组合群论中的一个定理

Let F= F(X) be a free group of rand n, A be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of A if x is in the normal closure of A in F(X). Finally, an application of the theorem in Heegaard splitting of 3manifolds is given.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.     

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.     

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the L p Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L_0 = div A~0(x)? + B~0(x) · ? is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the LpDirichlet problem for the operator L_0 is solvable in the upper half-space Rn+. In this paper we prove that the Lpsolvability is stable under small perturbations of L_0. That is if L_1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L_0 and L_1 are sufficiently close in the sense of Carleson measures, then the LpDirichlet problem for the operator L_1 is solvable for the same value of p. As a corollary we obtain a new result on Lpsolvability of the Dirichlet problem for operators of the form L = div A(x)? + B(x) · ? where the matrix A satisfies weaker Carleson condition(expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoˇs,Petermichl and Pipher.     

关于M/G/l随机服务系统中L.Takacs引理的证明

L. Takacs' Lemma (ef [1] PP47-48) is a very important tool for the investigation of the transient behavior of the M/G/1 System. But the original proof contains a gap when γ(s, w) is expanded into Lagrange series by means of Lagrange theorem as given in the appendix of [1]. This is due to the fact that φ(s)=∫₀^∞e^-sxdH(x), the Laplace transform of the service time distribution , may not be analytic at s=0. An example of such a distrbution function H(x) is gives and a specific Lagrange theorem is proved By using this specific theorem, the gap in the original proof of Takacs′ Lemms is eliminated.     

EXISTENCE OF PERIODIC SOLUTIONS TO A p-LAPLACIAN NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION WITH VARIABLE PARAMETER

By the generalized Borsuk theorem in coincidence degree theory, a p-Laplacian neutral functional differential equation is studied. A new result on the existence of periodic solution is obtained. The interest is that some coeffcient in it is not a constant function and its sign can be changeable, which is different from that in the known literatures.     

On Laguerre Isopararmetric Hypersurfaces in R^7

Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.     

Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded （DG for short） algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] （see also Proposition 2.2）, Ak is not compact if H（A） is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H（A） is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.     

The Jacobson Relational Radical and the Jacobson Radical

In this paper it is shown that the intersection of all w-relations of a hemiring R is exactly the intersection of all primitive relations of this hemiring, and is called J-relational radical of the himiring R. Several properties of the J-relational radical are described. The radical R of a hemiring R is defined by the set {x\in R|x\tau 0}, where \tau is the J-relational radical of R. We obtain independently following results: 1.R is a right quasi-regular ideal which contains every right quasi-regular right ideal. 2.R=\cap {(0:M)|M,, irreducible cancellative right R-semimodule. The term "Jacobson semisimple” in ring theory is generalized to hemirings by defining “J-relational semisimple.” It is proved that if \tau is the Jacobson relational radical of a hemiring R, then R/\tau is J-relational semisimple. Finally the structure theorem of the hemirings is given. A hemiring is J-relational simisimple if and only if it is isomorphic to a subdirect sum of completely primitive hemirings.     

Finite Groups in Which Each Irreducible Character has at Most Two Zeros

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gG the conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem 3.15]) states that every nonlinear X ∈ Irr(G) has a zero on G, that is, an element x (or a conjugacy class xG) of G with X(x) = 0. So, if the number of zeros of character table is very small, we may expect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimes that G is a Frobenius group with a complement of order 2 if each row in charcter table has at most one zero (its proof uses the classification of simple groups). In this note, we characterize the finite group G satisfying the following hypothesis:     

Semigroups satisfying variable identities

The purpose of this note is to generalize a theorem of Tamura’s [3] providing a self-contained and, we think, more elementary proof than Tamura’s in that it avoids using the theory of contents. Tamura’s result states that a semigroup S satisfies an identify xy=f(x,y) with f(x,y) a word of length greater than 2 which starts with y and ends in x if and only if S is an inflation of a semilattice of groups satisfying the same identity. We investigate semigroups as in Tamura’s Theorem, except that we permit f(x,y) to vary with x and y.     

关于一个次可乘定理

本文证明了次可乘定理：设A是一个*代数,A上范数||.||使得（A,||.||）成为完备的赋范线性空间,而且,（I）对于X∈A,||x*x||||x||2;（II）对于正规元x∈A,||x*x||=||x||2,则（A,||.||）是C*-代数．同时,也给出了其对应的C*-等价定理．     

Herstein定理的推广

1955年Herstein将著名的Jacobson定理推广为:定理A.如果对R中任意x,y,存在可依赖于x,y的整系数多项式p(t),使[x-x2p(x),y]=0,则R是交换的.本文利用多项式的系数和定义了n元多项式,f(x1,x2…,xn)的Fk性质,并以此证明了一个环的交换性定理,当n=1时,即得到定理A.     

一类整除性定理的推广

[1 ]给出复数域C上多元多项式环 C[x1 ,x2 ,… ,xn]的一类整除性定理 ,本文把它推广为任意代数闭域 k上多元多项式环 k[x1 ,x2 ,… ,xn]的情形 .     

Liénard方程极限环的存在唯一性定理

<正> 的极限环的存在唯一性问题[1,2],给出了定理1,此定理的一个推论即已包含了熟知的Lienard定理以及Levinson-Smith[3],Sansone[2],Barbalat[4],余澍祥[5]的存在唯一性定理.作为定理1推论的直接应用,还对方程     

Виноградов中值定理的一个改进和它的某些应用

<正> 正如大家所熟知的,И.М.Виноградов院士的中值定理在解析数论的新研究中是一个很基本的工具.本文作者之一曾经在1941年与1949年先后二次改进过Виноградов中值定理,在本文中更将证胆如下的中值定理.     

积分中值ξ=ξ(x)的渐近性

对一类不满足g(n)≠0的函数g讨论了第一积分中值定理中ξ=ξ(x)在x→+∞时的渐近性质,并对第二积分中值定理的中值ξ=ξ(x)的渐近性进行了探讨,给出一些相关的结果.     

Improvement of Hartman''''s linearization theorem

Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) is bounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphism of Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x = Ax. In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we prove the result of global topological linearization without any special limitation and adding any condition. Thus, Hartman's linearization theorem is improved essentially.     

A GENERALIZATION OF A PROPOSITION ON EXPONENTIAL DICHOTOMY

1 Introduction and Statement of TheoremConsider systemx' = A(t)x f(t, x), (l.1)where x E R", A(t) is a continuous matrix function, f: R x R" - R" is acontinuous function.We say that the linear differential equation X' = A(t)x admits an exponential dichotomy, if it has a fundamental matrix X(t) such thatIX(t)PX--'(s)l 5 K' e--a(t--s) for s S t,(1.2)IX(t)(I -- P)X--'(s)I 5 K' e--a(s--t) for s 2 t, (1'2)where P is a projection (P' = P), K and a are positive constants.Remark Without …