Separation properties forMW-fractals

Corresponding to the irreducible 0–1 matrix (a _ij ) _n×n , take similitude contraction mappingsϕ _ij for eacha _ij =1, ina _ij =1, in R ^d with ratio 0<r _ij <1. There are unique nonempty compact setsF ₁,…,F _n satisfying for each1≤i≤n, F _i. We prove that open set condition holds if and only ifF _i is ans-set for some1≤i≤n, wheres is such that the spectral radius of matrix (r _ij ³ ) _{n x n} is 1. Partly supported by Natural Science Foundation of China, and partly by Natural Science Foundation of Hubei Province     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

On Some Optimal Constructions of Chess Tournament Combinations

An optimal solution for the following “chess tournament” problem is given. Let n, r be positive integers such that r<n. Put N=2n, R=2r+1. Let X_N,R be the set of all ordered pairs (T, A) of matrices of degree N such that T=(t_ij) is symmetric, A=(a_ij) is skew-symmetric, t_ij ∈,{0, 1, 2,…, R), a_ij ∈{0,1,–1}. Moreover, suppose t_ii=a_ii=0 (1?i?N). t_ij = t_ik>0 implies j=k, t_ij=0 is equivalent to a_ij=0, and |a_i1|+|a_i2|+…+|a_iN|=R (1?i?N). Let p(T, A) be the number of i such that 1?i?N and a_i1 + a_i2 + … + a_iN >0. The main result of this note is to show that max p(T, A) for (T, A)∈X_{N, R} is equal to [n(2r+1)/(r+1)], and a pair (T₀, A₀) satisfying p(T₀, A₀)=[n(2r+1)/(r+1)] is also given.     

Positivity Preserving Hadamard Matrix Functions

For every positive real number p that lies between even integers 2(m − 2) and 2(m − 1) we demonstrate a matrix A = [a_ij] of order 2m such that A is positive definite but the matrix with entries |a_ij|^p is not.     

Asymptotic normality of double-indexed linear permutation statistics

The paper provides sufficient conditions for the asymptotic normality of statistics of the form a _ijb_RiRj, wherea _ijandb _ijare real numbers andR _iis a random permutation.     

Symmetric matrices with given row sums

We identify the extreme points of the set of matrices (a_ij) with a_ij ? 0, a_ij = a_ji and ∑_ja_ij = r_i under the condition that ∑r_i is finite. We also give an example to show that the condition that ∑r_i is finite is essential.     

Certain complexes associated to a sequence and a matrix

Let R be a commutative noetherian ring with unit. To a sequencex:=x₁,...,x_n of elements of R and an m-by-n matrix α:=(α_ij) with entries in R we assign a complex D*(x;α), in case that m=n or m=n?1. These complexes will provide us in certain cases with explicit minimal free resolutions of ideals, which are generated by the elements a_i:=∑α_ijx_j and the maximal minors of α.     

Minimum permanents of doubly stochastic matrices with One fixed entry

For n  3 and sufficiently small a  0, the minimum value of the permanent function restricted on n × n doubly stochastic matrices with at least one entry equal to a is obtained. For n = 3, the explicit form of the function p is derived where p(a) = min{per(C):C = (c_ij )?Ω₃ c ₁₁ = a}a? [0, 1].     

Enumeration of rectangular arrays by length and coincidences

Summary Explicit formulas are obtained for the number of p-line arrays of integers (a_ij) (i=1, 2, ..., p; j=1, 2, ..., n) satisfying 1=a_p1=...=a₁₁⩽a_p2⩽...a₁₂⩽...⩽a_pn⩽...⩽a_1n and a_1j⩽j (j=1, 2, ..., n) and having k coincidences. (A coincidence is a column in which a_1j=...=a_pj.) Entrata in Redazione il 20 giugno 1972. Supported in part by NSF grant GP-17031.     

Zur Inversmonotonie diskreter Probleme

Summary This paper describes sufficient conditions for a real square matrixA=(a _ij ) to have a nonnegative inverse. It is not assumed thata _ij 0 forij. We indicate several applications to matricesA that occur in finite-difference and finiteelement methods for boundary-value problems.

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Diese Arbeit enthält Ergebnisse aus der Dissertation des Verfassers, die von Prof. Dr. E. Bohl, Universität Münster, angeregt und unterstützt wurde.     

