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

continuity and identity laws in the axiomatic basis of fuzzy set operations abstract t norms and t cornorms are considered as the suitable axiomatic basis for the aggregate functions used for the definition of fuzzy set intersection and union operations when the domain of the function is infinite as in most cases continuity has also to be included as an axiom this paper proves that the identity law axioms in t norm and t cornorm are implied by continuity and the other axioms a simple example is given to show that in the absence of continuity the identity law condition is necessary for a plausible axiomatic basis of fuzzy set operations we also proved that the only idempotent t norm t cornorm functions is min max

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

a measure theoretic axiomatization of fuzzy sets abstract using measurement theory this paper examines three empirical structures that underlie the representation of fuzzy sets the fuzzy membership structure the fuzzy component structure and the fuzzy system structure these qualitative structures justify the use of the standard min max system to represent fuzzy sets the results of this study facilitate the development of a sound measurement theoretic axiomatization of fuzzy systems

green s function calculation from equipartition theorem a method is presented to calculate the elastodynamic green s functions by using the equipartition principle the imaginary parts are calculated as the average cross correlations of the displacement fields generated by the incidence of body and surface waves with amplitudes weighted by partition factors the real part is retrieved using the hilbert transform the calculation of the partition factors is discussed for several geometrical configurations in two dimensional space the full space a basin in a half space and for layered media for the last case it results in a fast computation of the full green s functions additionally if the contribution of only selected states is desired as for instance the surface wave part the computation is even faster its use for full waveform inversion may then be advantageous

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

local and global existence of solutions for the cauchy problem of a generalized boussinesq equation abstract we consider the local and global existence of solutions for a generalized boussinesq equation u t t u x x u x x x x u k 1 x x 0 k 4 with initial data in some homogenous besov type space

reconstruction possibilities for fuzzy sets towards a new level of understanding abstract using the methodological approach of rational reconstruction some possible explications of the notion of fuzzy set are critically discussed with respect to their intuitive meaning plausibility and consistency together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory based on the approaches of 39 and 18 we show that zadeh s original intuition of a fuzzy set and its formalization as generalized membership function can be reinterpreted in a plausible and consistent way and thus are supported not only on a formal level by mathematical results but can also be justified in principle on a semantical level hence both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality and it is shown how the latter can be given a sound foundation based upon the former

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

fuzzy shell clustering algorithms in image processing fuzzy c rectangular and 2 rectangular shells objective function based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors although there are special algorithms to discover lines the detection of rectangles needs further treatment a simple line detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary in this paper a special fuzzy shell clustering algorithm for rectangular contours is developed the principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles ellipses and lines

fuzzy hypothesis testing for a population proportion based on set valued information abstract in this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one tailed hypergeometric test with fuzzy non complementary hypotheses and incorporate the conventional hypergeometric test with crisp hypotheses into its framework in particular we formulate hypotheses as ontic or epistemic fuzzy sets since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading further we suggest modeling of membership functions in terms of cost or frequency due to considerations of user s priorities or incomplete knowledge in relation to hypotheses formulation considering a hypothesis as union of its crisp and fuzzy areas we illustrate via real life examples that in contrast to classical test theory fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non complementary hypotheses the test concept is introduced for both test methods a test of significance and an alternative test and leads to a crisp test decision considering further cost aspects like the sample size the generalized error criteria are derived and interpreted in relation to both hypotheses additionally a corresponding sensitivity analysis is presented for one of the most promising test types

a parametric model for fusing heterogeneous fuzzy data presented is a model that integrates three data types numbers intervals and linguistic assessments data of these three types come from a variety of sensors one objective of sensor fusion models is to provide a common framework for data integration processing and interpretation that is what our model does we use a small set of artificial data to illustrate how problems as diverse as feature analysis clustering cluster validity and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model the effects of reparameterization on computational outputs are discussed numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types

a framework for linguistic modelling a new framework for linguistic reasoning is proposed based on a random set model of the degree of appropriateness of a label labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set valued function from a population of individuals into the set of subsets of labels appropriateness degrees are then evaluated relative to the distribution on this random set where the appropriateness degree of a label corresponds to the probability that it is contained in the set of appropriate labels this interpretation is referred to as label semantics a natural calculus for appropriateness degrees is described which is weakly functional while taking into account the logical structure of expressions given this framework it is shown that a bayesian approach can be adopted in order to infer probability distributions on the underlying variable given constraints both in the form of linguistic expressions and mass assignments in addition two conditional measures are introduced for evaluating the appropriateness of a linguistic expression given other linguistic information

additional criteria in fuzzy two level pattern recognition abstract pattern recognition with fuzzy sets is considered with regard to medical diagnostic problems an alternative interpretation of membership grade is suggested which expresses the degree of illness of a patient with respect to a certain disease a formal statement of a two level fuzzy classifier is presented the main emphasis is put on additional criteria designed to evaluate a fuzzy diagnostic decision for an unknown object two groups of criteria are formed decision making connected and domain dependent the decision making group of criteria can be viewed as related to the soft consensus measure in group decisions theory interconnections between criteria are discussed the notions proposed are illustrated with real medical data from the fields of aviation medicine and neonatology

fuzzy set connectives as combinations of belief structures abstract consonant belief structures provide a representation for fuzzy sets owing to the fact that their plausibility measures are essentially possibility measures we note that two belief structures are equivalent if their plausibility and belief functions are equal this observation leads us to provide a multiple number of equivalent representations for any belief structure commensurate representations can be induced for two different belief structures by forcing the same number of focal elements with the same weights we show that if we represent two consonant belief structures in a commensurate manner then their aggregations are closed with respect to consonance provided that the additional requirement that the underlying probability distributions satisfy a condition of correlation is imposed the results of this work allow us to use belief structure representations for the manipulation of fuzzy subsets under various logical combinations all basic fuzzy set connectives can thus be interpreted in the framework of the theory of evidence

a behavioral model for linguistic uncertainty abstract the paper discusses the problem of modelling linguistic uncertainty which is the uncertainty produced by statements in natural language for example the vague statement mary is young produces uncertainty about mary s age we concentrate on simple affirmative statements of the type subject is predicate where the predicate satisfies a special condition called monotonicity for this case we model linguistic uncertainty in terms of upper probabilities which are given a behavioural interpretation as betting rates possibility measures and probability measures are special types of upper probability measure we evaluate zadeh s suggestion that possibility measures should be used to model linguistic uncertainty and the bayesian claim that probability measures should be used our main conclusion is that when the predicate is monotonic possibility measures are appropriate models for linguistic uncertainty we also discuss several assessment strategies for constructing a numerical model

interpolation and partial differential equations one of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations pdes nowadays interpolation theory has been developed in an almost unbelievable way see the bibliography of maligranda interpolation of operators and applications 1926 1990 2nd ed lulea university lulea 1993 p 154 in this article some model examples are presented which display how powerful this theory is when dealing with pdes one main aim is to point out when it suffices to use classical interpolation theory and also to give concrete examples of situations when nonlinear interpolation theory has to be applied some historical remarks are also included and the relations to similar results are pointed out

fuzzy sets and membership functions based on probabilities abstract since the introduction of fuzzy set theory in 1965 several attempts to establish the relationship between the grades of membership and the classical probability measures have been made it turns out that there are different sources of fuzziness that must be dealt with differently in the present paper we examine in detail two types of fuzziness namely the fuzziness due to classification in an under or overdimensioned universe and the fuzziness due to the intersubject differences in opinion for the former case the membership function is defined to be equal to the normalized distance from the point to the boundary of the set in a specific metric it is shown that this definition of the membership function is fully consistent with the max min operations for the union intersection however the membership function of the complement is defined differently from the usual one minus rule the fuzziness due to the intersubject differences turns out to be a simple averaging process and the explicit formulas for this case were derived several examples that illustrate the notions of fuzzy intervals and fuzzy numbers are given and the interpretations of the derived membership curves are presented the formulas for calculation of the membership function of a sum of fuzzy numbers and of a product of fuzzy numbers times a constant are derived an extended definition of the measure of fuzziness is presented and applied to the defined membership functions

