Polynomial Research Articles

Possible regression models for the municipal finances of the municipal corporations of various Indian states II ∗

In continuation of the ongoing long term research work of the first author on the detailed mathematical and statistical analysis of thebudgetoftheBangaloreMunicipalCorporation(see,A.Sathyavathihttps://ssrn.com/abstract=4403863,https:/ssrn.com/abstract=4461494, https://ssrn.com/abstract=4518454 and https://ssrn. com/abstract=4568499), we carry ahead our previous study (A. Sathyavathi and L. M. Upadhyaya, Possible regression models for the municipal finances of the municipal corpo rations of various Indian states, Bull. Pure Appl. Sci. Sect. E Math. Stat. 42E(1), 72–93 (2023))to propose some possible regression models to account for the observed trends of the revenue receipts and the revenue expenditures of the municipal corporations of the various Indian States including the Union Territory of Chandigarh for the 2018-19 Bud get Estimates, which pertains to the data of finances of municipal corporations of the various Indian States during the financial year 2018-2019. We also present the com parisons of our different models for each of the categories of our present explorations. The entire data used by us in this study is gratefully taken by us from the Report on Municipal Finances which was published by the Reserve Bank of India recently on 10 November, 2022 (see,https://m.rbi.org.in/scripts/AnnualPublications.aspxhead=Report%20on%20Municipal%20Finances,https://rbidocs.rbi.org.in/rdocs/Publications/ PDFs/RMF101120223A34C4F7023A4A9E99CB7F7FEF6881D0.PDF). We find that a Polyno mial Degree 4 model best describes the financial data and in addition we also give a Rational Model and an Exponential Association 2 model for describing the trend of fi nances for the municipal corporations of India during the year 2018-19 in the Budget Estimates category and we also present our computational results of the Polynomial De gree 4 Model and the Rational Model using MATLAB.

Enhanced neural network model for regional ionospheric modeling and evaluation under different solar-geomagnetic conditions

Monitoring spatiotemporal variations of ionospheric vertical total electron content (VTEC) is crucial for space weather and satellite positioning. In the present study, an enhanced neural network (ENN) model is proposed to capture the changing characteristics of ionospheric VTEC and compared with the traditional mathematical models, i.e. the POLYnomial (POLY) model, generalized trigonometric series function and spherical harmonic function (SHF) model. The ionospheric VTEC data obtained from 31 permanent global positioning system stations in the southwest region of China on 26 August and 8 September, 2017, were used to test the performance of the mentioned models under different Solar-geomagnetic conditions. The ENN model is derived from the ensemble learning method, and the disadvantage that simple backpropagation neural network learners that are not robust enough is weakened by the ENN model. After statistical analysis and single-frequency precise point positioning (SF-PPP) experiments, it is demonstrated that the ENN model is superior to the above three mathematical models, regardless of the solar-geomagnetic conditions. In terms of mean absolute error, root mean square error, standard deviation, and mean absolute percentage error, the ENN model outperforms the SHF model, which is the best mathematical model in the analysis, by 40.7%, 30.20%, 29.88%, 38.04% under quiet solar-geomagnetic conditions, and by 37.66%, 29.93%, 30.96%, 32.01% under active solar-geomagnetic conditions. In addition, the accuracy of the SF-PPP is greatly affected by the error caused by ionosphere. In the static SF-PPP experiment of this study, the ENN model can better correct ionospheric error. Under quiet and active solar-geomagnetic conditions, the SF-PPP accuracy can be improved by 85.1% and 85.2% with the ionosphere delay correction from the ENN model.

Implementation of reactive compensator for voltage balancing using AI based models and novel performance index

Voltage-unbalance is one of the power quality deficiencies that degrades electrical power systems performance. In this work, voltage unbalance problem is tackled through two stages; evaluation using a novel performance index and mitigation using a thyristor-controlled reactor (TCR) compensator with artificial intelligent (AI) based models. Unlike standard performance indices that rely on voltages' root mean square (RMS) values, the proposed index depends on the space vector (SV) signal amplitude for voltage unbalance evaluation. This signal depends on the instantaneous values of the three-phase voltages and has twice the system frequency. Therefore, the proposed index entitled as space vector unbalance factor (SVUF) reflects the amount of voltage unbalance and reduces the time necessary for evaluation by half. Subsequently, advanced models based on several algorithms are proposed to generate the required firing angles for TCR compensator to restore voltage balance, including radial basis functions networks (RBFNs), hybrid-RBFNs (H-RBFNs), polynomials (PNs), and simplified neural networks (NNs). Models' structure, prediction capability, and response time are analyzed. Results show that the time required for voltage unbalance mitigation is reduced. Moreover, the models used to generate the firing angles are simplified significantly while maintaining high accuracy.

