Extremals of the cone of norms and the cohomology of \(c^ 3\)- stratifications
type: Item
instance of: scholarly article
title: Extremals of the cone of norms and the cohomology of \(c^ 3\)- stratifications
zbMATH Open document ID: 0769.15009
DOI: 10.1016/0196-8858(92)90002-E
author: Eberhard Triesch
author: Andreas W. M. Dress
author: Ernst Ruch
published in: Advances in Applied Mathematics
publication date: 1992-09-26T00:00:00Z
review text: The problem considered is the following: Given real \(m\times n\) matrices \(X\) and \(Y\), when does there exist a stochastic matrix \(S\) such that \(SX=Y\)? The answer is given in terms of a cohomology theory for fans which are stratifications of \(R^ n\) by closed convex cones. There are links to a classical result of Hardy, Littlewood and Pólya, who described when two probability distributions on a finite set can be mapped one onto the other by a bistochastic matrix. There is a section devoted to cones and stratifications by closed convex cones. A cohomology theory is developed, where the first cohomology group measures to what extent the above mentioned problem can be solved. The most satisfactory result is obtained when \(n=2\). Finally a counterexample due to \textit{A. Horn} [cf. \textit{S. Sherman}, Proc. Am. Math. Soc. 3, 511-513 (1952; Zbl 0048.250) and ibid 5, 998-999 (1954; Zbl 0058.011)] to a conjecture by S. Kakutani is explained in the light of the present theory.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 55N99
zbMATH DE Number: 56225
zbMATH Keywords: extremals of cones
zbMATH Keywords: stochastic matrix
zbMATH Keywords: cohomology theory
zbMATH Keywords: convex cones
zbMATH Keywords: stratifications
zbMATH Keywords: counterexample
reviewed by: Tage Bai Andersen
MaRDI profile type: Publication
cites work: Q3943083
cites work: Q3952665
cites work: Q4039929
cites work: On a Result of S. Sherman Concerning Doubly Stochastic Matrices
cites work: On a Conjecture Concerning Doubly Stochastic Matrices
cites work: A Correction to "On a Conjecture Concerning Doubly Stochastic Matrices"
full work available at URL: 0196-8858(92)90002-e
OpenAlex ID: W2093340679
description: scientific article; zbMATH DE number 56225