subjective valuations in a production model abstract this paper is devoted to a model of economic dynamics first a definition of fuzzy mapping some of its properties and economic interpretations are presented next a model of economic dynamics is defined and problems of optimization are considered

linear objective function optimization with fuzzy relation equation constraints regarding max av composition in this paper an optimization model with a linear objective function subject to a system of the fuzzy relation equations with max av composition is presented the solution set of such a fuzzy relation equations is a non convex set in this paper firstly we discuss the feasible solution set with two schemes and secondly study relationship between maximum and minimum points and also the feasible points as well furthermore an algorithm and few concrete examples are presented in order to optimize linear objective function

matrix chebyshev polynomials and continued fractions in the first part we expose the notion of continued fractions in the matrix case in this paper we are interested in their connection with matrix orthogonal polynomials in the second part matrix continued fractions are used to develop the notion of matrix chebyshev polynomials in the case of hermitian coefficients in the recurrence formula we give the explicit formula for the stieltjes transform the support of the orthogonality measure and its density as a corollary we get the extension of the matrix version of the blumenthal theorem proved in j approx theory 84 1 1996 96 the third part contains examples of matrix orthogonal polynomials

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

on the properties of fuzzy production correspondence abstract this paper represents an exploratory effort to develop and to characterize a fuzzy production correspondence deterministic analysis of production has been generalized to a fuzzy environment zadeh s extension principle is utilized to derive the conditions for divisibility multiplicability additivity convexity and super additivity of fuzzy production sets furthermore several results concerning the relationships between the basic properties of fuzzy production sets are established

rational choice under fuzzy preferences the orlovsky choice function abstract there have been many different suggestions in the literature as to how to generate exact choices on the basis of fuzzy preferences this paper seeks to provide axiomatic support to one of these choice functions due to orlovsky it is shown that under a transitivity restriction independent of max min transitivity this choice function implies and is implied by a set of simple conditions

a note on the maximization of matrix valued hankel determinants with applications in this note we consider the problem of maximizing the determinant of moment matrices of matrix measures the maximizing matrix measure can be characterized explicitly by having equal matrix valued weights at the zeros of classical one dimensional orthogonal polynomials the results generalize classical work of schoenberg indag math 62 1959 282 to the case of matrix measures as a statistical application we consider several optimal design problems in linear models which generalize the classical weighing design problems

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

convexly combined fuzzy relational equations and several aspects of their application to fuzzy information processing abstract this paper presents some theoretical results on a set of the convexly combined fuzzy relational equations represented in the form x k 2 x k r y k k 1 2 k where x k and y k denote fuzzy sets r and 2 fuzzy relations the sup min compositional operator the inf max one and a fuzzy convex combinator whose complement is furthermore we propose a method to formulate the fuzzy information processing approximately by means of the convexly combined fuzzy relational equations focusing our attention on the modeling of fuzzy input output systems normality of fuzzy sets characterizing fuzzy input fuzzy logical aspects of systems and association between fuzzy information finally it is suggested that the proposed method is suitable for the modelling of several types of fuzzy systems

fuzzy relational equations with generalized connectives and their applications the idea of fuzzy relational equation with generalized connectives is introduced and algorithms of resolution of this class equations are presented in details it is clearly shown that some types of well known fuzzy relational equations sup min inf max can be treated as special cases of a wise class of equations under discussion applicational aspects are carefully investigated presenting the advantages of the proposed approach

functioning by analogy abstract the paper discusses some problems arising in the development of a mathematical apparatus enabling one to describe functioning of analogy based systems as well as ways to their practical application

solving a linear programming problem with the convex combination of the max min and the max average fuzzy relation equations abstract in this paper we introduce a fuzzy operator constructed by the convex combination of two known operators max min and max average compositions h j zimmermann fuzzy set theory and it s application kluwer academic publishers boston dordrecht london 1999 this operator contains some properties of the two known compositions when it generates the feasible region for linear optimization problems we investigate linear optimization problems whose feasible region is the fuzzy sets defined with this operator thus firstly the structure of these fuzzy regions is considered and then a method to solve the linear optimization problems with fuzzy equation constraints regarding this operator is presented

