Remarks on B（D,λ）-refinability

In this paper, some equivalent versions of B(D,λ)-refinability are given. One of these equivalent versions, is that a space X is B(D, ωo)-refinable if and only if X is strongly quasi-paracompact. As an application of the above result, the author shows that weak θ-refinability is strictly weaker than strong quasi-paracompactness in T4-spaces, which answers a question posed by Jiang. In addition, the author proves that a weak version of B(D,λ) always implies weak θ-refinability for any λ<ω1, and also give a T4, B(D,ωo)-refinable (=strongly quasi-paracompact) space which is not θ-refinable.     

关于θ-加细性

In this paper the preimage problem and hereditarily of θ-refinability are discussed.The main results are as followsi (1) The preimage of weakly δθ-refinable (δθ-refinable, weakly θ-refinable) space under a continuous closed mapping with f^-1(y) Lindelof is weakly δθ-refinable (δθ-refinable, weakly θ-refinable) space; (2) The θ-refinable space with closed G_δ is a hereditarily θ-refinable space.     

度量空间的弱开cs-映象的注记

In this paper, we prove that a space with a compact countable weak base if and only if it is a weak open cs-image of a metric space.     

HANKEL OPERATORS ON HARDY SPACES AND SCHATTEN CLASSES

For a holomorphic function f on the unit ball B~N of C~N, it is proved that the reduced Hankel oporator R_f on Hardy space H~2(B~N) is of Schatten class S_p for p≥1 if and only if f is in a corresponding Sobolev space.     

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).     

COMPLETE ACCUMULATION POINTS AND COMPACTNESS OF FUZZY SETS

Let A be a fuzzy set in a fuzzy topological space (X,τ) and β∈(0, 1]. We denote by ζ_x~a the fuzzy point defined by ζ_x~a(x)=a and ζ_x~a(y)=0 for eaoh y≠x, where a is called the height of ζ_x~a. A subfamily u of τ is called an open β-cover of A if each fuzzy point in A with height β is quasi-coincident with ([1]) some member of u. By a β-subcover of the open β-cover u of A is meant any subfamily of u that is also an open β-cover of A. A is called β-compact in (X, τ) if every open β-cover     

Ball-covering property of Banach spaces that is not preserved under linear isomorphisms

By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.     

Some Characterizations of the Properties ■~∞ and LB_∞

The aim of this paper is to give some properties of the linear topological invariant .Using these results we show that a nuclear Fréchet space F has the property LB_∞ if and only if everyseparately holomorphic function on an open subset U×V of E×F~* has a local Dirichlet representation,where E is a nuclear Fréchet space with the property ~∞ having a basis.     

Every Banach space with a w*-separable dual has a 1+ε-equivalent norm with the ball covering property

A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every ε>0 every Banach space with a w*-separable dual has a 1+ε-equivalent norm with the ball covering property.     

一些超空间的非双Lipschitz 齐次性

A metric space(X, d) is called bi-Lipschitz homogeneous if for any points x, y ∈ X,there exists a self-homeomorphism h of X such that both h and h-1are Lipschitz and h(x) = y.Let 2(X,d)denote the family of all non-empty compact subsets of metric space(X, d) with the Hausdorff metric. In 1985, Hohti proved that 2([0,1],d)is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2([0,1],e)is not bi-Lipschitz homogeneous for an admissible metric e satisfying some conditions. Another is that 2(X,d)is not bi-Lipschitz homogeneous if(X, d) has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space Rm.     

Banach空间中闭算子的多项式的谱分解性

Suppose X is a Banach space. The main results of this paper is as follows: Theorem. A closed operator T has the SDP if and only if for any complex polynomial p(λ) that is not constant, p(T) has the SDP.     

On P-sober Spaces

In this paper,we focus on p-sober spaces and prove that(1) the To space X is p-sober if and only if the Smyth power space of X is p-sober;(2) the space X has a p-sober dcpo model if and only if X is T₁ and p-sober;(3) every non-p-sober T₀ space does not have a p-sobrification;(4) the T₀ space X is sober if and only if X is p-sober and PD.     

On a Non-linear Open Mapping Theorem

Let c be a positive number.A continuous linear map F from a Banachspace X into a Banach space Y is said to be c-open if Y_1 F(X_c)where Y_1denotes the closed unit ball in Y and X_c the closed ball in X with centre at ori-gin and radius c.This is equivalent to require that the equation F_x=y has a so-lution x with ||x||≤||y|| whenever y∈Y. F is c-open for some c>0 if and only ifit is an open map.In this note we concern with a non-linear situation.Thus let     

关于单调亚紧空间

In the first part of this note, we mainly prove that monotone metacompactness is hereditary with respect to closed subspaces and open Fó-subspaces. For a generalized ordered (GO)-space X, we also show that X is monotonically metacompact if and only if its closed linearly ordered extension X* is monotonically metacompact. We also point out that every non-Archimedean space X is monotonically ultraparacompact. In the second part of this note, we give an alternate proof of the result that McAuley space is paracompact and metacompact.     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

On complete submanifolds with parallel mean curvature in negative pinched manifolds

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1).     

BASISITY PROBLEM AND WEIGHTED SHIFT OPERATORS

We investigate a basisity problem in the space ■p A D and in its invariant subspaces. Namely, let W denote a unilateral weighted shift operator acting in the space ■p A D,1 ≤ p ∞, by W zn= λnzn+1, n ≥ 0, with respect to the standard basis zn n≥0. Applying the so-called "discrete Duhamel product" technique, it is proven that for any integer k ≥ 1the sequence wi+nk-1 W | Ei knf n≥0is a basic sequence in Ei := span zi+n: n ≥ 0 equivalent to the basis zi+n n≥0if and only if f i = 0. We also investigate a Banach algebra structure for the subspaces Ei, i ≥ 0.     

A Note on Weighted Comp osition Op erators on the Fo ck Space

Based on a new characterization of bounded and compact weighted composition operators on the Fock space obtained by Le T(Le T. Normal and isometric weighted composition operators on the Fock space. Bull. London. Math. Soc., 2014,46: 847–856), this paper shows that a bounded weighted composition operator on the Fock space is a Fredholm operator if and only if it is an invertible operator, and if and only if it is a nonzero constant multiple of a unitary operator. The result is very different from the corresponding results on the Hardy space and the Bergman space.     

Convergent Sequences in Spaces with Finite Compactness Numbers

§ 1.Introduction　 For a compact Hausdoff space X,Bell and van Mill[1 ] defined the compactness numbercompn(X) of X in the following manner:　 compn(X)≤ k if X has an open subbase(that is a usual subbase;dually,we can de-fine a closed subbase) Usuch that each cover of X by elements of Uhas a subcover of at most k members;　 compn(X) =k if compn(X)≤ k holds but compn(X)≤ k-1 does not hold;　 compn(X) =ω if compn(X) \≤k for all k∈ ω.In particular,X is called supercompact if compn(…     

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.