an axiomatic approach to fuzzy cardinalities of finite fuzzy sets an axiomatic definition of fuzzy cardinalities for finite fuzzy sets defined by means of a convex fuzzy set on the natural numbers is presented in such a way that it includes the fuzzy cardinalities defined by authors like zadeh ralescu dubois wygralak and characterizes the cardinalities that fulfill the additivity property by means of the extended addition of fuzzy numbers such cardinalities result defined by two functions one nondecreasing and the other nonincreasing in a similar way to the scalar cardinality fuzzy sets and systems 110 2000 175

questions of cardinality of finite fuzzy sets abstract in this paper we focus our attention on finite fuzzy sets a complete simple and easily applicable cardinality theory for them is presented questions of equipotency and non classically understood cardinal numbers of finite fuzzy sets are discussed in detail also problems of arithmetical operations addition subtraction multiplication division and exponentiation on as well as ordering relation for those cardinals are carefully investigated

a probabilistic definition of a nonconvex fuzzy cardinality the existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality in particular the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities in order to preserve some properties based on the addition of cardinalities such as the additivity property we have found that in order to solve many real world problems such as the induction of fuzzy rules in data mining convex cardinalities are not always appropriate in this paper we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support these cardinalities are not convex in general but they are most suitable for solving problems and contrary to the generalizing opinion they are found to be more intuitive for humans their suitability relies mainly on the fact that they assume dependency among objects with respect to the property to be in a fuzzy set the cardinality measures are generalized to relative ones among pairs of fuzzy sets we also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures finally a fuzzy ranking of the cardinality of fuzzy sets is proposed and a definition of graded equipotency is introduced

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

nonconvex fuzzy mappings and the fuzzy pre variational inequality in this paper we introduce the concepts of fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping from r n to the set of fuzzy numbers and put forward the concepts of fuzzy extended directional differential subgradient and subdifferential some properties are obtained for fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping relations are discussed between the fuzzy pre variational inequality and the fuzzy vector optimization problem

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

topology and shape optimization for elastoplastic structural response abstract it is a common practice to base both material topology optimization as well as a subsequent shape optimization on linear elastic response however in order to obtain a realistic design it might be essential to base the optimization on a more realistic physical behavior i e to consider geometrically or and materially nonlinear effects in the present paper an elastoplastic von mises material model with linear isotropic hardening softening for small strains is used the objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed with respect to the specific features of either topology or shape optimization for example the number of optimization variables or their local global influence on the structural response different methods are applied for topology optimization problems the gradient of the ductility is determined by the variational adjoint approach in shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach the topology optimization problem is solved by an optimality criteria oc method and the shape optimization problem by a mathematical programming mp method in topology optimization a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities the procedures are verified by 2d design problems under plane stress conditions

generalized cardinal numbers and operations on them abstract in this paper we present a cardinality theory for so called vaguely defined objects which are mild generalizations of fuzzy sets obtained by introducing lower and upper approximations of the membership functions the theory makes use of the sentential calculus in the infinite valued ukasiewicz logic and can be applied to fuzzy sets with arbitrary supports some kinds of twofold fuzzy sets partial sets etc the notion of equipotency of vaguely defined objects is studied in detail the resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects in the second part of the paper basic operations on the generalized cardinals are defined and carefully investigated similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated

minimum weight design of truss structures with geometric nonlinear behavior the paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior the nonlinear critical load is determined by finding the load level at which the hessian of the potential energy ceases to be positive definite a recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables number of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements element stresses system stability etc 1 2 the optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables the optimality criterion was derived by differentiating the lagrangian with respect to the design variables the displacements were assumed to be small and in the finite element analysis linear equilibrium equations were solved to determine the response of the structure to the ap plied loads in the case of system stability the constraints were defined see refs 3 6 by the associated linear eigenvalue problem this definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections the correct procedure for these structures would be to analyze the structure by using nonlinear equilibrium equations this will be particularly true for large space structures lss in this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior in the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode this design tends to become im perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub stantially the imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal see ref 7 a real structure has geometric im perfections and these structures tend to have nonlinear

