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http://dx.doi.org/10.4134/CKMS.c210127

HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS  

Im, Bokhee (Department of Mathematics Chonnam National University)
Smith, Jonathan D.H. (Department of Mathematics Iowa State University)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.2, 2022 , pp. 371-384 More about this Journal
Abstract
Fix an integer n ≥ 1. Then the simplex Πn, Birkhoff polytope Ωn, and Latin square polytope Λn each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ωn are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.
Keywords
Birkhoff polytope; stochastic matrix; quasigroup; Latin square; approximate quasigroup; generalized doubly stochastic; tristochastic tensor;
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