Experimental Mathematics: Recent Developments and Future Outlook

While extensive usage of high-performance computing has been a staple of other scientific and engineering disciplines for some time, research mathematics is one discipline that has heretofore not yet benefited to the same degree. Now, however, with sophisticated mathematical computing tools and environments widely available on desktop computers, a growing number of remarkable new mathematical results are being discovered partly or entirely with the aid of these tools. With currently planned improvements in these tools, together with substantial increases expected in raw computing power, due both to Moore’s Law and the expected implementation of these environments on parallel supercomputers, we can expect even more remarkable developments in the years ahead.

[1]  Jonathan M. Borwein,et al.  Empirically Determined Apéry-like Formulae for Zeta(4n+3) , 1997 .

[2]  P. Borwein,et al.  Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point , 1997, Canadian Journal of Mathematics.

[3]  D. J. Broadhurst,et al.  Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops , 1996, hep-th/9609128.

[4]  F. Gill,et al.  Advice to a Young Scientist , 1982 .

[5]  D. Zagier.  Values of Zeta Functions and Their Applications , 1994 .

[6]  Neil J. A. Sloane,et al.  The encyclopedia of integer sequences , 1995 .

[7]  David H. Bailey,et al.  On the Random Character of Fundamental Constant Expansions , 2001, Exp. Math..

[8]  Jonathan M. Borwein,et al.  Experimental Evaluation of Euler Sums , 1994, Exp. Math..

[9]  Jonathan M. Borwein,et al.  Empirically Determined Apéry-Like Formulae for ζ(4n+3) , 1997, Exp. Math..

[10]  J. Borwein,et al.  Emerging Tools for Experimental Mathematics , 1999 .

[11]  Jonathan M. Borwein,et al.  Central Binomial Sums, Multiple Clausen Values, and Zeta Values , 2001, Exp. Math..

[12]  Jonathan M. Borwein,et al.  Some Remarkable Properties of Sinc and Related Integrals , 2001 .

[13]  David H. Bailey,et al.  A Fortran 90-based multiprecision system , 1995, TOMS.

[14]  David H. Bailey,et al.  A seventeenth-order polylogarithm ladder , 1999 .

[15]  J. Holbrook,et al.  Statistical sampling and fractal distributions , 1996 .

[16]  Jonathan M. Borwein,et al.  Special Values of Multidimensional Polylogarithms , 2001 .

[17]  D. H. Lehmer.  Factorization of Certain Cyclotomic Functions , 1933 .

[18]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[19]  D. J. Broadhurst.  Massive 3-loop Feynman diagrams reducible to SC$^*$ primitives of algebras of the sixth root of unity , 1999 .

[20]  David Bailey,et al.  On the rapid computation of various polylogarithmic constants , 1997, Math. Comput..

[21]  David H. Bailey,et al.  Parallel integer relation detection: Techniques and applications , 2001, Math. Comput..

[22]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[23]  David H. Bailey,et al.  Analysis of PSLQ, an integer relation finding algorithm , 1999, Math. Comput..

[24]  J. Borwein,et al.  Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity , 1998 .

[25]  D. J. Broadhurst,et al.  Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots , 1996 .

[26]  David J. Broadhurst,et al.  Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ?(3) and ?(5) , 1998 .

[27]  George Polya,et al.  Mathematical discovery : on understanding, learning, and teaching problem solving , 1962 .

[28]  Reuben Hersh,et al.  FRESH BREEZES IN THE PHILOSOPHY OF MATHEMATICS , 1995 .

[29]  J. Borwein,et al.  Strange series and high precision fraud , 1992 .

[30]  Jonathan M. Borwein,et al.  Making Sense of Experimental Mathematics , 1996 .

[31]  Isaac Asimov,et al.  Isaac Asimov's Book of Science and Nature Quotations , 1990 .

[32]  Gert Almkvist,et al.  Borwein and Bradley's Apérv-Like Formulae for ζ(4n + 3) , 1999, Exp. Math..

[33]  R. Forcade,et al.  Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two , 1979 .

[34]  Marshall Missner,et al.  Who Got Einstein's Office? , 1987 .