Hermite Functions Research Articles

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

A quadrilateral inverse plate element for real-time shape-sensing and structural health monitoring of thin plate structures

The inverse finite element method (iFEM) emerged as a powerful tool in shape-sensing and structural health monitoring (SHM) applications with distinct advantages over existing methodologies. In this study, a quadrilateral inverse-plate element is formulated via a sub-parametric approach using bi-linear and non-conforming cubic Hermite basis functions for engineering structures, which can be modeled as thin plates. Numerical validation involves dense and assumed sparse sensor arrangements for in-plane, out-of-plane, and mixed general loading conditions. iFEM analysis reveals efficient monotonic convergence to analytical and high-fidelity finite element reference solutions. After successful numerical validation, defect detection analysis is performed considering minute geometric discontinuities and structural stiffness reduction because of latent subsurface defects under tensile and transverse loading conditions. The inverse formulation successfully resolves the presence of simulated defects under a sparse sensor arrangement. The proposed inverse-plate element is accurate in the full-field reconstruction of shape-sensing profiles and reliable in defect identification and quantification in thin plate structures.

On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

The frame set of a window ϕ∈L2(R) is the subset of all lattice parameters (α,β)∈R+2 such that G(ϕ,α,β)={e2πiβm⋅ϕ(⋅−αk):k,m∈Z} forms a frame for L2(R). In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters α,β>0 with αβ<1, there exists a γ>0 depending on αβ such that G(ϕ(γ⋅),α,β) forms a frame for L2(R). Our results give explicit frame bounds.

Comparison of Fourier and Trigonometric Transform based Multicarrier Modulations for Visible Light Communication

It is predicted that the radio frequency spectrum will be insufficient in the near future due to the increase in wireless data. Visible Light Communication (VLC) is an alternative solution, which promises high speeds. Similar to other wireless communication systems, VLC systems prefer Multicarrier Modulation (MCM), but the signals are converted to be real and unipolar before transmission for optical communication. In this paper, two optical MCM groups that utilize Discrete Fourier Transform (DFT) and Discrete Trigonometric Transform (DTT) are questioned with respect to Bit Error Rate (BER), spectral efficiency and complexity. DFT based techniques use complex mapped signals together with their Hermitian symmetries to obtain real output signals, while DTT based techniques already output real signals when the input signal is real mapped. It is seen that DFT based techniques have lower BERs because of used mapping. DTT based techniques improve spectral efficiency, but they are limited to real mappings with higher error rates. For both transformations, the real signals are made unipolar by adding a bias (DCO-MCM), by asymmetrically clipping (ACO-MCM) or by sending positive and negative values separately (UnO-MCM). It is shown that, adding a dc bias (DCO-MCM) increases BERs, where ACO-MCM and UnO-MCM have close performances with lower BERs.

Finite element approach for free vibration and transient response of bi-directional functionally graded sandwich porous skew-plates with variable thickness subjected to blast load

At the first time, the finite element method was used to model and analyze the free vibration and transient response of non-uniform thickness bi-directional functionally graded sandwich porous (BFGSP) skew plates. The whole BFGSP skew-plates is placed on a variable visco-elastic foundation (VEF) in the hygro-thermal environment and subjected to the blast load. The BFGSP skew-plate thickness is permitted to vary non-linearly over both the length and width of the skew-plate, thereby faithfully representing the real behavior of the structure itself. The analysis is based on a four-node planar quadrilateral element with eight degrees of freedom per node, which is approximated using Lagrange Q4 shape function and C1 level non-conforming Hermite shape function based on refined higher-order shear deformation plate theory. The forced vibration parameters of the non-uniform thickness BFGSP skew-plate are fully determined using Hamilton's principle and the Newmark-β direct integration technique. Accuracy of the calculation program is validated by comparing its numerical results with those from reputable sources. Furthermore, a thorough assessment is conducted to determine the impact of various parameters on the free and forced vibration responses of the non-uniform thickness BFGSP skew-plate. The findings of the paper may be used in the development of civil and military structures in situations that are prone to exceptional forces, such as explosions and impacts load.

Almost everywhere convergence of Bochner–Riesz means for the twisted Laplacian

Let L \mathcal L denote the twisted Laplacian in C d \mathbb {C}^d . We study almost everywhere convergence of the Bochner–Riesz mean S t δ ( L ) f S^\delta _{t}(\mathcal L) f of f ∈ L p ( C d ) f\in L^p(\mathbb C^d) as t → ∞ t\to \infty , which is an expansion of f f in the special Hermite functions. For 2 ≤ p ≤ ∞ 2\le p\le \infty , we obtain the sharp range of the summability indices δ \delta for which the convergence of S t δ ( L ) f S^\delta _{t}(\mathcal L) f holds for all f ∈ L p ( C d ) f\in L^p(\mathbb {C}^d) .

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Biorthogonal Majorana zero modes, ELC waves and soliton-fermion duality in non-Hermitian sl(2) affine Toda coupled to fermions

We study a non-Hermitian (NH) sl(2) affine Toda model coupled to fermions through soliton theory techniques and the realizations of the pseudo-chiral and pseudo- Hermitian symmetries. The interplay of non-Hermiticity, integrability, nonlinearity, and topology significantly influence the formation and behavior of a continuum of bound state modes (CBM) and extended waves in the localized continuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalar field topological charges and winding numbers in the spectral topology are uncovered. The biorthogonal Majorana zero modes, dual to the NH Toda solitons with topological charges 2πargz=±i=±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{2}{\\pi}\\arg \\left(z=\\pm i\\right)=\\pm 1 $$\\end{document}, appear at the complex-energy point gap and are pinned at zero energy. The zero eigenvalue λ(z = ± i) = 0, besides being a zero mode, plays the role of exceptional points (EPs), and each EP separates a real eigenvalue A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-symmetric and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-symmetry broken regimes for an antilinear symmetry A∈PTγ5PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A}\\in \\left\\{\\mathcal{PT},{\\gamma}_5\\mathcal{PT}\\right\\} $$\\end{document}. Our findings improve the understanding of exotic quantum states, but also paves the way for future research in harnessing non-Hermitian phenomena for topological quantum computation, as well as the exploration of integrability and NH solitons in the theory of topological phases of matter.

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.

Orthonormal expansions for translation-invariant kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R). This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

Linear finite element formulation for free vibration and buckling analyses of multi-directional FGP doubly curved shallow shells in thermal environment

ABSTRACT This paper presents the free vibration and buckling analyses of multi-directional functionally graded porous doubly curved shallow shells resting on Pasternak elastic foundations in a thermal environment. For the first time, a linear finite element formulation using the refined high-order shear deformation theory, combined with the Hermitian function, is developed to compute the vibration and buckling behaviour of doubly curved shallow shells. Mechanical properties of MFGP material change according to the length, width and thickness directions and the porosity distribution including even and uneven. The doubly curved shallow shell is operated in uniform, linear and non-linear temperature environments. The convergence and accuracy of the proposed method are verified by comparing numerical results with published works. In addition, a comprehensive numerical investigation was carried out to evaluate the influence of parameters on the natural frequency and critical buckling behaviour of MFGP doubly curved shallow shells.

Dynamic response of magneto-electro-elastic composite plates lying on visco-Pasternak medium subjected to blast load

This is the first endeavor to present a finite element approach to predict free oscillation and transient characteristics of magneto-electro-elastic (MEE) composite rectangular and elliptical plates resting on the visco-Pasternak medium in a hygro-thermal environment subjected to blast load. The variation of electric and magnetic potentials along the thickness direction of the MEE plate is determined via the Maxwell equation and magneto-electric boundary condition. The governing equations of motion for the MEE plate are obtained using refined higher-order shear plate theory and Hamilton's principle. Stiffness matrices, mass matrices, damping matrices, and force vectors of the plate are derived using a four-node quadrilateral element with eight degrees of freedom per node approximated using C1-order non-conforming Hermite and Lagrange functions. In addition, the Navier close-form solutions for rectangular plates with simply supported boundary conditions are used as a useful comparison tool for the numerical solutions. Dynamic response results of plates were obtained using Newmark's direct integration written by Matlab's programming. The accuracy of the method is verified through numerical comparison with confidence statements. The results show that the applied magnetic and electric potential highly inspires the transient responses. These results can be used in vibration control studies of structures and structures subjected to explosive or low-velocity impact loads.

Enhancing spectral efficiency with low complexity filtered‐orthogonal frequency division multiplexing in visible light communication system

AbstractThe filtered‐orthogonal frequency division multiplexing (F‐OFDM) scheme has gained attention as a promising solution in the field of visible light communication (VLC) systems. One crucial aspect in VLC is the conversion of the complex F‐OFDM signal into a real signal that corresponds with direct detection and intensity modulation. Traditionally, achieving a real F‐OFDM signal has involved imposing Hermitian symmetry (HS) on the samples of the Inverse Fast Fourier transform (IFFT), which requires 2N‐point IFFT and obtains an N‐point FFT, thus adding complexity. In this study, a novel approach is presented and implemented, aiming to enhance spectral efficiency and reduce system complexity by generating a real F‐OFDM signal without relying on HS. This approach is then compared with HS‐free (HSF)‐OFDM, direct current biased optical OFDM, and asymmetrically clipped optical OFDM. The suggested method offers a remarkable improvement of ~50% in the required IFFT/FFT volume. Consequently, this method reduces hardware complexity and power usage compared with the traditional F‐OFDM method. Moreover, regarding error rates, the proposed method demonstrates better spectral efficiency than HSF‐OFDM.

Correcting Hardening Artifacts of Aero-Engine Blades with an Iterative Linear Fitting Technique Framework.

Aero engines are the key power source for aerospace vehicles. Cermet turbine blades are the guarantee for the new-generation fighters to improve aero-engine overall performance. X-ray non-destructive reconstruction can obtain the internal structure and morphology of cermet turbine blades. However, the beam hardening effect causes artifacts in objects and affects the reconstruction quality, which is an issue that needs to be solved urgently. This study proposes a hardening-correction framework for industrial computed tomography (ICT) images based on iterative linear fitting. First, an iterative binarization was performed to improve the penetration length accuracy of the forward projection. Then, the proposed linear fitting technology combined with the Hermite function model is derived and analyzed to obtain suitable parameters of blade data. Finally, the fitting curves of the blade data, using the proposed method and the traditional polynomial fitting method, were analyzed and compared and were used to correct the engine turbine blade projection data to reconstruct different groups of tomographic images. Different groups of tomographic images were analyzed using three quantitative image quality evaluation indicators. The results show that the root-mean-square error (RMSE) of the tomographic image obtained by the proposed framework is 0.0133, which is lower than that of the compared method. The peak signal-to-noise ratio (PSNR) is 37.7050 dB and the feature structural similarity (FSIM) is 0.9881, which are both higher than that of the compared method. The proposed method improves the hardening-artifact-correction capability and can obtain higher-quality images, which provides new ideas for the development of imaging and detection of new-generation aero-engine turbine blades.

Critical scaling of a two-orbital topological model with extended neighboring couplings

Extended-range models are the interesting systems, which has been widely used to understand the non-local properties of the fermions at quantum scale. We aim to study the interplay between criticality and extended range couplings under various symmetry constraints. Here, we consider a two orbital Bernevig–Hughes–Zhang model in one dimension with longer (finite neighbor) and long-range (infinite neighbor) couplings. We study the behavior of model using scaling laws and universality class for models with Hermitian, parity-time (PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}) symmetric and broken time-reversal symmetries. We observe the interesting results on multi-criticalities, where the universality class of critical exponent is different than the normal criticalities. Also, the results can be generalized by considering the interplay between criticalities and different symmetry classes of Hamiltonian. Also, with the introduction of extended-range of coupling, there occurs different criticalities, and we provide the analogy to characterize their universality classes. We also show the violation of Lorentz invariance at multi-criticalities and evaluation of short-range limit in long-range models as the highlights of this work.

Discrete Quantum Harmonic oscillator and Kravchuk transform

Abstract. We consider a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the almost second-order convergence of these discrete functions towards Hermite functions, uniformly for large numbers of modes. We then describe an efficient way to simulate these eigenfunctions and the corresponding transformation. We finally show some numerical experiments corroborating our different results.

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).

Weighted Hermite Variable Projection Networks for Classifying Visually Evoked Potentials.

The occipital cortex responds to visual stimuli regardless of a patient's level of consciousness or attention, offering a noninvasive diagnostic tool for both ophthalmologists and neurologists. This response signal manifests as a unique waveform referred to as the visually evoked potential (VEP), which can be extracted from the electroencephalogram (EEG) activity of a human being. We propose a trainable VEP representation to disentangle the underlying explanatory factors of the data. To enhance the learning process with domain knowledge, we present an innovative parameterization of classical Hermite functions that effectively captures VEP pattern variations arising from patient-specific factors, disorders, and measurement setup influences. Then, we introduce a differentiable variable projection (VP) layer to fuse Hermite basis function expansions (BFEs) of VEP signals with machine learning (ML) approaches. We prove the existence of an optimal set of parameters in the least-squares sense, assess the representation power of such layers, and calculate their analytical derivatives, which allows us to utilize backpropagation for training. Finally, we evaluate the effectiveness of the proposed learning framework in VEP-based color classification. To achieve this, we have designed a novel measurement system dedicated to intraoperative clinical use cases, which presents new ways for patient monitoring during neurosurgical procedures.

Variable Projection Support Vector Machines and Some Applications Using Adaptive Hermite Expansions.

In this paper, we develop the so-called variable projection support vector machine (VP-SVM) algorithm that is a generalization of the classical SVM. In fact, the VP block serves as an automatic feature extractor to the SVM, which are trained simultaneously. We consider the primal form of the arising optimization task and investigate the use of nonlinear kernels. We show that by choosing the so-called adaptive Hermite function system as the basis of the orthogonal projections in our classification scheme, several real-world signal processing problems can be successfully solved. In particular, we test the effectiveness of our method in two case studies corresponding to anomaly detection. First, we consider the detection of abnormal peaks in accelerometer data caused by sensor malfunction. Then, we show that the proposed classification algorithm can be used to detect abnormalities in ECG data. Our experiments show that the proposed method produces comparable results to the state-of-the-art while retaining desired properties of SVM classification such as light weight architecture and interpretability. We implement the proposed method on a microcontroller and demonstrate its ability to be used for real-time applications. To further minimize computational cost, discrete orthogonal adaptive Hermite functions are introduced for the first time.

Generalized Heisenberg-Weyl groups and Hermite functions

A generalisation of Euclidean and pseudo-Euclidean groups is presented, where the Weyl-Heisenberg groups, well known in quantum mechanics, are involved. A new family of groups is obtained including all the above-mentioned groups as subgroups. Symmetries, like self-similarity and invariance with respect to the orientation of the axes, are properly included in the structure of this new family of groups. Generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of these new groups, are introduced.