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.     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

Reznichenko families of trees and their applications

Examples of Talagrand, Gul'ko and Corson compacta resulting from Reznichenko families of trees are presented. The K_σδ property for weakly -analytic Banach spaces with an unconditional basis is proved.     

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}.     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

Gradient estimates and Jackson’s theorem in spaces related to measures

Jackson’s theorem is established in a new kind of holomorphic function space Q_μ related to measures in any starlike circular domain in . Particularly, the result covers many spaces including BMOA, Q_p, Q_K, and F(p,q,s) spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem.     

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.     

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.     

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 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.     

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)

     

On partial sums of certain meromophic -valent functions

In this paper, we study the ratio of meromorphic p-valent functions in the punctured disk U^*={z:0<|z|<1} of the form to its sequence of partial sums of the form . Also, we determine sharp lower bounds for and .     

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)

     

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].     

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.     

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.     

Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows

Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ_*>0 such that the problem has at least two positive solutions if λ(0,λ_*), has at least one positive solution if λ=λ_*<+∞ and has no positive solution if λ>λ_*. To prove the results, we prove a norm on W^1,p(x)(Ω) without the part of ||_L^p(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.     

Weakly sequentially continuous differentiable mappings

It is well known that a (linear) operator between Banach spaces is completely continuous if and only if its adjoint takes bounded subsets of Y^* into uniformly completely continuous subsets, often called (L)-subsets, of X^*. We give similar results for differentiable mappings. More precisely, if UX is an open convex subset, let be a differentiable mapping whose derivative is uniformly continuous on U-bounded subsets. We prove that f takes weak Cauchy U-bounded sequences into convergent sequences if and only if f^′ takes Rosenthal U-bounded subsets of U into uniformly completely continuous subsets of . As a consequence, we extend a result of P. Hájek and answer a question raised by R. Deville and E. Matheron. We derive differentiable characterizations of Banach spaces not containing ℓ₁ and of Banach spaces without the Schur property containing a copy of ℓ₁. Analogous results are given for differentiable mappings taking weakly convergent U-bounded sequences into convergent sequences. Finally, we show that if X has the hereditary Dunford–Pettis property, then every differentiable function as above is locally weakly sequentially continuous.     

Linear information versus function evaluations for -approximation

We study algorithms for the approximation of functions, the error is measured in an L₂ norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n^-b.Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether p>b is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Woźniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that implies . Here we prove that the case and b=0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case is quite different, it is still open whether b=p always holds in that case.