Upper eigenvalue bounds for pencils of matrices

Eigenvalue bounds are obtained for pencils of matrices A ? vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A. The upper bounds obtained depend on the “connectivity” structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations.     

Optimal block diagonal scaling of block 2-cyclic matrices

We describe a class of optimal block diagonal scalings (preconditionings) of a symmetric positive definite block 2-cyclic matrix, generalizing a result of Forsythe and Strauss [1] for (point) 2-cyclic matrices.     

Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices

New upper and lower bounds are given for the arithmetic,geometric and harmonic means of a set of positive definite matrices     

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

Lower bounds for small diagonal ramsey numbers

Let p = 4r + 1 be a prime. Let G be the graph on the p points 0, 1,…, p−1 formed by connecting two points with an edge iff their difference is a quadratic residue mod p. Let k be the size of the largest clique contained in G. Then it is well known that the diagonal Ramsey number R₂(k + 1) > p. We show R₂(k + 2) > 2p + 2. We also compute k for all p < 3000.     

关于B-Nekrasov矩阵线性互补问题最优误差界的注记

研究了B-Nekrasov矩阵线性互补问题的含有参数误差界的最优值问题,利用函数的单调性,在_0_(i_1)···_n···_(i_(n-1))≥0且0_n1的情况下,得到了该误差界的最优值.     

Upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices

Upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained. We provide a unified approach that yields or improves upon most of the bounds that have been obtained so far.     

Upper bounds for ruin probabilities under stochastic interest rate and optimal investment strategies

In this paper, we study the upper bounds for ruin probabilities of an insurance company which invests its wealth in a stock and a bond. We assume that the interest rate of the bond is stochastic and it is described by a Cox-Ingersoll-Ross (CIR) model. For the stock price process, we consider both the case of constant volatility (driven by an O-U process) and the case of stochastic volatility (driven by a CIR model). In each case, under certain conditions, we obtain the minimal upper bound for ruin probability as well as the corresponding optimal investment strategy by a pure probabilistic method.     

Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices

Three equivalence relations are considered on the set of n × n matrices with elements in F₀, an abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field. But only multiplication is involved. Thus our formulation in terms of an abelian group with o is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartie graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the group F play a role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F. Thus, our method of reducing propositions concerning the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some n, w or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.     

Upper bounds for Ramsey numbers

The Ramsey number R(G₁,G₂,…,G_k) is the least integer p so that for any k-edge coloring of the complete graph K_p, there is a monochromatic copy of G_i of color i. In this paper, we derive upper bounds of R(G₁,G₂,…,G_k) for certain graphs G_i. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854.     

Upper bounds for harmonious colorings

A harmonious coloring of a simple graph G is a coloring of the vertices such that adjacent vertices receive distinct colors and each pair of colors appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colors in such a coloring. We improve an upper bound on h(G) due to Lee and Mitchem, and give upper bounds for related quantities.     

New permanental bounds for Ferrers matrices

We survey the most recent results on permanental bounds of a nonnegative matrix. Some older bounds are revisited as well. Applying refinements of the arithmetic mean-geometric mean inequality leads to sharp bounds for the permanent of a fully indecomposable Ferrers matrix. In the end, several relevant examples comparing the bounds are discussed.     

Lyapunov diagonal semistability of acyclic matrices

It is shown that an acyclic matrix is Lyapunov diagonally semistable if and only if the matrix has the weak principal submatrix rank property. This result completes the solution of the problem of characterizing the various types of matrix stability for acyclic matrices. Also, those acyclic matrices which have a unique Lyapunov scaling factor are characterized.     

Eigenvalue bounds for block preconditioned matrices

In the symmetric positive definite case, two-sided eigenvalue bounds for block Jacobi scaled matrices and upper eigenvalue bounds for matrices preconditioned with an incomplete block factorization are derived. A quantitative characterization of block matrix partitionings is also suggested, which can be used when analyzing various block preconditioning methods. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 5–41.     

Spectral variation bounds for diagonalisable matrices

Summary This note is related to an earlier paper by Bhatia, Davis, and Kittaneh [4]. For matrices similar to Hermitian, we prove an inequality complementary to the one proved in [4, Theorem 3]. We also disprove a conjecture made in [4] about the norm of a commutator. This work was done when the first author visited the SFB 343 at University of Bielefeld in May and June 1994.