Hermite Orthogonal Polynomials Research Articles

The symmetric Dunkl-classical orthogonal polynomials revisited

ABSTRACT We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach in connection with the Dunkl operator. The main aim of this technique is first and foremost to determine the recurrence coefficients. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and confirm that the two families of generalized Hermite orthogonal polynomials and Gegenbauer orthogonal polynomials are the only ones belonging to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114. doi: 10.1016/j.amc.2006.08.108].

Higher Order Orthogonal Polynomials as Activation Functions in Artificial Neural Networks

Activation functions are used in Artificial Neural Networks to provide non-linearity to the system. Several different activation functions in use are very well known by almost any AI practitioner however this is not the case for polynomial activation functions. Increasing attention to these valuable mathematical functions can encourage more research and help to fill the gap. During this work, Chebyshev and Hermite orthogonal polynomials were used as activation functions. Calculations were conducted on 3 different datasets with different hyperparameters. According to the results, calculations done by Chebyshev activation functions take less time, but Chebyshev can be more fragile depending on the solved problem. On the other hand, Hermite shows a more robust and generalized behavior, it is less dependent on the problem type, and it improves by necessary adjustments.

Life cycle assessment of greenhouse gas emissions with uncertainty analysis: A case study of asphaltic pavement in China

There are a large number of complex uncertainties in the road life cycle assessment (LCA) analysis process and ignoring these uncertainties will lead to biased results and decision-making. To improve the accuracy of the quantification of carbon emissions (CEs) over the life cycle of pavements, a system for quantifying uncertainty was developed and a case study was conducted in conjunction with sensitivity analysis. The data quality index (DQI) combined with the beta distribution method was implemented to assess parameter uncertainty. The uncertainty of the model parameters was characterized using the results of slice sampling. The uncertainty in the shape of the model was described by the difference before and after the correction of the Hermite orthogonal polynomials. Various uncertainties are conveyed by the Monte Carlo simulation sampling method. To provide reliable emissions data, VISSIM and MOVES were used to calculate additional CEs from congestion. This study assesses the uncertainty of a specific road LCA case. The results show that uncertainty in model form has the greatest impact on results. The use and maintenance phases are key stages for improving the reliability of pavement life cycle CEs, with CEs and uncertainty contributions of around 90% and 94% respectively. The time-triggered conservation strategy has a 75% probability of dominance. This study provides a comprehensive theoretical basis for improving the quality of road LCA CE results, and also makes recommendations for the development of green and low-carbon road engineering.

Intelligent Terminal Sliding Mode Control of Active Power Filters by Self-Evolving Emotional Neural Network

In this article, a control system based on evolutionary emotional neural network is proposed for active power filters (APFs) to improve power quality. First, the dynamic model of the APF containing external disturbances and component parameter perturbations is introduced. The global fast terminal sliding mode (GFTSM) control method is proposed for the APF and its finite-time convergence and global robustness are demonstrated. In addition, an emotional neural network based on Hermite orthogonal polynomials as the activation function is constructed and combined with an evolutionary mechanism to form a self-evolving emotional neural network (SEENN). Then, a model-free control system based on SEENN is designed to address the model dependence of the GFTSM controller design. The parameter update law is designed under the Lyapunov framework to ensure stability. Finally, the results of prototype experiments show the excellent performance of the proposed control algorithm.

Fock space associated with quadrabasic Hermite orthogonal polynomials

This paper introduces a new idea for constructing operators associated with a certain class of probability measures. Special cases include known classical ones and noncommutative probability. The main example, derived from Feller’s book, is the hyperbolic

Plancherel–Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations

We study the asymptotic properties of multiple orthogonal Hermite polynomials which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptotic expansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel–Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of bases of homogeneous difference equations, and the second on reducing difference equations to pseudodifferential ones and using the theory of the Maslov canonical operator. The results of these approaches agree.

Hydrological Modelling of the Mono River Basin at Athiémé

Abstract. The objective of this study is to model the Mono River basin at Athiémé using stochastic approach for a better knowledge of the hydrological functioning of the basin. Data used in this study consist of observed precipitation and temperature data over the period 1961–2012 and future projection data from two regional climate models (HIRHAM5 and REMO) over the period 2016–2100. Simulation of the river discharge was made using ModHyPMA, GR4J, HBV, AWBM models and uncertainties analysis were performed by a stochastic approach. Results showed that the different rainfall-runoff models used reproduce well the observed hydrographs. However, the multi-modelling approach has improved the performance of the individual models. The Hermite orthogonal polynomials of order 4 are well suited for the prediction of flood flows in this basin. This stochastic modeling approach allowed us to deduce that extreme events would therefore increase in the middle of the century under RCP8.5 scenario and towards the end of the century under RCP4.5 scenario.

Analysis of the error distribution density convergence with its orthogonal decomposition in navigation measurements

Modern trends towards the expansion of online services lead to the need to determine the location of customers, who may also be on a moving object (vessel or aircraft, others vehicle – hereinafter the “Vehicle”). This task is of particular relevance in the fields of medicine – when organizing video conferencing for diagnosis and/or remote rehabilitation, e.g., for post-infarction and post-stroke patients using wireless devices, in education – when organizing distance learning and when taking exams online, etc. For the analysis of statistical materials of the accuracy of determining the location of a moving object, the Gaussian normal distribution is usually used. However, if the histogram of the sample has “heavier tails”, the determination of latitude and longitude’s error according to Gaussian function is not correct and requires an alternative approach. To describe the random errors of navigation measurements, mixed laws of a probability distribution of two types can be used: the first type is the generalized Cauchy distribution, the second type is the Pearson distribution, type VII. This paper has shown that it’s possible obtaining the decomposition of the error distribution density using orthogonal Hermite polynomials, without having its analytical expression. Our numerical results show that the approximation of the distribution function using the Gram-Charlier series of type A makes it possible to apply the orthogonal decomposition to describe the density of errors in navigation measurements. To compare the curves of density and its orthogonal decomposition, the density values were calculated. The research results showed that the normalized density and its orthogonal decomposition practically coincide.

Spin and pseudospin symmetries in radial Dirac equation and exceptional hermite polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Asymptotics of Multiple Orthogonal Hermite Polynomials $$H_{n_1,n_2}(z,\alpha)$$ Determined by a Third-Order Differential Equation

Asymptotics of Multiple Orthogonal Hermite Polynomials $$H_{n_1,n_2}(z,\alpha)$$ Determined by a Third-Order Differential Equation

Deep Learning Based Uncertainty Analysis in Computational Micromechanics of Composite Materials

Design of new materials is quite a difficult task owing to various time and length scales and affiliated uncertainties. The major challenge is to include all these in a conventional model. Hyperparameter models in machine learning can be used to overcome these issues. In this paper, an artificial neural network (ANN) model is developed to estimate the effective elastic parameters of unidirectional fiber reinforced composites using representative volume elements (RVE) considering uncertainty in the fiber diameter. The diameter probability distribution is constructed from the acquired gray images by employing image processing operations. The generalized Polynomial Chaos (gPC) expansion is then used to represent the distribution as a random input parameter for finite element analysis, from where the effective parameters are realized. Similarly, the outputs of the FE model, i.e., elastic parameters, are approximated by gPC expansions having unknown deterministic coefficients and random orthogonal Hermite polynomials. A set of collocation points are generated from roots of the random polynomials; from there, the unknown coefficients are estimated. The realization samples are utilized to train an ANN algorithm based on supervised deep learning. The developed ANN model is later tested and validated for a new sample set of data. It is shown that the ANN model with few hidden layers and neurons has a high accuracy for estimation of the elastic parameters directly from the information on the distribution of fiber diameters.

The Adelic Grassmannian and Exceptional Hermite Polynomials

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson's adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in its own right, but also proves useful in two ways: it leads to new algorithms for effectively computing the associated differential and difference operators and it also answers some open questions about them.

