Mathematics by Experiment

Bailey, David C.

doi:10.1201/b10704

Cited by 85 publications

(28 citation statements)

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“…Fact 9 is often a fine way of proving convexity of a function g by showing g arises as a conjugate, see [7,10,25], even by computer [3]. A particularly good tool is Proposition 10 (Smoothness and biconjugacy, [20,28]) If f * * is proper in a Banach space and f * is everywhere Fréchet differentiable then f is convex.…”

Section: Agrees With F Exactly When F Is Convex Proper and Lower-semmentioning

confidence: 99%

Proximality and Chebyshev sets

Borwein

2006

Optimization Letters

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Section: Agrees With F Exactly When F Is Convex Proper and Lower-semmentioning

confidence: 99%

Proximality and Chebyshev sets

Borwein

2006

Optimization Letters

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“…Proof: Clearly, by the original definitions (1) and 2, B n (s), ∆ n (s) are both finite for nonnegative (s) so it suffices to look at negative (s) in any case. For B n (s), relation (11) has a pole factor at s = −n.…”

Section: Quadrature Formulae For All Complex Powersmentioning

confidence: 99%

Advances in the theory of box integrals

Bailey

Borwein

2009

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Box integrals-expectations | r| s or | r − q| s over the unit n-cube (or nbox)-have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for n ≤ 4 dimensions the box integrals are for any integer s hypergeometrically closed in a sense we clarify herein. For n = 5 dimensions, we show that a single unresolved integral we call K 5 stands in the way of such hyperclosure proofs. We supply a compendium of exemplary closed forms that naturally arise algorithmically from this theory.

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“…In the remaining sections we describe various representative applications of PSLQ. More detail about these examples is given in [8] and the references therein.…”

mentioning

confidence: 99%

“…Since 1996, numerous formulas of this same type have been found for various constants [1,8]. Here, for instance, is a formula for π 2 that permits base-3 digits to be computed beginning at an arbitrary starting position:…”

mentioning

confidence: 99%