Some classes of integral matrices

Integral matrices A for which the system Ax=b will have an integral basic solution are considered. Results for these matrices are presented which parallel results concerning other matrices for which the system has integral solutions.     

Condition number of matrices derived from two classes of integral equations

We investigate some integral equations, i. a. the so-called Kupradze functional equations, where the two variables of the kernel belong to two different point sets. An extensive survey of the literature shows the various applications of these equations. By a discretization of the integral equations they are replaced by systems of linear algebraic equations. The condition number of the corresponding matrices is investigated, analytically and numerically. It is thereby quantitatively found in which way the condition of the matrices deteriorates when the two point sets are moved away from each other.     

Majorization for certain classes of analytic functions using multiplier transformation

In the present paper we investigate the majorization properties for certain classes of multivalent analytic functions defined by multiplier transformation. Moreover, we point out some new or known consequences of our main result.     

Composition of binary integral quadratic forms via integral 2 × 2 matrices and composition of matrix classes

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices AB with irrational eigen values which generate the maximal order. The most general integral similarity between AB is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a "compositum" of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.     

Majorization for certain classes of analytic functions defined by fractional derivatives

Here we investigate a majorization problem involving starlike function of complex order belonging to a certain class defined by means of fractional derivatives. Relevant connections of the main results obtained in this paper with those given by earlier workers on the subject are also pointed out.     

Sums of integral matrices

A study is begun of the behavior of invariant factors when integral matrices are added.     

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.     

Distribution of second-order integral matrices

A refinement of the known results regarding the uniform distribution of the integral solutions of the equation ad – bc=n is obtained in domains of special form. It is shown that the general case can be investigated in the same manner.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 49–55, 1990.     

Some infinite classes of Hadamard matrices

Some results of Williamson [Duke Math. J., 11 1944, Bull. Amer. Math. Soc., 53 1947] and Wallis (J. Combinatorial Theory, 6 1969] are considerably improved to establish that in each case referred to, the same stated condition or conditions, which according to either of the authors give rise to one Hadamard matrix, actually imply the existence of an infinite series of Hadamard matrices. Also proved is the existence of some infinite series of Williamson's matrices, which coupled with the interesting findings of Turyn [J. Combinatorial Theory Ser. A, 16 1974] establish the existence of infinitely many more series of Hadamard matrices than those known so far.     

A result concerning classes of matrices

This paper is concerned with the n-dimensional integral representations of an order in an algebraic number field of degree n, and their composition.     

Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

Some special classes of infinite matrices

In this paper, we present a simple method to construct examples of infinite matrices belonging to different classes of matrix spaces. Moreover, we introduce a scale of matrix spaces which extends the well-known scale of classical Lebesgue spaces.     

Two theorems on integral matrices

The following two results are proved: (1) For a positive definite integral symmetric matrix S of rank (S) < 7 or when rank (S) = 8, S has an odd entry in its diagonal, there is an integral matrix A satisfying AA^t = Sif there is a rational matrix R with RR^t = S (2) Given an integral matrix A of size r×n such that AA^t = mI_r there is then always an integral completion matrix B of size n×n satisfying BB^t = mI_r whenever n-r is less than or equal to 7. This threshold number 7 is the best possible. (Here m, n satisfy the obvious necessary conditions.)     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .