Properties of Hadamard product of inverse M‐matrices

This paper concerns with the properties of Hadamard product of inverse M‐matrices. Structures of tridiagonal inverse M‐matrices and Hessenberg inverse M‐matrices are analysed. It is proved that the product A ○ A^T satisfies Willoughby's necessary conditions for being an inverse M‐matrix when A is an irreducible inverse M‐matrix. It is also proved that when A is either a Hessenberg inverse M‐matrix or a tridiagonal inverse M‐matrix then A ○ A^T is an inverse M‐matrix. Based on these results, the conjecture that A ○ A^T is an inverse M‐matrix when A is an inverse M‐matrix is made. Unfortunately, the conjecture is not true. Copyright © 2004 John Wiley Sons, Ltd.     

Primitive non-powerful sign pattern matrices with base 2

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M ^k ?=?J _#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M)?=?2, the minimum number of non-zero entries of M is 5n???8 or 5n???7 depending on whether n is even or odd.     

A bound on the scrambling index of a primitive matrix using Boolean rank

The scrambling index of an n×n primitive matrix A is the smallest positive integer k such that A^k(A^t)^k=J, where A^t denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M=AB for some m×b Boolean matrix A and b×n Boolean matrix B. In this paper, we give an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M). Furthermore we characterize all primitive matrices that achieve the upper bound.     

Moore-Penrose Inverse of a Gram Matrix and Its Nonnegativity

Let M=[A  a] be a matrix of order m×n, where A∈ℝ ^m×(n−1) and a∈ℝ ^m is an m×1 vector. In this article, we derive a formula for the Moore-Penrose inverse of M ^* M and obtain sufficient conditions for its nonnegativity. The results presented here generalize the ones known earlier. The authors thank the anonymous referee for suggestions on an earlier version that have resulted in an improved presentation.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.     

Some <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces defined by Orlicz functions

In this article we introduce the sequence spaces cI(M),c0I(M),mI(M) and m0I(M) using the Orlicz function M.We study some of the properties like solid,symmetric,sequence algebra,etc and prove some inclusion relations.     

Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

Groups of Congruences and Restriction Matrices

Let be a domain of the Euclidean space R ^m sent onto itself by a finite group G of congruences. In this paper we first define M elementary restriction matrices related to the group G and to a system of irreducible matrix representations of G. We then describe a general procedure to generate M restriction matrices for any finite-dimensional space V() of real functions defined on , when V() is invariant with respect to G. The number M depends only on the group G. Restriction matrices for the space V() have a block structure and all blocks can be obtained as from an elementary restriction matrix. Restriction matrices related to V() define a decomposition of V() as the sum of M subspaces. Finally, owing to restriction matrices, we propose a result of decomposition for linear systems. Several examples are presented.This revised version was published online in October 2005 with corrections to the Cover Date.     

Automatic differentiation: Reduced gradient and reduced Hessian matrix

This paper is concerned with automatic differentiation methods for computing the reduced gradient M ^t G and the reduced Hessian matrix M ^t HM. Hereby G is the gradient and H is the Hessian matrix of a real function F of n variables, and M is a matrix with n rows and k columns where kn. The reduced quantities are of particular interest in constrained optimization with objective function F. Two automatic differentiation methods are described, a standard method that produces G and H as intermediate results, and an economical method that takes a shortcut directly to the reduced quantites. The two methods are compared on the basis of the reqired computing time and storage. It is shown that the costs for the economical method are less than (k ²+3k+2)/(n ²+3n+2) times the expenses for the standard method.     

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

Trigonometric approximation in reflexive Orlicz spaces

The Lipschitz classes Lip(α, M) , 0 α≤ 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f ∈ Lip (α, M) is estimated by n-α .     

Some dimension-free features of vector-valued martingales

Summary Given any local maringaleM inR ^d orl ², there exists a local martingaleN inR ², such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR ² which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|² and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144     

On Finite Complete Presentations and Exact Decompositions of Semigroups

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation.

It is also proved that a semigroup M ⁰[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.     

Square-zero derivations of matrix algebras over commutative rings

Let R be a commutative ring with identity, M _n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M _n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x ² = 0. We call such maps as square-zero derivations.     

Strong Stability of the Batch Arrival Queueing Systems

In this work, we use the strong stability method to study the batch arrival queue after a perturbation of the batch size distribution. We show that, under some hypotheses, the characteristics of the batch arrival queueing system M ^X /M/1 may be approximated by the correspondent characteristics of the system M ^Geo /M/1.

After clarifying the conditions of approximation, we obtain stability inequalities with an exact computation of constants.     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

Let M be an incidence matrix for a projective plane of order n. The eigenvalues of M are calculated in the Desarguesian case and a standard form for M is obtained under the hypothesis that the plane admits a (P,L)-transitivity G, |G| = n. The study of M is reduced to a principal submatrix A which is an incidence matrix for n ² lines of an associated affine plane. In this case, A is a generalized Hadamard matrix of order n for the Cayley permutation representation R(G). Under these conditions it is shown that G is a 2-group and n = 2^r when the eigenvalues of A are real. If G is abelian, the characteristic polynomial |xI – A| is the product of the n polynomials |x – (A)|, a linear character of G. This formula is used to prove n is a prime power under natural conditions on A and spectrum(A). It is conjectured that |xI – A| x ⁿ² mod p for each prime divisor p of n and the truth of the conjecture is shown to imply n = |G| is a prime power.     

Microlocal properties of scattering matrices

We consider the scattering theory for a pair of operators H₀ and H = H₀ + V on L²(M, m), where M is a Riemannian manifold, H₀ is a multiplication operator on M, and V is a pseudodifferential operator of order ? μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.     

Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.