Generalizations of Gonçalves' inequality

Let be a polynomial with complex coefficients and roots , ..., , let denote its norm over the unit circle, and let denote Mahler's measure of . Gonçalves' inequality asserts that

We prove that

for , where is an explicit constant, and that

for . We also establish additional lower bounds on the norms of a polynomial in terms of its coefficients.

     

Hand-to-hand combat with thousand-digit integrals

In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multi-step uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K′, E and E′. We find that in many cases such integrals can be “experimentally” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with large-scale runs of the “PSLQ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.     

Generalized linear complementarity problems treated without fixed-point theory

We study the (monotone) linear complementarity problem in reflexive Banach space. The problem is treated as a quadratic program and shown to satisfy appropriate constraint qualifications. This leads to a theory of the generalized monotone linear complementarity problem which is independent of Brouwer's fixed-point theorem. Certain related results on linear systems are given. The final section concerns copositive operators.This research was partially supported by NSERC Grant No. A-5516.The author thanks the referee for his painstaking and thorough comments on this paper.     

Approximate subgradients and coderivatives in R n

We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.