quantum monte carlo methods recent developments abstract recent developments of the quantum monte carlo methods are reviewed briefly with emphasis on basic ideas namely the transformation of a d dimensional quantum system to the corresponding d 1 dimensional classical system some recent applications to quantum spin systems are summarized in particular numerical evidence of the haldane gap is given the quantum transfer matrix method is also presented for it is also based on the st transformation the quantum monte carlo cam is explained briefly fractal path integrals are also presented namely a general theory of decomposition of exponential operators and symplectic integrators are explained the negative sign problem is also discussed

product formula methods for time dependent schrodinger problems this paper introduces a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time dependent hermitian operator rigorous upper bounds are derived for two different time ordered approximation schemes and for errors resulting from approximating a time ordered exponential by an ordinary exponential operator the properties and the usefulness of the product formula algorithms are examined by applying them to the problem of zener tunnelling the most efficient algorithm is employed to solve the time dependent schrodinger equation for a wavepacket incident on a time modulated rectangular barrier

the wigner smith matrix in acoustic scattering application to fluid loaded elastic plates the wigner smith matrix q is built up by differentiation of the unitary condition of the scattering matrix s the matrices q and s both contain the same information but with different points of view for structures with simple geometrical shapes such as plates or cavities the acoustic scattering is a two channel scattering represented by a 2 2 s matrix the elements of the q matrix can be described i by means of the phase derivatives of the elements of the matrix s ii by means of the phase derivatives of the eigenvalues of the s matrix the equivalence of these two descriptions allows one to express the phase derivatives of i in terms of the phase derivatives of ii the wigner smith matrix concept enables one to unify and to improve both the phase gradient method and the eigenvalue method in the frame of the multichannel scattering it obviously incorporates the resonance scattering theory approximate resonant formulas and numerical results are given for the case of fluid loaded elastic isotro

order conditions for canonical runge kutta schemes when numerically integrating hamiltonian systems of differential equations it is often advantageous to use canonical methods i e methods that preserve the symplectic structure of the phase space thus reproducing an important feature of the hamiltonian flow an s stage runge kutta rk method without redundant stages is canonical if and only if with a standard notation b i a ij b j a ji b i b j 0 1 leqq i j leqq s it is shown that for canonical rk methods there are many redundancies in the standard order conditions for a canonical method to have order p it is sufficient that the b i s a ij s satisfy a system of algebraicequations that has roughly speaking an equation per nonrooted tree of order leqq p furthermore a new methodology is presented for the investigation of the order of canonical integration methods not necessarily rk methods when applied to hamiltonian systems in the new approach consistency is studied by taylor expanding a suitable scalar function in ter

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

generalization in a multi state neural network the generalization ability of an extremely dilute feedback neural network with multi state neurons is studied by means of a deterministic noiseless parallel dynamics the overlap with any one of a macroscopic number of binary full activity concepts is determined when the network is trained with examples of variable activity according to a hebbian learning algorithm that favours stable symmetric mixture states explicit results about the phase diagram and the generalization error are obtained for a network with three state neurons which remain inactive below a threshold it is shown that the generalization ability can be considerably enhanced either by training the network with low activity examples or by means of a moderate increase in

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

lie algebraic approach to the coupled mode oscillator the coupling of electromagnetic fields for the construction of light amplifiers is described by a certain hamiltonian which can be expressed as the sum of three differential operators following the method of steinberg 1977 and dattoli et al 1990 the solution of schrodinger s equation for the hamiltonian of the system is constructed from the solutions of the corresponding equations for the components of the hamiltonian

fer s factorization as a symplectic integrator in this paper we analyze the main features of the so called fer expansion as a new method for integrating the ordinary differential equations derived from explicitly time dependent hamiltonian dynamical systems this method is based on a factorization of the evolution operator as an infinite product of exponentials of lie operators and thus exactly preserves the poincare integral invariants third and fourth order expansions are considered and numerical results are presented for a quadratic hamiltonian with various time dependent frequencies comparison is done with other numerical integration schemes

solution of linear partial differential equations by lie algebraic methods a new algorithm is proposed for obtaining explicit solutions of the cauchy problem defined by a certain class of partial differential equations pde of parabolic type the algorithm exploits the algebraic structure of the problem to transform the pde into an ordinary matrix differential equation which is then solved by lie algebraic techniques

