Every exponential group supports a positive harmonic function

Every Abelian Group Is a Class Group

Let T be the set of minimal primes of a Krull domain A. If S is a subset of T9 we form B = n AP for PeS and study the relation of the class group of B to that of A. We find that the class group of B is always a homomorphic image of that of A. We use this type of construction to obtain a Krull domain with specified class group and then alter such a Krull domain to obtain a Dedekind domain with t...

A new subclass of harmonic mappings with positive coefficients

‎Complex-valued harmonic functions that are univalent and‎ ‎sense-preserving in the open unit disk $U$ can be written as form‎ ‎$f =h+bar{g}$‎, ‎where $h$ and $g$ are analytic in $U$‎. ‎In this paper‎, ‎we introduce the class $S_H^1(beta)$‎, ‎where $1<betaleq 2$‎, ‎and‎ ‎consisting of harmonic univalent function $f = h+bar{g}$‎, ‎where $h$ and $g$ are in the form‎ ‎$h(z) = z+sumlimits_{n=2}^inf...

Harmonic Exponential Families on Manifolds

In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation...

Supports of a Convex Function

Almost Every Set in Exponential Time is P-Bi-Immune

A set A is P-bi-immune if neither A nor its complement has an innnite subset in P. We investigate here the abundance of P-bi-immune languages in linear-exponential time (E). We prove that the class of P-bi-immune sets has measure 1 in E. This implies thatàlmost' every language in E is P-bi-immune, that is to say, almost every set recognizable in linear exponential time has no algorithm that rec...