• Title/Summary/Keyword: stochastic matrices

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ON DOUBLY STOCHASTIC ${\kappa}-POTENT$ MATRICES AND REGULAR MATRICES

  • Pyo, Sung-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.401-409
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    • 2000
  • In this paper, we determine the structure of ${\kappa}-potent$ elements and regular elements of the semigroup ${\Omega}_n$of doubly stochastic matrices of order n. In connection with this, we find the structure of the matrices X satisfying the equation AXA = A. From these, we determine a condition of a doubly stochastic matrix A whose Moore-Penrose generalized is also a doubly stochastic matrix.

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MINIMUM PERMANENTS ON DOUBLY STOCHASTIC MATRICES WITH PRESCRIBED ZEROS

  • Song, Seok-Zun
    • Honam Mathematical Journal
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    • v.35 no.2
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    • pp.211-223
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    • 2013
  • We consider permanent function on the faces of the polytope of certain doubly stochastic matrices, whose nonzero entries coincide with those of fully indecomposable square (0, 1)-matrices containing identity submatrix. We determine the minimum permanents and minimizing matrices on the given faces of the polytope using the contraction method.

INFINITESIMALLY GENERATED STOCHASTIC TOTALLY POSITIVE MATRICES

  • Chon, In-Heung
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.269-273
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    • 1997
  • We show that each element in the semigroup $S_n$ of all $n \times n$ non-singular stochastic totally positive matrices is generated by the infinitesimal elements of $S_n$, which form a cone consisting of all $n \times n$ Jacobi intensity matrices.

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CONVEX POLYTOPES OF GENERALIZED DOUBLY STOCHASTIC MATRICES

  • Cho, Soo-Jin;Nam, Yun-Sun
    • Communications of the Korean Mathematical Society
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    • v.16 no.4
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    • pp.679-690
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    • 2001
  • Doubly stochastic matrices are n$\times$n nonnegative ma-trices whose row and column sums are all 1. Convex polytope $\Omega$$_{n}$ of doubly stochastic matrices and more generally (R,S), so called transportation polytopes, are important since they form the domains for the transportation problems. A theorem by Birkhoff classifies the extremal matrices of , $\Omega$$_{n}$ and extremal matrices of transporta-tion polytopes (R,S) were all classified combinatorially. In this article, we consider signed version of $\Omega$$_{n}$ and (R.S), obtain signed Birkhoff theorem; we define a new class of convex polytopes (R,S), calculate their dimensions, and classify their extremal matrices, Moreover, we suggest an algorithm to express a matrix in (R,S) as a convex combination of txtremal matrices. We also give an example that a polytope of signed matrices is used as a domain for a decision problem. In this context of finite reflection(Coxeter) group theory, our generalization may also be considered as a generalization from type $A_{*}$ n/ to type B$_{n}$ D$_{n}$. n/.

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PERMANENTS OF DOUBLY STOCHASTIC FERRERS MATRICES

  • Hwang, Suk-Geun;Pyo, Sung-Soo
    • Journal of the Korean Mathematical Society
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    • v.36 no.5
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    • pp.1009-1020
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    • 1999
  • The minimum permanent and the set of minimizing matrices over the face of the polytope n of all doubly stochastic matrices of order n determined by any staircase matrix was determined in [4] in terms of some parameter called frame. A staircase matrix can be described very simply as a Ferrers matrix by its row sum vector. In this paper, some simple exposition of the permanent minimization problem over the faces determined by Ferrers matrices of the polytope of n are presented in terms of row sum vectors along with simple proofs.

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ON TRIANGULAR STOCHASTIC MATRICES

  • Chon, In-Heung
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.629-639
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    • 1994
  • Loewner ([13]) showend the semigroup of the real non-singular totally positive matrices is generated by itsinfinitesimal elements, that is,s the set of all Jacobi matrices with non-negative off-diagonal elements. In general, a semigroup is not completely recreated from the set of its infinitesimal elements. We extend Loewner's work and show that the semi-group of all invertible upper (or lower) triangular stochastic matrices is generated by the set of its infinitesimal elements.

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FERMAT'S EQUATION OVER 2-BY-2 MATRICES

  • Chien, Mao-Ting;Meng, Jie
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.609-616
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    • 2021
  • We study the solvability of the Fermat's matrix equation in some classes of 2-by-2 matrices. We prove the Fermat's matrix equation has infinitely many solutions in a set of 2-by-2 positive semidefinite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational matrices and stochastic quadratic field matrices.

HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS

  • Im, Bokhee;Smith, Jonathan D.H.
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.371-384
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    • 2022
  • 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.

MINIMUM PERMANENTS OF DOUBLY STOCHASTIC MATRICES WITH k DIAGONAL p×p BLOCK SUBMATRICES

  • Lee, Eun-Young
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.199-211
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    • 2004
  • For positive integers $\kappa$ and p$\geq$3, let(equation omitted) where $J_{p}$ is the p${\times}$p matrix whose entries are all 1. Then, we determine the minimum permanents and minimizing matrices over (1) the face of $\Omega$(D) and (2) the face of $\Omega$($D^{*}$), where (equation omitted).