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.     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Periodicity in a Nonlinear Predator-prey System with State Dependent Delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear state dependent delays predator-prey system where a_i(t),c_j(t),d_i(t) are continuous positive periodic functions with periodic ω>0, b_1(t),b_2(t) are continuous periodic functions with periodic ωand ∫_0~ωbi(t)dt>0. T_i,σ_j, p_i (i=1,2,…,n, j=1, 2,…,m) are continuous and ω-periodic with respect to their first arguments, respectively, α_i, β_j,γ_i(i=1,2,…,n, j=1,2, …, m) are positive constants.     

Results on Local Times of a Class of Multiparameter Gaussian Processes

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.     

Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

In this paper, we consider the following second order three-point boundary value problem u″（t）＋a（t）f（u（t））=0,0〈t〈1,u（0）-u（1）=0,u＇（0）-u＇（1）=u（1/2）,where a ： （0, 1） → [0, ∞） is symmetric on （0, 1） and may be singular at t = 0 and t = 1, f ： [0, ∞） → [O, ∞） is continuous. By using Krasnoselskii＇s fixed point theorem ia a cone, we get some existence results of positive solutions for the problem. The associated Green＇s function for the three-point boundary value problem is also given.     

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.     

Strong Converse Inequality for Left Gamma Quasi-Interpolants

The rate of convergence for the Gamma operators cannot be faster than O(1/n). In order to obtain much faster convergence, quasi-interpolants in the sense of Sablonniere are considered. For the first time in the theory of quasi-interpolants, the strong converse inequality is solved in sup-norm with the K-functional K_λ~α(f, t~(2r)) (0≤λ≤1, 0<α< 2r).     

Exponential Attractor for a Nonlinear Boussinesq Equation

This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H0^2（0, 1） × L^2（0, 1）. The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H0^3（0, 1） × H0^1（0, 1）.     

Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales

Let T be a time scale such that 0, T ∈ T. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales where α ∈ Ctd（（O,T,[0,∞））,f∈ Ckd（[0,∞）×[0,∞））,β,γ ∈[0,∞）,ξi ∈（0,p（T）.b,ai∈ （0,∞） （for i = 1,..., m - 2） are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b 〉 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation （T = R） and difference equation （T = Z）.     

Existence of Positive Solutions for a Class of Sigular Boundary Value Problem of Fourth Order

Some results concerning existence of positive solutions for the singular boundary value problems u^(4)(t)=f(t,u(t)) t∈（0,1) u(0)=u(1)=0 u‘(0)=u‘(1)=0 have been given, where f(t, x) may be singular at t = 0, 1.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Fundamental Solution of the Anisotropic Porous Medium Equation

We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1（u^mi）xixi in R^n × （O,∞）, where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n（2 ＋ ∑i^n=1 mi）.     

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

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.     

A Nontrivial Product of Filtration s ＋ 5 in the Stable Homotopy of Spheres

In this paper, some groups Ext A^s.t （Zp, Zp） with specialized s and t are first computed by the May spectrM sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s＋5 in the stable homotopy groups of spheres πpnq＋（s＋3）pq＋（s＋1）q-5S which is represented （up to a nonzero scalar） by β＋2bohh∈ExtA^s＋5,P^nq＋（n＋3）pq＋（n＋1）q＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 5 is a prime number, n ≥3, 0≤ s 〈 p - 3, q = 2（p - 1）.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

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]}