frequency domain methods for the solutions of n order fuzzy differential equations abstract the solutions of the fuzzy differential equations in refs 8 10 12 14 16 have so far been obtained by integration or fuzzy integration in the time domain it is often more convenient particularly in the fuzzy random vibration problems 12 and the fuzzy stochastic dynamic systems 14 to obtain the solutions by integration or fuzzy integration in the frequency domain this is accomplished by the use to generalized fuzzy harmonic analysis 12 14 in the second paper of a series of reports on fuzzy differenrial equations we continue studying the n th order fuzzy differential equation x n t a n 1 t x n 1 t a 0 t x t f t where x n t x n 1 t x 1 t are n th n 1 th 1st same oorder or reverse order derived functions of an unknown fuzzy set valued function x t respectively x t is known fuzzy set valued function a i t i 0 1 n 1 are deterministic functions of parameter t the solving processes of frequency domain for n th order fuzzy differential equations are put forward one example is considered in order to demonstrate the rationality and validity of the methods the work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations 8 fuzzy dynamical systems 2 fuzzy random vibration 12 fuzzy stochastic dynamical systems 14 18 20 and fuzzy stochastic systems 21 23

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

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

representation of uncertain multichannel digital signal spaces and study of pattern recognition based on metrics and difference values on fuzzy n cell number spaces in this paper we discuss the problem of characterization for uncertain multichannel digital signal spaces propose using fuzzy n cell number space to represent uncertain n channel digital signal space and put forward a method of constructing such fuzzy n cell numbers we introduce two new metrics and concepts of certain types of difference values on fuzzy n cell number space and study their properties further based on the metrics or difference values appropriately defined we put forward an algorithmic version of pattern recognition in an imprecise or uncertain environment and we also give practical examples to show the application and rationality of the proposed techniques

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

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

iterated fuzzy set systems a new approach to the inverse problem for fractals and other sets abstract images with grey or colour levels admit a natural representation in terms of fuzzy sets but without the usual probabilistic interpretation of the latter we introduce a fuzzy set approach which incorporates in part the technique of iterated function systems ifs for the construction analysis and or approximation of typically fractal sets and images the method represents a significant departure from ifs especially in the interpretation of the resulting image the introduction of grey level maps i 0 1 0 1 associated with the contractive maps w i of the ifs affords much greater flexibility in the generation of images as well as in the inverse problem

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

lp boundedness of the wave operator for the one dimensional schrodinger operator given a one dimensional perturbed schrodinger operator h d2 dx2 v x we consider the associated wave operators w defined as the strong l2 limits lim s to pm infty e ish e ish 0 we prove that w are bounded operators on lp for all 1 p provided 1 x 2 v x in l 1 or else 1 x v x in l 1 and 0 is not a resonance for p we obtain an estimate in terms of the hilbert transform some applications to dispersive estimates for equations with variable rough coefficients are given

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

monotone method for fuzzy differential equations we develop the monotone iterative technique to approximate the extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval to our procedure it is crucial to have at our disposal fuzzy comparison results and several properties of sets of fuzzy functions and fuzzy numbers including a compactness result in spaces of fuzzy functions which are useful from the point of view of applications

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

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

first order linear fuzzy differential equations on the space of linearly correlated fuzzy numbers abstract the present paper is concerned with the solutions of first order linear fuzzy differential equations where differentiability of fuzzy functions is defined under an assumption of linear correlation between fuzzy numbers motivated by the fact that the structure of the space of linearly correlated fuzzy numbers strongly depends on the symmetry of the basic fuzzy number here we address first order linear fuzzy differential equations by distinguishing whether the basic fuzzy number is symmetric or not in the non symmetric case a first order linear fuzzy differential equation may be transformed into an equivalent system of ordinary differential equations related to the representation functions of the linearly correlated fuzzy number valued function in the symmetric case according to the monotonicity of the diameter of the fuzzy solution a first order linear fuzzy differential equation may be transformed into a system of ordinary differential equations associated with the representation functions of the canonical form of the linearly correlated fuzzy number valued function in addition using our extension method one may obtain solutions with either increasing or decreasing diameters several examples are provided in order to illustrate the proposed method

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

on the restudy of fuzzy complex analysis part 1 the sequence and series of fuzzy complex numbers and their convergences this paper studies the convergence of the sequence and series of fuzzy complex numbers we define the metric convergence and levelwise convergence of sequences on the metric space of fuzzy complex numbers and discuss the relationship between two convergences all this may be a foundation for researching fuzzy complex analysis

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

a note on fuzzy differential equations and the extension principle in this note we show by an example that the main result in a paper by bassanezi et al m t mizukoshi l c barros y chalco cano h roman flores r c bassanezi fuzzy differential equations and the extension principle information sciences 177 2007 3627 3635 is incorrect

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

on random fuzzy functional differential equations in this paper we prove the existence and uniqueness result for random fuzzy functional differential equations for the existence and uniqueness we use the method of successive approximations some kind of boundedness of the solution is established

on random fuzzy differential equations we consider a cauchy problem in a random fuzzy setting under the condition of lipschitzean right hand side the existence and uniqueness of the solution is proven also the continuous dependence on the right hand side and initial condition is shown some kind of boundedness of the solution is established

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

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

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of 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 studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

an analytical method for solving linear fredholm fuzzy integral equations of the second kind in this paper we use the parametric form of a fuzzy number and convert a linear fuzzy fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case for fuzzy fredholm integral equations with kernels the sign of which is difficult to determine a new parametric form of the fuzzy fredholm integral equation is introduced we use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solutions of the linear fuzzy fredholm integral equation of the second kind the proposed method is illustrated by solving some examples using the ham it is possible to find the exact solution or the approximate solution of the problem in the form of a series

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

inertial range and the finite reynolds number effect of turbulence the kolmogorov law which is the unique exact relationship of inertial range statistics is applied to investigate the finite reynolds number effect in particular to study how the width of the inertial range of finite reynolds number turbulence changes with the taylor microscale reynolds number mathrm r ensuremath lambda it is found that there is no inertial range when mathrm r ensuremath lambda ensuremath leqslant 2000 and within tolerance of 1 error mathrm r ensuremath lambda should be higher than 10 4 in order to have an inertial range wider than one decade the so called inertial range found in experiments and simulations is just a scaling range and is not the same as kolmogorov s inertial range the finite reynolds number effect cannot be neglected within such a scaling range and should be considered in comparing experiments or simulations with theories of the inertial range statistics

on the theory of statistical and isotropic turbulence the statistical theory of turbulence initiated by taylor 1935 and v karman howarth 1938 has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given in fact kolmogoroff 1941 a b and independently onsager 1945 and v weizsaecker 1948 have introduced a similarity hypothesis which allows a determination of the spectrum for eddies with large reynolds numbers and the author heisenberg 1948 has extended these calculations to include those frequency components which have small reynolds numbers since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem one may say that the statistical part of the spectrum is now well understood recently batchelor townsend 1947 1948 a b have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel and the following discussions will apply the statistical theory to this problem

revisiting fuzzy differential equations abstract we revisit the original formulation of fuzzy differential equations fdes and provide examples to show that the theory of fdes is richer than the corresponding theory of ordinary differential equations odes and therefore need not be rejected the alternative formulation in terms of differential inclusions does not always have advantage over odes as claimed we also discuss a stability result via lyapunov functions in this setup

fixed point theorems in fuzzy real line abstract some sufficient conditions for the existence of fixed points of increasing operators in fuzzy real line r l are given we also establish some coupled quasi fixed point theorems of mixed monotone operators in fuzzy real line r l

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

structure of wave operators for a scaling critical class of potentials we prove a structure formula for the wave operators in bbb r 3 w pm mathop hbox s lim limits t to pm infty e it delta v p c e it delta and their adjoints for a scaling invariant class of scalar potentials v in b b left v mid sum k in bbb z 2 k 2 big chi x in 2 k 2 k 1 x v x big l 2 infty right under the assumption that zero is neither an eigenvalue nor a resonance for delta v the formula implies the boundedness of wave operators on l p spaces 1 leq p leq infty on weighted l p spaces and on sobolev spaces as well as multilinear estimates for e ith p c when v decreases rapidly at infinity we obtain an asymptotic expansion of the wave operators the first term of the expansion is of order langle y rangle 4 commutes with the laplacian and exists when v in langle x rangle 3 2 epsilon l 2 we also prove that the scattering operator s w w is an integrable combination of isometries the proof is based on an abstract version of wiener s theorem applied in a new function space

contributions to fuzzy analysis in some sense fuzzy analysis deviates highly from classical analysis needing its own specific notions the main reason which causes this situation consists in the different algebraic and structure theoretic properties of the usual real numbers and fuzzy numbers the paper which is an extended version of 4 is devoted to this problem presenting some of these specific notions in particular examples of proper fuzzy normed vector spaces and proper fuzzy normed algebras are given

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

fuzzy approximation by fuzzy convolution type operators here we introduce and study four sequences of naturally arising fuzzy integral operatorsof convolution type that are integral analogs of known fuzzy wavelet type operators defined via a scaling function their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity also their high order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the n t h order n 1 h fuzzy derivative of the engaged fuzzy number valued function the fuzzy global smoothness preservation property of these operators is demonstrated also

instability of bound states of a nonlinear schrodinger equation with a dirac potential abstract we study analytically and numerically the stability of the standing waves for a nonlinear schrodinger equation with a point defect and a power type nonlinearity a major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves this is overcome by a perturbation method and continuation arguments among others in the case of a repulsive defect we show that the standing wave solution is stable in h rad 1 r and unstable in h 1 r under subcritical nonlinearity further we investigate the nature of instability under critical or supercritical nonlinear interaction we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints in the subcritical radial case unstable bound states cannot collapse but rather narrow down until they reach the stable regime a finite width instability in the nonradial repulsive case all bound states are unstable and the instability is manifested by a lateral drift away from the defect sometimes in combination with a finite width instability or a blowup instability

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

wave operator bounds for one dimensional schrodinger operators with singular potentials and applications boundedness of wave operators for schrodinger operators in one space dimension for a class of singular potentials admitting finitely many dirac delta distributions is proved applications are presented to for example dispersive estimates and commutator bounds

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

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

a note on numerical solutions of fuzzy differential equations by extended runge kutta like formulae of order 4 in this note we show that the example presented in a recent paper by ghazanfari et al is incorrect namely the exact solution suggested by the authors is not solution of the given fuzzy differential equation fde indeed the authors have proposed an exact solution which is independent from the initial condition so we obtain the correct exact solution using the characterization theorem proposed by bede et al under seikkala differentiability also some details are given for the mentioned example

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

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

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

set valued and fuzzy stochastic differential equations driven by semimartingales abstract in the paper we present set valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties then we study the existence of solutions to set valued and fuzzy set valued stochastic differential equations driven by semimartingales the stability of solutions is also established

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

kolmogorov constants for the second order structure function and the energy spectrum we examine the behavior of the kolmogorov constants c 2 c k and c k1 which are respectively the prefactors of the second order longitudinal structure function and the three dimensional and one dimensional longitudinal energy spectrum in the inertial range we show that their ratios c 2 c k1 and c k c k1 exhibit clear dependence on the microscale reynolds number r implying that they cannot all be independent of r in particular it is found that c k1 c 2 0 25 1 95r 0 68 the study further reveals that the widely used relation c 2 4 02c k1 holds only asymptotically when r 10 5 it is also found that c 2 has much stronger r dependence than either c k or c k1 if the latter indeed has a systematic dependence on r we further show that the varying dependence on r of these three numbers can be attributed to the difference of the inertial range in real and wave number space with the inertial range in real space known to be much shorter than that in wave number space

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

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

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