time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

extended double complex linear systems and new multiple infinite dimensional hidden symmetries for the general symplectic gravity models by using a so called extended double ed complex method the previously found doubleness symmetry for each member of the class of general symplectic gravity models is further exploited and extended a 2 n 1 2 n 1 matrix double complex h potential is constructed for any non negative integer n and the motion equations in two dimensions are written in a double complex formulation a double duality mapping is proposed and two pairs of ed complex hauser ernst type linear systems j math phys 21 1126 1980 are established based on these linear systems explicit formulations of new multiple hidden symmetry transformations for the studied theories are given for any fixed n these symmetry transformations are verified to constitute multiple infinite dimensional lie algebras each of which is a semidirect product of the kac moody sp 2 n 1 r and virasoro algebras without center charges these results demonstrate that the ed complex method is necessary and more effective and the general symplectic g

on the convergence and representation of random fuzzy number integrals in this paper a new kind of convergence w s convergence is introduced for certain type of fuzzy numbers taking values in separable reflexive banach spaces the convergence theorem of random fuzzy number integrals in the w s sense is given and a condition under which a fuzzy number function can be represented by random fuzzy number integrals is obtained

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

the s differentiability of a fuzzy valued mapping several approaches to define the differential of fuzzy valued mappings can be found in the literature in this paper we introduce a new concept of differential of a fuzzy valued mapping which is based on the notion of differential of the support function associated with such a mapping properties of the new concept as well as the relation between it and previous approaches are also analyzed in detail

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

two point boundary value problems of undamped uncertain dynamical systems in this paper we study two point boundary value problems of a class of uncertain dynamical systems i e undamped uncertain dynamical systems we introduce the concept of large solutions and small solutions for two point boundary value problems of undamped uncertain dynamical systems and under some conditions establish the existence and uniqueness of large solutions and small solutions and settle the relationship between large solutions and small solutions

a stacking theorem approach for fuzzy differential equations abstract this paper presents new existence results for fuzzy differential equations our analysis combines the stacking theorem with new solution set results for ordinary differential equations

a simplified derivation of the geroch group in two dimensional reduced gravity by generalizing our previous treatment of hidden symmetry in two dimensional chiral models a simple and explicit approach is proposed to the geroch group for the vacuum einstein equations with the metric tensor depending only on two variables two dimensional reduced gravity an infinite number of infinitesimal transformations for the metric tensor preserving the equations of motion are summarized by an explicit parametric transformation and the commutators among them are calculated in a simple and straightforward way these transformations and their group structure are further identified to be those of the well known geroch group

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

on set valued stochastic integrals and fuzzy stochastic equations in this paper we extend the notion of set valued and fuzzy stochastic integrals to semimartingale integrators we present their main properties and finally we establish the existence of solutions of a fuzzy integral stochastic equation driven by a brownian motion the approach is based on the existence of solutions of an appropriately formulated martingale problem for a system of stochastic inclusions and the theorem of negoita and ralescu

improved predictor corrector method for solving fuzzy initial value problems in this paper an improved predictor corrector ipc method is proposed to solve the fuzzy initial value problem the ipc method is generated by combining an explicit three step method and an implicit two step method the methods are compared with the methods discussed in t allahviranloo n ahmady e ahmady numerical solution of fuzzy differential equations by predictor corrector method information sciences 177 7 2007 1633 1647 and are shown to be more accurate the convergence and stability of the proposed methods are also presented in detail these methods are illustrated by solving some examples

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

numerical solution of fuzzy differential equations by predictor corrector method in this paper three numerical methods to solve the fuzzy ordinary differential equations are discussed these methods are adams bashforth adams moulton and predictor corrector predictor corrector is obtained by combining adams bashforth and adams moulton methods convergence and stability of the proposed methods are also proved in detail in addition these methods are illustrated by solving two fuzzy cauchy problems

some topological properties of solutions to fuzzy differential systems in this paper the initial value problems of fuzzy differential systems are investigated by using a kind of new methods at first the concept of relatively open sets is introduced then some basic topological properties of solutions for the initial value problems of fuzzy differential systems are obtained

a continuous time dynamic fuzzy system ii fuzzy potentials abstract this paper is a sequel of yoshida 9 fuzzy potentials are introduced in a continuous time dynamic fuzzy system and a fuzzy relational differential equation is given to characterize the fuzzy potentials

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

global existence of solutions to fuzzy differential equations abstract the local existence and uniqueness theorems are investigated by wu et al j math anal appl 202 1996 629 644 for the cauchy problem of the fuzzy valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper we point out a variety of results which assure global existence of solutions to fuzzy differential equations

caratheodory solutions of fuzzy differential equations in this paper the existence of solutions for fuzzy differential equations with right hand side satisfying a caratheodory condition on a class of fuzzy sets is established

extended double riemann hilbert problem and multiple representations of the infinite dimensional symmetric group for the stationary axisymmetric vacuum field equations the double complex function method suggested in previous papers is extended and the extended method is used to establish two mutually dual extended double hauser ernst equations and related riemann hilbert problems then quadruple representations of the semidirect product of the kac moody and virasoro groups are given these show that for the stationary axisymmetric vacuum space times there are more and richer symmetry structures than previously expected the multiple forms of some well known transformations can be obtained as some special cases

the structure stability of periodic solutions for first order uncertain dynamical systems abstract this paper studies the structural stability of periodic solutions for first order fuzzy differential equations fdes understood as differential inclusions i e first order uncertain dynamical systems the existence and uniqueness of periodic solutions for this first order fuzzy problems have been obtained on general fuzzy number space when the forcing function has specific perturbations the structural stability of the periodic solutions are discussed by using the support function the dini theorem and the convergence theorem in the differential inclusion theory

two point boundary value problems of uncertain dynamical systems abstract recent research shows that in many cases there are no solutions periodic solutions etc to fuzzy two point boundary value problems in the sense of h derivatives to overcome the disadvantage of h derivatives in studying fuzzy two point boundary value problems in this paper we study fuzzy two point boundary value problems in the sense of differential inclusions i e two point boundary value problems of uncertain dynamical systems we obtain the existence and uniqueness of solutions and the existence of periodic solutions to two point boundary value problems of uncertain dynamical systems after introducing the concept of big solutions to two point boundary value problems of undamped uncertain dynamical systems we establish the existence and uniqueness of big solutions discuss the relationship between solutions and big solutions and further get a conclusion concerning the solutions of two point boundary value problems of uncertain dynamical systems

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

fuzzy differential equations for universal oscillators abstract this study considers universal oscillator fuzzy differential equations the differential inclusion method is applied to this problem by considering the universal oscillator fuzzy differential equations as universal oscillator uncertain dynamical systems three types of damped oscillators are considered comprising underdamped critically damped and overdamped oscillators the existence and uniqueness of the solutions to each case are discussed and proofs are given under some certain conditions for universal oscillator uncertain dynamical systems with an undamped forcing function the existence and uniqueness of the solutions and big solutions are also verified while the scopes of the trajectories to the solutions are obtained based on the inclusion relationship between solutions and big solutions we also prove the existence and uniqueness of solutions to universal oscillator uncertain dynamical systems with a damped forcing function by using other more direct conditions

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

stability in possibilistic linear programming with continuous fuzzy number parameters abstract we prove that possibilistic linear programming problems introduced by buckley in 2 with continuous fuzzy number parameters are well posed i e small changes of the membership function of the parameters may cause only a small deviation in the possibilistic distribution of the objective function

periodic problems of first order uncertain dynamical systems it is well known that there are no periodic solutions to fuzzy differential equations in the sense of hukuhara derivatives i e h derivatives so fuzzy differential equations in the sense of h derivatives cannot be used to describe periodic phenomena in the real world to overcome this disadvantage this paper studies periodic problems of first order fuzzy differential equations in the sense of differential inclusions i e periodic problems of first order uncertain dynamical systems and obtains the existence of periodic solutions for first order uncertain dynamical systems

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

regular double riemann hilbert problems and double kac moody algebraic structures for two dimensional reduced gravity by using the double complex function method two regular double riemann hilbert problems are established therefore two double kac moody algebras are given these structures show that in two dimensional reduced gravity in fact there is more exquisite hidden symmetry than common nonlinear systems

the riemann hilbert problem approach to a representation of the virasoro group ii the development of a riemann hilbert rh problem approach to the representation of the virasoro group introduced in our earlier paper w li and b y hou j math phys 30 1198 1989 is continued the rouche theorem is employed to prove that there exists a set of scalar functions in the complex plane in terms of which the structure of the virasoro group can be decided the general form of the rh problem will also be presented and the uniqueness of its solution will be proven for future use a standard form of a matrix fredholm equation equivalent to our rh problem will be deduced this may help to establish in a future paper the existence of a solution of the rh problem

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

membership induced on manifolds by vector fields and flows abstract in many engineering and scientific applications the need arises to perform differential calculations on manifolds if such calculations involve imprecise data methods for computing with such data must be determined definitions and results are given for memberships on manifolds induced by differentiation along vector fields and by flows of vector fields with uncertain parameters examples are presented which illustrate the concepts

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

impulsive functional differential inclusions and fuzzy population models in this paper we shall establish some existence results for the impulsive functional differential inclusion and the fuzzy impulsive functional differential equation with some conditions and study the properties of the solution set and the attainable set finally the results will be used to fuzzy population models

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

computationally efficient methods for analysis and synthesis of real signals using fft and ifft the discrete fourier transform dft and the inverse discrete fourier transform idft are used in a wide variety of signal processing applications even with the increased speed of modern processors there is an ongoing need to further develop more efficient methods for computing dft and idft with a particular effort to reduce the number of complex multiplications the properties of certain complex sequences are extraordinarily useful in the sense that they lead to data manipulation schemes that result in the sequences to which traditional but much shorter fast fourier transform fft algorithms may be applied this is achieved by exploiting a certain regularity in the complex data the index reversed complex conjugate sequence and the mirror symmetric complex conjugate sequence were defined a significant reduction in the number of complex computations is achieved if a sequence in either domain exhibits such symmetry

initial value problem for fuzzy differential equations under dissipative conditions a new concept of inner product on the fuzzy space e n d is introduced studied and used to prove several theorems stating the existence uniqueness and boundedness of solutions of fuzzy differential equations a stability result is also proved in the same context

a nonlinear time frequency analysis method an alternative method of time frequency signal analysis is introduced and is compared with the conventional discrete fourier transfer dft based methods the structure of the proposed algorithm and its mathematical properties are presented the proposed algorithm has a nonlinear structure that provides a frequency adaptivity feature it can be implemented both in analog and in digital form and is particularly suitable for real time applications where computational efficiency is important a comparison is made with the dft in terms of algorithm efficiency sensitivity and complexity it is shown that compared with the dft the algorithm is more efficient for real time applications and is less sensitive to noise and variations in the frequency and sampling rate while maintaining simplicity of the structure and computational efficiency slower convergence rate of the algorithm compared with the dft constitutes its main shortcoming

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

comparation between some approaches to solve fuzzy differential equations in this paper we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via zadeh s extension principle we get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented we also give some properties and we show the relationships with others interpretation finally we propose a numerical procedure for obtaining the fuzzy solution

existence and interrelation between set and fuzzy differential equations abstract an existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general then the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it is studied

remarks on peano like theorems for fuzzy differential equations the validity of the classical peano theorem is noted for differential equations on the metric space en dp of normal fuzzy convex sets in rn where dp is the p th mean of the hausdorff distances between corresponding level sets this contrasts with differential equations on the space en d where d is the max metric which is not locally compact so more than just continuity is required to ensure the existence of solutions some additional properties differing from that given by kaleva are mentioned

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

existence and uniqueness of solutions to cauchy problem of fuzzy differential equations we studied the cauchy problem of fuzzy dierential equations on the basis of introducing the concept of dierentiation which is a generalization of the denition of h dierentiability due to puri and ralescu 1983 for fuzzy set valued mappings of real variables whose values are in the fuzzy number space e n d the existence and uniqueness theorem is obtained for the cauchy problem x 0 f t x x t 0 x0 when f satises the generalized lipschitz condition c 2000

new infinite dimensional quadruple symmetry groups for the general symplectic gravity model the symmetry structures of the general symplectic gravity model are further studied by using the so called extended double ed complex function method the usual riemann hilbert rh problem is extended to an ed complex formulation for any fixed non negative integer n two pairs of ed rh transformations are constructed and they are verified to give infinite dimensional quadruple symmetry groups of the general symplectic gravity model each of these symmetry groups has the structure of semidirect product of kac moody group sp 2 n 1 r and virasoro group moreover the infinitesimal forms of these rh transformations are calculated out and they are found to give exactly the same results as previous these demonstrate that the two pairs of ed rh transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper the finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generatio