A polynomial chaos method for arbitrary random inputs using B-splines

Isogeometric analysis which extends the finite element method through the usage of B-splines has become well established in engineering analysis and design procedures. In this paper, this concept is considered in context with the methodology of polynomial chaos as applied to computational stochastic mechanics. In this regard it is noted that many random processes used in several applications can be approximated by the chaos representation by truncating the associated series expansion. Ordinarily, the basis of these series are orthogonal Hermite polynomials which are replaced by B-spline basis functions. Further, the convergence of the B-spline chaos is presented and substantiated by numerical results. Furthermore, it is pointed out, that the B-spline expansion is a generalization of the Legendre multi-element generalized polynomial chaos expansion, which is proven by solving several stochastic differential equations.

Indefinite integrals of special functions from hybrid equations

ABSTRACTElementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.

Series estimation for single‐index models under constraints

SummaryIn this paper, a semi‐parametric single‐index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pending issue in the literature. Meanwhile, the link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimisation with the constraint of identification condition for the index parameter, which addresses an important problem in the literature of single‐index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimates are demonstrated through an extensive Monte Carlo study and an empirical example.

Sequential Stochastic Response Surface Method Using Moving Least Squares-Based Sparse Grid Scheme for Efficient Reliability Analysis

The present work demonstrates an efficient method for reliability analysis using sequential development of the stochastic response surface. Here, orthogonal Hermite polynomials are used whose unknown coefficients are evaluated using moving least square technique. To do so, collocation points in the conventional stochastic response surface method (SRSM) are replaced by the sparse grid scheme so as to reduce the number of function evaluations. Moreover, the domain is populated sequentially by the sparse grid based on the outcome of the optimization to find out the most probable failure point. Hence, the support points are generated based on a coupled effect of the optimization for failure region and the sub-grids hierarchy. Continuous and differentiable penalty function is imposed to determine multiple failure points, if any, by repeating the optimization. Once the response surface is developed, reliability analysis is carried out using importance sampling. Five different benchmark examples are presented in this study to validate the performance of the proposed modeling. As the accuracy of the method is established, two reliability-based design examples involving nonlinear finite element (FE) analysis of plates are demonstrated. Numerical study shows the efficiency of the proposed sequential SRSM in terms of accuracy and number of time-exhaustive evaluation of the original performance function, as compared to other methods available in the literature. Based on these results, it may be concluded that the proposed method works satisfactorily for a large class of reliability-based design problems.

Coordinating analytic and visual approaches: Math majors’ understanding of orthogonal Hermite polynomials in the inner product space [formula omitted] in a technology-assisted learning environment

The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Hermite polynomials as vectors of the inner product space ℙnℝ in a Dynamic Geometry Software (DGS) – MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches (Zazkis, Dubinsky, & Dautermann, 1996) for visualizing inner products of Hermite polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis. Research participants not only produced such creative inner product space visualizations of the Hermite polynomials with the induced improper integral inner product 〈f,g〉=∫−∞∞e−x2f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.

Asymptotic Expansion of Risk-Neutral Pricing Density

A new method for pricing contingent claims based on an asymptotic expansion of the dynamics of the pricing density is introduced. The expansion is conducted in a preferred coordinate frame, in which the pricing density looks stationary. The resulting asymptotic Kolmogorov-backward-equation is approximated by using a complete set of orthogonal Hermite-polynomials. The derived model is calibrated and tested on a collection of 1075 European-style ‘Deutscher Aktienindex’ (DAX) index options and is shown to generate very precise option prices and a more accurate implied volatility surface than conventional methods.

On the extreme eigenvalues of certain matrices of non-standard inner products of Hermite polynomials

We study Hermite orthogonal polynomials and Gram matrices of their non-standard inner products. The weight function of the non-standard inner product is obtained from the Gauss probability density function by its horizontal shift by a real parameter. We are interested in the spectral properties of these matrices and some of their modifications. We show how the largest and smallest eigenvalues of the matrices depend on the parameter.

Series Estimation for Single-Index Models Under Constraints

In this paper, a semiparametric single-index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pending issue in the literature. Meanwhile, the link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimization with the constraint of an identification condition for the index parameter, which addresses an important problem in the literature of single-index models. In addition, making the best use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimators are demonstrated through an extensive Monte Carlo study and an empirical example.