根值判别法的两个推广

讨论正项级数的根值判别法．若将判别极限lim↓n→∞ n√an更改为lim↓n→∞或lim↓n→∞^n√am^n＋i,则相应结果在一定条件下将比原判别方法更为精细,且应用范围也有所推广．     

Two counterexamples in normalization

In this paper, we offer a correction to Commutative Algebra by Oscar Zariski and Pierre Samuel by noting two erroneous theorems and offering counterexamples for each.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Partitions and sums and products—Two counterexamples

A negative answer is provided to a question of Erdös. Specifically, a two celled partition of N is provided with the property that neither cell includes an infinite set together with all finite products and pairwise sums from that set. Also, a seven celled partition of N is provided with the property that no cell includes an infinite set together with all pairwise products and pairwise sums from that set.     

Two counterexamples related to the Kreiss matrix theorem

The modern version of the Kreiss matrix theorem states that A ⁿ esK (for alln0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the unit disk with constantK. In this paper we show for any fixedK + 1 that the upper boundesK is sharp in the sense that a linear dependence on the dimensions is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that exp(tA) esK (for allt0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the left half plane with constantK.The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.) and was carried out at the Department of Mathematics and Computer Science, University of Leiden, The Netherlands.     

Two finitely generated varieties having no infinite simple members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote's negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

     

Two counterexamples to the spectral mapping theorem for semigroups of positive operators

LetX be anL ¹- orL ²-Banach lattice. Let (W(t)) _t0 be aC ₀-semigroup of positive operators onX with generator–T. Two examples ofC ₀-semigroups are presented for which the identity fails for almost allt>0.     

On the Stieltjes moment problem on semigroups

We characterize finitely generated abelian semigroups such that every completely positive definite function (a function all of whose shifts are positive definite) is an integral of nonnegative miltiplicative real-valued functions (called nonnegative characters).     

有限余生成维数

引入模的有限余生成维数的概念，研究了它的性质．同时，我们探讨了模的有限余生成维数、有限余表现维数和内射维数三之间的关系。     

两个函数的拓扑极大极小定理

A topological minimax theorem involving two functions is derived. It generalizes Greco and Horvath＇minimax theorem given by them in 2002.     

The Extremism of Two Variable Function Based on the Positive Definite Property

In this paper we give out a sufficient and solution of quadratic function's maximum with theory of quadratic form and give out the definition of the positive definite property of the following homogeneous polynomials of degree In two variables function, based on the definite of local maximum of two variables function.     

拟Raabe判别法与拟对数判别法的强弱关系

拟Raabe判别法是新近提出的关于正项级数收敛性的一种比较细致的判别法.对通项递减的正项级数来说,此判别法强于传统的Raabe判别法与Gauss判别法.通过对拟Raabe判别法与另一个细致的判别法——拟对数判别法强弱关系的探讨,得出了后一判别法强于前者的结论.     

两个一元t-分布之间的Kullback-Leibler距离

该文首次求出了两个一元t-分布之间的Ｋｕｌｌｂａｃｋ Ｌｅｉｂｌｅｒ距离的精确表达式．     

判定级数敛散性的两种初等方法

借助实例介绍针对某类级数敛散性的两种初等判定方法,即由级数通项构造相关不等式后运用比较判别法,或对级数恒等变换后再进行拆项求和.     

The characters of some 2-blocks of the babymonster,its covering group and the monster

The double of a free nonabelian group over a subgroup of finite index greater than two is shown not to be coherent, answering a question of C. T. C. Wall.     

The Constrained Solutions of Two Matrix Equations

We study the symmetric positive semidefinite solution of the matrix equation AX ₁ A ^T + BX ₂ B ^T = C, where A is a given real m×n matrix, B is a given real m×p matrix, and C is a given real m×m matric, with m, n, p positive integers; and the bisymmetric positive semidefinite solution of the matrix equation D ^T XD = C, where D is a given real n×m matrix, C is a given real m×m matrix, with m, n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions. Received December 17, 1999, Revised January 10, 2001, Accepted March 5, 2001     

两个多元t分布之间的Kullback Leibler距离

该文将文［1］的结果推广到多元t分布的情形.结果表明, 两个多元t分布之间的Kullback Leibler距离与分布中的刻度矩阵之比的最大特征根密切相关.作为应用, 文章建立了一种在估计未知正定阵时的熵损失函数.