Polynomial Basis Research Articles

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.

Conservative discontinuous Galerkin interpolation: Sheared boundary conditions

Local studies of accretion disks and laboratory magnetized plasmas employ analytical coordinate mappings that introduce sheared boundary conditions (BCs). We present a discontinuous Galerkin (DG) algorithm to apply such BCs based on projections and quadrature-free integration. The procedure is high-order accurate, preserves moments exactly and works in multiple dimensions. Tests of increasing complexity are provided, beginning with translations of one and two dimensional fields, followed by 3D and 5D simulations with sheared (twist-shift) BCs. Results show that the algorithm is (p+1)-order accurate in the DG representation and (p+2)-order accurate in the cell averages, with p being the order of the polynomial basis. Quantification of the algorithm's hyperdiffusion and discussion of aliasing errors are given. This technique enables conservative local simulations of plasma turbulence with DG, not possible until now.

The effect of basis polynomial degree on the performance of compressible flow simulations employing a split-form DG method

High order methods have garnered significant interest recently for simulating compressible flows due to their low dissipation nature. In this regard, the use of discontinuous Galerkin method has been actively pursued due to the advantages ensuing from its compact interpolation. However, these methods generally require employing a limiter for stable computation of supersonic compressible flows that contain shocks. Many limiters are designed to incorporate ideas from robust second order implementations of the Finite Volume Method for shock capturing. It is therefore prudent to examine the effectiveness of high order discontinuous Galerkin solutions when simulating flows containing discontinuities. To this end, the present work investigates the variation of accuracy per degrees of freedom as the basis polynomial degree is changed. Several standard one-dimensional and two-dimensional compressible flow cases are simulated for multiple basis polynomial degrees while employing different meshes to keep the degrees of freedom fixed. It is found that high order solutions are more efficient than second order solution as regards the accuracy per degrees of freedom. The improvement in this metric with increasing basis polynomial degree plateaus at cubic basis degree for cases containing strong discontinuities (shock Mach numbers greater than 3), and shifts to a further higher basis in simpler problems. However, in problems containing slowly moving shocks and or weak discontinuities, high order solutions generate increased numerical oscillations, suggesting the need for improvement in anti-aliasing techniques.

The general Bernstein function: Application to χ‐fractional differential equations

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

Raman spectroscopy assisted support vector machine: a steadfast tool for noninvasive classification of urinary glucose of diabetes mellitus

The currentstudy presents the application of a Raman spectroscopy-assisted Support Vector Machine (SVM) algorithm for the analysis and classification of urine samples of diabetes mellitus and healthy control subjects. Raman spectra of pathological as well as control samples were acquired in the spectral range of 600 to 1800 cm−1. Spectral differences were observed in Raman spectra of diabetic and non-diabetic urine samples based on variations in their chemical composition. Effective machine-learning techniques were utilized to highlight the spectral differences between diabetic and normal urine samples. Data classification was carried out by using SVM models with different kernel functions: radial basis function (RBF), polynomial function, linear function, and sigmoid. The algorithm classifies data sets into different classes based on even subtle variations in the spectral features. The classification performance of the model was evaluated by employing a 10-fold cross-validation method. The best performance was obtained with RBF having a diagnostic accuracy, precision, sensitivity, specificity, F-score, and AUC of approximately 94%, 100%, 90%, 100%, 0.94, and 0.99 respectively.

Feature Extraction of a Planetary Gearbox Based on the KPCA Dual-Kernel Function Optimized by the Swarm Intelligent Fusion Algorithm

The feature extraction problem of coupled vibration signals with multiple fault modes of planetary gears has not been solved effectively. At present, kernel principal component analysis (KPCA) is usually used to solve nonlinear feature extraction problems, but the kernel function selection and its blind parameter setting greatly affect the performance of the algorithm. For the optimization of the kernel parameters, it is very urgent to study the theoretical modeling to improve the performance of kernel principal component analysis. Aiming at the deficiency of kernel principal component analysis using the single-kernel function for the nonlinear mapping of feature extraction, a dual-kernel function based on the flexible linear combination of a radial basis kernel function and polynomial kernel function is proposed. In order to increase the scientificity of setting the kernel parameters and the flexible weight coefficient, a mathematical model for dual-kernel parameter optimization was constructed based on a Fisher criterion discriminant analysis. In addition, this paper puts forward a swarm intelligent fusion algorithm to increase this method’s advantages for optimization problems, involving the shuffled frog leaping algorithm combined with particle swarm optimization (SFLA-PSO). The new fusion algorithm was applied to optimize the kernel parameters to improve the performance of KPCA nonlinear mapping. The optimized dual-kernel function KPCA (DKKPCA) was applied to the feature extraction of planetary gear wear damage, and had a good identification effect on the fuzzy damage boundary of the planetary gearbox. The conclusion is that the DKKPCA optimized by the SFLA-PSO swarm intelligent fusion algorithm not only effectively improves the performance of feature extraction, but also enables the adaptive selection of parameters for the dual-kernel function and the adjustment of weights for the basic kernel function through a certain degree of optimization; so, this method has great potential for practical use.

Detection of oral mucosal lesions by the fluorescence spectroscopy and classification of cancerous stages by support vector machine.

Detection of oral mucosal lesions has been performed by an in-house developed fluorescence-based portable device in the present study. A laser diode of 405 nm wavelength and a UV-visible spectrometer are utilized in the portable device as excitation and detection sources. At the 405 nm excitation wavelength, the flavin adenine dinucleotide (FAD) band at 500 nm and three porphyrin bands at 634, 676, and 703 nm are observed in the fluorescence spectrum of the oral cavity tissue. We have conducted this clinical study on a total of 189 tissue sites of 36 oral squamous cell carcinoma (OSCC) patients, 18 dysplastic (precancerous) patients, and 34 volunteers. Analysis of the fluorescence data has been performed by using the principal component analysis (PCA) method and support vector machine (SVM) classifier. PCA is applied first in the spectral data to reduce the dimension, and then classification among the three groups has been executed by employing the SVM. The SVM classifier includes linear, radial basis function (RBF), polynomial, and sigmoid kernels, and their classification efficacies are computed. Linear and RBF kernels on the testing data sets differentiated OSCC and dysplasia to normal with an accuracy of 100% and OSCC to dysplasia with an accuracy of 95% and 97%, respectively. Polynomial and sigmoid kernels showed less accuracy values among the groups ranging from 48 to 88% and 51 to 100%, respectively. The result indicates that fluorescence spectroscopy and the SVM classifier can help to identify early oral mucosal lesions with significant high accuracy.

An adaptive Ritz formulation for progressive damage modelling in variable angle tow composite plates

In this work, an adaptive Ritz model for the analysis of variable angle tow composite plates featuring damage initiation and evolution under progressive loading is proposed, developed, implemented and tested. The plate kinematics is represented employing a first-order shear deformation theory, while the plate equilibrium equations at a given load step are obtained by minimizing the structure potential energy. The constitutive behaviour is modelled within the framework of continuum damage mechanics. In particular the initiation and evolution of damage, up to failure, are tracked by defining irreversible damage indices related to both fibres and matrix, both in tensile or compression loading. The discrete equations are then obtained by assuming a polynomial Ritz approximation of the primary kinematic variables in the energy minimization. Preliminary tests show how the application of the method as a single-domain approach induces the emergence of problematic spurious effects, related to Gibbs artefacts due to the inability of the selected polynomial basis to represent damage localization. An adaptive multi-domain technique is thus proposed to circumvent such issues, which has been successfully validated by benchmark tests. Eventually, original results about variable angle tow plates featuring damage evolution under progressive loading are presented.

Research on structural parameter optimization of a new axial inlet hydrocyclone separator based on response surface optimization method.

Aiming at the problem that the current hydrocyclone separator is affected by multiple structural parameters and there is interaction between the multiple structural parameters, it is difficult to determine the optimal structure. Taking the new axial inlet hydrocyclone separator as the research object, a fast parametric optimization method based on response surface optimization method is proposed. The overflow outlet diameter, overflow tube depth and small cone length, which have significant influence on the separation efficiency of the axial inlet hydrocyclone separator, are selected as the optimal variables, and the cyclone separation efficiency is selected as the response index. A mathematical driving model between the response index and the optimal variables is constructed by using the second-order polynomial basis function. The optimal structural parameters of the new axial inlet hydrocyclone separator are obtained through the response optimization of the parameter variables in the global response range through the mathematical driving model, and the numerical simulation method and laboratory test are double verified. The results demonstrated that the axial inlet hydrocyclone achieved the highest separation efficiency within the studied operational parameter range when the overflow pipe diameter was 6mm, the overflow pipe depth was 20mm, and the small cone length was 60mm. The separation efficiency improved from 89% to 93%. The rapid optimization of the structural parameters of the axial inlet hydrocyclone was successfully accomplished using the response surface optimization method.

Enhancing curve and surface applications with trigonometric polynomial basis functions.

This article primarily focuses on the utilization and importance of parametric curves in surface design. It delves into the construction and applications of parametric curves, exploring the implementation of trigonometric polynomial basis functions that possess two shape parameters. Initially, these basis functions are employed in constructing both rational and non-rational curves. Later, they are employed to define the surfaces generated by these curves. The discussion includes rational surfaces, tensor product surfaces, and various specialized surfaces. The aim is to provide a comprehensive understanding of the role and potential of parametric curves in surface design.

On the approximation of analytic functions by infinite series of fractional Ruscheweyh derivatives bases

<abstract><p>This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the $ \mathbb{T}_{\rho} $-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results.</p></abstract>

An application for predicting phishing attacks: A case of implementing a support vector machine learning model

The imminent threat that phishing websites poses is a major concern for internet users worldwide. These fraudulent websites are crafted by cyber attackers to appear trustworthy and deceive vulnerable users into divulging confidential data like medical health records, credit card details, passwords, and Personal Identifiable information (PII). To bait their victims, cybercriminals employ tactics such as social engineering, spear-phishing attacks, and email phishing scams. As a result, unsuspecting individuals may be enticed to visit these websites, putting their sensitive information at risk. This work presents an application designed to predict phishing attacks after comparing polynomial and radial basis function of support vector machine (SVM). The proposed application leverages a dataset of known legitimate, suspicious and phishing attacks stored in a database and employs an SVM algorithm for classification based on user input. The application provides a user-friendly graphical user interface (GUI) that allows reporting of new phishing incidents based on the features that have strong relationship in determining if a website is phishing or not. The proposed application utilizes the inherent scalability of database technology to support record expansion whenever there is an instance of a user initiating phishing prediction thereby, making it suitable for use in a wide range of organizational settings.

Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures

<abstract><p>In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $ L_{\infty} $ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.</p></abstract>

Full Breit Hamiltonian in the Multiwavelets Framework.

New techniques in core-electron spectroscopy are necessary to resolve the structures of oxides of f-elements and other strongly correlated materials that are present only as powders and not as single crystals. Thus, accurate quantum chemical methods must be developed to calculate core spectroscopic properties in such materials. In this contribution, we present an important development in this direction, extending our fully adaptive real-space multiwavelet basis framework to tackle the four-component Dirac-Coulomb-Breit Hamiltonian. We show that multiwavelets can reproduce one-dimensional grid-based approaches. They are however a fully three-dimensional approach which can later be extended to molecules and materials. Our multiwavelet implementation attained precise results irrespective of the chosen nuclear model, provided that the error threshold is tight enough and that the chosen polynomial basis is sufficiently large. Furthermore, our results confirmed that in two-electron species, the magnetic and Gauge contributions from s-orbitals are identical in magnitude and can account for the experimental evidence from K and L edges.

Accurate polynomial fitting and evaluation via Arnoldi

Given sample data points $ \{(x_j,f_j)\}_{j = 1}^N $, in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415], an Arnoldi-based procedure is proposed to accurately evaluate the best fitting polynomial, in the least squares sense, at new nodes $ \{s_j\}_{j = 1}^M $, based on the Vandermonde basis. Numerical tests indicated that this procedure can in general achieve high accuracy. The main purpose of this paper is to perform a forward rounding error analysis in finite precision. Our result establishes sensitivity factors regarding the accuracy of the algorithm, and provides a theoretical justification for why the algorithm works. For least-squares approximation on an interval, we propose a variant of this Arnoldi-based evaluation by using the Chebyshev polynomial basis. Numerical tests are reported to demonstrate our forward rounding error analysis.

Full-F turbulent simulation in a linear plasma device using a gyro-moment approach

Simulations of plasma turbulence in a linear plasma device configuration are presented. These simulations are based on a simplified version of the gyrokinetic (GK) model proposed by Frei et al. [J. Plasma Phys. 86, 905860205 (2020)], where the full-F distribution function is expanded on a velocity-space polynomial basis allowing us to reduce its evolution to the solution of an arbitrary number of fluid-like equations for the expansion coefficients, denoted as the gyro-moments (GM). By focusing on the electrostatic and neglecting finite Larmor radius effects, a full-F GM hierarchy equation is derived to evolve the ion dynamics, which includes a nonlinear Dougherty collision operator, localized sources, and Bohm sheath boundary conditions. An electron fluid Braginskii model is used to evolve the electron dynamics, coupled to the full-F ion GM hierarchy equation via a vorticity equation where the Boussinesq approximation is used. A set of full-F turbulent simulations are then performed using the parameters of the LArge Plasma Device (LAPD) experiments with different numbers of ion GMs and different values of collisionality. The ion distribution function is analyzed illustrating the convergence properties of the GM approach. In particular, we show that higher-order GMs are damped by collisions in the high-collisional regime relevant to LAPD experiments. The GM results are then compared with those from two-fluid Braginskii simulations, finding qualitative agreement in the time-averaged profiles and statistical turbulent properties.

Orthogonal polynomial bases for data-driven analysis and control of continuous-time systems

Orthogonal polynomial bases for data-driven analysis and control of continuous-time systems

Automatic Determination of Set of Multivariate Basis Polynomials Based on Recursion and Application of LSPCR to High-Dimensional Uncertainty Quantification of Multi-Conductor Transmission Lines

Automatic Determination of Set of Multivariate Basis Polynomials Based on Recursion and Application of LSPCR to High-Dimensional Uncertainty Quantification of Multi-Conductor Transmission Lines

Comparison Of Kernel Support Vector Machine In Stroke Risk Classification (Case Study:IFLS data)

Stroke s a disability main source and main disability source to lost years of disability-adjusted life. Currently the information technology development, especially the field of machine learning has an important role in early warning of various diseases, such as strokes. One of the methods used for stroke classifying is Support Vector Machine (SVM). In this study, we aim to compare several kernel functions in SVM such as linear, radial basis function(RBF), polynomial, and sigmoid for classifying stroke risk. We determine the best kernel based on accuracy, sensitivity, and specificity values. The result of this study shows that linear kernel function gives the best performance in classifying with values of classification accuracy 99.0%, specificity 100.0%, ,and sensitivity 97.0%. Those scores are the highest scores among the other kernel , that means the linear kernel function is the best method for classifying strokes risk.

Comparative analysis of the approximation of derivatives of Caputo and Riemann-Liouville fractional orders in the bases of biorthogonal polynomials

On the basis of the model problem, a comparative analysis of the use of the spectral method in the basis of orthogonal polynomials for calculating the derivatives of fractional orders of Caputo and Riemann-Liouville was carried out. The rate of convergence of the approximation was analyzed