Hermite-Chebyshev pseudospectral method for inhomogeneous superconducting strip problems and magnetic flux pump modeling

Numerical simulation of superconducting devices is a powerful tool for understanding the principles of their work and improving their design. Usually, such simulations are based on a finite element method but, recently, a different approach, based on the spectral technique, has been presented for very efficient solution of several applied superconductivity problems described by one-dimensional integro-differential equations or a system of such equations. Here we propose a new pseudospectral method for two-dimensional magnetization and transport current superconducting strip problems with an arbitrary current-voltage relation, spatially inhomogeneous strips, and strips in a nonuniform applied field. The method is based on the bivariate expansions in Chebyshev polynomials and Hermite functions. It can be used for numerical modeling magnetic flux pumps of different types and investigating AC losses in coated conductors with local defects. Using a realistic two-dimensional version of the superconducting dynamo benchmark problem as an example, we showed that our new method is a competitive alternative to finite element methods.

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.

Resolving of Overlapping Asymmetrical Chromatographic Peaks by Using Wavelet-Transform and Gram-Charlier Peak Model

The paper describes a method for resolving overlapping asymmetric peaks that make up a chromatogram. The presented method uses the Gram-Charlier model in the form of the first three terms of the Gram-Charlier series as a basis. Using the wavelet transform, the parameters of this model are determined, which is used to describe a single or overlapping chromatographic peak. Hermitian wavelets of the first four orders are used in the computation of the wavelet transform. To speed up the computation of multiple wavelet transforms, the possibility of coding a signal using the Chebyshev-Hermite functions is considered in order to further restore the set of wavelet transforms simultaneously. According to the presented method, the parameters of the peaks are determined by analytical expressions without using the numerical approximation of the chromatogram by the peak model, which avoids the disadvantages of the numerical approach. The resulting method is used to resolve overlapping asymmetric peaks. The advantage of the method over others is shown by calculating the area of each of the resolved peaks.

Hilbert transform using a robust geostatistical method

In this paper, we introduced an efficient inversion method for Hilbert transform calculation which can be able to eliminate the outlier noise. The Most Frequent Value method (MFV) developed by Steiner merged with an inversion-based Fourier transform to introduce a powerful Fourier transform. The Fourier transform process (IRLS-FT) ability to noise overthrow efficiency and refusal to outliers make it an applicable method in the field of seismic data processing. In the first part of the study, we introduced the Hilbert transform stand on a efficient inversion, after that as an example we obtain the absolute value of the analytical signal which can be used as an attribute gauge. The method depends on a dual inversion, first we obtain the Fourier spectrum of the time signal via inversion, after that, the spectrum calculated via transformation of Hilbert transforms into time range using a efficient inversion. Steiner Weights is used later and calculated using the Iterative Reweighting Least Squares (IRLS) method (efficient inverse Fourier transform). Hermite functions in a series expansion are used to discretize the spectrum of the signal in time. These expansion coefficients are the unknowns in this case. The test procedure was made on a Ricker wavelet signal loaded with Cauchy distribution noise to test the new Hilbert transform. The method shows very good resistance to outlier noises better than the conventional (DFT) method.

A Low-Latency, Low-Power FPGA Implementation of ECG Signal Characterization Using Hermite Polynomials

Automatic ECG signal characterization is of critical importance in patient monitoring and diagnosis. This process is computationally intensive, and low-power, online (real-time) solutions to this problem are of great interest. In this paper, we present a novel, dedicated hardware implementation of the ECG signal processing chain based on Hermite functions, aiming for real-time processing. Starting from 12-bit ADC samples of the ECG signal, the hardware implements filtering, peak and QRS detection, and least-squares Hermite polynomial fit on heartbeats. This hardware module can be used to compress ECG data or to perform beat classification. The hardware implementation has been validated on a Field Programmable Gate Array (FPGA). The implementation is generated using an algorithm-to-hardware compiler tool-chain and the resulting hardware is characterized using a low-cost off-the-shelf FPGA card. The single-beat best-fit computation latency when using six Hermite basis polynomials is under 1 s with a throughput of 3 beats/s and with an average power dissipation around 28 mW, demonstrating true real-time applicability.

Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schrodinger operators

In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schr\"{o}dinger operators in R^d. As a generalisation of the G. Szego's family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|^{2\mu} e^{-|x|^2} (resp. |x|^{2\mu}) in R^d. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product [u,v]_{H^s(R^d)}=((-\Delta)^{s/2}u, (-\Delta)^{s/2} v)_{R^d} associated with the IFL of order s>0. As an immediate consequence, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient in solving PDEs with the fractional Schrodinger operator: (-\Delta)^s +|x|^{2\mu} with s\in (0,1] and \mu>-1/2 in R^d. We construct the second family of multivariate nontensorial Muntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrodinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrodinger eigenvalue problems.

Heisenberg–Weyl Groups and Generalized Hermite Functions

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.