calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

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

differentiability of multivalued functions on time scales and applications to multivalued dynamic equations abstract in this paper a class of new derivatives of multivalued functions on a time scale called the hukuhara hilger derivative suited to our study of multivalued dynamic equations is developed moreover some corresponding properties of the hukuhara hilger derivative are discussed the proposed approach forms the appropriate environment within which the study of multivalued dynamic inclusions on time scales can be developed a first example of multivalued dynamic equations is included

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

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

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

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

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

is the dimension of chaotic attractors invariant under coordinate changes several different dimensionlike quantities which have been suggested as being relevant to the study of chaotic attractors are examined in particular we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points it is found that some are and some are not it is suggested that the word dimension be reversed only for those quantities have this invariance property

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

almost periodicity of set valued functions and set dynamic equations on time scales in this paper we first propose the concept of the almost periodicity of set valued functions on almost periodic time scales then based on the exponential dichotomy we discuss the existence and uniqueness of almost periodic solutions of set dynamic equations on time scales as an application several examples are presented to exhibit that one can obtain the existence of almost periodic solutions to interval valued dynamic equations or fuzzy dynamic equations and their discrete counterparts in a united means

a reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

a reduced order extrapolated finite difference iterative scheme based on pod method for 2d sobolev equation abstract in this study we devote ourselves to the reduced order extrapolated finite difference iterative roefdi modeling and analysis for the two dimensional 2d sobolev equation to this end we first establish the reduced order extrapolated finite difference iterative roefdi scheme holding sufficiently high accuracy but containing very few degrees of freedom for the 2d sobolev equation via the proper orthogonal decomposition pod technique and then we analyze the stability and convergence of the roefdi solutions finally we use the numerical experiments to verify the feasibility and effectiveness of the roefdi scheme

stability for time varying linear dynamic systems on time scales we study conditions under which the solutions of a time varying linear dynamic system of the form x t a t x t are stable on certain time scales we give sufficient conditions for various types of stability including lyapunov type stability criteria and eigenvalue conditions on slowly varying systems that ensure exponential stability finally perturbations of the unforced system are investigated and an instability criterion is also developed

stability of moving invariant sets and uncertain dynamic systems on time scales abstract in this paper we study the stability of moving invariant sets and uncertain dynamic systems on time scale a control application is considered

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

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition the proper orthogonal decomposition pod is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations in this paper we make an initial effort to investigate problems related to pod reduced modeling of a large scale upper ocean circulation in the tropic pacific domain we construct different pod models with different choices of snapshots and different number of pod basis functions the results from these different pod models are compared with that of the original model the main findings are 1 the large scale seasonal variability of the tropic pacific obtained by the original model is well captured by a low dimensional system of order 22 which is constructed using 20 snapshots and 7 leading pod basis functions 2 the rms errors for the upper ocean layer thickness of the pod model of order 22 are less than 1m that is less than 1 of the average thickness and the correlation between the upper ocean layer thickness with that from the pod model is around 0 99 3 retaining modes that capture 99 energy is necessary in order to construct pod models yielding a high accuracy

reduced order modeling of time dependent incompressible navier stokes equation with variable density based on a local radial basis functions finite difference lrbf fd technique and the pod deim method abstract the main propose of this investigation is to introduce a rapid and impressive numerical procedure to simulate the time dependent incompressible navier stokes equation with variable density the developed formulation is constructed by using the meshfree rbf fd technique also to improve the numerical results an extra diffusion term has been added to the density equation with a small coefficient on the other hand a reduced order technique e g proper orthogonal decomposition pod has been employed to get a fast numerical method and to decrease the elapsed cpu time on the other hand since the considered problem is fully nonlinear thus in the reduced order model based on the pod idea there are some nonlinear terms that they take more computational time hence we employ the discrete empirical interpolation method deim to overcome the nonlinear terms four test problems have been proposed to show the ability of the numerical simulations

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

calculus of fuzzy vector valued functions and almost periodic fuzzy vector valued functions on time scales abstract in this paper we establish an embedding theorem for the fuzzy multidimensional space and introduce the new types of multiplications of fuzzy vectors determined by the determinant algorithm then some fundamental results of calculus of fuzzy vector valued functions on time scales are established based on these results we introduce the concept of shift almost periodic fuzzy vector valued functions and develop a theory of almost periodic fuzzy functions on time scales this function theory can be used to study almost periodic fuzzy dynamic equations on time scales including a new type of fuzzy dynamic equations called fuzzy q dynamic equation i e quantum fuzzy dynamic equation

