Order Polynomial Research Articles

Propagation of spatiotemporal Airy-Laguerre complex-variable-function Gaussian wave packets in a chiral medium

We analyze the propagation of spatiotemporal Airy-Laguerre complex-variable-function Gaussian (ALCG) wave packets using phase modulation. These wave packets can split into left circularly polarized ALCG (LCP-ALCG) and right circularly polarized ALCG (RCP-ALCG) components due to circular birefringence in a chiral medium. Our discussion primarily focuses on the effects of polynomial order, angle parameters, distribution factor, cross-phase factor, first-order chirp factor, and chiral parameter on the ALCG wave packets. The distinct LCP-ALCG and RCP-ALCG wave packet characteristics are obtained, and the radiation forces on Rayleigh dielectric particles are investigated. Our results emphasize the influence of various parameters in the propagation properties of ALCG wave packets, unveiling their potential for advancements in optical communication and optical trapping domains.

Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.

Numerical Solutions of Some Weak Singular Nonlinear Integral Equations of The First Type Using The Spectral Collocation Method

In this paper, a collocation-spectral approximation is proposed for weakly nonlinear and neutral singular Volterra integral-differential equations with rough solutions. We used some appropriate transformations to transform the equation of the equation into a new equation, so that the solution of the new equation has a better order (smoothness) and Jacobi's orthogonal polynomial theory can be easily used. Under appropriate assumptions on the nonlinear part, we were able to perform an acceptable error analysis on the soft and the weighted soft. To obtain a numerical approximation, some numerical examples (linear and non-linear) with uneven solutions are considered and numerical results are also presented. Also, a comparison between the proposed method and some existing numerical methods is also provided.

An approach of Finding Second Derivative Finite Differences Compact Schemes

In solving problems from Computational Fluid Dynamics (CFD) and physics, huge efforts should be afforded to obtain accurate and applicable schemes for the derivatives. Based on the idea of high order polynomials, many sets of second derivative schemes are derived in this paper. These sets are grouped according to the order of the accuracy of the approximations from order three to order seven. Different types of second derivative forward, central, and backward compact with some traditinal approximations are inrtoduced at each set by the proposed method. The order of accuracy is verified of each scheme using the technique of finding the values of the coefficients for the error terms by matching both sides of the given scheme. Many schemes that are introduced in this article are applicable to some problems from science and engineering.

A corrected Raviart–Thomas Spectral Difference scheme stable for arbitrary order of accuracy on triangular and tetrahedral meshes

A theoretical stability estimate is presented for the Raviart–Thomas Spectral Difference scheme for triangular and tetrahedral meshes. It reveals the difficulty of guaranteeing the existence of stable flux points for an arbitrary order of accuracy. To obtain a numerical method stable for any order of accuracy and any space dimension, the present article introduces a new correction term of the flux approximation to cancel the numerical error caused by the Riemann solver exactly. This modification enables the introduction of a new family of so-called Corrected Raviart–Thomas Spectral Difference stable for any polynomial order of approximation. Stability estimates are provided for the L2 norm and a weighted broken Sobolev norm. Numerical experiments in 2D and 3D are conducted for solving linear and non-linear hyperbolic equations with a polynomial order up to p=8 for triangular elements and up to p=5 for tetrahedral elements. They confirm the accuracy, stability, and robustness of the newly introduced numerical scheme. The new method is shown to have a computational cost similar to the Raviart–Thomas Spectral Difference with – for a given mesh – a higher accuracy.

Converting an HLA-B27 flow assay from the BD FACSCanto to the BD FACSLyric.

HLA-B27 is a major histocompatibility complex (MHC) class I antigen which exhibits strong association (90%) with ankylosing spondylitis. HLA-B27 detection in patients by flow cytometry is a widely used clinical test, performed on many different flow cytometer models. We sought to develop and validate a test conversion protocol for the HLA-B27 test performed on the BD FACSCanto to BD's newer FACSLyric flow cytometers. The development and validation experiments were performed using anti-HLA-B27*FITC/CD3*PE antibody-stained whole blood patient specimens. The anti-HLA-B27*FITC logarithmic median fluorescence (LMF) results on the BD FACSCanto were converted to median fluorescence intensity (MFI) values on the BD FACSLyric. Clustering of the HLA-B27 positive and negative values, using a 3rd order polynomial equation, resulted in a conversion of the BD FACSCanto cutoff values, negative (<150 LMF) and positive (≥160 LMF), to negative (<4530 MFI) and positive (≥6950 MFI) on the BD FACSLyric. Accuracy was assessed by comparing the flow results obtained on the BD FACSCanto and BD FACSLyric to a molecular PCR based assay. Additional validation parameters (compensation verification, intra- and inter-assay precision, and instrument comparison) were performed per the recommendations outlined in the Clinical and Laboratory Standards Institute (CLSI) H62 guidelines for validation of flow cytometry assays.

