the reduction of binary fuzzy relations and its applications this paper introduces the concept of t reduction of binary fuzzy relations and establishes its properties it presents an approach to providing the one valued restoration of any boundary transitive and t asymmetric binary fuzzy relation by its t reduction a theoretical basis for this approach is established showing that t reduction is the smallest fuzzy relation whose t transitive closure coincides with the given binary fuzzy relation and where an nxn fuzzy relation on a finite universe with cardinality n can be represented by n 1 values this approach opens up an original way for constructing the membership functions of certain binary fuzzy relations important examples for the application of this concept are given

inverse wave scattering with discontinuous wave speed the inverse scattering problem on the line is studied for the generalized schrodinger equation d2 dx2 k2h x 2 q x where h x is a positive piecewise continuous function with positive limits h as x this equation in the frequency domain describes the wave propagation in a nonhomogeneous medium where q x is the restoring force and 1 h x is the variable wave speed changing abruptly at various interfaces a related riemann hilbert problem is formulated and the associated singular integral equation is obtained and proved to be uniquely solvable the solution of this integral equation leads to the recovery of h x in terms of the scattering data consisting of q x a reflection coefficient either of h and the bound state energies and norming constants some explicitly solved examples are provided

on approximate reasoning with graded rules this contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy if then rules formalized by approximating formulas of a special type two alternatives that follow from the dual character of approximating formulas are developed in parallel the link to the theory of fuzzy control systems is also explained

an algorithm for computing the transitive closure of a fuzzy similarity matrix abstract up to now many algorithms for computing the transitive closure of a fuzzy similarity matrix have been proposed this paper presents a new algorithm which possesses all the advantages of these previous algorithms moreover its complexity is smaller

on the riemann hilbert problem for the one dimensional schrodinger equation a matrix riemann hilbert problem associated with the one dimensional schrodinger equation is considered and the existence and uniqueness of its solutions are studied the solution of this riemann hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual faddeev class some examples of explicit solutions of the riemann hilbert problem are given and the connection with ambiguities in the inverse scattering problem is established

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

l fuzzy covering relation we investigate lattice valued l fuzzy covering or fuzzy neighboring relation arising from a given lattice valued order we show that every l fuzzy covering is obtained by synthesis of crisp coverings arising from the corresponding cut orderings as a converse we introduce and investigate lattice valued relations which turn out to be fuzzy coverings of fuzzy orders

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

conservative nonlinear difference scheme for the cahn hilliard equation ii abstract numerical solutions for the cahn hilliard equation is considered using the crank nicolson type finite difference method existence of the solution for the difference scheme has been shown by brouwer fixed point theorem stability convergence and error analysis of the scheme are shown we also show that the scheme preserves the discrete mass even though the linearized scheme in 1 is conditionally stable and does not preserve the mass

intentional possibilistic approach to dealing with completeness vagueness and uncertainty abstract the paper outlines a possibilistic intentional framework suitable for dealing with incompleteness vagueness and ambiguity which captures their global as well as local aspects after discussing three levels of intentions in possibilistic modelling we present a formalization of an intentional possibilistic scheme for capturing the statics and dynamics of interaction of participants and observers of a modelling process we continue with a detailed discussion of perspectives in fuzzy logics formalization of perspectival models by means of fuzzy relational structures presents the foundations for discussion of the relationship of observation and participation the paper is concluded with exploratory examination of interrelationship between vagueness ambiguity incompleteness and inexactness within the framework of our possibilistic perspectival approach

on a reflexivity preserving family of cardinality based fuzzy comparison measures in this paper we generalize and extend previous work on a triparametric family of cardinality based fuzzy similarity measures more specifically we i generalize this family to a broader family of fuzzy comparison measures ii summarize previously obtained results regarding this family and fill up a few gaps by adding some novel results iii compare our parametric family with an already existing one and iv elaborate on the ranking invariance between certain members of the presented triparametric family of cardinality based fuzzy comparison measures

