are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

separation of variables for quantum integrable systems on elliptic curves we extend sklyanin s method of separation of variables to quantum integrable models associated to elliptic curves after reviewing the differential case the elliptic gaudin model studied by enriquez feigin and rubtsov we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables these transfer matrices are associated to representations of the elliptic quantum group e sl2 by difference operators one model of statistical mechanics to which this method applies is the interaction round a face model with antiperiodic boundary conditions the eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher order theta functions

the phase transition of the directed polymer on disordered hierarchical lattices the directed polymer on disordered hierarchical lattices is studied using an exact renormalization scheme the phase transition is studied and a hyperscaling relation is derived the fixed distributions of the renormalized energies are obtained numerically the specific heat and associated exponents are computed from the fixed distributions

fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

alternative characterizations for the representation of families of sets by fuzzy sets we investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets this concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set negoita and ralescu obtained two theorems both giving necessary and sufficient conditions one called the representation theorem and the second called the generalized representation theorem our aim in this paper is to take a closer look at both theorems hence establishing some equivalent forms and several constructive remarks including some directions for future investigations

differential migdal kadanoff renormalization group for disordered systems phase transitions in quenched random systems are studied within the migdal kadanoff approximation exact differential recursion relations for the distribution of coupling are derived the q state potts model with random ferromagnetic couplings is studied in two and three dimensions in three dimensions there is a non perturbative random fixed point which exists when the specific heat exponent is either positive or slightly negative if the specific heat exponent is slightly negative a multicritical point separates strong and weak disorder regimes the analytic results are confirmed by numerical simulations of the recursion relations

interpretations of various uncertainty theories using models of modal logic a summary abstract this paper summarizes our efforts to establish the usual semantics of propositional modal logic as a unifying framework for various uncertainty theories interpretations for fuzzy set theory dempster shafer theory probability theory and possibility theory are discussed some properties of these interpretations are also presented as well as directions for future research

critical behaviour of dyson s hierarchical model with a random field the critical behaviour of dyson s hierarchical ising model is studied in the presence of a random field simple scaling relations determine the exponents eta eta exactly for random fields with either short or long range correlations the schwartz soffer inequality eta or 2 eta is satisfied as an equality provided the random field correlations are not too long ranged for short range random fields the exponent v is calculated to order in lambda 4 3 where lambda is the range parameter of the dyson model the exponent v is also computed to o in 2 with in 2 lambda 3 for the hierarchical o n model in zero field the results reveal striking similarities with the exponents of the corresponding one dimensional model with long range interactions

weighted mean field theory for the random field ising model we consider the mean field theory of the random field ising model obtained by weighing the many solutions of the mean field equations with boltzmann like factors these solutions are found numerically in three dimensions and we observe critical behaviour arising from the weighted sum we estimate the exponents

a fuzzy neural network for pattern classification and feature selection a fuzzy neural network with memory connections for classification and weight connections for selection is introduced thereby solving simultaneously two major problems in pattern recognition pattern classification and feature selection the proposed network attempts to select important features from among the originally given plausible features while maintaining the maximum recognition rate the resulting value of weight connection represents the degree of importance of feature moreover the knowledge acquired by the network can be described as a set of interpretable rules the effectiveness of this new method has been validated by using anderson s iris data the results are first the use of two features selected by our method from among the original four in the proposed network results in virtually identical classifier performance and second the constructed classifier is described by three simple rules that are of if then form

critical behaviour of the random field ising model we present an analytic real space renormalization group calculation for the random field ising model we apply the migdal kadanoff approximation for the renormalization of a cubic cell in dimensions d introducing a new field partitioning scheme which allows us to treat the random field fluctuations in a coherent manner our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality we expand analytically in further we show that and the magnitude of the inequality go to zero exponentially with we calculate the crossover exponent from pure to the random field system and find surprisingly good agreement with experimental values we find that satisfies the schwartz soffer inequality the susceptibility exponent of the pure system we expand in and find that the magnitude of the inequality varies exponentially in finally we find that dimensional reduction is satisfied to first order in with the reduced dimension

on the missing scaling relation in random field systems abstract a detailed proof is given that for random field systems the susceptibility and correlation function are related at the transition this yields an additional scaling relation 2

a review and comparison of six reasoning methods abstract five fuzzy reasoning methods are reviewed and their performance is compared with respect to a fuzzy control system model developed by an objective method based on three sets of real life data it is found from the investigation that the reasoning precision the calculation time and the number of possible input states to which a given reasoning method responds differ according to each reasoning method generally the point valued reasoning methods which are based on the assumption that connectives are crisp give better precision and shorter calculation time when the connectives are assumed to be linguistic reasoning with interval valued fuzzy sets are more appropriate to represent linguistic uncertainty for this reason type ii fuzziness generated with interval valued fuzzy sets are better handled with interval valued reasoning methods

interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference information systems which contain only crisp data precise and unique attribute values for all objects have been widely investigated due to the fact that in realworld applications imprecise data are abundant uncertainty is inherent in real information systems in this paper information systems are called fuzzy information systems and formalized by objects attributes f in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values to interpret and extract fuzzy decision rules from fuzzy information systems the meta theory based on modal logic proposed by resconi et al is modified the modified meta theory not only expresses uncertainty between objects and their attributes but also uncertainty in the process of recognizing fuzzy information systems in addition according to perception computing proposed by zadeh granules of fuzzy information systems can be represented by fuzzy decision rules so that fuzzy inference methods can be used to obtain the decision attribute of a new object finally a novel way of combining evidences based on the modified meta theory is introduced which extends the concept of combining evidences based on dempster shafer theory

fuzzy expert systems for ie or ms abstract fuzzy expert systems can be developed for the effective use of management within the domains of concern associated with industrial engineering ie operations research or and management science ms these models are designed with 1 expressive powers of representation embedded in linguistic variables and their linguistic values in natural language expressions and 2 improved methods of inference based on fuzzy logic which is a generalization of multi valued logic the results of these fuzzy expert system models are either 1 approximately good in comparison with their classical counterparts i e within a few percentage points of the optimal results of classical models as for example for the cases of production and spare parts planning or 2 much better than their counterparts as for example for the cases of job shop scheduling and consumer preference prediction in market share analysis in order to establish a sound basis for the preceding claims basic issues of representation and inference are first reviewed and then several prototype case studies are presented to support these claims

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

fixed point distributions of short range ising spin glasses on hierarchical lattices fixed point distributions for the couplings of ising spin glasses with nearest neighbor interactions on hierarchical lattices are investigated numerically hierarchical lattices within the migdal kadanoff family with fractal dimensions in the range 2 58 ensuremath le d ensuremath le 7 as well as a lattice of the wheatstone bridge family with fractal dimension d ensuremath approx 3 58 are considered three initial distributions for the couplings are analyzed namely the gaussian bimodal and uniform ones in all cases after a few iterations of the renormalization group procedure the associated probability distributions approached universal fixed shapes for hierarchical lattices of the migdal kadanoff family the fixed point distributions were well fitted either by stretched exponentials or by q gaussian distributions both fittings recover the expected gaussian limit as d ensuremath rightarrow ensuremath infty in the case of the wheatstone bridge lattice the best fit was found by means of a stretched exponential distribution

constructing fuzzy measures in expert systems abstract this paper is an overview of results regarding various representations of fuzzy measures and methods for constructing fuzzy measures in the context of expert systems which were obtained by the authors and their associates during the last three years included are methods for constructing fuzzy measures by various transformations by extension by statistical inference and by various data driven methods based either on the sugeno integral or the choquet integral and using neural networks genetic algorithms or fuzzy relation equations

membership function elicitation in possibilistic programming problems in this paper we propose an elicitation method for membership functions in the setting of possibilistic programming each possibility distribution is defined by several possible domains with different possibility degrees taking advantage of the properties of possibility and necessity measures a nested target table for the objective function and for each constraint is elicited from a decision maker to aggregate all the goals an additional nested target table is established through those nested target tables membership functions can be determined it is emphasized that the information of nested target tables is much more important than that of membership values in the setting of possibilistic programming the membership functions in possibilistic programming are not assumed to be cardinal but ordinal in this sense the decision making via possibilistic programming can be regarded as a qualitative one

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

the measurement of membership by subjective ratio estimation when we perform algebraic operations on membership degrees we must be sure that they are compatible with the kind of scale on which these degrees are measured for example we will not add in a weighted average membership degrees that are measured on an ordinal scale an important question is therefore how to determine on which kind of scale we measure the membership there are several techniques for measuring membership in this paper we present some of them based on stevens technique of ratio estimation and we characterize them thereby providing a sound way to determine the level of measurement we also discuss the problem of the representation of the union or intersection by a t conorm or t norm

the symbol grounding problem there has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling this paper describes the symbol grounding problem how can the semantic interpretation of a formal symbol system be made intrinsic to the system rather than just parasitic on the meanings in our heads how can the meanings of the meaningless symbol tokens manipulated solely on the basis of their arbitrary shapes be grounded in anything but other meaningless symbols the problem is analogous to trying to learn chinese from a chinese chinese dictionary alone a candidate solution is sketched symbolic representations must be grounded bottom up in nonsymbolic representations of two kinds 1 iconic representations which are analogs of the proximal sensory projections of distal objects and events and 2 categorical representations which are learned and innate feature detectors that pick out the invariant features of object and event categories from their sensory projections elementary symbols are the names of these object and event categories assigned on the basis of their nonsymbolic categorical representations higher order 3 symbolic representations grounded in these elementary symbols consist of symbol strings describing category membership relations e g an x is a y that is z connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations thereby connecting names to the proximal projections of the distal objects they stand for in this way connectionism can be seen as a complementary component in a hybrid nonsymbolic symbolic model of the mind rather than a rival to purely symbolic modeling such a hybrid model would not have an autonomous symbolic module however the symbolic functions would emerge as an intrinsically dedicated symbol system as a consequence of the bottom up grounding of categories names in their sensory representations symbol manipulation would be governed not just by the arbitrary shapes of the symbol tokens but by the nonarbitrary shapes of the icons and category invariants in which they are grounded

maximal mean field solutions in the random field ising model the pattern of the symmetry breaking we study the mean field equations for the 3d random field ising model we discuss the phase diagram of the model and we address the problem of finding if such equations admit more than one solution we find two different critical values of the temperature t one where the magnetization takes a non zero expectation value and one where we start to have more than one solution to the mean field equation we find that inside a given solution there are no divergent correlation lengths

probably approximately correct learning in fuzzy classification systems an efficient method for learning trapezoidal membership functions for fuzzy predicates is presented positive and negative examples of one class are given together with a system of classification rules the learned membership functions can be used for the fuzzy predicates occurring in the given rules to classify further examples we show that the obtained classification is approximately correct with high probability this justifies the obtained fuzzy sets within one particular classification problem instead of relying on a subjective meaning of fuzzy predicates as normally done by a domain expert

meta linguistic axioms as a foundation for computing with words abstract meta linguistic axioms are proposed in analogy to the axioms of classical theory as a foundation for computing with words consequences of these meta linguistic expressions are explored in the light of interval valued type 2 fuzzy sets the mapping of meta linguistic axioms to fuzzy disjunctive and conjunctive canonical forms generate two set theoretic axioms for each meta linguistic axiom that is composed of two elements such as commutativity and four set theoretic axioms for each meta linguistic axiom that is composed of three elements such as associativity as opposed to one axiom that is generated in the classical theory the set theoretic axioms double or quadruple because the basic equivalences of the classical theory break down in fuzzy theory in this setting we become aware that first these new axioms hold as a matter of degree and secondly there is an upper and lower limit with which they hold as a matter of degree

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

approximate reasoning for production planning abstract approximate reasoning for production planning could effectively be based on an interval valued representation of linguistic variables an interval valued representation provides a framework that i incorporates a dispersion about membership values and ii produces a robust result compared to classical results the proposed representation and inference is applied to a set of production problems in prototype studies the result of one such prototype study is summarized

a context model for fuzzy concept analysis based upon modal logic in this paper we present interesting relationships between the context model modal logic and fuzzy concept analysis it has been shown that the context model proposed by gebhardt and kruse int j approx reason 9 1993 283 can be semantically extended and considered as a data model for fuzzy concept analysis within the framework of the meta theory developed by resconi el al in 1990s consequently the context model provides a practical framework for constructing membership functions of fuzzy concepts and gives the basis for a theoretical justification of suitably use of well known t norm based connectives such as min max and product sum rules in applications furthermore an interpretation of mass assignments of fuzzy concepts within the context model is also established

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

the dynamical elliptic quantum gaudin models and their solutions in this paper we construct the hamiltonians of both periodic and open elliptic quantum gaudin models and show their relations with the elliptic quantum group and the boundary elliptic quantum group respectively we define the eigenstates of these two models to be the bethe vectors with 0 of the elliptic quantum group and the boundary elliptic quantum group respectively then the hamiltonians are exactly diagonalized by using the algebraic bethe ansatz method

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

uniqueness results in the representation of families of sets by fuzzy sets the important concept of identifying a fuzzy subset of a set r with its corresponding chain of level sets is widely used in the literature we investigate in this paper the problem of characterization of all fuzzy subsets of r that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 although our main focus is on the problem of uniqueness we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets we continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed to further support our results we present some examples which we hope will shed more light on the above mentioned problem of identification we finish by raising a few natural questions that are left as open problems for further investigation

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

real space renormalization group analysis of the random field ising model we present an analytic migdal kadanoff renormalization group analysis of the random field ising model the renormalization flows to a zero temperature critical point from which we calculate independently three critical exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality expanding analytically in we find that and the distance from the upper bound of the equality go to zero exponentially with

non specificity and interval valued fuzzy sets abstract non specificity is a characteristic of interval valued fuzzy sets vagueness is represented by fuzzy sets which capture the meaning representation of linguistic terms or linguistic variables this is type i semantic uncertainty combinations of two or more vague linguistic terms with vague linguistic operators generate semantic uncertainty when disjunctive and conjunctive combinations of concepts are formed in particular disjunctive and conjunctive normal form expressions of combined concepts generate an interval of uncertainty this is type ii semantic uncertainty the size i e the cardinality of this interval provides a foundation for the fuzzy measure theoretical investigation of the combination of vagueness associated with two or more linguistic concepts and the operators that combine them in particular the length of the interval provides the non specificity measure for interval valued fuzzy sets thus non specificity generated by interval valued fuzzy sets forms an essential linkage for the development of evidence theory with fuzzy sets

fuzzy sets spl minus a convenient fiction for modeling vagueness and possibility this paper is a reply to laviolette and seaman s critical discussion of fuzzy set theory rather than questioning the interest of the bayesian approach to uncertainty some reasons why bayesian find the idea of a fuzzy set not palatable are laid bare some links between fuzzy sets and probability that laviolette and seaman seem not to be aware of are pointed out these links suggest that contrary to the claim sometimes found in the literature probability theory is not a special case of fuzzy set theory the major objection to laviolette and seaman is that they found their critique on as very limited view of fuzzy sets including debatable papers while they fail to account for significant works pertaining to axiomatic derivation of fuzzy set connectives possibility theory fuzzy random variables among others

some exact inequalities for the random field ϕ4 system exact lower bound on the correlation function and susceptibility are obtained in terms of the quantity r r0 12u 2 d ddl l l where l l are the exact averages further lower bounds are obtained on r relating it to an exactly soluble model

analysis of critical paths in a project network with fuzzy activity times abstract this paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers in that the membership function of the fuzzy total duration time is constructed the basic idea is based on the extension principle and linear programming formulation a pair of linear programs parameterized by possibility level is formulated to calculate the lower and upper bounds of the fuzzy total duration time at by enumerating different values of the membership function of the fuzzy total duration time is constructed and the fuzzy critical paths are identified at the same time moreover by applying the yager ranking method definitions of the most critical path and the relative degree of criticality of paths are developed and these definitions are theoretically sound and easy to use in practice two examples with activity times being fuzzy numbers of l r and l l types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach since the total duration time is completely expressed by a membership function rather than by a crisp value the fuzziness of activity times is conserved completely and more information is provided for critical path analysis

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

on the harris criterion for hierarchical lattices the authors discuss the condition necessary for the harris criteria to be valid on hierarchical lattices they prove that disorder is always relevant when the specific heat exponent alpha p is positive and show how to construct lattices for which disorder becomes relevant while alpha p is still negative

a fuzzy c means variant for the generation of fuzzy term sets a fuzzy c means fcm variant is proposed for the generation of fuzzy term sets with overlap the proposed variant differs from the original mainly in two areas the first modification ensures that two end terms take the maximum and minimum domain values as their centers the second modification prevents the generation of non convex fuzzy terms that often occurs with the original algorithm the optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion the exponential weight m used in the algorithm is found to greatly affect the shape of the membership function the effect of data size used for the generation of fuzzy terms is also discussed a generalized shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the fcm variant using various m values

an empirical test of some measurement theoretic axioms for fuzzy sets in the previous years some authors have been elaborating on the measurement theoretic foundations of fuzzy set theory a well known problem in this approach is the difficult applicability of the deterministically formulated axioms on data obtained from an expert or social science data in general in this paper a statistical method is proposed which can be used for the testing of measurement axioms in a pairwise comparison design consequently we apply this method for the testing of two of these axioms i e transitivity and order of operations used in the context of fuzzy set theory the results clearly indicate that subjects act in agreement with transitivity but for order of operations there is no such vivid evidence which may have repercussions for some of the earlier theoretical work

lack of self averaging in random systems liability or asset abstract the study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings provided that the system is large enough careful examination suggests that this idea might have a flaw when the correlation length becomes of the order of the size of the system we find that certain bounded quantities are not self averaging when the correlation length becomes of the order of the size of the system this suggests that the lack of self averaging expressed in terms of properly chosen signal to noise ratios may serve to identify phase boundaries this is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase of the random field ising system and compare the findings with more traditional measures

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

wess zumino witten model on elliptic curves at the critical level we construct a gaudin type lattice model as the wess zumino witten model on elliptic curves at the critical level bethe eigenvectors are obtained by the bosonization technique

a self consistent ornstein zernike approximation for the random field ising model we extend the self consistent ornstein zernike approximation scoza first formulated in the context of liquid state theory to the study of the random field ising model within the replica formalism we treat the quenched random field just as another spin variable thereby avoiding the usual average over the random field distribution this allows us to study the influence of the distribution on the phase diagram in finite dimensions the thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization temperature and disorder strength as independent variables a preliminary analysis based on high temperature and 1 d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength no sharp transition however occurs for d 4 for the bimodal distribution we find a tricritical point which moves to weaker fields as the dimension is reduced for the gaussian distribution a tricritical point may appear for d around 4

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

canonical forms of fuzzy truthoods by meta theory based upon modal logic in this paper we redefine with the meta theory based on modal logic the operations between fuzzy sets of verity i e fuzzy sets of truthood with this new approach we can unify the different formulas for the operations and or and not between the fuzzy sets of verity i e truthood the operations between the fuzzy sets of verity become sensitive to the logic value true or false that agents persons sensors assign to the worlds contexts the operations are also sensitive to the difference of the worlds and time of synchronisation it should be pointed out that when we use the logic operations as and or and not we generally assume that the worlds are the same and change their truth value at the same time but it is known that there are cases where this synchronic situation and identity is not always valid that is there exist transformations that change one world for one proposition to another world for another proposition in conclusion the linguistic and or not operations become dependent on the particular truth value of a world on the synchronisation and on the worlds assigned to the two propositions via transformations thus all of these possible changes in the structure of the worlds in the modal logic cause the gradation of the linguistic operations e g and or and not an individual world person agent sensor assigns to an atomic sentence either a true or false value and uses the classical two value logic operations of and or not that is the crisp true or false responses assignments of worlds generate gradation of truthood value the uncertainty in a fuzzy set is represented with sets of worlds in a conflict situation i e the same proposition may be true in one world and false in another consonant or dissonant relations between sets of worlds that depends on the synchronisation and or transformation of worlds cause the generation of gradation by the linguistic operations and or and not that is the linguistic operations change for different concrete situations set of worlds caused by the generation of a representation in each world where the descriptive set membership assignment d may be two or infinite valued but the verity assignment v is two valued in a particular world representation whereas membership values of the fuzzy set of truth verifications associated with the set of worlds are in 0 1 furthermore the combination of fuzzy membership values generates a type ii fuzzy set that is captured by fdcf and fccf formulas in this paper we show that there exist a deeper connection between the set membership assignment and the verity assignment of truthood by modal logic with suitable extensions

an efficient and flexible mechanism for constructing membership functions abstract this paper introduces a bezier curve based mechanism for constructing membership functions of convex normal fuzzy sets the mechanism can fit any given data set with a minimum level of discrepancy in the absence of data the mechanism can be intuitively manipulated by the user to construct membership functions with the desired shape some numerical experiments are included to compare the performance of the proposed mechanism with conventional methods

the measure of the degree of truth and the grade of membership abstract a method is presented to define the meaning of a fuzzy proposition and of a fuzzy set through the construction of fuzzy propositions with non ambiguous intermediate truth values

treating fuzziness in subjective evaluation data this paper proposes a technique to deal with fuzziness in subjective evaluation data and applies it to principal component analysis and correspondence analysis in the existing method or techniques developed directly from it fuzzy sets are defined from some standpoint on a data space and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares in this paper we try to map the variation in evaluation data into the parameter space while preserving information as much as possible and thereby define fuzzy sets in the parameter space clearly it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle however with a fuzzy model which uses the extension principle the possibility distribution spreads out as the explanatory variable values increase this does not necessarily make sense for subjective evaluations such as a 5 level evaluation for instance instead of doing so we propose a method for explicitly expressing the vagueness of evaluation using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread as a numerical example we present an analysis of subjective evaluation data on local environments

efficient refinement of uncertain data by fuzzy integrity constraints abstract fuzzy data models were proposed to handle uncertain data based on a set of fuzzy functional dependencies the fuzzy values in a fuzzy relation may be refined into clearer ones in this paper we develop efficient algorithms for the refinement process a dependency graph is defined for a set of fuzzy functional dependencies by this graph an efficient refinement sequence can be determined these algorithms are proved correct and efficient

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

hierarchical scheduling based on approximate reasoning a comparison with isis abstract a hierarchical scheduling problem is considered the hierarchy consists of five levels each level has an associated rule base there are six basic input factors 1 the priority of a job 2 the earliness of the due date of a job 3 the likelihood of the availability of resources for the work station w s s 4 the capability of w s s to meet physical constraints 5 the capability of w s s to meet quality requirements and 6 work in progress inventory costs there are five consequents one for each level 1 suggested place of a job in the order of jobs 2 the suggested place of w s s in the order of w s s 3 the current suggested place of a w s s in the order of w s s 4 the next current suggested place of w s s in the order of w s s and 5 the suggested place of the queue arrangements both the input factors antecedents and the output factors consequents take on linguistic terms to express the qualitative assessments of a scheduling expert within the context of the specific factor an approximate reasoning framework based on interval valued fuzzy sets and a search tree heuristic with similarity measures is executed for the analysis of the hierarchical scheduling problem using a fuzzy tool box coded in lisp that interacts with kee for the representation of the knowledge base in a frame structure the software model is tested with real life data used in m fox s isis model a comparison of the two models based on tardiness measure shows that there are slight advantages to fuzzy set representation and approximate reasoning with respect to tardiness measure above and beyond the user friendly nature of knowledge representation available with fuzzy sets

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

a fuzzy set model for market share and preference prediction abstract this paper presents a new method of modelling preferences using fuzzy sets a fuzzy set conjoint model is developed by incorporating fuzzy measurement of evaluations into the vector preference model a comparison test of rank prediction is conducted on data gathered from consumer subjects by an adaptive software program a methodology is developed to generate aggregate crisp market share estimates from individual fuzzy set preference predictions the result is a substantial improvement in individual predictive validity using the fuzzy set model and more accurate aggregate market share prediction for use in product design demand forecasting and other management functions

real space renormalization group approach to the random field ising model the authors discuss results from two types of real space renormalization group rsrg calculations applied to the random field ising model in three dimensions starting from a lattice of size l the rsrg is used to reduce the lattice to a size l 2 on which the trace is done exactly in this way thermodynamic properties such as the magnetization and susceptibility can be determined approximately they find that for a given size the susceptibility increases as the temperature t is reduced down to the transition temperature tc and becomes essentially independent of temperature below tc both in the vicinity of tc and at lower temperatures there are large sample to sample fluctuations in the susceptibility which grow with increasing system size they interpret these results in terms of the droplet theory of the transition

type i and type ii fuzzy system modeling abstract two system modeling approaches are proposed with a unification of fuzzy methodologies type i and type ii knowledge representation are reviewed together with their proposed approximate reasoning schemes type i representation and inference are based on boolean normal forms type i models are myopic because in these models either the disjunctive or the conjunctive boolean normal forms are selected rather arbitrarily for knowledge representation and inference and then they are fuzzified whereas in type ii models representation and inference are based on fuzzy disjunctive and conjunctive normal forms which are derived from the fuzzy truth tables