White Noise Approach Construction of Ф4 4 Quantum to The Fields （Ⅱ）

In this paper a complete proof for the existence of generalized operators satisfying abstract 02 dynamical equations of quantum motions δ^2/δt^2Ф（t, x） ＋ （ △- m^2）Ф（t, x） = -λ ：Ф^3（t, x）, subject to a suitable initial condition, is given under the framework of white noise analysis. Also some important commutation relations related to Ф44 quantum fields are discussed and proved in detail.     

广义算子意义下的量子随机微分方程

本文研究广义算子及其Wick积意义下的最子随机微分方程。主要结果为：建立了解的存在唯一性定理；证明了解的连续性及解对初值的连续依赖性，本文给出了两个例子。     

THE LERAY-SCHAUDER CONDITION FOR 1-SET WEAKLY-CONTRACTIVE AND （ws）-COMPACT OPERATORS

Abstract The main purpose of this article is to prove a collection of new nxea point theorems for （ws）-compact and so-called 1-set weakly contractive operators under Leray- Schauder boundary condition. We also introduce the concept of semi-closed operator at the origin and obtain a series of new fixed point theorems for such class of operators. As consequences, we get new fixed point existence for （ws）-compact （in particular nonexpansive） self mappings unbounded closed convex subset of Banach spaces. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. Later on, we give an application to generalized Hammerstein type integral equations.     

LIE IDEALS, MORITA CONTEXT AND GENERALIZED （α β）-DERIVATIONS

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.     

I THE （μ＋ κ）oORBITS OF ESSENTIALLY NORMAL OPERATORS AND THEIR STRONG IRREDUCIBILITY

The authors characterize the (μ +κ)-orblts of a class essmntially normal operators and provethat some essentially normal operators with connected spectrum are strongly irreducible aftera small compact perturbation. This partially answers a question of Domigo A. Herrero.     

THE （u＋К）-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let Н be a complex,separable,infinite dimensional Hilbert space,T∈L(Н),(U+κ)(T) denotes the (U+κ)-orbit of T,i.e.,(U+κ)(T)={R^-1 TR:R is invertible and of the form unitary plus compact}.Let Ω be an analytic and simply connected Cauchy domain in C and n∈N,A（Ω,n）denotes the class of operators,each of which satisfies (i) T is essentially normal;(ii)σ(T)=Ω^-,ρF（T）∩σ（T）=Ω;（iii）ind(λ-T)=-n,nul(λ-T)=0,(λ∈Ω）。it is proved that given T1,T2∈A(Ω,n）and c>0,there exists a compact operator K with ||K||<ε such that T1+K∈(u+κ)(T2),this result generalizes a result of P.S.Guinand and L.Marcoux[6,15],Furthermore,the authors give a character of the norm closure of (u+κ)(T),and prove that for each T∈А(Ω,n）,there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.     

（1.2） INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

1.(1.2)InverseandLocalFinePropertyofaFamilyofOperatorsTxLetEandFbebothBanachspaces,andB(E,F)thesetofallboundedlinearoperatorsfromEintoFAnoperatorT B(F,E)issaidtobea(1.2)inverseofTifTT T=TandT TT =T .IfT satisfiesonlythefirstcondition,thenT issaidtobe...     

C2（H）上广义导算子

本文主要给出了 C_2(H)上广义导算子δ_(A,B)、τ_(A,B)为弱正规算子、Jordon 类算子的充要条件。     

方程φe（n）= nd（e=1,2,4）的可解性

利用已有的广义欧拉函数的准确计算公式来研究方程φe(n)的可解性,其中n为正整数,d为n的正因子.并利用初等的方法和技巧给出方程φe(n)=n/d(e=1,2,4)的全部正整数解(n,d).     

QUANTUM COMPLEXITY OF THE APPROXIMATION FOR THE CLASSES B（Wpr（[0, 1）d）） AND B（Hpr（[0, 1]d））

We study the approximation of functions from anisotropic Sobolev classes B(W_p~r([0,1]~d)) and H¨older-Nikolskii classes B(W_p~r([0,1]~d)) in the L q([0,1] d) norm with q ≤ p in the quantum model of computation.We determine the quantum query complexity of this problem up to logarithmic factors.It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.     

W2^2（D）空间第一类算子方程近似解

０引言本文在具有再生核的Ｓｏｂｏｌｅｖ空间Ｗ＿２~２（Ｄ）中研究第一类算子方程的近似求解问题,本文利用再生核空间技巧得到的方程（１）的近似解ｕｎ（Ｍ）有如下特点：１近似解ｕｎ（Ｍ）的构成只需用到ｆ在有限个节点｛Ｗｉ｝１ｎ　Ｄ上的值;２当｛Ｍｉ｝１在Ｄ中稠密时,可以得到方程解析解ｕ（Ｍ）的表达式,并且而且误差随节点个数ｎ的增加接空间范数单调下降．数值算例表明,该方法是有效的．1Ｗ22（Ｄ）空间及其再生核记Ｄ=［ａ,ｂ］×［ｃ,ｄ］为有限区域,设是Ｄ上一组互异节点系．在［１］中,设Ｗ２２［ａ,ｂ］＝｛ｕ…     

（2）——逆的特性及若干分块问题

（２）－－逆近年来才开始引起学术界广泛的关注，虽然已有不少的工作，但是是在一定背景下讨论的，而关于其本身的研究甚少，本文主要对（２）－逆相应于传统广义逆的表征作了一些研究，它具有一些独特的性质，最后考虑了若干分块阵的（２）－逆。     

某些广义插值在L^2πp和Lp（R）空间的逼近

给出了两种插值算子，并研究了它们在L^2πp和Lp(R)空间上的逼近阶，用实数α阶的积分模给出逼近度。     

方程φ2(φ3（n）)=3ω（n）的可解性

φ_e（n）为广义Euler函数,探讨了含有e=2和e=3时的广义复合Euler函数的不定方程φ₂(φ₃（n）)=3^w（n）的可解性问题.基于广义Euler函数φ_e（n）的性质,借助初等方法给出方程φ₂(φ₃（n）)=3^w（n）的全部20组解.     

关于空间（k1,k2）—拟正则映射

本文考虑空间（ｋ１,ｋ２）－拟正则映射的Ｌ＾ｐ（ｐ〉ｎ）可积性,以及当ｋ１→１,ｋ２→０时ｐ的渐近行为。     

Generalized Integral Representations for Functions with Values in C（V_（3,3））

By using the solution to the Helmholtz equation u-λu = 0(λ ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.     

DENSENESS OF RADIAL-BASIS FUNCTIONS IN L^2（R^n） AND ITS APPLICATIONS IN NEURAL NETWORKS

The authors discuss problems of approximation to functions in L2(Rn) and operators fromL2(Rn1) to L2(Rn2) by Radial-Basis Functions. The results obtained solve the problem ofcapability of RBF neural networks, a basic problem in neural networks.     

Matching Realization of Uq（sln＋1） of Higher Rank in the Quantum Weyl Algebra Wq（2n）

In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.     

LIE SYMMETRY ANALYSIS AND PAINLEVE ANALYSIS OF THE NEW （2＋1）-DIMENSIONAL KdV EQUATION

Lie point symmetries associated with the new （2＋1）-dimensional KdV equation ut ＋ 3uxuy ＋ uxxy = 0 are investigated. Some similarity reductions are derived by solving the corresponding characteristic equations. Painlevé analysis for this equation is also presented and the soliton solution is obtained directly from the Bǎcklund transformation.