Restricted p-Centers for Sets in Real Locally Convex Spaces

ABSTRACT

Let p be a continuous seminorm on a real locally convex space X. The current paper is concerned with the problem of restricted p-centers in X. Characterizations of restricted p-centers and strongly unique restricted p-centers with respect to a “sun” are provided, and then some recent results due to Laurent and Pai [9 P. J. Laurent and D. Pai ( 1998 ). On simultaneous approximation . Numer. Funct. Anal. Optim. 19 : 1045 – 1064 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] are extended and improved.     

Locally Completely Regular Epigroups

The object of the article is to characterize epigroups which are locally completely regular. Some subclasses of such epigroups are also described.     

Extensions and biextensions of locally constant group schemes,tori and abelian schemes

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objects. In particular, we prove that if G _i (for i = 1, 2, 3) is an extension of an abelian S-scheme A _i by an S-torus T _i , the category of biextensions of (G ₁, G ₂) by G ₃ is equivalent to the category of biextensions of the underlying abelian S-schemes (A ₁, A ₂) by the underlying S-torus T ₃.      

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.     

Centralizer sizes and nilpotency class in Lie algebras and finite -groups

In this work we solve a conjecture of Y. Barnea and M. Isaacs about centralizer sizes and the nilpotency class in nilpotent finite-dimensional Lie algebras and finite -groups.

     

Sur les sous-groupes nilpotents du groupe de Cremona

Résumé. Nous décrivons les sous-groupes nilpotents non virtuellement abéliens du groupe des transformations birationnelles du plan projectif complexe: un tel groupe est d'ordre fini ou virtuellement métabélien.      

Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups

For nilpotent quantum groups [as introduced by Franz et al. ⁽⁷⁾], we show that (in sharp contrast to the classical case) the symmetrization of a probability distribution and the first moments of together determine uniquely the original distribution .     

Zero-Dimensional Imbeddability

In this article, we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen–Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen–Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.