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In Quest of Birkhoff's Theorem in Higher Dimensions

Xin Li, P. Mikusiński, H. Sherwood and M. D. Taylor
Lecture Notes-Monograph Series
Vol. 28, Distributions with Fixed Marginals and Related Topics (1996), pp. 187-197
Stable URL: http://www.jstor.org/stable/4355892
Page Count: 11
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In Quest of Birkhoff's Theorem in Higher Dimensions
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Abstract

A doubly stochastic matrix is a non-negative function defined on {1,..., m} × {1,..., m} such that all row and column sums are 1. A hypermatrix is a non-negative function defined on a set of the form $X=\{1,\ldots ,m_{1}\}\times \ldots \times \{1,\ldots ,m_{n}\}$. A hypermatrix is called multiply stochastic if it satisfies a suitably generalized version of the row and column condition for ordinary doubly stochastic matrices. Note that this is a double generalization of doubly stochastic matrices: we not only consider higher dimensions but allow "non-square matrices". Our goal is to describe extremal multiply stochastic hypermatrices in terms of their support, in terms of transfer vectors, and as local minima of the entropy function and to characterize the set of such extremals for a certain class of 3 × 3 × 3 hypermatrices.

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