Multidimensional Polynomial Research Articles

Achieving High Precision and Productivity in Laser Machining of Ti6Al4V Alloy: A Comprehensive Study using a n-predictor polynomial regression model and PSO algorithm

Ti-6Al-4V, the Titanium alloy, has significant utilizations in aerospace, automotive, and marine sectors for its low density and high strength at elevated temperature. But its chemical activity and low thermal conductivity inhibits its machining by conventional method. Nd: YAG laser beam machining (LBM) finds extensive use in rapid and precise cutting of Ti6Al4V. This study has examined the influences of various LBM machining variables, including laser power, gas pressure and stand-off distance, in cutting 5mm thick Ti-6Al-4V plate. In assessing the effectiveness and performance of the LBM process, three response functions—surface roughness, angle of kerf, and material removal rate—have been designated. From the experimental data, different regression models have been established to estimate these response functions in terms of the machining parameters. Based on R2 score and RMSE, multi-dimensional polynomial regression is decided as the most suitable regression model. Subsequently, the Particle Swarm Optimization technique has been applied to identify the optimal machining parameters for reducing angle of kerf and surface roughness, while increasing material removal rate. Three individual single-objective functions underwent optimization, along with a multi-objective function. Furthermore, experimental verification was conducted for the optimal input parameters in the single-objective as well as the multi-objective optimization scenarios, resulting in an accuracy of 97% and 98%, respectively. Such a close agreement emphasizes the accuracy of the developed regression model as well as it signifies the reliability and efficacy of the optimization technique.

[formula omitted]-optimal designs for a multidimensional second-degree polynomial model with no intercept

The paper investigates the problem of constructing D-optimal designs for the multidimensional second-degree polynomial model without an intercept term. On a hyperparallelepiped of the given dimensionality and symmetric with respect to the origin, D-optimal designs are found in explicit analytical form.

Multidimensional polynomial patterns over finite fields: Bounds, counting estimates and Gowers norm control

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument.

Monopole harmonics on $\mathbb{CP}^{n-1}$

We find the spectra and eigenfunctions of both ordinary and supersymmetric quantum-mechanical models describing the motion of a charged particle over the \mathbb{CP}^{n-1}ℂℙn−1 manifold in the presence of a background monopole-like gauge field. The states form degenerate SU(n)SU(n) multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the well-known isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.

Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees

Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees

Data-driven polynomial chaos-interval metamodel for dynamics and reliability analysis under hybrid uncertainty

As the complexity of engineering systems increases, the probability distribution information of model parameters becomes limited owing to the influence of environmental or natural phenomena. To solve this problem, a data-driven polynomial chaos-interval metamodel (DDPCIM) is proposed herein to quantify random and interval uncertainties in an engineering system. The orthogonal polynomial basis corresponding to each dimensional random variable was based on the statistical moments of the random variable. The multidimensional orthogonal polynomial basis was obtained by performing a tensor-product operation on a one-dimensional orthogonal polynomial basis. Legendre (or Chebyshev) polynomials can be selected as the polynomial basis functions corresponding to the interval variables. In addition, the interval sampling technique was improved to make the bounds of the output response closer to realistic results. The proposed DDPCIM has higher computational accuracy and efficiency than traditional methods, such as the polynomial Chaos–Legendre metamodel (PCLM) and the method combining Monte Carlo and scanning tests (CMCST). Finally, the DDPCIM was applied to achieve the desired level of reliability for engineering systems with hybrid uncertainty. Numerical examples show that the DDPCIM is a robust and efficient method for analysing the reliability of engineering structures with hybrid uncertainty.

Methods of Partial Differential Equation Discovery: Application to Experimental Data on Heat Transfer Problem

The paper presents two effective methods for discovering process models in the form of partial differential equations based on an evolutionary algorithm and an algorithm for the best subset selection. The methods are designed to work with sparse and noisy data and implement various numerical differentiation techniques, including piecewise local approximation using multidimensional polynomial functions, neural network approximation, and an additional algorithm for selecting differentiation steps. To verify the algorithms, the experiment is carried out on pulsed heating of a viscous liquid (glycerol) by a submerged horizontal cylindrical heat source. Temperature measurements are taken only at six points, which makes the data very sparse. The noise level ranges from 0.2 to 1% of the observed maximum temperature. The algorithms can successfully restore the structure of the heat transfer equation in cylindrical coordinates and determine the thermal diffusivity coefficient with an error of 2.5–20%, depending on the algorithm type and heating mode. Additional synthetic setups are employed to analyze the dependence of accuracy on the noise level. Results also demonstrate the algorithms’ ability to identify underlying processes such as convective motion.

Information and Theoretical Foundations for Control of the State of Agrocenoses in Real Time

Objective: Presentation of a complete theory of management of the state of agrocenoses, including sowing crops and weeds. This theory makes it possible to obtain a given crop yield with the required reliability. Such management is formed on the basis of estimates of the parameters of the state of crops, soil environment and weeds, formed according to the data of Earth remote sensing in real time. Methods: The presented theory is based on new mathematical models of the parameters of the state of agricultural crops, soil environment, weeds, as well as models of the relationship of these parameters with remote sensing (RS) data. Control factors in agricultural technology are doses of mineral fertilizers, herbicides and irrigation rates. Therefore, the parameters of the state in the management system under consideration, in addition to the parameters of the biomass of agricultural crops and weeds, are the chemical parameters of the soil environment. All technological operations are carried out at the onset of certain phenological phases of the development of crop sowing. RS data are entered precisely at such moments of time when the parameters of the state of crops and weeds are estimated on their basis. The presented theory is based on classical control principles used in modern dynamic systems. The most important feature of the proposed approach to solving the problem of agrocenoses management is the development of the principle of spatial distribution of agricultural technology parameters. It consists in using models with spatial variables, which are complemented by models of spatial approximation of controls in the form of multidimensional power polynomials. The parameters of the approximation models are evaluated according to the optimal parameters of technological operations obtained on 15-20 test sites of the field. Results: In accordance with the proposed theory, mathematical models of agrocenoses, a real-time control algorithm using RS data and a specialized software package were developed, with the help of which the control system was tested on the example of spring wheat sowing. Conclusion: As a result of testing a specialized software package for the implementation of the proposed theory, the operability of the proposed theory of agrocenoses control in real time and the possibility of using it in practical systems for automated control of agricultural technologies were established.

Parallel Kalman filter group integrated particle filter method for the train nonlinear operational status high-precision estimation under non-Gaussian environment

For the problem of multi-mode state estimation in actual train operation, this paper proposes a nonlinear non-gaussian high-precision parallel Kalman filter group (NN-HEKFG) integrated Particle Filter. A multi-model Gaussian decomposition of the probability density function for state equations and measurement equations is performed, and each local state model is represented by a multi-dimensional high-order polynomial to establish the expanded dimensional state model. Then, by updating the mean and variance of the local state expanded dimensional model and in turn solving the particle filtering posterior probability density distribution function, the global estimation results are obtained. In reducing the number of Gaussian terms, a new parameter reduction criterion is established, which can effectively carry out the re-identification of parameters such as weights and means, so as to avoid the problem of parameter explosion. The superiority of NN-HEKFG over particle filters and Gaussian sum filters and its effectiveness for train running state estimation are verified by simulating the multi-model running state of trains.

A numerical calculation method of laser illumination starting time suitable for air-to-ground missiles

Focusing on the problem of the precise determination of laser illumination starting time for air-to-ground missiles (AGMs), a numerical calculation method based on multidimensional polynomial fitting is proposed in this paper. The presented method takes various factors including temperature, altitude, and platform velocity into consideration. Influence by the parameters is delivered step by step through feature points-based polynomial fitting, and high-precision laser illumination starting time can be achieved. Helicopter-borne AGM flight data have verified the effectiveness of the proposed method.

In orbit calibration of the non-orthogonality of the two fluxgate sensors onboard CSES

With the accumulation of data, it is possible to study the long-term variation of Fluxgate Magnetometers (FGM) of High Precision Magnetometer (HPM) onboard the China Seismo-Electromagnetic Satellite (CSES). The calibration method of HPM depends on an assumption that the linear parameters of FGM, including the gain factors, the offsets, and non-orthogonal angles are stable. However, HPM exhibits some unexpected trends in the scalar residuals when processed according to the assumption. Study on the residual finds out that under changing space thermal conditions, the non-orthogonal angles of FGM change periodically, suggesting that the data calibration method we used before should be modified. A multi-dimensional polynomial model is established for defining the change of non-orthogonal angles of FGM as a function of the sun incidence angles, geographic latitude and sensor temperature. Apply the polynomial model to data from August 2018 to May 2021, the standard deviation of the scalar residuals is reduced to around 0.5 nT and tends to be more random and in line with the normal distribution. Meanwhile, the variation trend in gain factors and offset factors are eliminated. Results show that the model can correctly reflect the period variation of the non-orthogonal angles of FGM with the space thermal conditions.Graphical

