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

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

Nearest points and delta convex functions in Banach spaces

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.Comment: To appear in Bull. Aust. Math. So

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

The Cyclic Douglas-Rachford Method for Inconsistent Feasibility Problems

We analyse the behaviour of the newly introduced cyclic Douglas-Rachford algorithm for finding a point in the intersection of a finite number of closed convex sets. This work considers the case in which the target intersection set is possibly empty.Comment: 13 pages, 2 figures; references updated, figure 2 correcte