Third-degree Polynomial Research Articles

Growth indexes of Pennisetum purpureum cv. Roxo cultivated in different seasons under rainfed conditions

ABSTRACT The objective was to estimate the growth indexes of elephant grass cv. Roxo managed under rainfed conditions in different growing seasons. Seven growth ages (9, 18, 27, 36, 45, 54, and 63 days) were evaluated in three growing seasons (rainy, transition and dry). A completely randomised design was adopted, with split-plot arrangement with time repeated measures, with the ages being the plots and the seasons the subplots. Interaction of age x season was observed. The specific leaf area mean was 0.0286 ± 0.0039 during the rainy season. In the transition season, it adjusted to a third-degree polynomial model; in the dry season it declined linearly (0.0003 m2 g-1). The leaf weight ratio adjusted to the third-degree polynomial model in the rainy and dry seasons, and in the transition season it was maximised at 28 DAC (0.694 g g-1). The leaf area ratio continuously declined in the rainy season. In the transition and dry seasons, it adjusted to the third-degree polynomial model. The net assimilation rate presented averages of 3.73 ± 2.50 and 3.63 ± 1.48 g m-2 day-1 in the rainy and transition seasons, respectively. In the dry season, it was minimised at 48.4 days. The relative growth rate mean was 0.0075 ± 0.0037 during the rainy season. In the transition and dry seasons, it adjusted to the third-degree polynomial model. The growth index rates of elephant grass cv. Roxo are modified by growing seasons, and the pattern and magnitude of the response varies throughout the growth cycle.

A Speech Emotion Recognition Model Based on Multi-Level Local Binary and Local Ternary Patterns

Interpreting a speech signal is quite challenging because it consists of different frequencies and features that vary according to emotions. Although different algorithms are being developed in the speech emotion recognition (SER) domain, the success rates vary according to the spoken languages, emotions, and databases. In this study, a new lightweight effective SER method has been developed that has low computational complexity. This method, called 1BTPDN, is applied on RAVDESS, EMO-DB, SAVEE, and EMOVO databases. First, low-pass filter coefficients are obtained by applying a one-dimensional discrete wavelet transform on the raw audio data. The features are extracted by applying textural analysis methods, a one-dimensional local binary pattern, and a one-dimensional local ternary pattern to each filter. Using neighborhood component analysis, the most dominant 1024 features are selected from 7680 features while the other features are discarded. These 1024 features are selected as the input of the classifier which is a third-degree polynomial kernel-based support vector machine. The success rates of the 1BTPDN reached 95.16%, 89.16%, 76.67%, and 74.31% in the RAVDESS, EMO-DB, SAVEE, and EMOVO databases, respectively. The recognition rates are higher compared to many textural, acoustic, and deep learning state-of-the-art SER methods.

A Study of QSAR based on Polynomial Modeling in Matlab

Mu-opioid receptor (MOR) is an attractive target for in silico docking experiments. Many potent analgesics currently in use act through the MOR. The main objective of the present work was to find the polynomial function for modelling of the structure-activity relationship of a series of MOR analogues and the results of the molecular docking with MOR (PDBid:4dkl). The relationship of the biological activity of the ligands with the ChemScore function and with the total energy (MolDock function) was modelled with first- to third-degree polynomials and surface fitted method, assessed by least squares method. The finding, established in the paper, suggests that the third order polynomial could be successfully used for modelling of the relationship between the biological effect of the MOR analogues and results from docking procedure. Analysis and comparison of the data from in vitro tests and docking studies could help to understand better the relationship between in vitro biological effects and docking studies and to answer whether the models of the biological macromolecules (in our case MOR) correspond to the real 3D structure.

Study on the Lightfastness of Ink Based on Visible and Infrared Spectral Analysis

A method based on visible reflection and infrared absorption spectroscopy is proposed to elucidate the colour variation law and mechanism of gravure ink in daylight. Three kinds of gravure spot colour ink, blue, red and cyan, are used for a xenon arc ageing experiment. The spectral reflectance of the samples is collected every 2 hours to analyse the variation in the chromaticity. A polynomial regression model of the colour difference and ageing time is established by the nonlinear least squares method. The coloration composition of the inks is analysed according to the infrared spectrum of the samples. The results show that in a 36-hour experiment with an illumination intensity of 550 W/m2, the lightfastness of the different colours of gravure spot ink is significant. The integrated value of the differential reflectance spectrum analysis shows that blue ink exhibits the largest change in chromaticity, followed by the red and cyan inks. The maximal colour differences of the three samples are 12.37, 4.78 and 1.34, respectively. A third-degree polynomial regression of the colour difference of the ink and ageing time achieves a good fitting result, with regression coefficients of 0.9700, 0.9980, and 0.9646, respectively. The infrared spectroscopy results show that the change in the chromaticity corresponds to changes in the molecular composition and absorption intensity of the pigments. The blue ink, with the largest change in chromaticity, exhibits the largest change in the absorption spectrum. The cyan ink, with inconspicuous changes in the chromaticity, exhibits the smallest change in the absorption spectrum. This research provides a more accurate theoretical basis for an analysis of the colour variation law and mechanism of gravure ink in daylight and provides a reference for lightfastness analysis of textiles, dyes and coatings.

Effect of Physiological Fluids Contamination on Selected Mechanical Properties of Acrylate Bone Cement.

This study analyses the degradation rate of selected mechanical properties of bone cement contaminated with human blood and saline solution. During the polymerisation stage, the PMMA cement specimens were supplemented with the selected physiological fluids in a range of concentrations from 0% to 10%. The samples were then subjected to the standardised compression tests, as per ISO 5833: 2002, and hardness tests. The obtained results were analysed statistically to display the difference in the degradation of the material relative to the degree of contamination. Subsequently, numerical modelling was employed to determine the mathematical relationship between the degree of contamination and the material strength degradation rate. The introduction of various concentrations of contaminants into the cement mass resulted in a statistically significant change in their compressive strength. It was shown that the addition of more than 4% of saline and more than 6% of blood (by weight) causes that the specimens exhibit lower strength than the minimum critical value of 70 MPa, specified in the abovementioned International Standard. It was further revealed that the cement hardness characteristics degraded accordingly. The mathematical models showed a very good fit with the results from the experiments: The coefficient of determination R2 was 0.987 in the case of the linear hardness model for blood and 0.983 for salt solution; secondly, the values of R2 for the third-degree polynomial model of compressive strength were 0.88 for blood and 0.92 for salt. From the results, it can be seen that there is a quantitative/qualitative relationship between the contamination rate and the drop in the tested mechanical characteristics. Therefore, great effort must be taken to minimise the contact of the bone cement with physiological fluids, which naturally occur in the operative field, particularly when the material cures, in order to prevent the cement material strength declining below the minimum threshold specified in the ISO standard.

Development of a new technique for quantitative PCR analysis

A new technique for automatic determination of the quantification cycle value Cq is proposed. The technique is based on the third-degree polynomial with a limited range of approximation, which is determined automatically. A reduction in the random error of constructing the calibration dependence in quantitative analyzes of genetically modified soy has been achieved in comparison with the well-known ANK_Shell software of the nucleic acid analyzers ANK.

Processing of videos of the fatigue cracks propagation in the rotor blades of the helicopter

The article describes the processing of videos of growing fatigue cracks in the main rotor blades of a helicopter, which are formed during their fatigue tests on the stand. A specially created computer program allows to determine the geometrical parameters of a crack based on video recording (length, width, extreme points, center of gravity). Each frame of the video is binarized, after which a binary matrix is created on its basis, which is subject to element-by-element processing. The obtained data are presented in the form of a graph showing the dependence of the crack growth rate on the time of its growth. The graph is traced in the Mathcad program, as a result, two data vectors are created — the time and crack growth rate, which undergo an interpolation procedure with the construction of a third-degree polynomial. According to the results of the integration of the polynomial, the critical length of the crack is determined, after reaching which its rapid growth begins.

The study of the dependence of spring crops yield on the abiotic environmental factors using nonlinear interpolation

The article discusses the effect of abiotic environmental factors on crop yields. Abiotic factors cannot be changed by human activity; therefore, considering their impact on yield is very important for predicting crop yields. The article analyzes the dependence of spring wheat yield on such abiotic environmental factors as average ten-day air temperatures during the growing season, ten-day amounts of precipitation during the growing season and the Selyaninov hydrothermal coefficient during the growing season. To identify dependencies, the authors applied a non-linear approximation method using a third-degree polynomial dependence with one variable. The study was conducted during the most important growing season of spring wheat, in May and June. The authors show the equations of the third-degree polynomial dependences and graphs, as well as their graphic interpretation. Based on the obtained graphs, the authors have analyzed favorable intervals of abiotic factors for the formation of high yields of spring wheat. The authors indicate the most optimal temperature and humidity conditions for each ten days of May and June. These dependences are suitable not only for spring wheat but also for other spring grain crops, such as oats and barley.

Development of implicit method for numerical modeling of turbomachine blade thermoelastic vibrations

There was a tendency in recent decades to increase the combustion temperature in gas-turbine engines (GTE). This allows increasing both the efficiency of the engine and the output power. In modern engines, the temperature of the exhaust gas is already significantly higher than the melting temperature of blade material. In this regard, in the design of GTE turbines there is a need to use numerical methods that allow the most reliable modeling of unsteady aerothermoelastic effects. One of the components of the aerothermoelastic problem is to integrate the unsteady equations of thermoelasticity together with the equations of aerodynamics. As these equations must be solved together with a single step in time, implicit numerical integration methods should be preferred. The object of research is the unsteady interaction of thermoelastic vibrations of the turbine blades and gas flow. This paper presents an implicit numerical method for modeling thermoelastic vibrations of the GTE turbine flow parts, including turbine blades with cooling channels. The method is based on equations of linear thermoelasticity, which are integrated by the finite element method. The investigated volume is divided into cells, forming a calculation grid with hexahedrons with additional nodes. The compute nodes are selected so that one element has 20 nodes. The approximation of the parameters in the element is performed using third-degree polynomials. Time integration is also performed with third-order accuracy. The results of testing the method on test problems, as well as comparing the results of the vibrations simulation of the standard configuration blades with the results of other authors are shown. The discrepancy of the results does not exceed 0.4 % for the test problem and 0.7 % for the blade vibrations. The obtained results indicate that the presented method can be used for numerical simulation of the unsteady thermoelastic vibrations of the gas-turbine engine flow parts.

Finite element implementation based on explicit, Galerkin and Crank–Nicolson methods to phase field theory for thermal- and surface- induced martensitic phase transformations

In this paper, a nonlinear finite element approach is used to solve the time-dependent phase field or Ginzburg–Landau (GL) equation for phase transformations (PTs) at the nanoscale. The utilized Helmholtz energy includes a third-degree polynomial of the phase variable which results in a nonlinear dependence on the order parameter in the GL equation. The method of weighted residuals is used to derive the corresponding finite element formulation. Using the divergence theorem, the Laplace term in the GL equation is reduced to the first-order term which can represent both the isolated and variable surface energy boundary conditions, allowing for surface-induced phenomena. In order to apply the Dirichlet boundary condition, i.e., constant phase order parameter, the penalty method is used. Quadrilateral elements have been used in the Cartesian coordinate system, and irregular elements for complex geometries are transformed to regular ones by using natural coordinates. Four-point Gaussian integral has been used for integral calculations. The alpha family and both the explicit and implicit methods (Crank–Nicolson and Galerkin methods) are used for time discretization. Thus, the Picard and Newton–Raphson methods are used to linearize the nonlinear equation and their computational efficiencies are compared. In the explicit method, in order to calculate the inverse of the stiffness matrix, the lumping technique has been used which transforms it into a modified diagonal matrix, and this significantly reduced the computation time. In the implicit methods of Picard and Newton–Raphson, both the exact and numerical approaches were used to calculate the tangential matrix for the unknown order parameter and no difference in results between them was found. The convergence for both the Picard and Newton–Raphson methods was studied and compared. Cubic to tetragonal, thermal- induced transformation in NiAl is considered. The order parameter profile and the width and velocity of austenite (A)–martensite (M) interface are calculated and verified with previous works. Examples of thermal- and surface-induced PTs in nanosized samples without hole and with it are studied, presenting planar and nonplanar interfaces. The developed algorithm and code present a proper tool to solve more advanced phase field problems for PTs at the nanoscale with complex geometries and including mechanics effects and with considering complex interactions.

Vector-Matrix Method of Numerical Implementation of the Polynomial Integral Volterra Operators

The article deals with the quadrature method for the numerical implementation of polynomial integral operators. With the computer implementation of Volterra-type integral models, the typical problem is the accumulation of calculations at each step of the computational process. For its acceleration it is suggested to apply the vector-matrix approach. The suggested approach is based on quadrature methods: rectangles, trapezoids, and Simpson's. For homogeneous polynomial integral Volterra operators of the first-, second- and third-degree, respectively, the objects in the form of vectors, matrices, and three-dimensional structures containing the coefficients of the corresponding quadrature formulas have been constructed. The suggested vector-matrix approach involves the reduction of computational operations to the elementary multiplication of elements of the corresponding structures and allows efficient use of parallel algorithms, which significantly accelerates the execution of computational tasks for the implementation of integral operators. In the research work the complexity of implementation is estimated depending on the number of possible parallel flows. The estimation of the suggested approximations of integral representations is researched by model examples, in which there are models in the form of second- and third-degree polynomial integrals of Volterra. The results of computational experiments showed that among the considered quadrature methods, the trapezoidal method is optimal in terms of «precision — complexity of implementation». The accuracy of the numerical implementation of integral models depends on the chosen method, the simulation step, the type of kernel, and does not depend on the dimensionality of the operator. The vector-matrix approach allows building of efficient algorithms for the numerical implementation of integral models and greatly simplifies their software implementation, as it allows easy scaling to a multidimensional case. Such representation allows to use advantages of matrix-oriented packages of applications (Matlab, Octave, Scilab), the peculiarity of which is the high speed of execution of matrix operations.

Permeation characterization and modelling of polyethylene/clay nanocomposites for packaging

The approach of upgrading barrier properties of polymer with nanofillers is effectual and well established now-a-days to manufacture films and quality packaging. This research deals with enhancement of barrier attributes of clay-based high density polyethylene nanocomposites against oxygen and water permeability. Na-Montmorillonite and two grades of kaolin clays were used for HDPE/clay nanocomposites preparation in twin-screw extruder with temperature profile of 160, 170, 180, 190 and 200 °C along the extruder at 110 rpm with clay loading up to 10 wt%. The extent of maximum reduction in water and oxygen vapors permeation was more prominent at 5 wt% sample of each clay. XRD analysis revealed no exfoliation in kaolin clay samples but indicated presence of exfoliation in Na-MMT samples. SEM and TEM micrographs revealed nano-level dispersion for both kaolin clays and Na-MMT and agglomerate formation at high wt% of clays. Experimental data for oxygen and water vapors permeation through nanocomposites were fitted on Nielsen, Cussler and Gusev–Lutsi models. Modelling of barrier performance was utilized for acquiring aspect ratio of filler for nanocomposites. Maximum and average aspect ratio of fillers and third-degree polynomial equation for nanocomposite, through curve fitting, were determined to predict improved barrier properties of nanocomposites.

Calculation of Spectroscopic Parameters of Diatomic Molecules with an Open Electron Shell

The spectroscopic parameters of the BeH and CH molecules with an open electron shell are calculated in the basis of the Slater functions. An analytical expression for the total electron energy of molecules with an open electron shell is obtained. The total electron energy of the BeH and CH molecules was calculated by the Hartree–Fock–Roothaan (HFR) method for seven internuclear distances, including the equilibrium internuclear distance. A table of total energy values is constructed depending on the internuclear distance R. The total energy of a diatomic molecule is approximated as a third-degree polynomial in R. Applying the least squares method to energy values from the above-mentioned table, an analytical formula for the total energy of the BeH and CH molecules is obtained as a function of R. The analytical formula for E(R) has allowed the spectroscopic parameters ωe (harmonic vibrational frequency), ωexe (anharmonicity constant), Be (rotational constant), and αe (rovibrational interaction constant) to be calculated.

Self-diffusion and structure of monovalent ions in two dimensions: A molecular dynamics study

The knowledge of the self-diffusion coefficient is indispensable for different processes in chemical engineering and in the design of new materials, which sometimes occur in restricted or confined geometries. In this work we studied the self-diffusion and structural properties of monovalent ions moving in two dimensions. The two-dimensional soft primitive model was used to describe the interaction between ions and molecular dynamics simulations of the electroneutral mixture were performed to calculate the properties. Radial distribution functions, density-density and charge-charge correlation functions as well as the mean square displacement, the velocity autocorrelation function and the diffusion coefficient were evaluated at supercritical conditions. For the lowest reduced temperature, the formation of clusters, chains and rings of ions influenced the cation-anion radial distribution function, particularly at low densities, where very high values of the first maximum were observed. Clusters and free ions were evaluated. These arrays also influence the density-density and charge-charge correlation functions. Concerning the self-diffusion, it increases as temperature is higher and decreases when density increases, as physically expected. The behavior of the self-diffusion coefficient was analyzed by trying different fitting models. We found that the density dependence is well described by a third-degree polynomial or a linear-logarithmic fit. When the simultaneous dependence with density and temperature is considered, we found that the diffusion is a unique function of the variable ργ/T, where the exponent γ is about 1.7, in contrast with three-dimensional m − 6 Lennard-Jones fluids, where it has been found to be in the range 3.4–13.5. We also investigated the type of diffusive motion. The features observed in both, the velocity autocorrelation function and the mean square displacement, indicated that the diffusion of the ions takes place as normal diffusion, in contrast with other two-dimensional fluids, where superdiffusive and subdiffusive motion have been observed.

New Compound Open Channel Section with Polynomial Sides: Improving Cost and Aesthetics

Previous research on compound trapezoidal cross sections has mainly focused on improving the prediction of the discharge (flow rate) because of its inherent challenges. This paper focuses on two other important aspects: Section shape and optimal construction cost. First, the paper proposes a new compound section with third-degree polynomial sides of main channel with horizontal bottom (HB) that allows its top corners to be smooth, called herein compound polynomial section. The special cases of this versatile section include the simple polynomial section, polygonal section, trapezoidal-rectangular section, two-segment linear-side section, and parabolic bottom-trapezoidal section. The simple polynomial section, which is the bank-full part of the compound polynomial section, can further produce parabolic (with or without HB), trapezoidal, rectangular, and triangular sections. Second, an optimization model that minimizes construction cost (excavation and lining) of the compound (or simple) polynomial section is developed. The model includes discharge and physical constraints. Theoretical and empirical methods of discharge prediction were used in the model. The results show that the simple polynomial section was more economical than the popular parabolic section by up to 8.6% when the side slopes were restricted. The new polynomial-based sections not only reduced construction cost, but also improved maintenance and aesthetics. As such, the new sections should be of interest to researchers and practitioners in hydraulic engineering.

Dendritic growth of the Pt–Cu islands on Cu(111) surface: Self-learning kinetic Monte Carlo simulations

Growth of the Pt–Cu islands on the Cu(111) surface at different deposition fluxes, relative amounts of Pt atoms and surface temperatures is investigated on the atomic scale by performing the self-learning kinetic Monte-Carlo simulations. The shape transition of the islands from sixfold symmetry to threefold symmetry with increasing of the relative amount of Pt atoms nPt/nCu in clusters at room temperature was found and explained by the corner diffusion anisotropy. Nonmonotonic dependence of the fractal dimension of the dendritic islands on the ratio nPt/nCu is observed. It is shown that this effect can be interpreted in the framework of the generalized diffusion limited aggregation model if we assume that the dependence of the effective diffusion barrier on the ratio nPt/nCu has the third-degree polynomial function form. This dependence is in qualitative agreement with the analysis of the edge diffusion barriers. The dendritic cluster obtained with the simulations at room temperature looks very similar to the experimentally observed one (Soy et al. (2015) [33]).

Development of an SMA-based camber morphing UAV tail core design

This paper presents a new design for the core of a lightweight tail section of an unmanned aerial vehicle (UAV) with camber-morphing horizontal and vertical stabilizers. The core of each stabilizer is composed of an aluminum spar, two active end ribs, and multiple inactive ribs. Each active end rib is composed of a solid leading compartment connected to a flexible corrugated trailing segment. Thermally activated shape memory alloy (SMA) wire actuators along the length of the corrugated segment are used to control the camber of each active rib. The SMA wires are extended through the hollow spar to increase the amount of actuation and are guided using polycarbonate pulleys. A parametric CAD model was created to automatically regenerate the corrugated trailing segment geometry based on trough height, width and angle, as well as the used NACA airfoil. The locations of the vertical webs are adjusted with each parameter set so that the webs are always at the trough centers for consistency. COMSOL’s LiveLink was used to pass the generated CAD geometry to COMSOL, where finite element structural analysis is performed to study the effect of the geometric parameters on the camber deformation under SMA wire actuation and applied loads. The deformed shape of the trailing segment is then approximated as a third-degree polynomial and used to modify the four-digit NACA airfoil equation, generating a deformed shape for the airfoil. 2D computational fluid dynamics simulations are then performed to compute the lift-to-drag ratio for each structural configuration, from which the geometric parameters that maximize the performance of the stabilizer at the design speed can be selected. The proposed UAV SMA-based camber-morphing rear control section has been successfully manufactured and tested. Camber morphing up to 10.7° was successfully achieved and showed very good agreement with the numerical prediction.

O3o3: A Variant of Spectral Elements with a Regular Collocation Grid

AbstractIn this study, an alternative local Galerkin method (LGM), the o3o3 scheme, is proposed. o3o3 is a variant or generalization of the third-order spectral element method (SEM3). It uses third-order piecewise polynomials for the representation of a field and piecewise third-degree polynomials for fluxes. For the discretization, SEM3 uses the irregular Legendre–Gauss–Lobatto grid while o3o3 uses a regular collocation grid. o3o3 can be regarded as an inhomogeneous finite-difference scheme on a uniform grid, which means that the finite-difference equations are different for each group with three points. A particular version of o3o3 is set as an example of many possibilities to construct LGM schemes on piecewise polynomial spaces in which the basis functions used are continuous at corner points and function spaces having continuous derivatives are shortly discussed. We propose a standard o3o3 scheme and a spectral o3o3 scheme as alternatives to the standard method of using the quadrature approximation. These two particular schemes selected were chosen for ease of implementation rather than optimal performance. In one dimension, compared to standard SEM3, o3o3 has a larger CFL condition benefiting from the use of a regular collocation grid. While SEM3 uses the irregular Legendre–Gauss–Lobatto collocation grid, o3o3 uses a regular grid. This is considered an advantage for physical parameterizations. The shortest resolved wave is marginally smaller than that with SEM3. In two dimensions, o3o3 is implemented on a sparse grid where only a part of the points on the underlying regular grid are used for forecasting.

Spectral deformation in a problem of singular perturbation theory

Quasi-classical asymptotic behavior of the spectrum of a non-self-adjoint Sturm–Liouville problem is studied in the case of a one-parameter family of potentials being third-degree polynomials. For this problem, the phase-integral method is used to derive quantization conditions characterizing the asymptotic distribution of the eigenvalues and their concentration near edges of the limit spectral complex. Topologically different types of limit configurations are described, and critical values of the deformation parameter corresponding to type changes are specified.

A Geometric Model to Determine Patient-Specific Cup Anteversion Based on Pelvic Motion in Total Hip Arthroplasty.

Introduction Cup position is critical to stability in total hip arthroplasty and is affected by pelvis motion during positions of daily life. The purpose of this study was to explicitly define the relationship between sagittal pelvic motion and resultant cup functional anteversion and create a tool to guide the surgeon to a patient-specific intra-operative anteversion. Materials and Methods 10,560 combinations of inclination, anteversion, and pelvic tilt were generated using a geometric model. Resultant functional anteversion was calculated for each iteration and variables were correlated. An electronic mobile tool was created that compares inputted patient-specific values to population-based averages to determine pelvic positions and dynamics that may lead to instability. Results A third-degree polynomial equation was used to describe the relationship between variables. The freely downloadable mobile tool uses input from pre-operative plain radiographic measurements to provide the surgeon a quantitative correction to intra-operative cup anteversion based on differences in functional anteversion compared to population-based averages. Conclusion This study provides a geometric relationship between planned cup position, pelvic position and motion, and the resultant functional anteversion. This mathematical model was applied to an electronic tool that seeks to determine an individualized intra-operative cup anteversion based on measured patient-specific pelvic dynamics.