Factorization Index for Completely Positive Graphs

In this paper, we prove that the set of all factorization indices of a completely positive graph has no gaps. In other words, we give an affirmative answer to a question raised by N. Kogan and A. Berman [8] in the case of completely positive graphs. Received December 9, 1997, Accepted May 16, 2002     

图中具有某种性质的子图

设ｇ和ｆ是定义在图Ｇ的顶点集合Ｖ（Ｇ）上的整数值函数且对每个ｘ∈Ｖ（Ｇ）都有０≤ｇ（ｘ）≤ｆ（ｘ）且ｇ（ｘ）和ｆ（ｘ）为偶数。本文证明了：若Ｇ是一个（ｍｇ＋ｋ－１,ｍｆ－ｋ＋１）－图,１≤ｋ≤ｍ,Ｈ是Ｇ中一个给定的有ｋ条边的子图,则Ｇ存在一个子图Ｒ使得Ｒ有一个（ｇ,ｆ）－因子分解与Ｈ正交。     

二元整系数多项式因式分解的一种理论和算法

我们发现可以把二元多项式盾成系数为一元多项式的一元多项式来进行分解，据此，本文建立了二元整系数多项式因式分解的一种理论，提出了一个完整的分解二元整系数多项式的算法。这个算法还能很自然地推广成分解多元整系数多项式的算法。     

A bump triangular dynamic factorization algorithm for the simplex method

A dynamic factorization algorithm is developed which algebraically partitions the basis inverse in such a manner so that the simplex method can be executed from a series of small inverses and the basis itself. This partition is maintained dynamically so that the additional memory required to represent the basis inverse reduces to this series of small inverses for in-core implementations.The algorithm is intended for use in solving general large-scale linear programming problems. This new method of basis representation should permit rather large problems to be solved completely in-core.Preliminary computational experience is presented and comparisons are made with Reid's sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases. The computational experience indicated that a significant reduction in memory requirements can usually be obtained using the dynamic factorization approach with only a slight (up to about 20%) degradation of execution time.This research was supported in part by the Air Force Office of Scientific Research, Air Force System Command, USAF, under AFOSR Contract/Grant Number AFOSR-74-2715.     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.     

An unusual decomposition of a complete 7-partite graph of order 28

A decomposition of the complete 7-partite graph on 28 vertices where each set in the partition has four vertices is given. Several unusual properties of this decomposition are discussed, giving rise to several natural questions.     

Decidability for modules over a group ring - III

ABSTRACT

Representations of simple Jordan superalgebras of Hermitian 3?×?3 matrices over the exceptional simple alternative superalgebras B (1,2) and B (4,2) of characteristic 3 are studied. Every irreducible bimodule over these superalgebras up to isomorphism is either a regular bimodule or its opposite. As corollaries,some analogues of the Kronecker factorization theorem are proved for Jordan superalgebras that contain H₃(B (1,2)) and H₃(B(4,2)).     

关于隐式LU分解算法分解算法求矩阵主子阵逆的注记（英）

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