Hermite Polynomial Expansion Research Articles

Configurational studies on amylose. II. Moments and distribution functions

AbstractStatistical mechanical averages of vectors and tensors characterizing the configuration of amylose chains have been calculated. These quantities are expressed in the reference frame affixed to the first glucose unit, the x axis being situated along the O(4)—O(1) virtual bond vector, and the Y axis in the plane of the virtual bond and the O(4)—C(4) bond. The persistence vector a as defined by the average of the end‐to‐end vector r converges slowly to the limiting persistence vector a∞ with increasing chain length. Configurational averages of the Cartesian tensors formed from the displacement vector ρ = r − a have been computed up to seventh rank according to the generator matrix method. The density distribution functions Wα (ρ) evaluated for xu = 40 by using the three‐dimensional Hermite polynomial expansion truncated at the term involving the tensor of seventh rank are approximately cylindrically symmetric about one of the principal axes of the second‐moment tensor <ρ×2. The density distribution function Wα(ρ) is slightly asymmetric even for xu = 80.

A solution-scheme for the convective-diffusion equation

This paper presents a versatile solution-scheme for the convective-diffusion equation. A small-time, asymptotic, solution for an instantaneous point source of scalar contaminant is expressed as a three-dimensional, Hermite polynomial expansion and manipulated, using superposition, to generate the contaminant concentration field that results at larger times and for arbitrary, continuous or instantaneous, source contaminant distribution. This equation is commonly used to model contaminant dispersion in complex environmental flows so that the considerable degree of generality, flexibility and efficiency of this solution-scheme highly commends it to this application. The off-diagonal terms in the diffusivity tensor and the non-zero gradient of this term and the mean-velocity field are shown to make a significant contribution to the evolution of the contaminant concentration field resulting from the instantaneous release of contaminant from a point source.

Solution of the Fokker Plank-Kolmogorov Equation by Using The Maximum Entropy Principle

Abstract The state probability density p(x, t) of a stochastic markovian system is generally given as the solution of the so-called Fokker–Plank-Kolmogorov equation, but at date, the explicit form of the solution of this equation is not known, and only numerical approximations are available, mainly that one which involves Hermite polynomial expansions. In the present paper an alternative is proposed which uses the “maximum entropy principle”. The system is described by the two first moments of its transition probability therefore one can obtain the time derivatives of the moments of its states probability. There is a complete equivalence between these derivatives and the F-P-K equation; and then p(x, t) is determined by applying the maximum entropy principle subject to the knowledge of the derivatives of the first moments. The method is described for a one-dimensional system, but it can be generalized to the vector case, and it could be of interest in other problems as non-linear filtering, for instance. Suggestions for further research are given.

An integro-differential formula on the Wiener kernels and its application to sandwich system identification

It is well known that a nonlinear system with a white Gaussian noise input can be characterized in terms of kernels using the celebrated Wiener theory. In a practical use of the method, however, one may encounter difficulty in obtaining higher order kernels except for the first few because of, for instance, the excessive computational requirement. In this paper, we give an integro-differential formula on the kernels and as its application, an algorithm to identify a cascade system of a linear, a memoryless nonlinear, and linear subsystems, which we call a sandwich system as a whole. According to the formula, kernels up to the second order for different power levels of the input noise are required to identify the subsystems. Impulse response functions of the two linear subsystems can be obtained under appropriate normalization conditions, while the nonlinear subsystem is estimated in the form of a truncated Hermite polynomial expansion. As illustrated examples, two such systems are identified using the algorithm.

An analytical method for reflected flux density calculations

Conception, evaluation and real time control of solar “power tower” systems require the use of fast and accurate computer programs for calculating the flux density distributions on the receiver. Since the classical methods of “cone optics” and “hermite polynomial expansion” have some limitations of speed and accuracy, we have built an analytical model for calculating the convolution of the solar brightness distribution with the principal image of a heliostat (i.e. the fictive image for a “point sun”). We first characterize a principal image of a focusing heliostat by its shape and its geometrical concentration factor. Then this image is projected back onto the central plane (which passes through the center of the mirror), and considered as a flat reflecting surface. And the problem is reduced to density calculation for a flat heliostat. For each point of the receiver, the density of flux reflected by a heliostat is obtained by direct resolution of a convolution integral. The different formulations used to express the density function correspond to the various types of intersections between the image of the solar disk for the considered point and the principal image of the heliostat. Confrontation of this method with a program based on “cone optics” shows a good concordance of results and a strong decrease of computation time. We want to apply this method to the existing “THEMIS” solar plant built in France and to compare our results with real observations. Our density calculation programs will help conceiving fields of focusing heliostats for a new generation of power systems (gaz turbine systems).

Representation of breakthrough curves for fixed-bed adsorbers and reactors using moments of the impulse response

An approximate expression for breakthrough curves (step responses) in isothermal packed-bed adsorbers and reactors is obtained by using the first two moments of the impulse response in a leading term of a Laguerre polynomial series expansion. The approximation is expressed in terms of an incomplete gamma function which can readily be obtained either by Romberg integration or from tabulated values. The approximation agrees well with analytical solutions for both long and short packed-beds. It is shown that the leading term of the Laguerre series is a better approximation to analytical solutions for short beds than the Hermite polynomial expansion and is at least equal to the latter in case of long columns.

On a general storage problem and its approximating solution

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn+1(y) = ∫ l(y, w)gn(w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

On a general storage problem and its approximating solution

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn +1(y) = ∫ l(y, w)gn (w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

Neutral particle transport in plasmas

A kinetic theory description of neutral hydrogen transport in a hydrogen plasma halfspace is discussed. It is shown that singular eigenfunctions can be obtained for a large class of plasma temperature profiles. A singular integral equation is obtained for the surface neutral distribution, and the F-N collocation scheme is used as a solution method for this equation. An approximate solution method of the kinetic equation is given in terms of a truncated Hermite polynomial expansion in the velocity variable.

Adaptive estimation procedures for multi-parameter Monte Carlo computations

This paper describes further developments of a Monte Carlo variance reduction technique which was originally devised by Chorin in 1971. The basic idea of expanding the estimator using orthonormal Hermite polynomials as basis functions has now been extended by the authors to include: (i) multi-parameters, (ii) symmetrized estimators of increased efficiency, and (iii) an adaptive series selection algorithm so that only those coefficients determined with sufficient precision are retained in the Hermite polynomial expansion of the estimator. The so-called Tri-Symmetric Chorin (TSC) estimator is described in detail and computational results obtained using the adaptive algorithm are presented for 6- and 12-parameter problems.

Splitting schemes for the numerical solution of a two-dimensional Vlasov equation

Splitting schemes are applied to the numerical solution of a two-dimensional Vlasov equation. Results obtained when solving the equation in configuration space, by treating the convective term and the acceleration term separately, are compared with results previously obtained using a different method where the two-dimensional Vlasov equation was transformed in velocity space using Hermite polynomials expansion.

Numerical solution of a two-dimensional Vlasov equation

By means of a Hermite polynomials expansion in velocity space of the distribution function, the two-dimensional Vlasov equation is solved numerically by applying a multistep technique. We first study the case of a two-dimensional strongly nonlinear Landau damping, where a large amplitude oscillation is initially applied to the system. A second case is considered where an initially stable oscillation is applied to two-dimensional counter-streaming plasmas. The time evolution of these systems is studied and the performance of the numerical code is analyzed.

Statistical mechanics of helical wormlike chains. III. Scattering functions

The scattering function for the helical wormlike chain is evaluated by the use of the Hermite polynomial expansion of the distribution function, and its general behavior is studied. The convergence of this series is first discussed; upper and lower Tchebychev bounds for the exact scattering function are also examined. Recalling that the model may be regarded as a hybrid of the three extreme forms of rod, random coil, and regular helix, the results are then compared with the scattering functions for these forms. Thus, it is understood that the scattering function for the helical wormlike chain exhibits first coil-like and then helical damped-oscillatory or rodlike behavior as the scattering angle is increased. A comparison of theory with experiment is made for the small-angle scattering of x-rays and neutrons by tactic and atactic poly(methylmethacrylate) chains.

Forecasting Transformed Series

Summary Suppose that a forecasting model is available for the process Xt but that interest centres on the instantaneous transformation Yt = T(Xt). On the assumption that Xt is Gaussian and stationary, or can be reduced to stationarity by differencing, this paper examines the autocovariance structure of and methods for forecasting the transformed series. The development employs the Hermite polynomial expansion, thus allowing results to be derived for a very general class of instantaneous transformations.

Hydrodynamic equations for anisotropic plasmas in magnetic fields. II. Transport equations including collisions

For Pt.I see abstr. A27985 of 1969. A set of hydrodynamic equations is derived for an arbitrary anisotropic plasma allowing for Coulomb collisions among plasma particles. Using a generalized expansion in Hermite polynomials for the distribution functions of plasma constituents collision integrals for electrical conductivity, viscosity, and heat conductances of parallel and perpendicular thermal energy are calculated. The latter are given only for an electron-ion plasma; the general result for arbitrary mass-ratio of the plasma constituents is given elsewhere (Chodura et al., 1970). Finally explicit results for the coefficients of ion-viscosity and heat conductivity of electrons and ions are derived.

Summability of Hermite polynomial expansions of generalized functions

1. Introduction: The well-known result that every function f(x) ∈ L2(–∞, ∞) can be expanded in L2(– ∞,∞) by where and was recently followed by an expansion theorem for generalized functions. In papers by Wildlund(8), Giertz(4) and Zemanian(9) it was shown that a tempered distribution f∈S′ can be expanded by Hermite functions that isfor all ψ∈S (the space of rapidly decreasing test functions).

A Note on Gaussian Measures in a Banach Space

Previous article Next article A Note on Gaussian Measures in a Banach SpaceA. V. SkorokhodA. V. Skorokhodhttps://doi.org/10.1137/1115052PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"A Note on Gaussian Measures in a Banach Space." Theory of Probability & Its Applications, 15(3), p. 508 Previous article Next article FiguresRelatedReferencesCited ByDetails Wiener–Hermite polynomial expansion for multivariate Gaussian probability measuresJournal of Mathematical Analysis and Applications, Vol. 454, No. 1 | 1 Oct 2017 Cross Ref On a majorization inequality for sums of independent random vectorsStatistics & Probability Letters, Vol. 19, No. 2 | 1 Jan 1994 Cross Ref Remarks on Inequalities for Large Deviation ProbabilitiesI. F. Pinelis and A. I. SakhanenkoTheory of Probability & Its Applications, Vol. 30, No. 1 | 28 July 2006AbstractPDF (539 KB)Summary of Reports Presented at Sessions of the Seminar in the Theory of Probability and Mathematical Statistics at the Kiev Shevchenko State University 1981Theory of Probability & Its Applications, Vol. 27, No. 2 | 17 July 2006AbstractPDF (671 KB)Grandes deviations et applicationsEcole d’Eté de Probabilités de Saint-Flour VIII-1978 | 3 October 2006 Cross Ref Gaussian characterizations of certain Banach spacesJournal of Multivariate Analysis, Vol. 7, No. 1 | 1 Mar 1977 Cross Ref On the Boundedness of Stable ProcessesTransactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the 1974 European Meeting of Statisticians | 1 Jan 1977 Cross Ref The Density of the Distribution of the Maximum of a Gaussian ProcessV. S. Tsirel’sonTheory of Probability & Its Applications, Vol. 20, No. 4 | 17 July 2006AbstractPDF (914 KB)Asymptotic evaluation of certain Markov process expectations for large time—IIICommunications on Pure and Applied Mathematics, Vol. 29, No. 4 | 1 Jul 1976 Cross Ref Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (February–October 1974)Theory of Probability & Its Applications, Vol. 20, No. 2 | 17 July 2006AbstractPDF (1740 KB)Mappings of Gaussian Cylindrical Measures in Banach SpacesW. Linde and A. PietschTheory of Probability & Its Applications, Vol. 19, No. 3 | 28 July 2006AbstractPDF (1075 KB)On the Lévy Inequality for Random Variables in a Banach SpaceV. V. BuldyginTheory of Probability & Its Applications, Vol. 19, No. 1 | 28 July 2006AbstractPDF (409 KB)On Gaussian measures in certain locally convex spacesJournal of Multivariate Analysis, Vol. 2, No. 3 | 1 Sep 1972 Cross Ref Volume 15, Issue 3| 1970Theory of Probability & Its Applications359-556 History Submitted:16 December 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1115052Article page range:pp. 508-508ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Contribution to the theory of chromatography : Linear non-equilibrium elution chromatography

The model of elution chromatography is solved using a Hermite polynomials expansion. The column is infinitely long and filled along the whole length, the detector being placed at a distance L from the feeding point. The following factors were considered: longitudinal diffusion in the mobile phase, finite rate of mass transfer through the surface of porous grains, radial diffusion inside the grains and finite rate of sorption on the internal surface of the grains. Uniform shape of the grains and uniform distribution of the filling in the column are assumed. The effect of the eddy diffusion may be included in the longitudinal diffusion coefficient. The effect of the pressure gradient and non-uniform concentration in the cross-section of the column and along the boundary of individual grains is neglected. A method of describing chromatographic peaks by means of statistical moments was used and the central moments up to the fifth order were calculated. The first moment is of basic significance for the determination of retention time. It is shown that it depends only on the partition coefficients and on the longitudinal diffusion, while it is not affected by transport phenomena in the grain and across the surface film or by the shape and size of the grain. The effect of the longitudinal diffusion on the determination of retention time from the first moment (i.e. the center of gravity) of the peak is not significant. The second central moment has a significance for the determination of the peak width and like all higher moments it depends on all factors characterizing theoretically the optimum velocity of the carrier for which the relative width of the peak reaches a minimum value.

Density of Electronic Energy Levels for a One-Dimensional Liquid

Preliminary to a three-dimensional calculation, the density of one-electron energy levels for a one-dimensional array of square wells whose positions are Gaussianly distributed independent random variables has been calculated. The Hamiltonian matrix elements are calculated using the tight-binding approximation and the Montroll moment-trace method is used to compute the first six moments of the density of states curve. Hermite and Legendre polynomial expansions are then used to obtain the density of states curve itself. This method is, of course, valid independent of dimension. The integrated density of states curve as here obtained agrees nicely with the computer solution given by Landauer and Helland.