Rank Properties in Finite Semigroups II: The Small Rank and the Large Rank

This paper is concerned with the application of two possible definitions of rank to certain well-known semigroups.AMS Subject Classification (2000), 20M10     

The Direct Factor Problem for Modular Group Algebras of Isolated Direct Sums of Torsion-Complete Abelian Groups

Let R be an arbitrary commutative unitary ring of prime characteristic p and G an arbitrary abelian group whose p-component G_p is an isolated direct sum of torsion-complete abelian groups. Then G_p is a direct factor of S(RG). As a consequence, the same holds when G is a direct sum of groups for which their p-components are torsion-complete groups. In particular when G is p-mixed, it is a direct factor of V(RG) provided R is a field. The formulated results extend a classical theorem of May (Contemp. Math., 1989) for direct sums of cyclic groups and its generalization due to the author (Proc. Amer. Math. Soc., 1997).AMS Subject Classification (2000): Primary 16 U60, 16 S34; Secondary 20 K10, 20 K20, 20 K21.     

The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

不确定型AHP中单一准则下增加一组元素的保序性条件

本对区间数集定义了两种合理的偏序关系，给出了不确定型AHP中单一准则下增加一组元素弱保序的一个充分条件和强保序的一个充要条件。     

AHP中增加一组元素的严格保序性条件

给出了左主特征向量法（LEM）下增加一组元素严格保序的充要条件,通过研究在右主特征向量法（REM）,左主特征向量法（LEM）,几何平均特征向量法（GMEM）下增加一组元素严格保序的条件之间的关系,从而给出GMEM下增加一组元素严格保序的充分条件。     

一类变换半群的秩

设自然数n≥4,X_n={1,2,…,n},证明了X_n上的保序压缩全变换半群W_n的秩为n-1.     

有限秩Toeplitz算子的(半)换位子

获得了Bergman空间上形如(T_f,T_g]-T_(h~n)和[T_f,T_g]-T_(h~n)的算子为有限秩时的充要条件,其中f,g,h均为有界调和函数,n为非负自然数.     

一类矩阵多项式的秩特征

给出了一类矩阵多项式的秩特征定理及它的多种证法.     

The Fixed Point Group of a Free Group Automorphism

It is proved that the fixed point group of an arbitrary automorphism of a free group of finite rank has an algorithmically computable basis.     

矩阵的秩与零化多项式

从一道考研数学试题出发,深入探讨了矩阵的秩与零化多项式之间的内在联系,推广了已知的相关结果,给出了该类问题的一般处理技巧.     

Rank 0 Quadratic Twists of a Family of Elliptic Curves

In this paper, we consider a family of elliptic curves over with 2-torsion part ₂. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0.     

一类矩阵多项式秩的恒等式

得到了一类矩阵多项式秩的恒等式,改进和推广了近期的一些结果.     

单的迹稳定秩一C*-代数的消去性质

本文证明了一个单的有单位元的迹稳定秩一的C*-代数具有消去律,利用此结果证明了单的有单位元的迹稳定秩一的C*-代数是稳定秩一的.最后讨论了迹稳定秩一的C*-代数的K0群的性质.     

Rank One Discrete Valuations of Power Series Fields

In this article we study rank one discrete valuations of the field k((X ₁,…, X _n )) whose center in k[[X ₁,…, X _n ]] is the maximal ideal. In Sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote Section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in Section 5.

The constructions given in these sections are not effective in the general case, because we need either to use Zorn's lemma or to know explicitly a section σ of the natural homomorphism R _v  → Δ _v between the ring and the residue field of the valuation v.

However, as a consequence of this construction, in Section 7, we prove that k((X ₁,…, X _n )) can be embedded into a field L((Y ₁,…, Y _n )), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function.     

The Axiomatic Rank of the Quasivariety of Orderable Groups Is Infinite

We prove that the axiomatic rank of the quasivariety of orderable groups and that of the quasivariety of -torsion-free groups are infinite.     

分层有限偏序集的直积

A new concept Graded Finite Poset is proposed in this paper.Through discussing some basic properties of it,we come to that the direct product of graded finite posets is connected if and only if every graded finite poset is connected.The graded function of a graded finite poset is unique if and only if the graded finite poset is connected.     

用模糊集方法计算工程质量等级确定质量系数

采用工程模糊集理论方法,用科学合理的数字来衡量工程质量.提出将专家定量评分法与定性评议法科学地统一起来,使权重的确定更为合理与科学.同时,用实例进行实证分析.     

两个矩阵秩等式的推广

给出文[1]建立的两个矩阵秩等式的统一推广.