多元多项式齐次方程组的一个解法

本文对多元多项式齐次方程组给出一种解法，并给出其解子模的基所含向量的个数在最坏情况时的下界。     

SOLUTION OF A RESEARCH PROBLEM

ＳＯＬＵＴＩＯＮＯＦＡＲＥＳＥＡＲＣＨＰＲＯＢＬＥＭＷＵＳＨＩＱＵＡＮ（巫世权）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｔｉｏｎａｌＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｃｈａｎｇｓｈａ４１００７３，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ...     

The Siegel norm,the length function and character values of finite groups

In this paper we present some new results on the connection between the Siegel norm, the length function and irreducible character values of finite groups. In addition, we provide algorithms to compute the length of a cyclotomic integer and the set of cyclotomic integers with Siegel norm bounded by a given positive real number.     

The Formula of General Term of the Sum of Equal Powers of Natural Numbers

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｏｆｉｎｄｔｈｅｇｅｎｅｒａｌｔｅｒｍｆｏｒｍｕｌａｏｆｃａｌｃｕｌａｔｉｎｇｗａｓａｎｏｌｄｅｓｔｐｒｏｂｌｅｍｉｎＮｕｍｂｅｒＴｈｅｏｒｙ．Ｅａｒｌｙｉｎ３００Ｂ．Ｃ．,Ａｒｃｈｉｍｅｄｏ,ｔｈｅｍａｔｈｅｍａｔｉｃｉａｎｏｆａｎｃｉｅｎｔＧｒｅｅｃｅ,ｗｏｒｋｅｄｏｕｔｔｈｅｆｏｒｍｕｌａ∑ｎｋ＝１ｋ＝ｎ（ｎ＋１）２ａｎｄ∑ｎｋ＝１ｋ２＝ｎ（ｎ＋１）（２ｎ＋１）６．Ｓｏｆａｒ,ｐｅｏｐｌｅｈａｖｅｇｏｔｍａｎｙｆｏｒｍｓｏｆｅｐｒｅｓｓｉｏｎｏｆｔｈｅｇｅ…     

The diffeomorphic types of the complements of arrangements in ??3 II

For any arrangement of hyperplanes in ℂℙ³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ³. This work was partially supported by NSA grant and NSF grant     

用AHP法建立教学质量评估体系

以AHP为理论基础,结合Delphi方法,对高校教师教学质量评估体系进行尝试性建构。通过构建结构→个体判断→群判断处理→量化指标体系生成的分步操作,来达到科学、民主、公正、客观的评价目标。     

The diffeomorphic types of the complements of arrangements in CP~3Ⅱ

For any arrangement of hyperplanes in CP~3,we introduce the soul of this arrangement. The soul,which is a pseudo-complex,is determined by the combinatorics of the arrangement of hyper- planes.In this paper,we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other.In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected.This generalizes our previous result on hyperplane point arrangements in CP~3.     

A generalization of Alon–Aydinian–Huang theorem

Let $S_{n,k}^m$ be the collection of sets of real numbers of size n, in which every subset of size larger than k has a sum less than m, where $n\ge k+1$, and m is some real number. Denote by $a_{n,k}^m$ the maximum number of nonempty subsets of a set in $S_{n,k}^m$ with a sum at least m. In particular, when $m=0$, Alon, Aydinian, Huang ((2014) [1]) proved that $a_{n,k}^0={\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)$, where two technical proofs, based on a weighted version of Hall's theorem and an extension of the nonuniform Erd?s–Ko–Rado theorem, were presented. In this note, we extend their elegant result from $m=0$ to any real number m, and show that $a_{n,k}^m=\{\begin{array}{cc}{\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)\hfill & \text{ if }m\ge 0\hfill \\ {\sum }_{i=1}^k\left(\begin{array}{c}n\\ i\end{array}\right)\hfill & \text{ if }m<0\hfill \end{array}$. Our proof is obtained by exploring the recurrence relation and initial conditions of $a_{n,k}^m$.     

A Note on the Theorems of Church-Turing and Trachtenbrot

We sketch proofs of the theorems of Church-Turing and Trachtenbrot using a semi-monomorphic axiomatization. Mathematics Subject Classification : 03B10, 03B25.