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

some specific types of fuzzy relation equations in this paper we study some specific types of fuzzy equations more specifically we focus on analyzing equations involving two fuzzy subsets of the same referential and a fuzzy relation defined over fuzzy subsets

an alternative approach to the resolution of fuzzy relation equations abstract in this paper we propose a solution procedure by means of a quasi characteristic matrix the matrix is so defined that not only the solution set can be determined but also the type of solution set such as the non empty set full set unique minimal solution and unique solution can be identified before the solution set is determined therefore it provides an insight into the structure of fuzzy relation equations

solving interval valued fuzzy relation equations solving systems of fuzzy relation equations is an important topic in fuzzy set theory this paper studies the composite interval valued fuzzy relation equations after analyzing the properties of its solution set we convert the fuzzy relation equations into a fuzzy relation inequality system and propose an efficient computational procedure to generate the whole solution set examples are included to illustrate the idea and algorithm

the bilateral union intersection operator abstract the theory of right union or right intersection preserving operator has been discussed by cheng lichun and peng boxing 1 here we introduce the theory of anti union or anti intersection operator and based on the four kinds of operators mentioned above the theory of the bilateral union intersection operators including the left or right inverse operator and its properties are proposed

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

fuzzy choice functions revealed preference and rationality abstract this paper investigates the conditions under which a fuzzy set valued choice function is rationalisable by a fuzzy revealed preference relation satisfying certain regularity conditions attention is confined to the case where the domain of the choice function consists of all crisp finite subsets of the universal set of alternatives two sets of weak and strong axioms of fuzzy revealed preference wafrp and safrp are stated in both cases it turns out that wafrp is not equivalent to safrp and that safrp does not characterise rationality in the special case where every set of avialable alternatives has at least one element which is unambiguously chosen wafrp is equivalent to safrp but safrp still does not characterise rationality a fuzzy congruence condition stronger than safrp is proposed and is shown to be necessary and sufficient for rationality

fast solving method of fuzzy relational equation and its application to lossy image compression reconstruction a fast solving method of the solution for max continuous t norm composite fuzzy relational equation of the type g i j r sup t spl square a sub i sup t spl square b sub j i 1 2 i j 1 2 j where a sub i spl isin f x x x sub 1 x sub 2 x sub m bj spl isin f y y y sub 1 y sub 2 y sub n r spl isin f x spl times y and spl square max continuous t norm composition is proposed it decreases the computation time ijmn l t p to jm i n l p where l t and p denote the computation time of min t norm and relative pseudocomplement operations respectively by simplifying the conventional reconstruction equation based on the properties of t norm and relative pseudocomplement the method is applied to a lossy image compression and reconstruction problem where it is confirmed that the computation time of the reconstructed image is decreased to 1 335 6 the compression rate being 0 0351 and it achieves almost equivalent performance for the conventional lossy image compression methods based on discrete cosine transform and vector quantization

the theory of approximate prices analytical foundations of experimental cost benefit analysis in a fuzzy decision space abstract the essay presents a theory of efficient prices for cost benefit analysis in a fuzzy space the approach proceeds by taking consumers income and producers outputs and costs as given the price preferences of consumers and producers are elicited and then embedded in a fuzzy space through fuzzy mappings to obtain a fuzzy compact price space where fuzzy price decisions are constructed solutions to the fuzzy price decision problems are abstracted through fuzzy mathematical programming to obtain fuzzy equilibrium prices from the fuzzy price space measures of price disagreement fuzzy consumer surplus and fuzzy producer surplus are advanced theorems of existence and uniqueness are stated the total result is a theory of fuzzy prices for cost benefit analysis for decision problem in general including cases where market imputations of prices may not be available to us as well as those cases where market failure may yield price distortions the theory is not only compactible with either contigent variation method direct information elicitation and revealed preference method market based evaluation but provides a direction for cases where problems may exist in both a computational example is provided to illustrate the working mechanism of the theory

convex combinations of generalized fuzzy relational equations abstract convex combinations of two fuzzy relational equations are studied these equations are built with compositions of the type conorm norm and norm conorm this study is a generalization of the works of ohsato and sekigushi previous results concerning usual fuzzy relational equations lead to the development of convexly combined equations first conditions for the existence of solutions are studied then the resolution of equations is discussed finally the concept of pseudo solutions of a system of several equations is introduced

solving fuzzy relation equations with a linear objective function abstract an optimization model with a linear objective function subject to a system of fuzzy relation equations is presented due to the non convexity of its feasible domain defined by fuzzy relation equations designing an efficient solution procedure for solving such problems is not a trivial job in this paper we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0 1 integer programming with a branch and bound solution technique after presenting our solution procedure a concrete example is included for illustration purpose

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

spatial fuzzy consumer s decision making a multicriteria analysis abstract this paper is devoted to a multicriteria analysis of the consumer s behavior when the decision maker is acting in a fuzzy space and manifesting an imprecise attitude at first the process of decision making is described with the help of three relationships between the set of goods which are supplied in several locations the set of their characteristics and the set of the consumer s a priori possible behaviors all these relations are fuzzy the model applies the theory of fuzzy relations equations then the stages of the decision process are analyzed often fuzzy behavior relations are like black boxes the mathematical solution of the model indicates in which conditions their valuations are possible the main interest of this method is not to use additive operations on subjective items and to use operators which are coherent with the fuzzy nature of the variables

adjointness of fuzzy systems this paper deals with two adjoint fuzzy systems based on and compositions which correspond to the normal use of and in l fuzzy logic respectively the resolution of composite fuzzy relational inequations for these adjoint fuzzy systems can be skillfully treated by the gentzen type sequent calculus as well as by the lattice theoretical way owing to the adjointness of and compositions

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

positive solutions for nonhomogeneous elliptic equations involving critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for the dirichlet problem u f z u q 2 u g z u 2 2 u in where 0 1 q 2 2 2 n n 2 0 r n n 3 is a bounded domain with smooth boundary and f g are nonnegative continuous functions on

optimization of fuzzy relation equations with max product composition abstract an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max product composition is considered since the feasible domain is non convex traditional linear programming methods cannot be applied we study this problem and capture some special characteristics of its feasible domain and the optimal solutions some procedures for reducing the original problem are presented the problem is transformed into a 0 1 integer program which is then solved by the branch and bound method for illustration purpose an example of the procedures is provided

on fuzzy multi valued functions part 1 introduction and general properties abstract in this paper a concept of fuzzy multi valued function is introduced and other related objects are defined next their properties that assume no topological structure are presented

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

a new look at solving a system of fuzzy relational equations abstract solving a system of fuzzy relational equations exactly has been studied for many years a usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint such understanding is reexamined in this paper we show that the usual pairwise disjoint condition is too conservative and a system of max t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi overlapped a condition commonly found in most rule based system applications in addition the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived if it cannot be satisfied we show that the system of equations could still be solved almost exactly by specifying the t norm as drastic product the results have been applied to study the capacity of fuzzy relations

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

from fuzzy set theory to non additive probabilities how have economists reacted abstract in this paper we are going to try to conduct a comparative review of the two programmes devoted to the application of fuzzy sets theory in economics for the first and to nonadditive probabilities for the second while putting forward some explanations and conjecture as to their development and their future in section 1 we defend the thesis which claims that the first programme has only had a global focus on the approach in terms of generalization and we shall go deeper into the analysis of the particular area of general equilibrium as it is the programme on which ponsard worked the most in section 2 we shall seek to show that the second programme devoted to the introduction of nonadditive probabilities in decision theory on the contrary has attempted to come up with independent findings this would explain why this last programme has managed to fit into topics at a standstill or in crisis easily which is undeniably the case of the standard theory of decision making while the first programme was only envisaged as a kind of mathematical game involving the application of new rules to an old model in the case of the first programme one might thus speak of an extension principle and in the second case of a resolution principle

