on viewing fuzzy measures as fuzzy subsets we introduce the concept of a fuzzy measure discuss some of their basic properties and look at some notable examples we introduce a correspondence between a fuzzy measure on x and a fuzzy subset over the power set of x these fuzzy subsets are required to have special properties on the space 2x and we refer to these as measure type fuzzy sets we take advantage of this correspondence between a fuzzy measure and a fuzzy set to suggest the formulation of various set operations on fuzzy measures

modeling interaction phenomena using fuzzy measures on the notions of interaction and independence interaction phenomena among elements of a finite set can be modeled by a discrete fuzzy measure the notion of marginal interaction between two elements is at the root of the definition of interaction we first extend this notion by defining that of marginal interaction among pairwise disjoint subsets and we study some of its properties then we generalize the notion of interaction index to pairwise disjoint subsets finally we formalize the intuitive notion of independence and we propose the concept of measure of marginal amount of interaction in order to fill the gap between the notions of interaction and independence

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

discrete choquet integral and some of its symmetric extensions classical extensions of the choquet integral defined on 0 1 to 1 1 are the asymmetric and the symmetric choquet integral the second one being called also the sipos integral recently the balancing choquet integral was introduced as another kind of a symmetric extension of the discrete choquet integral we introduce and discuss a new type of such extension the fusion choquet integral and discuss its properties and relationship to the balancing and the symmetric choquet integral the symmetric maximum introduced by grabisch is shown to be a special case of the fusion and the balancing choquet integral several extensions of owa operators are also discussed

componentwise decomposition of some lattice valued fuzzy integrals lattice valued fuzzy measures are lattice valued set functions which assign the bottom element of the lattice to the empty set the top element of the lattice to the entire universe and satisfy the property of monotonicity if the lattice is complete then a lattice valued fuzzy integral of sugeno type with similar properties such as the sugeno integral in its original form can be introduced in a natural way the main result of the paper is a componentwise decomposition theorem of an l valued fuzzy integral to its l valued fuzzy integrals components where l is a complete lattice with negation and l a b a email protected l a email protected is organized as a complete lattice too this result is useful to obtain the properties of l valued fuzzy integrals from the properties of l valued fuzzy integrals and to calculate in a simple way the values of some integrals from the values of the components the important case l 0 1 when l becomes the lattice of the intuitionistic fuzzy values is distinctly discussed an idea of application to synthetic evaluation of objects is also suggested

weighted minimum and maximun operations in fuzzy sets theory abstract this short note comments on the interpretation of weighted minimum and maximum operations introduced in a previous paper as medians and corrects a misleading statement about it

interactive determination of a utility function represented as a fuzzy integral abstract in this paper we propose an alternative method for interactive assessment of a utility function when this function is represented by a fuzzy integral it is generally agreed that the fuzzy integral provides a good representation of the human subjective structure however this type of integral may not be convenient because the assessment of the underlying fuzzy measure requires a lot of data in this paper we adopt an alternative point of view for determining the values of the fuzzy measure from few data by interactively changing equation constraints our method will contribute to increasing the applicability of the fuzzy measure and of the fuzzy integral in the field of decision support systems

pictorial representations of fuzzy connectives part ii cases of compensatory operators and self dual operators abstract as the continuation of part i in which pictorial representations of t norms t conorms and averaging operators were made with the aid of a computer this paper discusses compensatory operators and proposes generalized compensatory operators which can be obtained from averaging operators self dual operators are discussed which can be defined by using t norms t conorms and averaging operators finally symmetric sums are reviewed which are also self dual operators the pictorial representations of these operators are also made

non abelian quantum statistics on graphs we show that non abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces in particular we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space x the framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of x which can be achieved by determining its homology groups we apply this methodology for configuration spaces of graphs as a conclusion we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non abelian anyons on networks that are used in quantum computing these conclusions are based on our solution of the so called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs

on equivalence classes of fuzzy connectives the case of fuzzy integrals an equivalence relation between connectives or operators in the framework of multiple criteria decision making is introduced two operators are said to be equivalent if they lead to the same ranking of alternatives some general results to find equivalence classes of operators are given through the concept of level surfaces in a second part these results are applied to the case of discrete fuzzy integrals which are considered here as n place operators first some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals

characterization of the ordered weighted averaging operators this paper deals with the characterization of two classes of monotonic and neutral mn aggregation operators the first class corresponds to mn aggregators which are stable for the same positive linear transformations and presents the ordered linkage property the second class deals with mn idempotent aggregators which are stable for positive linear transformations with the same unit independent zeroes and ordered values these two classes correspond to the weighted ordered averaging operator owa introduced by yager in 1988 it is also shown that the owa aggregator can be expressed as a choquet integral

on the extension of pseudo boolean functions for the aggregation of interacting criteria abstract the paper presents an analysis on the use of integrals defined for non additive measures or capacities as the choquet and the sipos integral and the multilinear model all seen as extensions of pseudo boolean functions and used as a means to model interaction between criteria in a multicriteria decision making problem the emphasis is put on the use besides classical comparative information of information about difference of attractiveness between acts and on the existence for each point of view of a neutral level allowing to introduce the absolute notion of attractive or repulsive act it is shown that in this case the sipos integral is a suitable solution although not unique properties of the sipos integral as a new way of aggregating criteria are shown with emphasis on the interaction among criteria

quantum graphs with singular two particle interactions we construct quantum models of two particles on a compact metric graph with singular two particle interactions the hamiltonians are self adjoint realizations of laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions in order to find such hamiltonians closed and semi bounded quadratic forms are constructed from which the associated self adjoint operators are extracted we provide a general characterization of such operators and furthermore produce certain classes of examples we then consider identical particles and project to the bosonic and fermionic subspaces finally we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate weyl asymptotic law

generalizations of owa operators owa operators can be seen as symmetrized weighted arithmetic means as choquet integrals with respect to symmetric measures or as comonotone additive functionals following these three different looks on owas we discuss several already known generalizations of owa operators including gowa iowa oma operators as well as we propose new types of such generalizations

fuzzy measure of fuzzy events defined by fuzzy integrals abstract we define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by murofushi called fuzzy t conorm integral encompassing previous definitions zadeh defined the probability measure of a fuzzy event and later the possibility measure of fuzzy event using a duality property of fuzzy t conorm integral we propose a general definition of fuzzy measure of fuzzy events which is compatible with previous definitions of zadeh and possesses all properties of a fuzzy measure in particular the duality property using our definition we examine the case of decomposable measures and belief functions a comparison with previous works is provided

n particle quantum statistics on graphs we develop a full characterization of abelian quantum statistics on graphs we explain how the number of anyon phases is related to connectivity for 2 connected graphs the independence of quantum statistics with respect to the number of particles is proven for non planar 3 connected graphs we identify bosons and fermions as the only possible statistics whereas for planar 3 connected graphs we show that one anyon phase exists our approach also yields an alternative proof of the structure theorem for the first homology group of n particle graph configuration spaces finally we determine the topological gauge potentials for 2 connected graphs

analytic investigations of the effects of near surface three dimensional galvanic scatterers on mt tensor decompositions an outcropping hemispherical inhomogeneity embedded in a two dimensional 2 d earth is used to model the effects of three dimensional 3 d near surface electromagnetic em static distortion analytical solutions are first derived for the galvanic electric and magnetic scattering operators of the heterogeneity to represent the local distortion by 3 d structures of fields which were produced by a large scale 2 d structure these 3 d scattering operators are applied to 2 d electric and magnetic fields derived by numerical modeling to synthesize an mt data set synthetic noise is also included in the data these synthetic data are used to study the parameters recovered by several published methods for decomposing or parameterizing the measured mt impedance tensor the stability of these parameters in the presence of noise is also examined the parameterizations studied include the conventional 2 d parameterization swift 1967 eggers s 1982 and spitz s 1985 eigenstate formulations latorraca et al

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

a genetic fuzzy linguistic combination method for fuzzy rule based multiclassifiers fuzzy set theory has been widely and successfully used as a mathematical tool to combine the outputs provided by the individual classifiers in a multiclassification system by means of a fuzzy aggregation operator however to the best of our knowledge no fuzzy combination method has been proposed which is composed of a fuzzy rule based system we think this can be a very promising research line as it allows us to benefit from the key advantage of fuzzy systems i e their interpretability by using a fuzzy linguistic rule based classification system as a combination method the resulting classifier ensemble would show a hierarchical structure and the operation of the latter component would be transparent to the user moreover for the specific case of fuzzy multiclassification systems the new approach could also become a smart way to allow standard fuzzy classifiers to deal with high dimensional problems avoiding the curse of dimensionality as the chance to perform classifier selection at class level is also incorporated into the method we conduct comprehensive experiments considering 20 uci datasets with different dimensionality where our approach improves or at least maintains accuracy while reducing complexity of the system and provides some interpretability insight into the multiclassification system reasoning mechanism the results obtained show that this approach is able to compete with the state of the art multiclassification system selection and fusion methods in terms of accuracy thus providing a good interpretability accuracy tradeoff

remarks on non standard statistics the theory of non standard statistics is reviewed and placed in a co homological setting some general mathematical results are shown to be relevant the specific heat of a two particle molecular gas is calculated as a function of temperature and the statistics determining parameter it shows strong behaviour close to the fermion case the second virial coefficient is calculated by a slightly improved method and its behaviour is compared with that of the vacuum energy of a quantum field interacting with a flux tube

the wowa operator and the interpolation function w chen and otto s interpolation method revisited abstract chen and otto introduced in 1995 an interpolation method for defining membership functions from a set of points in this paper we report some errors found in that algorithm and introduce some new conditions on the boundaries that are more general than the ones in the original work we construct in this way a new interpolation method and prove that it has a smooth shape on the boundaries this study is related to the wowa operator an aggregation function that generalizes both the weighted mean and the owa operator wowa uses an interpolation method to calculate a vector of weights as a combination of other two vectors the new method is introduced to implement this combination

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

on the moments and distribution of discrete choquet integrals from continuous distributions we study the moments and the distribution of the discrete choquet integral when regarded as a real function of a random sample drawn from a continuous distribution since the discrete choquet integral includes weighted arithmetic means ordered weighted averaging functions and lattice polynomial functions as particular cases our results encompass the corresponding results for these aggregation functions after detailing the results obtained in j l marichal i kojadinovic distribution functions of linear combinations of lattice polynomials from the uniform distribution statistics probability letters 78 2008 985 991 in the uniform case we present results for the standard exponential case show how approximations of the moments can be obtained for other continuous distributions such as the standard normal and elaborate on the asymptotic distribution of the choquet integral the results presented in this work can be used to improve the interpretation of discrete choquet integrals when employed as aggregation functions

orthogonality in csamt and mt measurements the electric and magnetic fields from a single plane wave source on a one dimensional 1 d earth or a plane wave source polarized parallel or perpendicular to strike on a two dimensional 2 d earth are orthogonal on a layered earth and in the far field of a controlled source the electric and magnetic fields are also orthogonal therefore orthogonality of e and h data is a necessary condition to justify the application of 1 d or 2 d modeling algorithms having a plane wave source a strict criterion to prove orthogonality and thus provide a rationale for the choice of interpretation methods can be defined directly in terms of field data however field data acquired in the intermediate and near field of any electromagnetic em source are generally not orthogonal even on a plane layered earth representing these nonorthogonal data in an orthogonal coordinate system can be misleading particularly for the minor axis components of the polarization ellipses nonorthogonality also arises because of 3

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

continuous generated associative aggregation operators continuous generated associative aggregation operators are discussed the only asymmetric solutions are the trivial operators of the projection to the first or the last coordinate respectively the only nontrivial convenient aggregation operators are necessarily symmetric and they are either continuous archimedean t norms and t conorms or generated nullnorms the latter may combine any nilpotent t conorm and any t norm with any prescribed annihilator k 0 1

decomposable and k additive multi capacities and multi polar capacities extensions of several concepts for definition of capacities given by a small number of parameters in the unipolar and bipolar framework to the case of multi polar input spaces are studied the paper covers decomposable symmetric category symmetric additive reduced additive and k additive multi polar capacities and discusses its properties the multi polar mobius transform of multi polar capacities is introduced and its properties for coordinate wisely given additive and k additive multi polar capacities are described

2 additive capacity identification methods from multicriteria correlation preference information the essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity the 2 additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity besides the preference with respect to the decision criteria most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set which is a time consuming task for the decision maker in this paper we propose some models to identify 2 additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information mccpi the mccpi is a group of 2 d preference information which can be described and obtained by the refined diamond diagram the common principle of the proposed models is to minimize the different kinds of deviations between the mccpi and the most desired 2 additive capacity ies a multicriteria decision making example is presented to show the feasibility of the proposed methods and a 2 d scale of the mccpi is also introduced in the further discussion of the illustrative example

fuzzy measure analysis of public attitude towards the use of nuclear energy abstract this paper is concerned with applying fuzzy measures and fuzzy integrals to analyze public attitude towards the use of nuclear energy to this end a questionnaire on the use of nuclear energy is set up and data are collected in japan the philippines and the frg factor analysis is performed to get the primary structure of public attitude it is shown that the attitude of the responders to the questionnaire in each country is well explained with its hierarchical structure obtained by fuzzy measure analysis

hierarchical madm with fuzzy integral for evaluating enterprise intranet web sites in the real world most criteria have inter dependent or interactive characteristics so they cannot be evaluated by conventional additive measures thus to approximate the human subjective evaluation process it would be more suitable to apply a fuzzy integral model in which it is not necessary to assume additivity and independence this research proposes an effective algorithm to determine the value using the input data of fuzzy densities and the fuzzy integral based on fuzzy measure to determine the overall evaluation this research also gives an example of evaluating enterprise intranet web sites with illustrations of the hierarchical structure of fuzzy measure for a choquet integral model the results show that the fuzzy integral is more suitable than a traditional multicriteria evaluation method for human subjective evaluation or when criteria are not mutually independent

implications of the born approximation for the magnetotelluric problem in three dimensional environments a first order born approximation is obtained for the integral equations governing the surface magnetotelluric response over a three dimensional earth although accurate only in cases of low resistivity contrasts the resulting expressions 1 exhibit a linear relationship between a spatial perturbation in subsurface resistivity and the ensuing perturbation on the surface field response and more importantly 2 allow arbitrary degrees of complexity in the geometrical characteristics of the subsurface the linear system solutions derived from the born approximation are studied by examining the properties of their associated kernels these kernels may be thought of as a suite of horizontal magnetotelluric wavelets weighting the subsurface resistivity distribution at different depth levels analytical expressions for the wavelets are obtained in the wavenumber domain thus generating a suite of magnetotelluric transfer functions expressions for the latter are particularized to the cases of one and t

controlled source tensor magnetotellurics impedance tensor and tipper vectors known to connect the electromagnetic surface components for plane wave excitation are shown to be uniquely defined for horizontal electric or horizontal magnetic dipole sources two independent source polarizations are needed for their estimation in practice analysis of impedance tensors and tipper vectors for a layered half space shows that the impedance tensor can be antidiagonalized by rotating the measurement system so that one of the measurement directions coincides with the direction to the transmitter dipole the tipper vector points towards the transmitter dipole in the static limit all transfer functions are real and simple analytic results for a homogeneous half space show that impedance elements are proportional to the inverse of the product of conductivity and distance between source and receiver while the tipper vector is a unit vector pointing towards the transmitter dipole

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

fuzzy t conorm integral with respect to fuzzy measures generalization of sugeno integral and choquet integral abstract this article discusses integrals with respect to sugeno s fuzzy measures sugeno integrals and choquet integrals with respect to general fuzzy measures are generalized in the setting of a t conorm algebraic system first t conorm integrals with respect to decomposable measures based on t conorm algebraic system are defined they are a modification of the weber integral secondly they are extended to fuzzy t conorm integrals with respect to general fuzzy measures the extension is made in the same way as the extension of the lebesgue integral to the choquet integral the class of fuzzy t conorm integrals includes choquet integrals and sugeno integrals properties of the integrals especially their representation are discussed

homology groups for particles on one connected graphs we present a mathematical framework for describing the topology of configuration spaces for particles on one connected graphs in particular we compute the homology groups over integers for different classes of one connected graphs our approach is based on some fundamental combinatorial properties of the configuration spaces mayer vietoris sequences for different parts of configuration spaces and some limited use of discrete morse theory as one of the results we derive the closed form formulae for ranks of the homology groups for indistinguishable particles on tree graphs we also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles our motivation is the search for new kinds of quantum statistics

complex structured decision making model a hierarchical frame work for complex structured data we introduce a hierarchical framework we call complex structured decision making model for complexly structured knowledge representation in intelligent decision making we show that our model extends non hierarchical flat decision making models to hierarchical decision making models that are similar to comprehensible human decision making processes further we make an argument that hierarchial representation of knowledge in a complex structured decision making model simplifies the approximation of aggregation functions to easily adapt to the underline relation of the system additionally using a real world complex structured data set we show that hierarchically organized fuzzy integrals e g choquet integral and sugeno integral and fuzzy signatures outperform these non hierarchical fuzzy integrals

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented