applications of fuzzy chaos to fuzzy simulation in this paper we wish to use fuzzy chaos to generate fuzzy simulation we first discuss how fuzzy chaos can produce a chaotic sequence of fuzzy numbers we apply this technique to fuzzily simulate a solution to a simple fuzzy optimization problem other applications of fuzzy simulation to fuzzy mathematical programming fuzzy max flow fuzzy regression and tuning a fuzzy controller are also presented

exact solutions to nonlinear nonautonomous space fractional diffusion equations with absorption we analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions in particular the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation we also discuss some implications by considering a diffusion coefficient d x t d t x theta theta in r and a drift force f k 1 t x k alpha x x alpha 1 in both situations we relate our solutions to those obtained within the maximum entropy principle by using the tsallis entropy

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

remarks on nonlinear stochastic partial differential equations an application of the splitting up method the objective of this article is to apply the splitting up method to the existence theorem for nonlinear stochastic partial differential equations with use of this method an approximating sequence is constructed by the compactness argument a convergent subsequence can be extracted and this fact provides the solution

approximation of the zakai equation by the splitting up method the objective of this article is to apply an operator splitting method to the time integration of zakai equation using this approach one can decompose the numerical integration into a stochastic step and a deterministic one both of them much simpler to handle than the original problem a strong convergence theorem is given in the spirit of existing results for deterministic problems

convergence rates of the splitting scheme for parabolic linear stochastic cauchy problems we study the splitting scheme associated with the linear stochastic cauchy problem du t au t dt dw t t in 0 t u 0 x here a is the generator of an analytic c 0 semigroup s s t t geqslant0 on a banach space e and w w t t geqslant0 is a brownian motion with values in a fractional domain space e beta associated with a we prove that if alpha beta gamma theta geqslant0 are such that gamma theta 1 and alpha beta theta gamma frac 1 2 then the approximate solutions u n converge to the solution u in the space c gamma 0 t e alpha both in l p means and almost surely with rate 1 n theta

interval random sets and entropy abstract the problem of proper choice from the class of interval random sets generated by some fuzzy set is considered this problem is determined for the maximum entropy criterion the class of interval random sets is studied

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

a parametric representation of fuzzy numbers and their arithmetic operators abstract direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex implementation of the extension principle is equivalent to solving a nonlinear programming problem to overcome this difficulty many applications limit the membership functions to certain shapes usually either triangular fuzzy numbers tfn or trapezoidal fuzzy numbers trfn then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers thus making the calculations trivial unfortunately the tfn shape is not closed under multiplication and division the result of these operators is a polynomial membership function and the triangular shape only approximates the actual result the linear approximation can be quite poor and may lead to incorrect results when used in engineering applications we analyze this problem and propose six parameters which define parameterized fuzzy numbers pfn of which tfns are a special case we provide the methods for performing fuzzy arithmetic and show that the pfn representation is closed under the arithmetic operations the new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as tfns the new method has better accuracy and similar computational speed to using tfns and appears to have benefits when used in engineering applications

a mild ito formula for spdes this article introduces a new somehow mild ito type formula for the solution process of a stochastic partial differential equation of evolutionary type

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

single value simulation of fuzzy variable some further results abstract the problem of a proper choice from the class of random sets equivalent to a fuzzy set under one point coverages is considered the selected random variable is used for construction of a random variable probability distribution serving as a random generator of a value of the fuzzy variable with values restricted by the fuzzy set the problem is determined for two selection criteria i maximum probability possibility consistency and ii maximum entropy in the case of the entropy criterion the problem has been solved first by goodman and here only another proof of the result is presented

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

a numerical technique for solving a class of fractional variational problems this paper presents a numerical method for solving a class of fractional variational problems fvps with multiple dependent variables multi order fractional derivatives and a group of boundary conditions the fractional derivative in the problem is in the caputo sense in the presented method the given optimization problem reduces to a system of algebraic equations using polynomial basis functions an approximate solution for the fvp is achieved by solving the system the choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed we extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique

a probabilistic approach to the arithmetics of fuzzy numbers an alternative look on fuzzy numbers based on two random variables and their summation and multiplication is introduced this approach to summation covers the standard zadeh s extension principle and triangular norm based approach however it is more general we illustrate it on some examples moreover copula based summation preserving the class of triangular trapezoidal fuzzy numbers is discussed

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

similarity preserving t norm based additions of fuzzy numbers abstract in this paper the notions of similar fuzzy numbers and similarity preserving additions of fuzzy numbers are introduced using similar fuzzy numbers enables to investigate preserving the type of fuzzy numbers with respect to the t norm based arithmetical operations especially in the case of fuzzy numbers with unbounded supports the conditions for preserving similarity of fuzzy numbers with unbounded supports with respect to the additions based on continuous t norms are given since the fuzzy numbers we deal with are different from fuzzy numbers considered in a previous paper busefal 69 1997 43 54 there are more possible cases described the introduced notions are illustrated with several examples

order of convergence of splitting schemes for both deterministic and stochastic nonlinear schrodinger equations we first prove the second order convergence of the strang type splitting scheme for the nonlinear schrodinger equation the proof does not require commutator estimates but crucially relies on an integral representation of the scheme it reveals the connection between strang type splitting and the midpoint rule we then show that the integral representation idea can also be used to study the stochastic nonlinear schrodinger equation with multiplicative noise of stratonovich type even though the nonlinear term there is not globally lipschitz we prove the first order convergence of a splitting scheme of it both schemes preserve the mass they are very efficient because they use explicit formulas to solve the subproblems containing the nonlinear or the nonlinear plus stochastic terms

random sets and fuzzy interval analysis abstract a general expression of function with random set valued arguments is stated which encompasses zadeh s extension principle as well as functions of random variables and interval analysis a monotonicity property is derived for algebraic operations performed between random set valued variables this property extends the monotonicity of functions with set valued arguments in the sense of inclusion the application of this result to the approximate calculation of functions of random variables is suggested and illustrated by an example

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

convergence of numerical schemes for the solution of parabolic stochastic partial differential equations we consider the numerical solution of the stochastic partial differential equation u t 2 u x 2 u w x t where w is space time white noise using finite differences for this equation gyongy has obtained an estimate of the rate of convergence for a simple scheme based on integrals of w over a rectangular grid we investigate the extent to which this order of convergence can be improved and find that better approximations are possible for the case of additive noise u 1 if we wish to estimate space averages of the solution rather than pointwise estimates or if we are permitted to generate other functionals of the noise but for multiplicative noise u u we show that no such improvements are possible

power and exponentials laws theory and application abstract in this paper we present novel derivatives based on exponential and power law kernel important properties and numerical approximations via riemann liouville caputo for first and second order were obtained we applied the new operators to describe different attractors and the advection diffusion equation interesting simulations were obtained for different values of the fractional orders and we showed that the new operator is useful in practical applications and the numerical simulations show the efficiency and the accuracy of the new derivative to modeling real world problems

single value simulation of fuzzy variables abstract the basic ideas of the present paper arise from relations between random sets and fuzzy ones the suggested approach which may be called fuzzy numerical simulation allows for ascribing a precise numerical value to a fuzzy variable by generating a value of a random variable related in some way to the fuzzy variable the so called generative characteristics of a fuzzy variable being the expected value and variance of the generating random variable are introduced the problem of simultaneous generating values of two fuzzy variables is also discussed finally methods of generating when a fuzzy variable is described with a complex body of evidence are presented

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

some new results concerning random sets and fuzzy sets abstract this paper is a continuation of past work showing direct connections between fuzzy set theory and classical probability theory through the use of random sets this includes characterizations of random intervals one point coverage equivalent to fuzzy sets determination of all nested random sets equivalent to fuzzy sets the solution of the one and multiple point coverage problems for random sets in finite spaces and the use of entropy for ordering random sets within the class of one point coverage equivalent ones for any given fuzzy set finally a connection between random variables random sets and fuzzy sets is pointed out

on the splitting up method and stochastic partial differential equations we consider two stochastic partial differential equations du varepsilon t l ru varepsilon t f r t dv varepsilon t r m k u varepsilon t g k t circ dy t k qquad hspace 5pt varepsilon 0 1 driven by the same multidimensional martingale y yk and by different increasing processes vr0 vr1 r 1 2 d1 where lr and mk are second and first order partial differential operators and stands for the stratonovich differential we estimate the moments of the supremum in t of the sobolev norms of u1 t u0 t in terms of the supremum of the differences break vr0t vr1t hence we obtain moment estimates for the error of a multistage splitting up method for stochastic pdes in particular for the equation of the unnormalized conditional density in nonlinear filtering

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

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

ranking fuzzy interval numbers in the setting of random sets abstract a new approach to the problem of extending interval order relations to the case of fuzzy interval numbers is presented in the approach the connectedness between fuzzy sets and random sets is used finally eight different fuzzy order relations are given some formal properties of these relations are investigated and the problem of computing the values of their membership functions discussed for trapezoidal fuzzy interval numbers the formulae depend on the parameters from which these numbers are derived

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

numerical approach of fokker planck equation with caputo fabrizio fractional derivative using ritz approximation abstract in this manuscript a type of fokker planck equation fpe with caputo fabrizio fractional derivative is considered we present a numerical approach which is based on the ritz method with known basis functions to transform this equation into an optimization problem it leads to a nonlinear algebraic system then we obtain the coefficients of basis functions by solving the algebraic system the convergence of this technique is discussed extensively three examples are included to show the applicability and validity of this method