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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.
Automatic Computation of Conservation Laws in the Calculus of Variations and
 Optimal Control, Comput. Methods Appl. Math
"... We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether’s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematica ..."
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Cited by 8 (6 self)
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We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether’s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.
Computers in mathematical inquiry
 in The Philosophy of Mathematical Practice
, 2008
"... Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, ..."
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Cited by 5 (0 self)
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Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character,
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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Cited by 3 (1 self)
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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
Seized opportunities
 Notices of the AMS
, 2010
"... that integrals were fun. Every undergraduate student encounters the evaluation of integrals at an early stage of his/her education. In my case this happened in a class at the Universidad ..."
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that integrals were fun. Every undergraduate student encounters the evaluation of integrals at an early stage of his/her education. In my case this happened in a class at the Universidad
Two Catalantype Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind
"... Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalantype Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2)(2x) n and (1/2)(4x) n ..."
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Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalantype Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2)(2x) n and (1/2)(4x) n in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representations of the central binomial coefficients and Catalan numbers. 1
HighPrecision Arithmetic: Progress and Challenges
"... For many scientific calculations, particularly those involving empirical data, IEEE 32bit floatingpoint arithmetic produces results of sufficient accuracy, while for other applications IEEE 64bit floatingpoint is more appropriate. But for some very demanding applications, even higher levels of p ..."
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For many scientific calculations, particularly those involving empirical data, IEEE 32bit floatingpoint arithmetic produces results of sufficient accuracy, while for other applications IEEE 64bit floatingpoint is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of highprecision computation and presents a sample of applications, including some new results from the past year or two. This article also discusses what facilities are required to support future highprecision computing, in light of emerging applications and changes in computer architecture, such as highly parallel systems and graphical processing units.
Quadratic Dirichlet LSeries
, 2005
"... Let D = 1 or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: where n = p e1 1 p e2 2 · · · p ek k D ..."
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Let D = 1 or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: where n = p e1 1 p e2 2 · · · p ek k D
DOUGALL’S BILATERAL 2H2SERIES AND RAMANUJANLIKE πFORMULAE
"... Abstract. The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall’s bilateral 2H2series. Several unusual transformations into fast convergent series are established. They lead surprisingly to numerous infinite series expressions for π, including several f ..."
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Abstract. The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall’s bilateral 2H2series. Several unusual transformations into fast convergent series are established. They lead surprisingly to numerous infinite series expressions for π, including several formulae discovered by Ramanujan (1914) and recently by Guillera (2008). Roughly speaking, hypergeometric series is defined to be a series ∑ Cn with term ratio C1+n/Cn a rational function of n. In general, it can be explicitly written as pFq a1, a2, ·· ·,ap b1, b2, ·· ·,bq ∣ z where the rising shifted factorial is given by ∞ ∑ (a1)n(a2)n ···(ap)n z