feynman path integral for dirac system with analytic potential the cauchy problem for the dirac equation for a particle in the presence of an external electromagnetic field in 3 1 space time dimensions is considered the potential of the external field is supposed to be an analytic function of the first order of growth the local solution in the generalized sense is constructed in the form of rigorously defined feynman path integral in phase space

a unified derivation for fast estimation algorithms by the conjugate direction method abstract fast estimation algorithms inspired by the classical method of levinson have been developed in the areas of time series analysis system identification and signal processing this paper provides a unified derivation for the levinson durbin whittle wiggins robinson fast recursive least squares rls ladder or lattice and fast cholesky algorithms as special cases of the conjugate direction method cdm this gives a novel derivation and interpretation for all these methods

a general orthogonalization technique with applications to time series analysis and signal processing a new orthogonalization technique is presented for computing the qr factorization of a general n x p matrix of full rank p n 2 p the method is based on the use of projections to solve increasingly larger subproblems recursively and has an o np2 operation count for general matrices the technique is readily adaptable to solving linear least squares problems if the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so called lattice algorithm for solving linear prediction problems from this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least squares problems by orthogonalization as opposed to solving the normal equations by fast toeplitz algorithms

a probabilistic approach to semi classical approximations abstract we study in this paper the semi classical expansion of the schrodinger equation using a probabilistic approach based on the wiener measure using almost analytic extensions we exhibit a probabilistic ansatz for the wave function we show that this ansatz approximates very well the wave function in the semi classical regime and gives the semi classical expansion under mild hypothesis on the potential at infinity and no analyticity conditions in this paper the study takes place before the caustics

minimal dynamical systems from pdes using sobolev eigenfunctions abstract a modified karhunen loeve transform is derived using a weighted sobolev norm this approach optimizes the approximation of the higher derivative terms in numerical simulations of partial differential equation and reduces the sizes of the associated system of ordinary differential equations the kuramoto sivashinsky equation is used as an example to demonstrate how a galerkin procedure based on eigenfunctions determined by the sobolev optimality criterion improves the performance of the standard karhunen loeve procedure we present an example of a simulation which required a 19 mode expansion using the k l approach and show how this is reduced to a 7 mode simulation using the sobolev eigenfunctions

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

on the formulation of the feynman path integral through broken line paths we study the formulation of the feynman path integral through broken line paths in non relativistic quantum mechanics this formulation is very familiar to us and well known to be useful but its rigorous meaning is given little except for special cases in the present paper using the ideas in the theory of difference methods and the theory of pseudo differential operators we show rigorously for some class of potentials that this formulation is well defined and that this feynman path integral gives the probability amplitude i e the solution of the schrodinger equation

noncommutative operations on wiener functionals and feynman s operational calculus the authors recently introduced a family att 0 of banach algebras of functionals on wiener space and showed that for every f in at the functional integrals k t f exist and are given by a perturbation expansion which serves to disentangle in the sense of feynman s operational calculus the operator k t f when is purely imaginary k t f is the analytic operator valued feynman integral of f in this paper noncommutative operations and on wiener functionals are defined which enable the authors to provide a precise and rigorous interpretation of certain aspects of feynman s operational calculus for noncommuting operators in particular the authors show that if f at1and g at2then f g at1 t2 and k t1 t2 f g k t1 f k t2 g hence the product of operators which can be disentangled using their recent results can itself be disentangled the noncommutative operations the disentangling algebras at and the functional integrals possess a rich interlocking structure

systolic implementation of the lattice algorithm for least squares linear prediction problems abstract we present a linear systolic array of o n cells that solves the least squares linear prediction problem in time o n via an algorithm based on the so called lattice algorithm the total storage required is o n words i e only a constant number of words are needed at each cell

a probabilistic formula for the quantum n body problem and the non linear schrodinger equation in operator algebra on donne une formule probabiliste simple du propagateur pour le probleme a n corps quantique en utilisant un certain type de processus de collision introduit pour traiter l equation de boltzmann linearisee on applique la formule a l equation de schrodinger non lineaire dans la version de seconde quantification introduite par jordan et klein

modal reduction of pde models by means of snapshot archetypes abstract a new method for constructing low dimensional reduced models of dissipative partial differential equations is proposed the original pde u t f u is projected onto a linear subspace spanned by the so called snapshot archetypes that are selected spatial profiles of u x t the selection rule of the snapshot archetypes characterizes the method two different selection methods are proposed we provide an energetic criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics this approach is tested for several pde systems such as the kuramoto sivashinsky equation the arneodo elezgaray reaction diffusion model and the self ignition dynamics of a coal stockpile the latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations

stochastic interpretation of feynman path integral a class of functions hnf was chosen which are boundary values of holomorphic functions it was shown that if the potential v hnf and the initial wave function hnf then the solution of the schrodinger equation is t hnf the unitary evolution t can be expressed by an integral with respect to the wiener measure to each t hnf is associated a complex markov process qt fulfilling stochastic hamilton equations the time evolution of as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of feynman s sum over trajectories are discussed

a fast but unstable orthogonal triangularization technique for toeplitz matrices abstract d sweet s clever qr decomposition algorithm for toeplitz matrices is considered it requires only o n 2 flops to factor an n n matrix we analyze the algorithm to point out its deficiencies and compare it against two other toeplitz solvers

numerical performances of autoregressive spectrum estimators based on three term recurrences the numerical performance of two fast algorithms for estimating the power spectral density of an autoregressive process is studied the algorithms perform similarly to the burg algorithm but require only two thirds as many multiplications as the most efficient implementation of the burg algorithm this allows the high resolution associated with the burg algorithm to be obtained using many fewer computations one algorithm is the deterministic form of the split lattice algorithm adjoined to the split levinson recursions however its resolution is relatively poor the other algorithm corrects a bias in the first algorithm and has resolution similar to the burg algorithm

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

probabilistic solutions of generalised birth and death equations and applications to field theory the author gives probabilistic solutions to the equations describing non relativistic quantum electrodynamical systems these solutions involve besides the usual diffusion processes also birth and death processes corresponding to the photon number variables some inequalities are stated and in particular bounds to the ground state energy of systems composed of a non relativistic particle interacting with a field are established the result is general and it is applied as an example to the polaron problem

a qr type algorithm for fitting the delta ar model to autocorrelation windowed data a qr algorithm is developed to fit the delta autoregressive dar model of vijayan et al see ieee trans automat cont vol 36 p 314 1991 to autocorrelation windowed sampled data to obtain the dar model parameters one must solve a linear system in the matrix called q sub n unfortunately there is presently no way to obtain good estimates of q sub n the proposed qr type algorithm overcomes this problem by computing the dar model parameters without the need for estimating q sub n directly the qr algorithm proposed is a simple modification of the classical qr algorithm for the conventional ar model due to c p rialan and l l scharf 1986

split burg and covariance lattice algorithms the number of multiplications in split lattice algorithms related to the autocorrelation matrix is reduced by 30 percent in comparison with classical lattice algorithms split version of the burg method bm and the covariance lattice method clm are presented as in previously published split algorithms the number of multiplications in the proposed algorithms is reduced

sliding windows and lattice algorithms for computing qr factors in the least squares theory of linear prediction the authors pose a sequence of linear prediction problems that differ a little from those previously posed the solutions to these problems introduce a family of sliding window techniques into the least squares theory of linear prediction by using these techniques it is possible to perform qr factorization of the toeplitz data matrices that arise in linear prediction the matrix q is an orthogonal version of the data matrix and the matrix r is a cholesky factor of the experimental correlation matrix the qr and cholesky algorithms generate generalized reflection coefficients that may be used in the usual ways for analysis synthesis or classification

schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

the reduction of complex dynamical systems using principal interaction patterns abstract a method of constructing low dimensional nonlinear models capturing the main features of complex dynamical systems with many degrees of freedom is described the system is projected onto a linear subspace spanned by only a few characteristic spatial structures called principal interaction patterns pips the expansion coefficients are assumed to be governed by a nonlinear dynamical system the optimal low dimensional model is determined by identifying spatial modes and interaction coefficients describing their time evolution simultaneously according to a nonlinear variational principle the algorithm is applied to a two dimensional geophysical fluid system on the sphere the models based on principal interaction patterns are compared to models using empirical orthogonal functions eofs as basis functions a pip model using 12 patterns is capable of capturing the long term behaviour of the complete system monitored by second order statistics while in the case of eofs 17 modes are necessary

fast qr factorization of vandermonde matrices abstract we introduce a fast algorithm for computing the qr factors of a complex column vandermonde matrix v the complexity of the algorithm is 5 mn 7 n 2 2 o m where m is the number of rows in v and n is the number of columns we assume that m n the matrices q and r may be computed independently if desired two special cases for the row vandermonde matrix with real elements or unit magnitude elements are also studied and similar results are obtained

archetypal analysis of spatio temporal dynamics abstract a comparison is made between the principal component or karhunen loeve decomposition of two sets of spatio temporal data one numerical the other experimental and a new procedure called archetypal analysis cutler and breiman 1994 archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values archetypes may be more appropriate than kl when the data do not have elliptical distributions and are often well suited to studying regimes in which the system spends a long time near a steady state punctuated with quick excursions to outliers in the data set which may represent intermittency other advantages and disadvantages of each method are discussed

representation formulas of the cauchy problem for hyperbolic systems generalizing dirac system on considere l equation de dirac dans un potentiel electromagnetique externe qui ne depend pas du temps et on donne une version rigoureuse de l integrale de parcours de feynman on considere l energie cinetique comme une perturbation du potentiel et on effectue une transformee de fourier sur des variables d espace de facon que le potentiel electromagnetique devienne un operateur de convolution a valeur matricielle et l energie cinetique devient une multiplication par une fonction a valeur matricielle de p on obtient la solution du probleme de cauchy comme une esperance mathematique sur un processus de poisson construit avec le potentiel

