on the mean field and classical limits of quantum mechanics the main result in this paper is a new inequality bearing on solutions of the n body linear schrodinger equation and of the mean field hartree equation this inequality implies that the mean field limit of the quantum mechanics of n identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation this result applies to the case of c1 1 interaction potentials the quantity measuring the approximation of the n body quantum dynamics by its mean field limit is analogous to the monge kantorovich or wasserstein distance with exponent 2 the inequality satisfied by this quantity is reminiscent of the work of dobrushin on the mean field limit in classical mechanics func anal appl 13 115 123 1979 our approach to this problem is based on a direct analysis of the n particle liouville equation and avoids using techniques based on the bbgky hierarchy or on second quantization

from the hartree dynamics to the vlasov equation we consider the evolution of quasi free states describing n fermions in the mean field limit as governed by the nonlinear hartree equation in the limit of large n we study the convergence towards the classical vlasov equation for a class of regular interaction potentials we establish precise bounds on the 0rate of convergence

h reinforcement learning control of robot manipulators using fuzzy wavelet networks in this paper an h reinforcement learning controller based on a fuzzy wavelet network fwn is proposed to perform a position tracking task for a robot manipulator the proposed controller adopts the actor critic reinforcement learning control scheme the primary reinforcement is generated by a performance measurement unit the learning unit of the controller consists of an associative search network asn and an adaptive critic network acn the asn is employed to approximate unknown nonlinear functions in the robot dynamics and the acn is utilized to construct a more informative signal than the primary reinforcement alone to tune the asn since the fwn can provide accurate function approximation both the asn and acn are implemented by the fwn in addition the proposed controller requires no prior knowledge about the dynamics of the robot manipulators and no off line learning phase moreover by employing the h control theory it is possible to attenuate the effects of the approximation errors of the fwns and external disturbances to a prescribed level in contrast to the general h problem only simple equations rather than riccati equations should be solved computer simulations on a scara robot with 3 degrees of freedom confirm the effectiveness of the fwn based controller with h stabilization

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

the quintic nls as the mean field limit of a boson gas with three body interactions we investigate the dynamics of a boson gas with three body interactions in dimensions d 1 2 we prove that in the limit of infinite particle number the bbgky hierarchy of k particle marginals converges to a limiting gross pitaevskii gp hierarchy for which we prove existence and uniqueness of solutions factorized solutions of the gp hierarchy are shown to be determined by solutions of a quintic nonlinear schrodinger equation our proof is based on and extends methods of erdos schlein yau klainerman machedon and kirkpatrick schlein staffilani

explicit periodic solitary wave solutions for the 2 1 dimensional boussinesq equation in this paper 2 1 dimensional boussinesq equation is investigated by using homoclinic test method with the aid of maple new explicit periodic solitary wave solutions are obtained moreover mechanical feature of wave is exhibited

a doob h transform of the gross pitaevskii hamiltonian we perform a doob h transformation of the n body hamiltonian for n trapped interacting bose particles starting from the unique real strictly positive ground state this identifies the unitarily equivalent markov generator describing the n interacting diffusion processes by applying the convergence results coming from the gross pitaevskii gp scaling limit to the energy form associated with this markov generator we obtain the asymptotic generator uniquely corresponding to the gp hamiltonian which has been rigorously derived from the n body hamiltonian by performing the same scaling limit our asymptotic generator is a non linear diffusion generator with a killing rate governed by the gp order parameter the associated stochastic process properly describes the single particle motion in the bose einstein condensate including the self interaction which acts here as a probability density dependent killing rate finally in a one dimensional setting we discuss under which conditions the process conditioned to survival converges to a quasi stationary distribution having density with respect to the speed measure given by the principal eigenfunction of the diffusion generator with killing

a reinforcement learning adaptive fuzzy controller for robots in this paper a new reinforcement learning scheme is developed for a class of serial link robot arms traditional reinforcement learning is the problem faced by an agent that must learn behavior through trial and error interactions with a dynamic environment in the proposed reinforcement learning scheme an agent is employed to collect signals from a fixed gain controller an adaptive critic element and a fuzzy action generating element the action generating element is a fuzzy approximator with a set of tunable parameters and the performance measurement mechanism sends an error metric to the adaptive critic element for generating and transferring a reinforcement learning signal to the agent moreover a tuning algorithm of the proposed scheme that can guarantee both tracking performance and stability is derived from the lyapunov stability theory therefore a combination of adaptive fuzzy control and reinforcement learning scheme is also concerned with algorithms for eliminating a sequence of decisions from experience simulations of the proposed reinforcement adaptive fuzzy control scheme on the cart pole balancing problem and a two degree of freedom 2dof manipulator scara robot arm verify the effectiveness of our approach

on the extended applications of homogenous balance method in this paper we study the travelling wave reductions for certain 2 1 and 3 1 dimensional physically important nonlinear evolutionary equations by using the recently proposed homogenous balance method hbm through this analysis we explore certain new solutions for the equations we have studied

new families of exact solutions to the integrable dispersive long wave equations in 2 1 dimensional spaces the integrable dispersive long wave equations especially the higher dimensional ones are of current interest in both physics and mathematics obtained in this paper via a symbolic computation based method are new families of exact solutions to the 2 1 dimensional integrable dispersive long wave equations sample solutions from those families are presented solitary waves are merely a special case in one family

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

new soliton like solutions to the potential kadomstev petviashvili pkp equation in this paper new soliton like solutions are obtained for 2 1 dimensional potential kadomstev petviashvili pkp equation by using the symbolic computation method developed by gao and tian solitary wave solutions obtained in appl math comput 123 2001 29 are merely a special case in this paper the method can also be extended to other types of nolionear evolution equations in mathematical physics

simulation based optimization of markov reward processes this paper proposes a simulation based algorithm for optimizing the average reward in a finite state markov reward process that depends on a set of parameters as a special case the method applies to markov decision processes where optimization takes place within a parametrized set of policies the algorithm relies on the regenerative structure of finite state markov processes involves the simulation of a single sample path and can be implemented online a convergence result with probability 1 is provided

rigorous derivation of the cubic nls in dimension one we derive rigorously the one dimensional cubic nonlinear schrodinger equation from a many body quantum dynamics the interaction potential is rescaled through a weak coupling limit together with a short range one we start from a factorized initial state and prove propagation of chaos with the usual two step procedure in the former step convergence of the solution of the bbgky hierarchy associated to the many body quantum system to a solution of the bbgky hierarchy obtained from the cubic nls by factorization is proven in the latter we show the uniqueness for the solution of the infinite bbgky hierarchy

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

uniqueness of solutions to the 3d quintic gross pitaevskii hierarchy abstract in this paper we study solutions to the three dimensional quintic gross pitaevskii hierarchy we prove unconditional uniqueness among all small solutions in the critical space h 1 which corresponds to h 1 on the nls level with slight modifications to the proof we also prove unconditional uniqueness of solutions to the hartree hierarchy without a smallness condition our proof uses the quantum de finetti theorem and is an extension of the work by chen hainzl pavlovic and seiringer 9 and our previous work 32

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

a central limit theorem in many body quantum dynamics we study the many body quantum evolution of bosonic systems in the mean field limit the dynamics is known to be well approximated by the hartree equation so far the available results have the form of a law of large numbers in this paper we go one step further and we show that the fluctuations around the hartree evolution satisfy a central limit theorem interestingly the variance of the limiting gaussian distribution is determined by a time dependent bogoliubov transformation describing the dynamics of initial coherent states in a fock space representation of the system

systems control with generalized probabilistic fuzzy reinforcement learning reinforcement learning rl is a valuable learning method when the systems require a selection of control actions whose consequences emerge over long periods for which input output data are not available in most combinations of fuzzy systems and rl the environment is considered to be deterministic in many problems however the consequence of an action may be uncertain or stochastic in nature in this paper we propose a novel rl approach to combine the universal function approximation capability of fuzzy systems with consideration of probability distributions over possible consequences of an action the proposed generalized probabilistic fuzzy rl gpfrl method is a modified version of the actor critic ac learning architecture the learning is enhanced by the introduction of a probability measure into the learning structure where an incremental gradient descent weight updating algorithm provides convergence our results show that the proposed approach is robust under probabilistic uncertainty while also having an enhanced learning speed and good overall performance

rigorous derivation of the gross pitaevskii equation with a large interaction potential consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 hn n 0i cn and that the one particle density matrix converges to a projection as n then we prove that the k particle density matrices of n t factorize in the limit n moreover the one particle orbital wave function solves the time dependent gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant proportional to the scattering length of the potential v in 12 we proved the same statement under the condition that the interaction potential v is sufficiently small in the present work we develop a new approach that requires no restriction on the size of the potential

quantitative derivation of the gross pitaevskii equation starting from first principle many body quantum dynamics we show that the dynamics of bose einstein condensates can be approximated by the time dependent nonlinear gross pitaevskii equation giving a bound on the rate of the convergence initial data are constructed on the bosonic fock space applying an appropriate bogoliubov transformation on a coherent state with expected number of particles mathit n the bogoliubov transformation plays a crucial role it produces the correct microscopic correlations among the particles our analysis shows that on the level of the one particle reduced density the form of the initial data is preserved by the many body evolution up to a small error that vanishes as mathit n 1 2 in the limit of large mathit n

robust adaptive critic control of nonlinear systems using fuzzy basis function networks an lmi approach abstract this paper proposes an adaptive critic tracking control design for a class of nonlinear systems using fuzzy basis function networks fbfns the key component of the adaptive critic controller is the fbfn which implements an associative learning network aln to approximate unknown nonlinear system functions and an adaptive critic network acn to generate the internal reinforcement learning signal to tune the aln another important component the reinforcement learning signal generator requires the solution of a linear matrix inequality lmi which should also be satisfied to ensure stability furthermore the robust control technique can easily reject the effects of the approximation errors of the fbfn and external disturbances unlike traditional adaptive critic controllers that learn from trial and error interactions the proposed on line tuning algorithm for aln and acn is derived from lyapunov theory thereby significantly shortening the learning time simulation results of a cart pole system demonstrate the effectiveness of the proposed fbfn based adaptive critic controller

mean field evolution of fermionic mixed states in this paper we study the dynamics of fermionic mixed states in the mean field regime we consider initial states that are close to quasi free states and prove that under suitable assumptions on the initial data and on the many body interaction the quantum evolution of such initial data is well approximated by a suitable quasi free state in particular we prove that the evolution of the reduced one particle density matrix converges as the number of particles goes to infinity to the solution of the time dependent hartree fock equation our result holds for all times and gives effective estimates on the rate of convergence of the many body dynamics towards the hartree fock evolution 2015 wiley periodicals inc

exact solutions of the two dimensional burgers equation soliton like solutions of the two dimensional burgers equation are obtained some new solutions from soliton like solutions are also presented

adaptive fuzzy controller application to the control of the temperature of a dynamic room in real time this paper presents a direct adaptive fuzzy controller for unknown monotonic nonlinear systems thus not requiring the system model but only a little information about it the plant monotonicity and its delay without any off line pre training the algorithm achieves very high control performance through a three stage algorithm 1 output scale factor 2 adaptation of the fuzzy rule consequents and 3 optimization of the position of the membership functions the design is simple in the sense that both the membership functions and the rule base can be initialized from arbitrary values it can be applied to a large class of monotonic dynamic or static plants due the fact that the system is able to modify its behaviour in real time i e during the control process

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

existence of dynamics for a 1d nls equation perturbed with a generalized point defect in the present paper we study the well posedness for the one dimensional cubic nls perturbed by a generic point interaction point interactions are described as the 4 parameter family of self adjoint extensions of the symmetric 1d laplacian defined on the regular functions vanishing at a point and in the present context can be interpreted as localized defects interacting with the nls field a previously treated special case is given by an nls equation with a defect which we generalize and extend as far as well posedness is concerned to the whole family of point interactions we prove existence and uniqueness of the local cauchy problem in strong form initial data and evolution in the operator domain of point interactions weak form initial data and evolution in the form domain of point interactions and conservation laws of mass and energy are proved for finite energy weak solutions of the problem which imply global existence of the dynamics a technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions to overcome it a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow

mean field dynamics singular potentials and rate of convergence we consider the time evolution of a system of n identical bosons whose interaction potential is rescaled by n 1 we choose the initial wave function to describe a condensate in which all particles are in the same one particle state it is well known that in the mean field limit n the quantum n body dynamics is governed by the nonlinear hartree equation using a nonperturbative method we extend previous results on the mean field limit in two directions first we allow a large class of singular interaction potentials as well as strong possibly time dependent external potentials second we derive bounds on the rate of convergence of the quantum n body dynamics to the hartree dynamics

approximate dynamic programming with a fuzzy parameterization dynamic programming dp is a powerful paradigm for general nonlinear optimal control computing exact dp solutions is in general only possible when the process states and the control actions take values in a small discrete set in practice it is necessary to approximate the solutions therefore we propose an algorithm for approximate dp that relies on a fuzzy partition of the state space and on a discretization of the action space this fuzzy q iteration algorithm works for deterministic processes under the discounted return criterion we prove that fuzzy q iteration asymptotically converges to a solution that lies within a bound of the optimal solution a bound on the suboptimality of the solution obtained in a finite number of iterations is also derived under continuity assumptions on the dynamics and on the reward function we show that fuzzy q iteration is consistent i e that it asymptotically obtains the optimal solution as the approximation accuracy increases these properties hold both when the parameters of the approximator are updated in a synchronous fashion and when they are updated asynchronously the asynchronous algorithm is proven to converge at least as fast as the synchronous one the performance of fuzzy q iteration is illustrated in a two link manipulator control problem

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

new solitonic solutions to a 3 1 dimensional jimbo miwa equation abstract performing computerized symbolic computation we are able to present new analytic solutions of a 3 1 dimensional jimbo miwa equation which does not pass any of the conventional integrability tests applying the generalized tanh method we have found such exact solutions as both the nontraveling solitonic and traveling solitary waves

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

gross pitaevskii equation as the mean field limit of weakly coupled bosons we consider the dynamics of n boson systems interacting through a pair potential n 1va xi xj where va x a 3v x a we denote the solution to the n particle schrodinger equation by n t recall that the gross pitaevskii gp equation is a nonlinear schrodinger equation and the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices open image in new window solves the gp hierarchy under the assumption that a n for 0 3 5 we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy with the coupling constant in the nonlinear term of the gp equation given by v x dx the uniqueness of the solutions of this hierarchy remains an open question

adaptive control of robot manipulators using fuzzy logic systems under actuator constraints in this paper a stable fuzzy adaptive controller for trajectory tracking is developed for robot manipulators without velocity measurements taking into account the actuator constraints the controller is based on structural knowledge of the dynamics of the robot and measurements of link positions only the gravity torque including system uncertainty like payload variation etc is estimated by a fuzzy logic system fls the adaptive controller represents an amalgamation of a filtering technique to eliminate velocity measurements and the theory of function approximation using fls to estimate the gravity torque the proposed controller ensures the local asymptotic stability and the convergence of the position error to zero the proposed controller is robust not only to structured uncertainty such as payload parameter variation but also to unstructured one such as disturbances the validity of the control scheme is shown by simulations on a two link robot manipulator

the decomposition method for solving 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation we study the solitary wave solutions of the 2 1 dimensional boussinesq equation u t t u x x u y y u 2 x x u x x x x 0 and 3 1 dimensional kp equation u x t 6u x 2 6uu x x u x x x x u y y u z z 0 in this paper by considering the decomposition scheme we first obtain the exact solitary wave solutions of the 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation for the initial conditions without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation