Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.     

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and ^∗〈A^∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ^∗〈A^∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(^∗〈A^∗A〉) of ^∗〈A^∗A〉 are obtained, and a characterization of Zt(^∗〈A^∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(^∗〈A^∗A〉) are also answered.     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

Polynomial least squares fitting in the Bernstein basis

The problem of polynomial least squares fitting in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.     

Combinatorial structures in loops I. Elements of the decomposition theory

Difference sets have been extensively studied in groups, principally in Abelian groups. Here we extend the notion of a difference set to loops. This entails considering the class of 〈υ, k〉 systems and the special subclasses of 〈υ, k, λ〉 principal block partial designs (PBPDs) and 〈υ, k, λ〉 designs. By means of a certain permutation matrix decomposition of the incidence matrices of a system and its complement, we can isomorphically identify an abstract 〈υ, k〉 system with a corresponding system in a loop. Special properties of this decomposition correspond to special algebraic properties of the loop. Here we investigate the situation when some or all of the elements of the loop are right inversive. We identify certain classes of 〈υ, k, λ〉 designs, including skew-Hadamard designs and finite projective planes, with designs and difference sets in right inverse property loops and prove a universal existence theorem for 〈υ, k, λ〉 PBPDs and corresponding difference sets in such loops.     

On Smital properties

Let A be an algebra and J⊂A an ideal of subsets of a group 〈X,+〉 with an invariant topology τ. We say that a triple 〈A,J,τ〉 has the Smital property if, for any set A∈A?J and any set D dense in τ, the set c(A+D) belongs to J. In the paper we compare this property and similar ones with the well-known Steinhaus type properties. We consider several weak and strong versions of Smital properties.     

A theorem of the transversal theory for matroids of finite character

Let M = (S, I) be a matroid of finite character on the infinite set S. Let $\text{A}\text{= 〈A}{}_1\text{:i ∈ I〉}$ be any system of subsets of S each having finite rank and let $\text{B}\text{= 〈B}{}_1\text{: j ∈ J〉}$ be a finite system of sets of arbitrary rank. Necessary and sufficient conditions are given for the system $\text{A}$ ? $\text{B}$ to have an independent system of distinct representatives.     

Wijsman convergence: A survey

A net 〈A _λ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=lim_λ d(x, A). Interest in this convergence notion originates from the seminal work of R. Wijsman, who showed in finite dimensions that the conjugate map for proper lower semicontinuous convex functions preserves convergence in this sense, where functions are identified with their epigraphs. In this paper, we review the attempts over the last 25 years to produce infinite-dimensional extensions of Wijsman's theorem, and we look closely at the topology of Wijsman convergence in an arbitrary metric space as well. Special emphasis is given to the developments of the past five years, and several new limiting counterexamples are presented.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

On Buzyakova's example; quotients of the Cantor trees

Let T be the Cantor tree and let A be a subset of the ωth level of T (= Cantor set C). Buzyakova considered the quotient space TAT^′ obtained from T×2 by identifying two points 〈a,0〉 and 〈a,1〉 for each a∈A to construct an example of a non-submetrizable space of countable extent with a Gδ-diagonal. We prove that the space TAT^′ is submetrizable if and only if C?A is an Fσ-set in C with the Euclidean topology. This improves Buzyakova's Lemma.     

Hopf algebras constructed by the FRT-construction

Given any bialgebra A and a braiding product 〈|〉 on A, a bialgebra U_〈|〉 was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”, Comm. Algebra 19 (1991) 3295-3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97-110]. In the present paper it is proved that when U_〈|〉 is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra.     

Stabilization of unbounded bilinear control systems in Hilbert space

We consider bilinear control systems of the form y^′(t)=Ay(t)+u(t)By(t) where A generates a strongly continuous semigroup of contraction (etA)_t?0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. We suppose that this system is unbounded in the sense that the linear operator B is unbounded from the state Y into itself. Tacking into account eventual control saturation, we study the problem of stabilization by (possibly nonquadratic) feedback of the form u(t)=−f(〈By(t),y(t)〉). Applications to the heat equation is considered.     

On the product of vector spaces in a commutative field extension

Let K⊂L be a commutative field extension. Given K-subspaces A,B of L, we consider the subspace 〈AB〉 spanned by the product set . If dimKA=r and dimKB=s, how small can the dimension of 〈AB〉 be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on dimK〈AB〉 turns out, in this case, to be provided by the numerical function

     

A gradient method for the matrix eigenvalue problemAx=λBx

LetA andB ben×m matrices. A gradient method for the minimization of the functionalF(x)=‖Ax?(〈Ax, Bx〉/〈Bx, Bx〉)Bx‖² is developed. The minima ofF are the eigenvectors of the eigenproblemAx=λBx. The concept of a non-defective eigenvalue for this generalized eigenvalue problem is developed. It is then shown that geometric convergence is attained for non-defective eigenvalues. A convergence rate analysis is given where it is shown that the rapidity of convergence of the gradient method to an eigenvalue λ depends on the degree of non-defectiveness of λ and the singular values ofA?λB.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

Generalized symmetric accelerated over relaxation method for solving absolute value complementarity problems

In this paper, we suggest and analyze a symmetric accelerated over relaxation (SAOR) method for absolute complementarity problems of finding x ∈ R ⁿ , such that x ≥ 0, Ax ? |x| ? b ≥ 0, 〈x, Ax ? |x| ? b〉 = 0, where A ∈ R ^n×n and b ∈ R ⁿ . We discuss the convergence of SAOR method when the system matrix A is an L-matrix. Several examples are given to illustrate the implementation and efficiency of the method. The results proved in this paper may stimulate further research in this fascinating and interesting field.     

Inverses of tridiagonal Toeplitz and periodic matrices with applications to mechanics

The linear algebraic equation Ax = b with tridiagonal coefficient matrix of A is solved by the analytical matrix inversion. An explicit formula is known if A is a Toeplitz matrix. New formulas are presented for the following cases: (1) A is of Toeplitz type except that A(1, 1) and A(n, n) are different from the remaining diagonal elements. (2) A is p-periodic (p > 1), by which is meant that in each of the three bands of A a group of p elements is periodically repeated. (3) The tridiagonal matrix A is composed of periodic submatrices of different periods. In cases (2) and(3) the problem of matrix inversion is reduced to a second-order difference equation with periodic coefficients. The solution is based on Floquet's theorem. It is shown that for p = 1 the formulae found for periodic matrices reduce to special forms valid for Toeplitz matrices. The results are applied to problems of elastostatics and of vibration theory.     

A characterization of linearly semisimple groups

Let G = SpecA be an affine K-group scheme and à = {w ∈ A*: dim _K A*·w < ∞, dim _K w· A* < ∞}. Let 〈?,?〉: A* × Ã → K, 〈w, \(\tilde w\)〉:=tr(w~w), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.