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.     

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

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.     

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.     

Nonoscillation for a Second Order Linear Delay Differential Equation with Impulses

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delayequation with impulses(r(t)u')'=-p(t)u(t-τ)are obtained in this paper,where p(t)=sum from ∞to n=1 a_n δ(t-t_n),δ(t) is a Dirac δ-unction,and for each n∈N,a_n>0,t_n→∞as n→∞.Furthermore,the boundedness of the solutions is also investigated if the equationis nonoscillatory.An example is given to illustrate the use of the main theorems.     

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     

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

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.     

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.     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

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.     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

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,

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.