Multivariate Hermite Polynomials Research Articles

A gradient method for high-dimensional BSDEs

Abstract We develop a Monte Carlo method to solve backward stochastic differential equations (BSDEs) in high dimensions. The proposed algorithm is based on the regression-later approach using multivariate Hermite polynomials and their gradients. We propose numerical experiments to illustrate its performance.

Multivariate Hermite polynomials and information matrix tests

The information matrix test for a normal random vector is shown to coincide with the sum of the moment tests for all third- and fourth-order multivariate Hermite polynomials. The statistic is decomposed as the sum of the marginal information matrix test for a subvector, the conditional information matrix test for the complementary subvector, and a third leftover component. It is also shown that exact finite sample distributions can be obtained by drawing spherical Gaussian vectors and orthogonalising them using sample moments. These tests are applied to assess the implications of Gibrat’s law for US city sizes using the three most recent censuses.

Properties of Multivariable Hermite Polynomials in Correlation with Frobenius–Genocchi Polynomials

The evolution of a physical system occurs through a set of variables, and the problems can be treated based on an approach employing multivariable Hermite polynomials. These polynomials possess beneficial properties exhibited in functional and differential equations, recurring and explicit relations as well as symmetric identities, and summation formulae, among other examples. In view of these points, comprehensive schemes have been developed to apply the principle of monomiality from mathematical physics to various mathematical concepts of special functions, the development of which has encompassed generalizations, extensions, and combinations of other functions. Accordingly, this paper presents research on a novel family of multivariable Hermite polynomials associated with Frobenius–Genocchi polynomials, deriving the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified, as well as establishing the series representations, summation formulae, operational and symmetric identities, and recurrence relations satisfied by these polynomials. This proposed scheme aims to provide deeper insights into the behavior of these polynomials and to uncover new connections between these polynomials, to enhance understanding of their properties.

Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials.

Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

Some Families of Differential Equations Associated with Multivariate Hermite Polynomials

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered.

Certain Properties of Δh Multi-Variate Hermite Polynomials

The research described in this paper follows the hypothesis that the monomiality principle leads to novel results that are consistent with past knowledge. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite polynomials ΔhHm(q1,q2,⋯,qr;h). We obtain their recurrence relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families, and thus the operational principles satisfied by these polynomials are derived and will prove beneficial for future observations.

Calculus: High-Dimensional Numerical and Symbolic Calculus in R

The R package calculus implements C++ optimized functions for numerical and symbolic calculus, such as the Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators and numerical integration in arbitrary orthogonal coordinate systems. The library applies numerical methods when working with R functions or symbolic programming when working with characters or expressions. The package handles multivariate numerical calculus in arbitrary dimensions and coordinates and implements the symbolic counterpart of the numerical methods whenever possible, without depending on external computer algebra systems. Except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

Multivariate holomorphic Hermite polynomials

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Ito polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Ito polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Ito polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Ito polynomials are established.

A Witness Function Based Construction of Discriminative Models Using Hermite Polynomials

In machine learning, we are given a dataset of the form $\{(x_j,y_j)\}_{j=1}^M$, drawn as i.i.d. samples from an unknown probability distribution $\mu$; the marginal distribution for the $x_j$'s being $\mu^*$, and the marginals of the $k^{th}$ class $\mu_k^*(x)$ possibly overlapping. We address the problem of detecting, with a high degree of certainty, for which $x$ we have $\mu_k^*(x)>\mu_i^*(x)$ for all $i\neq k$. We propose that rather than using a positive kernel such as the Gaussian for estimation of these measures, using a non-positive kernel that preserves a large number of moments of these measures yields an optimal approximation. We use multi-variate Hermite polynomials for this purpose, and prove optimal and local approximation results in a supremum norm in a probabilistic sense. Together with a permutation test developed with the same kernel, we prove that the kernel estimator serves as a ``witness function'' in classification problems. Thus, if the value of this estimator at a point $x$ exceeds a certain threshold, then the point is reliably in a certain class. This approach can be used to modify pretrained algorithms, such as neural networks or nonlinear dimension reduction techniques, to identify in-class vs out-of-class regions for the purposes of generative models, classification uncertainty, or finding robust centroids. This fact is demonstrated in a number of real world data sets including MNIST, CIFAR10, Science News documents, and LaLonde data sets.

Multimode Bogoliubov transformation and Husimi’s Q-function

In this paper, we present numerical schemes for evaluating the matrix elements of Gaussian/non-Gaussian operators in the Fock state basis, which are identified as multivariate Hermite polynomials (MHPs). Using the integral transformation operator to perform the multimode Bogoliubov transformation, Husimi’s Q-functions of Gaussian/non-Gaussian operators are easily derived as the generating functions of MHPs.

Construction of Multivariate Interpolation Hermite Polynomials for Finite Element Method

A new algorithm for constructing multivariate interpolation Hermite polynomials in analytical form in a multidimensional hypercube is presented. These polynomials are determined from a specially constructed set of values of the polynomials themselves and their partial derivatives with continuous derivatives up to a given order on the boundaries of the finite elements. The effciency of the finite element schemes, algor thms and programs is demonstrated by solving the Helmholtz problem for a cube.

Deconvolution of 3-D Gaussian kernels

Ulmer and Kaissl formulas for the deconvolution of one-dimensional Gaussian kernels are generalized to the three-dimensional case. The generalization is based on the use of the scalar version of the Grad's multivariate Hermite polynomials which can be expressed through ordinary Hermite polynomials.

Multivariate generalized Gram–Charlier series in vector notations

The article derives multivariate Generalized Gram-Charlier (GGC) series that expands an unknown joint probability density function (\textit{pdf}) of a random vector in terms of the differentiations of the joint \textit{pdf} of a reference random vector. Conventionally, the higher order differentiations of a multivariate \textit{pdf} in GGC series will require multi-element array or tensor representations. But, the current article derives the GGC series in vector notations. The required higher order differentiations of a multivariate \textit{pdf} in vector notations are achieved through application of a specific Kronecker product based differentiation operator. Overall, the article uses only elementary calculus of several variables; instead Tensor calculus; to achieve the extension of an existing specific derivation for GGC series in univariate to multivariate. The derived multivariate GGC expression is more elementary as using vector notations compare to the coordinatewise tensor notations and more comprehensive as apparently more nearer to its counterpart for univariate. The same advantages are shared by the other expressions obtained in the article; such as the mutual relations between cumulants and moments of a random vector, integral form of a multivariate \textit{pdf}, integral form of the multivariate Hermite polynomials, the multivariate Gram-Charlier A (GCA) series and others.

Stochastic spectral projection of electrochemical thermal model for lithium-ion cell state estimation

Abstract A novel approach for integrating a pseudo-two dimensional electrochemical thermal (P2D-ECT) model and data assimilation algorithm is presented for lithium-ion cell state estimation. This approach refrains from making any simplifications in the P2D-ECT model while making it amenable for online state estimation. Though deterministic, uncertainty in the initial states induces stochasticity in the P2D-ECT model. This stochasticity is resolved by spectrally projecting the stochastic P2D-ECT model on a set of orthogonal multivariate Hermite polynomials. Volume averaging in the stochastic dimensions is proposed for efficient numerical solution of the resultant model. A state estimation framework is developed using a transformation of the orthogonal basis to assimilate the measurables with this system of equations. Effectiveness of the proposed method is first demonstrated by assimilating the cell voltage and temperature data generated using a synthetic test bed. This validated method is used with the experimentally observed cell voltage and temperature data for state estimation at different operating conditions and drive cycle protocols. The results show increased prediction accuracy when the data is assimilated every 30s. High accuracy of the estimated states is exploited to infer temperature dependent behavior of the lithium-ion cell.

Some integrals involving multivariate Hermite polynomials: Application to evaluating higher-order local powers

We present the formula for a certain integral with respect to multivariate Hermite polynomials. Such integrals are used for deriving higher-order local power functions of asymptotically chi-squared tests. As an example, we provide asymptotic expansion for the local power function of Rao’s score test.

OPERATIONAL CALCULUS ASSOCIATED WITH CERTAIN FAMILIES OF GENERATING FUNCTIONS

In this paper, we discuss how the operational calculus can be exploited to the theory of mixed generating functions. We use operational methods associated with multi-variable Hermite polynomials, Laguerre polynomials and Bessels functions to drive identities useful in electromagnetism, fluid mechanics etc. Certain special cases giving bilateral generating relations related to these special functions are also discussed.

The dual multivariate Charlier and Edgeworth expansions

Abstract The Charlier differential series for distribution and density functions is the foundation for the Edgeworth expansions of distribution and density functions of sample estimators. Here, we give two forms of these expansions for multivariate distributions using multivariate Bell polynomials. Two forms arise because the multivariate Hermite polynomials have a dual form. These dual forms for the multivariate Charlier and Edgeworth expansions appear to be new.

Tomographic probability representation in the problem of transitions between Landau levels

The problem of quantum motion of a charged particle in a magnetic field is considered in the framework of the tomographic probability representation. The coherent and Fock states of a charge moving in a time-dependent homogeneous magnetic field are studied in the tomographic probability representation. These states are expressed in terms of the quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. These results are then generalized and applied to calculate the transition probabilities between the Landau levels for the case of transitions between stationary states of different magnetic fields, when the initially constant field becomes time dependent and then stays at some constant value again. The problem is studied in terms of wave functions and symplectic tomograms as well.

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.