Hermite Polynomials Research Articles

Dynamics of pollutant dispersion and solid–fluid interfacial layer in Jeffrey nanofluid flow subjected to waste discharge concentration: Implementation of probabilists’ Hermite polynomial collocation method

The current study explores the flow of a Jeffrey nanofluid with the impact of a non-uniform heat source/sink and magnetic effects over a stretching sheet by accounting for a pollutant’s transient dispersion after being released from a source outside the system. The need to enhance the efficiency of prospective future technologies, including heat exchangers, solar collectors with nanoscale solid particles suspended in the base fluid, powered engines, pharmaceutical procedures, and hybrid microelectronics, has grown recently. Graphene is treated as a nanoparticle and NaA lg as a base liquid. To emphasize the thermal integrity of the flow that is now taken into consideration, the influence of the diameter of the nanoparticle and the liquid–solid interfacial layer is also shown at the molecular level. The modeled equations are transformed into ordinary differential equations by means of the appropriate similarity transformations. Probabilists’ Hermite polynomial collocation technique is employed to solve these nonlinear dimensionless ODEs, and the graphical representations are created for certain significant values of the underlying physical parameters in the flow model. The result reveals that as Deborah number increases, velocity upsurges but temperature declines. A rise in concentration is observed when the local pollutant external source variation parameter rises. The temperature is enhanced with the advanced values of non-uniform heat source/sink parameters.

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.

Hermite, Higher order Hermite, Laguerre type polynomials and Burgers like equations

The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous non-linear partial differential equations can be derived for Laguerre polynomials. The starting point of this extension is the Laguerre diffusive equation, whose nonlinear extension reveals interesting implications involving families of mixed polynomials. In this way we have a general scheme to obtain new exact explicit solutions for nonlinear PDEs by using Laguerre, Hermite and other families of polynomials of Appèl and non-Appèl type.

SEMIPARAMETRIC ESTIMATION AND VARIABLE SELECTION FOR SPARSE SINGLE INDEX MODELS IN INCREASING DIMENSION

This paper considers semiparametric sieve estimation in high-dimensional single index models. The use of Hermite polynomials in approximating the unknown link function provides a convenient framework to conduct both estimation and variable selection. The estimation of the index parameter is formulated from solutions obtained by the routine penalized weighted linear regression procedure, where the weights are used in order to tackle the unbounded support of the regressors. The resulting index parameter estimator is shown to be consistent and sparse, and the asymptotic normality for the estimators of both the index parameter and the link function is established. To perform variable selection in the ultra-high dimension case, we further suggest a forward regression screening method, which is shown to enjoy the sure independence screening property. This screening procedure can be used before the penalized variable selection to reduce the burden of dimensionality. Numerical results show that both the variable selection procedures and the associated estimators perform well in finite samples.

Transient dispersion of a reactive solute in an oscillatory Couette flow through an anisotropic porous medium

This research discusses the significance of reactive solute dispersion relevant to ecological, biological, and geological contexts. It examines solute movement under oscillatory Couette flow through anisotropic porous media between parallel plates under the effect of heterogeneous boundary reactions. The flow is driven by the combined effect of upper plate oscillation in its plane and time-dependent pressure gradient. The lower plate is assumed rough, which introduces slip velocity. A semi-analytical approach, with the method of moments and finite difference scheme, is utilized to explore the transient dispersion in steady and oscillatory flows with or without a non-zero mean. Using the Hermite polynomial, the mean concentration for purely oscillatory and combined flow is obtained, highlighting notable variations based on flow factors. The results suggest that increment in anisotropic angle ϕ reduces dispersion and enhances mean concentration for permeability ratio K < 1, but this reverses for K > 1. Three dispersion phases emerge: diffusive, anomalous, and Taylor's regimes. Gaussian cloud distribution occurs at small and large times, with intermediate stages displaying anomalous dispersion and asymmetric longitudinal distribution. Effects of boundary absorption stabilize over time. The research focuses on the practical significance of different permeabilities of porous media, emphasizing applications of anisotropic porous media in fields such as chemical engineering and industrial processes.

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.

A New Extension of Combination of Linear and Bilinear Generating Relations in Function Spaces Associated with Hypergeometric Polynomials; Some New Applications of Two Signals in Special Functions

For a certain class of generalized hypergeometric polynomials, the first derive some special cases on linear and bilinear generating functions and then apply these generating functions in order to reduce the corresponding results for the classical Jacobi, Hermite, Laguerre and Gegenbauer Polynomials, hypergeometric functions of Gauss and functions of Bessel and Kelvin. They also consider several linear generating functions for these polynomials as well as for some multivariable Jacobi and multivariable Laguerre polynomials which were investigated in recent years. Some of the linear and bilinear generating functions, presented in this paper, are associated with the hypergeometric polynomials.

An efficient methodology for simulating multivariate non-Gaussian stochastic processes

Simulation of multivariate non-Gaussian stochastic processes is still a challenging task, where the efficiency and accuracy may not be balanced. In this paper, an efficient methodology is presented to address this challenge. This methodology makes use of a probability weighted moments-based Hermite polynomial model and a wavenumber–frequency spectrum density-based spectral representation approach. First, a probability weighted moments-based Hermite polynomial model is employed to establish a transformation relationship between Gaussian and non-Gaussian samples. Importantly, this model only requires the solution of a solvable system of equations. Second, the target non-Gaussian wavenumber–frequency spectrum density can be promptly transformed into the Gaussian one, where a fast root matching strategy and fast Fourier transform are utilized. Finally, the underlying Gaussian samples are generated by the spectral representation method, and the non-Gaussian samples are obtained via the proposed transformation model. The computational results from numerical examples demonstrate that the proposed methodology is considered to be an accurate, applicable, and efficient approach for simulating multivariate non-Gaussian stochastic processes.

Spectral properties of weighted Cauchy singular integral transform on S-poly-Barmgann spaces

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the k-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô–Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman projection leading to the explicit determination of its singular values. The provided results extend to the quaternionic setting those obtained by A. Intissar and A. Intissar in (2006) [32] for the weighted Cauchy transform on the complex plane.

Post-quantum complex Itô–Hermite polynomials

Post-quantum complex Itô–Hermite polynomials

Wick–Fourier–Hermite Series in the Theory of Linear and Nonlinear Transformations of Gaussian Distributions

This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.

Curve fitting for seismic waves of earthquake with Hermite polynomials

We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil).

Спектральные методы полиномиальной интерполяции и аппроксимации

Classical problem of interpolation and approximation of functions with polynomials is considered here as a special case of spectral representation of functions. We developed this approach earlier for Legendre and Chebyshev orthogonal polynomials. Here we use Newton's fundamental polynomials as basis functions. We demonstrate that the spectral approach has computational advantages over the divided differences method. In a number of problems, Newton and Hermite interpolations are indistinguishable in our approach and computed by the same formulas. Also, computational algorithms that we constructed earlier with the use of orthogonal polynomials are adapted without modifications for use of Newton and Hermite polynomials.

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>

Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent

<p>We provide new insights into the solvability property of a Hamiltonian involving the fourth Painlevé transcendent and its derivatives. This Hamiltonian is third-order shape invariant and can also be interpreted within the context of second supersymmetric quantum mechanics. In addition, this Hamiltonian admits third-order lowering and raising operators. We have considered the case when this Hamiltonian is irreducible, i.e., when no special solutions exist for given parameters $ \alpha $ and $ \beta $ of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $. This means that the Hamiltonian does not admit a potential in terms of rational functions (or the hypergeometric type of special functions) for those parameters. In such irreducible cases, the ladder operators are as well involving the fourth Painlevé transcendent and its derivative. An important case for which this occurs is when the second parameter (i.e., $ \beta $) of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $ is strictly positive, $ \beta > 0 $. This Hamiltonian was studied for all hierarchies of rational solutions that come in three families connected to the generalized Hermite and Okamoto polynomials. The explicit form of ladder, the associated wavefunctions involving exceptional orthogonal polynomials, and recurrence relations were also completed described. Much less is known for the irreducible case, in particular the excited states. Here, we developed a description of the induced representations based on various commutator identities for the highest and lowest weight type representations. We also provided for such representations a new formula concerning the explicit form of the related excited states from the point of view of the Schrödinger equation as two-variables polynomials that involve the fourth Painlevé transcendent and its derivative.</p>

The altered Hermite matrix: implications and ramifications

<p>Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.</p>

Modified Ito Generalization of the Hermite Polynomials for the Linearization of Radio Over Fiber Links With Increased Numerical Stability

Modified Ito Generalization of the Hermite Polynomials for the Linearization of Radio Over Fiber Links With Increased Numerical Stability

A Unified Approach to Computing the Zeros of Orthogonal Polynomials

We present a unified approach to calculating the zeros of the classical orthogonal polynomials and discuss the electrostatic interpretation and its connection to the energy minimization problem. This approach works for the generalized Bessel polynomials, including the normalized reversed variant, as well as the Viet\'e--Pell and Viet\'e--Pell--Lucas polynomials. We briefly discuss the electrostatic interpretation for each aforesaid case and some recent advances. We provide zeros and error estimates for various cases of the Jacobi, Hermite, and Laguerre polynomials and offer a brief discussion of how the method was implemented symbolically and numerically with Maple. In conclusion, we provide possible avenues for future research.

Generating superpositions of quantum states via a beam splitter with position measurement

We use the quadrature measurement to generate the novel nonclassical states via the beam splitter with two input states, i.e., a Fock state and a vacuum state. It is interesting to find that the desired target states are the Hermite polynomial excited vacuum states. Our results have shown that the zero-position detection for the position detector, the little photon number in the input state, and the high transmittance of the beam splitter (BS) are beneficial to improve the detection efficiency of finding the output states. The proposed states quantum statistical properties and squeezing effects are also studied in detail via different criteria. Our numerical analysis demonstrates that the output quantum states are new nonclassical states. Compared with the method of photon catalysis, position detection is easier to realize in experiments. Therefore, the results in this paper shall provide theoretical support for the experimental generation of several new nonclassical states.

A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

In this paper, we consider the following \(\mathcal{L}\)-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.