拟线性常微分方程组边值问题的奇摄动

本文研究拟线性常微分方程组边值问题x′=f(t,x,y,ε),εy″=g(t,x,y,ε)y′+h(t,x,y,ε), x(0,ε)=A(ε),y(0,ε)=B(ε),y(1,ε)=C(ε)的奇摄动.其中x,f,y,h,A,B和C均属于Rn和g是对角矩阵.在适当的假设下,利用对角化技巧和微分不等式理论获得了解的存在和它的按分量逐个一致有效的估计.     

非线性椭圆方程自由边值问题球对称解的存在性

给出了如下的非线性椭圆方程自由边值问题-Δu=λu+(1+ε)u+p,x∈B Rn,u|Ω=μ,∫Ωnu=-M(1)在C[0,1]中的球对称解的存在性.并得到比上述问题更一般的非线性椭圆方程自由边值问题-Δu=h(u),x∈B Rn,u|Ω=μ,∫Ωun=-M,在C[0,1]中的球对称解的存在性,其中B为Rn中的单位球,p>1,λ>0,μ<0,M>0,ε>0;λ,μ,M,ε均为常数,n为正整数.     

Nichtlineare Eigenwertprobleme mit zwei Parametern

In this paper we are searching for solutions (x,μdepending on (ε,τ)-of the nonlinear eigenvalue problem (A?λ)x+B(ε,x)+C(τ,x)=μx. Thus we are interested mainly in imposing such conditions on the analytic operators B(ε,x) and C(τ,x) that will ensure us to obtain solutions. Also we will discuss how these solutions depend onε andτ nearε=τ=0.     

一阶系统非线性边值问题的奇摄动*

本文应用比较定理研究了一类非线性边界条件的向量非线性奇摄动问题εx='f(t,x,y,e)εy'=g(t,x,y,ε)x(0)=A(ξ₁,ξ₂,x(1)-x(0),y(1)-y(0),ε)y(0)=B(ξ₁,ξ,x(1)-x(0),y(1)-y(0),ε)这里ξ₁,ξ₂为ε的函数。0<ε<<1,在适当的条件下,作出了任意次精度的渐近展式。并得出余项估计。     

半线性系统的内层现象

本文研究半线性系统
εy″=f(t,y,ε) (ay(a,ε)=A(ε), y(b,ε)=B(ε)
的奇摄动边值问题的内层现象,其中ε>0是一个小参数,y、f、A和B是n-维向量函数。这种向量边值问题似乎还没有研究过,尽管纯量的边值问题已广泛地研究过。在适当地假设下如同纯量问题一样我们得到解的存在、同时使用适当的不等式,这个解的估计也得到。     

微分多项式的例外集

设ｆ为任一超越整函数，为ｆ的任一微分多项式；本文证明了在满足｜ａ＿ｎ－ａ＿ｍ｜＞ε｜ａ＿ｎ｜（ｎ≠ｍ）和的无穷个圆盘的并集之外取任意有限非零复数无穷次。     

变系数四阶边值问题正解存在性

该文结合算子谱论,应用锥不动点定理,建立了四阶边值问题\[\left\{ {\begin{array}{l}u^{(4)} + B(t){u}' - A(t)u = f(t,u),0 < t < 1 ,\\u(0) = u(1) = {u}'(0) = {u}'(1) = 0 \end{array}} \right.\]正解存在性定理,这里$A(t),B(t) \in C[0,1]$,$f(t,u):[0,1]\times[0,\infty ) \to [0,\infty )$连续.     

An algebraic construction of the generic singularities of Boardman-Thom

In this paper we study some of the functorial properties of the infinite jet space in order to give a coordinate free algebraic definition of the generic singularities of Boardman-Thom. More precisely, suppose thatk is a commutative ring with an identity and suppose that A is a commutative ring with an identity which is ak-algebra. An A-k-Lie algebra L is ak-Lie algebra with ak-Lie algebra map ? from L to the algebra ofk-derivations of A to itself such that ford, d′εL anda, a′εA, then $$[ad'],a 'd'] = a(\varphi (d')ad' - a'(\varphi (d')a)d' + aa'[d',\;d'].$$ . There is a universal enveloping algebra for such Lie algebras which we denote by E(L). Denote byL-alg the category of A-algebras B which have L and hence E(L) acting as left operators such that foraεA,dεL, (da)i _B=d(a.i _B). If F is the forgetful functor fromL-alg to the category of A-algebras, we show that F has a left adjoint J(L, ·) which is the natural algebraic translation of the infinite jet space. In the third section of this paper we construct a theory of singularities for a derivation from a ring to a module and then we apply this construction to J(L, C) where C is an A-algebra. These singularities are subschemas with defining sheaf of ideals given by Fitting invariants of appropriately chosen modules when A and B are polynomial rings over a fieldk and C=A? _k B; these are the generic singularities of Boardman-Thom. Finally we show that, under some rather general conditions on the structure of C as an A-algebra, the generic singularities are regular immersions in the sense of Berthelot.     

多元线性模型中的有偏估计

刘金山在1999年给出了多元线性模型参数分量βi和参数矩阵B的有偏估计β1=(X'X)-1YCi,i=1,…,m和B=(β1,β2,…,βm)以及βi的性质.本文从另一角度得到了同样的估计,证明了刘金山文中所给的两个检验是UMP检验,估计βi是βi的线性可容许估计,证明了B不是B的可容许估计,由此给出了两种改进估计.     

