模糊集贴近度表示的一般规律

分析了模糊集贴近度理论,得到模糊集贴近度表示的几种形式,为贴近度的实际应用提供了极大的方便.     

模糊集贴近度的一般表示形式

根据模糊集贴近度理论,研究了模糊集贴近度的各种一般表示形式,推广了常见的各种贴近度表示形式.     

L-拓扑空间的相对β-紧性

在L-拓扑空间中引入相对β-紧性的概念,得到了它的一些性质,如它是L-好的推广,对β-闭子集遗传,被Mβ-连续的广义Zadeh型函数所保持等.     

Attribute dependency functions considering data efficiency

Pawlak’s attribute dependency degree model is applicable to feature selection in pattern recognition. However, the dependency degrees given by the model are often inadequately computed as a result of the indiscernibility relation. This paper discusses an improvement to Pawlak’s model and presents a new attribute dependency function. The proposed model is based on decision-relative discernibility matrices and measures how many times condition attributes are used to determine the decision value by referring to the matrix. The proposed dependency degree is computed by considering the two cases that two decision values are equal or unequal. A feature of the proposed model is that attribute dependency degrees have significant properties related to those of Armstrong’s axioms. An advantage of the proposed model is that data efficiency is considered in the computation of dependency degrees. It is shown through examples that the proposed model is able to compute dependency degrees more strictly than Pawlak’s model.     

Computations via Newtonian and relativistic kinematic systems

We are developing a rigorous methodology to analyse experimental computation, by which we mean the idea of computing a set or function by experimenting with some physical equipment. Here we consider experimental computation by kinematic systems under both Newtonian and relativistic kinematics. An experimental procedure, expressed in a language similar to imperative programming languages, is applied to equipment, having the form of a bagatelle, and is interpreted using the two theories. We prove that for any set A of natural numbers there exists a two-dimensional kinematic system B_A with a single particle P whose observable behaviour decides n∈A for all n∈N. The procedure can operate under (a) Newtonian mechanics or (b) relativistic mechanics. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on the methodology, which seeks a formal theory for performing abstract experiments with physical restrictions on the construction of systems. We conclude with some open problems.     

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n=|F|?2 and for a∈A let s(a) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a∈A with n−2s(a)?0. Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280-288] we verify the conjecture for the particular case when m?3 and n?m²−²m/2. Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n−2s(a), for all a∈A, is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.     

On Codes Identifying Sets of Vertices in Hamming Spaces

A code is called (t, 2)-identifying if for all the words x, y(x y) and the sets (B _t (x) B _t (y)) C and are nonempty and different. Constructions of such codes and a lower bound on the cardinality of these codes are given. The lower bound is shown to be sharp in some cases. We also discuss a more general notion of -identifying codes and introduce weakly identifying codes.     

Level Function Method for Quasiconvex Programming

In this paper, we introduce the notion of level function for a continuous real-valued quasiconvex function. The existence, construction, and application of level functions are discussed. Further, we propose a numerical method based on level functions for the solution of quasiconvex minimization problems. Several versions of the algorithms are presented. Also, we apply the idea of the level function method to the solution of a class of variational inequality problems. Finally, the results of numerical experiments on the proposed algorithms are reported.     

A Result on Vector Variational Inequalities with Polyhedral Constraint Sets

In this note, by using some well-known results on properly efficient solutions of vector optimization problems, we show that the Pareto solution set of a vector variational inequality with a polyhedral constraint set can be expressed as the union of the solution sets of a family of (scalar) variational inequalities.     

关于自相似集的Hausdorff测度

得到了 Ｈａｕｓｄｏｒｆｆ容度与 Ｈａｕｓｄｏｒｆｆ测度相等的集的充分必要条件．对于满足开集条件的自相似集,验证了它的Ｈａｕｓｄｏｒｆｆ容度与Ｈａｕｓｄｏｒｆ测度相等并给出了它的Ｈａｕｓｄｏｒｆｆ测度的一个便于应用的公式．作为例子,给出了均匀康托集的Ｈａｕｓｄｏｒｆｆ测度的一种新的计算方法,对于Ｋｏｃｈ曲线的Ｈａｕｓｄｏｒｆｆ测度的上限也作了讨论．