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

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

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

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

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

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

dyson s ranks and appell lerch sums denote by p n the number of partitions of n and by n a m n the number of partitions of n with rank congruent to a modulo m we find and prove a general formula for dyson s ranks by considering the deviation of the ranks from the average begin aligned d a m sum n 0 infty left n a m n frac p n m right q n end aligned using appell lerch sum properties we decompose d a m into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions whose modulus we can control using our decomposition we show how our formula gives as a straightforward consequence atkin and swinnerton dyer s results on ranks as well as celebrated work of bringmann ono and rhoades s on maass forms we also give an example of how our methods apply to work of lovejoy and osburn on rank differences

a recursive model reduction method for approximate inference in gaussian markov random fields this paper presents recursive cavity modeling a principled tractable approach to approximate near optimal inference for large gauss markov random fields the main idea is to subdivide the random field into smaller subfields constructing cavity models which approximate these subfields each cavity model is a concise yet faithful model for the surface of one subfield sufficient for near optimal inference in adjacent subfields this basic idea leads to a tree structured algorithm which recursively builds a hierarchy of cavity models during an ldquoupward passrdquo and then builds a complementary set of blanket models during a reverse ldquodownward pass rdquo the marginal statistics of individual variables can then be approximated using their blanket models model thinning plays an important role allowing us to develop thinned cavity and blanket models thereby providing tractable approximate inference we develop a maximum entropy approach that exploits certain tractable representations of fisher information on thin chordal graphs given the resulting set of thinned cavity models we also develop a fast preconditioner which provides a simple iterative method to compute optimal estimates thus our overall approach combines recursive inference variational learning and iterative estimation we demonstrate the accuracy and scalability of this approach in several challenging large scale remote sensing problems

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

texture analysis through a markovian modelling and fuzzyclassification application to urban area extraction fromsatellite images herein we propose a complete procedure to analyze and classify the texture of an image we apply this scheme to solve a specific image processing problem urban areas detection in satellite images first we propose to analyze the texture through the modelling of the luminance field with eight different chain based models we then derived a texture parameter from these models the effect of the lattice anisotropy is corrected by a renormalization group technique coming from statistical physics this parameter which takes into account local conditional variances of the image is compared to classical methods of texture analysis afterwards we develop a modified fuzzy cmeans algorithm that includes an entropy term the advantage of such an algorithm is that the number of classes does not need to be known a priori besides this algorithm provides us with further information i e the probability that a given pixel belongs to a given cluster finally we introduce this information in a markovian model of segmentation some results on spot5 simulated images spot3 images and ers1 radar images are presented these images are provided by the french national space agency cnes and the european space agency esa

mock modularity of the m d rank of overpartitions abstract we investigate the modular properties of a new partition rank the m d rank of overpartitions in fact this is an infinite family of ranks indexed by the positive integer d that gives both the dyson rank of overpartitions and the overpartition m 2 rank as special cases the m d rank of overpartitions is the holomorphic part of a certain harmonic maass form of weight 1 2 we give the exact transformation of this harmonic maass form along with a few identities for the m d rank

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

tenth order mock theta functions in ramanujan s lost notebook ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved the first six of ramanujan s tenth order mock theta function identities it is the purpose of this paper to prove the seventh and eighth identities of ramanujan s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral l j mordell s transformation formula for the definite integral plays a key role in the proofs of these identities also the properties of modular forms are used for the proofs of theta function identities

estimating gaussian markov random field parameters in a nonstationary framework application to remote sensing imaging in this paper we tackle the problem of estimating textural parameters we do not consider the problem of texture synthesis but the problem of extracting textural features for tasks such as image segmentation we take into account nonstationarities occurring in the local mean we focus on gaussian markov random fields for which two estimation methods are proposed and applied in a nonstationary framework the first one consists of extracting conditional probabilities and performing a least square approximation this method is applied to a nonstationary framework dealing with the piecewise constant local mean this framework is adapted to practical tasks when discriminating several textures on a single image the blurring effect affecting edges between two different textures is thus reduced the second proposed method is based on renormalization theory statistics involved only concern variances of gaussian laws leading to cramer rao estimators this method is thus especially robust with respect to the size of sampling moreover nonstationarities of the local mean do not affect results we then demonstrate that the estimated parameters allow texture discrimination for remote sensing data the first proposed estimation method is applied to extract urban areas from spot images since discontinuities of the local mean are taken into account we obtain an accurate urban areas delineation finally we apply the renormalization based on method to segment ice in polar regions from avhrr data

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

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

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

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

discrete markov image modeling and inference on the quadtree noncasual markov or energy based models are widely used in early vision applications for the representation of images in high dimensional inverse problems due to their noncausal nature these models generally lead to iterative inference algorithms that are computationally demanding in this paper we consider a special class of nonlinear markov models which allow one to circumvent this drawback these models are defined as discrete markov random fields mrf attached to the nodes of a quadtree the quadtree induces causality properties which enable the design of exact noniterative inference algorithms similar to those used in the context of markov chain models we first introduce an extension of the viterbi algorithm which enables exact maximum a posteriori map estimation on the quadtree two other algorithms related to the mpm criterion and to bouman and shapiro s 1994 sequential map smap estimator are derived on the same hierarchical structure the estimation of the model hyper parameters is also addressed two expectation maximization em type algorithms allowing unsupervised inference with these models are defined the practical relevance of the different models and inference algorithms is investigated in the context of image classification problem on both synthetic and natural images

sixth order mock theta functions two new mock theta functions of the sixth order are defined the main theorem in this paper theorem 1 1 provides four transformation formulas relating the new mock theta functions with ramanujan s mock theta functions of the sixth order two further representations of the new mock theta functions are established lastly a hitherto unproved entry from ramanujan s lost notebook related to sixth order mock theta functions is proved

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

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

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

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

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

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

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

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

eulerian series as modular forms in 1988 hickerson proved the celebrated mock theta conjectures a collection of ten identities from ramanujan s lost notebook which express certain modular forms as linear combinations of mock theta functions in the context of maass forms these identities arise from the peculiar phenomenon that two different harmonic maass forms may have the same non holomorphic parts using this perspective we construct several infinite families of modular forms which are differences of mock theta functions

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

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