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

hukuhara differentiability of interval valued functions and interval differential equations on time scales abstract using the concept of the generalized hukuhara difference in this paper we introduce and study the differentiability and the integrability for the interval valued functions on time scales some illustrative examples to interval differential equations on time scales are presented

the henstock stieltjes integral for fuzzy number valued functions in this paper we firstly define and discuss the henstock stieltjes integral for fuzzy number valued functions which is an extension of the usual fuzzy riemann stieltjes integral in addition several necessary and sufficient conditions of the integrability for fuzzy number valued functions are given by means of the henstock stieltjes integral of real valued functions and henstock integral of fuzzy number valued functions secondly the continuity and the differentiability of the primitive for the fuzzy henstock stieltjes integral are discussed we find that there exists a fuzzy number valued function which is fuzzy henstock stieltjes integrable but whose primitive is not a differentiable almost everywhere thirdly we introduce some quadrature rules for the fuzzy henstock stieltjes integral by giving error bounds for the mappings of bounded variation and of lipschitz type we also consider the generalization of classical quadrature rules such as midpoint type trapezoidal and simpson s quadrature finally we propose the concept of weak equi integrability for sequences of fuzzy henstock stieltjes integrable functions under this concept we prove two convergence theorems for sequences of the fuzzy henstock stieltjes integrable functions at the same time the formula of integration by parts is also studied

set valued functions in frechet spaces continuity hukuhara differentiability and applications to set differential equations abstract the set k c f of compact convex subsets of a frechet space f is studied in detail and is realized as a projective limit of metric spaces the notion of hausdorff metric on it is replaced by a family of corresponding semi metrics which provide the necessary background for the support of continuity and lipschitz continuity finally the notion of hukuhara derivative suited to our study is developed the proposed approach forms the appropriate environment within which the study of set differential equations for frechet spaces can be developed a first example on r is included

existence results for extremal solutions of interval fractional functional integro differential equations abstract in this study we introduce interval fractional functional integro differential equations iffides under the caputo generalized hukuhara fractional differentiability we establish some necessary comparison results and use the monotone iterative technique combined with the method of upper and lower solutions to investigate the existence of extremal solutions for iffides an example is presented to illustrate the results

existence of solutions to fuzzy differential equations with generalized hukuhara derivative via contractive like mapping principles abstract we study the existence and uniqueness of solution for fuzzy initial value problems in the setting of a generalized hukuhara derivative and by using some recent results of fixed point of weakly contractive mappings on partially ordered sets

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

estimating bounds for quadratic assignment problems associated with hamming and manhattan distance matrices based on semidefinite programming quadratic assignment problems qaps with a hamming distance matrix for a hypercube or a manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems in this paper we consider the issue of how to obtain lower bounds for those two classes of qaps based on semidefinite programming sdp by exploiting the data structure of the distance matrix b we first show that for any permutation matrix x the matrix y alpha e xbx t is positive semidefinite where alpha is a properly chosen parameter depending only on the associated graph in the underlying qap and e ee t is a rank 1 matrix whose elements have value 1 this results in a natural way to approximate the original qaps via sdp relaxation based on the matrix splitting technique our new sdp relaxations have a smaller size compared with other sdp relaxations in the literature and can be solved efficiently by most open source sdp solvers experimental results show that for the underlying qaps of size up to n 200 strong bounds can be obtained effectively

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

some properties of strong solutions to stochastic fuzzy differential equations we consider stochastic fuzzy differential equations driven by m dimensional brownian motion such equations can be useful in modeling of hybrid dynamic systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously under a boundedness condition which is weaker than linear growth condition and the lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations solutions which are fuzzy stochastic processes and their uniqueness are considered to be in a strong sense an estimation of error of the picard approximate solution is established we give a boundedness type result for the solution defined on finite time interval also the stabilities of solution on initial condition and coefficients of the equation are shown the existence and uniqueness of a solution defined on infinite time interval is proven finally some applications of fuzzy stochastic differential equations are considered all the results presented in this paper apply to set valued stochastic differential equations

some results on sheaf solutions of sheaf set control problems abstract in this paper we study the existence and some properties of solutions of the so called set control differential equations scde and sheaf solutions of sheaf set control problems

algebra of generalized hukuhara differentiable interval valued functions review and new properties abstract this article deals with the algebra of gh differentiable interval valued functions specifically we give conditions for the gh differentiability of the sum and the gh difference of two gh differentiable interval valued functions we also consider the product and the composition of a differentiable real function and a gh differentiable interval valued function we briefly review existing results on the topic and show that some of them are not correct new properties are also presented therefore this article corrects complements and generalizes previously obtained results some examples and counterexamples illustrate the theory

wavelets associated with representations of the affine weyl heisenberg group the affine weyl heisenberg group generated by time and frequency translations and time dilations is considered and some associated resolutions of the identity are derived as a result it will be shown that they can all be obtained from those associated with the affine group and the weyl heisenberg group by frequency translations of the analyzed function and the analyzing and reconstructing wavelets

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

stability of fuzzy differential equations with the second type of hukuhara derivative in this paper we consider fuzzy differential equations with the second type of hukuhara derivative with the help of lyapunov like functions we establish stability uniform stability and uniformly asymptotical stability theorems respectively for a class of fuzzy differential equations with the second type of hukuhara derivative besides a very useful comparison principle is derived which connects the stability relationships between fuzzy differential equations with the second type of hukuhara derivative and scalar differential equations in addition some examples are shown to illustrate the effectiveness of the results

relationship between bede gal differentiable set valued functions and their associated support functions in this study we adapt the concept of the bede gal derivative which was initially suggested for fuzzy number valued functions to set valued functions we use an example to demonstrate that this concept overcomes some of the shortcomings of the hukuhara derivative we prove some properties of bede gal differentiable set valued functions we also study the relationship between a bede gal differentiable set valued function and its value s support function which we call the associated support function we provide examples of set valued functions that are not bede gal differentiable whereas their associated support functions are differentiable we also present some applications of the bede gal derivative to solving set valued differential equations

nonlinear variation of parameters formula for set differential equations in a metric space nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values other needed results are also proven

fuzzy delay differential equations under generalized differentiability abstract we interpret a fuzzy delay differential equation using the concept of generalized differentiability in this setting we prove the existence of two fuzzy solutions each one corresponding to a different type of differentiability uniqueness is understood in the sense that the solutions considered do not have switching points the applicability of the theoretical results is illustrated with some real world examples

on set differential equations in banach spaces a second type hukuhara differentiability approach in this paper we consider set differential equations in banach spaces and with the second type hukuhara derivative we show that a sequence of successive approximations converges uniformly to a unique local solution of a cauchy problem some comparison results are formulated and a continuity of solution with respect to equation data is shown the existence of a global solution is also established

variation of constant formula for first order fuzzy differential equations in this paper we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions we also correct and complete some previous results finally some examples are given to illustrate our results

numerical solutions of fuzzy differential equations by an efficient runge kutta method with generalized differentiability in this paper an extended fourth order runge kutta method is studied to approximate the solutions of first order fuzzy differential equations using a generalized characterization theorem in this method new parameters are utilized in order to enhance the order of accuracy of the solutions using evaluations of both f and f instead of using the evaluations of f only the proposed extended runge kutta method and its error analysis which guarantees pointwise convergence are given in detail furthermore the accuracy and efficiency of the proposed method are demonstrated in a series of numerical experiments

stability criteria for set control differential equations abstract the existence and comparison results on solutions of set control differential equation were studied in n d phu t t tung some results on sheaf solutions of sheaf set control problems nonlinear analysis 67 2007 1309 1315 in this paper we present the stability criteria for solutions of set control differential equation

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

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

interval cauchy problem with a second type hukuhara derivative in this paper we consider interval differential equations such the equations can be appropriate in modeling of dynamical systems under presence of uncertainty of parameters we study an interval initial value problem with a second type hukuhara derivative by an example of real world application we indicate the advantages of the usage of such a kind of interval valued derivative a continuous dependence of the solution on initial value and right hand side of the equation is shown the existence of approximate local solutions is proven and then it is used in the derivation of existence of at least one local solution to interval cauchy problem with second type hukuhara derivative the compactness of solutions set is also stated finally the explicit formulae for local solutions to linear interval differential equations are provided

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

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

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

stability of set differential equations and applications abstract the paper introduces some results on stability of sheaf solutions exponential stability of solutions and sheaf solutions of set differential equations and their applications to set control differential equations

the initial value problem for interval valued second order differential equations under generalized h differentiability in this paper the interval valued second order differential equations under generalized hukuhara differentiability isdes are introduced under suitable conditions we obtain the existence and uniqueness results of solutions to isdes to prove this assertion we use idea of contraction principle and successive approximations furthermore we use the method based on properties of linear transformations ltm to find the explicit solution of initial value problem for linear second order differential equation with interval valued forcing function and with interval initial values iivp we apply the linear transformations method to two example problems including a vibrating mass and an electrical circuit finally the method based on the analysis of a solution to real valued second order differential equation is investigated to solve interval valued second order differential equations with interval valued coefficients interval valued forcing function and interval initial values some examples are presented to illustrate applicability of the proposed method

numerical solution of a fuzzy stochastic single species age structure model in a polluted environment this paper presents an investigation of a fuzzy stochastic single species age structure model in a polluted environment both the fuzziness of the initial condition and the stochastic disturbance of the environment are incorporated into the model by using the theory of fuzzy stochastic differential equation fsde and the successive approximation the global existence and uniqueness of solutions of the model are proved in addition the error estimation and stability of the numerical solutions are obtained furthermore making use of euler maruyama em method the convergence of the em numerical approximation is established numerical simulations are carried out to verify the theoretical results our results show that the technique of numerical solution of fsde can be used to estimate the evolution tendency of the population density in a polluted environment

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented