Parametric Euler Sum Identities

We consider some parametrized classes of multiple sums first studied by Euler. Identities between meromorphic functions of one or more variables generate reduction formulae for these sums.Comment: 12 page

Log-sine evaluations of Mahler measures, II

We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in "Log-sine evaluations of Mahler measures" and "Special values of generalized log-sine integrals" by two of the authors. This motivates a detailed study of various multiple polylogarithms and worked examples are given. Our techniques enable the reduction of several multiple Mahler measures, and supply an easy proof of two conjectures by Boyd.Comment: 35 page

Densities of short uniform random walks

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.Comment: 32 pages, 9 figure

Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A, B$ are maximally monotone and $A$ is a linear relation, as soon as Rockafellar's constraint qualification holds: \dom A\cap\inte\dom B\neq\varnothing. Moreover, $A+B$ is of type (FPV).Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1010.4346, arXiv:1005.224

Thirty-two Goldbach Variations

We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory material added and material on inequalities, Hilbert matrix and Witten zeta functions. Errors in the second section on Complex Line Integrals are corrected. To appear in International Journal of Number Theory. Title change

Log-sine evaluations of Mahler measures

We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then applied to evaluation of further families of multiple Mahler measures.Comment: 25 page

A Cyclic Douglas-Rachford Iteration Scheme

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas-Rachford scheme, are promising.Comment: 22 pages, 7 figures, 4 table