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

categorization in a layered neural network abstract the layered feedforward network of domany meir and kinzel is extended to investigate the ability to recognize any one of a macroscopic number of concepts ancestors when the network is trained through a finite number of examples descendents of each concept by means of a generalized hebbian learning rule between cells on two consecutive layers learning curves describing the generalization error are obtained as well as phase diagrams and basins of attraction that exhibit the possible coexistence of a generalization phase a retrieval phase of examples and a spin glass phase

categorization ability in a biologically motivated neural network abstract in this work we study the categorization properties of an extreme and asymmetrically diluted version of the hopfield model for associative memory when the effect of the refractory period of the neurons is taken into account in the dynamics of the system the simplest way of modeling these refractory periods is by means of a time dependent threshold that acts only on those neurons that emit a signal and favors them to be at rest during a given time interval the dynamic equations are derived in the limit of extreme dilution using an approach that explicitly preserves the dependence of the system on its whole history the categorization error is analyzed for different values of the parameters in particular we confront our analytical results with numerical simulations for the noiseless case t 0 when the number of examples or their correlations increases the system always categorizes independently of the amplitude of the potential that mimics the effect of the refractory period

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

symmetry classification and exact solutions of the kramers equation the complete classification of lie symmetries for the kramers equation is described group invariance under infinitesimal transformations is used to generate a wide class of exact solutions some of the known solutions are obtained as particular cases the general results can also be used to solve some types of moving boundary problems we researched some q conditional symmetry properties of this equation

solving linear ordinary differential equations by exponentials of iterated commutators a sequence of transformations of a linear system of ordinary differential equations is investigated it is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations the transformations are implemented as a basis of a numerical method order stability and error control of this method are analyzed numerical examples demonstrate the potential of this approach

multi step tangential versus transversal linearizations in non linear dynamics a family of multi step tangential linearization mtnl techniques is developed for numeric analytic and geometric integration of non linear oscillators under deterministic loading a family of multi step transversal linearization mtrl schemes is also explored and a comparison between the two linearization families is provided with a view of bringing out their relative advantages and deficiencies the mtnl methodology essentially works through an application of the concept of mtrl to non linear rate equations different variants of the mtnl are obtained and it turns out that they are specific cases of a generalized mtnl method local error estimates are also provided moreover higher order mtrl and mtnl methods are successfully implemented through chebyshev nodes instead of equidistant nodes this helps to improve accuracy even with larger step sizes a comparison of relative errors through the mtrl and mtnl techniques consistently indicates a superior quality of approximations via the mtrl route whilst the usage of the tangential linearization is very common in non linear computational mechanics it is demonstrated that the transversal version offers in addition to higher numerical accuracy an easier implementation and avoids computation of jacobian matrices this allows the mtrl family to be applicable even to non linear oscillators having c0 vector fields copyright 2007 john wiley sons ltd

existence of short time approximations of any polynomial order for the computation of density matrices by path integral methods in this paper i provide significant mathematical evidence in support of the existence of direct short time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials while for theorem 2 which is experimental i only provide a physicist s proof i believe the present development is mathematically sound as a verification i explicitly construct two short time approximations to the density matrix having convergence orders 3 and 4 respectively furthermore in appendix b i derive the convergence constant for the trapezoidal trotter path integral technique the convergence orders and constants are then verified by numerical simulations while the two short time approximations constructed are of sure interest to physicists and chemists involved in monte carlo path integral simulations the present paper is also aimed at the mathematical community who might find the results interesting and worth exploring i conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real time feynman path integral simulations

lie algebraic solutions of linear fokker planck equations nonstationary solutions of linear fokker planck equations with arbitrary drift and diffusion coefficients are derived using lie algebraic techniques the time evolution operator is given as a product of exponentials of the differential operators appearing in the equations

study of correction terms for higher order decompositions of exponential operators many kinds of higher order decompositions are applied to an exactly solvable model and the trotter number dependence of the correction terms is evaluated systematically practically useful decompositions are explored

categorization in a hopfield network trained with weighted examples i finite number of concepts abstract we consider the categorization problem in a hopfield network with a finite number of concepts and trained with s examples of weight 1 s we find that the retrieval capacity of an example with weight 1 and the corresponding categorization error depends also on the arithmetic mean m 1 s 1 2 s of the other weights for 1 m 1 we find that the line of first order transitions between the retrieval and categorization phases ends at a critical point in the s t plane when two solutions are present the global minimum of the free energy corresponds to the solution with the highest weight

symplectic phase flow approximation for the numerical integration of canonical systems new methods are presented for the numerical integration of ordinary differential equations of the important family of hamiltonian dynamical systems these methods preserve the poincare invariants and therefore mimic relevant qualitative properties of the exact solutions the methods are based on a runge kutta type ansatz for the generating function to realize the integration steps by canonical transformations a fourth order method is given and its implementation is discussed numerical results are presented for the henon heiles system which describes the motion of a star in an axisymmetric galaxy

generalization in a hopfield network with noise the generalization ability of the hopfield model of neural networks trained with examples is studied in mean field theory in the presence of synaptic noise although the latter improves the generalization ability for a finite number of concepts it does not for a macroscopic number of them nevertheless the network performance is still robust against synaptic noise numerical simulations are performed to verify the mean field theory predictions

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

fractal path integrals with applications to quantum many body systems abstract a general formulation of the fractal decomposition of exponential operators is reviewed briefly and its mathematical structure is discussed together with some applications to quantum many body systems more explicitly the operator e x a b is decomposed into a product of the form ex a b f m x o x m 1 with f m x e t 1 xa e f 2 xb e f 3 xa e t 4 xb e t m xb the parameters t j show a fractal structure as shown in the text

on representations of lie algebras for quantized hamiltonians abstract we prove that the lie algebra ls k k sk0 k0 k k where s is a real number k0 is a hermitian diagonal operator and k k has nontrivial matrix representations if and only if s 0

compact exponential product formulas and operator functional derivative a new scheme for deriving compact expressions of the logarithm of the exponential product is proposed and it is applied to several exponential product formulas a generalization of the dynkin specht wever dsw theorem on free lie elements is given and it is used to study the relation between the traditional method based on the dsw theorem and the present new scheme the concept of the operator functional derivative is also proposed and it is applied to ordered exponentials such as time evolution operators for time dependent hamiltonians

evolution of su 2 and su 1 1 states a further mathematical analysis the problem of the evolution of su 2 and su 1 1 states is analyzed from a unified point of view a generalized rabi matrix for a time dependent su 2 su 1 1 coherence preserving hamiltonian is also derived

cayley klein parameters and evolution of two and three level systems and squeezed states in this paper the time behavior of quantum states ruled by hamiltonians linear in the su 2 su 1 1 and su 3 generators in terms of the cayley klein parameters originally introduced in classical mechanics is analyzed also pointed out is the link between the cayley klein parameters and the wei norman ordering functions exploited in the context of the schrodinger representation

generalized bracket formulation of constrained dynamics in phase space a generalized bracket formalism is used to define the phase space flow of constrained systems the generalized bracket naturally subsumes the approach to constrained dynamics given by dirac some time ago the dynamical invariant measure and the linear response of systems subjected to holonomic constraints are explicitly derived in light of previous results it is shown that generalized brackets provide a simple and unified view of the statistical mechanics of non hamiltonian phase space flows with a conserved energy

on the overlapping resonances concept of acoustic transmission through an elastic plate ii numerical examples and physical implications abstract in a companion paper 1 the fiorito madigosky and uberall approximate theory of transmittivity of an infinite isotropic elastic plate in terms of lamb mode resonances 2 is examined and extended by validating coherent addition of these overlapping resonances the present paper provides numerical and heuristic tests which yield convincing proof of the applicability of the overlapping resonances concept at various thicknesses of a water immersed steel plate including the half space and zero thickness limits the behaviour is shown to be compatible with known exact behaviour overlapping resonances explain the reduction towards zero of the transmittivity near the coincidence angles of the zero order lamb modes as these converge to the rayleigh angle contrary to earlier interpretations both zero order modes survive at the half space limit the a 0 mode is shown to contribute below fd critical becoming then highly damped and effectively non propagating the interpretation provided of the transition of the a 0 mode in this region bridges the gap between conventional lamb mode theory and thin plate theory

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

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

singular invariant equation for the 1 1 fokker planck equation semi invariants for the linear parabolic equations with two independent variables time variable t and space variable x and one dependent variable u are derived under the transformation of the independent variables by using the infinitesimal method we also obtain the joint invariant equation for the above mentioned equation under equivalence transformation in fact we prove a necessary and sufficient condition for a 1 1 parabolic equation to be reducible via a local equivalence transformation to the one dimensional classical heat equation this result provides practical criteria for reduction finally examples of 1 1 fokker planck equations from applications are given to verify the results obtained

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

continuous numerical solutions of coupled mixed partial differential systems using fer s factorization in this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time dependent systems of the type ut a t uxx b t u 0 0 0 u 0 t u p t 0 u x 0 f x 06 x6p after truncation of an exact series solution the numerical solution is constructed using fer s factorization given 0 and t0 t1 with 0 t0 t1 and d t0 t1 f x t 06x6p t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in d t0 t1 an algorithm is also included c 1999 elsevier science b v all rights reserved ams classication 65m15 34a50 35c10 35a50

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

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

karlin mcgregor like formula in a simple time inhomogeneous birth death process algebraic discussions are developed to derive transition probabilities for a simple time inhomogeneous birth death process algebraic probability theory and lie algebraic treatments make it easy to treat the time inhomogeneous cases as a result an expression based on the charlier polynomials is obtained which can be considered as an extension of a famous karlin mcgregor representation for a time homogeneous birth death process

lie algebraic discussions for time inhomogeneous linear birth death processes with immigration analytical solutions for time inhomogeneous linear birth death processes with immigration are derived while time inhomogeneous linear birth death processes without immigration have been studied by using a generating function approach the processes with immigration are here analyzed by lie algebraic discussions as a result a restriction for time inhomogeneity of the birth death process is understood from the viewpoint of the finiteness of the dimensionality of the lie algebra

quantum analysis non commutative differential and integral calculi a new scheme of quantum analysis namely a non commutative calculus of operator derivatives and integrals is introduced this treats differentiation of an operator valued function with respect to the relevant operator in a banach space in this new scheme operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly derivatives of hyperoperators are also defined some possible applications of the present calculus to quantum statistical physics are briefly discussed

optimization of extrapolated cayley transform with non hermitian positive definite matrix abstract for the extrapolated cayley transform we give necessary and sufficient conditions for guaranteeing its convergence and contraction in the euclidean norm we derive upper bounds for the convergence and the contraction factors and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated cayley transform respectively

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

convergence of general decompositions of exponential operators a general theorem is proved concerning the convergence of decompositions of exponential operators in a banach space or normed space as a corollary the convergence of fractal decompositions is proved the convergence of generalized trotter like formulas is also shown to result from the general theorem

limits on inferring the past here we define and study the properties of retrodictive inference we derive equations relating retrodiction entropy and thermodynamic entropy and as a special case show that under equilibrium conditions the two are identical we demonstrate relations involving the kullback leibler divergence and retrodiction probability and bound the time rate of change of retrodiction entropy as a specific case we invert various langevin processes inferring the initial condition of n particles given their final positions at some later time we evaluate the retrodiction entropy for langevin dynamics exactly for special cases and find that one s ability to infer the initial state of a system can exhibit two possible qualitative behaviors depending on the potential energy landscape either decreasing indefinitely or asymptotically approaching a fixed value we also study how well we can retrodict points that evolve based on the logistic map we find singular changes in the retrodictivity near bifurcations counterintuitively the transition to chaos is accompanied by maximal retrodictability

high order non unitary split step decomposition of unitary operators we propose a high order numerical decomposition of exponentials of hermitian operators in terms of a product of exponentials of simple terms following an idea which has been pioneered by m suzuki implementing it for complex coefficients we outline a convenient fourth order formula which can be written compactly for an arbitrary number of non commuting terms in the hamiltonian and which is superior to the optimal formula with real coefficients both in complexity and accuracy we show asymptotic stability of our method for a sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models

an interpretation and derivation of the lattice boltzmann method using strang splitting the lattice boltzmann space time discretisation as usually derived from integration along characteristics is shown to correspond to a strang splitting between decoupled streaming and collision steps strang splitting offers a second order accurate approximation to evolution under the combination of two non commuting operators here identified with the streaming and collision terms in the discrete boltzmann partial differential equation strang splitting achieves second order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps and the other operator is applied once for a full timestep we show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second order accurate and explicit scheme from an integration of the discrete boltzmann equation along characteristics this approach extends easily to include general matrix collision operators and also body forces finally we show that the validity of the lattice boltzmann discretisation for grid scale reynolds numbers larger than unity depends crucially on the use of a crank nicolson approximation to discretise the collision operator replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive due to the splitting error unless the grid scale reynolds number remains well below unity

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

lie algebraic methods and solutions of linear partial differential equations in this paper an algebraic method to obtain the solution of linear partial differential equations of the evolution type is discussed the proposed method exploits the lie differential operators and their matrix realization to reduce the equation to an easily solvable generalized matrix form some applications to problems of specific interest are also discussed

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

layered neural networks with non monotonic transfer functions abstract we investigate storage capacity and generalization ability for two types of fully connected layered neural networks with non monotonic transfer functions random patterns are embedded into the networks by a hebbian learning rule one of them is a layered network in which a non monotonic transfer function of even layers is different from that of odd layers the other is a layered network with intra layer connections in which the non monotonic transfer function of inter layer is different from that of intra layer and inter layered neurons and intra layered neurons are updated alternately we derive recursion relations for order parameters for those layered networks by the signal to noise ratio method we clarify that the storage capacity and the generalization ability for those layered networks are enhanced in comparison with those with a conventional monotonic transfer function when non monotonicity of the transfer functions is selected optimally we also point out that some chaotic behavior appears in the order parameters for the layered networks when non monotonicity of the transfer functions increases

new evolution equations for the joint response excitation probability density function of stochastic solutions to first order nonlinear pdes by using functional integral methods we determine new evolution equations satisfied by the joint response excitation probability density function pdf associated with the stochastic solution to first order nonlinear partial differential equations pdes the theory is presented for both fully nonlinear and for quasilinear scalar pdes subject to random boundary conditions random initial conditions or random forcing terms particular applications are discussed for the classical linear and nonlinear advection equations and for the advection reaction equation by using a fourier galerkin spectral method we obtain numerical solutions of the proposed response excitation pdf equations these numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi element probabilistic collocation methods it is found that the response excitation approach yields accurate predictions of the statistical properties of the system in addition it allows to directly ascertain the tails of probabilistic distributions thus facilitating the assessment of rare events and associated risks the computational cost of the response excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the pde is subject to high dimensional random boundary or initial conditions the question of high dimensionality for evolution equations involving multidimensional joint response excitation pdfs is also addressed

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

sixth order lie group integrators abstract in this paper we present the coefficients of several sixth order symplectic integrators of the type developed by r ruth to get these results we fully exploit the connection with lie groups these integrators as well as all the explicit integrators of ruth may be used in any equation where some sort of lie bracket is preserved in fact if the lie operator governing the equation of motion is separable into two solvable parts the ruth integrators can be used

quantum transfer matrix method and thermo quantum dynamics abstract the quantum transfer matrix qtm method is reviewed and thermo quantum dynamics is formulated on the basis of the qtm the free energy of the relevant quantum system e g the one dimensional heisenberg model is expressed only in terms of the maximum eigenvalue max of the qtm even the correlation length t is expressed in terms of the ratio of max over the next largest eigenvalue 2 as t 1 log max 2 this new formulation has the great merit that the thermal average q for any local observable q in the thermodynamic limit is expressed as an expectation value over a temperature dependent state vector in the single conjugate hilbert space in contrast to the usage of the double hilbert space in the thermo field dynamics

magnus and fer expansions for matrix differential equations the convergence problem approximate solutions of matrix linear differential equations by matrix exponentials are considered in particular the convergence issue of magnus and fer expansions is treated upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained the very few previously published bounds are improved bounds to the error of approximate solutions are also reported all results are based just on algebraic manipulations of the recursive relation of the expansion generators

symmetry properties of one and two dimensional fokker planck equations the authors study symmetry properties of some fokker planck fp equations in the one dimensional case when symmetry groups turn out to be six parameter ones this allows us to find changes of variables to reduce such fp equations to the one dimensional heat equation they present seven well known fp equations which are reduced to the heat equation they also study the symmetry and obtain some exact solutions of the kramers equation

convergence of unitary and complex decompositions of exponential operators the convergence of the fractal unitary and complex decompositions of exponential operators is proved for hermitian operators in a banach space

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

numerical integrators based on the magnus expansion for nonlinear dynamical systems abstract explicit numerical integration algorithms up to order four based on the magnus expansion for nonlinear non autonomous ordinary differential equations are presented and tested on problems possessing qualitative very often geometric features that is convenient to preserve under numerical discretization the range of applications covers augmented dynamical systems highly oscillatory problems and nonlinear non autonomous partial differential equations of evolution previously discretized in space

symmetry classification and exact solutions of the one dimensional fokker planck equation with arbitrary coefficients of drift and diffusion symmetry properties of the one dimensional fokker planck equations with arbitrary coefficients of drift and diffusion are investigated it is proved that the group symmetry of these equations can be one two four or six parametric and corresponding criteria are obtained the changes of the variables reducing fokker planck equations to the heat equation and schrodinger equation with certain potentials are determined using the substructure of the invariance algebra of the rayleigh process equation we obtain the differential invariants and construct some classes of the exact solutions of this equation

computation of nonlinear behavior of hamiltonian systems using lie algebraic methods lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general hamiltonian systems a procedure is presented for the computation of nonlinear effects of arbitrarily high degree and explicit formulas are given through effects of degree 5 expected applications include accelerator design charged particle beam and light optics other problems in the general area of nonlinear dynamics and perhaps with suitable modification the area of s matrix expansions in quantum field theory

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

positive definiteness in the numerical solution of riccati differential equations in this work we address the issue of integrating symmetric riccati and lyapunov matrix differential equations in many cases typical in applications the solutions are positive definite matrices our goal is to study when and how this property is maintained for a numerically computed solution there are two classes of solution methods direct and indirect algorithms the first class consists of the schemes resulting from direct discretization of the equations the second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy we show first that using a direct algorithm a one step scheme or a strictly stable multistep scheme explicit or implicit limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite then we show two ways to obtain positive definite higher order approximations by using indirect algorithms the first is to apply a symplectic integrator to an associated hamiltonian system the other uses stepwise linearization

