OnQ-matrices

In a recent paper [1], Aganagic and Cottle have established a constructive characterization for aP ₀-matrix to be aQ-matrix. Among the principal results in this paper, we show that the same characterization holds for anL-matrix as well, and that the symmetric copositive-plusQ-matrices are precisely those which are strictly copositive.     

OnQ-matrices

Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ Q , is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive.     

Q-matrices and boundedness of solutions to linear complementarity problems

This paper is concerned with the existence and boundedness of the solutions to the linear complementarity problemw=Mz+q,w0,z0,w ^T z=0, for eachq ⁿ . It has been previously established that, ifM is copositive plus, then the solution set is nonempty and bounded for eachq ⁿ iffM is aQ-matrix. This result is shown to be valid also forL ₂-matrices,P ₀-matrices, nonnegative matrices, andZ-matrices.     

A note on sufficient conditions forQ 0 andQ 0 ⋂P 0 matrices

In this note we settle an open problem posed by Al-Khayyal on a condition being sufficient for a matrix to belong to the class ofQ ₀-matrices. The answer is in the affirmative and we further relax the condition and obtain a sufficient condition forQ ₀-matrices. The results yield a class of matrices for which the linear complementarity problems can be solved as simple linear programs.     

A note onQ-matrices

This paper concerns three classes of matrices that are relevant to the linear complementarity problem. We prove that within the class ofP ₀-matrices, theQ-matrices are precisely the regular matrices.Research supported by Department of Energy, Contract EY-76-S-03-0326 PA # 18.     

On a decomposition of conditionally positive-semidefinite matrices

A symmetric matrix C is said to be copositive if its associated quadratic form is nonnegative on the positive orthant. Recently it has been shown that a quadratic form x'Qx is positive for all x that satisfy more general linear constraints of the form Ax?0, x≠0 iff Q can be decomposed as a sum Q=A'CA+S, with Cstrictly copositive and S positive definite. However, if x'Qx is merely nonnegative subject to the constraints Ax?0, it does not follow that Q admits such a decomposition with C copositive and S positive semidefinite. In this paper we give a characterization of those matrices A for which such a decomposition is always possible.     

Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems

We consider semidefinite monotone linear complementarity problems (SDLCP) in the space ⁿ of real symmetric n×n-matrices equipped with the cone ⁿ₊ of all symmetric positive semidefinite matrices. One may define weighted (using any Mⁿ₊₊ as weight) infeasible interior point paths by replacing the standard condition XY=rI, r>0, (that defines the usual central path) by (XY+YX)/2=rM. Under some mild assumptions (the most stringent is the existence of some strictly complementary solution of (SDLCP)), these paths have a limit as r0, and they depend analytically on all path parameters (such as r and M), even at the limit point r=0.Mathematics Subject Classification (1991): 90C33, 65K05     

A Filtration on the Chow Groups of a Complex Projective Variety

Let X/ _C be a projective algebraic manifold, and further let CH ^k (X) _Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F ^l}_{l 0} on CH ^k (X) _Q of Bloch-Beilinson type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that F ^k+1 = 0.     

On co-positive,semi-monotoneQ-matrices

In this paper we consider not necessarily symmetric co-positive as well as semi-monotoneQ-matrices and give a set of sufficient conditions for such matrices to beR ₀-matrices. We give several examples to show the sharpness of our results. Construction of these examples is based on the following elementary proposition: IfA is a square matrix of ordern whose first two rows are identical and bothA ₁₁ andA ₂₂ areQ-matrices whereA _ii stands for the principal submatrix ofA obtained by deleting rowi and columni fromA, thenA is aQ-matrix.Dedicated to our colleague and friend B. Ramachandran on his sixtieth birthday.Corresponding author.     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

Spectral and Computational Analysis of Block Toeplitz Matrices Having Nonnegative Definite Matrix-Valued Generating Functions

It is well known that the generating function f L ¹([–, ], ) of a class of Hermitian Toeplitz matrices A _n(f) _n describes very precisely the spectrum of each matrix of the class. In this paper we consider n × n Hermitian block Toeplitz matrices with m × m blocks generated by a Hermitian matrix-valued generating function f L ¹([–, ], C ^m×m ). We extend to this case some classical results by Grenander and Szegö holding when m = 1 and we generalize the Toeplitz preconditioning technique introduced in the scalar case by R. H. Chan and F. Di Benedetto, G. Fiorentino and S. Serra. Finally, concerning the spectra of the preconditioned matrices, some asymptotic distribution properties are demonstrated and, in particular, a Szegö-style theorem is proved. A few numerical experiments performed at the end of the paper confirm the correctness of the theoretical analysis.     

