climate modeling and experimental mathematics, a few of which are described in this article. Such developments suggest that in the future, the numeric precision used for a scientific computation may be as important to the program design as are the algorithms and data structures.
HighPrecision Computation and Mathematical Physics
"... At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highpreci ..."
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Cited by 16 (3 self)
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At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highprecision software packages that include highlevel language translation modules to minimize the conversion effort. This paper presents a survey of recent applications of these techniques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb nbody atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that highprecision arithmetic facilities are now an indispensable component of a modern largescale scientific computing environment.
Closed forms: what they are and why we care
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not cl ..."
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Cited by 14 (7 self)
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.
A generalization of Wigner’s law
 Comm. Math. Phys
"... We present a generalization of Wigner's semicircle law: we consider a sequence of probability distributions (p1; p2; : : :), with mean value zero and take an N N real symmetric matrix with entries independently chosen from pN and consider analyze the distribution of eigenvalues. If we normali ..."
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We present a generalization of Wigner's semicircle law: we consider a sequence of probability distributions (p1; p2; : : :), with mean value zero and take an N N real symmetric matrix with entries independently chosen from pN and consider analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N!1 for certain pN the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the kth moment of pN (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when pN does not depend on N we obtain Wigner's law: if all moments of a distribution are nite, the distribution of eigenvalues is a semicircle. 1
A Remarkable Sequence of Integers
, 2008
"... A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and numbertheoretical nature. ..."
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Cited by 10 (5 self)
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A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and numbertheoretical nature.
Experimental Determination of ApéryLike Identities for ζ(2n + 2)
, 2006
"... We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1 ..."
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Cited by 9 (1 self)
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We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1
On the bits counting function of real numbers
 J. Austral. Math. Soc
"... Abstract. Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational number x> 0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting number numbers like 2, e or pi: their conjectural simple normality in base 2 is equivalent to Bn(x) ∼ n ..."
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Cited by 6 (0 self)
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Abstract. Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational number x> 0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting number numbers like 2, e or pi: their conjectural simple normality in base 2 is equivalent to Bn(x) ∼ n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x, y> 0, which we prove to be sharp up to a multiplicative constant. As a byproduct, we provide an answer to a question raised by Bailey et al. (On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number. 1.
Expressions for values of the gamma function
 Kyushu J. Math
, 2005
"... This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of π and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to eva ..."
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This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of π and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to evaluate, for example, algebraic Gauss hypergeometric functions by the Gauss identity. Also, algebraic independence of gamma values and their relation to the elliptic K function are briefly discussed. 1