On the Loewy Length and the Noetherian Dimension of Artinian Modules

The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions.     

New Dimensions on Translations Between Logics

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: (conservative) translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another. Dedicated to the memory of Mário Tourasse Teixeira and Antonio Mário Sette     

Gaussian measure of sections of dilates and translations of convex bodies

In this paper we give a solution for the Gaussian version of the Busemann–Petty problem with additional information about dilates and translations. We also discuss the size of the Gaussian measure of the hyperplane sections of the dilates of the unit cube.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.     

落影空间——模糊集合的概率描述

模糊集合论与概率论都是在描述客观现实中的不确定现象。二者之间既有联系又有区别,它们的相互关系一直被人们探讨着。近年来,国外有些学者从二者的同一性方面着眼,作了一些工作。I.R.Goodman指出:对于任一模糊子集,都可以构造一个随机集合以该模糊子集为其落影(“落影”一词是笔者后取的);H.T.Nguyen从信任函数的角度出发,也论证了随机集合与模糊集合的关系;K.Hirota、H.Kwakernaak、E.P.Klement等人从另外一些角度试图建立模糊集合论与概率论之间的联系。     

A generalized notion of quotient for finite Abelian groups

A finite Abelian group G is partitioned into subsets which are translations of each other. A binary operation is defined on these sets in a way which generalizes the quotient group operation. Every finite Abelian group can be realized as such a generalized quotient with G cyclic.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

带共轭且带平移的变系数奇异积分方程组的封闭形式解

本文讨论了一类带轭且带平和多的变系数奇异积分方程组，在一些补充假定下，利用化为四元素边值问题的方法，得到了所求方程级数形式的解和积分形式的可解条件。     

Translations and construction of generalised nets

A theory of certain types of translations for generalised nets is developed, and the structure of nets constructed by a special method is analysed, especially with regard to subsets and extensions. This enables nonisomorphism results to be established, along with theorems that the number of nonisomorphic solutions of a certain type (for example, complete or maximal) tends to infinity with k.     

Size of subsets of groups and Haar null sets

This is a study of several notions of size of subsets of groups. The first part (sections 3–5) concerns a purely algebraic setting with the underlying group discrete. The various notions of size considered there are similar to each other in that each of them assesses the size of a set using a family of measures and translations of the set; they differ in the type of measures used and the type of translations allowed. The way these various notions relate to each other is tightly and, perhaps, unexpectedly connected with the algebraic structure of the group. An important role is played by amenable, ICC (infinite conjugacy class), and FC (finite conjugacy class) groups.The second part of the paper (section 6), which was the original motivation for the present work, deals with a well-studied notion of smallness of subsets of Polish, not necessarily locally compact, groups – Haar null sets. It contains applications of the results from the first part in solving some problems posed by Christensen and by Mycielski. These applications are the first results detecting connections between properties of Haar null sets and algebraic properties (amenability, FC) of the underlying group.Received: October 2003 Revision: January 2004 Accepted: January 2004     

EXTENDING SCHEMES TO A CARTESIAN CLOSED CATEGORY

Abstract

Our objective was to embed schemes of finite type over a field k in a suitably small Cartesian closed category. Two types of globalized versions of an ind-affine scheme were proposed: locally ind-affine ringed spaces and ind-schemes obtained by taking the inductive limit of closed subschemes of a locally ind-affine ringed space in ringed spaces. First in ε case some reasonably general conditions implying that translations, basic open subsets and closed subsets of an ind-affine scheme are again ind-affine schemes were obtained. Certain immersive properties of locally ind-affine ringed spaces are shown. As an adjunct we then determine a class of locally ind-affine ringed spaces which since they patch appropriately are ind-schemes. A restriction of locally ind-affine ringed spa1 leads to the category of locally ind-affine schemes (containing the category of schemes of finite type over k) which is see1 to be Cartesian closed with respect to the contravariant variable. Possible extensions to the covariant variable are studied.     

Some aspects of intuitionistic fuzzy sets

We first discuss the significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets. We then consider the extension to intuitionistic fuzzy subsets of a number of ideas from standard fuzzy subsets. In particular we look at the measure of specificity. We also look at the problem of alternative selection when decision criteria satisfaction is expressed using intuitionistic fuzzy subsets. We introduce a decision paradigm called the method of least commitment. We briefly look at the problem of defuzzification of intuitionistic fuzzy subsets.     

局部环上矩阵广义逆半群的子集及其性质

设R是一个局部环,A是一个可相似对角化的n阶矩阵.利用矩阵方法研究了环R上矩阵A的广义逆半群的子集,得到了其做成正规子群的条件和其中元素可逆的条件,也得到了矩阵广义逆半群的一些性质.     

N-fuzzy Filters in Ordered Semigroups

In this paper we define N-fuzzy filters,N-fuzzy bi-ideal subsets and N-fuzzy bi-filters of ordered semigroups and characterize ordered semigroups in terms of N-fuzzy filters, N-fuzzy bi-ideal subsets and N-fuzzy bi-filters.We establish relationship of N-fuzzy filters and prime N-fuzzy ideals of ordered semigroups. Also we discuss the relationship of N-fuzzy bi-filters and prime N-fuzzy bi-ideal subsets of ordered semigroups.     

Average densities of frequency bands of wavelets

A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L ²(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero.     

On a Remarkable Class of Left-symmetric Algebras and its Relationship with the Class of Novikov Algebras

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x, y]·z = 0. We derive some basic characterizations of such left-symmetric algebras, and we highlight their relationships with the so-called Novikov algebras and derivation algebras.     

Frame definability in finitely valued modal logics

In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics.     

Orthomorphisms and the construction of projective planes

We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four self-dual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the ``sporadic' (not translation) plane of Mathon.

     

半拓扑子集的一些性质

本文引入半边界,并且进一步研究了半拓扑子集的一些性质,例如十集定理(定理2.6).另外也讨论了子空间,乘积空间和商空间中的半开集.     

Diameter partitioning

We discuss the problem of partitioning a set of points into two subsets with certain conditions on the diameters of the subsets and on their cardinalities. For example, we give anO(n ² logn) algorithm to find the smallestt such that the set can be split into two equal cardinality subsets each of which has diameter at mostt. We also give an algorithm that takes two pairs of points (x, y) and (s, t) and decides whether the set can be partitioned into two subsets with the respective pairs of points as diameters.