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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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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.
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 9 (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.
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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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.
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 5 (3 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.
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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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
Incrementally Inferring ContextFree Grammars for DomainSpecific Languages
 In Proceedings of the Eighteenth International Conference on Software Engineering and Knowledge Engineering (SEKE'06
, 2006
"... Grammatical inference (or grammar inference) has been applied to various problems in areas such as computational biology, and speech and pattern recognition but its application to the programming language problem domain has been limited. We propose a new application area for grammar inference which ..."
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Grammatical inference (or grammar inference) has been applied to various problems in areas such as computational biology, and speech and pattern recognition but its application to the programming language problem domain has been limited. We propose a new application area for grammar inference which intends to make domainspecific language development easier and finds a second application in renovation tools for legacy software systems. We discuss the improvements made to our core incremental approach to inferring contextfree grammars. The approach affords a number of advancements over our previous geneticprogramming based inference system. We discuss the beam search heuristic for improved searching in the solution space of all grammars, the Minimum Description Length heuristic to direct the search towards simpler grammars, and the righthandside subset constructor operator. 1.
Experimental Mathematics and Mathematical Physics
 CONTEMPORARY MATHEMATICS
"... One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathemat ..."
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One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multidimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.
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 4 (2 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.
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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Cited by 3 (1 self)
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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