ON APPROXIMATION PROPERTIES OF NON-CONVOLUTION TYPE NONLINEAR INTEGRAL OPERATORS-Dedicated to Professor Paul L. Butzer on his 80th Birthday

In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]     

Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Existence of Solutions to a Singular Initial Value Problem

Under the sign assumptions we investigate the global existence of solutions of the initial value problem x＇ =f（t, x, x＇）, x（0） = A, where the scalar function f（t, x,p） may be singular at x = A.     

LOWER BOUNDS FOR SUP＋INF AND SUP＊INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

We give two results about Harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.     

The commutant of analytic Toeplitz operators on Bergman space

In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.     

Strong entropy for system of isentropic gas dynamics

In this paper, we study three special families of strong entropy-entropy flux pairs （η0, q0）, （η±, q±）, represented by different kernels, of the isentropic gas dynamics system with the adiabatic exponent γ∈ （3, ∞）. Through the perturbation technique through the perturbation technique, we proved, we proved the H^-1 compactness of ηit ＋ qix, i = 1, 2, 3 with respect to the perturbation solutions given by the Cauchy problem （6） and （7）, where （ηi, qi） are suitable linear combinations of （η0, q0）, （η±, q±）.     

Realizing enveloping algebras via varieties of modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.     

Linearly McCoy Rings and Their Generalizations

A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.     

群逆的仿射组合表示

In this paper, we first give two equalities in the operation of determinant. Using the expression of group inverse with full-rank factorization Ag = F（GF）^-2G and the Cramer rule of the nonsingular linear system Ax = b, we present a new method to prove the representation of group inverse with arlene combination
Ag=∑（I,J）∈N（A） 1/υ^2det（A）IJ ajd AJI.
A numerical example is given to demonstrate that the formula is efficient.     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

Properties on solutions of some q-difference equations

We consider the properties on solutions of some q-difference equations of the form ∑ n j=0 aj（z）f（qj z）=an＋1（z）, where a0（z）,..., an＋1（z） are meromorphic functions, a0（z）an（z） ≠ 0 and q ∈ C such that 0 〈 ｜q｜ ≤ 1. We give estimates on the upper bound for the length of the gap in the power series of entire solutions of （＊） when the coefficients a0（z）,..., an＋1（z） are polynomials and 0 〈 ｜q｜ 〈 1. For some special cases, we give estimates of growth of f（z）. And we also show that the case 0 〈 ｜q｜ 〈 1 is different from the case ｜q｜=1.     

On Gliding Hump Properties of Matrix Domains

In this note, we establish several results concerning the gliding hump properties of matrix domains. In order to discuss F-WGHP, we introduce the UAK-property and find that this sort of property has close relationship with F-WGHP. In the course of discussing F-WGHP and WGHP of （C0）cn, we discuss the F-WGHP and WGHP of the almost-null sequence space f0.     

一类奇异三阶三点边值问题的正解

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.     

Sz\`{a}sz-Mirakjan 算子拟中插式的点态逼近等价定理

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. A.T. Diallo investigated some approximation properties of Szasz-Mirakjan Quasi-Interpolants, but he obtained only direct theorem with Ditzian-Totik modulus wφ^2r （f, t）. In this paper, we extend Diallo＇s result and solve completely the characterization on the rate of approximation by the method of quasi-interpolants to functions f ∈ CB[0, ∞） by making use of the unified modulus wφ^2r（f, t） （0≤λ≤ 1）.     

The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrdinger Equations

This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.