stochastic light cone ctmrg a new dmrg approach to stochastic models we develop a new variant of the recently introduced stochastic transfer matrix dmrg which we call stochastic light cone corner transfer matrix dmrg lctmrg it is a numerical method to compute dynamic properties of one dimensional stochastic processes as suggested by its name the lctmrg is a modification of the corner transfer matrix dmrg adjusted by an additional causality argument as an example two reaction diffusion models the diffusion annihilation process and the branch fusion process are studied and compared with exact data and monte carlo simulations to estimate the capability and accuracy of the new method the number of possible trotter steps of more than 105 shows a considerable improvement on the old stochastic tmrg algorithm

phase space metric for non hamiltonian systems we consider an invariant skew symmetric phase space metric for non hamiltonian systems we say that the metric is an invariant if the metric tensor field is an integral of motion we derive the time dependent skew symmetric phase space metric that satisfies the jacobi identity the example of non hamiltonian systems with linear friction term is considered

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

generalization and chaos in a layered neural network the generalization performance of a multi state and a graded response layered attractor neural network trained with examples of low activity is established exactly for monotonic and non monotonic input output functions complex behaviour is found which goes from fixed point attractors to chaos through a cascade of bifurcations depending on an appropriate threshold or cut off parameter the effect of the irregular behaviour on the generalization curves is explicitly demonstrated and phase diagrams for the recognition ratio of concepts in terms of the threshold cut off exhibit ordered generalization disordered paramagnetic or self sustained activity and chaotic phases

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

long time symplectic integration the example of four vortex motion detailed comparisons are made between long time numerical integration of the motion of four identical point vortices obtained using both a fourth order symplectic integration method of the implicit runge kutta type and a standard fourth order explicit runge kutta scheme we utilize the reduced hamiltonian formulation of the four vortex problem due to aref pomphrey initial conditions which give both fully chaotic and also quasi periodic motions are considered over integration times of order 10 6 10 7 times the characteristic time scale of the evolution the convergence as the integration time step is decreased of the poincare section is investigated when smoothness of the section compared to the converged image and the fractional change in the hamiltonian h are used as diagnostic indicators it is found that the symplectic scheme gives substantially superior performance over the explicit scheme and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution it is concluded that this performance derives from the symplectic rather than the implicit character of the method

lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field the system is assumed to be anisotropic and the magnetic field is applied along z axis a canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators two classes of real and complex quadratic invariants constants of motion are obtained we employ the lie algebraic technique to find the most general solution for the wave function for both real and complex invariants the quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave functions in the coherent states and number states are obtained some discussion related to the advantage of using the quadratic invariants to solve the cauchy problem instead of the direct use of the hamiltonian itself is also given

on the overlapping resonances concept of acoustic transmission through an elastic plate i an examination of properties abstract the fiorito madigosky and uberall approximate representation 1 of the acoustic transmission coefficient of an infinite isotropic elastic plate by resonances associated with lamb modes is examined and clarification of its properties is obtained some theoretical extension is provided which validates coherent addition of these overlapping resonances minor limitations shown are that the theory is inapplicable near grazing incidence and that it gives only a partial solution at incidence beyond the second critical angle when the frequency thickness product is the variable existing overlapping resonances theory is shown not to yield a unique relationship between the incident wave and a set of physical motions within the plate and an approach to overcome this non uniqueness is suggested an equivalence is demonstrated between the 3 db resonance half width of each lamb mode and the imaginary part of the modal wavenumber per unit wavenumber of the fluid and this facilitates the numerical checking of the theory which is carried out in a companion paper 2 and yields convincing proof of the applicability of the approach

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

quantum monte carlo methods and general decomposition theory of exponential operators and symplectic integrators abstract the present paper reviews briefly the basic ideas of the quantum monte carlo methods the general decomposition theory of ordered exponentials and symplectic integrators

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells