Hermite Functions Research Articles

Novel finite element modeling for free vibration and buckling analysis of non-uniform thickness 2D-FG sandwich porous plates using refined Quasi 3D theory

A novel four-node quadrilateral element with eleven degrees of freedom per node is approximated using the C1 -order non-conforming Hermite and Lagrange functions and the novel refined Quasi-3D plate hypothesis for buckling and free vibration analysis of non-uniform thickness bi-directional functionally graded sandwich porous (BFGSP) plates resting on variable elastic foundations (VEF) in a hygro-thermal environment. Material and mechanical properties that change in both the length and thickness directions with three different laws of porosity, which are made up of a fully ceramic core layer and two bi-directional functionally graded (2D-FG) material ones, may be used a lot in the aerospace engineering and military industries. In order to analyze the buckling and free oscillation behaviors shown by the plate with arbitrary boundary conditions, a series of mathematical methods created in Matlab’s software are used. The computation program’s correctness is verified by comparing numerical findings to dependable assertions. In addition, a comprehensive analysis of the impact of factors on the buckling and free oscillation responses is conducted. The findings reveal that the novel porosity patterns, the hygro-thermal environment, the elastic medium characteristics, and the boundary conditions have a substantial impact on the mechanical behaviors of the non-uniform thickness 2D-FG sandwich porous plates. The results of this article can be used as a useful reference for engineers when calculating and designing structures of this type in engineering practice.

Research on alleviating nonlinear problem by data feature extraction and fusion based on DFT-spread

Coherent optical orthogonal frequency division multiplexing (CO-OFDM) system is susceptible to nonlinear effects because of its intrinsic high peak-to-average ratio (PAPR). In this paper, two methods based on discrete Fourier transform spread (DFT-S) are introduced to improve the nonlinear tolerance of CO-OFDM system by extracting and fusing data features, which extract the features of communication data and perform feature fusion to get the important significance of the data. The first one is based on the 2-ary amplitude shift keying (2-ASK) modulation and the Hermitian symmetry of DFT, and the other one is based on the advantages of selective mapping algorithm and characteristics of data subcarrier mapping mode which has a significant impact on PAPR. Simulation results show that compared with the traditional CO-OFDM and DFT-S CO-OFDM systems, the improved methods have better performance in terms of PAPR and BER.

Analysing spectral lines in Gaia low-resolution spectra

Context. With its third data release, European Space Agency’s Gaia mission has published the first set of low-resolution spectra for a large number of celestial objects. However, these spectra differ in their nature from typical spectroscopic data, as they do not consist of wavelength samples with associated flux values. Instead, they are represented by a linear combination of Hermite functions. Aims. We derive an approach to studying spectral lines that is robust and efficient for spectra that are represented as a linear combination of Hermite functions. Methods. For this purpose, we combined established computational methods for orthogonal polynomials with the peculiar mathematical properties of Hermite functions and basic properties of the Gaia spectrophotometers. In particular, we made use of simple computatios for the derivatives of linear combinations of Hermite functions and their roots. Results. A simple and efficient computational method for deriving the position in wavelength, statistical significance, and line strengths is presented for spectra represented by a linear combination of Hermite functions. The derived method is fast and robust enough to be applied to large numbers of Gaia spectra without the need for high-performance computing resources or human interference. We present example applications to hydrogen Balmer lines, He I lines, and a broad interstellar band in Gaia DR3 low-resolution spectra.

An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.

A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions.

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments.

A correspondence between the free and interacting field

We discover a correspondence between the free field and the interacting states. This correspondence is firstly given from the fact that the free propagator can be converted into a tower of propagators for massive states, when expanded with the Hermite function basis. The equivalence of propagators reveals that in this particular case the duality can naturally be regarded as the equivalence of one theory on the plane wave basis to the other on the Hermite function basis. More generally, the Hermite function basis provides an alternative quantization process with the creation/annihilation operators that correspond directly to the interacting fields. As an illustration, we apply this basis to the 3+1 dimensional Yang–Mills theory, where the three-dimensional space being reduced through the Hermite function basis, and an auxiliary parameter omega denotes for string tension. At large omega limit, with then considering only the lowest order Hermite function (Lowest Landau Level), the equivalent action becomes the Banks–Fischler–Shenker–Susskind (BFSS) matrix model. At small omega limit, the perturbative series summed over all orders of Hermite function gives a massive gluon propagator.

Uncertainty Principle for Hermite Functions and Null-Controllability with Sensor Sets of Decaying Density

We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset Ssubset mathbb {R}^d ensuring that the L^2-seminorm associated to S is equivalent to the full L^2-norm on mathbb {R}^d when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of S. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from S. In all our results, the set S may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.

