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Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
Belyi Functions for Archimedean Solids
 Discrete Math
, 1996
"... The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which co ..."
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The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which correspond to the socalled Archimedean solids. R#sum# La notion de fonction de Belyi est un outil technique qui relie la combinatoire des cartes (c'est#dire, des graphes plong#s sur des surfaces) avec la th#orie de Galois, la th#orie des nombres alg#briques et la th#orie des surfaces de Riemann. Dans cet article nous calculons les fonctions de Belyi pour une classe des cartes semireguli#res, correspondant # ce qu'on appelle les solides d'Archim#de. 1 Introduction The title of the present paper attempts to link together traditional and contemporary mathematics. The name of Archimedes represents tradition; that of Belyi, one of the most recent advances in Galois theory, known (even i...
Implications of Experimental Mathematics for the Philosophy of Mathematics,” chapter to appear
 Current Issues in the Philosophy of Mathematics From the Viewpoint of Mathematicians and Teachers of Mathematics, 2006. [Ddrive Preprint 280
"... Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute emba ..."
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Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. ” (Christopher Koch, 2004) That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics. In my view it is now both necessary and possible to admit quasiempirical inductive methods fully into mathematical argument. In doing so carefully we will enrich mathematics and yet preserve the mathematical literature’s deserved reputation for reliability—even as the methods and criteria change. What do I mean by reliability? Well, research mathematicians still consult Euler or Riemann to be informed, anatomists only consult Harvey 3 for historical reasons. Mathematicians happily quote old papers as core steps of arguments, physical scientists expect to have to confirm results with another experiment. 1 Mathematical Knowledge as I View It Somewhat unusually, I can exactly place the day at registration that I became a mathematician and I recall the reason why. I was about to deposit my punch cards in the ‘honours history bin’. I remember thinking “If I do study history, in ten years I shall have forgotten how to use the calculus properly. If I take mathematics, I shall still be able to read competently about the War of 1812 or the Papal schism. ” (Jonathan Borwein, 1968) The inescapable reality of objective mathematical knowledge is still with me. Nonetheless, my view then of the edifice I was entering is not that close to my view of the one I inhabit forty years later. 1 The companion web site is at www.experimentalmath.info
Pure product polynomials and the ProuhetTarryEscott problem
 Math. Comp
, 1997
"... Abstract. An nfactor pure product is a polynomial which can be expressed in the form Qn i=1 (1−xαi) for some natural numbers α1,...,αn. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every nfactor pure product has norm at least 2n. We ..."
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Abstract. An nfactor pure product is a polynomial which can be expressed in the form Qn i=1 (1−xαi) for some natural numbers α1,...,αn. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every nfactor pure product has norm at least 2n. We describe three algorithms for determining the least norm an nfactor pure product can have. We report results of our computations using one of these algorithms which include the result that every nfactor pure product has norm strictly greater than 2n if n is 7, 9, 10, or 11. 1.
OUTLINE of PRESENTATION
, 2004
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt ..."
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt
Experimental Mathematics: ApéryLike Identities for ζ(n)
, 2005
"... We wish to consider one of the most fascinating and glamorous functions of analysis, the Riemann zeta function. (R. Bellman) Siegel found several pages of... numerical calculations with... zeroes of the zeta function calculated to several decimal places each. As Andrew Granville has observed “So muc ..."
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We wish to consider one of the most fascinating and glamorous functions of analysis, the Riemann zeta function. (R. Bellman) Siegel found several pages of... numerical calculations with... zeroes of the zeta function calculated to several decimal places each. As Andrew Granville has observed “So much for pure thought alone. ” (JB & DHB) www.cs.dal.ca/ddrive AK Peters 2004 Talk Revised: 03–29–05ApéryLike Identities for ζ(n) The final lecture comprises a research level case study of generating functions for zeta functions. This lecture is based on past research with David Bradley and current research with David Bailey. One example is Z(x): = 3 k=1 n=1