A reciprocal relation for Hermite polynomials

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable. We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.

Nonclassicality of photon-modulated spin coherent states in the Holstein-Primakoff realization

Two new photon-modulated spin coherent states (SCSs) are introduced by operating the spin ladder operators J ± on the ordinary SCS in the Holstein-Primakoff realization and the nonclassicality is exhibited via their photon number distribution, second-order correlation function, photocount distribution and negativity of Wigner distribution. Analytical results show that the photocount distribution is a Bernoulli distribution and the Wigner functions are only associated with two-variable Hermite polynomials. Compared with the ordinary SCS, the photon-modulated SCSs exhibit more stronger nonclassicality in certain regions of the photon modulated number k and spin number j, which means that the nonclassicality can be enhanced by selecting suitable parameters.

Helium isotope separation by bi-layer membranes of g-C3N4

The problem of helium isotope separation via bi-layer membranes of graphitic carbon nitride g-C3N4 has been studied. The probability of passing isotopes through the membrane is derived from solving the Schrödinger integral equation using Hermite polynomials. The potential energy of the membrane is calculated based on modified Lennard-Johnes potential. The separation degree of the 3He/4He reaches the value of 1045 due to the resonant effect.

Path Planning for an HEXA Parallel Mechanism using a Modified PSO Algorithm

This paper presents a method for determining the optimal trajectory of the characteristic point based on the kinematic analysis of a HEXA parallel mechanism. The optimization was performed based on a modified PSO algorithm based on Hermite polynomials (MH-PSO). The change made to the initial algorithm consists in restricting the search space of the solutions by using the Hermite polynomial expressions of the geometric parameters as time functions for defining the movements of the end-effector. The MH-PSO algorithm, from its inception, ensures a faster convergence of solutions and ease of computational effort and is the main advantage of the method presented. During the optimization process, the function followed was the length of the trajectory described by the sequence of positions of the characteristic point, belonging to the end effector element, in compliance with additional conditions imposed. The use of the Hermite functions and PSO algorithm leads to minimal effort for analysis and mathematical formulation of the optimization problem.

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.

Signal processing of nonlinear dynamic systems

The paper considers Hermite polynomials that act as a self-similar basis for the decomposition of functions in phase space. It is shown that the equations of behavior of nonlinear dynamical systems are simplified. It is also noted that the wavelet decomposition over Hermite polynomials reduces the number of approximation coefficients and improves the quality of approximation.

A computationally efficient technique for the solution of pulp washing models

An efficient numerical technique for the solution of the pulp washing model is proposed in this study. Two linear and one nonlinear model are explained with quintic Hermite collocation method. In this technique, quintic Hermite polynomials (C2 continuous) are used as a basis function and orthogonal collocation method is applied within each element of the partitioned domain. For accuracy and applicability of the method, a comparison of the numerical results with analytic ones is made. The method is found to be stable using stability analysis and convergence criteria. The effect of Peclet number on exit solute concentration and other parameters is presented in the form of breakthrough curves. The results are derived for a broad range of parameters and the present method is found to be more useful and refined for solving the two-point boundary value problems.

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14