On the enumeration of certain sets of planted plane trees

Using the definition of planted plane trees given by D. A. Klarner (“A correspondence between sets of trees,” Indag. Math.31 (1969), 292–296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of (r + 1)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.     

模糊粗糙集的表示及应用

一个模糊粗糙集是一对模糊集,它可以用一簇经典粗糙集表示出来.本文研究了模糊粗糙集的表示问题,利用模糊集的分解定理证明了一个模糊粗糙集可以用一簇粗糙模糊集表示出来,利用这个结果可以证明模糊粗糙集的一些重要性质.     

Classification of non‐well‐founded sets and an application

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti‐well‐founded (awf) sets, i. e., non‐well‐founded sets whose all maximal ∈‐paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈‐sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbá?ek concerning nonstandard Boffa universes of sets is considerably extended.     

Codes projectifs à deux poids, “caps” complets et ensembles de différences

Using certain sets of points of a finite projective geometry some results are obtained from properties of two-weight projective codes. A problem concerning complete caps is solved. A deep connection between binary uniformily packed codes and difference sets over elementary Abelian 2-groups is established and a characterization of these difference sets is given.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

自相似集边界的结构及维数

本文研究了一类由平面上点的表示系统所生成的内部非空的自相似集,证明其边界曲线的一半是三个A－完备集的并集,并给出计算这类完备集的结构矩阵的简单方法,从而利用Marion定理得出这类自相似集边界的Hausdorff维数。     

函数S-粗集与规律双向分离

函数S-粗集(function singular rough sets)是用R-函数等价类定义的,函数是一个规律,函数S-粗集具有规律特征.函数S-粗集推广了Z.Pawlak粗集.利用函数S-粗集,给出规律生成,规律分离的讨论,提出规律分离定理.给出的结果在投资分险规律估计中得到了应用.     

关于等差数列型素数集的分解

设C为非负整数集.若存在非负整数集A和B,使得C=A+B,其中|A|,|B|≥2,则称C为可分解的,否则称C为不可分解的或本原的.讨论特殊的等差数列型素数集合,证明了4k+1型素数集不可分解,以及4k+3型素数集不可分解.     

Nonuniform Multiwavelet Sets and Multiscaling Sets

In this article, we study the theory of nonuniform minimally supported frequency multiwavelets and nonuniform multiscaling sets. A characterization of nonuniform multiwavelet sets is obtained which generalizes a result of Yu and Gabardo. After introducing a notion of generalized nonuniform scaling set, we obtain a characterization of nonuniform multiscaling sets associated with nonuniform multiresolution analysis having finite multiplicity. In addition, we provide a geometric construction to find families of symmetric nonuniform multiwavelet sets.     

Functional Erdös-Rényi laws for processes indexed by sets

We consider a process X(A) indexed by some sets A∈A, where A is a collection of sets. We prove a functional form of the Erdös-Rényi laws. The result may be specialized to get back and sometimes improve previous versions of the classical Erdös-Rényi laws.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

A compact labeling scheme for series-parallel graphs

A linear time labeling algorithm is presented for series-parallel graphs. The labels enable us to efficiently implement dynamic programming algorithms for sequencing problems with series-parallel precedence constraints. The labeling scheme can also be used to efficiently count and generate the initial sets, terminal sets and independent sets in transitive series-parallel digraphs and to provide a characterization of the maximal independent sets in transitive digraphs.     

Optimization of the Hausdorff distance between sets in Euclidean space

Set approximation problems play an important role in many areas of mathematics and mechanics. For example, approximation problems for solvability sets and reachable sets of control systems are intensively studied in differential game theory and optimal control theory. In N.N. Krasovskii and A.I. Subbotin’s investigations devoted to positional differential games, one of the key problems was the problem of identification of solvability sets, which are maximal stable bridges. Since this problem can be solved exactly in rare cases only, the question of the approximate calculation of solvability sets arises. In papers by A.B. Kurzhanskii and F.L. Chernous’ko and their colleagues, reachable sets were approximated by ellipsoids and parallelepipeds.In the present paper, we consider problems in which it is required to approximate a given set by arbitrary polytopes. Two sets, polytopes A and B, are given in Euclidean space. It is required to find a position of the polytopes that provides a minimum Hausdorff distance between them. Though the statement of the problem is geometric, methods of convex and nonsmooth analysis are used for its investigation.One of the approaches to dealing with planar sets in control theory is their approximation by families of disks of equal radii. A basic component of constructing such families is best n-nets and their generalizations, which were described, in particular, by A.L. Garkavi. The authors designed an algorithm for constructing best nets based on decomposing a given set into subsets and calculating their Chebyshev centers. Qualitative estimates for the deviation of sets from their best n-nets as n grows to infinity were given in the general case by A.N. Kolmogorov. We derive a numerical estimate for the Hausdorff deviation of one class of sets. Examples of constructing best n-nets are given.     

A property of creative sets

A new approach to the study of creative sets using the notion of a table is offered. Making use of tables conforming to recursively enumerable sets, novel properties of creative sets are established. Harrington's theorem on the definability of creative sets in the lattice of recursively enumerable sets is proved, and we reprove Lachlan's theorem which states that one of the factors in a direct product of creative sets is again creative. Supported by RFFR grant No. 93-01-16014. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 294–307, May–June, 1996.     

基于依赖空间的不协调决策信息系统的分配约简

从属性集角度研究不协调决策信息系统的分配约简问题。定义了一种决策分配二元关系,并利用这种关系建立了属性集幂集上的等价关系,由此产生依赖空间。同时利用决策分配二元关系和依赖空间给出了不协调决策信息系统分配协调集的判定定理,进而得到了一种保持不协调决策信息系统分配不变的属性约简方法。同时通过实例验证方法的有效性。     

An Interactive Approach for Project Selection

This paper describes an interactive method for presenting a sequence of feasible sets of indivisible projects to a decision-maker. For each set as a whole, the decision-maker evaluates its utilities with respect to each of several attributes; the utilities are then combined to give a single utility for the set.A sequence of zero-one programmes is used to ensure that the only sets presented are those which are feasible and which are not contained in larger feasible sets. The sets are presented in descending order of size, and the presentation can be terminated by the decision-maker or by supplementary rules.The method for presenting sets can be used in other contexts in which complete enumeration of possible sets would otherwise be required.     

On Radon’s recipe for choosing correct sites for multivariate polynomial interpolation

A class of sets correct for multivariate polynomial interpolation is defined and verified, and shown to coincide with the collection of all correct sets constructible by the recursive application of Radon’s recipe, and a recent concrete recipe for correct sets is shown to produce elements in that class.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

A decomposition of continuity via ideals

A new class of sets in ideal topological spaces is introduced and using these sets, a decomposition of continuity is given.      

An approach to spectral sets of certain zeta-functions

A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of those by trigonometric polynomials. The author studied spectral sets of the Riemann zeta-function and zeta-functions belonging to a certain class. The purpose of the present paper is to improve the results in the previous studies for spectral sets. The improvement is based on the relation between spectral sets and support of distributions.