Convex decompositions of fuzzy partitions

In this paper we investigate some algebraic and geometric properties of fuzzy partition spaces (convex hulls of hard or conventional partition spaces). In particular, we obtain their dimensions, and describe a number of algorithms for effecting convex decompositions. Two of these are easily programmable, and each affords a different insight about data structures suggested by the fuzzy partition decomposed. We also show how the sequence of partitions in any convex decomposition leads to a matrix for which the norm of the corresponding coefficient vector equals a scalar measure of partition fuzziness used with certain fuzzy clustering algorithms.     

Linear inequalities concerning partitions into distinct parts

The Ramanujan Journal - Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n)...     

Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof

The number c_n of weighted partitions of an integer n, with parameters (weights) b_k, k1, is given by the generating function relationship . Meinardus (1954) established his famous asymptotic formula for c_n, as n→∞, under three conditions on power and Dirichlet generating functions for the sequence b_k. We give a probabilistic proof of Meinardus' theorem with weakened third condition and extend the resulting version of the theorem from weighted partitions to other two classic types of decomposable combinatorial structures, which are called assemblies and selections.     

Linear fuzzy regression

This paper presents an approach which is useful for regression analysis in the case of heterogeneity of a set of observations, for which regression is evaluated. The proposed procedure consists of two stages. First, for a set of observations, fuzzy classification is determined. Due to this, homogenous classes of observations which are of hyperellipsoidal shape, are obtained. Then for each fuzzy class, the so called linear fuzzy regression is evaluated.

In the paper the method of calculating linear fuzzy regression coefficients is given. It is a generalized version of the least squares method. The formula for the values of coefficients is given. Some properties of linear fuzzy regression are analyzed. It is proved that in one- and two-dimensional cases, the formulae are analogous to those for usual regression. A measure of goodness-of-fit and the method of determination of the number of fuzzy classes are also given.

Presented examples indicate the superiority of fuzzy regression in comparison to usual regression in the case of heterogenous observations.     

Some aspects of synthesis of probabilistic fuzzy controllers

This paper deals with some aspects of analysis and design problems of a class of control algorithms called probabilistic fuzzy controllers.The notion of random intersection coefficient is proposed here.The decomposition of a multi-dimensional probabilistic fuzzy controller is discussed as well. Such decomposition provides uniform expressions for both single-input, single-output and multi-dimensional probabilistic fuzzy controllers and it improves the computational efficiency.As an example, a two-input and two-output control system is presented as well.     

Linear operators in finite dimensional probabilistic normed spaces

In this paper, we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces. Then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete.     

On a correspondence between probabilistic and fuzzy choice functions

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to be sufficient to ensure that the associated fuzzy choice function is rational.     

A study on the transitivity of probabilistic and fuzzy relations

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables.     

Linear programming with multiple fuzzy goals

This paper describes the use of fuzzy set theory in goal programming (GP) problems. In particular, it is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions. Models are then presented for the use of fuzzy goal programming with preemptive priorities, with Archimedean weights, and with the maximization of the membership function corresponding to the minimum goal. Examples are also provided.     

The aggregation operations for decision making in probabilistic fuzzy environment

This paper deals with some important classes of aggregation operations on various kinds of sets applied to decision making problems.These operations are mainly based on general concepts such as triangular norms (t- and s-norms). In this paper we focus particularly on operations on probabilistic sets and their distribution function representation. The considerations are illustrated by means of numerical examples.     

区间值模糊概率集及其在模式识别中的应用

首次提出区间值模糊概率集(IvFPS)的概念.区间值模糊概率集的矩和其他一些重要性质,以及区间值模糊概率集间的相关性也将得到研究和探讨.最后通过一个实际数例来阐述区间值模糊概率集的相关系数在模式识别中的应用.     

Comparison of the probabilistic approximate classification and the fuzzy set model

It is shown that the generalized notion (probabilistic approximate classification) of rough sets can be conveniently described by the concept of fuzzy sets. A discussion of the proper choice of the definition for the membership function of the intersection (union) of fuzzy sets is also presented. However, from the point of view of the probabilistic approximation space, it is argued that there does not exist a universal definition for the fuzzy intersection (union) operation.     

Sequentially congruent partitions and partitions into squares

The Ramanujan Journal - In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentiallyc congruent” partitions: the mth part is...     

Linear matrix differential dynamical systems with fuzzy matrices

In this paper, we investigate linear first-order fuzzy matrix differential dynamical systems where the coefficients matrix is described by a fuzzy matrix. We show some properties of the matrix differential dynamical systems, and their phase portraits are described by means of examples.