Algebras of fuzzy sets

In this paper we investigate two kinds of algebras of fuzzy sets, which are obtained by using Zadeh's extension principle. We give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets. We prove that if the structure of truth values is a complete residuated lattice, the induced algebra of a subalgebra of an algebra can be embedded into the induced algebra of fuzzy sets of . For direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product. In the case of homomorphisms and direct products, the two kinds of algebras of fuzzy sets behave in different ways.     

On the direct product of fuzzy topological groups

The connections between some countability properties of a topological space (X, $\text{T}$) and its generated fuzzy topological space (X, ω($\text{T}$)) are investigated.     

On some results of analysis for fuzzy metric spaces

A necessary and sufficient condition for a fuzzy metric space to be complete is given. We prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable. Uniform limit theorem is generalized to fuzzy metric spaces.     

On the topological equivalence of flows of Brouwer homeomorphisms

We prove that the set of all regular points of a flow of Brouwer homeomorphisms is invariant under topological equivalence of flows. We also show that a similar result holds for the first prolongational limit set.     

On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies

In this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies. Necessary and sufficient conditions for the existence of a solution are given, and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique.     

On the two-dimensional cauchy problem of a hyperbolic system

In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may involve delta-wave.     

Topological classification of supertransitive flows on closed nonorientable surfaces. Part I. Necessary conditions for topological equivalence

In this paper a topological invariant is introduced for the class of supertransitive flows on closed nonorientable surfacesM of negative Euler characteristic. We describe properties of this invariant and prove that it provides necessary conditions for the topological equivalence of flows belonging to the above-mentioned class of supertransitive flows. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 461–470, September, 2000.     

The solutions of linear fuzzy stochastic differential systems

As a completion of previous work of Feng (Fuzzy Sets and Systems 115 (2000) 351) on the general theory of fuzzy stochastic differential systems, the solutions of linear fuzzy stochastic differential systems for general coefficient matrix are discussed.     

On the restudy of fuzzy complex analysis: Part II. The continuity and differentiation of fuzzy complex functions

We consider functions mapping complex numbers into fuzzy complex numbers or mapping fuzzy complex numbers into complex numbers. We study the continuity and differentiation of such functions.     

On the fuzzy solution of LR fuzzy linear systems

In this paper, the solution of fuzzy linear system (FLS) is investigated based on a 1$1$-level expansion. To this end, 1$1$-level of FLS is solved for calculating the core of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Using our proposed method, if the computed solution satisfies the FLS, it can be as exact or approximated solution. Finally, the existence of solution of FLS is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method.     

On the algebraic solution of fuzzy linear systems based on interval theory

In this paper, fuzzy linear systems involving a crisp square matrix and a fuzzy right-hand side vector are considered. A new approach to solve such systems based on interval theory and the new concept “interval inclusion linear system” is proposed. Also, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Numerical examples are given to illustrate the efficiency of the proposed method.     

On the calculation of extended max and min operations between convex fuzzy sets of the real line

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.     

On strong consistency of the fuzzy generalized nearest neighbor rule

The k nearest neighbor rule (k-NNR) is a well-known nonparametric decision rule in pattern classification. Fuzzy set theory has been widely used to represent the uncertainty of class membership. Several researchers extended conventional k-NNR to fuzzy k-NNR, such as Bezdek et al. [Fuzzy Sets and Systems 18 (1986) 237–256], Keller et al. [IEEE Trans. Syst. Man, and Cybern. 15(4) (1985) 580–585], Béreau and Dubuisson [Fuzzy Sets and Systems 44 (1991) 17–32]. In this paper, we describe a fuzzy generalized k-NN algorithm. This algorithm is a unified approach to a variety of fuzzy k-NNR's. Then we create the strong consistency of posterior risk of the fuzzy generalized NNR.     

Some properties of the equations governing a two-dimensional quasi-gasdynamic model of traffic flows

A recently proposed two-dimensional quasi-gasdynamic model of traffic flows is considered. Its Petrovskii parabolicity is analyzed, and the stability of small perturbations against a constant background is investigated. In a nonlinear setting, an energy equality is derived and an energy estimate of the solution is obtained.     

On the loss of information due to fuzziness in experimental observations

The absence of exactness in the observation of the outcomes of a random experiment always entails a loss of information about the experimental distribution. This intuitive assertion will be formally proved in this paper by using a mathematical model involving the notions of fuzzy information and fuzzy information system (as intended by Tanaka, Okuda and Asai) and Zadeh's probabilistic definition. On the basis of this model we are first going to consider a family of measures of information enclosing some well-known measures, such as those defined by Kagan, Kullback-Leibler and Matusita, and then to establish methods for removing the loss of information due to fuzziness by increasing suitably the number of experimental observations.