Exploring the versatile properties and applications of multidimensional degenerate Hermite polynomials

<abstract><p>In this study, we develop various features in special polynomials using the principle of monomiality, operational formalism, and other qualities. By utilizing the monomiality principle, new outcomes can be achieved while staying consistent with past knowledge. Furthermore, an explicit form satisfied by these polynomials is also derived. The emphasis of this study is to introduce the degenerate multidimensional Hermite polynomials (DMVHP) denoted as $ \mathbb{H}^{[r]}_n(j_1, j_2, j_3, \cdots, j_r; \vartheta) $, which are closely related to the classical Hermite polynomials and are a significant class of orthogonal polynomials. The fundamental properties, such as symmetric identities for these polynomials are also established. An operational framework is also established for these polynomials.</p></abstract>

Generalized Savitzky–Golay filter for smoothing triangular meshes

In this paper, the well-known Savitzky–Golay filter is generalized for three-dimensional triangular meshes. Savitzky–Golay filter is designed for smoothing noisy measurement signals, initially, a locally polynomial model is fitted to approximate the discrete measurement values. We will show, that the original idea can be naturally adapted for smoothing functions defined on an irregular two-dimensional triangular mesh. Primarily surfaces represented by three-dimensional triangular meshes are in the focus of this paper, which are by definition locally homeomorphic to the two-dimensional topological disk. As the generalized filter is meaningful on the plane, we will define local embeddings that send vertices and their neighborhoods to the disk. The points of the two-dimensional disk and the three-dimensional surface can be paired to each other using these embeddings, thereafter low-degree multidimensional polynomials can be fitted to the point pairs obtaining a similar continuous model, just like in the one-dimensional case. Finally, we show an application of our method for mesh smoothing, and a comparison to other mesh smoothing methods, e.g. the Laplacian smoothing and the Taubin smoothing.

A class of linearization-based collocation methods for initial value and boundary value engineering problems

Nonlinear initial value and boundary value problems are generally classified as two different categories of problems and the methods utilized to solve them are also different. This paper proposed a uniform linearization-collocation framework through which the initial value and boundary value problems can be solved in a similar way. An optional pre-conditioning procedure is provided to guarantee the computational accuracy. Rigorous analysis about the convergence rate is given. Different from the commonly seen convergence rate analysis based on one-dimensional system and mean-value theorem, this analysis is based on multidimensional system and polynomial approximation. A perturbed orbit propagation problem, a radiation heat transfer problem and an orbit optimization problem are solved to verify the validity and efficiency of the proposed methods. The proposed methods not only outperform the highly optimized MATLAB built-in integrator ODE45 in orbit propagation, but also outperform the bvp4c and bvp5c in solving the boundary value problem. Most importantly, the proposed methods are capable of solving some problems where the bvp4c and bvp5c are invalid. The quasi-linearization-collocation method proved to be a highly accurate, efficient and stable solver for ordinary differential equations with various types of constraints.

Direct multi-dimensional Chebyshev polynomial based reconstruction for magnetic particle imaging

Magnetic Particle Imaging is a tomographic imaging technique that measures the voltage induced due to magnetization changes of magnetic nanoparticle distributions. The relationship between the received signal and the distribution of the nanoparticels is described by the system function. A common method for image reconstruction is using a measured system function to create a system matrix and set up a regularized linear system of equations. Since the measurement of the system matrix is time-consuming, different methods for acceleration have been proposed. These include modeling the system matrix or using a direct reconstruction method in time, known as X-space reconstruction. In this work, based on the simplified Langevin model of paramagnetism and certain approximations, a direct reconstruction technique for Magnetic Particle Imaging in the frequency domain with two- and three-dimensional Lissajous trajectory excitation is presented. The approach uses Chebyshev polynomials of second kind. During reconstruction, they are weighted with the frequency components of the voltage signal and additional factors and then summed up. To obtain the final nanoparticle distribution, this result is rescaled and deconvolved. It is shown that the approach works for both simulated data and real measurements. The obtained image quality is comparable to a modeled system matrix approach using the same simplified physical assumptions and no relaxation effects. The reconstruction of a 31 × 31 × 31 volume takes less than a second and is up to 25 times faster than the state-of-the-art Kaczmarz reconstruction. Besides, the derivation of the proposed method shows some new theoretical aspects of the system function and its well-known observed similarity to tensor products of Chebyshev polynomials of second kind.

Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

Processing of finely dispersed technogenic raw materials for aluminum production in order to extract valuable components

The paper provides the results of experiments on the hydrometallurgical processing of finely dispersed technogenic raw materials for primary aluminum production in Soderberg cells (case study of the Irkutsk Aluminum Smelter) – aged sludge. The components of this sludge are dust from electrostatic precipitators (79.7 %), wet gas cleaning sludge (4.4 %) and coal froth flotation tailings (15.8 %). According to the grain-size analysis carried out, aged sludge sample particles have a size of –50 μm. According to the chemical composition analysis of the sludge sample, main components in it are carbon, cryolite, chiolite with a small amount of other compounds (corundum, ralstonite, spodumene, fluorite). Fluorine leaching experiments were carried out with a 2 % sodium hydroxide solution at a stirrer speed of ~1020 rpm. Using the mathematical planning of a three-factor experiment, it was found that the maximum concentration of fluorine in the solution (15.844 g/dm3) is achieved with the following optimal parameters of fluorine alkaline leaching: temperature of 90 °C, liquid-to-solid ratio of 9 : 1, and time of 90 min. The multidimensional polynomial equation was obtained for fluorine alkaline leaching from aged sludge. Cryolite was obtained from fluorine-containing solutions (by the reaction of sodium fluoride interaction with sodium bicarbonate and an aluminate solution), which was confirmed by X-ray phase analysis data.

High‐Order Extended Kalman Filter Design for a Class of Complex Dynamic Systems with Polynomial Nonlinearities

A novel High-order extended Kalman Filter (HEKF) is designed for a class of complex dynamic systems with polynomial nonlinearities. The state and measurement models are represented by multi-dimensional high-order polynomials, respectively. All high-order polynomials in the state model are defined as implicit variables. By combining original variables with implicit variables, the state model is equivalently formulated to be a pseudo-linear form. By modeling dynamic relationship between implicit variables and combining original variables with all implicit variables, a new linear augmented state model is established correspondingly. The measurement model can be equivalently formulated as a linear form. On the basis of the new linear state and linear measurement models, the HEKF is designed and derived in detail. Simulation results demonstrate the effectiveness of the proposed estimator.

Behavioral Modeling of Memristor-Based Rectifier Bridge

In electrical engineering, radio engineering, robotics, computing, control systems, etc., a lot of nonlinear devices are synthesized on the basis of a nanoelement named memristor that possesses a number of useful properties, such as passivity, nonlinearity, high variability of parameters, nonvolatility, compactness. The efficiency of this electric element has led to the emergence of many memristor technologies based on different physical principles and, as a result, to the occurrence of different mathematical models describing these principles. A general approach to the modeling of memristive devices is represented. The essence is to construct a behavioral model that approximates nonlinear mapping of the input signal set into the output signal set. The polynomials of split signals, which are adaptive to the class of input signals, are used. This adaptation leads to the model’s simplification important in practice. Multi-dimensional polynomials of split signals are built for the rectifier bridge at harmonic input signals. The modeling error is estimated in the mean-square norm. It is shown that the accuracy of the modeling is increased in the case of using the piecewise polynomial with split signals.

Quantification of uncertainty in steel plates subject to fatigue with variable load via chaos expansion polynomial

This investigation analyzed the propagation of the uncertainty of the cycle number to fatigue failure, taking into account the strain life method, considering as random variables some physical parameters of the material and load vector in a hole-in-center; plate composed of an aluminum alloy 2024-T351 under fatigue, with variable amplitude loading through multidimensional Hermite polynomials. The application of series of multidimensional Hermite polynomials allowed the prediction of the randomness of the output vector. This research demonstrated that a 2nd degree multidimensional Hermite polynomial adequately estimates the propagation of uncertainty with 45 points per random variable, while the use of Monte Carlo simulation requires 1E + 06 points per random variable. This generates significant computational savings and time. The results showed that variations in the material parameters are not important compared to the variations in the load parameters, which generate an increase in the probability of failure in aluminum alloy parts subjected to fatigue.