Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

Nonnegative iterations with asymptotically constant coefficients

Let A_k,k=0,1,2,…, be a sequence of real nonsingular n×n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties Au=λu and u=1. Under further additional conditions on the spectrum of A, it is shown that if x₀≠0 and the iterates

are nonnegative, then converges to u and converges to λ as k→∞.     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

-Linked planar graphs

A graph G is (m,n)-linked if for any two disjoint subsets R,BV(G) with |R|m and |B|n, G has two disjoint connected subgraphs containing R and B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if G is 4-connected and maximal.     

Norm of weighted composition operators from -Bloch spaces to weighted-type spaces

We calculate in an elegant way operator norm of the weighted composition operator from the α-Bloch space, with α(0,∞){1}, to a weighted-type space on the unit ball. This result can be regarded as a complement to our recent result regarding the same problem for the case α=1.     

Succinct data structures for flexible text retrieval systems

We propose succinct data structures for text retrieval systems supporting document listing queries and ranking queries based on the tf*idf (term frequency times inverse document frequency) scores of documents. Traditional data structures for these problems support queries only for some predetermined keywords. Recently Muthukrishnan proposed a data structure for document listing queries for arbitrary patterns at the cost of data structure size. For computing the tf*idf scores there has been no efficient data structures for arbitrary patterns.Our new data structures support these queries using small space. The space is only 2/ times the size of compressed documents plus 10n bits for a document collection of length n, for any 0<1. This is much smaller than the previous O(nlogn) bit data structures. Query time is O(m+qlogn) for listing and computing tf*idf scores for all q documents containing a given pattern of length m. Our data structures are flexible in a sense that they support queries for arbitrary patterns.     

-stability of discrete hybrid systems via variational Lyapunov method

The aim of this paper is to give some results on ₀-stability for discrete hybrid systems via a variational cone-valued Lyapunov method.     

Hyperbolic topology of normed linear spaces

In a previous paper [H. Tsuiki, Y. Hattori, Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, Theoret. Comput. Sci. 405 (2008) 198–205], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces L_p(Ω,Σ,μ) on which the hyperbolic topology induced by the norm _p coincides with the norm topology. We show the following:

(1) The hyperbolic topology and the norm topology coincide for 1<p<∞.

(2) They coincide on L₁(Ω,Σ,μ) if and only if μ(Ω)=0 or Ω has a finite partition by atoms.

(3) They coincide on L_∞(Ω,Σ,μ) if and only if μ(Ω)=0 or there is an atom in Σ.

Asymptotic structure and the existence of noncompact operators between Banach spaces

We investigate the problem of the existence of a noncompact operator T:X₀X→Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for and , let T_ξ,η be the linear map which sends each x_i to y_i. We prove that if for some then every T:X₀X→Y is compact. If for n=2 all such maps have norm 1 we show the existence of a noncompact T:X₀X→Y.     

Über ein Turánsches problem für radiale, positiv definite Funktionen, II

Turán's problem is to determine the greatest possible value of the integral for positive definite functions f(x), , supported in a given convex centrally symmetric body , . We consider the problem for positive definite functions of the form f(x)=(x₁), , with supported in [0,π], extending results of our first paper from two to arbitrary dimensions.Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, ℓ-1 summability of the inverse Fourier integral on . Their investigations gave rise to a pair of transformations (h_d,m_d) on which they studied using special functions, in particular spherical Bessel functions.To study the d-dimensional Turán problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turán's problem.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.