Interfacial strength of joints between fibers and dispersedly filled epoxy matrices

The concentration dependences of adhesive strength are investigated for fiber/dispersedly filled epoxy matrix systems. The measurements were carried out using an improved model of adhesiometer under normal conditions at a constant rate of loading. It is shown that the adhesive strength as a function of filler concentration has a maximum, which is more or less pronounced. The location of the maximum depends on the nature of filler and particle geometry. The increase in the adhesive strength at the maximum reaches 20–30% in comparison with that for the unfilled epoxy matrix. Since the interfacial strength between steel wire and all the mineral powders investigated is zero, the growth in the adhesive strength upon introduction of a finely divided filler in polymer binders is rather un expected. The possible reasons for the phenomenon observed are discussed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 3–14, January–February, 2007.     

Clean matrices and unit-regular matrices

An element in a ring R is said to be clean (respectively unit-regular) if it is the sum (respectively product) of an idempotent element and an invertible element. If all elements in R are unit-regular, it is known that all elements in R are clean. In this note, we show that a single unit-regular element in a ring need not be clean. More generally, a criterion is given for a matrix to be clean in a matrix ring M₂(K) over any commutative ring K. For K=Z, this criterion shows, for instance, that the unit-regular matrix is not clean. Also, this turns out to be the “smallest” such example.     

Embedding matrices in Hadamard matrices

It is shown that if A is any n×n matrix of zeros and ones, and if k is the smallest number not less than n which is the order of an Hadamard matrix, then A is a submatrix of an Hadamard matrix of order k².     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.     

Characterizations of EP matrices and weighted-EP matrices

A square complex matrix A is said to be EP if A and its conjugate transpose A^∗ have the same range. In this paper, we first collect a group of known characterizations of EP matrix, and give some new characterizations of EP matrices. Then, we define weighted-EP matrix, and present a wealth of characterizations for weighted-EP matrix through various rank formulas for matrices and their generalized inverses.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences).     

关于径向矩阵与谱矩阵

§1.引言与记号 设A∈C~(s×n),则称 ‖A‖=‖AX‖/‖X‖ 为A的谱模(谱范数),其中‖X‖表示向量X∈C~(n×1)的Euclid范数。即当X=(x_1,…,x_n)~(?)时,‖X‖=(XX)~1/2=sum from i=1 to n(|X_1|~2)~1/2;‖AX‖为向量AX的Euclid范数。 如众周知,我们有如下结论: 引理 1[1]、设A、B∈C~(n×n),则谱模满足范数的三个条件: 1>.恒正性:‖A‖≥0且‖A‖=0 A=0; 2>.齐次性:若α∈C,则‖αA‖=|α|·‖A‖; 3>.三角不等式:‖A+B‖≤‖A‖+‖B‖。     

Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

.     

Embedding rectangular matrices in hadamard matrices

It is shown that any m×n±1 matrix may be embedded in a Hadamard matrix of order kl, where k and l are the least orders greater than or equal to m and nrespectively in which Hadamard matrices exist.     

Factoring matrices into the product of two matrices

Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix. The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of these ideas are outlined. Several examples on factoring are presented. AMS subject classification (2000)  15A23, 65F30     

Extending subpermutation matrices in regular classes of matrices

We determine precise conditions in order that every n × n matrix of 0's and 1's with exactly k 1's in each row and column has the property that each subpermutation matrix of rank d can be extended to a permutation matrix. An application is given to completing partial latin squares.     

Mixtures of order matrices and generalized order matrices

This paper deals with matrix representations of linear orders, mixtures of order matrices and the non-integral solutions of the linear systems defining them.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

Approximation of high-dimensional kernel matrices by multilevel circulant matrices

Kernels are important in developing a variety of numerical methods, such as approximation, interpolation, neural networks, machine learning and meshless methods for solving engineering problems. A common problem of these kernel-based methods is to calculate inverses of kernel matrices generated by a kernel function and a set of points. Due to the denseness of these matrices, finding their inverses is computationally costly. To overcome this difficulty, we introduce in this paper an approximation of the kernel matrices by appropriate multilevel circulant matrices so that the fast Fourier transform can be applied to reduce the computational cost. Convergence analysis for the proposed approximation is established based on certain decay properties of the kernels.