a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

canonical orthonormal wigner supermultiplet basis the explicit construction of an orthonormal basis for states of good spin isospin and su 4 wigner supermultiplet symmetry is given in a bargmann representation space a complete set of quantum labels is provided by a sp 3 r contains implies u 3 complementary symmetry

boson realization of sp 4 r ii the generating kernel formulation in a previous paper of this series the matrix elements were discussed with respect to boson states of an operator k2 required for the boson realization of the sp 4 r lie algebra in the present paper it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp 4 r coherent states the results have relevance for the determination of the matrix elements of the generators of the sp 4 r lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group and are in principle generalizable to symplectic algebras of higher dimensions

fuzzy logic based algorithms for maximum covering location problems this paper concerns a class of maximum covering location problems in networks in uncertain environments it is assumed that a relative weights of demand nodes are either deterministic or imprecise described by linguistic expressions and b potential facility site locations are limited to network nodes the concept of coverage is extended to include a degree of node coverage which means that the borders between the subset of covered demand nodes and the subset of uncovered demand nodes are inexact the acceptable service distance travelling times from a facility site to demand nodes are modelled by fuzzy sets three new algorithms for choosing the best facility locations are developed which assume that 1 demands at all nodes are equally important 2 relative weights of demand at nodes are deterministic and 3 weights of demand at nodes are imprecise and described by linguistic terms respectively the algorithms are based on searching among potential facility nodes by applying comparison operations on discrete fuzzy sets it is shown how to extend the proposed algorithms from one site to multi site covering problems illustrative examples of selecting locations for logistics centres in a distribution company are given

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

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

a probabilistic approach to rank complex fuzzy numbers abstract techniques for ranking simple fuzzy numbers are abundant in the literature however we lack efficient methods for comparing complex fuzzy numbers that are induced by arithmetic operations in this paper a probabilistic approach is taken membership functions of fuzzy numbers are first converted into probability density functions the mellin transform is then introduced to compute the mean and the variance of a complex fuzzy number the fuzzy number with the higher mean is ranked higher if the means are equal the one with the smaller variance is judged higher rank two numerical examples and a fuzzy multiple attribute decision analysis are illustrated

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

the maximal distance between imprecise point objects in this paper we propose a new mathematical model of imprecise point objects and a new algorithm for determining the maximal distance between imprecise point objects results obtained with our model can be used in various applications such as gis imprecise spatial object modelling robotics environment models and image analysis imprecise feature extraction imprecise point objects are modelled as fuzzy points in linear fuzzy space linear fuzzy space is defined over r n where the membership function is convex upper semi continuous symmetric w r t the core and linearly decreasing w r t the core distance where the core is a single point a fuzzy convex hull is defined in linear fuzzy space the distance between objects points is modelled as an imprecise point object in linear fuzzy space defined over r 1 fuzzy ordering relations are defined in distance space the maximal distance is a fuzzy set over a set of distances the maximal distance between a set of imprecise point objects is then determined as the maximal distance of fuzzy convex hull edge points an algorithm for determining the maximal distance between imprecise points belonging to one and two dimensional linear fuzzy space is given for an exact calculation of this maximal distance computational complexity of the algorithm is of polynomial growth of the fourth degree however if the maximal distance is determined approximately then the algorithms computational complexity is of power growth of the quadratic degree with respect to the cardinality of the convex edge

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

conditioned weighted l r approximations of fuzzy numbers we compute the extended weighted l r approximation of a given fuzzy number by a method based on general results in hilbert spaces the weighted average euclidean distance being considered the metric properties of the extended weighted l r approximation of a fuzzy number are proved we elaborate on a general method to study the existence uniqueness and to calculate the l r approximations of fuzzy numbers under the preservation of some parameters we apply the results to find the weighted l r approximations preserving ambiguity and value and respectively width in the general and unimodal case

ranking the sequences of fuzzy values when ranking fuzzy values there is an ambiguity in selecting the most preferred ranked sequence from a collection of candidate sequences because every sequence has a possibility to which the preference relations in the sequence are correct in this paper we propose a method for ranking the sequences of fuzzy values that assigns a preference degree to each ranked sequence each preference degree indicates the relative preference of the given sequence over the other sequences the proposed method gives information regarding which sequence could be classified as the most preferred sequence and which sequences as alternatives

