Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.     

Sign-nonsingular matrices and even cycles in directed graphs

A sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (a_ij) such that at least one of a_ij and a_ij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and a_ij=0 iff a_ji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects.     

Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

For a graph matrix M, the Hoffman limit value H(M) is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of M(H_n), where the graph H_n is obtained by attaching a pendant edge to the cycle C_n-1 of length n-1. In spectral graph theory, M is usually either the adjacency matrix A or the Laplacian matrix L or the signless Laplacian matrix Q. The exact values of H(A) and H(L) were first determined by Hoffman and Guo, respectively. Since H_n is bipartite for odd n, we have H(Q)=H(L). All graphs whose A-index is not greater than H(A) were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed H(Q). The results obtained are determinant to describe all graphs whose L-index is not greater then H(L). This is done precisely in Wang et al. (in press) [21].     

The Banach-Lie algebra of multiplication operators on a JBW-triple

The Banach-Lie algebra L(A) of multiplication operators on the JB^∗-triple A is introduced and it is shown that the hermitian part Lh(A) of L(A) is a unital GM-space the base of the dual cone in the dual GL-space ^∗(Lh(A)) of which is affine isomorphic and weak^∗-homeomorphic to the state space of L(A). In the case in which A is a JBW^∗-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space Lh(A) satisfy

0?D(u,u)+D(v,v)?idA,     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Q-matrices and spherical geometry

A real n × n matrix M is a Q-matrix if the linear complementarity problem w ? Mz=q, w ? 0, z ? 0, w^tz=0 has a solution for all real n-vectors q. M is nondegenerate if all its principal minors are nonzero. Spherical geometry is applied to the problem of characterizing nondegenerate Q-matrices. The stability of 3 × 3 nondegenerate Q-matrices and a generalization of the partitioning property of P-matrices are rather easily proved using spherical geometry. It is also proved that the set of 4 × 4 nondegenerate Q-matrices is not open.     

Some representations of solutions of elliptic equations

The Dirichlet problem for the region of the plane inside closed smooth curve C for second-order elliptic equations is considered. It is shown that under certain circumstances the solution u can be written uniquely in the form u(P) = ∝_cF(P, Q) g(Q) ds_Q, where F(P, Q) is the fundamental solution of the elliptic equation, and g?L² if the boundary value function f is absolutely continuous with square integrable derivative (f?W); and u(P) = p(F(P, ·)) where p is a unique bounded linear functional on W if f?L². These representations are valid in the exterior of C also. As special cases with slight modifications, the exterior Dirichlet problems for the Helmholtz and Laplace equations are mentioned.It is shown also that if kernel F(P′, Q), with P′ and Q on C, has a complete set of eigenfunctions {ψ_k(P′)} then u(P) can be expanded in a series of their extensions {ψ_k(P)}, where ψ_k(P) = λ_k ∝_cF(P, Q) ψ_k(Q) ds_Q.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).     

Existence theory and Q-matrix characterization for the generalized linear complementarity problem: Revisited

We provide conditions under which a vertical block matrix is a Q-matrix if one or all representative sub-matrices are Q-matrices and vice versa. It is also shown, by means of counterexamples, that Eq. (3) of [A.A. Ebiefung, Existence theory and Q-matrix characterization for the generalized linear complementarity problem, Linear Algebra Appl. 223/224 (1995) 155-169] is incorrect.     

A companion matrix analogue for orthogonal polynomials

Given a set of orthogonal polynomials {P_i(x)}, it is shown that associated with a polynomial a(x)=∑a_ip_i(x) there is a matrix A which possesses several of the properties of the usual companion form matrix C. An alternative and possibly preferable form A' is also suggested. A similarity transformation between A [orA'] and C is given. If b(x) is another polynomial then the matrix b(A) [or b(A')] has properties like those of b(C), relating to the greatest common divisor of a(x) and b(x).     

Notes on the products of the lower topology and Lawson topology on posets

In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω(S×T)=Ω(S)×Ω(T) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ(S) or σ(T) being continuous, then Λ(S×T)=Λ(S)×Λ(T) iff S and T satisfy property K, where for a poset L, Ω(L) means the lower topological space, Λ(L) means the Lawson topological space, and L is said to satisfy property K if for any x∈L, there exist a Scott open U and a finite F⊆L with x∈U⊆↑F.     

Simple image set of linear mappings in a max-min algebra

For a given linear mapping, determined by a square matrix A in a max-min algebra, the set S_A consisting of all vectors with a unique pre-image (in short: the simple image set of A) is considered. It is shown that if the matrix A is generally trapezoidal, then the closure of S_A is a subset of the set of all eigenvectors of A. In the general case, there is a permutation π, such that the closure of S_A is a subset of the set of all eigenvectors permuted by π. The simple image set of the matrix square and the topological aspects of the problem are also described.     

Some Structure Theorems for Atomistic Algebraic Lattices

We prove that any atomistic algebraic lattice is a direct product of subdirectly irreducible lattices iff its congruence lattice is an atomic Stone lattice. We define on the set A(L) of all atoms of an atomistic algebraic lattice L a relation R as follows: for a, b A(L), (a, b) R ? θ(0, a) ∧ θ(0, b) ≠ ?_{Con L} . We prove that Con L is a Stone lattice iff R is transitive and we give a characterization of Cen (L) using R. We also give a characterization of weakly modular atomistic algebraic lattices.     

Sign determinacy of M-matrix minors

We show that if A is an M-matrix for which the length of the longest simple cycle in its associated undirected graph G(A) is at most 3, then every minor of A has determined sign (nonnegative or nonpositive), independent of the magnitudes of the matrix entries. Consequently, if A and B are M-matrices such that G(A) and G(B) are subgraphs of an undirected graph with longest simple cycle at most 3, then all principal minors of AB are nonnegative.     

On sesquilinear forms over fields with involution

Let k be a field of characteristic ≠2 with an involution σ. A matrix A is split if there is a change of variables Q such that (Q^σ)^TAQ consists of two complementary diagonal blocks. We classify all matrices that do not split. As a consequence we obtain a new proof for the following result. Given a square matrix A there is a matrix S such that (S^σ)^TAS=A^T and S^σS=I.     

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ?3, then T(A) = λQAQ^t + Q(A)I_n for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.