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.     

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)

     

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.     

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rⁿ, let be the family of all fuzzy sets ofRⁿ, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^ω iff Y is compact; and the space is homeomorphic to {(x_n)Q:sup|x_n|<1} iff Y is non-compact and locally compact.     

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.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

Solvability of the -Laplacian with nonlocal boundary conditions

Rather mild sufficient conditions are provided for the existence of positive solutions of a boundary value problem of the form

which unify several cases discussed in the literature. In order to formulate these conditions one needs to know only properties of the homeomorphism and have information about the level of growth of the response operator F. No metric information concerning the linear operators L₀,L₁ in the boundary conditions is used, except that they are positive and continuous and such that L_j(1)<1 j{0,1}.     

Instance-optimality in probability with an -minimization decoder

Let Φ(ω), ωΩ, be a family of n×N random matrices whose entries _i,j are independent realizations of a symmetric, real random variable η with expectation and variance . Such matrices are used in compressed sensing to encode a vector by y=Φx. The information y holds about x is extracted by using a decoder . The most prominent decoder is the ℓ₁-minimization decoder Δ which gives for a given the element which has minimal ℓ₁-norm among all with Φz=y. This paper is interested in properties of the random family Φ(ω) which guarantee that the vector will with high probability approximate x in to an accuracy comparable with the best k-term error of approximation in for the range kan/log₂(N/n). This means that for the above range of k, for each signal , the vector satisfies

with high probability on the draw of Φ. Here, Σ_k consists of all vectors with at most k nonzero coordinates. The first result of this type was proved by Wojtaszczyk [P. Wojtaszczyk, Stability and instance optimality for Gaussian measurements in compressed sensing, Found. Comput. Math., in press] who showed this property when η is a normalized Gaussian random variable. We extend this property to more general random variables, including the particular case where η is the Bernoulli random variable which takes the values with equal probability. The proofs of our results use geometric mapping properties of such random matrices some of which were recently obtained in [A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523].     

Schurian vector space categories, hereditary algebras and Roiter's norm

We associate a positive real number to any vector space K-category over a field K. Generalizing a result of Nazarova and Roiter, we show that a schurian vector space K-category is representation-finite if and only if is finite and . Such vector space categories are quasilinear, i.e. its indecomposables are simple modules over their endomorphism ring. Recently, Nazarova and Roiter introduced the concept of -faithful poset in order to clarify the structure of critical posets. Their conjecture on the precise form of -faithful posets was established by Zeldich. We generalize these results and characterize -faithful quasilinear vector space K-categories in terms of a class of hereditary algebras H_ρ(D) parametrized by a skew-field D and a rational number ρ1.     

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.     

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.     

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).     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

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.     

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.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

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.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.