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

Differentiability of functions: approximate,global, and differentiability along curves over non-archimedean fields

This paper is devoted to the study of approximate and global smoothness and smoothness along curves of functions f(x ₁,...,x _m ) of variables x ₁,...,x _m in infinite fields with nontrivial non-Archimedean valuations and relations between them. Theorems on classes of smoothness C ⁿ or of functions with partial difference quotients continuous or bounded uniformly continuous on bounded domains up to order n are investigated. We prove that from f ○ u ∈ C ⁿ (K, K ^l) or f ○ u ∈ (K, K ^l) for each C ^∞ or curve u: K → K ^m it follows that f ∈ C ⁿ (K ^m , K ^l) or f ∈ (K ^m , K ^l), where m ≥ 2. Then the classes of smoothness C ^n,r and and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved. Moreover, the approximate differentiability of functions relative to measures is defined and investigated. Its relations with the Lipschitzian property and almost everywhere differentiability are studied. Non-Archimedean analogs of classical theorems of Kirzsbraun, Rademacher, Stepanoff, and Whitney are formulated and proved, and substantial differences between two cases are found. Finally, theorems about relations between approximate differentiability by all variables and along curves are proved. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

A Taylor–Wiles System for Quaternionic Hecke Algebras

Let &ell >3 be a prime. Fix a regular character of F&^{2 ×} of order &–1, and an integer M prime to &. Let fS ₂(₀(M&²)) be a newform which is supercuspidal of type at &. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor–Wiles system for M, and prove that T is the universal object for a deformation problem (of type at & and semi-stable outside) of the Galois representation ¯ _f over F¯_& associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type for f and the classical one.     

D.C. Versus Copositive Bounds for Standard QP

The standard quadratic program (QPS) is min_xεΔx^TQx, where is the simplex Δ = {x ⩽ 0 ∣ ∑_i=1ⁿ x_i = 1}. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One class of ‘d.c.’ (for ‘difference between convex’) bounds for QPS is based on writing Q=S−T, where S and T are both positive semidefinite, and bounding x^T Sx (convex on Δ) and −x^Tx (concave on Δ) separately. We show that the maximum possible such bound can be obtained by solving a semidefinite programming (SDP) problem. The dual of this SDP problem corresponds to adding a simple constraint to the well-known Shor relaxation of QPS. We show that the max d.c. bound is dominated by another known bound based on a copositive relaxation of QPS, also obtainable via SDP at comparable computational expense. We also discuss extensions of the d.c. bound to more general quadratic programming problems. For the application of QPS to bounding the stability number of a graph, we use a novel formulation of the Lovasz ϑ number to compare ϑ, Schrijver’s ϑ′, and the max d.c. bound.     

On the extreme spectral properties of Toeplitz matrices generated byL 1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue ₁ ⁽ⁿ⁾ of symmetric (Hermitian)n ×n Toeplitz matricesT _n (f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that ₁ ⁽ⁿ⁾ tends to essinff =m _f in the following way: ₁ ⁽ⁿ⁾ –m _f 1/n ^2k . These authors use three assumptions:A1)f –m _f has a zero inx =x ₀ of order 2k.A2)f is continuous and at leastC ^2k in a neighborhood ofx ₀.A3)x =x ₀ is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L ¹[–, ]. In this paper we further extend this theory to the case of a functionf L ¹[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m _f is the only parameter which characterizes the rate of convergence of ₁ ⁽ⁿ⁾ tom _f .