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．     

More on Bounding Introspection in Modal Nonmonotonic Logics

Abstract By we denote the set of all propositional formulas. Let be the set of all clauses. Define . In Sec. 2 of this paper we prove that for normal modal logics , the notions of -expansions and -expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of -expansions for I and -expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of -expansions for I is the same as that of -expansions for I. The project is supported by NSFC     

Analysis of an Age Structured SEIRS Epidemic Model with Varying Total Population Size and Vaccination

This article focuses on the study of an age structured SEIRS epidemic model with a vaccinationprogram when the total population size is not kept at constant.We first give the explicit expression of thereproduction number R(ψ,(?)) in the presence of vaccine ((?) is the exponent of growth of total population),andshow that the infection-free steady state is linearly stable if R(ψ,(?))<1 and unstable if R(ψ,(?))>1,then weapply the theoretical results to vaccination policies to determine the optimal age or ages at which an individualshould be vaccinated.It is shown that the optimal strategy can be either one- or two-age strategies.     

A Representation of the Lorentz Spin Group and Its Application

For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj （m）, （j = 0, 1,..., m - 1） which satisfy γj （m）γk （m） ＋γk （m）γj （m） = 2ηjk （m）I[m/2], where （ηjk （m）） 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation （m） of the Lorentz Spin group is known. For m≥ 6, we prove that （i） when m = 2n, （n ≥ 3）, （m） is the group generated by the set of matrices {T｜T=1/√ξ（（I＋k） 0 ＋ 0 I-K） （ U 0 0 U）, （ii） when m = 2n ＋ 1 （n≥ 3）, （m） is generated by the set of matrices {T｜T=1/√ξ（I -k^- k I）U,U∈ （m-1）,ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj（m-2）＋a^（m-2） In],K^-=i[m-3∑j=0 a^j γj（m-2）-a^（m-2） In]}     

Asymptotic Properties of the Maximum Likelihood Estimate in Generalized Linear Models with Stochastic Regressors

For generalized linear models （GLM）, in case the regressors are stochastic and have different distributions, the asymptotic properties of the maximum likelihood estimate （MLE） β^n of the parameters are studied. Under reasonable conditions, we prove the weak, strong consistency and asymptotic normality of β^n     

Characterization of Compact Support of Fourier Transform for Orthonormal Wavelets of L^2（R^d）

Let{ψμ} be an orthonormal wavelet of L^2（R^d） and the support of a whole of its Fourier transform be Uμsupp{ψμ}=Пi=1^d[Ai, Di]-Пi=1^d（Bi, Ci）, Ai≤Bi≤Ci≤Di. Under the weakest condition that each │ψμ│, is continuous for ω ∈ δ（Пi=1^d[Ai, Di]）, a characterization of the above support of a whole is given.     

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.     

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.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

Counting SL2（F2^s） Representations of Torus Knot Groups

In this paper,we count the number of SL2(F2^s)-representations of torus knot groups up to a conjugacy.For the finite field F2^s with character 2,the counting method is similar to that in out previous work[1].Explicit formulae of the effective counting are given in this paper.Twisted Alexander polynomials related to those reprsentations are discussed.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Strongly Singular Integral Operators on Weighted Hardy Space

In this paper, we obtain that a strongly singular integral operator is bounded on space for 1 < p < ∞. We also obtain that a strongly singular integral operator is a bounded operator from to for some weight w and 0 < p ≤ 1. And by an atomic decomposition, we obtain that a strongly singular integral operator is a bounded operator on for some w and 0 < p ≤ 1. Supported by National 973 Program of China (Grant No. 19990751)     

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

Exponential Stability of Traveling Pulse Solutions of a Singularly Perturbed System of Integral Differential Equations Arising From Excitatory Neuronal Networks

We establish the exponential stability of fast traveling pulse solutions to nonlinear singularly per-turbed systems of integral differential equations arising from neuronal networks.It has been proved that expo-nential stability of these orbits is equivalent to linear stability.Let (?) be the linear differential operator obtainedby linearizing the nonlinear system about its fast pulse,and let σ((?)) be the spectrum of (?).The linearizedstability criterion says that if max{Reλ:λ∈σ((?)),λ≠0}(?)-D,for some positive constant D,and λ=0 is asimple eigenvalue of (?)(ε),then the stability follows immediately (see [13] and [37]).Therefore,to establish theexponential stability of the fast pulse,it suffices to investigate the spectrum of the operator (?).It is relativelyeasy to find the continuous spectrum,but it is very difficult to find the isolated spectrum.The real part ofthe continuous spectrum has a uniformly negative upper bound,hence it causes no threat to the stability.Itremains to see if the isolated spectrum is safe.Eigenvalue functions (see [14] and [35,36]) have been a powerful tool to study the isolated spectrum of the as-sociated linear differential operators because the zeros of the eigenvalue functions coincide with the eigenvaluesof the operators.There have been some known methods to define eigenvalue functions for nonlinear systems ofreaction diffusion equations and for nonlinear dispersive wave equations.But for integral differential equations,we have to use different ideas to construct eigenvalue functions.We will use the method of variation of param-eters to construct the eigenvalue functions in the complex plane C.By analyzing the eigenvalue functions,wefind that there are no nonzero eigenvalues of (?) in {λ∈C:Reλ(?)-D} for the fast traveling pulse.Moreoverλ=0 is simple.This implies that the exponential stability of the fast orbits is true.     

Singularly perturbed nonlinear elliptic problems on manifolds

Let be a connected compact smooth Riemannian manifold of dimension with or without smooth boundary We consider the following singularly perturbed nonlinear elliptic problem on

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

Let be a unit sphere of the d–dimensional Euclidean space ℝ ^d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E _j (f,H ^p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H ^p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for

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.     

Invariant Mean—Value Property and M—Harmonicity in the Unit Ball of R^n

In 1993,Ahern,Flores and Rudin showed that,if f is integrable over the unit ball BC^n of C^n and satisfies∫BC^nfoφdv=f（φ（0)) for every φ∈Aut(BC^n),then f is M-harmonic if and only if n≤11.The present paper is about an analogous question in the context of the unit ball Bn of R^n as well as in the weighted setting.