φ形式Khintchine不等式不成立

本文论证了龙顺潮、王键建立的φ形式的Khintchine 不等式不成立,他们的不等式如下:φ- 1 ∑n1φ(｜aj｜) ≤C 12n ∑ε1…∑εn｜ε1a1+ …+ εnan｜,其中φ(x)为满足φ(a+ b)≥φ(a)+ φ(b) (a,b≥0),其反函数φ- 1(x)单调增加的函数,C为常数,εi= ±1,∑ε1…∑εn对所有(ε1…εn)取值,a1,…,an 为任意复数.     

求带释放时间的半导体煅烧排序的最短交付时间的一个高效PTAS

本文研究一个目标是最小化最大交付时间的能分批处理的非中断单机排序问题．这个问题来源于半导体制造过程中对芯片煅烧工序的排序．煅烧炉可以看成一个能同时最多加工B（〈n）个工件的处理机．此外，每个工件有一个可以允许其加工的释放时间和一个完成加工后的额外交付时间．该问题就是将工件分批后再依批次的排序加工，使得所有工件都交付后所需的时间最短．我们设计了一个用时O（f（l／ε）n^5/2）的多项式时间近似方案，其中关于1／ε的指数函数厂（1／ε）对固定的ε是个常数.     

Banach空间的正规结构与对径点

假设S（X）是Banach空间X的单位球面,作引进了四个新的几何参数：Jε（X）=sup{βε（x）,x∈S（X）},jε（X）=inf{βε（x）,x∈S（X）},Gε（X）=sup{αε（x）,x∈S（X）},gε（X）=inf{αε（x）,x∈S（S）},其中≤ε≤1,βε（x）=sup{min{‖x εy‖,‖x-εy‖,y∈S（X）}},αε（x）=inf{max{‖x εy‖,‖x-εy‖,y∈S（X）}},讨论了这些参数的性质,本主要结果是：如果主要结果是：如果有一个ε,0≤ε≤1,使得Jε（X）＜1 ε/2或gε（X）＞1 ε/3,那末X有一至正规结构。     

當若阿-富理埃級數的係數

<正> 不能用蔡查羅(Cesaro)的平均法求它的和;這是哈戴和立篤耳武德老早指出過的,他們的解析是依靠着(?)函數的理論。 其後,鐵起馬虛用初等的方法作成如下的實例:     

Asymptotic estimates for the principal eigenvalue of the laplacian in a geodesic ball

LetM be a compact Riemannian manifold and letB _ε be a geodesic ball of radiusε with center0 ∈ M. We investigate the asymptotic behavior ofλ _ε , the principal eigenvalue of the Laplace-Beltrami operator on \(M/\bar B_\varepsilon\) with homogeneous Dirichlet boundary conditions. We prove thatλ _ε ~Cφ _n (ε) wheren = dimM, φ ₂ (ε)=(logε ^?1)^?1 andφ _n (ε) = (n-2)ε ^n-2 (n>2). In the case whereM is a model the constantC is explicitly evaluated.     

Longest cycles in 3-connected graphs

In this paper, we get the following result: Let G be a 3-connected graph with n vertices. Then , where

Full-size image

. This is a new lower bound for the circumference c(G) of a 3-connected graph G.     

A note on unsolvable systems of max–min (fuzzy) equations

For unsolvable systems of linear equations of the form Ax=b over the max–min (fuzzy) algebra

Full-size image

we propose an efficient method for finding a Chebychev-best approximation of the matrix in the set

Full-size image

.     

Directed triangles in directed graphs

We show that each directed graph on n vertices, each with indegree and outdegree at least n/t, where

Full-size image

, contains a directed circuit of length at most 3.     

常差分方程奇异摄动问题的渐近方法

在本文中,我们讨论如下差分方程问题(P_ε):(L.y)_k≡εy(k+1)+a(k,ε)y(k)+b(k,ε)y(k-1)=f(k,ε)(1≤k≤N-1)B₁y≡-y(0)+c₁y(1)=a,B₂y≡-c₂y(N-1)+y(N)=β这里ε是一个小参数,c₁,c₂,a,β为常数,a(k,ε),b(k,ε),f(k,ε)(1≤k≤N)是k和ε的函数.首先,我们讨论了常系数的情形;接着引进伸长变换对变系数的情形进行了讨论,给出了解的一致渐近展开式;最后给出了一个数值例子.     

一类向量四阶非线性微分方程边值问题的奇摄动

我们研究伴有边界摄动的向量边值问题:
ε2y(4)=f(x,y,y″,ε,μ)(μy(x,ε,μ)|_x=μ=A₁(ε,μ),y(x,ε,μ)|_x=1-μ=B₁(ε,μ)
y″(x,ε,μ)|_x=μ=A₂(ε,μ),y″(x,ε,μ)|_x=1-μ=B₂(ε,μ)
其中y,f,A_j和B_j(j=1,2)是n维向量函数和ε,μ是两个正的小参数.虽然纯量边值问题曾有人研究过,但这样的向量边值问题尚未被研究.在适当的假设下,利用微分不等式方法,我们找到向量边值问题的一个解和获得一致有效的渐近展开式.