fuzzy expected utility abstract the aim of this paper is to state the foundations of fuzzy decision theory under uncertainty attention is focused on the analysis of a fuzzy expected utility given a fuzzy decision problem which does not rest on a borel field but on a weak borel field and according to the information available we shall use successively the probability and the possibility of a fuzzy subset when the imprecise preference structure of the decision maker verifies four axioms we can determine a function of fuzzy expected utility this latter is a numerical translation of the imprecise preferences of the decision maker in an uncertain world

fuzzy mathematical models in economics abstract this paper is a survey of three fuzzy mathematical models dealing with economic choice economic calculation and general economic equilibrium they are the elements of a fuzzy theory of value the aim is to test the efficiency of the application of fuzzy subsets theory to the analysis of the foundations of economics and thus to legitimate its use in the area

an approach to the analysis of fuzzy systems this paper deals with a formal description of ill defined processes fuzzy systems by the use of fuzzy relational equations it is pointed out that fuzzy relational equations form a generalized version of the difference equations widely considered in control theory some equivalence between these two kinds of description is presented basic problems of fuzzy systems e g identification prediction sensitivity and stability are shown and numerical algorithms are given indices of each method are introduced especially the degree of fuzziness the sensitivity index which makes it possible to express the quality of each of them

on dyukarev s resolvent matrix for a truncated stieltjes matrix moment problem under the view of orthogonal matrix polynomials abstract the main result of this paper is a new representation of yu m dyukarev s resolvent matrix for the non degenerate truncated matricial version of the classical stieltjes moment tmcsm problem this new representation is based on the use of a quadruple of q q matrix polynomials which are characterized by certain orthogonality properties in this way useful new interrelations between the moment problems of stieltjes and hamburger which are associated with a stieltjes positive definite sequence s j j 0 2 n of complex q q matrices are found this includes an explicit formula connecting the resolvent matrices used by yu m dyukarev and i v kovalishina respectively furthermore the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a stieltjes positive definite sequence are expressed in terms of the dyukarev stieltjes parameters additionally we obtain two different representations for each extremal solution of the tmcsm problem via matrix continued fraction expansions

a small set of axioms for residuated logic we show that a set of three axioms together with modus ponens suffices to build up a formal system for the class r of all quadruples l t j in which l is a partially ordered set with a top element and t is a triangular norm on l that has a residuated implication j we show that this r is not a variety

the use of fuzzy outranking relations in preference modelling this paper deals with preference modelling in the context of decision aid in this framework conflicting systems of logic uncertain knowledge ambiguous positions are always present in order to tackle this problem a multiple criteria methodology is proposed mainly based on fuzzy outranking relations introduced both at one dimensional and multi dimensional levels some properties of outranking relations are investigated such relations are then combined using fuzzy logical connectives to generate relational systems of fuzzy preferences that are shown to be very useful to reflect the vagueness of information in the various preference situations that may be considered in the modelling process

issues on adjointness in multiple valued logics we contribute to the theory of implications and conjunctions related by adjointness in multiple valued logics we suggest their use in zadeh s compositional rule of inference to interpret generalized modus ponens inference schemata we provide new complete characterizations of implications that distinguish left arguments implications that satisfy the exchange principle divisible conjunctions commutative conjunctions associative conjunctions and triangular norms we also introduce and characterize pseudo strict and pseudo continuous implications and conjunctions and we explore the close relationship between these two notions

solving linear optimization problems with max star composition equation constraints in this paper an optimization model with a linear objective function subject to a system of fuzzy relation composition equations is presented since the non empty feasible solution set of the fuzzy relation equations is generally a non convex set the conventional linear programming method will not be suitable for solving such a problem therefore an efficient solution procedure for such problems appears to be necessary in this paper firstly we characterize the feasible solution set and then introduce two efficient procedures for solving the problem finally two concrete examples are included for more details

an algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max prod composition abstract fuzzy sets as the feasible region for optimization problems is an interesting and on going research topic s c fang g li solving fuzzy relations equations with a linear objective function fuzzy sets syst 103 1999 107 113 7 j lu s c fang solving nonlinear optimization problems with fuzzy relation constraints fuzzy sets syst 119 2001 1 20 16 e khorram a ghodpusian linear objective function optimization with fuzzy relation constraints regarding max av composition appl math comput in press doi 10 1016 j amc 2005 04 021 in this paper we focus on these kind problems in which the solutions region is the fuzzy relation equation with max prod composition and the objective function is linear whereas one of the major difficulties in such problems is non convexity of the feasible region it is preferable to study these regions in the first step hence we have primarily investigated two methods and their relationship and then we have determined the feasible region via them after determining the feasible set we have given an algorithm to optimize the linear objective function on such these regions finally we have presented two examples to illustrate the methods and algorithms

s norm aggregation of infinite collections abstract the aggregation of membership values by means of the iterate application of an s norm is of cardinal relevance for the development of any general theory of approximate reasoning which relies on a sup t model for relation composition here we present a definition of s norm aggregation for possibly infinite collections which can be demonstrated to generalize previous approaches consistently it provides a common framework in which classical sup t and finite s t compositions can be analyzed as well as other significant cases general properties of the s norm aggregation concept its specialization for the class of continuous s norms and its relationship with fuzzy measure theory are discussed

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

multiple positive solutions for a class of kirchhoff type problems involving critical sobolev exponents abstract in this paper we study the multiplicity results of positive solutions for a class of kirchhoff type problems involving critical growth terms with the help of nehari manifold and ljusternik schnirelmann category theory we investigate how the coefficient g x of the critical nonlinearity affects the number of positive solutions of that problem furthermore we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g

multiple positive solutions for indefinite semilinear elliptic problems involving a critical sobolev exponent in this paper we study the decomposition of the nehari manifold by exploiting the combination of concave and convex nonlinearities the result is subsequently used in conjunction with the ljusternik schnirelmann category and variational methods to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical sobolev exponent

extending the contraposition property of propositional logic for fuzzy abduction abduction deals with assumption based reasoning to explain an observation in the context of fuzzy reasoning abduction attempts to determine the membership function of the fuzzy propositions present in the antecedent of a rule when the membership functions for the propositions in the consequent of the rule are given currently available models of fuzzy abduction are capable of inferring the membership function of the antecedent clause accurately when the antecedent includes single fuzzy proposition however when the antecedent clause of a rule contains multiple fuzzy propositions these models fail to determine the independent membership function of the individual propositions present in the antecedent this paper presents a new formulation to handle the above problem by fuzzy extension of the well known contraposition property of propositional logic several interesting properties due to the fuzzy extension of the classical contraposition have been derived an algorithm for automated abduction using the extended contraposition property has been developed to demonstrate the principle of abduction with rules containing one or more fuzzy propositions in the antecedent consequent the time complexity of the proposed fuzzy abduction for a sequence of n chained rules where each rule has m fuzzy propositions is o mn considering a uniform cost for composition operation and t norm computation of the antecedent

an efficient procedure for solving a fuzzy relational equation with max archimedean t norm composition in the literature a necessary condition for minimal solutions of a fuzzy relational equation with max product composition shows that each of its components is either zero or the corresponding component s value of the greatest solution in this paper we first extend this necessary condition to the situation with max archimedean triangular norm t norm composition based on this necessary condition we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently furthermore rather than work with the actual equations we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions numerical examples with specific cases of the max archimedean t norm composition are provided to illustrate the procedure

compensatory fuzzy multiple level decision making abstract fuzzy set theory has been shown to be an effective tool to overcome the computational difficulties encountered in solving large multiple level programming problems shih et al 1996 in this paper compensatory operators are introduced for adjusting the decision making process between the different levels and also between the decision makers of the same level after a brief consideration of the bi level and three level systems the large decentralized organizations with both equal and unequal goals are investigated various numerical examples are given to compare the influences of compensation and to illustrate the approaches

