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Experimental computation with oscillatory integrals

Description: A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.
Date: June 26, 2009
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Experimental Mathematics and Computational Statistics

Description: The field of statistics has long been noted for techniques to detect patterns and regularities in numerical data. In this article we explore connections between statistics and the emerging field of 'experimental mathematics'. These includes both applications of experimental mathematics in statistics, as well as statistical methods applied to computational mathematics.
Date: April 30, 2009
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

High-Precision Computation and Mathematical Physics

Description: 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. Such calculations are facilitated by high-precision software packages that include high-level 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 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 computing environment.
Date: November 3, 2008
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Exploratory Experimentation and Computation

Description: We believe the mathematical research community is facing a great challenge to re-evaluate the role of proof in light of recent developments. On one hand, the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to data-mine on the Internet, has provided marvelous resources to the research mathematician. On the other hand, the enormous complexity of many modern capstone results such as the Poincare conjecture, Fermat's last theorem, and the classification of finite simple groups has raised questions as to how we can better ensure the integrity of modern mathematics. Yet as the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished.
Date: February 25, 2010
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

PROPOSED SIAM PROBLEM

Description: A recent paper by the present authors, together with mathematical physicists David Broadhurst and M. Larry Glasser, explored Bessel moment integrals, namely definite integrals of the general form {integral}{sub 0}{sup {infinity}} t{sup m}f{sup n}(t) dt, where the function f(t) is one of the classical Bessel functions. In that paper, numerous previously unknown analytic evaluations were obtained, using a combination of analytic methods together with some fairly high-powered numerical computations, often performed on highly parallel computers. In several instances, while we were able to numerically discover what appears to be a solid analytic identity, based on extremely high-precision numerical computations, we were unable to find a rigorous proof. Thus we present here a brief list of some of these unproven but numerically confirmed identities.
Date: August 12, 2008
Creator: BAILEY, DAVID H. & BORWEIN, JONATHAN M.
Partner: UNT Libraries Government Documents Department

Computer-Assisted Discovery and Proof

Description: With the advent of powerful, widely-available mathematical software, combined with ever-faster computer hardware, we are approaching a day when both the discovery and proof of mathematical facts can be done in a computer-assisted manner. his article presents several specific examples of this new paradigm in action.
Date: December 10, 2007
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

The Greatest Mathematical Discovery?

Description: What mathematical discovery more than 1500 years ago: (1) Is one of the greatest, if not the greatest, single discovery in the field of mathematics? (2) Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? (3) Was fiercely resisted in Europe for hundreds of years after its discovery? (4) Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, together with the basic arithmetic computational schemes, which were discovered in India about 500 CE.
Date: May 12, 2010
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Elliptic integral evaluations of Bessel moments

Description: We record what is known about the closed forms for variousBessel function moments arising in quantum field theory, condensed mattertheory and other parts of mathematical physics. More generally, wedevelop formulae for integrals of products of six or fewer Besselfunctions. In consequence, we are able to discover and prove closed formsfor c(n,k) := Int_0 inf tk K_0 n(t) dt, with integers n = 1, 2, 3, 4 andk greater than or equal to 0, obtaining new results for the even momentsc3,2k and c4,2k . We also derive new closed forms for the odd momentss(n,2k+1) := Int_0 inf t(2k+1) I_0(t) K_0n(t) dt,with n = 3, 4 and fort(n,2k+1) := Int_0 inf t(2k+1) I_02(t) K_0(n-2) dt, with n = 5, relatingthe latter to Green functions on hexagonal, diamond and cubic lattices.We conjecture the values of s(5,2k+1), make substantial progress on theevaluation of c(5,2k+1), s(6,2k+1) and t(6,2k+1) and report more limitedprogress regarding c(5,2k), c(6,2k+1) and c(6,2k). In the process, weobtain 8 conjectural evaluations, each of which has been checked to 1200decimal places. One of these lies deep in 4-dimensional quantum fieldtheory and two are probably provable by delicate combinatorics. Thereremains a hard core of five conjectures whose proofs would be mostinstructive, to mathematicians and physicists alike.
Date: January 6, 2008
Creator: Bailey, David H.; Borwein, Jonathan M.; Broadhurst, David & Glasser, M.L.
Partner: UNT Libraries Government Documents Department

Experimental Mathematics and Mathematical Physics

Description: 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 multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.
Date: June 26, 2009
Creator: Bailey, David H.; Borwein, Jonathan M.; Broadhurst, David & Zudilin, Wadim
Partner: UNT Libraries Government Documents Department

Problem Proposed for the American Mathematical Monthly

Description: The problem is to define: P(x) := {Sigma}{sub k = 1}{sup {infinity}} arctan (x - 1/(k + x + 1) {radical}(k + 1) + (k + 2) {radical}(k + x)). (1) (a) Find explicit, finite-expression evaluations of P(n) for all integers n {ge} 0. (b) Show {tau} := lim{sub x {yields} -1{sup +}} P(x) exists, and find an explicit evaluation for {tau}. (c) Are there a more general closed forms for P, say at half-integers? Solution with the abbreviations: r := {radical} (k + 1), s := {radical} (k + x) the argument of arctan in (1) becomes s{sup 2} - r{sup 2}/(s{sup 2} + 1) r + (r{sup 2} + 1) s = s - r/r s + 1 = 1/r - 1/s / 1 + 1/r 1/s.
Date: November 14, 2008
Creator: Bailey, David H.; Borwein, Jonathan M. & Waldvogel, Jorg
Partner: UNT Libraries Government Documents Department

