任意基数集上的拟阵的圈及连通性

通过研究任意基数集上的拟阵圈的性质,继而得出当此拟阵连通时,它可由含任一给定元素的圈集唯一确定的结论.     

超拟阵的独立集公理、基公理和圈公理

Dunstan等在1972年首先提出了超拟阵的概念,用以将定义拟阵的承载集合从有限集推广到偏序集.Barnabei等在1998年研究了另一种偏序集上的拟阵结构,即偏序集拟阵.由有限分配格和有限偏序集之间的对应关系可知,偏序集拟阵就是分配格上的超拟阵.本文研究超拟阵的公理系统,建立模格上的超拟阵的独立元公理,证明模格上超拟阵的中间基性质和基的交换性质并用这两个性质分别刻画了模超拟阵.最后指出了Barnabei等给出的分配超拟阵圈公理中的一个错误,重新提出并证明分配超拟阵的圈消去性质并建立了分配超拟阵的圈公理.作为圈消去性质的一个应用,本文证明了分配超拟阵中覆盖基的元素包含唯一的圈.     

由上近似数诱导的拟阵及其特征

拟阵理论与粗糙集理论之间有很多相似之处,近年来,探讨这二者之间的联系成为一个研究热点.首先利用基于等价关系的上近似数诱导了一系列拟阵结构,然后刻画了这类拟阵的独立集、基集、秩函数以及闭包等,讨论了与其它拟阵之间的一些联系.     

拟阵基图的连通性

设 E 是有限元素的集合,M 是 E 上的拟阵,B 是M 的基集,记 M=(E,B).对任意的S_1、S_2■E,令 S_1-S_2={e|e∈S_1,e■S_2},S_1+S_2={e|e∈S_1或 e∈S_2},若 S={e},则简记为 S=e.图 G 的顶点及边集合分别记为 V(G)、E(G).拟阵 M 的基图 G=B(M)使 V(G)={b|b∈B},对任意的 b、b′∈V(G),bb′∈E(G)当且仅当|b-b′|-1.拟阵的基图是图的树图概念的推广,它在实际中有重要应用.文献[1]证明了:任意一个拟阵的基图如果至少     

关于模糊拟阵的圈子集套

本文主要采用通过导出拟阵来研究模糊拟阵的方法,探讨模糊拟阵模糊圈的性质和构造。这种方法的基本原理是两条:闭模糊拟阵可以由其导出拟阵序列和基本序列唯一确定,而模糊圈可以被分解为导出拟阵的圈和独立子集套。借助这种方法,本文主要做了三方面工作:一是讨论了模糊拟阵的模糊圈集和导出拟阵圈集之间的关系。比如模糊圈、初等模糊圈和最大初等模糊圈与导出拟阵圈之间的关系等;二是基于模糊圈和导出拟阵圈之间的关系,定义了导出拟阵圈函数和导出拟阵圈子集套两个概念。然后,详细研究了利用这两个概念来构造模糊圈的方法。同时,分析了在圈子集套和数列满足什么条件时,这种方法有效;三是分别用导出拟阵圈和圈子集套给出了准模糊图拟阵和精细模糊拟阵的充要条件。     

关于一致模糊横贯拟阵的研究

"模糊横贯拟阵'的反例"~([1])一文指出不是所有模糊集族的模糊部分横贯都能构成一个模糊拟阵的模糊独立集族。本文找到一类满足"一致性"条件的模糊集族,其模糊部分横贯全体一定能组成一个模糊拟阵的模糊独立集族(称这类模糊拟阵为一致模糊横贯拟阵);然后详细讨论了一致模糊横贯拟阵的基本序列、导出拟阵序列和模糊基等许多性质;还讨论了一致模糊横贯拟阵与准模糊图拟阵的关系,与正规模糊拟阵的关系;最后证明了一致模糊横贯拟阵是一类准模糊图拟阵。     

