The Average Widths of Sobolev-Wiener Classes and Besov-Wiener Classes

This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively.     

Littlewood-Paley theorem for arbitrary intervals: Weighted estimates

Let 1 < r < 2 and let b is a weight on ℝ such that satisfies the Muckenhoupt condition A_r′/2 (r′ is the exponent conjugate to r). If f_j are functions whose Fourier transforms are supported on mutually disjoint intervals, then

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model ,where yi's are responses, xi = (xi1, xi2,…,xip)' and ti ∈T are known and nonrandom design points, T is a compact set in the real line is an unknown parameter vector, g(·) is an unknown function and {Ei} isa linear process, i.e., random variables with zeromean and variance o2e. Drawing upon B-spline estimation of g(·) and least squares estimation of 0, we construct estimators of the autocovariances of {Ei}- The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {Ei} are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

Homological Connectivity Of Random 2-Complexes

Let Δ_n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δ_n−1 obtained by starting with the full 1-dimensional skeleton of Δ_n−1 and then adding each 2−simplex independently with probability p. Let denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems

Abstract   Let ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < ··· < ξ _m−2 < 1, a _i , b _i ∈ [0,∞) with and . We consider the m-point boundary-value problem

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

Wavelets and Geometric Structure for Function Spaces

Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)     

CONSTRUCTIVE PROPERTIES OF A KIND OF UNIFORMLY ALMOST PERIODIC FUNCTIONS

In this paper,we have discussed constructive properties of a kind of uniformly almost periodic functions, of which the sequence of its Fourier exponents has unique limit point at infinity. \[\begin{gathered} f(x) \sim \sum\limits_{k = - \infty }^\infty {{A_k}} {e^{i{\Lambda _k}x}} \hfill \ {\Lambda _0} = \alpha ,0 < \alpha \leqslant {\Lambda _k} < {\Lambda _{k + 1}}(k = 0,1,2,...) \hfill \ \mathop {\lim }\limits_{k \to \infty } {\Lambda _k} = \infty ,{\Lambda _k} = - {\Lambda _k} \hfill \ |{\Lambda _k}| + |{\Lambda _{ - k}}| > 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (k \ne 0) \hfill \\ \end{gathered} \] Analogons to the approximation theory of periodic functioiis, we get some theorems similar to the Jackson theorem, Bernstein theorem and Zygmund theorem of periodio functions.     

Polynomial Trace Estimates for Semigroups

For semigroups (e ^tA ) _t0 of operators on a Hilbert space, we give conditions guaranteeing trace estimates of the polynomial type 0$$ " align="middle" border="0"> , where denotes the trace class. As an application we present higher order analogues of results due to E.B. Davies, B. Simon and M. van den Berg of the type 0$$ " align="middle" border="0"> , for certain unbounded domains , e.g. spiny urchin domains.     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model

By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition modelwhere ri and r2 are continuous w-periodic functions in R+=[0,∞) with ,ai,ci(i =1,2) are positive continuous w-periodic functions in R+=[0,∞),bi (i = 1,2) is nonnegative continuous w-periodic function in R+=[0,∞), w and T are positive constants. Ki,Lt ∈ C([-T,0], (01 88)) and Ki(s)ds = 1,ds - 1. i = 1,2. Some known results are improved and extended.     

Rigidity Theorem of Hypersurfaces with Constant Scalar Curvature in a Unit Sphere

In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations     

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

Pentavalent vertex-transitive diameter two graphs

We classify the family of pentavalent vertex-transitive graphs Γ with diameter 2. Suppose that the automorphism group of Γ is transitive on the set of ordered distance 2 vertex pairs. Then we show that either Γ is distance-transitive or Γ is one of \(\overline {{C_8}} ,{\kern 1pt} {K_5}\square {K_2},{\kern 1pt} {C_5}\left[ {{K_2}} \right],{\kern 1pt} \overline {2{C_4}} ,{\kern 1pt} or{\kern 1pt} {K_3}\square {K_4}\).     

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

Some generalizations of Knopp's identity*

For integers a, b and n > 0, define

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation