At the present time, IEEE 64-bit floating-point 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 high-precision software packages that include high-level 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.

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

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 domain-specific 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 context-free 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 right-hand-side subset constructor operator. 1.

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 number-theoretical nature.

Abstract. 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. 1. Closed Forms: What They Are Mathematics abounds in terms which are in frequent use yet which are rarely made precise. Two such are rigorous proof and closed form (absent the technical use within differential algebra). If a rigorous proof is “that which ‘convinces ’ the appropriate audience ” then a closed form is “that which looks ‘fundamental ’ to the requisite consumer. ” In both cases, this is a community-varying and epoch-dependent notion. What was a compelling proof in 1810 may well not be now; what is a fine closed form in 2010 may have been anathema a century ago. In the article we are intentionally informal as befits a topic that intrinsically has no one “right ” answer. Let us begin by sampling the Web for various approaches to informal definitions of “closed form.” 1.0.1. First approach to a definition of closed form. The first comes from MathWorld [55] and so may well be the first and last definition a student or other seeker-after-easy-truth finds. An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is

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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

{ddoty,lutz,satyadev} at cs dot iastate dot edu Abstract. We use entropy rates and Schur concavity to prove that, for every integer k ≥ 2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal. 1

At the present time, IEEE 64-bit floating-point arithmetic is sufficiently accurate for most sci-entific applications. However, for a rapidly growing body of important scientific computing ap-plications, a higher level of numeric precision is required. Such calculations are facilitated by high-precision software packages that include high-level language translation modules to min-imize the conversion effort. This paper presents a survey of recent applications of these tech-niques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb n-body atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that high-precision arithmetic facilities are now an indispensable component of a modern large-scale scientific com-puting environment.

Our community (appropriately defined) is facing a great challenge to reevaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages and of the growing capacity to data-mine on the internet. Add to that the enormous complexity of many modern mathematical results such as the Poincaré conjecture, Fermat’s last theorem, and the classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the need to ensure the role of proof is properly founded remains undiminished. I share with Polya the view that “[I]ntuition comes to us much earlier and with much less outside influence than formal arguments · · · Therefore, I think that in teaching (high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. ” — George Polya (1887-1985) [15, 2 p. 128] He goes on to reaffirm, nonetheless, that proof should certainly be taught