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

uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

on the notion of concept ii it is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks several qualitative approaches were thereafter tempted that aimed at modeling membership through ordinal scales or lattice fuzzy sets most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory we propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set a f equipped with a total order this function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity similarly the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f once the basic notions of membership and typicality are set the study of compound concepts is possible and leads to interesting results in particular we investigate the internal structure of concepts and obtain the characterization of all smooth subconcepts of a given concept

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

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

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

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

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

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

exact scaling behavior of partially convex vesicles we solve analytically for the perimeter area generating functions for two models of vesicles while from the solution of the first model staircase polygons one can easily extract the asymptotic scaling behavior the exact solution of the second column convex polygons is difficult to analyze this leads us to apply a recently developed method for deriving the scaling behavior indirectly utilizing a set of nonlinear differential equations one result of this work is a nontrivial confirmation of the scaling universality hypothesis

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

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

interacting columns generating functions and scaling exponents a model of self interacting columns related to models of partially directed walks and to histogram polygons is considered the generating function of this model is found in terms of q deformed bessel functions using a functional recursion scheme a transition related to a deflation inflation transition seen in staircase polygons or to a rough smooth transition in a solid on solid model is found and its scaling exponents are found in the context of a tricritical scaling analysis if the columns are also interacting with the horizontal axis then the inflated or smooth phase is also found to undergo an adsorption transition a special point exists where the model is critical with respect to both an adsorption transition and a deflation inflation transition

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

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

stacking models of vesicles and compact clusters we investigate three simple lattice models of two dimensional vesicles these models differ in their behavior from the universality class of partially convex polygons which has been recently established they do not have the tricritical scaling of those models and furthermore display a surprising feature their perimeter free energy is discontinuous with an isolated value at zero pressure we give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq series solutions and obtain the scaling functions where applicable

inflating square and rectangular lattice vesicles in this paper two simple models of lattice vesicles in two and in three dimensions are examined lattice vesicles have a rich mathematical nature and the combinatorial and scaling properties of square and rectangular lattice vesicle models are investigated and then generalized to three dimensions the phase diagram of the vesicles also includes a multicritical point and scaling of the generating functions of these models are studied to test the models for tricritical scaling behaviour in the models of square vesicles in the square lattice and cubical vesicles in the cubic lattice the tricritical scaling forms are explicitly confirmed however rectangular vesicles in the square lattice and rectangular box vesicles in the cubic lattice appear not to conform to the expected tricritical scaling forms

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

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

ordinal decomposability and fuzzy connectives the benefit of computing with linguistic terms is now generally accepted fuzzy set theory provides us the conceptual tool for the interpretation and evaluation of linguistic concepts and expressions it constitutes a quantification of the compatibility degree of objects with the associated linguistic concept through a membership function when we make computation using fuzzy membership values such as in the evaluation of fuzzy rules confidence the implicit assumptions are that the membership values have quantitative semantics the extensive scale assumption and that the numeric values are commensurate among the different fuzzy sets generated by the different concepts involved the common scale assumption in most situations these assumptions are difficult to justify and may lead to various anomalies the membership values are more suitably interpreted only as ordinal scales where the numeric representations reflect compatibility orderings in this paper we examine the concept of fuzzy intersection and union from the perspective of decomposability and ordinal conjoint structure in measurement theory we determine conditions under which a weak order induced by a fuzzy set or otherwise can be decomposed into other weak orders we show particular cases of ordinal decomposability which correspond naturally to our concept of fuzzy intersection and union this perspective of fuzzy connectives help us resolve some of the difficulties related to the above assumptions

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

job search and employment abstract a multistage search method based on fuzzy preference is used to analyze the processes of job search an employment decision based on fuzzy preference results in an exact choice the incorporation of fuzzy interval preference into employment decision however may lead to multiple choices which provides a theoretical justification for the existence of fuzzy supply of labor thus fuzziness in addition to randomness offers an explanation to wage dispersion

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

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

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

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

vagueness and bayesian probability this paper is a response to michael laviolette and john w seaman jr s ibid vol 2 no 1 p 4 1994 position paper the efficacy of fuzzy representations of uncertainty which criticizes fuzzy representations of uncertainty and suggests that bayesian probability can do better the commenter argues that the author s make some misleading comments about bayesian probability and he briefly discusses the problem of giving a satisfactory interpretation of membership functions

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

a measurement theoretic axiomatization of trapezoidal membership functions in many applications of fuzzy set theory the membership of an object is not defined directly one of its attributes like height age weight etc is first mapped to a real number often by means of a physical instrument and a parametric function then maps this real number to a membership degree in some fuzzy set like tall old heavy etc a very common parametric function is the trapezoidal one this paper presents some conditions guaranteeing the existence of such a trapezoidal membership function representing the knowledge of an expert further experimental research is needed for testing whether these conditions are satisfied by human agents

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

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

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

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

damping of beam vibrations by means of a thin constrained viscoelastic layer evaluation of a new theory abstract in this paper a new one dimensional theory of beams damped by means of a thin viscoelastic film is presented based upon a prescribed piecewise linear longitudinal displacement field and a shear stress field satisfying continuity conditions at the interfaces the beam formulation of the proposed theory is derived in order to demonstrate the accuracy of this beam model numerical tests were carried out by using the finite element method the present results are compared with well known analytical finite element with the msc nastran computer program and experimental solutions under various material temperature and boundary conditions it is found that the proposed theory is very simple and efficient in predicting the dynamic response of beams that are damped by means of a thin viscoelastic film

measurement theoretic justification of connectives in fuzzy set theory abstract the problem of representing intersection and union in fuzzy set theory is considered there are various proposals in the literature to model these concepts the possibility of using continuous triangular norms and conorms including min and max are taken up in a measurement theoretic setting the conditions are laid out to arrive at cardinal scales on which addition and multiplication are meaningful and critically discussed these conditions must either be accepted on normative grounds or must be empirically verified before the modeling process in order to see which operations are meaningful it is emphasized that the archimedean axiom and the existence of natural bounds are crucial in arriving at ratio and absolute scale representations

on choosing rationally when preferences are fuzzy abstract in real life exact choices are induced by fuzzy preferences these choices eventually satisfy certain plausible rationality properties several alternative rules for generating exact choices from fuzzy weak preference relations are introduced and the extent to which exact choice sets generated by these rules satisfy fairly weak rationality conditions is studied max min transitivity is shown to be crucial for obtaining a positive result

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

ordinal fuzzy sets fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions in general a linguistic term is described by the compatibility ordering induced in some universe of discourse uod a membership function in fuzzy set theory serves to reflect this ordering by assignment of values in 0 1 for objects in uod when we compute the meaning of a linguistic expression such as young and tall using fuzzy membership functions two implicit assumptions are made first we assume the membership values have quantitative meaning so that they can be quantitatively manipulated for example by adding or subtracting the extensive scale assumption second we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same the common scale assumption in many cases these assumptions cannot be justified some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions however the second issue has not been properly investigated in this paper we propose a framework that does not depend on both of these assumptions such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions

monte carlo study of 3d vesicles abstract a model of vesicles on a cubic lattice is studied by monte carlo methods vesicles have spherical topology and are made of lattice plaquettes an osmotic pressure difference acts between interior and exterior results are given for critical properties in the deflated regime

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

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

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

bending rigidity driven transition and crumpling point scaling of lattice vesicles the crumpling transition of three dimensional 3d lattice vesicles subject to a bending fugacity ensuremath rho exp ensuremath kappa mathit k mathit bt is investigated by monte carlo methods in a grand canonical framework by also exploiting conjectures suggested by previous rigorous results a critical regime with scaling behavior in the universality class of branched polymers is found to exist for ensuremath rho ensuremath gtrsim mathrm ensuremath rho mathit c for ensuremath rho mathrm ensuremath rho mathit c the vesicles undergo a first order transition that has remarkable similarities to the line of droplet singularities of inflated 2d vesicles at the crumpling point ensuremath rho mathrm ensuremath rho mathit c which has a tricritical character the average radius and the canonical partition function of vesicles with n plaquettes scale as mathit n ensuremath nu mathit c and mathit n mathrm ensuremath mathrm ensuremath theta mathit c respectively with ensuremath nu mathit c 0 4825 ifmmode pm else textpm fi 0 0015 and mathrm ensuremath theta mathit c 1 78 ifmmode pm else textpm fi 0 03 these exponents indicate a new class distinct from that of branched polymers for scaling at the crumpling point textcopyright 1996 the american physical society

mechanism of complicated volume deformation in polymers and its fractional time based description abstract in tension and compression of polymers some abnormal phenomena of volume changes were often observed and its mechanism is still unclear in this paper based on the thermodynamics principle it is shown that the combined effects of internal energy and entropy are the basic reason of causing the anomaly in polymers moreover to describe the complicated deformation through a simple model and have a correct modelization of mechanical behaviors a volume strain model is presented in which the variation of volume strain resulted from the combined effects is characterized through a fractional integral expression comparison between fitting results and experimental results reveals the proposed model can describe the volume deformation of polymers

soft sets an ordinal formulation of vagueness with some applications to the theory of choice abstract in the theory of fuzzy sets it is usual to represent the degree of belongingness by a number lying between 0 and 1 this paper develops an alternative framework which dispenses with such cardinal numerical representations of the degree of belongingness and relies instead on an ordinal formulation the ordinal notion of a vague set is called a soft set soft sets are shown to have a tight logical relation with the concept of l sets introduced earlier by goguen the paper also discusses applications of the framework of soft sets to several problems in the theory of choice

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

application of fractional operators to the analysis of damped vibrations of viscoelastic single mass systems free damped vibrations of an oscillator whose viscoelastic properties are described in terms of the fractional calculus kelvin voight model maxwell model and standard linear solid model are determined the problem is solved by the laplace transform method when passing from image to pre image one is led to find the roots of an algebraic equation with fractional exponents the method for solving such equations is proposed which allows one to investigate the roots behaviour in a wide range of single mass system parameters a comparison between the results obtained on the basis of the three models has been carried out it has been shown that for all models the characteristic equations do not possess real roots but have one pair of complex conjugates i e the test single mass systems subjected to the impulse excitation do not pass into an aperiodic regime in none of magnitudes of the relaxation and creep times main characteristics of vibratory motions of the single mass system as functions of the relaxation time or creep time which are equivalent to the temperature dependencies are constructed and analyzed for all three models

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

rectangular vesicles in three dimensions the phase diagram of a model of rectangular vesicles in three dimensions is examined the scaling of the generating function is determined in the area perimeter volume area and volume perimeter ensembles the results are interpreted within the framework of tricritical scaling and the crossover exponents associated with the transitions are determined we identify three phases in the phase diagram a needle phase a disc phase and a cubical phase these phases are separated by three curves of transitions that meet in a multicritical point