conditional probability fuzzy sets and possibility a unifying view we propose an interpretation of fuzzy set theory both from a semantic and a syntactic point of view in terms of conditional events and coherent conditional probabilities during past years a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory in this paper we emphasize the role of conditioning in a proper framework i e de finetti s coherence to get rid of many controversial aspects moreover we introduce suitable operations between fuzzy subsets looked on as corresponding operations between conditional events endowed with the relevant conditional probability finally we show how the concept of possibility function naturally arises as a coherent conditional probability

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

finite element galerkin method for the good boussinesq equation recently it has been shown by manoranjan et al l that this class of equations possesses an interesting solitary wave interaction mechanism moreover using semi group theoretic approch they have established global existence of a generalized solution provided the initial data are in a potential well further it is shown that finite time blowup may occur for weak solution in r if the initial data are not in the potential well and the total energy at t 0 exceeds the depth of the well in the first part of this paper we shall discuss existence uniqueness and regularity results for the problem 1 i 1 3 using the faedo galerkin method in the literature very few results are available pertaining to the numerical study of 1 l 1 4 in 2 defrutos et al have analysed pseudo spectral fourier method for the equation i i i 2 with periodic boundary conditions they have also discussed the nonlinear stability and convergence results in the analytic framework introduced earlier by lopez marcos and sanz serna 3 for related results on finite difference schemes one may refer to ortega and sanz serna 4 recently manoranjan et al 5 have applied petrov galerkin finite element method for cauchy problem without rigorous error analysis and have presented some numerical results in the later part of the present paper we have derived optima1 error estimates for both semidiscrete

detection and separation of ring shaped clusters using fuzzy clustering a new fuzzy clustering algorithm designed to detect and characterize ring shaped clusters and combinations of ring shaped and compact spherical clusters has been developed this fkr algorithm includes automatic search for proper initial conditions in the two cases of concentric and excentric intersected combinations of clusters validity criteria based on total fuzzy area and fuzzy density are used to estimate the optimal number of substructures in the data set the fkr algorithm has been tested on a variety of simulated combinations of ring shaped and compact spherical clusters and its performance proved to be very good both in identifying the input shapes and in recovering the input parameters application of the fkr algorithm to an mri image of the heart s left ventricle was used to investigate the possibility of using this algorithm as an aid in image processing

fuzzy semi markov migration process in credit risk we explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory we prove that a fuzzy market is viable if and only if an equivalent martingale measure exists from which we construct the forward probability measure and under which the discounted price of a default free bond is a martingale we model the evolution of credit migration of a defaultable bond as an inhomogeneous semi markov process with fuzzy states we study the effects of changing the real probability measure to a forward probability measure in addition we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default finally we discuss parameter estimation and calibration of the inhomogeneous markov chain with fuzzy states

a behavioral foundation for fuzzy measures in savage 41 a behavioral foundation was given for subjective probabilities to be used in the maximization of expected utility this paper analogously gives a behavioral foundation for fuzzy measures to be used in the maximization of choquet expected utility this opens the way to empirical verification or falsification of fuzzy measures and frees them of their ad hoc character

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

fuzzy logic and the quality of assessment of portfolios abstract in this paper it is argued that competence like height is a fuzzy concept this paper attempts to show the advantages of using fuzzy logic in the domain of assessment of prior learning apl via portfolios of evidence this issue which presently is of great concern to vocational educators highlights the need for defining some procedure in order to rationalize decisions made by assessors dealing with such type of evidence the assessment of portfolios are very much perceived presently as the way forward in modern educational assessment as portfolios do not only address evidence for past achievement but also evidence for current achievement thus highlighting the relevance of fuzzy logic in this area may raise awareness among educators of the benefits of such methodology the paper argues that fuzzy logic could offer useful insight in the attempt to rationalize assessor s deliberations when facing portfolio types of evidence the paper also tried to refine the fuzzy logic methodology perceived as most adequate for portfolio assessment as well as raise some issues that need addressing the examples referred to are based on the british national council for vocational qualifications ncvq apl procedures

interpretative versus prescriptive fuzzy set theory both traditional fuzzy set theory and the theory of subjective probabilities postulate their formulas the former because it does not accept a probabilistic interpretation of grades of membership the latter because it makes no connection between subjective probabilities and relative frequencies the interpretational theory of the tee model ascribes a well defined meaning to a membership value spl mu sub spl lambda in terms of probabilities which are limits of frequencies spl mu sub spl lambda is interpreted as the estimate by the subject of the probability that a given object would be assigned the label x in an everyday situation of uncertainty in contrast to the max min formulas used in many applications of fuzzy sets the postulated summation to 1 formula of all fuzzy clustering algorithms follows from the interpretative tee model this agrees with the author s view that fuzzy set theory should be defined as a theory which allows partial membership values of an object in a class or cluster not as a theory which uses specific mathematical operators the operators must be derived from the well defined interpretation of partial membership values

generalized eigenfunctions of layered elastic media and application to diffuse fields the spectral decomposition of the elastic wave operator in a layered isotropic half space is derived by means of standard functional analysis methods particular attention is paid to the coupled p sv waves the problem is formulated directly in terms of displacements which leads to a 2 2 sturm liouville system the resolvent kernel green s function is expressed in terms of simple plane wave solutions application of stone s formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity the generalized eigenfunction expansion is employed to define a diffuse field as a white noise process in modal space by means of a wigner transform we calculate vertical to horizontal kinetic energy ratios in layered media as a function of depth and frequency several illustrative examples are considered including energy ratios near a free surface in the presence of a soft layer numerical comparisons between the generalized eigenfunction summation and

a clustering algorithm for fuzzy model identification the fuzzy model proposed by takagi and sugeno can represent highly nonlinear systems and is widely used for the representation of fuzzy rules in this paper the model is firstly modified to make its identification easier based on the fuzzy c partition space four criteria are proposed for optimization of the model parameters following that a clustering algorithm composed of fuzzy c linear functions clustering and like fuzzy c means clustering is developed for minimizing the four criteria an identification scheme for rule s premise and consequence parameters is deduced from the clustering algorithm in succession finally four examples are demonstrated to verify the effectiveness of the proposed algorithm

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

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

the efficacy of fuzzy representations of uncertainty advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments these are 1 the reality hypothesis which holds that imprecision is an inherent property of the world external to an observer 2 the subjectivity hypothesis which holds that probability is an exclusively objective measure of uncertainty and that therefore subjective uncertainty can only be represented with fuzzy sets 3 the behaviorist hypothesis which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty 4 the probability as fiction hypothesis which claims that probability does not comprise a field of study in its own right and 5 the superset hypothesis which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment we discuss and criticize all five we then criticize the argument that fuzziness represents a type of uncertainty distinct from probability and also the inordinate complexity of fuzzy methods we present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples

comparison of empirical and computed values of fuzzy conjunction abstract the necessity for further experiments such as in 4 for the foundation of fuzzy sets theory is shown generality of the experimental results on fuzzy conjunction in the paper by thole zimmermann and zisno 1979 4 for the other objects and language is verified for other conditions usually the minimum the product and the restricted sum operators are used as an intersection conjunction of fuzzy sets in fuzzy optimization f optimization on the basis of experiments it is shown that these operators are correct for f optimization their correctness is founded on the stability of the extremal point a decision in bellman zadeh s scheme under a positive linear mapping it is shown that these mappings exist between empirical conjunction and considered operators as a statistical result of linear regression analysis

