On Semi—tensor Product of Matrices and Its Applications

The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product.Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained.Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression. These applications demonstrate the usefulness of the new matrix products.     

随机半范模上等度连续的典则模同态族的特征

给出了随机半范模上等度连续的典则模同态族的一个待征——随机半范模上强凸的等度连续的典则模同态族可以被一个性质较好的连续随机半范数所控制．     

完备随机内积模上稠定模同态的共轭算子

对定义在完备随机内积模的稠子模上的模同态引入了共轭算子的概念并讨论其基本性质,尤其证明了共轭算子的闭性.     

Module Homomorphisms on Random Normed Modules

ＭｏｄｕｌｅＨｏｍｏｍｏｒｐｈｉｓｍｓｏｎＲａｎｄｏｍＮｏｒｍｅｄＭｏｄｕｌｅｓＧｕｏＴｉｅｘｉｎ（郭铁信）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉａｍｅｎＵｎｉｖｅｒｓｉｔｙ，Ｘｉａｍｅｎ，Ｆｕｊｉａｎ，３６１００５）Ａｂｓｔｒａｃｔ...     

Characterization of Quasi-Coherent Modules that are Module Schemes

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ? _R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.     

A Characterization of Homomorphisms Between Banach Algebras

We show that every unital invertibility preserving linear map from a yon Neumann algebraonto a semi-simple Banach algebra is a Jordan homomorphism;this gives an affirmative answer to aproblem of Kaplansky for all von Neumann algebras.For a unital linear map Φ from a semi-simplecomplex Banach algebra onto another,we also show that the following statements are equivalent:(1)Φ is an homomorphism;(2)Φ is completely invertibihty preserving;(3)Φ is 2-invertibility preserving.     

A Characterization of Homomorphisms Between Banach Algebras

We show that every unital invertibility preserving linear map from a yon Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism; this gives an affirmative answer to a problem of Kaplansky for all yon Neumann algebras. For a unital linear map Ф from a semi-simple complex Banach algebra onto another, we also show that the following statements are equivalent: (1)Ф is an homomorphism; (2) Ф is completely invertibility preserving; (3) Ф is 2-invertibility preserving.     

复随机内积模上一类算子的谱定理

本文给出复随机内积模上几乎处处有界线性算子谱的几个基本定理，这些定理不但为随机内积模上几乎处处有界线性算子的进一步讨论有基本重要性，而且也为Ｈｉｌｂｅｒｔ空间上连续随机算子的谱研究提供了一个新途径。     

On 𝒦-Nonsingular Modules and Applications

We introduce the notion of 𝒦-nonsingularity of a module and show that the class of 𝒦-nonsingular modules properly contains the classes of nonsingular modules and of polyform modules. A necessary and sufficient condition is provided to ensure that this property is preserved under direct sums. Connections of 𝒦-nonsingular modules to their endomorphism rings are investigated. Rings for which all modules are 𝒦-nonsingular are precisely determined. Applications include a type theory decomposition for 𝒦-nonsingular extending modules and internal characterizations for 𝒦-nonsingular continuous modules which are of type I, type II, and type III, respectively.     

Reflexivity and Ring Homomorphisms of Finite Flat Dimension

In this article we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for certain hypotheses and the strictness of many inequalities.     

A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.