• 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.

• Korean Journal of Mathematics
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• v.11 no.2
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• pp.155-160
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• 2003

In this paper we shall study the limit of the doubly stochastic matrices obtained from Boolean regular matrices.

• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.423-432
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• 1998

Let p, q be integers such that 2 $\leq$ p, q $\leq$ n, and let $D_{p, q}$ denote the matrix obtained from $I_{n}$, the identity matrix of order n, by replacing each of the first p columns by an all 1's vector and by replacing each of the first two rows and each of the last q-2 rows by an all 1's vector. In this paper the permanent minimization problem over the face, determined by the matrix $D_{p, q}$, of the polytope of all n $\times$ n doubly stochastic matrices is treated.d.

• 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).

• 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/.

• 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 Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n 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 Ω_n 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.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.857-867
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• 1995

Throughout this paper, let $M_{mn}(R)$ be the set of all $m \times n$ real matrices, and let $R^n$ the set of all real row vectors with n components.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.825-839
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• 1997

The permanent function on certain faces of the polytope of doubly stochastic matrices are studied. These faces are shown to be barycentric, and the minimum values of permanent are determined.