Hermite Polynomials Research Articles

Field theory of active Brownian particles with dry friction

Abstract We present a field theoretic approach to capture the motion of a particle with dry friction for one- and two-dimensional (2D) diffusive particles, and further expand the framework for 2D active Brownian particles. Starting with the Fokker–Planck equation and introducing the Hermite polynomials as the corresponding eigen-functions, we obtain the actions and propagators. Using a perturbation expansion, we calculate the effective diffusion coefficient in the presence of both wet and dry frictions in a perturbative way via the Green–Kubo relation. We further compare the analytical result with the numerical simulation. Our result can be used to estimate the values of dry friction coefficient in experiments.

Projection and modified projection methods for nonlinear Hammerstein integral equations on the real line using Hermite polynomials

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the real line. Therefore, this paper proposes polynomially based projection and modified projection methods to solve Hammerstein integral equations on the real line with sufficiently smooth kernels. The approximating operator employed is either the orthogonal projection or an interpolatory projection using Hermite polynomials as basis functions. We analyse the convergence of the proposed approaches and its iterated version and we establish superconvergence results. Through different numerical tests, the effectiveness of the proposed methods is presented to demonstrate the given theoretical framework.

Superposition and interference of Gaussian wave packets for laser transverse modes

We interpret the intrinsic mechanism of the transition from geometric modes to eigenmodes in the Gaussian wave-packet representation based on numerical simulation. We then construct multiaxial super-geometric modes and polygonal vortex modes based on Gaussian wave packets. We highlight that the superposition of Gaussian wave packets can build the geometric modes described by Hermite polynomials and that the eigenmodes are the result of the interference of Gaussian wave packets. We conclude that Gaussian wave packets might serve as a complete basis for constructing structured light generated in solid-state laser cavities. Our study may provide meaningful insight into Gaussian wave packets for laser transverse modes.

On solutions to the Cauchy problem for a class of forced Burgers equations

Summary A Cauchy problem for forced Burgers equation (FBE) with time-dependent coefficients is studied in this work. We find explicit solutions of the FBE in terms of Hermite polynomials subject to various classes of initial data. Further, we prove the existence of solutions of FBE with respect to the bounded, continuous and summable initial data. We then derive an approximate solution to the Cauchy problem and find the error estimations between the true and approximate solutions. We give asymptotic profiles of the solutions depending on the initial data.

Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials.

Seizure detection in EEG signal using Gaussian-stockwell transform and Hermite polynomial features

Electroencephalography (EEG) is a non-intrusive and powerful tool for understanding the complex nature of the brain. Epileptic seizures, a neurological disorder, are characterized by excessively synchronized neuronal activity within the brain. This abnormal synchronization disrupts the normal functioning of neural networks, leading to seizures. This work proposes to use the Gaussian-based Stockwell Transform (GST) to segment EEG signals into various frequency bands such as delta, theta, alpha, beta, and gamma band for analysis. The feature extraction process on these sub-band signals generates a feature set, encompassing the fluctuation index, relative amplitude, holoentropy, logarithmic band power, and the newly introduced Hermite polynomial-based delta amplitude modulation spectrogram. To differentiate between epileptic and non-epileptic events in EEG signals, the previous features have been applied to the Random Forest Classifier (RFC) algorithm. The effectiveness of our proposed method is validated using a benchmark EEG dataset, with GST and RFC demonstrating accurate results in distinguishing seizure activity. Experimental results show an optimal classification accuracy of 96.4 %, a sensitivity of 97 %, and a specificity of 96.4 %.

Several Characterizations of the Generalized 1-Parameter 3-Variable Hermite Polynomials

Several Characterizations of the Generalized 1-Parameter 3-Variable Hermite Polynomials

Exploring percolation features with polynomial algorithms for classifying Covid-19 in chest X-ray images

Covid-19 is a severe illness caused by the Sars-CoV-2 virus, initially identified in China in late 2019 and swiftly spreading globally. Since the virus primarily impacts the lungs, analyzing chest X-rays stands as a reliable and widely accessible means of diagnosing the infection. In computer vision, deep learning models such as CNNs have been the main adopted approach for detection of Covid-19 in chest X-ray images. However, we believe that handcrafted features can also provide relevant results, as shown previously in similar image classification challenges. In this study, we propose a method for identifying Covid-19 in chest X-ray images by extracting and classifying local and global percolation-based features. This technique was tested on three datasets: one comprising 2,002 segmented samples categorized into two groups (Covid-19 and Healthy); another with 1,125 non-segmented samples categorized into three groups (Covid-19, Healthy, and Pneumonia); and a third one composed of 4,809 non-segmented images representing three classes (Covid-19, Healthy, and Pneumonia). Then, 48 percolation features were extracted and give as input into six distinct classifiers. Subsequently, the AUC and accuracy metrics were assessed. We used the 10-fold cross-validation approach and evaluated lesion sub-types via binary and multiclass classification using the Hermite polynomial classifier, a novel approach in this domain. The Hermite polynomial classifier exhibited the most promising outcomes compared to five other machine learning algorithms, wherein the best obtained values for accuracy and AUC were 98.72% and 0.9917, respectively. We also evaluated the influence of noise in the features and in the classification accuracy. These results, based in the integration of percolation features with the Hermite polynomial, hold the potential for enhancing lesion detection and supporting clinicians in their diagnostic endeavors.

