The theme of this paper is the application of linear analysis to simplify and extend convex analysis. The central problem treated is the standard convex program -minimize a convex function subject to inequality constraints on other convex functions. The present approach uses the support planes of the constraint region to transform the convex program into an equivalent linear program. Then the d...

We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.

An analysis of the characteristic function of Gaussian quadratic forms is presented in [1] to study the performance of multichannel communication systems. This technical report reviews this analysis, obtaining alternative expressions to original ones in compact matrix format. Keywords—Quadratic forms, characteristic function, multichannel communication systems.

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and ...

We consider an inverse problem of identifying the support D of a source term in an elliptic equation −∆u(x) + q(x)χD(x)u(x) = 0, x ∈ Ω and u(x) = f(x), x ∈ ∂Ω. Here q is a given positive function and χD is the characteristic function of a subdomain D such that D ⊂ Ω. By using a Carleman estimate, we prove the global uniqueness in this inverse problem within convex hulls of polygons D’s.

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulæ for the distribution of the sum of n non-identically distributed uniform random variables in both the continuo...

We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to Mœglin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula t...

All references to definitions and results in this Supplement refer to Dillenberger, Krishna, and Sadowski (2017, henceforth DKS) unless otherwise specified. The supplement is organized as follows. Section 1 below provides details of the Information Cost representation in Section 6 as well as a proof of Theorem 4 in DKS. Section 2 reviews some relevant notions from convex analysis, which are use...