Hypergeometric Forms for Ising-Class Integrals

Description: We apply experimental-mathematical principles to analyzecertain integrals relevant to the Ising theory of solid-state physics. Wefind representations of the these integrals in terms of MeijerG-functions and nested-Barnes integrals. Our investigations began bycomputing 500-digit numerical values of Cn,k,namely a 2-D array of Isingintegrals for all integers n, k where n is in [2,12]and k is in [0,25].We found that some Cn,k enjoy exact evaluations involving DirichletL-functions or the Riemann zeta function. In theprocess of analyzinghypergeometric representations, we found -- experimentally and strikingly-- that the Cn,k almost certainly satisfy certain inter-indicialrelations including discrete k-recursions. Using generating functions,differential theory, complex analysis, and Wilf-Zeilberger algorithms weare able to prove some central cases of these relations.
Date: July 1, 2006
Creator: Bailey, David H.; Borwein, David; Borwein, Jonathan M. & Crandall,Richard E.
Partner: UNT Libraries Government Documents Department

Highly Parallel, High-Precision Numerical Integration

Description: This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for functions with singularities or infinite derivatives at endpoints. Such a scheme is of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where high-precision numerical values of definite integrals can be used to numerically discover new identities. This paper discusses techniques for a parallel implementation of this scheme, then presents performance results for 1-D and 2-D test suites. Results are also given for a certain problem from mathematical physics, which features a difficult singularity, confirming a conjecture to 20,000 digit accuracy. The performance rate for this latter calculation on 1024 CPUs is 690 Gflop/s. We believe that this and one other 20,000-digit integral evaluation that we report are the highest-precision non-trivial numerical integrations performed to date.
Date: April 22, 2005
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Experimental Mathemataics: Examples, Methods andImplications

Description: Recent years have seen the flowering of ''experimental'' mathematics, namely the utilization of modern computer technology as an active tool in mathematical research. This development is not limited to a handful of researchers, nor to a handful of universities, nor is it limited to one particular field of mathematics. Instead, it involves hundreds of individuals, at many different institutions, who have turned to the remarkable new computational tools now available to assist in their research, whether it be in number theory, algebra, analysis, geometry or even topology. These tools are being used to work out specific examples, generate plots, perform various algebraic and calculus manipulations, test conjectures, and explore routes to formal proof. Using computer tools to test conjectures is by itself a major time saver for mathematicians, as it permits them to quickly rule out false notions.
Date: January 31, 2005
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Ten Problems in Experimental Mathematics

Description: This article was stimulated by the recent SIAM ''100 DigitChallenge'' of Nick Trefethen, beautifully described in a recent book. Indeed, these ten numeric challenge problems are also listed in a recent book by two of present authors, where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent was to present ten problems that are characteristic of the sorts of problems that commonly arise in ''experimental mathematics''. The challenge in each case is to obtain a high precision numeric evaluation of the quantity, and then, if possible, to obtain a symbolic answer, ideally one with proof. Our goal in this article is to provide solutions to these ten problems, and in the process present a concise account of how one combines symbolic and numeric computation, which may be termed ''hybrid computation'', in the process of mathematical discovery.
Date: September 30, 2004
Creator: Bailey, David H.; Borwein, Jonathan M.; Kapoor, Vishaal & Weisstein, Eric
Partner: UNT Libraries Government Documents Department

Future Prospects for Computer-Assisted Mathematics

Description: The recent rise of ''computer-assisted'' and ''experimental'' mathematics raises intriguing questions as to the future role of computation in mathematics. These results also draw into question the traditional distinctions that have been drawn between formal proof and computationally-assisted proof. This article explores these questions in the context of the growing consensus among computer technologists that Moore's Law is likely to continue unabated for quite some time into the future, producing hardware and software much more powerful than what is available today.
Date: October 26, 2005
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

Integrals of the Ising Class

Description: From an experimental-mathematical perspective we analyze"Ising-class" integrals. Our experimental results involvedextreme-precision, multidimensional quadrature on intricate integrands;thus, highly parallel computation was required.
Date: June 1, 2006
Creator: Bailey, David H.; Borwein, Jonathan M. & Crandall, Richard E.
Partner: UNT Libraries Government Documents Department

Box Integrals

Description: By a "box integral" we mean here an expectation $\langle|\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube,with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray*&&\int_01 \cdots \int_01 \left((r_1 - q_1)2 + \dots+(r_n-q_n)2\right)^ s/2 \ dr_1 \cdots dr_n.\end eqnarray* The study ofbox integrals leads one naturally into several disparate fields ofanalysis. While previous studies have focused upon symbolic evaluationand asymptotic analysis of special cases (notably $s = 1$), we workherein more generally--in interdisciplinary fashion--developing resultssuch as: (1) analytic continuation (in complex $s$), (2) relevantcombinatorial identities, (3) rapidly converging series, (4) statisticalinferences, (5) connections to mathematical physics, and (6)extreme-precision quadrature techniques appropriate for these integrals.These intuitions and results open up avenues of experimental mathematics,with a view to new conjectures and theorems on integrals of thistype.
Date: June 1, 2006
Creator: Bailey, David H.; Borwein, Jonathan M. & Crandall, Richard E.
Partner: UNT Libraries Government Documents Department

On the binary expansions of algebraic numbers

Description: Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through bit position N satisfies {number_sign}(|y|, N) > CN{sup 1/D} for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals {summation}{sub n{ge}0} 1/2{sup f(n)} where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet we can also handle numbers with a substantially denser occurrence of 1's. Though the number z = {summation}{sub n{ge}0}1/2{sup n{sup 2}} has too high a 1's density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z{sup 2}.
Date: July 1, 2003
Creator: Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E. & Pomerance, Carl
Partner: UNT Libraries Government Documents Department