coherent states for the harmonic series irreducible representations of u p q i unitary operator coherent states unitary operator coherent states as defined by klauder 1963 perelomov 1972 and gilmore 1974 are considered for the harmonic series irreducible representations of u p q various properties of these states are reviewed including their reproducing kernel and measure explicit forms as well as the representation they lead to for the u p q generators starting from a contraction of the su p q lie algebra both dyson and holstein primakoff boson realisations of u p q are obtained in terms of pq pairs of boson creation annihilation operators and of the generators of a u p u q intrinsic group such boson realisations are applied to determine the matrix elements of the u p q generators between discrete bases classified according to the chain u p q contains implies u p u q finally the isomorphism between so 4 2 and su 2 2 is employed to derive the corresponding properties of the so 4 2 unitary operator coherent states for application in atomic physics

existence uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition abstract the main aim of this paper is to characterize the set of real parameters associated to a fuzzy number represented in a general form which include the most important characteristics with the following property for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter the uniqueness of the nearest trapezoidal fuzzy number with this property is proved the average euclidean distance being considered as an important property each resulting trapezoidal approximation operator is continuous the main results are illustrated by examples

a shortest path problem on a network with fuzzy arc lengths abstract we concentrate on a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length introducing an order relation between fuzzy numbers based on fuzzy min a nondominated path or pareto optimal path from the specified node to every other node is defined an algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path as a result a number of nondominated paths can be obtained and is offered to a decision maker however a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path due to this situation we propose a method to reduce the number of paths according to a possibility level the proposed algorithm is numerically evaluated on large scale random networks

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

solving linear systems via pfaffians abstract an explicit solution is given for the system of linear equations e e t where are alternating matrices of pfaffian 1 which at most differ in their first row and column and e is of the form i 2 2 r e 1 i x i if v v w r r 1 with v w 1 v w then a sequence of cohn transforms with respect to the fixed w which takes v w to v w is prescribed

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

dynamical group of microscopic collective states iii coherent state representations in d dimensions the present series of papers deals with various realizations of the dynamical group spc 2d r of microscopic collective states for an a nucleon system in d 1 2 or 3 dimensions when these collective states are assumed to be invariant under the orthogonal group o n associated with the n a 1 relative jacobi vectors in this paper we further study the barut girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to spc 2d r a class of sp 2 r coherent states introduced by barut and girardello for such purpose our starting point is another coherent state representation namely the perelomov one previously considered by kramer we also propose a third new class of coherent states leading to holstein primakoff representation in a straightforward way we review various properties of these three classes of coherent states such as their reproducing kernel and measure explicit forms their gener

alpha bounds of fuzzy numbers applications of operators of contrast modification of fuzzy sets to fuzzy numbers are studied it is shown that for a fuzzy number a and for 0 1 an upper bound and an lower bound of a can be created cardinality of the boundary region will be called flexibility of a relationship among bounds strong bounds and ultimate bounds of a is discussed when bounds are trapezoidal fuzzy numbers then flexibility of a can be used to assess roughness of approximation of a in the knowledge base consisting of trapezoidal fuzzy numbers

approximations by lr type fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated furthermore approximations in euclidean distance class can be divided into two subclasses one is to study approximations of fuzzy numbers without constraints the other one is to study approximations preserving some attributes in this paper we use lr type fuzzy numbers to approximate fuzzy numbers the proposed approximations will generalize all recent approximations without constraints in euclidean class also an efficient formula is provided

canonical solution of the state labelling problem for su n so n and littlewood s branching rule i general formulation the internal state labelling problem for the d row irreducible representations of su n where 2d or n when reduced with respect to so n is shown to amount to the external state labelling problem for u d the canonical solution of the latter due to biedenharn et al 1967 provides a canonical solution of the former which reflects the operation of littlewood s branching rule for u n contains implies o n in a very simple way

analytical expressions for the matrix elements of the non compact symplectic algebra analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp 3 r it is shown that the expressions are exact for representations of the type sigma 1 sigma 2 sigma 3 and for all states of an arbitrary representation that are multiplicity free with respect to the u 3 subalgebra furthermore they are remarkably accurate in general

generating kernel for the boson realisation of symplectic algebras the discussion of boson realisations of symplectic algebras requires as an essential ingredient an operator k needed for the passage from the dyson to the holstein primakoff realisation in previous papers the matrix form of k2 was derived through appropriate recursion relations in the present analysis the authors show that k2 can be determined in explicit and closed form as the overlap of coherent states of the symplectic group the matrix elements of k2 can then be obtained by expanding this overlap in terms of appropriate eigenstates

principal component analysis of fuzzy data using autoassociative neural networks this paper describes an extension of principal component analysis pca allowing the extraction of a limited number of relevant features from high dimensional fuzzy data our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as pca without explicit matrix diagonalization fuzzy input values are propagated through the network using fuzzy arithmetics and the weights are adjusted to minimize a suitable error criterion the inputs being taken as target outputs the concept of correlation coefficient is extended to fuzzy numbers allowing the interpretation of the new features in terms of the original variables experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

a generalized serre problem abstract a possible generalization of the serre problem quillen suslin theorem on the freeness of projective modules over polynomial rings which seems to be of interest in systems theory applications is shown to follow from the solution to the original serre problem

matrix representation of the generators of symplectic algebras i the case of sp 4 r because of their many physical applications there has been considerable interest lately in the basis for irreducible representations irreps of symplectic groups and in the matrix elements of the generators of these groups with respect to this basis in the present paper the authors carry out this programme for the irreps in the positive discrete series of sp 4 r by building up this basis from powers of the raising generators applied to the lowest weight state by using the dyson boson realisation of the generators their matrix representation can be obtained by direct differentiation while the overlap of the states of the basis can be determined with the help of coherent states the extension of the analysis to symplectic groups of arbitrary dimension is straightforward and will be implemented explicitly for sp 6 r in a later paper

generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites abstract the linguistic term set is an applicable and flexible technique in qualitative decision making qdm to further develop the linguistic term set this paper proposes a generalized asymmetric linguistic term set galts based on the asymmetric sigmoid semantics which belongs to an asymmetric and non uniform linguistic term set and can be used to address the qdm problems involving risk appetites of the decision maker dm then a value at risk fitting varf approach is designed for obtaining the risk appetite parameters of the galts and six desirable properties of the galts are analyzed i e asymmetry non uniformity generality variability range consistency and diminishing utility based on the above approaches and the generalized asymmetric linguistic preference relations galprs a qdm process involving risk appetites of the dm is designed because the galprs consist of subjective information provided by the dm the process is not perfectly consistent and is usually difficult to change or repeat thus a transitivity improvement approach is investigated and the corresponding calculation steps are provided finally an example dealing with the problem of investment decision making is provided and the results fully demonstrate the validity of the proposed methods for qdm involving risk appetites

a fuzziness measure for fuzzy numbers applications abstract extending the ideas of delgado et al 1995 to include the notion of fuzziness we obtain canonical representations of fuzzy numbers in particular we show that any fuzzy number can be represented by either a trapezoidal or a quasitrapezoidal number having the same basic attributes as the original fuzzy number

numerical methods for interval and fuzzy number comparison based on the probabilistic approach and dempster shafer theory there are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or boolean value in this paper we use the dempster shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number the complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach these relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the dempster shafer theory a natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples

a generalized α cut for a fixed value 0 1 we introduce the relation sharpness and a measure of fuzziness of fuzzy sets then we propose an approximation of a fuzzy set a by its generalized cut g a a generalized cut of a is a special type of fuzzy set that is sharper than a and less sharp than the crisp approximation of a by its ordinary cut a attention will be given to the generalized cuts of fuzzy numbers

median value and median interval of a fuzzy number median value and median interval of a fuzzy number are investigated effects of concentration and dilation of a trapezoidal fuzzy number on the location of the median value and the median interval are studied relationship between the median interval and the mean interval is explored central value and central interval of a fuzzy number are proposed

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

trapezoidal approximations of fuzzy numbers preserving the expected interval algorithms and properties fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed it is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number like its ambiguity width its value and weighted expected value

general approximation of fuzzy numbers by f transform in this paper we will prove that in most of the cases the extended inverse fuzzy transform preserves the quasi concavity of a fuzzy number and hence it can be used to generate fuzzy numbers by approximating the restriction of the membership function to its support in the case of continuous fuzzy numbers with cores containing more than one element the rate of uniform convergence is of linear type and the same holds when we approximate the important characteristics of a fuzzy number such as the value or the ambiguity moreover we have the preservation of the support and the convergence of the core which in addition can be determined precisely in the case of continuous fuzzy numbers with one element core it is in general necessary to normalize the approximation but the support is preserved again and the core can be determined exactly in this case too moreover the approximations have again linear rate of uniform convergence

on weighted possibilistic mean and variance of fuzzy numbers dubois and prade defined an interval valued expectation of fuzzy numbers viewing them as consonant random sets carlsson and fuller defined an interval valued mean value of fuzzy numbers viewing them as possibility distributions in this paper we shall introduce the notation of weighted interval valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval valued probabilistic mean we shall also introduce the notations of crisp weighted possibilistic mean value variance and covariance of fuzzy numbers which are consistent with the extension principle furthermore we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

design of capacitated minimum spanning tree with uncertain cost and demand parameters the classical capacitated minimum spanning tree problem cmstp deals with finding a minimum cost spanning tree so that the total demand of the vertices in each subtree does not exceed the capacity limitation in most of the cmstp models the edge costs and the demands of the vertices in the network are assumed to be known with certainty this paper considers the cmstp model where the edge costs and or the demands are only approximately known a fast approximate reasoning algorithm which is based on the esau williams savings heuristic and fuzzy logic rules is proposed the computational results of the study based on the proposed approach are also reported

a structure theorem for the elementary unimodular vector group given a pair of vectors v w r r 1 with v w t 1 a suslin constructed a matrix s r v w sl 2 r r we study the subgroup sum r r generated by these matrices and its elementary subgroup eum r r generated by the matrices s r e 1 e e 1 e t 1 for e e r 1 r the basic calculus for eum r r is developed via a key lemma and a fundamental property of suslin matrices is derived

dynamical group of microscopic collective states i one dimensional case in the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an a nucleon system defined as those a particle states invariant under the orthogonal group o n associated with the n a 1 jacobi vectors the present paper discusses the case of a hypothetical one dimensional space simple invariance considerations show that the dynamical group of collective states is then the group spc 2 r which is the restriction to the collective subspace of the group sp 2 r of linear canonical transformations in n dimensions conserving the o n symmetry in addition to the well known realization of the dynamical group in the schrodinger representation based upon the dzublik zickendraht transformation two new realizations are proposed the first acts in a barut hilbert space which is the subspace of a bargmann hilbert space of analytic functions left invariant by o n a unitary mapping is established between the ordi

generalizing the l fuzzy unit interval abstract given an arbitrary connected topological space an l fuzzy space is constructed when the original space is the real line unit interval the constructed space is the l fuzzy real line l fuzzy unit interval for some spaces including r n for all n the original space is embedded as a subspace of the constructed space lastly the construction yields a non trivial fuzzy topology on certain classically important spaces of monotone mappings

screening and vertex operators for u n screening and vertex operators for u n are given in terms of generators of the group left action on the flag manifold u n 1 su n their matrix elements in a basis of holomorphic sections of a vector bundle are computed

existence of interval triangular and trapezoidal approximations of fuzzy numbers under a general condition abstract recently many scholars studied approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators in fact these approximations may not exist for some linear operators the purpose of this paper is to study necessary and sufficient conditions of linear operators which are preserved by interval triangular symmetric triangular trapezoidal or symmetric trapezoidal approximations of fuzzy numbers in addition an effective method for solving such problems is proposed

weighted trapezoidal and triangular approximations of fuzzy numbers in 2007 zeng and li proposed a weighted triangular approximation of a fuzzy number unfortunately this approximation may fail to be a fuzzy number in this paper we improve this approximation and propose a generalization by the name of weighted trapezoidal approximation their algorithms are also presented finally some examples and relevant properties are discussed

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

separation axioms and the fuzzy real lines we show that for each lattice l which is completely distributive and equipped with an order reversing involution the canonical topology of the fuzzy real line l is uniformizable in the sense of hutton furthermore this uniformity induces a pseudometric which induces the canonical topology consequently l satisfies the higher order separation axioms of hutton and sarkar more generally we answer most of the open questions concerning l and its subspaces with respect to each of the separation axiom schemes of hutton and sarkar answer some of these same questions for the stratified fuzzy real line c l and its subspaces and analyze c l and its subspaces with respect to the separation axioms of rodabaugh

weighted trapezoidal approximation preserving cores of a fuzzy number recently various researchers have proved that approximations of fuzzy numbers may fail to be fuzzy numbers in this contribution we suggest a new weighted trapezoidal approximation of an arbitrary fuzzy number which preserves its cores we prove that the weighted trapezoidal approximation preserving core is always a fuzzy number we then discuss properties of the approximation strategy including translation invariance scale invariance identity and continuity the advantage is that the method presented here is simpler than other methods computationally

sums of fuzzy set of summands abstract we investigate the addition operation on fuzzy numbers with a fuzzy set of summands this is a natural generalization of the corresponding operation which involves a crisp set of summands it is shown that the resulting set is a type 2 fuzzy set we construct a type 2 membership function of this set the sum type 2 fuzzy set can be decomposed according to secondary membership grades into the corresponding collection of fuzzy numbers it helps to represent the addition operation on fuzzy numbers with a fuzzy set of summands in a form which is convenient for a proper understanding and applications illustrative examples are given

weighted semi trapezoidal approximations of fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated in the present paper we study a special class of non linear approximations with respect to a weighted euclidean distance we call it weighted semi trapezoidal approximations the proposed approximations generalize all recent approximations in euclidean distance class first we embed fuzzy numbers into a hilbert space then compute weighted semi trapezoidal approximations by means of best approximations in a closed convex subset of the hilbert space finally we propose formulas of matrix type which is more clear than the previous contributions

boson realisation of symplectic algebras the boson realisation of the two dimensional symplectic lie algebra sp 2 for a given irreducible representation irrep of the sp 2 group has been known for a long time more recently the boson realisation of the 2d dimensional symplectic lie algebras sp 2d have been derived for particular irreps of the sp 2d group the author outlines the corresponding result for an arbitrary irrep of sp 2d

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

defining the borda count in a linguistic decision making context different kinds of decision rules have been successfully implemented under a linguistic approach this paper aims the same goal for the borda count a well known procedure with some interesting features in order to this two ways of extension from the borda rule to a linguistic framework are proposed taking into account all the agents opinions or only the favorable ones for each alternative when compared with each other in the two cases both individual and collective borda counts are analyzed asking for properties as good as those of the original patterns

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

an approximation formula for the kappa matrix elements of the symplectic algebra sp 6 r a simple approximation formula valid to the high degree of accuracy of the toronto approximation is given for both the off diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of sp 6 r

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

hilbert spaces of analytic functions and representations of the positive discrete series of sp 6 r hilbert spaces of analytic functions are constructed which carry irreducible representations of the positive discrete series of sp 6 r

partially coherent states of the real symplectic group in the present paper we introduce partially coherent states for the positive discrete series irreducible representations d n 2 1 n 2 of sp 2d r encountered in physical applications these states are characterized by both continuous and discrete labels the latter specify the row of the irreducible representation 1 2 d of the maximal compact subgroup u d while the former parametrize an element of the factor space sp 2d r h where h is the sp 2d r subgroup leaving the 1 2 d representation space invariant we consider three classes of partially coherent states respectively generalizing the perelomov and barut girardello coherent states as well as some recently introduced intermediate coherent states we prove that each family of partially coherent states forms an overcomplete set in the representation space of d n 2 1 n 2 and study its generating function properties we show that it leads to a representation of the sp 2d r generators in the form of differential operato

trapezoidal approximations of fuzzy numbers using quadratic programs abstract in this paper we will prove that the nearest trapezoidal approximation of fuzzy numbers with respect to weighted l 2 type metrics with or without additional constraints can be obtained via quadratic programs actually the approach is even more general based on so called finite polyhedral subsets of fuzzy numbers which include most of the important special classes of fuzzy numbers available in the literature in particular we will recapture the algorithm to compute the nearest weighted trapezoidal approximation of a fuzzy number by a method which we believe that has the potential to be extended to more complex approximation problems then we will improve the lipschitz constant of the trapezoidal approximation operator preserving the ambiguity to achieve this improved result we will exploit the fact that we have an analytical expression for this operator however note that the same result is obtained if this solution function is described by quadratic programs therefore for similar problems we still can obtain lipschitz constants for the approximation operator even if an analytical expression of this operator is not available

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a procedure for ranking fuzzy numbers using fuzzy relations abstract this paper presents a method to give fuzzy order relations between fuzzy numbers these are founded on the concept of comparison function defined in the paper and the use of fuzzy measures related with the same numbers the link of such relations with fuzzy operations is investigated too

a fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making decision making is a process common to human beings the uncertainty and fuzziness of problems demand the use of the fuzzy linguistic approach to model qualitative aspects of problems related to decision the recent proposal of hesitant fuzzy linguistic term sets supports the elicitation of comparative linguistic expressions in hesitant situations when experts hesitate among different linguistic terms to provide their assessments the use of linguistic intervals whose results lose their initial fuzzy representation was introduced to facilitate the computing processes in which such expressions are used the aim of this paper is to present a new representation of the hesitant fuzzy linguistic term sets by means of a fuzzy envelope to carry out the computing with words processes this new fuzzy envelope can be directly applied to fuzzy multicriteria decision making models an illustrative example of its application to a supplier selection problem through the use of fuzzy topsis is presented

note on trapezoidal approximation of fuzzy numbers we show by two examples that trapezoidal approximation of fuzzy number a in grzegorzewski mrowka fuzzy sets and systems 153 2005 115 135 are not always trapezoidal

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism