Multivariate Hermite Polynomials Research Articles

Tomographic probability representation for states of charge moving in varying field

The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.

Approximation of the stochastic Galerkin matrix in the low‐rank canonical tensor format

AbstractIn this article we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log‐normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats

In this article, we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials.

Sensitivity analysis of piezoaeroelastic energy harvesters

We perform a sensitivity analysis of a piezoaeroelastic energy harvester consisting of a pitching and plunging rigid airfoil supported by flexural and torsional springs with a piezoelectric coupling attached to the plunge degree of freedom. We employ the nonintrusive formulation of the polynomial chaos expansion in terms of the multivariate Hermite polynomials to quantify the effects of variations in the load resistance, the eccentricity (distance between the center of mass and the elastic axis), and the nonlinear coefficients of the springs on the harvested power and the pitch and plunge amplitudes. As a first step, the normal form of the dynamics of the system near the Hopf bifurcation is used to select parameters that ensure a supercritical instability and maximize the generated power. The results show that the harvested power can be mostly affected by the eccentricity. Moreover, decreasing the nonlinear coefficient of the torsional spring results in a decrease in the pitch amplitude and an increase in the plunge amplitude and hence the harvested power. These results give guidance for optimizing and assessing the uncertainty in the performance of piezoaeroelastic energy harvesters.

On integrals involving Hermite polynomials

We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of gaussian functions and of multiple products of Hermite polynomials.

Non-Gaussian gravitational clustering field statistics

In this work we investigate the multivariate statistical description of the matter distribution in the nonlinear regime. We introduce the multivariate Edgeworth expansion of the lognormal distribution to model the cosmological matter field. Such a technique could be useful to generate and reconstruct three-dimensional nonlinear cosmological density fields with the information of higher order correlation functions. We explicitly calculate the expansion up to third order in perturbation theory making use of the multivariate Hermite polynomials up to sixth order. The probability distribution function for the matter field includes at this level the two-point, the three-point and the four-point correlation functions. We use the hierarchical model to formulate the higher order correlation functions based on combinations of the two-point correlation function. This permits us to find compact expressions for the skewness and kurtosis terms of the expanded lognormal field which can be efficiently computed. The method is, however, flexible to incorporate arbitrary higher order correlation functions which have analytical expressions. The applications of such a technique can be especially useful to perform weak-lensing or neutral hydrogen 21 cm line tomography, as well as to directly use the galaxy distribution or the Lyman-alpha forest to study structure formation.

Uncertainty Quantification of Piezoelectric Energy Harvesters from Aeroelastic Vibrations

A stochastic approach is presented to evaluate the uncertainties associated with variations in design parameters of a piezoaeroelastic energy harvester. The sensitivities of the harvested power to variations in the load resistance, the eccentricity (distance between the center of mass and the elastic axis), and the nonlinear coe cients are also determined. Moreover, the non-intrusive formulation of the polynomial chaos expansion in terms of the multivariate Hermite polynomials was employed to quantify the sensitivities in the harvested power and the plunge and pitch motions. The results show that the relationship between the input parameters and the harvested power is highly nonlinear. The results show also that the generated power is most sensitive to variations in the eccentricity and that the nonlinear coe cient of the plunge spring is less influential than the nonlinear coe cient of the torsional spring on the harvester's performance.

Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P (a ) for Y ~ Np (0,V ) and for repeated integrals of the density of Y . When V −1 y > 0 in R 3 the expansion for P (Y ) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of N p (0,V −1 ). This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials. These are given explicitly.

Positive-type functions on groups and new inequalities in classical and quantum mechanics

Out of any unitary representation of a group, positive-type functions on the group can be obtained. These functions allow one to construct positive semi-definite matrices that may be used to define new inequalities for higher moments of observables associated with classical probability distribution functions and density states of quantum systems. The inequalities stemming from the Heisenberg–Weyl group representations are considered in connection with Gaussian distributions. We obtain new inequalities for multi-variable Hermite polynomials.

Some conditional expectation identities for the multivariate normal

We give formulas for the conditional expectations of a product of multivariate Hermite polynomials with multivariate normal arguments. These results are extended to include conditional expectations of a product of linear combination of multivariate normals. A unified approach is given that covers both Hermite and modified Hermite polynomials, as well as polynomials associated with a matrix whose eigenvalues may be both positive and negative.

A stochastic approach for modeling incident gust effects on flow quantities

The uncertainty in gust loads on a rigid, flat-plate airfoil at zero angle-of-attack due to imprecise knowledge of the gust parameters is quantified. The loads are computed using the unsteady vortex lattice model, which includes temporal variations in wake vorticity and the associated downwash on the airfoil. The non-intrusive formulation of the polynomial chaos expansion in terms of the multivariate Hermite polynomials is employed to quantify the uncertainty in the predicted unsteady lift. The expansion coefficients were estimated through Latin hypercube sampling of the parameters in the vertical and streamwise gust spectra. The first-order chaos expansion in terms of the uncertain spectral parameters was found to be sufficient for representing the stochastic aerodynamic lift, which was found to be most sensitive to imprecision in the standard deviation of the vertical component of the gust. These conclusions were found to be unaffected by ignoring the effects of gusts on the locations of the shed vortices in the airfoil’s wake.

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘n of orthogonal polynomials of degree n for a radially symmetric weight on ℝd, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘n with the property that each f∈℘n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials pξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝd:|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘n.

Moment representation of Bernoulli polynomial, Euler polynomial and Gegenbauer polynomials

The Hermite polynomials can be represented to the moments of normal distribution by the work of Withers [2000. A simple expression for the multivariate Hermite polynomials. Statist. Probab. Lett. 47, 165–169]. This paper generally shows certain combinatorial polynomials and orthogonal polynomials are also the moments of random variables, such as Bernoulli polynomials, Euler polynomials, Gegenbauer polynomials.

Simple representations for Hermite polynomials

Hermite polynomials have applications in almost every area of electrical and electronic engineering. Simple (hitherto unknown) representations are derived for the Hermite polynomials. For simplicity, the univariate case is considered. The same result holds for the multivariate and complex Hermite polynomials.

Some statistical applications of Faa di Bruno

The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503–520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.

Normal moments and Hermite polynomials

Hermite polynomials are used to derive expressions for the moments about the origin of univariate and multivariate normal distributions. A recurrence relation derived for multivariate Hermite polynomials leads to a recurrence relation for the multivariate normal moments.

Seasonal Modeling of Multivariate Distributions of Metocean Parameters With Application to Marine Operations

Statistical information about the joint occurrence of metocean parameters is of importance for many offshore activities. For instance, in marine operations, environmental limitations may be brought about by both wind and wave conditions. Thus, knowledge of their joint occurrence is important as the persistence duration (i.e., the duration of the sea state persistence above or beneath a given level) and the seasonal dependence of wind and waves appear to be of large interest. However, such a modeling becomes difficult as the number of considered variables increases, especially when utilizing a common parameterization of some conditional distributions. This paper proposes a general methodology that aims at modeling seasonal joint distributions of n such parameters from their correlation structure and the n marginal distributions fitted by generalized gamma ones. Two methods are proposed in order to derive an approximate joint distribution from the modeled margins. The first one matches the correlation matrix only, whereas the second one, which is based on a multivariate Hermite polynomials expansion of the multinormal distribution, is able to match joint moments of order higher than two. However, more restrictive conditions are shown by the latter. An application to the simple example of the joint occurrence of significant wave height and the mean wind velocity at the 10m elevation is used to illustrate the methods. Eventually, examples of applications like simultaneous persistence of wind and wave conditions as well as seastate forecasting from statistics are given.

Photon-Number Tomography of Multimode States and Positivity of the Density Matrix

For one-mode and multimode light, the photon-number tomograms of Gaussian quantum states are explicitly calculated in terms of multivariable Hermite polynomials. Positivity of the tomograms is shown to be a necessary condition for positivity of the density matrix.

Multiindex Multivariable Hermite Polynomials

In the present paper multiindex multivariable Hermite polynomials in terms of series and generating function are defined. Their basic properties, differential and pure recurrence relations, differential equations, generating function relations and expansions have been established. Few deductions are also obtained.

An algebraic study on theAN-1- andBN-Calogero models with bosonic, fermionic and distinguishable particles

Through an algebraic method using the Dunkl--Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic non-symmetric multivariable Hermite and Laguerre polynomials corresponding to the distinguishable case are presented. Symmetric and anti-symmetric polynomials that respectively give the eigenstates for bosonic and fermionic particles are also presented by the symmetrization and anti-symmetrization of the non-symmetric ones. The norms of all the eigenstates for all cases are algebraically calculated in a unified way.