Persistence of large-scale coherent structures in a turbulent pipe flow through an improved lattice Boltzmann approach

We simulated a turbulent pipe flow within the lattice Boltzmann method using a multiple-relaxation-time collision operator with Maxwell–Boltzmann equilibrium distribution expanded, for the sake of a more accurate description, up to the sixth order in Hermite polynomials. The moderately turbulent flow (Reτ≈181.3) is able to reproduce up to the fourth statistical moment with great accuracy compared with other numerical schemes and with experimental data. A coherent structure identification was performed based on the most energetic streamwise turbulent mode, which revealed a surprising memory effect related to the large-scale forcing scheme that triggered the pipe's turbulent state. We observe that the existence of large-scale motions that are out of the pipe's stationary regime does not affect the flow's detailed single-point statistical features. Furthermore, the transitions between the coherent structures of different topological modes were analyzed as a stochastic process. We find that for finely resolved data, the transitions are effectively Markovian, but for larger decimation time lags, due to topological mode degeneracy, non-Markovian behavior emerges, in agreement with previous experimental studies.

Norm convergence rate for multivariate quadratic polynomials of Wigner matrices

We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials q converges to a deterministic limit with a rate of convergence of N−2/3+o(1). Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.

Short-time Fourier transform and superoscillations

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

Hermite polynomial based affine projection Blake Zisserman algorithm for identification of robust sparse nonlinear system

Hermite polynomial based affine projection Blake Zisserman algorithm for identification of robust sparse nonlinear system

Deformations and q-Convolutions. Old and New Results

This paper is the survey of some of our results related to q-deformations of the Fock spaces and related to q-convolutions for probability measures on the real line R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. The main idea is done by the combinatorics of moments of the measures and related q-cumulants of different types. The main and interesting q-convolutions are related to classical continuous (discrete) q-Hermite polynomial. Among them are classical (q=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=1$$\\end{document}) convolutions, the case q=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=0$$\\end{document}, gives the free and Boolean relations, and the new class of q-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of q-convolutions. The main result is the construction of Brownian motion related to q-Discrete Hermite polynomial of type I.

On some series of special functions

Series of the form ∑ k = 0 ∞ ∏ i = 1 p ( a i ) k ∏ j = 1 q ( b j ) k w k F k ( z ) are considered, where F k ( z ) are special functions of hypergeometric type. Various types of series involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials are represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables.

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

Generalized complex Hermite polynomials with associated nonlinear coherent states

Generalized complex Hermite polynomials with associated nonlinear coherent states

Unveiling Multivariable Hermite-Based Genocchi Polynomials: Insights from Factorization Method

This paper presents novel polynomials resulting from the convolution of generalized multivariable Hermite polynomials and Genocchi polynomials. Investigating their properties, such as recurrence relations, explicit formulas utilizing shift operators, and differential equations, forms the core of our exploration. Moreover, we derive integro-differential and partial differential equations for these polynomials, thereby enriching the comprehension and applicability of these hybrid polynomials across diverse mathematical domains.

Simplified calculation method of non-Gaussian wind pressure peak factor for large-span spatial roof structures

Based on the wind tunnel test data of a long-span spatial roof structure, the correlation between higher-order statistics and peak factors was obtained according to the time and frequency domain characteristics of wind pressure-time history, as well as the coincidence condition between wind pressure power spectra of measurement points and wind speed power spectra. A simplified peak factor calculation formula was proposed based on the moment-based Hermite polynomial model. The Gaussian part of the peak factor was calculated through the wind speed spectrum based on the zero-crossing theory, and the parameters related to higher-order statistics were considered as the non-Gaussian part of the peak factor. The simplified method was verified through the instantaneous extreme value method based on a coal shed structure with available long-term wind pressure-time history. It revealed that the results of the proposed simplified method correlate well with that of the instantaneous extreme value method, indicating its accuracy and potential value for engineering practice.

Investigation on the sampling distributions of non-Gaussian wind pressure skewness and kurtosis

Accurate estimation of skewness and kurtosis is crucial for addressing issues related to non-Gaussian wind pressure distribution fitting, signal simulation, and extreme estimation. The skewness and kurtosis determined from finite-length signals are inherent random variables characterized by obvious value fluctuations. Very limited research has been conducted on the sampling distributions of skewness and kurtosis for non-Gaussian wind pressure, motivating the present study. Firstly, the investigation focuses on the non-Gaussian white noise. The expressions of main statistical indicators such as the variance, covariance, correlation coefficient, skewness, and kurtosis of the two sampling distributions are derived and then verified using the Hermite polynomial model. The primary factors influencing the sampling distribution variance (SDV) of white noise are analyzed, including the signal length and marginal moments of the parent distribution. Secondly, the concentration shifts to colored non-Gaussian processes. The composition of SDV is analyzed through theoretical derivation. On this basis, a method for estimating SDV values using the Gaussian process regression model is proposed, with accuracy and feasibility verified based on the long-duration wind pressure data measured in wind tunnel. Furthermore, the relationship between the SDVs of wind pressure processes, signal length, higher-order correlation functions, and marginal moments of the parent process is discussed.

Polymeromorphic Itô–Hermite functions associated with a singular potential vector on the punctured complex plane

We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov–Bohm effect. The functions are defined by their β-modified Rodrigues type formula and extend the polyanalytic Itô–Hermite polynomials to the polymeromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.