positive definite toeplitz embedding based on the cyclic extension of a data matrix embedding in a block toeplitz matrix is one of the techniques for efficient inversion of arbitrary covariance matrices arising in least squares estimation delosme and morf have shown how the concept of toeplitz distance leads to a positive definite embedding via a mixed form representation of the given matrix in this paper a new method is proposed for the positive definite block toeplitz embedding which has the particular feature of taking the signal itself as a starting point it consists of a suitable cyclic extension and embedding of the given signal in a multichannel stationary signal the main advantage of the method is that given the signal the embedding can be performed by inspection

natural tangent dynamics with recurrent biorthonormalizations a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic chaotic limit sets abstract this article develops a geometric method referred to as natural tangent dynamics with recurrent biorthonormalizations ntdrb that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold associated with an equilibrium point or exhibiting nontrivial limit sets limit cycles and chaotic attractors for the first class of systems the main focus relies on the identification of the slow fast timescales while for the latter dynamics within nontrivial attractors the central issue is the temporal dichotomy between unstable stable components as described by the structure of the lyapunov spectrum the method is based on vector and covector dynamics in the tangent and cotangent bundles induced by the phase space flow and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable slow subspaces the ntdrb method provides the most convenient and natural vector and covector bases for i the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold ii the identification of the stable and unstable components of the dynamics on a limit set iii the construction of a low dimensional reduced model reproducing the asymptotic behavior of the original systems

the feynman measure as a limit of complex measures abstract the fundamental solution of the schrodinger equation for a free particle is a distribution such distribution is approximated by a sequence of smooth functions for each one of these functions it is defined a complex measure on the space of paths for certain test functions there exists the limit of integrals of a test function with respect to the complex measures we define the feynman measure of one such function by this limit

probabilistic representation formulas for the time evolution of quantum systemsa by using a new procedure representation formulas are obtained for the solution of the cauchy problem for hyperbolic systems generalizing schrodinger and dirac equations this procedure makes explicit the relation of the underlying poisson jump process with the perturbation series it allows one to find out a path integral representation formula for the two dimensional dirac equation further it can be applied in the four dimensional case with a coulombian potential this approach is also used to describe the time evolution of dissipative two level systems and estimate transition probabilities

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

fixed point error analysis of the lattice and the schur algorithms for the autocorrelation method of linear prediction levinson recursions schur recursions and lattice recursions provide three different ways of computing reflection coefficients for a stationary time series of these three approaches only the schur and lattice algorithms are scaled algorithms that can be implemented in fixed point arithmetic the authors study the finite arithmetic properties of the schur and lattice recursions they derive error variances for the reflection coefficients and present experimental results which agree very closely with the analytical results they show analytically and experimentally that lattice recursions have a significant accuracy advantage over schur recursions when the problem is ill conditioned or equivalently when the absolute values of the reflection coefficients are close to one this result is not surprising since the lattice recursions which compute reflection coefficients by qr factoring a toeplitz data matrix are square root recursions with respect to schur recursions which compute reflection coefficients by cholesky factoring a toeplitz correlation matrix that is quadratic in the data

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

two composition methods for solving certain systems of linear equations this note is concerned with the following problem given a systema x b of linear equations and knowing that certains of its subsystemsa 1 x 1 b 1 a m x m b m can be solved uniquely what can be said about the regularity ofa and how to find the solutionx fromx 1 x m this question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively 7 13 15 in dieser arbeit wird die folgende fragestellung behandelt wenn man wei da gewisse teilsystemea 1 x 1 b 1 a m x m b m eines linearen gleichungssystemsa x b eindeutig losbar sind was kann man uber die losbarkeit des systemsa x b sagen und wie kann man gegebenenfallsx ausx 1 x m berechnen die losung dieses problems erlaubt die rekursive berechnung gewisser linearer oder quasilinearer folgentransformationen 7 13 15

a class of least squares filtering and identification algorithms with systolic array architectures a unified approach is presented for deriving a large class of new and previously known time and order recursive least squares algorithms with systolic array architectures suitable for high throughput rate and vlsi implementations of space time filtering and system identification problems the geometrical derivation given is unique in that no assumption is made concerning the rank of the sample data correlation matrix this method utilizes and extends the concept of oblique projections as used previously in the derivations of the least squares lattice algorithms both the growing and sliding memory exponentially weighted least squares criteria are considered

hybrid algorithm for fast toeplitz orthogonalization new techniques for fast toeplitz qr decomposition are presented the methods are based on the shift invariance property of a toeplitz matrix the numerical properties of the algorithms are discussed and some comparisons are made with two other fast toeplitz orthogonalization methods

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

a rigorous real time feynman path integral using improper riemann integrals we will formulate a rigorous version of the real time time sliced feynman path integral for the l2 transition probability amplitude we will do this for nonvector potential hamiltonians with potential which has at most a finite number of discontinuities and singularities we will also provide a nonstandard analysis version of our formulation