second type hukuhara differentiable solutions to the delay set valued differential equations abstract in this paper we consider the delay set valued differential equations with a recently introduced notion of a second type hukuhara derivative under condition that the right hand side of the equation is lipschitzian in the functional variable we obtain the existence and uniqueness of the solution to such the equations some properties of the solutions are established existence of at least one solution is also proved application of the second type hukuhara differentiability concept to the problems of set valued differential equations implies that the diameter of the solution values is a function nonincreasing in time

properties of set valued integrals and set valued stochastic equations driven by two parameter martingales abstract in the paper we study properties of aumann s stochastic integral driven by a two parameter increasing process and set valued ito s integral with respect to the two parameter martingale both types of integrals are understood as set valued processes next the existence uniqueness and convergence properties of solutions to set valued stochastic integral equations with respect to such integrators are investigated we present new types of such equations that generalize those studied earlier the results obtained in the paper present a set valued counterpart dealing with this topic known both in single valued deterministic and stochastic cases

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

a reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations the reduced order extrapolating fd schemes are just developed by luo s research team since 2013 see 47 52 and a very new technique the reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations is first built by us we provide theoretical analysis and examples to show the advantage of our method this paper is concerned with establishing the reduced order extrapolation central difference roecd scheme based on proper orthogonal decomposition pod for two dimensional 2d fourth order hyperbolic equations for this purpose we first develop the classical central difference cd scheme for the 2d fourth order hyperbolic equations and analyze its stability and convergence then by making use of the pod method we build the roecd scheme with fewer degrees of freedom and sufficiently high accuracy and furnish the error estimates of the roecd solutions and the algorithm procedure for solving the roecd scheme finally we employ some numerical examples to confirm the correctness of theoretical conclusions this implies that roecd scheme is feasible and efficient for seeking the numerical solutions of the 2d fourth order hyperbolic equations

spectral properties of a class of operators associated with conformal maps in two dimensions iff is a rational map of the riemann sphere define the transfer operator by open image in new window

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

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

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

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

entropy spectrum of lyapunov exponents for nonhyperbolic step skew products and elliptic cocycles we study the fiber lyapunov exponents of step skew product maps over a complete shift of n n ge2 symbols and with c1 diffeomorphisms of the circle as fiber maps the systems we study are transitive and genuinely nonhyperbolic exhibiting simultaneously ergodic measures with positive negative and zero exponents examples of such systems arise from the projective action of 2 times 2 matrix cocycles and our results apply to an open and dense subset of elliptic mathrm sl 2 mathbb r cocycles we derive a multifractal analysis for the topological entropy of the level sets of lyapunov exponent the results are formulated in terms of legendre fenchel transforms of restricted variational pressures considering hyperbolic ergodic measures only as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent we show that the entropy of the level sets is a continuous function of the lyapunov exponent the level set of the zero exponent has positive but not maximal topological entropy under the additional assumption of proximality as for example for skew products arising from certain matrix cocycles there exist two unique ergodic measures of maximal entropy one with negative and one with positive fiber lyapunov exponent

equivalence of gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification letm be a compact metrizable space f m m a homeomorphism satisfying expansiveness and specification anda m a function such that left sum limits k 0 n 1 a f k x a f k y right mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x k varepsilon infty whenevern 1 andx y are e n close i e d fkx fky 0 under these conditions bowen has shown that there is a uniqueequilibrium state fora assuming thatk 0 when 0 we show that is also the uniquegibbs state fora we further definequasi gibbs states and show that is the uniquef invariant quasi gibbs state fora

lyapunov spectrum of asymptotically sub additive potentials for general asymptotically sub additive potentials resp asymptotically additive potentials on general topological dynamical systems we establish some variational relations between the topological entropy of the level sets of lyapunov exponents measure theoretic entropies and topological pressures in this general situation most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub additive case and in the case that the entropy map is not upper semi continuous

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

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

exponential stability for set dynamic equations on time scales abstract a new theory known as set dynamic equations on time scales has been built the criteria for the equistability equiasymptotic stability uniform and uniformly asymptotic stability were developed in hong 2010 1 in this paper we consider the exponential stability exponentially asymptotic stability uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using lyapunov like functions

renormalisation induced phase transitions for unimodal maps the thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential t log df multiple and indeed infinitely many phase transitions at positive t can occur for some quadratic maps all unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum

low temperature phase transitions in the quadratic family abstract we give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function this implies that several dimension spectra and large deviation rate functions associated to this map are not expected to be real analytic in contrast to the uniformly hyperbolic case the quadratic map we study has a non recurrent critical point so it is non uniformly hyperbolic in a strong sense

the monotone iterative technique for impulsive hybrid set valued integro differential equations abstract we develop the monotone method for impulsive hybrid set integro differential equations in all its generality some interesting observations are presented

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

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

phase spaces and periodic solutions of set functional dynamic equations with infinite delay abstract very recently a new theory known as set dynamic equations on time scales has been built in this paper a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set respectively moreover for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping our results show that one can unify the study of some continuous or discrete problems in the sense of set dynamic equations on general time scales

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

a self contained automated methodology for optimal flow control this paper describes a self contained automated methodology for active flow control which couples the time dependent navier stokes system with an adjoint navier stokes system and optimality conditions from which optimal states i e unsteady flow fields and controls e g actuators may be determined the problem of boundary layer instability suppression through wave cancellation is used as the intital validation case to test the methodology here the objective of control is to match the stress vector along a portion of the boundary to a steady base flow control is effected through the injection or suction of fluid through a single orifice on the boundary the results demonstrate that instability suppression can be achieved without any a priori knowledge of the flow unsteadiness such as frequencies instability type etc the present methodology has been extended to three dimensions and may potentially be applied to separation control relaminarization and turbulence control applications using one to many sensors and actuators

decay of correlations and the central limit theorem for meromorphic maps let f be a dominant meromorphic self map of large topological degree on a compact kahler manifold we give a new construction of the equilibrium measure of f and prove that is exponentially mixing as a consequence we get the central limit theorem in particular for holder continuous observables but also for noncontinuous observables 2005 wiley periodicals inc

on the lyapunov spectrum for rational maps we study the dimension spectrum for lyapunov exponents for rational maps on the riemann sphere

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

a reduced finite element formulation based on pod method for two dimensional solute transport problems abstract a proper orthogonal decomposition pod method is applied to a usual finite element fe formulation for two dimensional solute transport problems with real practical applied background such that it is reduced into a reduced fe formulation with lower dimensions and high enough accuracy the error estimates between the reduced pod fe solutions and the usual fe solutions are provided it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that this validates the feasibility and efficiency of pod fe method

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

on connections between stochastic differential inclusions and set valued stochastic differential equations driven by semimartingales abstract in the paper we study properties of solutions to stochastic differential inclusions and set valued stochastic differential equations with respect to semimartingale integrators we present new connections between their solutions in particular we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set valued stochastic equations we also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set valued stochastic equation the results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases

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

uncertain systems and fuzzy differential equations the study of fuzzy differential equations is initiated and some typical results are given in order to understand the intricacies involved in incorporating fuzziness in the terms of differential equations

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

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

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

proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical pacific ocean reduced gravity model in this paper the tropical pacific ocean reduced gravity model is studied using the proper orthogonal decomposition pod technique of mixed finite element mfe method and an error estimate of pod approximate solution based on mfe method is derived pod is a model reduction technique for the simulation of physical processes governed by partial differential equations e g fluid flows or other complex flow phenomena it is shown by numerical examples that the error between pod approximate solution and reference solution is consistent with theoretical results thus validating the feasibility and efficiency of pod method

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

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

dynamics and stability of impulsive hybrid setvalued integro differential equations with delay abstract we study the dynamics and stability theory for impulsive hybrid set integro differential equations with delay sufficient conditions for the stability of the null solution of impulsive hybrid set integro differential equations with delay are presented

instability of set differential equations abstract this paper is devoted to the instability of set differential equations sdes using the geometric inequalities of brunn minkowski and a d aleksandrov we propose new methods for constructing lyapunov functions in combination with the known methods of stability theory the lyapunov s direct method the comparison method and the vector function method we establish conditions for the collapse of the solutions of the sdes estimates of the collapse time of solutions are also obtained examples of sdes in spaces of dimension 2 and 3 illustrating general theorems are given

a pod reduced order finite difference time domain extrapolating scheme for the 2d maxwell equations in a lossy medium abstract this paper is concerned with establishing a reduced order finite difference time domain fdtd extrapolating scheme based on proper orthogonal decomposition pod method for the two dimensional 2d maxwell equations in a lossy medium for this purpose we first review the classical fdtd scheme and associated results for the 2d maxwell equations in a lossy medium and then we formulate three sets of pod bases by means of the solutions obtained from the classical fdtd scheme on the very short time interval establish the pod reduced order fdtd extrapolating scheme and furnish the error estimates among the pod reduced order fdtd solutions and the classical fdtd solutions as well as the accuracy solution for the 2d maxwell equations in a lossy medium finally we enumerate two numerical examples to confirm that the pod reduced order fdtd extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium

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

set differential equations versus fuzzy differential equations the study of fuzzy differential equations fdes forms a suitable setting for a mathematical modeling of real world problems in which uncertainties or vagueness pervades in recent years the theory of fdes has been investigated extensively in the original formulation as well as in an alternative framework which leads to ordinary multivalued differential inclusions it has recently been realized that initiating the study of set differential equations in a metric space has several advantages in addition to providing a natural setting for considering fdes in this paper we present some interesting results in this direction with the necessary background material