非正(负)定四维流形中的浸入球

一、引言设 M~(2n)是单连通光滑流形,ξ∈π_n(M~(2n)),则ξ由 S~n 到 M~(2n) 的连续象表示,一个熟知的事实是当 n>2时,可以用 Whitney 技巧找到一个光滑嵌入球表示ξ,但当 n=2时,Whitney技巧失效,由此产生一个问题:设 M 是单连通4-流形,则 π_2(M)的每个元由 S~2到 M 的连续映射表示,这种映射能否同伦于一个嵌入呢?由于 M 单连通,这个问题也可以说成是否每个二维同调类都可由光滑嵌入的 S~2表示?答案通常是否定的.但由于 S~2在 M~4中有较高的余维数,因而这样的连续映射一定同伦于一个只有有限个二重点的浸入 S~2→M~4.一个更广泛的问题是:对于一个给定的同调类 u∈H_2(M,Z),表示它的浸入球至少可以有几个二重点?当二重点个数为0时便转化为前一个问题.     

GENUS MINIMIZING IN SYMPLECTIC 4-MANIFOLDS

The authors show that a symplectically embedded surface in a symplectic 4-manifold with b ₂ ⁺ greater than one minimizes genus in its homology class. Project supported by the MOST through National R and D Program (98-N6-01-01-A-1) for Women’s University of Korea.     

CURVATURE COMPUTATIONS OF 2-MANIFOLDS IN IR~k*1

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k > 3.     

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.     

The k-point Exponent Set of Primitive Digraphs with Girth 2

Let D =(V,E)be a primitive digraph.The vertex exponent of D at a vertex v∈V,denoted by exPD(V),is the least integer p such that there is a v→u walk of length p for each u∈V.Following Brualdi and Liu,we order the vertices of D so that exPD(v_1)≤exPD(v_2)≤…≤exPD(v_n).Then exPD(v_k)is called the k- point exponent of D and is denoted by exP_D(k),1≤k≤n.In this paper we define e(n,k):=max{exp_D(k)|D∈PD(n,2)} and E(n,k):= {expD(k)|D∈PD(n,2)},where PD(n,2)is the set of all primitive digraphs of order n with girth 2.We completely determine e(n,k)and E(n,k)for all n,k with n≥3 and 1≤k≤n.     

L_p-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.     

再生核空间中的一类最佳逼近及其应用

在[1-2]中分别定义了具有再生极的Hilbert空间W_2~1[a,b]和W,并给出再生核的解析式。本文讨论再生核空间中线性算子的一类最佳逼近,给出逼近算子的表达式及误差估计,作为特例得到类似于[1-4]中的插值近似公式,数值积分公式和数值原函数公式,但本文的公式计算更简便。     

VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM

Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets.This ...     

Solution curves and exact multiplicity results for 2mth order boundary value problems

For any integer m?2, we consider the 2mth order boundary value problem

(−1)mu(2m)(x)=λg(u(x))u(x),x∈(−1,1),     

Z/(2e)上本原序列不同压缩映射的导出序列

设 f( x)是 Z/ ( 2 e)上 n次强本原多项式 ,对形如 xe- 1 +η( x0 ,… ,xe- 2 )的二个 e元布尔函数 Φ( x0 ,… ,xe- 1 )和 Ψ( x0 ,… ,xe- 1 )及二条序列 a,b∈G( f( x) ) e,若Φ( a0 ,… ,ae- 1 ) =Ψ ( b0 ,… ,be- 1 ) ,给出了函数Φ ( x0 ,… ,xe- 1 )和Ψ ( x0 ,… ,xe- 1 )之间的关系与序列 a和 b之间的关系 .所给出的结论进一步说明了导出的二元序列具有良好的密码性质     

Exact L 2-Small Ball Asymptotics of Gaussian Processes and the Spectrum of Boundary-Value Problems

We sharpen a classical result on the spectral asymptotics of boundary-value problems for self-adjoint ordinary differential operator. Using this result, we obtain the exact L ₂-small ball asymptotics for a new class of zero-mean Gaussian processes. This class includes, in particular, the integrated generalized Slepian process, integrated centered Wiener process, and integrated centered Brownian bridge. Partially supported by RFFR grant No.07-01-00159 and by grant NSh-227.2008.1.     

许—模型下误差方差的可容许估计

本文讨论了在许－模型下非负二次估计类中误差方差的可容许估计的条件。     

环Z/(2~e)上压缩序列的0,1分布

本文研究环 Z/( 2 e)上本原序列最高权位的 0 ,1分布 ,证明了当 e≥ 8,次数 n≥2 0时 ,本原序列 a的最高权位序列 ae- 1 在一个周期中 0 (或 1 )所占的比例λ( ae- 1 )满足 43.6 76 8 <λ( ae- 1 ) <5 6 .32 32     

An improvement on the positivity results for 2-stage explicit Runge-Kutta methods

In this paper, we investigate the positivity property for a class of 2-stage explicit Runge-Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.     

多孔介质驱动问题的混合元最小二乘特征有限元方法及其收敛分析

１．引言多孔介质二相驱动问题的数学模型是偶合的非线性偏微分方程组的初边值问题．该问题可转化为压力方程和浓度方程［１－４］．浓度方程一般是对流占优的对流扩散方程,它的对流速度依赖于比浓度方程的扩散系数大得多的Ｆａｒｃｙ速度．因此Ｄａｒｃｙ速度的求解精度直接影响着浓度的求解精度．为了提高速度的求解精度,７０年代Ｐ．Ａ．Ｒａｖｉａｔ和Ｊ．Ｍ．Ｔｈｏｍａｓ提出混合有限元方法［５］．Ｊ．ＤｏｕｇｌａｓＪｒ,Ｔ．Ｆ．Ｒｕｓｓｅｌｌ,Ｒ．Ｅ．Ｅｗｉｎｇ,Ｍ．Ｆ．Ｗｈｅｅｌｅｒ［１］－［４］,［９］,［１２］袁…     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.