Using parity to accelerate Hermite function computations: Zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summation

Although Hermite functions have been studied for over a century and have been useful for analytical and numerical solutions in a myriad of areas, the theory of Hermite functions has gaps. This article is a unified treatment of all the operations—quadrature, summation, differentiation, and rootfinding—that can be accelerated by exploiting parity. Any function u(y) can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. Suppose that the quadrature on y∈[−∞,∞] is symmetric in the sense that if yj is a quadrature abscissa with weight wj, then −yj is also a quadrature point with weight wj. The number of multiplications is halved by evaluating the quadrature as ∫−∞∞u(y)dy≈w0u(0)+∑n=1Mwnu(yn)+u(−yn). Parity is equally useful in computing the zeros, maxima and minima of a truncated Hermite series of degree N for the important special cases that the series terms are all either symmetric or antisymmetric. The zeros and critical points are the eigenvalues of a companion matrix whose dimension is N/2 instead of N. In addition, for Hermite functions, we show that parity exploitation halves the dimension of the Jacobi matrix (a special case of the companion matrix) whose eigenvalues are the abcissas of Hermite–Gauss quadrature. The number of floating point operations for the recursion for the weights can likewise be halved. Lastly, the same is true for Chenshaw summation of a Hermite series.

Hermite functions and inner product in Sobolev space

Hermite functions and inner product in Sobolev space

Time Synchronization Technique Hardware Implementation for OFDM Systems With Hermitian Symmetry for VLC Applications

Time Synchronization Technique Hardware Implementation for OFDM Systems With Hermitian Symmetry for VLC Applications

Applying the most frequent values assisted Hilbert transform into seismic attributes

In the present study, we developed a new method to calculate two of the most important seismic attributes that can able to give a huge impact in the interpretation field leading to a better geological and geophysical interpretation of the data. The first attribute was the Reflection Intensity with great improvement in detection of thin layers that show low reflection coefficient and enhance the ability to follow unconformity, the other one is the Instantaneous Phase to discover the discontinuity of events. The new method depends on combining the most frequent values (MFV) with inversion to calculate the Hilbert transform and applying hybrid Seismic Denoising, using also dual inversion and calculated Steiner weight from the iterative reweighted least square method (IRLS). Scaled Hermite function in a series expansion was used to discretize the spectrum of the time trace. The procedure followed with adjusting the process using multi-windows technical on the generated synthetic seismic traces loaded with Gaussian and Cauchy noise at the same time. The result of the method was shown significant effects on outliers or normal Gaussian noise when applied to two seismic attributes better than the old way using the Discrete Fourier Transform (DFT) method.

Approximation of Wave Packets on the Real Line

In this paper we compare three different orthogonal systems in textrm{L}_2({mathbb {R}}) which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist–Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary—this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the textrm{L}_2({mathbb {R}}) norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist–Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets.

Construction of sequences with high nonlinear complexity from a generalization of the Hermitian function field

For [Formula: see text] an odd integer, we provide a sequence from the function field [Formula: see text] of the maximal curve over [Formula: see text] defined by the affine equation [Formula: see text]. This sequence has high nonlinear complexity, and this fact comes from the existence of a rational function on [Formula: see text] with pole divisor of small degree, and support in certain [Formula: see text] rational places.

A Review of Collocation Approximations to Solutions of Differential Equations

This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results.

On multivariate orthogonal polynomials and elementary symmetric functions

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.

Non-Hermite Spinor Boltzmann Equation and Its Hermitization

It is known that the Wigner distribution function in the quantum Boltzmann equation is not positive elsewhere, which causes trouble for its application to the mesoscopic transport problem. Similarly, if we extend the quantum Boltzmann equation to the spinor Boltzmann equation, which includes the spin freedom in the usual Wigner distribution, and the spinor matrix distribution function therein is not Hermite, we must make this non-Hermite spinor Boltzmann equation Hermitization. So in this paper, we propose a spinor Boltzmann equation with Hermite distribution function and then obtain the equations of continuity satisfied by the charge density, charge current, spin accumulation, and spin current. The numerical results for these physical quantities are illustrated by an example of spin-polarized transport scattered by impurities in a system of spintronics. Their differences compared with the usual spinor Boltzmann equation are shown.

A modified Tikhonov regularization for unknown source in space fractional diffusion equation

Abstract In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the method.

Sampling Trajectories for the Short-Time Fourier Transform

We study the problem of stable reconstruction of the short-time Fourier transform from samples taken from trajectories in {mathbb {R}}^2. We first investigate the interplay between relative density of the trajectory and the reconstruction property. Later, we consider spiraling curves, a special class of trajectories, and connect sampling and uniqueness properties of these sets. Moreover, we show that for window functions given by a linear combination of Hermite functions, it is indeed possible to stably reconstruct from samples on some particular natural choices of spiraling curves.

Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials.

Adaptive Hermite spectral methods in unbounded domains

A novel adaptive spectral method has been recently developed to numerically solve partial differential equations (PDEs) in unbounded domains. To achieve accuracy and improve efficiency, the method relies on the dynamic adjustment of three key tunable parameters: the scaling factor, a displacement of the basis functions, and the spectral expansion order. In this paper, we perform the first numerical analysis of the adaptive spectral method using generalized Hermite functions in both one- and multi-dimensional problems. Our analysis reveals why adaptive spectral methods work well when a “frequency indicator” of the numerical solution is controlled. We then investigate how the implementation of the adaptive spectral methods affects numerical results, thereby providing guidelines for the proper tuning of parameters. Finally, we further improve performance by extending the adaptive methods to allow bidirectional basis function translation, and the prospect of carrying out similar numerical analysis to solving PDEs arising from realistic difficult-to-solve unbounded models with adaptive spectral methods is also briefly discussed.