M-模糊化相关集族及M-模糊化β-圈集族

本文在已有研究基础上,讨论了M-模糊化相关集族的若干性质,给出了M-模糊化β-圈集族的定义,研究了M-模糊化β-圈集族的基本性质,并借助于M-模糊化拟阵的层拟阵结构,得到由M-模糊化相关集族和M-模糊化β-圈集族可以等价刻画M-模糊化拟阵这一重要结论。     

关于模糊拟阵的独立子集套和独立集函数

本文的研究方法主要是将模糊拟阵问题转化为普通拟阵问题来研究的方法。本文首先建立独立子集套概念,并使用这个概念和独立集函数概念构建了闭模糊拟阵的充要条件和模糊独立集的充要条件;然后,本文仔细分析了模糊基的性质,找到了一个使用独立子集套和独立集函数来描述的模糊基的充要条件;最后,利用模糊基的这个充要条件提出并证明了闭正规模糊拟阵的充要条件。     

拟阵基图的k-因子

本文提出了拟阵中元素邻接的概念。从而证明了任意一个简单拟阵 M 的基图都有一个ρ-连通的2ρ-因子,其中ρ是 M 的余秩。     

闭模糊拟阵模糊圈的算法研究

用闭模糊拟阵的基本序列来研究和描述它的模糊圈,找到了从闭模糊拟阵的模糊相关集或模糊独立集计算模糊圈的方法,并给出了相应的算法.     

On the reconstruction of binary codes from the dimensions of their subcodes

We prove that to reconstruct an arbitrary binary code up to equivalence it suffices to know the dimensions of all subcodes of even cardinality. It is shown that this theorem cannot be improved.     

On the Largest Numbers of Congruences of Finite Lattices

Order - We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czédli’s...     

The Rogers Semilattices of Generalized Computable Enumerations

We study the cardinality and structural properties of the Rogers semilattice of generalized computable enumerations with arbitrary noncomputable oracles and oracles of hyperimmune Turing degree. We show the infinity of the Rogers semilattice of generalized computable enumerations of an arbitrary nontrivial family with a noncomputable oracle. In the case of oracles of hyperimmune degree we prove that the Rogers semilattice of an arbitrary infinite family includes an ideal without minimal elements and establish that the top, if present, is a limit element under the condition that the family contains the inclusion-least set.     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

有横贯的Hall-型判定定理

给出了任意集上的一个集族有横贯的一些Hall-型判定定理,回答了Welsh关于相应的无限情形的公开问题,即对于无限情况,找到一个集族有横贯的充要条件.     

On Zipf’s law and rank distributions in linguistics and semiotics

A number of formulas of linguistic statistics are refined. The notions of real and virtual cardinality of a sign are introduced. We show that a formula refining Zipf’s law for the occurrence frequencies in frequency dictionaries can be extended to arbitrary sign objects, i.e., semiotic systems.     

Realization of a polylinear controller as a second-order differential system in a Hilbert space

We study the solvability of the inverse nonlinear system analysis problem viewed as qualitative solvability (necessary and sufficient conditions) of a realization of a time-varying polylinear controller as a second-order differential system whose admissible solutions include a given nonlinear pencil of arbitrary (finite, countable, or continual) cardinality of dynamic processes in a separable Hilbert space.     

Linear transformations on matrices: rank preservers and determinant preservers (Note)

Characterizations of rank preserving and determinant preserving linear transformations on matrices over an algebraically closed field, due to Marcus and Frobenius respectively, are generalized to linear transformations on matrices over an arbitrary field (rank preservers) or over any field which satisfies a weak cardinality condition (determinant preservers).     

On the period length of the generalized Lagrange algorithm

The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number fields. This algorithm yields a system of fundamental units and the class number of an arbitrary algebraic number field by means of computing cycles of reduced ideals. In this paper we prove that the cardinality of a cycle of reduced ideals in an ideal class of an order of an algebraic number field is O(R), where R is the regulator of this order, and where the O-constant only depends on the degree of the field. We also give a lower bound on this cardinality.