r-循环阵的逆阵或广义逆阵

在应用数学和其他学科(如数理统计、固态物理等)中,都将遇到求循环阵的逆阵或广义逆阵的问题.如何求非奇异循环阵的逆阵?文[1]提出了一种算法而无证明,文[2]则给出了这种算法的一个证明,文[3]又提出一种新算法,但上述两种算法的计算量大,实际使用时是很繁的.针对这一情况,文[4]除了对[1]中提出的方法重新给了一个初等证明外,还导出了一些特殊循环阵的逆阵公式.关于求奇异循环阵的广义逆阵的问题,则除了[3]中给出了某类特殊的奇异循环阵的 Moor-Penrose 逆阵外,还未见到有文章论述求奇异循环阵的广义逆阵的一般方法.本文给出了 r-循环阵的逆阵或一个反射 g 逆阵的公式和具体算法.特别,这个公式可用来求通常的循环阵及反循环阵的逆阵和 Moor-Penrose 逆阵.文[3]、[4]中的各个公式可用本文的统一方法推广到 r-循环阵的情形.     

反对称阵的最优传递阵

L.A.Saaty提出的AHP方法虽然对于多目标决策很有价值,但这方法未能做到最优决策,而且计算量相当大。本文根据Saaty的思想,进一步提出了容差阵的概念,讨论了容差阵的性质,解决了在最小二乘意义下求反对称阵的最优近似容差阵的求法,得到用算术平均方法求出权重、做出决策的方法,此方法不仅做到了最优决策,而且计算也相当简便。     

幂等阵和幂零阵的伴随阵的若干性质

本文利用极限过程的方法,证明了幂等阵和幂零阵的伴随矩阵分别是幂等阵和幂零阵．所得到的结论比［１］丰富得多．     

泛正定阵与广义正定阵的关系

本文利用矩阵的稳定值,给出泛正定阵的等价条件,进一步得到泛正定阵与广义正定阵之间的一些关系．     

半幻阵

首先解释题目.我国古代传说中的九宫图把从1至9这最初九个自然数排列成一个3×3的方阵,使得三个行,三个列以及两个对角线的和都等于15.这一奇妙的事实引出了人们对“幻方”的研究:能否和如何把前n~2个自然数(或者更一般些,把预先指定的n~2个整数)排列成一个n×n方阵,使得n个行和、n个列和以及两个对角     

拟次酉阵与拟次Hermite阵

提出了拟次酉阵、拟(反)次Hermite阵概念，研究了它们的性质及其相互间的关系，将正交矩阵广义Gayley分解推广到拟次酉阵上。     

k-控制阵

给出k-控制阵的定义,讨论k-控制阵与k-圈控制阵的关系,指出k-控制阵的周期是k的一个因子,指数不大于(n-1)k m.     

相似变换阵与合同变换阵的初等变换求法

引入初等相似变换与初等合同变换 ,使化方阵为 Jordan标准形的同时求得相似变换阵 ,化实对称阵为对角阵的同时求得合同变换阵 .算法易于理解 ,计算量较小 .     

体上逆阵的子阵的一个注记

用群表示论研究有限群的子群的构造时,我们遇到了如下必需解决的问题:“找出体上方阵的逆阵的子矩阵的逆与原矩阵的子阵及其某个子阵的逆的关系”。这一问题的解决不仅对表示与子表示之间的联系、且对复矩阵论较深入的领域—矩阵的“子结构”亦是相当有用的。本短文将用一特殊技巧来解决上述问题。     

A boolean model of role discrimination

This article analyzes, based on Boolean approach by C.C. Ragin, how the subjective internal structure of social roles conditions the possibility of discrimination of role performance (role discrimination). I postulate, in an actor's image, a role consists of some role elements, which make up the internal hierarchical structure (role image) according to each element's importance for the role. After introducing a set of definitions and assumptions, the role discrimination is formulated as a Boolean equation, taking the role elements as independent variables and the role discrimination itself as a dependent variable. Suppose role X, and suppose a performer and an observer interact each other on role performing, each having his/her own image. In general, between such two given actors, the performance of X by the performer can be discriminated by the observer under the condition that the pattern of practicing role elements included in each image ("practical profile") coincides with each other, at least partially. Taking up a simple case of two elements and introducing the judgment rule, I examine the possibility of role dicrimination for all possible combinations of the images. After that, some general propositions as to the possibility of role discrimination, that hold regardless of the number of role elements, are discussed.     

Rank one preservers between spaces of Boolean matrices

In this paper, we characterize (i) linear transformations from one space of Boolean matrices to another that send pairs of distinct rank one elements to pairs of distinct rank one elements and (ii) surjective mappings from one space of Boolean matrices to another that send rank one matrices to rank one matrices and preserve order relation in both directions. Both results are proved in a more general setting of tensor products of two Boolean vector spaces of arbitrary dimension.     

Molecules of Post algebras

Following A. Abian, we define amolecule of a partially ordered set to be any nonzero elementm with the property that any two nonzero elements less than or equal tom have a nonzero lower bound. In a Boolean algebra an element is a molecule iff it is an atom; however, in an arbitrary distributive lattice the notion of a molecule is too general to permit explicit characterization. In this note we examine the molecules in a coproduct of two distributive lattices and use our results to show that molecules play the same role in Post algebras as atoms do in Boolean algebras.Presented by R. S. Pierce.I would like to thank the referee for helpful comments that resulted in simplifying a number of proofs in this paper.     

Prime, irreducible elements and coatoms in posets

In this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.     

Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

Stabilization and set stabilization of switched Boolean control networks via flipping mechanism

Stabilization and set stabilization of switched Boolean control networks are investigated by using flipping mechanism in this paper. Firstly, with the help of Warshall algorithm, an explicit criterion for the stabilization of switched Boolean control networks is derived. Secondly, the necessary and sufficient condition for the solvability of stabilization of switched Boolean control networks, by flipping some elements of perturbation set once, is presented. Thirdly, a search algorithm is proposed to calculate the minimum number of stabilization flipped nodes and what exactly they are. Furthermore, a necessary and sufficient condition is established for the solvability of set stabilization of switched Boolean control networks by flipping some elements of perturbation set once. Analogously, an algorithm is given to find the minimum number of set stabilization flipped nodes. Finally, examples are shown to demonstrate the feasibility of the above results.     

On strongly and weakly defined boolean terms

A problem of Gaifman about strongly and weakly defined Boolean terms is solved by finding a Boolean algebra ℱ with a complete subalgebra ℰ such that some element of ℱ not in ℰ can be obtained from elements of ℰ by meets and joins in the normal completion of ℱ.     

Generalizations of Boolean products for lattice-ordered algebras

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL_w-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.     

Finiteness conditions and distributive laws for Boolean algebras

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)