Systematic Analysis of Wind Resources for Eolic Potential in Bangladesh

Energy consumption in Bangladesh increased for economic, industrial, and digitalization growth. Reductions in conventional sources such as natural gas (54%) and coal (5.6%) are calls to enhance renewable resources. This paper aims to investigate the atmospheric variables for potential wind zones and develop a statistical power-forecasting model. The study-site is Bangladesh, focusing on eight divisions across two regions. First, the southern zone includes Dhaka (Capital), Chittagong, Barishal, and Khulna. The northern regions are Rajshahi, Rangpur, Mymensingh, and Sylhet. This investigation illustrates wind (m/s) speeds at various heights (m) and analyzes the boundary layer height (BLH) from the European Center for Medium Range Weather Forecast reanalysis 5th generation (ERA5). The data is from a period of 40 years from 1979 to 2018, assessing with a climatic base of 20 years (1979 to 2000). The climatological analysis comprises trends, time series, anomalies, and linear correlations. The results for the wind speed (BLH) indicate that the weakest (lower) and strongest (higher) regions are Sylhet and Barishal, respectively. Based on power-curve relationships, a simple power predictive model (SPPM) is developed using global wind atlas (GWA) data (sample: 1100) to estimate the power density (W/m2) and found an accuracy of 0.918 and 0.892 for Exponential (EXP) and Polynomial (PN) with mean absolute percentage errors (MAPE) of 22.92 and 21.8%, respectively. For validation, SPPM also forecasts power incorporating historical observations for Chittagong and obtains correlations of 0.970 and 0.974 for EXP and PN with a MAPE of 10.26 and 7.69% individually. Furthermore, calculations for annual energy production reveal an average megawattage of 1748 and 1070 in the southern and northern regions, with an MAPE of 15.71 and 5.85% for EXP and PN models, except Sylhet. The SPPM’s predictability can be improved with observed wind speeds and turbine types. The research wishes to apply SPPM for estimating energy in operational power plants.

Higher order TN approximation for the neutron diffusion problem in a slab reactor

Abstract Higher order approximations of the Chebyshev polynomials of first kind (TN) are used for the first time in calculation of the diffusion lengths of monoenergetic neutrons in a homogeneous slab. In the method, the diffusion lengths of the neutrons are calculated using various values of the c, the number of secondary neutrons per collision. First, the traditional Legendre polynomials (PN) approximation and then the present TN method are used separately. The numerical results for the diffusion lengths are tabulated in the tables up to an order of N = 9. A brief comparison is also done between the results obtained from the present method and the ones in literature. The advantages of the present method can easily be observed from the good accordance between results given in the tables for comparison and its easily executable equations. For many of the c values, the results obtained from TN method are better than the results obtained from PN method.

Correction to: Good polynomials for optimal LRC of low locality

Correction to: Good polynomials for optimal LRC of low locality

Vibration of inhomogeneous fibrous laminated plates using an efficient and simple polynomial refined theory

In this article, a reliable model for the vibration of cross-ply and angle-ply laminated plates that own inhomogeneous elastic properties is considered. The methodology includes a theoretical study of free vibration behavior of composite plates with the inhomogeneous fibrous distribution of the volume fraction using a sinusoidal model by the use of the advanced refined theory of shear deformation of nth-higher-order. The micromechanical typical is proposed to represent the elastic and physical properties of the inhomogeneous laminated composite plate. The effects of inhomogeneity, lamination schemes, aspect ratio, and the number and order of layers on dimensionless vibration frequencies are investigated.

A semi-analytical FE method for the 3D bending analysis of nonhomogeneous orthotropic toroidal shells

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridian-radial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

New Technique for Solving Lane-Emden Equation with Vieta-Fibonacci Polynomials

In the present work, new efficient iterative algorithm is presented for solving Lane-Emden equation with initial conditions. The solution is considered as a linear combination of Vieta-Fibonacci polynomials with unknown coefficients. First, the Vieta-Fibonacci polynomials are presented with some new important properties. A new exact formula for finding operation matrix of derivative for the Vieta-Fibonacci polynomials is constructed. Using such new properties the original Lane-Emden equation is transformed to the solution of algebraic equation with small-unknown coefficients. The obtained solution is in the form of a power series with easily computable coefficients. Comparison with some known exact and approximate solutions shows that the proposed solution is highly accurate.

Knot Polynomials from $$\mathcal{R}$$-Matrices: Wherefore This Mathematics?

Knot Polynomials from $$\mathcal{R}$$-Matrices: Wherefore This Mathematics?

Influence of porosity on thermal buckling behavior of functionally graded beams

The interest of this work is the analysis of the effect of porosity on the nonlinear thermal stability response of power law functionally graded beam with various boundary conditions. The modelling was done according to the Euler-Bernoulli beam model where the distribution of material properties is imitated polynomial function. The thermal loads are assumed to be not only uniform but linear as well non-linear and the temperature rises through the thickness direction. The effects of the porosity parameter, slenderness ratio and power law index on the thermal buckling of P-FG beam are discussed.

A quadrature-free Legendre polynomial approach for the fast modelling guided circumferential wave in anisotropic fractional order viscoelastic hollow cylinders

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin–Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin–Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin–Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides.

On the Coefficients of Okamoto polynomial

On the Coefficients of Okamoto polynomial

Dynamic responses of functionally graded and layered composite beams

This paper presents and compares the free and damped forced vibrations of layered and functionally graded composite beams. In the considered study, a cantilever beam subjected to a harmonic point load at the free end is investigated with layered and functionally graded materials. In the kinematics of the beam, the Timoshenko beam theory is used. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem, the Ritz method is used. Algebraic polynomials are used with the trial functions for the Ritz method. In the obtaining of free vibration results, the eigenvalue procedure is implemented. In the solution of the damped forced vibration problem, the Newmark average acceleration method is used in the time history. In the damping effect, the Kelvin-Voigt viscoelastic model is used with the constitutive relations. In the numerical examples, the effects of material distribution parameter and dynamic parameters on the natural frequencies and forced vibration responses of functionally graded beams are obtained and compared with the results of the layered composite beam. Also, comparison studies are performed in order to validate the used formulations.

Cyclic LRCs with Availability from Linearized Polynomials

Cyclic LRCs with Availability from Linearized Polynomials

Using Laguerre polynomials as a basis for a new differential quadrature methodology to solve magneto-hydrodynamic (MHD) fourth-grade fluid flow

The addition of an application of a new version of the Differential Quadrature Method is the purpose of this work. The new method, tracing Laguerre polynomials, is applicable to test functions whose purpose is to establish the DQM weighting coefficients, focussing on the use of the DQM in investigating solving nonlinear differential equations numerically for the representation of the steady incompressible flow problem of a fourth-grade non-Newtonian fluid magnetic field between two stationary parallel plates. A series of graphs are used to demonstrate the ways a range of important physical parameters influence the velocity profile. The level of agreement when comparing a small number of grid points in the new technique with analytical solutions is remarkably high.

基于切比雪夫多项式拟合函数的新冠疫情新增数据分析——以英国、印度和埃塞俄比亚为例

基于切比雪夫多项式拟合函数的新冠疫情新增数据分析——以英国、印度和埃塞俄比亚为例

A bit on the zeros of $D_\alpha f(z)$ of a polynomial $f(z)$

A bit on the zeros of $D_\alpha f(z)$ of a polynomial $f(z)$

$(s,t)$-Dominating polynomials in graphs

$(s,t)$-Dominating polynomials in graphs

Thermal frequency analysis of FG sandwich structure under variable temperature loading

The thermal eigenvalue responses of the graded sandwich shell structure are evaluated numerically under the variable thermal loadings considering the temperature-dependent properties. The polynomial type rule-based sandwich panel model is derived using higher-order type kinematics considering the shear deformation in the framework of the equivalent single-layer theory. The frequency values are computed through an own home-made computer code (MATLAB environment) prepared using the finite element type higher-order formulation. The sandwich face-sheets and the metal core are discretized via isoparametric quadrilateral Lagrangian element. The model convergence is checked by solving the similar type published numerical examples in the open domain and extended for the comparison of natural frequencies to have the final confirmation of the model accuracy. Also, the influence of each variable structural parameter, i.e. the curvature ratios, core-face thickness ratios, end-support conditions, the power-law indices and sandwich types (symmetrical and unsymmetrical) on the thermal frequencies of FG sandwich curved shell panel model. The solutions are helping to bring out the necessary influence of one or more parameters on the frequencies. The effects of individual and the combined parameters as well as the temperature profiles (uniform, linear and nonlinear) are examined through several numerical examples, which affect the structural strength/stiffness values. The present study may help in designing the future graded structures which are under the influence of the variable temperature loading.