symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

bidirectional approximate reasoning for rule based systems using interval valued fuzzy sets abstract in this paper we extend the work of chen et al fuzzy sets and systems 91 1997 339 353 to present a new method to deal with bidirectional approximate reasoning for rule based systems based on the direction of matching between interval valued fuzzy sets we also use some examples to illustrate the bidirectional approximate reasoning process because the proposed method can perform bidirectional approximate reasoning based on the direction of matching between interval valued fuzzy sets it is more reasonable and more powerful than the one presented in chen et al fuzzy sets and systems 91 1997 339 353 the proposed method can provide a useful way to deal with bidirectional approximate reasoning for rule based systems using interval valued fuzzy sets

bidirectional approximate reasoning based on interval valued fuzzy sets abstract this paper proposes a bidirectional approximate reasoning method based on interval valued fuzzy sets where fuzzy production rules are used for knowledge representation and the fuzzy terms appearing in the fuzzy production rules of a rule based system are represented by interval valued fuzzy sets the proposed method is more flexible than the one presented in the paper by bien and chun ieee trans fuzzy systems 2 1994 177 due to the fact that it allows the fuzzy terms appearing in the fuzzy production rules of a rule based system to be represented by interval valued fuzzy sets rather than general fuzzy sets furthermore because the proposed method requires only simple arithmetic operations and because it allows bidirectional approximate reasoning it can be executed much faster and more flexible than the single input single output approximate reasoning scheme presented in the paper by gorzalczany fuzzy sets and systems 21 1981 10

extended fuzzy logic sets and systems the concepts of sets and approximate reasoning within extended fuzzy logic fle provide a systematic procedure for transforming unprecisiated knowledge into a nonlinear mapping over what we define here as f sets an f set differs from a fuzzy set in that it is associated with the restriction of validity in addition to that of possibility therefore by f set we can simultaneously deal with two different types of uncertainties one that is related to ill known objects represented by incomplete information information with its one or more aspects being imprecise vague partial nonspecific undetermined and another that is related to truth values considering gradualness here we define new concepts of vartheta cuts and alpha vartheta cuts introduce the f extension principle and consider arithmetic computations within fle we then address other aspects of the proposed fle system such as fuzzification and validification operations in input processing stage set conversion and defuzzification in output processing stage and inferencing in fact in this paper we intend to develop fle theoretically and practically from the stands of sets and systems to extend the concept of approximate reasoning as a consequence of this development we assert that considering the validity degree of methods and information can lead to more reasonable and trustworthy results through capturing more uncertainty

inference with a single fuzzy conditional proposition abstract a topic in the area of approximate reasoning that has been extensively researched is the following given a single fuzzy conditional proposition if x then y is b with a and b being fuzzy sets defining particular values of the linguistic variables x and y respectively and given some a a then what can be inferred about the value of y when a us replaced by a in the works of l zadeh j baldwin e hisdal r yager m mizumoto et al different solutions to this problem were proposed though different they all share one common feature they ignore the fact that a single fuzzy conditional proposition represents only one specific manifestation of the causal influence exerted by x on y which is being described by binding the fuzzy set b to the fuzzy set a thus the general pattern of the causal relatinship between the two linguistic variables if forgotten as a result the inference made when the original value of x is replaced by a new one might in some cases contradict the general pattern of the causal relationship the present paper makes an attempt to specify these types of inference with a single fuzzy conditional proposition which do not depend on the general pattern of the causal relationship between x and y it also discusses some associated problems

computationally efficient reasoning using approximated fuzzy intervals in this paper we emphasize the use of sigmoid like membership functions which take values in the open unit interval and propose the membership driven inference mdi reasoning scheme with sigmoid like membership functions one can avoid the so called indetermination part of the conclusion which occur in reasoning with the compositional rule of inference cri moreover the mdi and the min and product based cri are closed under such membership functions the axiomatic properties of the mdi reasoning scheme are shown including not only the generalized modus ponens but also the generalized modus tollens the generalized chain rule and more as a special sigmoid like function we present the so called squashing function by which piecewise linear fuzzy intervals can be arbitrarily approximated we show that by utilizing approximated fuzzy intervals in rules and premises the mdi reasoning scheme can be efficiently calculated only by the parameters that define the fuzzy sets in the rule and the premise

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

applicability of the fuzzy operators in the design of fuzzy logic controllers a study of the different roles played by the fuzzy operators in fuzzy control is developed in this paper the behavior of a very large amount of fuzzy operators is analyzed and a comparison of the accuracy of many fuzzy logic controllers designed by means of these operators is carried out in order to do that a comparison methodology is defined and two fuzzy control applications are selected the inverted pendulum problem and the fuzzy modeling of the real curve y x

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

some considerations about a least squares model for fuzzy rules of inference abstract in this paper the method of least squares is applied to the fuzzy inference rules we begin studying the conditions in which from a fuzzy set we can build another through the method of least squares then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens we present different theorems and examples that demonstrate the fundamental advantages of the method studied we conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective and

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

the effects of membership function on fuzzy reasoning abstract fuzzy modus ponens has been considered a generalization of the classic modus ponens logic inference rule it has been expected to provide a more powerful and useful inference capability in the process of fuzzy reasoning the definitions of membership functions of fuzzy sets used in the reasoning process may however adversely affect the reasoning results if the insignificant parts of the membership functions are incompatible this paper explains these undesirable effects and proposes a method to resolve this problem in addition an algorithm for the compositional rule of inference is developed to perform the fuzzy modus ponens syllogism test results shown in this paper indicate that the algorithm and the underlying proposed methods are reasonable approaches to a robust fuzzy logic reasoning system

fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions in this article three methods are presented to perform the center of gravity cog defuzzification method in the context of linguistic fuzzy models with t norm based inference one well known method the discretisation method and two new methods the slope based method and the modified transformation function method the methods are worked out for trapezoidal membership functions forming a fuzzy partition in the sense of ruspini experimental results show that the newly introduced methods exhibit excellent accuracy at an extremely low computational cost compared to the widely applied discretisation method

numerical methods for the simulation of trapped nonlinear schrodinger systems we propose analyze and compare the efficiency and accuracy of different numerical schemes for the solution of the nonlinear schrodinger equation with a trapping potential in particular we study schemes of finite difference pseudospectral and spectral types for the space discretization together with explicit symplectic multistep split step and standard variable step integrators to solve the time evolution all of these schemes are evaluated comparatively and some recommendations based on their accuracy and computational efficiency are made

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

the relative variance of the transmission function of a reverberation room the transmission function of a reverberation room is defined to be the square of the modulus of the ratio of the reverberant field sound pressure to the volume velocity of the sound source the input impedance of a reverberation room is defined to be the ratio of the near field sound pressure to the volume velocity of the sound source lyon s formula for the relative variance of the transmission function is extended to cover the case when the transmission function is averaged over multiple source and receiver positions this extended formula reduces easily to lyon s formula for the relative variance of the real part of the input impedance and thus demonstrates the relationship which exists between the two formulae the above formulae are derived under the assumption that the modal frequencies are randomly distributed lyon also derived a formula for the relative variance of the real part of the input impedance when the modal frequencies tend to repel each other it is shown that this formula of lyon is in error and the correct formula is derived the formula for the relative variance of the transmission function is modified so that it will reduce to the formula for the relative variance of the real part of the input impedance derived under this assumption of modal frequency repulsion these formulae for the relative variance of the transmission function are compared with experimental values of relative variance there is reasonable agreement between these formulae and the experimental results the above formulae are extended to cover the case when the transmission function or the real part of the input impedance is averaged over a frequency band the formulae derived in this paper are important for predicting the precision of sound power measurements in a reverberation room

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

an investigation into the effect of product approximation in the numerical solution of the cubic nonlinear schro dinger equation abstract we study the effect of product approximation on the galerkin solutions of the one dimensional cubic schrodinger equation a crank nicolson scheme is used to discretize in time the paper describes two numerical experiments in the first we examine the approximation obtained by the standard galerkin method and discuss the possibility of enforcing discrete analogs of the conservation laws satisfied by the exact solution in the second experiment numerical results obtained by the product approximation version of the galerkin method are compared and the effectiveness of the method for different combinations of test and trial functions is also investigated

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

the ensemble variance of random noise in a reverberation room the literature contains conflicting results for the ensemble relative variance of the intensity of a frequency band of random noise which has been passed through a reverberation room some theoretical and experimental results suggest that passing a band of noise through a reverberation room has no effect on its ensemble relative variance other theoretical and experimental results show that the ensemble relative variance is doubled this paper shows that the magnitude of the effect depends on the ratio of the integrating time of the detector to the integrating time of the reverberation room if this ratio is small the reverberation room has no effect if the ratio is large the ensemble relative variance is doubled this explains why conflicting results have appeared in the literature

the effect of generalised force correlations on the response statistics of a harmonically driven random system if the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the gaussian orthogonal ensemble goe of random matrices previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation the derivation of these expressions employed random point process theory and in the theoretical development it was assumed that the modal generalised forces were uncorrelated although this assumption is often valid there are cases in which correlations between the generalised forces can significantly affect the response variance and in the present work the existing theory is extended to include correlations of this type the extended theory is applicable to both single frequency responses and to band average responses and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

handling the nonlinearity of a fuzzy logic controller at the transition between rules abstract a fuzzy logic controller approximates a desired control surface by using the outputs of its fuzzy control rules the shape of this control surface is mainly influenced by the linguistic variables of the rule base the basic operators of the fuzzy sets the implication the inference and the defuzzification method the linearity nonlinearity of this control surface is discussed in many studies however none of those studies has explicitly considered the linearity nonlinearity at the transition between fuzzy logic rules in this paper a simple approach to control the linearity nonlinearity at this transition by a modified center of gravity defuzzification method is proposed namely by introducing the so called defuzzification weights to the overlapping areas of the consequent those defuzzification weights can be determined heuristically or automatically both cases will be demonstrated for the latter a feedforward neural network is employed