a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

fredholm differential operators with unbounded coefficients abstract we prove that a first order linear differential operator g with unbounded operator coefficients is fredholm on spaces of functions on r with values in a reflexive banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both r and r and a pair of the ranges of the dichotomy projections is fredholm and that the fredholm index of g is equal to the fredholm index of the pair the operator g is the generator of the evolution semigroup associated with the evolution family in the case when the evolution family is the propagator of a well posed differential equation u t a t u t with generally unbounded operators a t t r the operator g is a closure of the operator d dt a t thus this paper provides a complete infinite dimensional generalization of well known finite dimensional results by palmer and by ben artzi and gohberg

persistence and imperfection of nonautonomous bifurcation patterns abstract for nonautonomous dynamical systems a bifurcation can be understood as topological change in the set of bounded entire solutions to a given time dependent evolutionary equation following this idea a fredholm theory via exponential dichotomies on semiaxes enables us to employ tools from analytical branching theory yielding nonautonomous versions of fold transcritical and pitchfork patterns this approach imposes the serious hypothesis that precise quantitative information on the dichotomies is required an assumption hard to satisfy in applications thus imperfect bifurcations become important in this paper we discuss persistence and changes in the previously mentioned bifurcation scenarios by including an additional perturbation parameter while the unperturbed case captures the above bifurcation patterns we obtain their unfolding and therefore the local branching picture in a whole neighborhood of the system using an operator formulation of parabolic differential caratheodory differential and difference equations this will be achieved on the basis of recent abstract analytical techniques due to shi 1999 and liu shi and wang 2007

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

benchmarking in linguistic importance weighted aggregations in this work we concentrate on the issue of weighted aggregations and provide a possibilistic approach to the process of importance weighted transformation when both the importances interpreted as benchmarks and the ratings are given by symmetric triangular fuzzy numbers we will show that using the possibilistic approach i small changes in the membership function of the importances can cause only small variations in the weighted aggregate ii the weighted aggregate of fuzzy ratings remains stable under small changes in the nonfuzzy importances iii the weighted aggregate of crisp ratings still remains stable under small changes in the crisp importances whenever we use a continuous implication operator for the importance weighted transformation we illustrate the proposed method by some simple examples

nonautonomous bifurcation patterns for one dimensional differential equations abstract although bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades the notion of a nonautonomous bifurcation is not yet established in this article two different approaches are discussed which are based on special notions of attractivity and repulsivity generalizations of the well known one dimensional transcritical and pitchfork bifurcation are obtained

the numerical computation of homoclinic orbits for maps transversal homoclinic orbits of maps are known to generate shift dynamics on a set with cantor like structure in this paper a numerical method is developed for computation of the corresponding homoclinic orbits they are approximated by finite orbit segments subject to asymptotic boundary conditions we provide a detailed error analysis including a shadowing type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment this approach is applied to several examples in some of them parameters appear and closed loops of homoclinic orbits are found by a path following algorithm

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

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

transverse homoclinic orbits in periodically perturbed parabolic equations on considere le systeme autonome x ax g x ou a est un operateur sectoriel sur un espace de banach x on suppose que l equation differentielle a une solution bornee t telle que son equation variationnelle a une dichotomie exponentielle sur des demi droites on utilise la methode de lyapunov schmidt pour obtenir la fonction de bifurcation pour des solutions bornees de l equation perturbee non autonome

the fredholm alternative and exponential dichotomies for parabolic equations abstract it is proved in this paper for parabolic equations that the fredholm alternative holds for bounded solutions if the homogeneous equation has exponential dichotomies on r and on r and moreover that the converse is also valid under the condition of bounded growth

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

an approach to computing with words based on canonical characteristic values of linguistic labels herrera and martinez initiated a 2 tuple fuzzy linguistic representation model for computing with words cw which offers a computationally feasible method for aggregating linguistic information that are represented by linguistic variables with equidistant labels through counting indexes of the corresponding linguistic labels lawry introduced an alternative approach to cw based on mass assignment theory that takes into account the underlying definitions of the words recently we provided a new proportional 2 tuple fuzzy linguistic representation model for cw that is an extension of herrera and martinez s model and takes into account the underlying definitions of linguistic labels of linguistic variables in the process of aggregating linguistic information by assigning canonical characteristic values ccvs of the corresponding linguistic labels in this paper we study further into cw based on ccvs of linguistic labels to provide a unifying link between lawry s framework and herrera and martinez s 2 tuple framework as well as allowing for computationally feasible cw our approach is based on a formal definition of ccv functions of a linguistic variable which is introduced under the context of its proportional 2 tuple linguistic representation model as continuation of our earlier works and on a group voting model that is for the probabilistic interpretation of the whole semantics of the ordered linguistic terms set of an arbitrary linguistic variable after the general framework developed in the former part of this paper we focus on a particular linguistic variable probability we show that for a linguistic probability description the expectation of its posterior conditional probability is a canonical characteristic value of the linguistic probability description then a calculus for reasoning with linguistic syllogisms and inference from linguistic information is introduced it is investigated under the context that linguistic quantifiers in such linguistic syllogisms can be arbitrary linguistic probability description and that the related linguistic information can be linguistic facts overfacts or underfacts intrinsically our approach to this calculus is with computing with words based on canonical characteristic values of linguistic labels

bifurcation from a degenerate simple eigenvalue it is proved that a symmetry breaking bifurcation occurs at a simple eigenvalue despite the usual transversality condition fails and this bifurcation from a degenerate simple eigenvalue result complements the classical one with the transversality condition the new result is applied to an imperfect pitchfork bifurcation in which a forward transcritical bifurcation changes to a backward one when the perturbation parameter changes several applications in ecological and genetics models are shown

bifurcations in non autonomous scalar equations abstract in a previous paper we introduced various definitions of stability and instability for non autonomous differential equations and applied these to investigate the bifurcations in some simple models in this paper we present a more systematic theory of local bifurcations in scalar non autonomous equations

on gravity torsion and the spectral action principle abstract we consider compact riemannian spin manifolds without boundary equipped with orthogonal connections we investigate the induced dirac operators and the associated commutative spectral triples in case of dimension four and totally anti symmetric torsion we compute the chamseddine connes spectral action deduce the equations of motions and discuss critical points

imperfect transcritical and pitchfork bifurcations imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on banach spaces in particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined and two parameter unfoldings of singularities are rigorously established applications include semilinear elliptic equations imperfect euler buckling beam problem and perturbed diffusive logistic equation

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

fuzzy linguistic pert a model for program evaluation and review technique pert under fuzzy linguistic contexts is introduced in this fuzzy linguistic pert network model each activity duration is represented by a fuzzy linguistic description aggregation and comparison of the estimated linguistic expectations of activity durations are manipulated by the techniques of computing with words cw to provide suitable contexts for this purpose we first introduce several variations of basic linguistic labels of a linguistic variable such as weighted linguistic labels generalized linguistic labels and weighted generalized linguistic labels etc and then based on the notion of canonical characteristic value ccv function of a linguistic variable we develop some related cw techniques for aggregation and comparison of these linguistic labels afterward using a computing technique of linguistic probability introduced by zadeh and based on the new developed cw techniques for weighted generalized linguistic labels we investigate the associated linguistic expectation pert network of a fuzzy linguistic pert network also throughout the paper several examples are used to illustrate related notions and applications