Asymptotic Property of Approximation to x~αsgnx by Newman Type Operators

The approximation of |x| by rational functions is a classical rational problem.This paper deals with the rational approximation of the function x~αsgnx, which equals |x| ifα=1.We construct a Newman type operator r_n(x) and show {|x~αsgnx-r_n(x)|}～Cn~(-(α/2)e-(2nα)~(1/2)), where C is a constant depending onα.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.     

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO ｜x｜ AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）     

Exponential sums involving Maass forms

We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.     

Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.     

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.     

$\alpha$-可分解4-圈系统

An m-cycle system of order v and index λ, denoted by m-CS（v,λ）, is a collection of cycles of length m whose edges partition the edges of λKv. An m-CS（v,λ） is α-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a design are m｜λv（v-1）/2,2｜λ（v -1）,m｜αv,α｜λ（v-1）/2. It is shown in this paper that these conditions are also sufficient when m = 4.     

复指数系统在$L_{\alpha}^{p}$空间的不完备性

A necessary and sufficient condition is obtained for the incompleteness of complex exponential system in the weighted Banach space Lαp = {f：∫＋∞∞ ｜f（t）e-α（t）｜pdt ＋∞},where 1 ≤ p ＋∞ and α（t） is a weight on R.     

THE NOTE ON MATRIX-VALUED RATIONAL INTERPOLATION

In [3], a kind of matrix-valued rational interpolants （MRIs） in the form of Rn（x） = M（x）/D（x） with the divisibility condition D（x） ｜ ｜｜M（x）｜｜^2, was defined, and the characterization theorem and uniqueness theorem for MRIs were proved. However this divisibility condition is found not necessary in some cases. In this paper, we re- move this restricted condition, define the generalized matrix-valued rational interpolants （GMRIs） and establish the characterization theorem and uniqueness theorem for GMRIs. One can see that the characterization theorem and uniqueness theorem for MRIs are the special cases of those for GMRIs. Moreover, by defining a kind of inner product, we succeed in unifying the Samelson inverses for a vector and a matrix.     

A class of oscillatory singular integrals on triebel-lizorkin spaces

The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω（x＇ ）∈Llog＋L（S^n-1 ）, the Fp^a,q（R^n） boundedness of the above operator is obtained. Meanwhile ,when Ω（x） satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 （R^n）.     

A globally uniform asymptotic expansion of the hermite polynomials

In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn（√2n＋1α）to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme （SIAM J.Math.Anal.,（1994）,25：304-321）.A new estimate for the remainder is given.     

Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter,II

We establish the existence of positive bound state solutions for the singular quasilinear Schrodinger equation iψ/t=-div（ρ（｜ψ｜^2） ψ）＋ω（｜ψ｜62）ψ-λρ（｜ψ｜^2）ψ,x∈Ω,t〉0,where ω（τ^2）τ→∞ as τ→ 0 and, λ 〉 0is a parameter and Ω is a ball in ^RN. This problem is studied in connection with the following quasilinear eigenvalue problem with Dirichlet boundary condition -div（ρ（｜ ψ｜^2） ψ）=λ1ρ（｜ψ｜^2）ψ=λ1ρ（｜ψ｜^2）ψ,x∈Ω.Indeed, we establish the existence of solutions for the above Schrodinger equation when A belongs to a certain neighborhood of the first eigenvahie λ1 of this eigenvalue problem. The main feature of this paper is that the nonlinearity ω｜ψ｜^2 is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter A combined with the nonlinear nonhomogeneous term div （ρ（｜ ψ｜^2） ψ） which leads us to treat this prob- lem in an appropriate Orlicz space. The proofs are based on various techniques related to variational methods and implicit function theorem.     

PERTURBED PERIODIC SOLUTIONFOR BOUSSINESQ EQUATION

We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.     

Asymptotic Approximations of Jacobi Functions and Their Zeros with Error Bounds

The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variable x（t）, we＇ve got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions φu^（α,β）（t）（α 〉 -1） as μ→∞. Besides, an accurate approximation with error bounds is also constructed correspondingly for the zeros tμ,s of φu^（α,β）（t）（α≥ 0） as μ→∞, uniformly with respect to s = 1, 2,....     

上密度无限制的指数多项式逼近

We take a new approach to obtaining necessary and sufficient conditions for the incompleteness of exponential polynomials in L p α ,where L p α is the weighted Banach space of complex continuous functions f defined on the real axis R satisfying ＋∞ ∞ ｜f（t）｜ p e -α（t） dt） 1/p ,1 p ∞,and α（t） is a nonnegative continuous function defined on the real axis R.In this paper,the upper density of the sequence which forms the exponential polynomials is not required to be finite.In the study of weighted polynomial approximation,consideration of the case is new.     

On an Anisotropic Equation with Critical Exponent and Non-Standard Growth Condition

Using variational methods, we prove the existence of a nontrivial weak solution for the problem
{-∑i=1^Nδxi（｜δxiu｜pi-2δxiu）=λα（x）｜u｜q（x）-2u＋｜u｜p＊-2u,in Ω,
u=0 inδΩ,
where Ω R^N（N≥3） is a bounded domain with smooth boundary δΩ,2≤pi〈N,i=1,N,q：Ω→（1,p＊）is a continuous function, p＊ =N/∑i=1^N 1/pi-1 is the critical exponent for this class of problem, and λ is a parameter.     

关于斜McCoy环

For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.