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- Jonathan M. Borwein, Jon D. Vanderwerff
- 2009

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Like differentiability, convexity is a natural and powerful property of functions that… (More)

In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov. Consider a polynomial p on R n of degree d (the maximum of the degrees of the monomials in the expansion of p). We call p homogeneous if… (More)

Historically, the polylogarithm has attracted specialists and non-specialists alike w i t h i t s l o vely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we h a ve been forced to consider multidimensional… (More)

From an experimental-mathematical perspective we analyze " Ising-class " integrals. These are structurally related n-dimensional integrals we call C n , D n , E n , where D n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyze C n := 4 n! ∞ 0 · · · ∞ 0 1 n j=1 (u j + 1/u j) 2 du 1 u 1 · · · du n u n. We… (More)

- J M Borwein, I J Zucker, J Boersma
- 2004

Euler considered sums of the form ∞ ∑ m=1 1 m s m−1 ∑ n=1 1 n t. Here natural generalisations of these sums namely [d 1 , d 2 ] := [d 1 , d 2 ](s, t) = ∞ ∑ m=1 χ d 1 (m) m s m−1 ∑ n=1 χ d 2 (n) n t , have been investigated, where χ d 1 and χ d 2 are characters, and s and t are positive integers. The cases when d 1 and d 2 were either 1, 2a, 2b or −4 have… (More)

By a " box integral " we mean here an expectation | r − q| s where r runs over the unit n-cube, with q and s fixed, explicitly:

- Jonathan M. Borwein, Claude Cheval
- 2004

'.. . where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began " My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy , viz. the Logarithms: but my Lord,… (More)

- HENG HUAT, PATRICK SOLÉ, J. M. Borwein, F. G. Garvan
- 2002

In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Bor-weins (1987), (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, and there are hints that they should be normal numbers, i.e., that their base b digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP… (More)