Simplexes of L-subdivisions of Euclidean spaces

It is shown that necessary and sufficient conditions for a basic simplex of a point lattice in Eⁿ space to be an L-simplex are equivalent to conditions imposed on the coefficients a_ij of the form _i,j=1 ⁿ a_ijx_ix_j– _i=1 ⁿ a_iix_i. namely, that it should assume positive values for all possible integer values of the variables x₁..., x_n (excluding the obvious n+1 cases when the form is equal to 0). These conditions are obtained for n 5.Translated from Matematicheskie Zametkij Vol. 10, No. 6, pp. 659–670, December, 1971.     

On the Determinantal Representation of Quaternary Forms

We prove that a general polynomial form of degree d in 4 variables, over the complex field, can be written as the sum of two determinants of 2 × 2 matrices of forms, with given degree matrix (a _ij ), for any choice of non-negative integers a _ij  ≤ d with a ₁₁ + a ₂₂ = a ₁₂ + a ₂₁ = d.     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.     

Asymptotic behavior of quasi-competitive species defined by Lotka-Volterra dynamics

Competitive systems defined by Lotka-Volterra equations (I) $$\dot x_i = x_i \left( {r_i - \sum\limits_{j = 1}^n {a_{ij} x_j } } \right),i = 1,2,...n,$$ wherer _i>0,a _ij>0, have been extensively studied in the literature. Much attention has been drawn to, among other things, the non-periodic oscillation phenomenon, or May-type trajectory as it is called by some authors, since the discovery of that kind of trajectories in competitive Lotka-Volterra systems made by May and Leonard^[2]. Recently, the same phenomenon was reported to be existing in prey-predator systems. In this paper it is clear that one can expect the appearance of such phenomenon in a broader class of Lotka-Volterra systems, namely quasi-competitive systems (i.e.r _i>0. (a _ij/a_jj)+(a _ji/a_ii)>0 in (I)), which cover both competitive and some prey-predator systems in addition to others. Conditions are established in terms of the parameters of the systems for the existence of stable equilibrium, periodic oscillation and non-periodic oscillation.     

Sequences of orthogonal vectors with a fixed norm

Let A = (a_ij) be an n × m matrix with a_ij ∈ K, a field of characteristic not 2, where Σ_i=1ⁿa_ij² = e, 1 ≤ j ≤ m, and Σ_i=1ⁿa_ija_ij′ = 0 for j ≠ j′. The question then is when is it possible to extend A, by adding columns, to obtain a matrix with orthogonal columns of the same norm. The question is answered for n ? 7 ≤ m ≤ n as well as for more general cases. Complete solutions are given for global and local fields, the answer depending on what congruence class modulo 4 n belongs to and how few squares are needed to sum to e.     

Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n‐queens solutions

In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n‐queens solutions. We prove that when these modular n‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = (a_ij), for all i and j it holds that a_ij + an?i?1,n?j?1 = c for a fixed c. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007     

Hadamard定理在四元数体上的推广

Let K be any skew-field with central field F. A matrix A=(a_ij)n×n over K is called centralized if the characteristic matrix λI-A can be reduced by some elementary transformations into the following diagonal form:such that are all manic polynomials over F. The determinant of a centralized matrix A=(a_ij)n×n may be defined by (1) asfollows: and then the famous theorem of Hadamard can be generalized as in the following: Theorem. If A=(a_ij)n×n is a non-singular centralized matrix over the skewfield of quaternions, thenand the equality sign holds if and only if the columns of A are all muturally orthogonal. This theorem may be proved by the following three lemmas: Lemma 1. If A is a centralized n-rowed square matrix of quaternions, then sois and Lemma 2. For any n-rowed square, matrix A of quaternions, the. following two matrices are always centralized:and Lemma 3. If A is a centralized matrix of quaternions, then we have     

On periodicity of perfect colorings of the infinite hexagonal and triangular grids

A coloring of vertices of a graph G is called r-perfect, if the color structure of each ball of radius r in G depends only on the color of the center of the ball. The parameters of a perfect coloring are given by the matrix A = (a _ij ) _i,j=1 ⁿ , where n is the number of colors and a _ij is the number of vertices of color j in a ball centered at a vertex of color i. We study the periodicity of perfect colorings of the graphs of the infinite hexagonal and triangular grids. We prove that for every 1-perfect coloring of the infinite triangular and every 1- and 2-perfect coloring of the infinite hexagonal grid there exists a periodic perfect coloring with the same matrix. The periodicity of perfect colorings of big radii have been studied earlier.