relational duals of cluster validity functions for the c means family clustering aims to identify groups of similar objects to evaluate the results of cluster algorithms an investigator uses cluster validity indices while the theory of cluster validity is well established for vector object data little effort has been made to extend it to relationship based data as such this paper proposes a theory of reformulation for object data validity indices so that they can be used to rank the results produced by the relational means clustering algorithms more specifically we create a class of relational validity indices which is called dual relational indices that are guaranteed under certain but easily met constraints to produce the same results and hence the same cluster counts as their object data counterparts

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

structural and functional quantization of vagueness abstract this paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by bayesian decision theory the methods are called structural quantization and functional quantization respectively the idea behind structural quantization is that for a comparative vague category like good scores there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is functional quantization adopts the conclusion of bayesian decision theory that a rational agent s belief can be represented by a probability to quantize the term good score the method first assesses the two probabilities that respectively represent a rational agent s beliefs about john s score before and after learning that john has got a good score it then derives a membership function by comparing those two probabilities the equivalence of the two quantization methods is hypothesized the logic connectives not and and or are discussed in the setting of structural quantization while aggregation of information is addressed in the setting of functional quantization

probabilities and fuzzy events an operational approach abstract this paper tries to give a redesign of a well established research area of traditional fuzzy set theory the notion of a probability of a fuzzy event as well as related results are reexamined and new definitions for them are given which are based on the operational viewpoint of f set theory and on some concepts of operational statistics a critical survey of existing concepts will show some advantages of the operational approach

unity and diversity of fuzziness spl minus from a probability viewpoint like the fields of probability and statistics fuzzy set theory is characterized by a variety of viewpoints adherents of fuzzy methods however consistently maintain that probability is not necessarily the optimal representation of uncertainty we rebut this view

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

the initial boundary value problem for the good boussinesq equation on the half line abstract studied here is an initial boundary value problem for the good boussinesq equation on the half line it is shown that this initial boundary value problem when the initial boundary data satisfy certain compatible conditions is well posed in the l 2 based sobolev space h s r for s 1 2 in addition the mapping that associated to appropriate initial boundary data the corresponding solution is analytic as a function between appropriate banach spaces

a semantics for possibility theory based on likelihoods abstract in this paper a semantic basis for possibility theory based on likelihood functions is presented in some cases possibilities have been considered as approximations of shafer plausibility measures this approximation exchanges exactness of plausibility values for the simplicity of use of possibility values in this paper a different direction is followed possibility measures are considered as the supremum of a family of likelihood functions this is an exact interpretation not an approximation the minimum rule to combine possibility distributions is justified in this framework under general conditions conditions under which other rules can be applied are also studied

well posedness and nonlinear smoothing for the good boussinesq equation on the half line abstract in this paper we study the regularity properties of the good boussinesq equation on the half line we obtain local existence uniqueness and continuous dependence on initial data in low regularity spaces moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining half derivative smoothing of the nonlinear term in some cases our paper improves the result in 17 being the first result that constructs solutions for the initial and boundary value problem of the good boussinesq equation below the l 2 space our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in 20 and 13

the generalized modus ponens considered as a fuzzy relation abstract the generalized modus ponens gmp is an inference rule in fuzzy logic with the following scheme if s1 is p1 then s2 is p2 but s1 is q1 hence s2 is q2 the objective is to determine q2 given p1 p2 and q1 thepredicate p1 in the antecedent of the if then rule is fuzzy hence the condition s1 is p1 to derive s2 is p2 is not strict therefore it should be possible to render a reasonable approximation of q2 when q1 diverges slightly from p1 a number of criteria for this inference rule will be presented in literature there are three basic assumptions for dealing with the gmp namely 1 the if then rule is implemented by an implication rule from some multivalued logic i e the if then rule is represented by a fuzzy relation 2 the premiss p1 can be strengthened or weakened to obtain q1 by taking a power of the membership function of p1 and 3 the conclusion q2 is derived by means of the max min composition rule it will be shown by a number of theorems that these three items are incompatible with the criteria furthermore it will be shown that it is arduous to construct an implication rule to meet the conditions 2 and 3 a way to restore this is to alter condition 2 instead of taking a power of the membership function it is also possible to change the parameters of this function but this also causes complications hence requirement 3 should be modified into some functional relation it will be shown how this can be accomplished in combination with the functions with parameters changed

modeling of wave propagation in inhomogeneous media we present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time reversal invariance and reciprocity in generally inhomogeneous media the approach relies on a representation theorem of the wave equation to express the green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium following a predictable initial computational effort green s functions between arbitrary points in the medium can be computed as needed using a simple cross correlation algorithm

a fuzzy based customer classification method for demand responsive logistical distribution operations in some cases customer classification is important for the development of advanced logistical distribution strategies in response to the growing complexity in business logistical markets this paper presents a new approach that can be employed to cluster customers before executing fleet routing in logistical operations the proposed approach is developed on the basis of fuzzy clustering techniques and involves three sequential mechanisms including 1 binary transformation 2 generation of a fuzzy correlation matrix and 3 customer clustering such a customer clustering method should be performed prior to vehicle dispatching and routing in the process of goods distribution the proposed methodology clusters customers on the basis of their demand attributes rather than the static geographic property which is considered extensively in most published vehicle routing algorithms in addition to methodology development a case study was conducted to demonstrate the potential advantages of the proposed fuzzy clustering based method it is expected that this study can stimulate more research on time based logistics control and management

a probabilistic and statistical view of fuzzy methods fuzzy set theory has primarily been associated with control theory and with the representation of uncertainty in applications in artificial intelligence more recently fuzzy methods have been proposed as alternatives to traditional statistical methods in statistical quality control linear regression and forecasting among other areas we review some basic concepts of fuzzy methods point out some philosophical and practical problems and offer simpler alternatives based on traditional probability and statistical theory applications in control theory and statistical quality control serve as our primary examples

detection and characterization of cluster substructure ii fuzzy c varieties and convex combinations thereof in part i siam j appl math 40 1981 pp 339 357 fuzzy c lines was introduced as an algorithm for detection and characterization of linearly clustered data in part ii we address two extensions of the theory in part i specifically we will first generalize the straight line prototype of a cluster developed in part i to any r dimensional linear variety of r s 0 leqq r s secondly we will consider a distance functional which utilizes convex combinations of the distance functionals developed here and in part i all of the notation and symbols used here are unchanged from part i

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

detection and characterization of cluster substructure i linear structure fuzzy c lines in part i a generalization of the fuzzy c means or fuzzy isodata clustering algorithms is developed necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a picard iteration scheme which generates simultaneously i c fuzzy clusters in the data ii a set of c prototypical straight lines in feature space which best fit the data in a well defined sense iii a set of c prototpyical centers of mass on the c lines which characterize the core of each linear fuzzy cluster theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices a convergence theorem for each algorithm in the infinite family is given the algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially linear substructure in part ii the fuzzy c means and fuzzy c lines algorithms are shown to be special cases of a more general class of fuzzy algorithms the

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

supplement to an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i abstract basic idea of part i of an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators 1992 mainly on the conceptual construction of membership functions is straightened out and illustrated by detailed examples the purpose of this article is to clarify unclear points in notations and concepts of mabuchi 1992 as a step toward the extension of the theory from type 1 to three valued fuzzy sets

when upper probabilities are possibility measures abstract a characteristic property is given for a pair of upper and lower probabilities induced by lower probability bounds on a finite set of events to coincide with possibility and necessity measures approximations of upper probabilities by possibility measures are discussed the problem of combining possibility distributions viewed as upper probabilities is investigated and the basic fuzzy set intersections are justified in this framework