The expected norm of random polynomials

The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial

let

Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then

and

as .

In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives

and

This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.

     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

     

Copulas with maximum entropy

We shall find a multi-dimensional checkerboard copula of maximum entropy that matches an observed set of grade correlation coefficients. This problem is formulated as the maximization of a concave function on a convex polytope. Under mild constraint qualifications we show that a unique solution exists in the core of the feasible region. The theory of Fenchel duality is used to reformulate the problem as an unconstrained minimization which is well solved numerically using a Newton iteration. Finally, we discuss the numerical calculations for some hypothetical examples and describe how this work can be applied to the modelling and simulation of monthly rainfall.     

Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems

We discuss recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems.     

On lattice sums and Wigner limits

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective.     

On monochromatic triangles

Let A and B be two disjoint finite sets in R². Simple conditions that guarantee the existence of a triangle with vertices in one of the sets and with no points from the other set in its interior are given. The analogous problem for d-simplices in R^d is treated. Conditions are derived that guarantee the existence of a triangle with vertices in one of the sets and with no points from either set on its boundary.     

Multi-Variable Sinc Integrals and Volumes of Polyhedra

We investigate multi-variable integrals of products of sinc functions and show how they may be interpreted as volumes of symmetric convex polyhedra. We then derive an explicit formula for computing such sinc integrals and so equivalently volumes of polyhedra.     

A note on the existence of subgradients

We describe an apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space.Research partially funded on NSERC grant A5116.