an axiomatic approach to scalar cardinalities of fuzzy sets abstract we present an axiomatic approach to scalar cardinalities of fuzzy sets which is based on a system of three simple postulates a characterization theorem for those cardinalities is given the infinite family of possible scalar cardinalities of a fuzzy set generated by the postulates contains as particular cases all standard concepts of scalar cardinality e g the sigma count of a fuzzy set the cardinality of its core or support and the cardinality of its t level set

a graded approach to cardinal theory of finite fuzzy sets part i in this article we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent which indicates that they have the same number of elements the concepts of fuzzy sets and fuzzy classes in the class of all sets in zfc are introduced and several standard relations and constructions such as the fuzzy power set and exponentiation are defined a functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown and a relation to generalized cardinals and wygralak s cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

on triangular norm based generalized cardinals and singular fuzzy sets there are three basic types of triangular norm based generalized cardinals of fuzzy sets namely generalized fgcounts flcounts and fecounts all of them are convex fuzzy sets of usual cardinal numbers our attention will be focused on generalized fecounts if nonstrict archimedean triangular norms are involved generalized fecounts of many fuzzy sets become the zero function those singular fuzzy sets being totally dissimilar to any set of any cardinality and their properties are the subject of this paper a relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed

cardinality quantifiers and the aggregation of fuzzy criteria abstract in this paper we are interested in counting the number of elements of a fuzzy set we study two concepts of fuzzy cardinality one which gives the answer as a fuzzy set or fuzzy number and the other which gives an ordinary integer relationships between these concepts and others which have been studied in the literature e g sigma count are also investigated some properties of the new concepts are studied such as fuzzy convexity monotonicity and additivity the concepts of fuzzy cardinality are important for a better understanding of fuzzy quantified rules of the form qx s are a or qa s are b where q is a fuzzy quantifier and a b are fuzzy sets we study such rules as well as aggregation rules of the form q of a1 ap contain x

an integrated approach to the synthesis of geometrically non linear structures an integrated approach to the minimum weight design of geometrically non linear three dimensional truss structures with geometric imperfections subject to inequality constraints on static displacements stresses local buckling and cross sectional areas is investigated the integrated structural synthesis problem involves design and response quantities as independent variables and equilibrium equations describing the finite element model as equality constraints the non linear structural analysis and the optimization are thus merged together into a single process a computer program developed to compute the contraint values and analytical gradients is coupled with a generalized reduced gradient algorithm to solve the integrated problem numerical results for a geometrically non linear shallow dome example problem are presented for various types of imperfections furthermore it is found that the algorithm is capable of detecting and guarding against system as well as element elastic instability using equilibrium information only that is without imposing system and local buckling inequality constraints

fuzzy sets with triangular norms and their cardinality theory the most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality the resulting convex fuzzy sets of usual cardinals of nonnegative integers in the finite case are then called generalized cardinal numbers three types of them are of special interest and importance namely fgcounts flcounts and fecounts in this paper first we show that their original forms are suitable only for fuzzy sets with the classical min and max operations second we propose an appropriate generalization to fuzzy sets with triangular norms and conorms further we investigate the resulting generalized fgcounts flcounts and fecounts from the viewpoint of the corresponding equipotency relations inequalities and arithmetical operations

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

a graded approach to cardinal theory of finite fuzzy sets part ii fuzzy cardinality measures and their relationship to graded equipollence abstract in this article we propose an axiomatic system for fuzzy cardinality measures referred to as fuzzy c measures for short assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains the system generalizes an axiomatic system introduced by j casasnovas and j torrens 2003 we show that each fuzzy c measure is determined by two appropriate homomorphisms between the reducts of residuated dually residuated rdr lattices for linearly ordered rdr lattices we prove that each fuzzy c measure is a product of a non decreasing and a non increasing fuzzy c measure which indicates that there is a close relation between fuzzy c measures and fgcount flcount and fecount provided by l a zadeh 1983 and generalized by m wygralak 2001 finally the relationship of fuzzy c measures to graded equipollence introduced in the first part of this contribution is analyzed