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

t partitions of the real line generated by idempotent shapes abstract the idea of generating fuzzy numbers as equivalence classes of particular t equivalences on the real line r is fully exploited scales or generators are used to define certain pseudo metrics on r by means of a shape function these pseudo metrics are then transformed into binary fuzzy relations on r shapes leading to t equivalences and hence to a class of fuzzy numbers forming a t partition of r are completely characterized in the case of a continuous generator this characterization problem is shown to be closely related to determining the idempotents w r t the t addition of fuzzy numbers

a frequentist view on cycle transitivity of reciprocal relations we establish a connection between two transitivity frameworks the transitivity of fuzzy relations based on a commutative quasi copula and the cycle transitivity of reciprocal relations w r t the dual quasi copula as an upper bound function loosely speaking it turns out that the latter can be characterized by imposing a lower bound on the relative frequency with which the former is fulfilled when applied to reciprocal relations we provide two compelling cases the 4 6 theorem expressing that the winning probability relation of a set of independent random variables is at least 66 66 product transitive and the 5 6 theorem expressing that the mutual rank probability relation associated with a given poset is at least 83 33 product transitive moreover these lower bounds turn out be rather conservative illustrating that from a frequentist point of view transitivity is abundant

algorithms for computing the min transitive closure and associated partition tree of a symmetric fuzzy relation abstract based on a previously derived weight driven algorithm for the computation of the t transitive closure of an arbitrary binary fuzzy relation on a finite universe with t a triangular norm we establish three algorithms for the computation of the min transitive closure of a symmetric matrix with elements in 0 1 as a by product these algorithms enable to generate the partition tree associated with the min transitive closure of the given matrix in descending order of the cutting parameter

on the existence and construction of t transitive closures in this paper it is shown that any fuzzy relation r on an arbitrary universe x has a t transitive closure the triangular norm t involved is not subject to any conditions this existential result can be turned into an explicit expression in the case of a left continuous triangular norm t furthermore this continuity restriction vanishes in the case of a finite universe x in that case the explicit expression can also be operationalized

on the compositional characterization of complete fuzzy pre orders complete pre orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations in this paper we investigate this characterization in a fuzzy setting we consider two types of completeness weak completeness and strong completeness and decompose a fuzzy pre order by means of an indifference generator in particular a frank t norm in the weakly complete case we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them in the strongly complete case we lay bare a stronger type of transitivity of the strict preference relation we conclude the paper with a rather negative result there is no hope to obtain a compositional characterization of weakly complete fuzzy pre orders and hence also not of fuzzy pre orders in general

a linearized finite difference method for the solution of some mixed concave and convex non linear problems in the present work a linearized finite difference scheme is proposed in order to approximate the solution of some non linear equation characterized by mixed concave and convex non linearities it is proved that the scheme is uniquely solvable and convergent the resulting method is also analyzed for consistency and stability some numerical examples illustrating the method described in our work are given

factoring fuzzy transitivity transitivity of a precise weak preference relation implies that its asymmetric component strict preference and its symmetric component indifference are transitive and conversely the transitivity of strict preference and indifference implies that the underlying weak preference relation is transitive this paper examines to what extent analogous results may be obtained for the different transitivity conditions and different methods for deriving strict preference and indifference being used in the literature of choice with fuzzy preferences

fuzzy rough sets the forgotten step traditional rough set theory uses equivalence relations to compute lower and upper approximations of sets the corresponding equivalence classes either coincide or are disjoint this behaviour is lost when moving on to a fuzzy t equivalence relation however none of the existing studies on fuzzy rough set theory tries to exploit the fact that an element can belong to some degree to several soft similarity classes at the same time in this paper we show that taking this truly fuzzy characteristic into account may lead to new and interesting definitions of lower and upper approximations we explore two of them in detail and we investigate under which conditions they differ from the commonly used definitions finally we show the possible practical relevance of the newly introduced approximations for query refinement

lattice valued f transforms and similarity relations abstract the relationships between lattice valued f transforms and similarity relations are investigated several similarity relations are naturally associated with f transforms three types of such similarity relations related to lattice valued f transforms are investigated namely similarity relationships that are related to spaces with fuzzy partitions as ground structures of the f transform relationships between extensional hulls with respect to similarity relations and f transforms and relationships between similarities of pairs of functions on the one hand and similarities of the f transforms of these functions on the other some examples of possible applications are also presented

a top down algorithm for generating the hasse tree of a fuzzy preorder closure this paper describes a new top down algorithm for the stepwise generation of the different levels or hasse diagrams of the hasse tree associated to the fuzzy preorder closure min transitive closure of a given reflexive binary fuzzy relation the algorithm is based upon a recently established weight driven method for computing the min transitive closure of a reflexive binary fuzzy relation the way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the hasse tree the newly proposed algorithm does not require the complete computation of this closure

a factorization of the scattering matrix for the schrodinger equation and for the wave equation in one dimension for the one dimensional schrodinger equation with a potential decaying at both ends of the real axis the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential a similar result also holds for the wave equation in a nonhomogeneous nondispersive medium where the wave speed has the same asymptotics at both ends of the real line

phase plane analysis and classification of solutions of a mixed sublinear superlinear elliptic problem abstract in this paper some mixed sublinear superlinear critical problem extending the famous problem of brezis nirenberg are analysed the existence of solutions is discussed a phase plane analysis is performed in order to transform the problem into an ordinary differential equation finally a full classification of radial solutions according to their behavior at the origin is performed for subcritical critical and supercritical cases

characterisation of main classes of fuzzy relations using fuzzy modal operators fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets each of these operators is determined by a binary fuzzy relation on a given universe and transforms one fuzzy set in this universe to another one in this paper we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators interpretation of these characterisations is presented in the context of fuzzy modal logics we show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics

on modeling discontinuous media three dimensional scattering a realistic model for the propagation of scalar waves in a medium must take into account discontinuities of the medium parameters which may act as hard reflectors discontinuities of their gradient which may act as soft reflectors and continuous variations of parameters the direct scattering problem is presented here the mixed potential impedance equation that characterizes the model depends on two arbitrary parameters one of them would be the potential if the equation reduces to the schrodinger equation the other one would be related to the impedance if the equation describes more classical waves after the mathematical tools are constructed green s functions etc a rigorous three dimensional scattering theory is described it encompasses the quantum three dimensional scattering theory and the theory of acoustical scattering for instance by systems of regular surfaces of arbitrary shape the main integral equations of the quantum scattering theory are generalized scattering amplitudes due to reflectors and scattering amplitudes due to diffuse scattering after reflectors have been taken into account are defined and constructed born and quadratic approximations are discussed the explicit formulas corresponding to the scattering by discontinuities and those corresponding to diffuse scattering are not reducible to each other exactly i e unless filtering and errors are allowed the results can also be used to described rigorously the three dimensional scattering by the wave equation in the frequency domain and in particular the response to an impulsive localized source further generalizations are in progress

the simplest method of ascending value to find fuzzy transitive closure abstract on the basis of 1 this paper presents a method to find the fuzzy transitive closure i e the simplest method of ascending value this method makes the number of times of ascending values least and it avoids verification by using the small rectangle condition

generalized morphisms a new tool for comparative evaluation of performance of fuzzy implications t norms and co norms in relational knowledge elicitation this paper describes a technique for identification of the knowledge structures implicit in scientific data and further develops the supporting methodology for analysis of fuzzy relational structures using the mathematical theory of fuzzy relations and generalized morphism initial knowledge elicited using repertory grid with semiotic descriptors is transformed to fuzzy relational knowledge structures which capture semantic relationships in a problem domain by means of bk product in analysis the fuzzy relational structures computed in the different sized domains or those computed by means of various fuzzy logics can be analyzed using generalized morphism namely global structure embedding or structure preserving as well as meta analysis of fuzzy relational structures the result of meta analysis is represented as the hasse diagram structure of meta relations by classifying equivalent classes of fuzzy relational structures the paper is concluded with a case study using real life engineering data from the aviation industry

computationally efficient sup t transitive closure for sparse fuzzy binary relations the property of transitivity is one of the most important for fuzzy binary relations especially in the cases when they are used for the representation of real life similarity or ordering information as far as the algorithmic part of the actual calculation of the transitive closure of such relations is concerned works in the literature mainly focus on crisp symmetric relations paying little attention to the case of general fuzzy binary relations most works that deal with the algorithmic part of the transitive closure of fuzzy relations focus only on the case of max min transitivity disregarding other types of transitivity in this paper after formalizing the notion of sparseness and providing a representation model for sparse relations that displays both computational and storage merits we propose an algorithm for the incremental update of fuzzy sup t transitive relations the incremental transitive update itu algorithm achieves the re establishment of transitivity when an already transitive relation is only locally disturbed based on this algorithm we propose an extension to handle the sup t transitive closure of any fuzzy binary relation through a novel incremental transitive closure itc algorithm the itu and itc algorithms can be applied on any fuzzy binary relation and t norm properties such as reflexivity symmetricity and idempotency are not a requirement under the specified assumptions for the average sparse relation both of the proposed algorithms have considerably smaller computational complexity than the conventional approach this is established both theoretically and verified via appropriate computing experiments

scattering and inverse scattering in one dimensional nonhomogeneous media the wave propagation in a one dimensional nonhomogeneous medium is considered where the wave speed and the restoring force depend on location in the frequency domain this is equivalent to the schrodinger equation d2 dx2 k2 k2p x q x with an added potential proportional to energy the scattering and bound state solutions of this equation are studied and the properties of the scattering matrix are obtained the inverse scattering problem of recovering the restoring force when the wave speed and the scattering data are known are also solved

construction of fuzzy automata from fuzzy regular expressions li and pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice ordered monoid and generalize the well known results of the classical theory of formal languages in particular they have shown that a fuzzy language over an integral lattice ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton however they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression in this paper we provide such an efficient method transforming scalars appearing in a fuzzy regular expression ainto letters of the new extended alphabet we convert the fuzzy regular expression ato an ordinary regular expression a r then starting from an arbitrary nondeterministic finite automaton a that recognizes the language a r represented by the regular expression a r we construct fuzzy finite automata a a and a a r with the same or even less number of states than the automaton a which recognize the fuzzy language a represented by the fuzzy regular expression a the starting nondeterministic finite automaton a can be obtained from a r using any of the well known constructions for converting regular expressions to nondeterministic finite automata such as glushkov mcnaughton yamada s position automaton brzozowski s derivative automaton antimirov s partial derivative automaton or ilie yu s follow automaton

hierarchical clustering problems and analysis of fuzzy proximity relation on granular space in this paper based on granular space some hierarchical clustering problems and analysis for fuzzy proximity relation are developed by using rigorous mathematical descriptions and four results are obtained first the granular representation of a fuzzy proximity relation is studied an algorithm is obtained to compute the granular space or hierarchical clustering structure and the min transitive closure derived from the same fuzzy proximity relation on a finite universe and the consistent clustering properties are discussed second a consistency index that is based on granular space is developed and a global optimizing mathematical model is established to obtain the optimal clustering of a fuzzy proximity relation furthermore for given two fuzzy proximity relations on different granulations of the same universe a collaborative clustering method is studied by the intersection operation and this method can provide a better hierarchical clustering structure finally the theory for hierarchical clustering analysis of fuzzy proximity relations on granular space is established for given two fuzzy proximity relations on the same universe two sufficient conditions theorems 6 1 and 6 2 of isomorphism are given and the necessary condition theorem 6 3 and the sufficient condition theorem 6 4 for e similarity are obtained we also illustrate the effectiveness of these theories and methods with some practical examples these results may help form a comprehensive theory and methodology for related potential application and will help us to obtain a deeper understanding of the essence of hierarchical clustering procedures

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

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

closing reciprocal relations w r t stochastic transitivity we equip the set of reciprocal relations with an appropriate order relation capturing the bipolar semantics turning this set into a complete join semilattice we also introduce the notion of a compatible family of reciprocal relations and show that such a family has an infimum moreover we discuss the one to one correspondence between the set of 3 valued reciprocal relations and the set of complete crisp relations we refine the theorem of bandler and kohout concerning the existence of closures of elements of a poset w r t some given property rendering it applicable to the above join semilattice the paper is solely dedicated to the transitivity property although uniquely defined for 3 valued reciprocal relations general reciprocal relations can exhibit various types of transitivity we characterize the mappings g and h for which the corresponding g stochastic and h isostochastic transitive closure exists for any reciprocal relation in particular it follows that weak and strong stochastic transitive closures always exist while this is not the case for moderate stochastic transitivity moreover max isostochastic transitivity turns out to be the only practically relevant type of isostochastic transitivity finally we provide algorithms realizing each of these transitive closures

a transitivity analysis of bipartite rankings in pairwise multi class classification many multi class classification algorithms in statistics and machine learning typically combine several binary classifiers in order to construct an overall classifier in the popular pairwise ensemble one classifier is built for each pair of classes resulting in pairwise bipartite rankings in contrast ordinal regression algorithms consider a single ranking function for several ordered classes it is known in the literature that pairwise ensembles can be useful for ordinal regression however can single ranking models make a contribution to multi class classification the answer to this question should be affirmative as supported by theoretical results presented in this article we conduct a formal analysis of the consistency of pairwise bipartite rankings by uncovering the conditions under which they can be equivalently expressed in terms of a single ranking similar to the utility representability of pairwise preference relations it turns out that transitivity plays a crucial role in the characterization of the ranking representability of pairwise bipartite rankings to this end we introduce the new concepts of strict ranking representability a restrictive condition that can be verified easily and auc ranking representability a practically more useful condition that is more difficult to verify however the link between pairwise bipartite rankings and dice games allows us to formulate necessary transitivity conditions for auc ranking representability a sufficient condition on the other hand is obtained by introducing a new type of transitivity that can be verified by solving an integer quadratic program

bell type inequalities for quasi copulas we present all bell type inequalities regarding at most four random events of which not more than two are intersected at the same time reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors we prove that some of these inequalities hold for quasi copulas or copulas additionally for the important parametric family of frank t norms which are also copulas we identify for each of the inequalities the parameter values for which it is fulfilled thereby completing the results of pykacz and d hooghe

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

lyapunov type operators for numerical solutions of pdes in the present paper numerical methods are developed to approximate the solutions of some evolutionary nonlinear problems the continuous problems are transformed into some lyapunov type equations and then analysed for existence uniqueness convergence stability and error estimates the main idea consists in applying fourier analysis and von neumann criterion acting translation and scaling parameter methods to obtain contractive operators leading to fixed point theory

computing a t transitive lower approximation or opening of a proximity relation since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation assuming that such a violation is mainly due to decision maker estimation errors in some way the more number of elicitations the more probable inconsistency is this is mostly the case within a fuzzy framework even when the number of alternatives or object to be classified is relatively small in this paper we propose a fast method to compute a t indistinguishability from a reflexive and symmetric fuzzy relation being t any left continuous t norm the computed approximation we propose will take o n 3 time complexity where n is the number of elements under consideration and is expected to produce a t transitive opening to the authors knowledge there are no other proposed algorithm that computes t transitive lower approximations or openings while preserving the reflexivity and symmetry properties

discontinuous media and underdetermined scattering problems the problem of ambiguities in trying to determine a shape by means of scattering experiments with one or a few illuminating angles and all directions of receivers is discussed by means of numerical experiments the model equation gives a good representation of scattering of scalar waves which can taken into account impedance discontinuities inside the scatterer physical problems include for instance acoustical waves in media where the density and the lame parameter lambda may vary continuously everywhere except across a finite number of smooth surfaces through which they jump

nodal solutions for some nonlinear elliptic equations abstract in this paper we propose to study nodal solutions of some nonlinear elliptic equations derived from the famous equation of brezis nirenberg and we analyse in some cases the possible singularities of radial solutions at the origin

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures ii the identity case abstract it has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de morgan triplet involved contains a continuous archimedean triangular norm having zero divisors or hence a transform of the ukasiewicz triangular norm in this second part additional arguments for this statement are supplied in what we call the identity case is the identity mapping and the involutive negator is the standard negator first it is shown that the use of a continuous non archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible secondly using such a triangular norm implies that at least in the square 0 4 5 2 it actually behaves as the ukasiewicz triangular norm furthermore an important transformation theorem indicates that any fuzzy preference structure with respect to a de morgan triplet containing such a continuous non archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard ukasiewicz triplet these additional arguments conclude in a convincing way our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

a comment on construction of fuzzy automata from fuzzy regular expressions the aim of this comment is to point out that the fuzzy language recognized by a nondeterministic finite automaton which is associated with a fuzzy regular expression in the context of stamenkovic iric s method stamenkovic and iric 2012 11 is recognized by a fuzzy finite automaton with e moves every fuzzy automaton with e moves is also equivalent to a fuzzy automaton and there are effective methods for removing e moves in order to obtain the equivalent fuzzy automaton in this way our proposal to convert a fuzzy regular expression to a fuzzy finite automaton is based on the construction of a fuzzy automaton with e moves for the fuzzy regular expression and the construction of an equivalent fuzzy finite automaton by means of e removal operation

efficient computation of transitive closures abstract we describe an efficient time space balanced algorithm for computation of the transitive max min closure of a proximity relation i e of a fuzzy relation that is reflexive and symmetric the algorithm creates a binary tree representation of the transitive closure in o m log 2 m time and o m space where m is the number of edges in the proximity graph a central idea in algorithm is to order the edges after decreasing strength and then to process them in that order for any pair of vertices their similarity i e their membership value in the transitive closure is looked up in the tree in o log 2 n time where n is the number of vertices in the proximity graph we compare the performance of the algorithm with the performances of two other presented algorithms that compute transitive closures of broader classes of fuzzy relations

fuzzy multi criteria analysis of cutting techniques in a nuclear reactor dismantling project abstract three different cutting techniques employed for dismantling the thermal shield of the belgian br3 nuclear reactor are analyzed within the framework of classical multi criteria utility theory this utilitarian approach allows to incorporate into the decision analysis factors which are difficult to quantify in monetary terms such as the environmental impact of the cutting technique a well known drawback of this approach is that it imposes a linear ordering on the alternatives in real world situations it is common practice to deal with alternatives towards which the decision maker is indifferent or which are incomparable to the decision maker in order to allow for indifference and incomparability in the decision process the problem is redefined in terms of a newly developed fuzzy methodology this methodology leads to a fuzzy quasi order relation in the set of alternatives that can be analyzed at different a cuts producing classical quasi order relations a remarkable asset is that this technique is equally applicable for the analysis of the criteria involved in the decision process in conclusion this approach results in a better understanding of the role of both alternatives and criteria in this mcdm problem

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

l fuzzy interior systems the aim of this work is to introduce the concept of l fuzzy interior systems and l fuzzy interior operators we start by establishing a connection between l fuzzy interior systems and l fuzzy interior operators it is shown that an l fuzzy interior system is precisely the fuzzy system in opposition to the crisp system and an l fuzzy interior operator is a suitable interior operator that has a close relation to an l fuzzy interior system it is also shown that there is a galois correspondence between the category of l fuzzy interior system spaces and that of l fuzzy interior spaces

integral equation methods for the inverse problem with discontinuous wave speed the recovery of the coefficient h x in the one dimensional generalized schrodinger equation d2 dx2 k2h x 2 q x where h x is a positive piecewise continuous function with positive limits h as x is studied the large k asymptotics of the wave functions and the scattering coefficients are analyzed a factorization formula is given expressing the total scattering matrix as a product of simpler scattering matrices using this factorization an algorithm is presented to obtain the discontinuities in h x and h x h x in terms of the large k asymptotics of the reflection coefficient when there are no bound states it is shown that h x is recovered from an appropriate set of scattering data by using the solution of a singular integral equation and the unique solvability of this integral equation is established an equivalent marchenko integral equation is derived and is shown to be uniquely solvable the unique recovery of h x from the solution of this marchenko equation is presented some expli

a novel hierarchical clustering combination scheme based on fuzzy similarity relations clustering combination methods have received considerable attentions in recent years and many ensemble based clustering methods have been introduced however clustering combination techniques have been limited to flat clustering combination and the combination of hierarchical clusterings has yet to be addressed in this paper we address and formalize the concept of hierarchical clustering combination and introduce an algorithmic framework in which multiple hierarchical clusterings could be easily combined in this framework the similarity based description matrices of input hierarchical clusterings are aggregated into a transitive consensus matrix in which the final hierarchy could be formed empirical evaluation by using popular available datasets confirms the superiority of combined hierarchical clustering introduced by our method over the standard single hierarchical clustering methods

inverse scattering in 1 d nonhomogeneous media and recovery of the wave speed the inverse scattering problem for the 1 d schrodinger equation d2 dx2 k2 k2p x q x is studied this equation is equivalent to the 1 d wave equation with speed 1 1 p x in a nonhomogeneous medium where q x acts as a restoring force when q x is integrable with a finite first moment p x 1 and bounded below and satisfies two integrability conditions p x is recovered uniquely when the scattering data and q x are known some explicitly solved examples are provided

finite difference approximate solutions for a mixed sub superlinear equation in ben mabrouk and ben mohamed 2006 a ben mabrouk m l ben mohamed on some critical and slightly super critical sub superlinear equations far east j appl math 23 1 2006 73 90 special volume of pdes a ben mabrouk m l ben mohamed nodal solutions for some nonlinear elliptic equations appl math comp in press doi 10 1016 j amc 2006 08 003 it has proved some theoretical results dealing with some boundary value problem du f u 0 in b and u 0 on b where b is a domain in r d with smooth boundary in the present paper we perform a difference scheme method to approximate the solution of a nonlinear evolutionary problem associated to the elliptic problem studied in ben mabrouk and ben mohamed 2006 we give at the end some numerical implementations based on a mixed sublinear superlinear term of the form f u u p email protected u q 1u

special properties closures and interiors of crisp and fuzzy relations this paper offers optimal definitions of the special properties which relations on a set may possess and which combine to define the classically important types of preorders both kinds of orders tolerances and equivalences the useful concept of local properties is introduced the definitions are shown to unify in a highly convenient way fuzzy and crisp considerations the concepts of closure and interior of a relation with respect to a property are defined necessary and sufficient conditions for the full existence of specific closures and interiors are given and formulae for all of them are presented and validated a widely important area of application of the closure and interior operators is discussed finally fast algorithms are given for the computer calculation of the more troublesome closures and a standardized procedure for the analysis of local orders is presented

bound states and inverse scattering for the schrodinger equation in one dimension the one dimensional schrodinger equation is considered when the potential and its first moment are absolutely integrable when the potential has support contained on the left right half line it is uniquely constructed by using only the reflection coefficient from the right left the bound state norming constants determine whether the potential has support contained on a half line or on the full line the bound state energies and the unique set of norming constants yielding the potential with support contained on the left right half line are completely determined by the reflection coefficient from the right left an explicit example is provided

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

algorithms for the computation of t transitive closures we present two weight driven algorithms for the computation of the t transitive closure of a symmetric binary fuzzy relation on a finite universe x with cardinality n or equivalently of a symmetric n spl times n matrix with elements in 0 1 with t a triangular norm the first algorithm is proven to be valid for any triangular norm t whereas the second algorithm is shown to be valid when t is either the minimum operator or an archimedean triangular norm furthermore we investigate how these algorithms can be appropriately adapted to generate the t transitive closure of nonsymmetric binary fuzzy relations or matrices as well

on the construction of adjunctions between a fuzzy preposet and an unstructured set in this work we focus on adjunctions also called isotone galois connections in the framework of fuzzy preordered sets hereafter fuzzy preposets specifically we present necessary and sufficient conditions so that given a mapping f a b from a fuzzy preposet a into an unstructured set b it is possible to construct a suitable fuzzy preorder relation on b for which there exists a mapping g b a such that the pair f g constitutes an adjunction

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures i general argumentation abstract the generalization of the concept of a classical or crisp preference structure to that of a fuzzy preference structure expressing degrees of strict preference indifference and incomparability among a set of alternatives requires the choice of a de morgan triplet i e of a triangular norm and an involutive negator the resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences more specifically one of the degrees of strict preference indifference or incomparability should always be unconstrained to the preference modeller this intuitive requirement is violated when choosing a triangular norm without zero divisors since in that case fuzzy preference structures reduce to classical preference structures and hence none of the degrees can be freely assigned furthermore it is shown that the choice of a continuous non archimedean triangular norm having zero divisors is not compatible with our basic requirement the sets of degrees of strict preference indifference and incomparability in 0 1 are always bounded from above by a value strictly smaller than 1 these fundamental results imply that when working with a continuous triangular norm only archimedean ones having zero divisors are suitable candidates these arguments sufficiently support our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

studies in fuzzy relations over fuzzy subsets in this paper a generalisation of fuzzy relations is introduced fuzzy relations are defined on fuzzy subsets properties like ordered reflexivity symmetry transitity transitive closures of such generalised relations and operations on them are discussed

computing the lattice of all fixpoints of a fuzzy closure operator we present a fast bottom up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees along with the partial order on the set of all fixpoints fuzzy closure operators appear in several areas of fuzzy logic and its applications including formal concept analysis fca that we use as a reference area of application in this paper several problems in fca such as computing all formal concepts from data with graded attributes or computing non redundant bases of all attribute dependencies can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data the development of a general algorithm that is applicable in particular to these problems is the ultimate purpose of this paper we present the algorithm its theoretical foundations and experimental evaluation

mi algebras the existing arithmetics of fuzzy numbers usually stemming from the cut arithmetic do not preserve some of the important properties of the standard arithmetics of real numbers although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties the most important properties should be preserved at least in a weakened form we present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves all the important algebraic properties of the arithmetic of real numbers and thus seems to be an important seed for further investigations on this topic the suggested approach to arithmetics of extensional fuzzy numbers is demonstrated by many examples and besides its algebraic properties it is also shown to carry some desirable practical properties the investigation leads to novel algebraic structures mi algebras mi monoids mi groups mi fields that abstract the discussed properties the main idea of these structures is based on a set of pseudoidentities that complements the only commonly used identity element in a monoid group structure these pseudoidentities are elements that generalize the identity like behavior and allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form appropriate properties of suggested mi structures are introduced and demonstrated by a number of practical examples

formal concept analysis based on fuzzy granularity base for different granulations this paper introduces granular computing grc into formal concept analysis fca it provides a unified model for concept lattice building and rule extraction on a fuzzy granularity base for different granulations one of the strengths of grc is that larger granulations help to hide some specific details whereas fca in a grc context can prevent losses due to concept lattice complexity however the number of superfluous rules increases exponentially with the scale of the decision context to overcome this we present some inference rules and maximal rules and prove that the set of all these maximal rules is complete and nonredundant thus users who want to obtain decision rules should generate maximal rules examples demonstrate that application of the method is valid and practicable in summary this approach utilizes fca in a grc context and provides a practical basis for data analysis and processing

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

weak and graded properties of fuzzy relations in the context of aggregation process we study graded properties a properties of fuzzy relations which are parameterized versions of properties of a fuzzy relation defined by l a zadeh namely we take into account fuzzy relations which are a reflexive a irreflexive a symmetric a antisymmetric a asymmetric a connected a transitive where a 0 1 we also pay our attention to the composed versions of these basic properties e g an a equivalence a orders we also consider the so called weak properties of fuzzy relations which are the weakest versions of the standard properties of fuzzy relations we take into account the same types of properties as in the case of the graded ones using functions of n variables we consider an aggregated fuzzy relation of given fuzzy relations we give conditions for functions to preserve graded and weak properties of fuzzy relations

on modeling discontinuous media one dimensional approximations distorted wave born series are derived for the impedance equation a 1 d dx a d dx k2 p 0 where a is a piecewise differentiable function the first and second orders are given explicitly they hold for impedances a which include discontinuities before and or after perturbation so that the problem could not be reduced to the schrodinger one the results are used for discussing homogenizations currently practiced in modeling discontinuous media

label ranking by learning pairwise preferences preference learning is an emerging topic that appears in different guises in the recent literature this work focuses on a particular learning scenario called label ranking where the problem is to learn a mapping from instances to rankings over a finite number of labels our approach for learning such a mapping called ranking by pairwise comparison rpc first induces a binary preference relation from suitable training data using a natural extension of pairwise classification a ranking is then derived from the preference relation thus obtained by means of a ranking procedure whereby different ranking methods can be used for minimizing different loss functions in particular we show that a simple weighted voting strategy minimizes risk with respect to the well known spearman rank correlation we compare rpc to existing label ranking methods which are based on scoring individual labels instead of comparing pairs of labels both empirically and theoretically it is shown that rpc is superior in terms of computational efficiency and at least competitive in terms of accuracy

inverse problems for nonabsorbing media with discontinuous material properties one dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties in addition an impedence mismatch between source and medium is allowed the measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem the solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media

on the role of acyclicity in the study of rationality of fuzzy choice functions the role of the acyclicity property of fuzzy preference relations is studied in the framework of rationality of fuzzy choice functions two standard ways of constructing a fuzzy choice function from a given fuzzy preference relation are considered and properties such as acyclicity and completeness are shown to be sufficient to ensure the rationality of the fuzzy choice function special attention is paid to the triangular norm used for modelling the conjunction the results obtained are compared to the classical results on rationality of crisp choice functions finally the well known richter theorem is investigated in the fuzzy setting

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets

how to make t transitive a proximity relation three ways to approximate a proximity relation r i e a reflexive and symmetric fuzzy relation by a t transitive one where t is a continuous archimedean t norm are given the first one aggregates the transitive closure r macr of r with a maximal t transitive relation b contained in r the second one computes the closest homotecy of r macr or b to better fit their entries with the ones of r the third method uses nonlinear programming techniques to obtain the best approximation with respect to the euclidean distance for t the lukasiewicz or the product t norm the previous methods do not apply for the minimum t norm an algorithm to approximate a given proximity relation by a min transitive relation a similarity is given in the last section of the paper

the stability of local properties of fuzzy relations under ordinal equivalence abstract we consider a notion of equivalence relation between fuzzy relations and its properties the relation is called ordinal equivalence since it partitions the family of fuzzy relations into equivalence classes by means of the order defined on elements of each relation we compare fuzzy relations by their membership values and as a consequence we propose two types of local properties of fuzzy relations new properties of fuzzy relations are more compatible with the ordinal equivalence relation namely they are stable in the equivalence classes moreover we study dependencies between these new properties of fuzzy relations finally notes on applications of the presented local properties and the ordinal equivalence relation are provided