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

generalized chebyshev polynomials and discrete schrodinger operators we investigate a solution of the difference equation with the boundary conditions u0a b i u 1a b 0 where a b are hermitian elements of a c algebra una b are usually called generalized chebyshev polynomials of the second kind the equation cannot be easily simplified as in the scalar case because a and b do not need to commute however we are able to compute the spectrum of the corresponding orthogonality measure which is very important for the investigation of the discrete schrodinger operator related to una b

the fuzzy relation equation with union or intersection preserving operator abstract in this paper we discuss the union or intersection preserving operator defined on 0 1 we introduce the fuzzy relation equation with union preserving operator and give a method that calculates all minimal solutions also we introduce the fuzzy relation equation with intersection preserving operator and give a method that calculates all maximal solutions

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

neurocomputational approach to solve a convexly combined fuzzy relational equation with generalized connectives abstract in this paper we present a method to solve a convexly combined fuzzy relational equation with generalized connectives for this we propose a neural network whose structure represents the fuzzy relational equation then we derive a learning algorithm by using the concept of back propagation learning since the proposed method can be used for a general form of fuzzy relational equations such fuzzy max min or min max relational equations can be treated as its special cases

fuzzy phoneme recognition abstract first a scheme for pattern recognition is given in a learning step a relation between acoustic characteristics and the corresponding phoneme is computed using an inference rule next the phonemes are recognised as result of a fuzzy relation equation then the recognition quality is analysed by means of uncertainty measures specially sugeno s and possibility measures

extensions of fuzzy relational compositions based on generalized quantifiers abstract the aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by bandler and kohout these compositions employ two quantifiers the universal quantifier and the existential quantifier as there exists a big gap between these two quantifiers that may be appropriately filled in by intermediate generalized quantifiers we take this as a motivation for our study in particular we introduce the concept of fuzzy relational compositions based on generalized quantifiers furthermore we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers yet sometimes in a weaken form apart from fuzzy relational compositions the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations

local superlinearity and sublinearity for indefinite semilinear elliptic problems in this paper the usual notions of superlinearity and sublinearity for semilinear problems like du f x u are given a local form and extended to indefinite nonlinearities here f x s is allowed to change sign or to vanish for s near zero as well as for s near infinity some of the well known results of ambrosetti bre zis cerami are partially extended to this context

relative asymptotics for orthogonal matrix polynomials in this paper we study sequences of matrix polynomials that satisfy a non symmetric recurrence relation to study this kind of sequences we use a vector interpretation of the matrix orthogonality in the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class we study the generalized matrix chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples the concept of a dirac delta functional is introduced we show how the vector model that includes a dirac delta functional is a representation of a discrete sobolev inner product it also allows to reinterpret such perturbations in the usual matrix nevai class finally the relative asymptotics between a polynomial in the generalized matrix nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a dirac delta functional is deduced

the nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition abstract using the nehari manifold and fibering maps we prove the existence theorem of the nonlinear boundary problem u u u p 2 u for x u n u q 2 u for x on a bounded domain r n we discuss how the nehari manifold changes as changes and show how the existence of solutions depends on the properties of the nehari manifold

multiple positive solutions for the critical kirchhoff type problems involving sign changing weight functions abstract this paper is concerned with the multiplicity of positive solutions for the critical kirchhoff type problems involving indefinite weight functions 0 1 m u 2 d x u q x u q 2 u k x u 2 2 u x u 0 x where is a smooth bounded domain in r n n 3 1 q 2 m s a b s with 0 a 0 and b 0 the weight functions q and k are continuous and changing sign using the nehari manifold fibering maps and ljusternik schnirelmann category we prove that at least two positive solutions for 0 1 exist provided that 1 and 2 4 furthermore by the mountain pass theorem and ekeland s variational principle it is shown that 0 1 possesses at least three positive solutions whenever 2 n 2 including the case that 1 and 2 4 our results generalize some recent results in the literature

multiple positive solutions for a dirichlet problem involving critical sobolev exponent abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the ljusternik schnirelmann category to prove that the existence of multiple positive solutions for a dirichlet problem involving critical sobolev exponent

positive solutions for critical inhomogeneous elliptic problems in non contractible domains in this paper we investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non contractible domains we show that for f epsilon h 1 satisfying a suitable condition and 0 f not equivalent to 0 the dirichlet problem

an application of fuzzy subsets theory to the analysis of the consumer s spatial preferences abstract this paper discards to usual assumptions of the classical theory of consumer behaviour in mathematical economics first that the consumer is capable of perfectly discriminating between different goods second that the goods capable of satisfying needs are all supplied at a unique point actually consumer behaviour is by nature fuzzy and goods are scattered in a supply space and are transportable in this model the consumer is located at a given point of demand first a general study is made of the structures of spatial preferences in a fuzzy context next the particular properties are introduced in order to define a behaviour of attainable choice then the conditions which guarantee the existence of a completely preordered topological space are imposed in order to verify that the consumer s fuzzy behaviour is coherent these conditions permit to define the existence and continuity of a fuzzy utility as a numerical representation of the imprecise preference the resulting economic theory is neither ordinal nor cardinal but a valuation theory of preference

producer s spatial equilibrium with a fuzzy constraint the classical theory of the producer s equilibrium rests on two sets of particularly restrictive hypotheses first it is implicitely assumed that all inputs and outputs are located in a single place where the producer is also implanted and where the production is carried out next it is assumed that the producer follows a precise behaviour pattern by this we mean that the producer has complete information concerning the conditions of hisproductive activity and he has perfect command over both the set of inputs and the set of outputs he realises the maximum profit allowed by the technological constraint which limits his possible actions and by the given price system the aim of this study is to discard these two familiesof hypotheses relating first to the economic space of goods and of the agent and second to the precision of the economic agent s behaviour

an efficient solution procedure for fuzzy relation equations with max product composition we study a system of fuzzy relation equations with max product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions instead of solving the problem combinatorially the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions

positive solutions for a kirchhoff type problem involving multiple competitive potentials and critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for a kirchhoff type problem involving several potentials and critical nonlinearities on unbounded domain the first main result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials the second one is that with the help of nehari manifold and ljusternik schnirelmann category the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established moreover we present a sufficient condition for the nonexistence of ground state solution

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm

the truncated hamburger matrix moment problems in the nondegenerate and degenerate cases and matrix continued fractions abstract the present paper deals simultaneously with the nondegenerate and degenerate truncated hamburger matrix moment problems in a unified way based on the use of the schur algorithm involving matrix continued fractions a full analysis of them together with a relative matrix moment problem on the real axis is given with the help of the correspondence between the moment problem on the real axis and the nevanlinna pick np interpolation the solutions of the nontangential np interpolation in the nevanlinna class are derived as an application

a motion compression reconstruction method based on max t norm composite fuzzy relational equations a motion compression reconstruction method based on max t norm composite fuzzy relational equations mcf is proposed where into intra pictures i pictures and predictive pictures p pictures of the original motion are compressed by uniform and non uniform coders respectively the non uniform coders of the proposed method can preserve edge information of p pictures on the compressed image to perform an effective compression reconstruction of the p pictures a design method of non uniform coders is proposed based on an overlap level of fuzzy sets and a fuzzy equalization an experiment using 10 p pictures confirms that the root mean square error of the reconstructed images obtained by the proposed non uniform coders is decreased to 89 4 of that one of the uniform coders under the condition that compression rate the ratio between the file size of compressed image and original one is 0 0057 two test motions tennis and woman 100 frames are compressed and reconstructed by the proposed mcf

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions