一类时滞Logistic差分方程的全局吸引性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｏｎａｕｔｏｎｏｍｏｕｓｄｅｌａｙｌｏｇｉｓｔｉｃｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｙｎ =ｐｎｙｎ( 1 - ｙτ(ｎ) )　 ,ｎ =0 ,1 ,2 ,...,( 1 1 )ｗｈｅｒｅｐｎ ∞ｎ =0 ｉｓａｓｅｑｕｅｎｃｅｏｆｐｏｓｉｔｉｖｅｒｅａｌｎｕｍｂｅｒｓ ,τ(ｎ) ∞ｎ =0 ｉｓａｎｏｎｄｅｃｒｅａｓｉｎｇｓｅｑｕｅｎｃｅｏｆｉｎｔｅｇｅｒｓ,τ(ｎ) <ｎａｎｄｌｉｍｎ→∞τ(ｎ) =∞ ,Δｙｎ=ｙｎ +1- ｙｎ.Ｍｏｔｉｖａｔｅｄｂｙｐｌａｕｓｉｂｌｅａｐｐｌｉｃａｔｉｏｎｓ…     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Oscillation of Second Order Nonlinear Delay Damped Difference Equations

Some new oscillation criteria for nonlinear delay difference equation with damping △^2xn＋pn△xn＋F（n,x（n-τ,△x（n-σ）=0,n=0,1,2,…,（＊）are given. Our results partially solve the open problem posed in [Math. Bohemica, 125 （2000）, 421- 430]. Also, we will establish some new oscillation criteria for special cases of （＊）, which improve some of the well-known results in the literature.     

Equi-Gaussian curvature folding

In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, f: M → P _n , of an orientable surface M onto a polygon P _n we prove that

A note on the Ramanujan τ-function

Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that

Oscillation for systems of nonlinear neutral type parabolic partial functional differential equations

This paper discusses the oscillation of solutions for systems of nonlinear neutral type parabolic partial fuctional differential equations of the form     

Uniform estimates of positive solutions to quasi-linear differential equations

One considers the differential inequality

Oscillation of even order nonlinear functional differential equations with damping

This paper is concerned with a class of even order nonlinear damped differential equations

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

On the generalized Cauchy-Riemann systems

The factorization of the Laplacian by means of first order systems and of second order operators was considered by several authors (see, e.g [2],[3],[4]). In the paper the definition of Cauchy-Riemann system (CR-system) of ordern is given by their symbols. We prove that ifD ⁽ⁿ⁾ is the symbol andD ^{(s, n)} is the sign-matrix of CR-system, then

The Yang-Mills fields on the Minkowski space

Let the coordinatex=(x ⁰,x ¹,x ²,x ³) of the Minkowski spaceM ⁴ be arranged into a matrix

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

离散Lasota-Wazewska模型的全局吸引性

Consider the discrete Lasota-Wazewska model Nn 1-Nn=-μNn pe^-rNn-k,n=0,1,2,……where μ∈(0,1),r,p∈(0,∞) and k∈N. A sufficient condition for all positive solutions of ( * )to be attracted to its equilibrium N^* is obtained. It improves correspondent result obtained by Chen and Yu in 1999.     

On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems

For γ≥1 we consider the solution u=u（x） of the Dirichlet boundary value problem Δu ＋ u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u（x）=p（δ（x））[1＋A（x）（log 1/δ（x）^-6], where p（r） ≈ r r√2 log（1/r） near r = 0,δ（x） denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A（x） is a bounded function. For 1 〈 γ 〈 3 we find u（x）=（γ＋1/√2（γ-1）δ（x））^2/γ＋[1＋A（x）（δ（x））2γ-1/γ＋1] For γ3= we prove that u（x）=（2δ（x））^1/2[1＋A（x）δ（x）log 1/δ（x）]     

Convergence to Diffusion Waves for Nonlinear Evolution Equations with Ellipticity and Damping, and with Different End States

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects： {ψt=-（1-α）ψ-θx＋αψxx, θt=-（1-α）θ＋νψx＋（ψθ）x＋αθxx（E） with initial data （ψ,θ）（x,0）=（ψ0（x）,θ0（x））→（ψ±,θ±）as x→±∞ where α and ν are positive constants such that α 〈 1, ν 〈 4α（1 - α）. Under the assumption that ｜ψ＋ - ψ-｜ ＋ ｜θ＋ - θ-｜ is sufficiently small, we show the global existence of the solutions to Cauchy problem （E） and （I） if the initial data is a small perturbation. And the decay rates of the solutions with exponential rates also are obtained. The analysis is based on the energy method.     

Periodic Solutions of a Class of Second Order Functional Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of the following second order nonlinear differential equation:ax(t) bx^2k-1(t) cx^2k-1(t) g(x(t-T1),x(t-T2) ) = p(t) = p(t 2π)Our approach is based on the continuation theorem of the coincidence degree, and the priori estimate of periodic solutions.     

Identities Involving Coefficients of Automorphic L-Functions

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2, ℤ) and let

Star number and star arboricity of a complete multigraph

Let G be a multigraph. The star number s(G) of G is the minimum number of stars needed to decompose the edges of G. The star arboricity sa(G) of G is the minimum number of star forests needed to decompose the edges of G. As usual λK _n denote the λ-fold complete graph on n vertices (i.e., the multigraph on n vertices such that there are λ edges between every pair of vertices). In this paper, we prove that for n ⩾ 2