Quasi-d-Koszul Modules and Applications

Let R be a Noetherian semiperfect algebra. A necessary and sufficient condition for a finitely generated R-module to be quasi-d-Koszul is given, which improves one of the main results in [1]. The authors also give a necessary and sufficient condition for the Minimal Horseshoe Lemma holding in mod（R）. As an application, it is proved that the ＂Minimal Horseshoe Lemma＂ is true in the category of quasi-d-Koszul modules under certain conditions.     

"Whitney引理的推广及应用"一文中的错误

本文用一个十分简单的例子说明[1]对整体的Borel定理的证明是错误的.为此, 还须介绍函数芽和函数芽序列一致收敛的概念,并给出一个判定引理.     

关于“数学规划的稳定性”的一个注记

[1]的作者薛声家同志改进了[2]和[3]的一个基本假设条件,通过他的改进,[2]和[3]中的数学规划稳定性定理的应用范围得到了扩大。作者仿照薛声家同志的工作。把他的假设条件作进一步的改进。并用实例指出通过这进一步的改进[1]的数学规划稳定性定理的应用范围再一次得到了扩大。     

非线性Hénon方程解的Liouville定理

该文研究了二阶和四阶非线性Henon-Lane-Emden方程有限Morse指标解的Liouville定理.利用一种新方法,即使用单调公式、Pohozaev恒等式和doubling引理等相结合证明了其结果.     

一个失误的反例

文[1]的作者对于文[2]中的定理2举了一个粗心的反例W_t(x,y)=xy(x-y)(x-ty)．为此,我们不得不与文[1]的作者商榷某些主要问题．     

拓扑流形上Fréchet空间的道路连通分支

本文将D.Landis和M.Morse建立的有关微分流形上Fréchet空间的道路连通分支的定理(见[1])推广到拓扑流形上。     

高阶Morse芽的存在性

本文研究了多元C∞函数芽环中高阶Morse芽的存在性问题.利用由函数芽的偏导数生成的理想和C∞函数芽上的右等价关系,获得了在C∞函数芽环中,除了二元C∞函数芽环中有三阶和四阶的Morse芽以后,不再存在其它的Morse芽.以致在三元以上的C∞函数芽环中Morse引理不能推广到较高阶的情形.     

SST—PM空间与拓扑空间之积上映射的不动点定理

本文利用作者[2]的结果给出 SST-PM 空间[1]与拓扑空间之积上映射的几个不动点定理,它们以张石生[3]中的某些主要结果为特例。     

最小质因数的一个定理

在数论书[1][2][3]上常介绍一个著名的Wilson定理:“P是质数,则(P-1)!+1≡0(modp)”作者在[4]中曾把这个定理推广为下面的定理,“设m是一个有元根存在的正整数,a_1,a_2,…a_k是同在模m中的一个简化剩余系中的全部n次剩余,则有a_1a     

关于极限定理的随机转移

<正> 许多作者([3]-[5])曾研究过随机变量序列的部分和极限定理的随机转移.D.J.Aldous在[1]中讨论了函数空间D[0,+∞)上一类较广泛的随机指标极限定理.本文考察有别于他们工作的一类函数型极限定理的随机转移;文中首先讨论函数空间D[0,1]上泛函正态极限定理的随机转移问题,接着讨论一些非正态随机极限定理.     

A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.     

关于p－拉普拉斯方程径向解的一点注记

本文研究方程ｄｉｖ（｜ｕ｜ｐ－２ｕ）＋｜ｘ｜ｌ｜ｕ｜τ－１ｕ＝０，ｘ∈Ｂ的Ｄｉｒｉｃｈｌｅｔ边值问题ｕ｜Ｂ＝０的径向解。应用山路引理，我们将文［１］的结果（ｐ＝２时）推广到一般情形，证明上述问题有一非平凡的径向解。     

A proof of the Morse-Bott Lemma

It is often said that the Morse-Bott Lemma can be viewed as a “parameterized” Morse Lemma, and its proof should follow from the differentiability of the methods used to prove the Morse Lemma. The goal of this expository paper is to fill in the details. We present Palais' proof of the Morse Lemma using Moser's path method, which yields the necessary differentiability.     

The Dedekind-Mertens lemma and the contents of polynomials

We prove a sharpening of the Dedekind-Mertens Lemma relating the contents of two polynomials to the content of their product. We show that for a polynomial the integer in the Dedekind-Mertens Lemma may be replaced by the number of local generators of the content of . We also raise a question concerning the converse.

     

关于Thompson的一个结果的注记

本文改进了J .G .Thompson在JournalofCombinatorialTheory ,1 (1 96 6 )中得到的一个有用的引理的结果 .     

How to avoid using the Regularity Lemma: Pósa’s conjecture revisited

In this paper we investigate how the use of the Regularity Lemma and the Blow-up Lemma can be avoided in certain extremal problems of dense graphs. We present the ideas for the following well-known Pósa conjecture on the square of a Hamiltonian cycle. In 1962 Pósa conjectured that any graph G of order n and minimum degree at least contains the square of a Hamiltonian cycle. In an earlier paper we proved this conjecture with the use of the Regularity Lemma-Blow-up Lemma method for n≥n₀ where n₀ is very large. Here we present another proof (and a general method) that avoids the use of the Regularity Lemma and thus the resulting n₀ is much smaller.     

A Uniqueness Theorem of Meromorphic Maps and a Generalization of the Borel’s Lemma

This paper gives a generalization of the classical Borel's Lemma. Then as an application of this generalized Borel's Lemma, a uniqueness theorem for two linearly non-degenerate meromorphic maps of C^m into Pⁿ(C) (n ≥ 2) sharing 2n + 2 hyperplanes in general position is proved.     

整数环上的Farkas引理及其应用

将实数域上的Farkas引理进行推广,提出整数环上的Farkas引理.之后,利用整数环上的Farkas引理证明了一个有关Petri网结构有界性的判定定理.     

Discrete morse theory and the cohomology ring

In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].

     

A note on the short algebraic proof of Farkas' Lemma

The author published ‘A Short Algebraic Proof of the Farkas Lemma’ [SIAM J. Optim. 19 (2008), pp. 234–239]. The author then found, in his opinion, a better exposition of the proof. He would therefore like to publish the new form of the proof in this note.