Polynomial Function Research Articles

Adaptive EWMA control chart by using support vector regression

AbstractTraditional control charts depend on the process parameters that are used to monitor the shifts in the process. The adaptive control charts are used to adapt a process parameter during the online monitoring. This research introduces a support vector regression (SVR) based adaptive exponentially weighted moving average control chat to enhance the monitoring of the mean in industrial processes. The study systematically investigates the comparative efficiency of linear, radial basis function (RBF), and polynomial functions within the SVR framework. The proposed SVR‐based AEWMA control chart leverages the strengths of the RBF kernel, providing a robust mechanism for detecting shifts in the process mean by adapting the smoothing constant according to the size of the shift. To validate the efficacy of the proposed methodology, a practical application is presented by using real‐life data. The application showcases the adaptability and reliability of the SVR‐based adaptive EWMA control chart in effectively monitoring location shifts.

Time series clustering based on polynomial fitting and multi-order trend features

Time series clustering serves as a potent data mining method, facilitating the analysis of an extensive array of time series data without the prerequisite of any prior knowledge. It finds wide-ranging use across various sectors, including but not limited to, financial and medical data analysis, and sensor data processing. Given the high dimensionality, non-linearity, and redundancy characteristics associated with time series, conventional clustering algorithms frequently fall short in yielding satisfactory results when directly applied to this kind of data. As such, there is a critical need to judiciously select suitable feature extraction methods and dimension reduction techniques. This paper introduces a time series clustering algorithm, drawing primarily from polynomial fitting derivative features as a wellspring for feature extraction to achieve effective clustering results. Initially, Hodrick Prescott (HP) filtering comes into play for the processing of raw time series data, thereby eliminating noise and redundancy. Subsequently, polynomial curve fitting (PCF) is applied to the data to derive a globally continuous function fitting this time series. Next, by securing multi-order derivative values via this function, the time series is transformed into a multi-order derivative feature sequence. Lastly, we designed a polynomial function derivative features-based dynamic time warping (PFD_DTW) algorithm for determining the distance between two equal or unequal granular length time series, and subsequently a hierarchical clustering method anchored on the PFD_DTW distances for time series clustering after computing interspecies distances. The effectiveness of this method is corroborated by experimental results obtained from several practical datasets.

Structural damage localization based on deflection influence surfaces calculated by the three-dimensional boundary face method

A structural damage localization method based on deflection influence surfaces (DIS) calculated by the three-dimensional (3D) boundary element method (BEM) is proposed. By combining the 3D BEM with Structural Properties Intelligent Analysis Programs (SPIAP), the static equilibrium method (SEM) is performed on a square plate. Then, all the deflection values are automatically extracted by the SPIAP for the purpose of using the optimized bivariate polynomial function (BPF) to fit the DIS. The accuracy of the SPIAP-fitted DIS is proved by the optimized BPF program. Finally, a structural damage localization module is developed and verified through an experimental example. The numerical and experimental results indicate that compared to the analytical solution, the deflection calculation error of the square plate is no more than 1.8‰. Moreover, the damage localization accuracy of the simply supported beam under single-damage and double-damage cases has been improved by 28.6% and 16.7%, respectively.

Association of longitudinal pet ownership with wheezing in 3-year-old children using the distributed lag model: the Japan Environment and Children’s Study

BackgroundTime-varying exposures like pet ownership pose challenges for identifying critical windows due to multicollinearity when modeled simultaneously. The Distributed Lag Model (DLM) estimates critical windows for time-varying exposures, which are mainly continuous variables. However, applying complex functions such as high-order splines and nonlinear functions within DLMs may not be suitable for situations with limited time points or binary exposure, such as in questionnaire surveys.Objectives(1) We examined the estimation performance of a simple DLM with fractional polynomial function for time-varying binary exposures through simulation experiments. (2) We evaluated the impact of pet ownership on childhood wheezing onset and estimate critical windows.Methods(1) We compared logistic regression including time-varying exposure in separate models, in one model simultaneously, and using DLM. For evaluation, we employed bias, empirical standard error (EmpSE), and mean squared error (MSE). (2) The Japan Environment and Children's Study (JECS) is a prospective birth cohort study of approximately 100,000 parent-child pairs, registered across Japan from 2011 to 2014. We applied DLM to the JECS data up to age 3. The estimated odds ratios (OR) were considered to be within critical windows when they were significant at the 5% level.Results(1) DLM and the separate model exhibited lower bias compared to the simultaneously model. Additionally, both DLM and the simultaneously model demonstrated lower EmpSEs than the separate model. In all scenarios, DLM had lower MSEs than the other methods. Specifically, where critical windows is clearly present and exposure correlation is high, DLM showed MSEs about 1/2 to 1/200 of those of other models. (2) Application of DLM to the JECS data showed that, unlike other models, a significant exposure effect was observed only between the ages of 0 and 6 months. During that periods, the highest ORs were 1.07 (95% confidence interval, 1.01 to 1.14) , observed between the ages of 2 and 5 months.Conclusions(1) A simple DLM improves the accuracy of exposure effect and critical windows estimation. (2) 0–6 months may be the critical windows for the effect of pet ownership on the wheezing onset at 3 years.

Correction: Continued fraction representation of the generating function of Bernoulli polynomials

Correction: Continued fraction representation of the generating function of Bernoulli polynomials

Modelado matemático con funciones polinomiales del perfil de una leva mediante el análisis cinemático del seguidor

Modelado matemático con funciones polinomiales del perfil de una leva mediante el análisis cinemático del seguidor

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

Optimised correction polynomial functions for the Flux Reconstruction method in time-harmonic electromagnetism

The Flux Reconstruction (FR) method is classically used in the Computational Fluid Dynamics field. However, its use for the simulation of electromagnetic wave propagation is not as developed yet. Following on from the development of a priori error estimates for the 1D wave equations, we introduce optimisation problems to allow an adaptation of the FR correction polynomial functions to the discretisation parameters. Showing notable accuracy gains in 1D, especially in the preasymptotic regime, we generalise this procedure to the 3D Maxwell’s equations, leading to similar interesting possibilities to reduce the computational cost for a given accuracy.

Investigation on optimization-oriented EPC method in analyzing the non-linear oscillations under multiple excitations

The optimization-oriented exponential-polynomial-closure (OEPC) method is extended and investigated to analyze stochastic nonlinear oscillators under multiple excitations, with the purpose of obtaining probabilistic solutions for the corresponding system. The presented method extends the original EPC projection procedure for solving the FPK equation by minimizing an objective function, which is defined as the spatial integration of the weighted square residual error. Using the exponential polynomial function, the parameters within it can be located by seeking the minimum of the objective function. In this paper, the novel method has been tested with four examples of strong nonlinearities raised by polynomial nonlinear terms and parametric excitations. The results provide adequate evidence that the OEPC approach delivers notably improved accuracy compared to the Gaussian closure method and demonstrates superior efficiency compared to Monte Carlo simulation. The OEPC method presents a viable alternative for investigating stochastic nonlinear oscillators.

Legendre Polynomials are Associated in the Schrodinger Polar Equation of Helium Ions

The helium ion is one of the ions classified as a hydrogenic atom because it has only one electron in its outer orbital so that the helium ion problem can be solved using the Schrodinger equation approach. The Schrodinger equation of the polar part of the helium ion can be solved by the associated legendary polynomial function, . The polar function expresses the angle formed by the vector of the electron's position relative to the z-axis of the helium ion's nuclear position. And it is produced that the polar function of the hydrogenic atom has the same polar function both in position space and in momentum space .

Support vector regression-based heteroscedastic models for cryptocurrency risk forecasting

In this study, we aimed to combine support vector regression (SVR) with an autoregressive (AR) model and a generalized autoregressive conditional heteroscedastic (GARCH) or asymmetric GARCH model. We compared the performances of these combined models with the maximum likelihood estimation (MLE)-based models under a normal, Student’s t, or skewed Student’s t distributional assumption. We employed such models to protect the heteroscedastic risks of four major cryptocurrencies, namely, Bitcoin, Ethereum, Tether, and Binance Coin, before and during COVID-19. We accomplished this task by forecasting their volatility, Value-at-Risk (VaR), and Expected Shortfall (ES). The empirical results revealed that in terms of the root-mean-square error (RMSE) and mean-absolute error (MAE) metrics, the SVR-based models with a polynomial or radial basis kernel function, which is nonlinear, resulted in a more accurate volatility forecast compared to the MLE-based models under any distributional assumption. The superiority of the former models was more apparent when applied to Tether exhibiting the most stable volatility. This supports evidence of nonlinear characteristics in the cryptocurrency return and volatility series. A combination between SVR and an asymmetric GARCH model also confirms the presence of an asymmetrically nonlinear relationship between the returns and volatility. In addition, the correct-VaR and Kupiec’s backtesting results showed that the SVR-based models tended to produce the VaR forecasts for the four cryptocurrencies with better accuracy at the 1%, 5%, and 10% significance levels compared to the MLE-based ones. The use of the correct-ES metric also leads to the superior ES forecasting performance of the former models at any level. However, McNeil and Frey’s backtesting results demonstrated their worse ES forecasting performance at the 1% level. As the COVID-19 pandemic progressed, we also found an increase in the Bitcoin and Binance Coin market risks as well as a decrease in the Ethereum and Tether market risks.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Ground state solutions for weighted biharmonic Kirchhoff problem via the Nehari manifold approach

In this article, we delve into the investigation of the following non-local problem within the unit ball B 1 of R 4 : g ( ∫ B 1 w β ( x ) | Δu | 2 ) Δ w β 2 u = | u | q − 2 u + f ( x , u ) in B 1 , u = ∂ n u = 0 on ∂ B 1 , where Δ w β 2 . = Δ ( w β ( x ) Δ . ) is the weighted Biharmonic operator, where w β ( x ) represents a logarithmically singular weight, and the non-linearity involves a combination of a reaction source f ( x , u ) , critical in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function g is characterized by being positive and continuous. Employing the Nehari manifold method, the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.

Prediction of Pre-Loading Relaxation of Bolt Structure of Complex Equipment under Tangential Cyclic Load.

Bolts have the advantages of simple installation and easy removal. They are widely applied in aerospace and high-speed railway traffic. However, the loosening of bolts under mixed loads can lead to nonlinear decreases in pre-loading. This affects the safety performance of the structure and may lead to catastrophic consequences. Existing techniques cannot be used to monitor the bolt performance status in time. This has caused significant problems with the safety and reliability of equipment. In order to study the relaxation law of bolt pre-loading, this paper carries out an experimental analysis for 8.8-grade hexagonal bolts and calibrates the torque coefficient. We also studied different loading waveforms, nickel steel plate surface roughnesses, tangential displacement frequencies, four different strengths and bolt head contact areas of the bolt, the initial pre-loading, and the effects of tangential cyclic displacement on pre-loading relaxation. This was done in order to accurately predict the degree of bolt pre-loading loosening under external loads. The laws are described using the allometric model function and the nine-stage polynomial function. The least squares method is used to identify the parameters in the function. The results show that bolts with a smooth surface of the connected structure nickel steel flat plate, high-frequency working conditions, half-sine wave, and a high-strength have better anti-loosening properties. Taking 5-10 cycles of cyclic loading as a boundary, the pre-loading relaxation is divided into two stages. The first stage is a stage of rapid decrease in bolt pre-loading, and the second stage is the slow decrease process. The performance prediction study shows that the allometric model function is the worst fitted, at 71.7% for the small displacement condition. Other than that, the allometric model function and the nine-stage polynomial function can predict more than 85.5% and 90.4%, which require the use of least squares to identify two and ten unknown parameters, respectively. The complexity of the two is different, but both can by better indicators than the pre-loading relaxation law under specific conditions. It helps to improve the monitoring of bolt loosening and the system use cycle, and it can provide theoretical support for complex equipment working for a long time.

Some interesting birational morphisms of smooth affine quadric 3-folds *

We study a family of birational maps of smooth affine quadric 3-folds, over the complex numbers, of the form x1x4−x2x3= constant, which seems to have some (among many others) interesting/unexpected characters: (a) they are cohomologically hyperbolic, (b) their second dynamical degree is an algebraic number but not an algebraic integer, and (c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on C4 preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

A fractal modification of the tropical algebra for noise removal and optimal control

This paper proposes a fractal modification of tropical algebra for noise removal and optimal control. Fractal addition, fractal multiplication, and fractal dot product are defined and explained in a fractal space. A fractal tropical polynomials function can model a coastline at any scale, providing a new tool for fractal analysis of any irregular curve. An example is given to solve an optimal control subject to a fractal tropical polynomials function. This paper provides a new window for tropical algebra, combining fractal geometry and algebra.

Identification of Wiener state–space models utilizing Gaussian sum smoothing

In this paper, we address the problem of system identification for Wiener state–space models. Our approach is based on the Maximum Likelihood method and the Expectation–Maximization algorithm. In the problem of interest, we model the output nonlinearity as a piecewise polynomial function and we jointly estimate the parameters of the linear system with the coefficients of each polynomial section. In our proposal, the computation of the cost function in the Expectation–Maximization algorithm requires the computation of the joint distribution of the state and the output of the linear system given the output of the nonlinear block. These quantities are obtained from an approximation that leads to a novel Gaussian sum smoothing algorithm. Additionally, we show that our method also addresses the identification of state–space systems in which the output is produced by a known quantizer. We present numerical examples to illustrate the benefits of the proposed identification technique.

Optimized deep neural network to estimate orientation angles for solar photovoltaics intelligent systems

Using a single hidden layer neural network in estimating orientation angles for solar photovoltaics lacks the complexity required to model nonlinear relationships between input variables and the optimal orientation angles for solar photovoltaics. It struggles to generalize well to new and unseen data. More sophisticated neural network architectures such as deep learning with multi-hidden perceptron (MLP) can solve these issues by changing the architecture by deepening the network. Deepening the network will increase complexity, energy consumption, and time complexity. The study uses a novel approach to outperform traditional MLP models with two, three, four, and five hidden layers. An innovative approach was proposed by enhancing a single hidden layer MLP with a quadratic polynomial function, utilizing two robust methodologies, Least Absolute Residuals (LAR) and Bisquare methods. The results demonstrate that these approaches yield significant improvements in Root Mean Square Error (RMSE) and coefficient of determination (R squared). LAR-based MLP showed superiority over both bisquare-based and conventional MLPs methods in R2 and RMSE, ranging from 1.13 to 1.18 and 2.53 to 3.06, respectively. The study outperformed conventional MLP architectures with five hidden layers regarding accuracy and efficiency. The proposed model offers a more effective and less complex solution for data prediction tasks.

Stress-Strain State of Steel Fiber-Reinforced Concrete under Compression Taking into Account Unloading from Inelastic Region

The purpose of the study is to examine the physical and mechanical characteristics of steel fiber-reinforced concrete under compression, including: modulus of elasticity, Poisson ratio, values of ultimate strains under compression, values of compressive strength with different percentages of dispersed reinforcement. An experimental investigation program, which included the production of cube samples measuring 100×100×100 mm, as well as a compression test under static loading, taking into account unloading from the region of inelastic deformations, was developed and carried out. Two types of steel fiber were chosen as dispersed reinforcement: hooked end and wave shape. The volume content of steel fiber in the cube samples was 0.5, 1.0, 1.5 and 2.0 %. As a result of the investigation, the strength and deformation characteristics of steel fiber reinforced concrete under compression were obtained. Based on the experimental data, actual strain diagrams of steel fiber reinforced concrete were constructed, taking into account the type of reinforcing fibers and the percentage of reinforcing fiber. Based on the obtained diagrams, a law of deformation of steel fiber reinforced concrete is proposed, which can be described by a polynomial function of the fourth order with constant coefficients that determine the shape of the stress-strain curve. The presented research results can be used in developing a methodology for physically nonlinear analysis of steel fiber reinforced concrete elements with a percentage of dispersed reinforcement from 0.5 to 2.0 %.

Equivalent single‐layer Mindlin theory of laminated piezoelectric plates and application

AbstractBased on the Mindlin first‐order shear deformation theory, this paper proposes an equivalent single layer (ESL) plate theory to analyze the electro‐mechanical coupling problem of laminated piezoelectric plates (LPPs). The main features of the proposed approach are: (i) It assumes that the electric potential across the thickness is a polynomial function, ensuring its continuity at the interface. (ii) The electric displacements are continuous at the interface, in line with the interface continuity condition between laminated plates. The theoretical solutions for the deformation and electric potential of LPPs are obtained. The validity and accuracy of the theoretical solutions are confirmed through comparison with results of two‐ and four‐layer LPPs obtained from the three‐dimensional finite element method (FEM). The numerical results discuss the influence of different series expansions and emphasize the necessity of high‐order expansion. Meanwhile, the range of application of three‐dimensional FEM is discussed. It is expected that such a new analytical method can be instructive to the optimal design of piezoelectric device.