associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

the cauchy integral analytic capacity and uniform rectifiability several explanations concerning notation terminology and background are in order first notation by 7hi we have denoted the one dimensional hausdorff measure i e length and a z r stands for the closed disc with center z and radius r a curve f is called ad regular that is ahlfors david regular if it satisfies 1 2 with e f since the lower bound is automatic for curves this means that hl r n a z r 0 general sets satisfying 1 2 are called ad regular it is simplest to define the l2 boundedness of the cauchy singular integral operator via the truncated integrals we say that ce is bounded in l2 e without really defining the operator ce itself if there is m ox such that

a generalized concept of a fuzzy probabilistic controller this paper deals with some theoretical aspects of generalization of the concept called fuzzy probabilistic controller some basic operations on probabilistic sets and their distribution function characterization are discussed here then the main idea of the generalized algorithm of fuzzy probabilistic controller is described the considerations are illustrated by means of several examples

on distribution function description of probabilistic sets and its application in decision making the paper deals with some problems of decision making described and solved by the use of the concept of a probabilistic set the distribution function description of probabilistic sets which is important from a theoretical point of view is discussed and its application in decision making is illustrated by means of numerical examples

uniform rectifiability calderon zygmund operators with odd kernel and quasiorthogonality in this paper we study some questions in connection with uniform rectifiability and the l boundedness of calderon zygmund operators we show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the jones numbers we also use these new coefficients to prove that n dimensional calderonzygmund operators with odd kernel of type c2 are bounded in l if is an n dimensional uniformly rectifiable measure

epigraphical and uniform convergence of convex functions we examine when a sequence of lsc convex functions on a banach space converges uniformly on bounded sets resp compact sets provided it converges attouch wets resp painlev e kuratowski we also obtain related results for pointwise convergence and uniform convergence on weakly compact sets some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions for example a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of ell 1

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

fuzzy sets in pattern recognition methodology and methods abstract an objective of the paper is to discuss a state of the art of methodology and algorithms of fuzzy sets in the field of pattern recognition in real world recognition and classification problems we are faced with fuzziness that is connected with diverse facets of cognitive activity of the human being an origin of sources of fuzziness is related to labels expressed in feature space as well as to labels of classes taken into account in classification procedures an evident difference between a way of information processing by means of probability and fuzzy set theory and a way of interpretation of results is explained in detail in sequel methods of pattern recognition are studied in two main streams namely supervised and unsupervised learning different approaches to designing of classification schemes as e g relation calculus decision making approach etc are put into account a method of feature selection with the aid of fuzzy measure and fuzzy integral is introduced in clustering techniques methods relying on minimization of objective function are analyzed in detail also a problem of cluster validity expressed in terms of clustering indices is addressed numerical examples are also provided

mosco and slice convergence of level sets and graphs of linear functionals given that most results of interest for mosco convergence seem to require reflexivity see especially bb our results here suggest that although mosco convergence is well known and widely applied in the convex analysis community its essence is poorly understood in fact it is our view that mosco convergence in its historical formulation as well as the topology of mosco convergence introduced in bel is a misdefined concept

principal values for riesz transforms and rectifiability let e rd with hn e where hn stands for the n dimensional hausdorff measure in this paper we prove that e is n rectifiable if and only if the limit lime 0 y e x y ex y x y n 1dhn y exists hn almost everywhere in e to prove this result we obtain precise estimates from above and from below for the l2 norm of the n dimensional riesz transforms on lipschitz graphs

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

cauchy singular integrals and rectifiability of measures in the plane abstract let be a finite non negative borel measure on the complex plane c we shall prove the following result if for almost all a c formula and the limit formula exists and is finite then is rectifiable in the sense that there are rectifiable curves 1 2 such that formula

geometrical fuzzy clustering algorithms fuzzy clustering algorithms are a basic tool for cluster analysis among these the geometrical fuzzy clustering algorithms are used when the clustering problem can be viewed as trying to find linear or ellipsoidal concentrations in data this paper provides a theoretical framework in which currently used geometrical fuzzy clustering algorithms become special cases also a family of functions called feasible are defined which can be used to construct such algorithms and convergence results are obtained

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

potential applications of fuzzy sets in industrial safety engineering abstract this paper discusses potential applications of fuzzy set theory to risk analysis in the area of industrial safety engineering vagueness and imprecision in mathematical quantification of risk are equated with fuzziness rather than randomness the concept of risk evaluation using linguistic representation of the likelihood of the occurrence of a hazardous event exposure and possible consequences of that event is proposed the approximate reasoning technique based on fuzzy logic is used to derive fuzzy values of risk

on the concept of fuzzy probabilistic controllers the paper deals with a new approach to the construction of a so called fuzzy probabilistic controller where input output data are treated as probabilistic sets it is shown that the notion of a fuzzy controller is embedded in the concept discussed here first of all the distribution characterization of probabilistic sets is discussed some operations on probabilistic sets are presented and next the construction of the controller is performed numerical examples form an illustration of theoretical investigations

the measures with an associated square function operator bounded in l2 abstract in this paper we provide an extension of a theorem of david and semmes 3 to general non atomic measures the result provides a geometric characterization of the non atomic measures for which a certain class of square function operators or singular integral operators are bounded in l 2 the description is given in terms of a modification of jones coefficients

on some properties of a linguistic classifier abstract a general structure of a linguistic classifier wherein sub subclassifiers are the building blocks is suggested some properties of these sub subclassifiers are proposed and proved the physical significances of these properties are explored and it is shown that a condensed measure of belongingness of patterns to any class can be obtained which can conveniently be used for forming classification rules

on the uniform rectifiability of ad regular measures with bounded riesz transform operator the case of codimension 1 we prove that if is a d dimensional ahlfors david regular measure in mathbb r d 1 then the boundedness of the d dimensional riesz transform in l2 implies that the non baup david semmes cells form a carleson family combined with earlier results of david and semmes this yields the uniform rectifiability of

some aspects of synthesis of probabilistic fuzzy controllers abstract 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