Narrow Results

In the “Many Interacting Worlds” (MIW) discrete Hamiltonian system approximation of Schrödinger’s wave equation, introduced in Ref. 11, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein’s method, in Wasserstein-$1$ distance with rate ${𝒪}(\sqrt{logN}/N)$ has been shown Refs. 5, 13, and 15. In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-1 distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the “differential equation” approach to Stein’s method, which allows to handle behavior near zeros of the higher energy states.

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

Steerability is a useful and important property of "kernel" functions. It enables certain complicated operations involving orientation manipulation on images to be executed with high efficiency. Thus, we focus our attention on the steerability of Hermite polynomials and their versions modulated by the Gaussian function with different powers, defined as the Hermite kernel. Certain special cases of such kernel, Hermite polynomials, Hermite functions and Gaussian derivatives are discussed in detail. Correspondingly, these cases demonstrate that the Hermite kernel is a powerful and effective tool for image processing. Furthermore, the steerability of the Hermite kernel is proved with the help of a property of Hermite polynomials revealing the rule concerning the product of two Hermite polynomials after coordination rotation. Consequently, any order of the Hermite kernel inherits steerability. Moreover, a couple sets of an explicit interpolation function and basis function can be directly obtained. We provide some examples to verify steerability of the Hermite kernel. Experimental results show the effectiveness of steerability and its potential applications in the fields of image processing and computer vision.

We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner–Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on ℝ^ℕ. It is shown that the weak solution can be represented as Wiener–Itô Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in ℝ¹.

We establish sufficient conditions on the random inputs for weighted sequence norms of the Wiener–Itô decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions' Wiener–Itô decomposition are shown to be p-summable for the same value of 0 < p < 1.

We prove rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as of Finite Element discretizations of these approximations from a dense, nested family V₀ ⊂ V₁ ⊂ V₂ ⊂ ⋯ V of finite element spaces of continuous, piecewise linear Finite Elements.

This paper considers the problem of the center of mass motion of three fermionic particles, e. g., equal nucleons (no spin), which can move only in the x-direction in the laboratory system. This system (with three degrees of freedom only) is transformed to a system where one coordinate is proportional to the center of mass coordinate. The center of mass moves in a harmonic oscillator potential. The basis set of wave functions is constructed with Hermite polynomials in part. As example we study the case of a quadratic intrinsic potential between the particles. The problem may be important for the motion of three nucleons in a crystal channel.

We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This requires the computation of moments and correlators of the underlying asset price which for a polynomial jump–diffusion process are given analytically; hence, no numerical simulation is required to evaluate the series. This allows to derive analytical expressions for the option Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale $b$. We find that the rate of convergence of the series depends on $b$, for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup ${\mathrm{\text{e}}}^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers $L^N$ of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

The Beauville–Fujiki relation for a compact Hyperkähler manifold $X$ of dimension $2k$ allows to equip the symmetric power ${\mathrm{\text{Sym}}}^k{H}^2(X)$ with a symmetric bilinear form induced by the Beauville–Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere ${𝕊}^d$, or equivalently, over ${ℝ}^{d+1}$ with a Gaussian kernel.

Based on the consistent couple stress theory (CCST), we develop a Hermite-family $C^1$ finite layer method (FLM) for the three-dimensional (3D) free vibration analysis of a simply-supported, exponentially graded (EG) piezoelectric microplate under open- and closed-circuit surface conditions. In the formulation of the FLM, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The Hermite-family $C^1$ FLM for analyzing EG piezoelectric microplates is reduced to the Hermite-family $C^1$ FLM for analyzing EG piezoelectric macroscale plates and functionally graded (FG) elastic microplates by assigning a value of zero to the material length scale parameter and by ignoring the piezoelectric and flexoelectric effects in the formulation, respectively. The accuracy and convergence rate of the FLM are assessed by comparing their solutions with the benchmark solutions of both the EG piezoelectric macroplates and the power-law-type FG elastic microplates that are available in the relevant literature. We examine and discuss some key effects on the free vibration characteristics of an EG piezoelectric microplate, including the impact of the material length scale parameter, the material-property gradient index, the length-to-thickness ratio, the piezoelectric effect, and the flexoelectric effect.

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.

Recent work of Griffin, Ono, Rolen, and Zagier shows that the Jensen polynomials for the Riemann xi function converge to the Hermite polynomials under a suitable normalization. We generalize this result, proving that the normalized Jensen polynomials for a large class of genus zero or one entire functions converge either to the Hermite polynomials, or to a class of polynomials which can be written as a confluent hypergeometric function.

Approximation formulas for the Genocchi polynomials of complex order are obtained using contour integration with the contour avoiding branch cuts. An alternative expansion is also obtained by expanding a function involving the generating function in a two-point Taylor expansion.

We consider n × n matrices whose (i, j)th entry is , where X₁, …, X_n are i.i.d. standard Gaussian in ℝ^p, and f is a real-valued function. The weak limit of the eigenvalue distribution of these random matrices is studied at the limit when p, n → ∞ and p/n = γ which is a constant. We show that, under certain conditions on the kernel function f, the limiting spectral density exists and is dictated by a cubic equation involving its Stieltjes transform. The parameters of this cubic equation are decided by a Hermite-like expansion of the rescaled kernel function. While the case that f is differentiable at the origin has been previously resolved by El Karoui [The spectrum of kernel random matrices, Ann. Statist.38 (2010) 1–50], our result is applicable to non-smooth f, such as the Sign function and the hard thresholding operator of sample covariance matrices. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marčenko–Pastur (M.P.) distribution and Wigner's semi-circle distribution as special cases.

The generalized spectral-analytical method is based on the use of orthogonal decompositions of signals and analytic transformations in the space of expansion coefficients. The theory of classical orthogonal bases is a generalization of the theory of Fourier series to algebraic orthogonal polynomials of continuous and discrete arguments. The choice of the optimal set of features is carried out on the basis of integral signal estimates, and the mathematical operations on the signals studied are performed in the space of the expansion coefficients. Within the framework of the generalized spectral-analytical method, the problems of adaptive analytical description of one-dimensional and multidimensional digital information arrays are effectively solved in order to reduce the redundancy of signals’ description, as well as recognition problems on the basis of pairwise comparison of objects in Fourier coefficients features space.

This chapter describes the application of this method in the following problems:

• image recognition (by using invariants when recognizing contour objects),
• bioinformatics (recognition of repeats in sequences and complex spatial configurations of biomacromolecules),
• biomedicine (analysis of the spatial and temporal organization of electrical and magnetic activity of the brain).

We consider the Sobolev inner product

In this work, we introduce and investigate the Hermite-Apostol-Bernoulli polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Furthermore, we prove Raabe relation and give some relations between 2-dimensional Hermite polynomials and Apostol-Bernoulli polynomials.

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.