SIGN PATTERNS REQUIRING ALL DISTINCT EIGENVALUES

A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs indicated by the corresponding entries of A.We say that a sign pattern matrix A requires a matrix property P if every real matrix in Q(A)has the property P.A matrix with all distinct eigenvalues has many nice properties     

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.     

A New Class AOR Preconditioner for $L$-Matrices

Hadjidimos(1978) proposed a classical accelerated overrelaxation(AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant, L-matrices, and consistently orders matrices. Several preconditioned AOR methods have been proposed to solve system of linear equations Ax = b, where A ∈ R~(n×n) is an L-matrix. In this work, we introduce a new class preconditioners for solving linear systems and give a comparison result and some convergence result for this class of preconditioners. Numerical results for corresponding preconditioned GMRES methods are given to illustrate the theoretical results.     

Stability Analysis Of Block ɵ-Methods For Neutral Multidelay-Differential-Algebraic Equations

This paper is concerned with the stability analysis of Block ɵ -met hods forsolving neutral multidelay-di®erential- algebraic equations. We shown that if the coefficient matrices of neutral multidelay-differential-algebraic equations satisfying somestability conditions and ɵ ϵ [ 1 2; 1], then the numerical solution of Block µ-methods forsolving neutral multidelay-differential-algebraic equations is asymptotically stable.     

$\mathcal{H}$-Stability of Runge-Kutta Methods with Variable Stepsize for System of Pantograph Equations

This paper deals with H-stability of Runge-Kutta methods with variable stepsize for the system of pantograph equations. It is shown that both Runge-Kutta methods with nonsingular matrix coefficient A and stiffly accurate Runge-Kutta methods are H-stable if and only if the modulus of stability function at infinity is less than 1.     

Sensitivity and regionally proximal relation in minimal systems

A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n 1)-sensitive(n≥2)is determined.     

Painlev′e Prop erty and Integrability of Polynomial Dynamical Systems

The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has P...     

SOME REMARKS ON COMPLETELY POSITIVE MATRICES AND COMPLETELY POSITIVE GRAPHS

The completely positive matrices are important in the study of block designs arizing incombinatorial analysis (see [11])and are related to copositive matrices,which have appli-cations in control theory and in multimobical programming (see [12]).An n×n matrix A is said to be completely positive if A can be factored as A=BB~T forsome n×m real matrix B with nonnegative entries for some m<∞.If A is a completelypositive matrix,then the smallest number of columns in a nonnegative matrix B such that     

THE ESTIMATE OF THE RANK FOR REGRESSION COEFFICIENT MATRIX IN A MEDIAN REGRESSION MODEL

It is discussed to infer the rank of regression coefficient matrix in a multivariate linear regression model. If the zero median vector is unique and the design matrices satisfy some weaker conditions, it is derived that the estimators of the rank of regression coefficient matrix under the minimum distance criterion by using model selection method is strongly consistent.     

COMBINATIVE PRECONDITIONERS OF MODIFIED INCOMPLETE CHOLESKY FACTORIZATION AND SHERMAN-MORRISON-WOODBURY UPDATE FOR SELF-ADJOINT ELLIPTIC DIRICHLET-PERIODIC BOUNDARY VALUE PROBLEMS

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems,by making use of the specialstructure of the coefficient matrix we present a class of combinative preconditioners whichare technical combinations of modified incomplete Cholesky factorizations and Sherman-Morrison-Woodbury update.Theoretical analyses show that the condition numbers of thepreconditioned matrices can be reduced to(?)(h~(-1)),one order smaller than the conditionnumber(?)(h~(-2))of the original matrix.Numerical implementations show that the resultingpreconditioned conjugate gradient methods are feasible,robust and efficient for solving thisclass of linear systems.     

Almost periodic homogeneous linear difference systems without almost periodic solutions

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

Calculating the Lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix

A stochastic dynamic system of second order is considered. The system evolution is described by a dynamic equation with a stochastic transition matrix, which is linear in the idempotent algebra with operations of maximum and addition. It is assumed that some entries of the matrix are zero constants and all other entries are mutually independent and exponentially distributed. The problem considered is the computation of the Lyapunov exponent, which is defined as the average asymptotic rate of growth of the state vector of the system. The known results related to this problem are limited to systems whose matrices have zero off-diagonal entries. In the cases of matrices with a zero row, zero diagonal entries, or only one zero entry, the Lyapunov exponent is calculated using an approach which is based on constructing and analyzing a certain sequence of one-dimensional distribution functions. The value of the Lyapunov exponent is calculated as the average value of a random variable determined by the limiting distribution of this sequence.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

Recognizing Balanceable Matrices

A 0/±1 matrix is balanced if it does not contain a square submatrix with exactly two nonzero entries per row and per column in which the sum of all entries is 2 modulo 4. A 0/1 matrix is balanceable if its nonzero entries can be signed ±1 so that the resulting matrix is balanced. A signing algorithm due to Camion shows that the problems of recognizing balanced 0/±1 matrices and balanceable 0/1 matrices are equivalent. Conforti, Cornuéjols, Kapoor and Vušković gave an algorithm to test if a 0/±1 matrix is balanced. Truemper has characterized balanceable 0/1 matrices in terms of forbidden submatrices. In this paper we give an algorithm that explicitly finds one of these forbidden submatrices or shows that none exists. Received: October 2004     

Intrinsic products and factorizations of matrices

We say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. We present several examples, together with important applications. These applications include companion matrices and sign-nonsingular matrices.     

Singular, nonsingular, and bounded rank completions of ACI-matrices

An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries.     

On the finiteness property for rational matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.     

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ A X + B Recursion with Triangular Matrices

Journal of Theoretical Probability - We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the...     

Structural properties and recognition of restricted and strongly unimodular matrices

A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation. Partially supported by a New York University Research Challenge Fund Grant.