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

exponential convergence and tractability of multivariate integration for korobov spaces in this paper we study multivariate integration for a weighted korobov space for which the fourier coefficients of the functions decay exponentially fast this implies that the functions of this space are infinitely times differentiable weights of the korobov space monitor the influence of each variable and each group of variables we show that there are numerical integration rules which achieve an exponential convergence of the worst case integration error we also investigate the dependence of the worst case error on the number of variables s and show various tractability results under certain conditions on weights of the korobov space tractability means that the dependence on s is never exponential and sometimes the dependence on s is polynomial or there is no dependence on s at all

a boundary value problem for second order fuzzy differential equations abstract in this paper we interpret a two point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept we present a new concept of solutions and utilizing the generalized differentiability we investigate the problem of finding new solutions some examples are provided for which the new solutions are found

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

good parameters for a class of node sets in quasi monte carlo integration for 2 s 12 we determine good parameters in a general construction of node sets for s dimensional quasi monte carlo integration recently introduced by the third author some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques

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

decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics decomposition formulas of general exponential operators in a banach algebra and in a lie algebra are presented that yield a basis of monte carlo simulation of quantum systems they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena a global approximation method of transient phenomena is also formulated on the basis of decomposition formulas it is applied to the laser model as a simple example

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability we study the existence of solutions to a class of first order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability the objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena by considering a combination of two types of derivatives with a switching point we provide sufficient conditions which guarantee that the piecewise defined solutions match adequately and illustrate through some examples the process of construction and calculation of solutions

construction algorithms for polynomial lattice rules for multivariate integration we introduce a new construction algorithm for digital nets for integration in certain weighted tensor product hilbert spaces the first weighted hilbert space we consider is based on walsh functions dick and pillichshammer calculated the worst case error for integration using digital nets for this space here we extend this result to a special construction method for digital nets based on polynomials over finite fields this result allows us to find polynomials which yield a small worst case error by computer search we prove an upper bound on the worst case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights we extend the results for the weighted hilbert space based on walsh functions to weighted sobolev spaces in this case we use randomly digitally shifted digital nets the construction principle is the same as before only the worst case error is slightly different again digital nets obtained from our search algorithm yield a worst case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights these results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules we conclude the article with numerical results comparing the expected worst case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules

on the existence of solutions to periodic boundary value problems for fuzzy linear differential equations in this work we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first order linear fuzzy differential equations by using generalized differentiability and switching points in comparison with some previous works we consider equations whose coefficient may change its sign a finite number of times in the interval of interest we also study the existence of solutions which are crisp or real at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character

on the transient stochastic dynamics driven by gaussian colored noise of systems with time dependent control parameters the effect of initial conditions abstract on a recent work we studied the transient stochastic dynamics driven by gaussian white noise gwn of linear systems with time dependent control parameters by means of the connection between the nonlinear relaxation times nlrt and the quasideterministic qd approach in the present study we make an extension of the analysis to the gaussian colored noise gcn problem here we first calculate the characteristic time associated with the decay of the unstable state of such linear systems when the control parameter is a linear function of time of the form a t bt a0 with b a0 0 which is continuously swept from below to above a threshold t a 0 b this type of linear modulation is known as the ramp model then going further we consider a general case where the control parameter is modulated by a family of functions a t bt a0 with 0 the effects of the coupling between the initial state at time t 0 of the system with the noise are specially emphasized the results of nlrt for the ramp model and the general case are compared

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

a construction of polynomial lattice rules with small gain coefficients in this paper we construct polynomial lattice rules which have in some sense small gain coefficients using a component by component approach the gain coefficients as introduced by owen indicate to what degree the method improves upon monte carlo we show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component by component decays at a rate of n 2 1 for all 0 assuming that the function under consideration has bounded variation of order for some 0 1 and where n denotes the number of quadrature points an analogous result is obtained for korobov polynomial lattice rules it is also established that these rules are almost optimal for the function space considered in this paper furthermore we discuss the implementation of the component by component approach and show how to reduce the computational cost associated with it finally we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets

on periodic solutions to first order linear fuzzy differential equations under differential inclusions approach in this work we study some properties of the solutions to first order linear fuzzy differential equations under differential inclusions approach we show that the differential inclusions approach generally adapts better to some properties of the solutions to ordinary differential equations sufficient conditions for the solutions under this approach to be periodic and bounded are presented we also give some examples to show the applicability of these results

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

constructing embedded lattice rules for multivariate integration lattice rules are a family of equal weight cubature formulae for approximating high dimensional integrals by now it is well established that good generating vectors for lattice rules having n points can be constructed component by component for integrands belonging to certain weighted function spaces and that they can achieve the optimal rate of convergence although the lattice rules constructed this way are extensible in dimension they are not extensible in n thus when n is changed the generating vector needs to be constructed anew in this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of n while still being extensible in dimension by using an adaptation of the fast component by component construction algorithm which makes use of fast fourier transforms we are able to obtain good generating vectors for thousands of dimensions and millions of points under both product weight and order dependent weight settings at the cost of calo d n log n 2 operations with a sufficiently large number of points and good overall quality these embedded lattice rules can be used for practical purposes in the same way as a low discrepancy sequence we show for a range of weight settings in an unanchored sobolev space that our embedded lattice rules achieve the same optimal rate of convergence calo n 1 delta delta 0 as those constructed for a fixed number of points and that the implied constant gains only a factor of 1 30 to 1 55

passage time statistics in exponential distributed time delay models noisy asymptotic dynamics the stochastic dynamics toward the final attractor in exponential distributed time delay non linear models is presented then the passage time statistic is studied analytically in the small noise approximation the problem is worked out by going to the associated two dimensional system the mean first passage time left langle t e right rangle from the unstable state for this non markovian type of system has been worked out using two different approaches firstly by a rigorous adiabatic markovian approximation in the small mean delay time epsilon lambda 1 secondly by introducing the stochastic path perturbation approach to get a non adiabatic theory for any lambda this first passage time distribution can be written in terms of the important parameters of the models we have compared both approaches and we have found excellent agreement between them in the adiabatic limit in addition using our non adiabatic approach we predict a crossover and a novel behavior for the relaxation scaling time as a function of the delay parameter which for lambda ll 1 goes as left langle t e right rangle sim 1 sqrt lambda

fuzzy differential equations an approach via fuzzification of the derivative operator abstract in this paper we study fuzzy differential equations fdes in terms of derivative for fuzzy functions in a different way from the traditional hukuhara derivative defined for set valued functions the derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions we discuss the relation of this approach to fuzzy differential inclusions fdis and hukuhara and strongly generalized derivatives a theorem of existence of a solution is studied with hypothesis similar to those assumed for fdis some examples are explored in order to illustrate the theory

toward the existence and uniqueness of solutions of second order fuzzy differential equations in this paper the existence and uniqueness of solutions for second order fuzzy differential equations with initial conditions under generalized h differentiability is proved to this end the concept of second order generalized differential equation is defined which is based on an enlargement of the class of differentiable fuzzy mappings

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

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

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

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

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

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

the construction of good extensible rank 1 lattices it has been shown by hickernell and niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n p p 2 for all integers p 2 this paper provides algorithms for the construction of generating vectors which are finitely extensible for n p p 2 for all integers p 2 the proofs which show that our algorithms yield good extensible rank 1 lattices are based on a sieve principle particularly fast algorithms are obtained by using the fast component by component construction of nuyens and cools analogous results are presented for generating vectors with small weighted star discrepancy

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

revisit to the scaling theory of transient phenomena generalization to correlated noise and singular perturbation expansion up to infinite order abstract a scaling theory is formulated in nonlinear langevin equations with colored noise near the instability point from which the center of the initial distribution is deviated slightly an intuitive derivation is given as well as perturbational calculations up to the infinite order of the noise moments of all orders are calculated in the approximation in which only the most dominant terms are retained in the scaling limit hence the scaling distribution function is also obtained the onset time t 0 is derived as a functional of the correlation function of the noise and it is shown to increase when the noise correlation length increases

the quasideterministic approach in the dynamical characterization of rotating unstable systems abstract in this work we report a new theoretical scheme to deal with the dynamical characterization of rotating unstable systems driven by both internal noise and constant external force we use the quasideterministic qd approach and the passage time pt distribution to characterize the dynamical relaxation of systems of two and three variables and show that in this new scheme the validity regime of qd approach is satisfied only if the intensity of external force is smaller than or of the same order as the internal noise for the systems of three variables a special mathematical approach is given for a better understanding of underlying physics numerical simulation results are given to support our proposal

multivariate formulation of transient stochastic dynamics abstract the objective of this paper is to generalize the formulation reported in previous works which we term as standard formulation of quasideterministic approach to multivariate unstable systems as usual the new formalism is connected with the nonlinear relaxation times to characterize the transient stochastic dynamics of those unstable systems which are governed by a multivariate langevin type equation the generalized method is applied to study two analytical models in two and three variables in order to obtain explicit results and calculate the time scale associated with the decay of the unstable state of such systems the theoretical formalism constitutes an alternative method to study some physical systems in two and three varibles the laser systems represent one possibility in this theoretical context

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

exponential distributed time delay nonlinear models monte carlo simulations abstract the stochastic dynamics toward the final attractor in an exponential distributed time delay nonlinear model is studied in the small noise approximation the passage time statistic for this non markovian type of system has been worked out using monte carlo simulations we report the mean first passage time t e m c from the unstable state as a function of the mean time delay 1 we have compared our monte carlo simulations for 1 against previous results caceres 2008 and we have found excellent agreement in the adiabatic regime the crossover for 1 and a power law behavior t e m c for 1 have also been found in agreement with recent theoretical predictions caceres 2014

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

search for the symmetry of the fokker planck equation abstract it is shown that the fokker planck equations with linear drift x and nonlinear drift x x i e the rayleigh process have exact symmetries on their own solution manifolds the sl 2 r weyl algebra and the sl 2 r i respectively the symmetries of these two types of processes for 1 2 dimensional cases are explicitly found out too

representation of functions as walsh series to different bases and an application to the numerical integration of high dimensional walsh series we will prove the following theorem on walsh series and we will derive from this theorem an effective and constructive method for the numerical integration of walsh series by number theoretic methods further concrete computer calculations are given theorem for base b 2 dimension s 1 and o 1 c 0 b s n c a e r let i e c be the class of all functions f 0 l s c which are representable by absolutely convergent walsh series to base b with walsh coefficients w h hs with the following property w h hs 2 provided that a 1 s where

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

comparison results for fuzzy differential equations in this paper we present several comparison results for the solutions of fuzzy differential equations by using the hukuhara derivative these results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations besides some side results about the hukuhara difference and the partial orderings are given we illustrate the applicability of the new results by showing several examples

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

a study of the dynamical relaxation of unstable states driven by gaussian exponentially correlated noise the coloured noise omstein uhlembeck process problem corresponding to the decay process of unstable states is studied using the nonlinear relaxation time and the quasideterministic approach the effect of the non markovian character in this problem as well as the statistics of the initial conditions are both addressed our general result is applied to the landau model as a typical example

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

solvable models of the fokker planck equation an approach based on the gel fand levitan method from a given solvable fokker planck equation one can construct a number of other solvable models for diffusion in a stable or bistable potential fields using the gel fand levitan method of the inverse scattering theory the simplest way of achieving this is to change the lowest eigenvalue and or the normalization of the lowest eigenfunction of the ordinary differential equation obtained by separating the time dependent part for these cases it is shown that the new probability distribution is expressible in terms of integrals involving the original probability distribution and the gel fand levitan kernel the possibility of changing the lowest eigenvalue enables one to find bistable potential fields which would correspond to a well defined long time relaxation rate for the probability current

nonlinear fluctuations separation of procedures and linearization of processes a concept of separation of procedures is introduced to study cooperative phenomena theoretically some typical important examples of this concept are presented to clarify its usefulness kubo s stochastic liouville equation some generalized diffusionlike equations van kampen s expansion kubo s extensivity prigogine s entropy production the scaling theory of transient phenomena based on the lie algebra and suzuki s cam theory of cooperative phenomena are discussed from the new viewpoint of separation of procedures

on the solutions of first order linear impulsive fuzzy differential equations abstract this paper gives a representation of solutions to first order linear impulsive fuzzy differential equations with constant coefficients using a generalized differentiability concept and an impulsive fundamental matrix we present ideas similar to those in linear fuzzy differential equations for impulsive fuzzy cauchy problems finally numerical examples are given to illustrate the theory

the construction of extensible polynomial lattice rules with small weighted star discrepancy in this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance the construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest we also show that the construction can be done component by component

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

lattice point sets for deterministic learning and approximate optimization problems in this brief the use of lattice point sets lpss is investigated in the context of general learning problems including function estimation and dynamic optimization in the case where the classic empirical risk minimization erm principle is considered and there is freedom to choose the sampling points of the input space here it is proved that convergence of the erm principle is guaranteed when lpss are employed as training sets for the learning procedure yielding up to a superlinear convergence rate under some regularity hypotheses on the involved functions preliminary simulation results are also provided

the characteristic times of the transient stochastic dynamics with time dependent control parameters distributed initial conditions a systematic method is developed for the calculation of characteristic times called nonlinear relaxation times nlrt to describe the dynamical relaxation of the linear transient stochastic systems whose control parameters are time dependent functions for those control parameters which are modulated by a family of functions of the form a t bt a0 with 0 the method is applied to calculate the nlrt associated with the decay of unstable states of the linear stochastic systems when these parameters are continuously swept from below to above threshold t a0b 1 the ramp modulation is a model for which 1 it is studied and formulated in terms of the time differences s t t with t a0b the time scales of both models are compared under certain requirements of the involved parameters

optimal polynomials for t m s nets and numerical integration of multivariate walsh series existence theorems for good parameters in a construction of low discrepancy point sets based on rational functions over finite fields are proved a special class of these point sets is introduced and good parameters in the construction of these special point sets are tabulated in this way point sets in the s dimensional unit cube for 3 leq s leq 15 are obtained with very small discrepancy these special point sets yield excellent numerical results when applied to the quasi monte carlo integration of multivariate walsh series

three variable models in multivariate transient stochastic dynamics the presence of constant external force abstract in this work we show that the generalized formalism of the quasideterministic qd approach and its connection with the nonlinear relaxation times nlrt is absolutely consistent with 3 3 matricial models whose elements of matrix are all different from zero in the context of both gaussian white and colored noise in the absence of external force more general results than those reported in 1 2 are given the consistency of the formalism with such models in the presence of constant external force is also verified for gaussian white noise

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 the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

matricial formalism of transient dynamics ii the presence of time dependent external forces abstract in this work the formalism of the preceding reference which we will refer to as paper i is now extended to the case of gaussian stochastic fluctuations of time dependent external forces for those general nonlinear systems of two variables the formalism is developed explicitly when the external forces are taken as complex functions with time dependent phase diffusion model as an example in two variables we describe the switch on process of a laser model when the injected external signal is weak and has a time dependent fluctuating phase

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

solutions of the fokker planck equation for a double well potential in terms of matrix continued fractions solutions of the fokker planck kramers equation in position velocity space for the double well potentiald 2x2 2 d4x4 4 in terms of matrix continued fractions are derived it is shown that the method is also applicable to a boltzmann equation with a bgk collision operator results of eigenvalues and of the fourier transform of correlation functions are presented explicitly the lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero friction limit result

matricial formulation of transient stochastic dynamics driven by gaussian colored noise abstract in physica a 237 1997 113 we have proposed in terms of gaussian white noise gwn a generalized method in terms of eigenfunctions and eigenvalues of the quasideterministic qd approach and its connection with the nonlinear relaxation times nlrt to describe the transient stochastic dynamics of multivariate systems contrary to what happens with the standard formulation of qd approach the generalized theory is focused on those unstable systems which are not necessarily derived from a potential function in the present work we extend the generalized method to the case of gaussian colored noise gcn the coupling between the noise and the initial state of the system as a natural effect in colored noise problem is also addressed to justify the theory we study the same two analytical models in two and three variables of the preceding reference and calculate the time scales associated with those models

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

non linear relaxation time for stochastic processes driven by non gaussian noises decay of unstable states abstract the non linear relaxation time nlrt technique is developed for langevin equations with non gaussian noises exact results are obtained for the cases of dichotomous markov noise and poisson white shot noise as an explicit application we study the decay of unstable states under non gaussian fluctuations

the cauchy problem of fuzzy differential equations under generalized differentiability the generalization of the concept of h differentiability can be of great help in the dynamic study of fuzzy differential equations in this paper the concept of generalized differentiability is described from a new perspective on the basis of this concept the class of differentiable fuzzy set valued mappings is enlarged the cauchy problem for fuzzy differential equations is investigated in this enlarged setting as a result some new solutions are obtained the lengths of the support sets of these solutions may be non monotonic several examples are also shown

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

quasi monte carlo finite element methods for elliptic pdes with lognormal random coefficients in this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in mathbb r d rd d 1 2 3 d 1 2 3 with diffusion coefficient a varvec x omega a x given as a lognormal random field i e a varvec x omega exp z varvec x omega a x exp z x where varvec x x is the spatial variable and z varvec x cdot z x is a gaussian random field the analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 0 or infty over all possible realizations of a a focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem we give a rigorous error analysis for methods constructed from 1 standard continuous and piecewise linear finite element approximation in physical space 2 truncated karhunen loeve expansion for computing realizations of a a leading to a possibly high dimensional parametrized deterministic diffusion problem and 3 lattice based quasi monte carlo qmc quadrature rules for computing integrals over parameter space which define the expected values the paper contains novel error analysis which accounts for the effect of all three types of approximation the qmc analysis is based on a recent result on randomly shifted lattice rules for high dimensional integrals over the unbounded domain of euclidean space which shows that under suitable conditions the quadrature error decays with mathcal o n 1 delta o n 1 with respect to the number of quadrature points n n where delta 0 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration

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

almost periodic fuzzy number valued functions in this paper we develop a theory of almost periodic fuzzy functions i e of the almost periodic functions of real variable and with values fuzzy real numbers although the class of fuzzy real numbers does not form a linear normed space the majority of main properties of almost periodic functions with values in banach spaces are extended to this case applications to fuzzy differential equations and to fuzzy dynamical systems are given

lattice rules for nonperiodic smooth integrands the aim of this paper is to show that one can achieve convergence rates of n alpha delta n for alpha 1 2 1 2 and for delta 0 0 arbitrarily small for nonperiodic alpha smooth cosine series using lattice rules without random shifting the smoothness of the functions can be measured by the decay rate of the cosine coefficients for a specific choice of the parameters the cosine series space coincides with the unanchored sobolev space of smoothness 1 we study the embeddings of various reproducing kernel hilbert spaces and numerical integration in the cosine series function space and show that by applying the so called tent transformation to a lattice rule one can achieve the almost optimal rate of convergence of the integration error the same holds true for symmetrized lattice rules for the tensor product of the direct sum of the korobov space and cosine series space but with a stronger dependence on the dimension in this case

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

random weights robust lattice rules and the geometry of the cbcrc algorithm in this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube 0 1 s from a weighted reproducing kernel hilbert space we assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications i it is usually not known in practice how to choose the weights thus by assuming that the weights are random variables we obtain robust constructions with respect to the weights of lattice rules this to some extend removes the necessity to carefully choose the weights ii in practice it is convenient to use the same lattice rule for many different integrands the best choice of weights for each integrand may vary to some degree hence considering the weights random variables does justice to how lattice rules are used in applications in this paper the worst case error is therefore a random variable depending on random weights we show how one can construct lattice rules which perform well for weights taken from a set with large measure such lattice rules are therefore robust with respect to certain changes in the weights the construction algorithm uses the component by component cbc idea based on two criteria one using the mean of the worst case error and the second criterion using a bound on the variance of the worst case error we call the new algorithm the cbc2c component by component with 2 constraints algorithm we also study a generalized version which uses r constraints which we call the cbcrc component by component with r constraints algorithm we show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights 1 r thus in applications instead of finding one set of weights it is enough to find a convex polytope in which the optimal weights lie the price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights

new approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces abstract this study considers the solvability of nonlocal problems for fuzzy differential systems under gh differentiability using some linear transforms we convert partial differential equations into fuzzy integral systems before some similar results are obtained for fuzzy wave equations with nonlocal conditions as a consequence we give some further estimates of the convergence rates of approximation schemes to exact solutions the main tool used in our proofs is based on perov s fixed point principle which is applied to vector valued integral operators these operators appear as the sum of two integral operators where one is fredholm type and the other is of a volterra type that depends on the restriction of the domain the novel feature of this study is that we combine the matrix convergent to zero technique with calculations of fuzzy valued functions in generalized fuzzy metric spaces some examples are presented to demonstrate our theoretical results