Two-Dimensional Legendre Polynomial Method for Internal Tide Signal Extraction

This study employs the two-dimensional Legendre polynomial fitting (2-D LPF) method to fit M2 tidal harmonic constants from satellite altimetry data within the region of 53°E–131°E, 34°S–6°N, extracting internal tide signals acting on the sea surface. The M2 tidal harmonic constants are derived from the sea surface height (SSH) data of the TOPEX/Poseidon (T/P), Jason-1, Jason-2, and Jason-3 satellites via t-tide analysis. We validate the 2-D LPF method against the 300 km moving average (300 km smooth) method and the one-dimensional Legendre polynomial fitting (1-D LPF) method. Through cross-validation across 42 orbits, the optimal polynomial orders are determined to be seven for 1-D LPF, and eight and seven for the longitudinal and latitudinal directions in 2-D LPF, respectively. The 2-D LPF method demonstrated superior spatial continuity and smoothness of internal tide signals. Further single-orbit correlation analysis confirmed generally higher correlation with topographic and density perturbations (correlation coefficients: 0.502, 0.620, 0.245; 0.420, 0.273, −0.101), underscoring its accuracy. Overall, the 2-D LPF method can use all regional data points, overcoming the limitations of single-orbit approaches and proving its effectiveness in extracting internal tide signals acting on the sea surface.

A matrix approach to generalized Bernoulli–Fibonacci polynomials of order m and applications

A matrix approach to generalized Bernoulli–Fibonacci polynomials of order m and applications

Using Hough transform to define track boundaries as part of machine vision by train traffic-tracking hardware and software systems

Aim. In the context of machine vision, the problem of detected object boundary definition is usually solved using semantic segmentation that requires a high computational resource. Its application increases the complexity and the cost of the implemented hardware and software systems. This paper proposes an alternative method for defining the boundary of a segmented object, i.e., a railway track, for a train traffic tracking system. Methods. Since a railway track in an image can be represented by an n-th order polynomial, it is suggested to solve the problem of track boundary detection by using approximations in the form of straight lines. It is suggested using the Hough transform to detect straight lines. The former’s parametric space will be arranged in accordance with the problem to be solved. Conclusions. The proposed approximation will allow abandoning semantic segmentation and reduce the computational complexity and load.

192 Prediction of fecal starch content of fattening cattle using near-infrared spectroscopy and machine learning

Abstract Starch content in cattle feces can be used to predict starch digestibility. Near-infrared spectroscopy (NIRS) can be used to rapidly and non-chemically predict fecal starch content after a calibration model has been consequentially developed. Recent advancements in machine learning have introduced potent methods for developing optimized models. In addition, diverse preprocessing methods, such as scatter correction and smoothing, are available to eliminate irrelevant variance sources. This study aimed to identify optimal spectral preprocessing and regression methods for predicting fecal starch content using NIR spectra. NIR spectra were measured for 196 fecal samples collected from 9 fattening cattle farms. Fecal starch content on a DM basis was measured using enzymatic reactions and colorimetry. The 196 samples were divided into a calibration set (n = 123), a validation set (n = 29) comprising four farms, and an external validation set (n = 44) comprising five farms that differed from the calibration set. Based on the calibration set, two preprocessing methods were selected using the partial least squares (PLS) model. In method 1, the calibration models were developed by a grid search 5-fold cross validation on the 1,152 preprocessed calibration sets, and preprocessing with the lowest root mean square error (RMSE) of cross-validation among these calibration sets was selected. In method 2, the original calibration set (n = 123) was divided into a calibration set (n = 116) and a preprocessing selection set (n = 7) with different neutral detergent fiber and similar starch contents. After preprocessing, calibration models were developed as described in method 1. The preprocessing selection sets were then predicted by the developed models, and preprocessing with the lowest RMSE among these preprocessing selection sets was selected. Using the preprocessing methods selected in methods 1 and 2, the validation and external validation sets were predicted using seven regression models. As a result, the preprocessing selected by method 1 was in the wavelength range of 400–2500 nm, with robust normal variate (RNV) scatter correction and first-order derivative Savitzky-Golay filtering (2nd order polynomial, 2-nm window size). The preprocessing selected by method 2 had a wavelength range of 1500–2500 nm, RNV scatter correction, and 2nd-order derivative Savitzky-Golay filtering (3rd order polynomial, 25-nm window size). Among the seven regression models, the Lasso model was the most accurate for prediction. The Lasso models based on the preprocessing methods selected by methods 1 and 2 had RMSE of external validation = 2.054% of DM and 1.540% of DM, respectively. The feature importance of the Lasso model based on preprocessing method selected by methods 2 was higher at wavelengths of 1775 nm and 2325 nm. In conclusion, seven models were developed and evaluated based on diverse preprocessing methods using NIRS. The Lasso model exhibited the highest accuracy.

Aflatoxin contamination level estimation in food using reflectance multispectral imaging based system

This study presents a non-invasive, non-destructive, and rapid method of classifying and estimating aflatoxin levels in food samples using a multispectral imaging. A cost effective, mass deployable in-house built multispectral imaging system, suitable for onsite testing and operating in range from 365 nm to 940 nm, is utilized to capture the multispectral images of the aflatoxin contaminated bread samples. Spatial and spectral corrections are performed to mitigate the imperfections of the multispectral imaging system, followed by image preprocessing steps to mitigate the effect of random noise. Linear discriminant analysis is used to reduce the dimensionality of the feature space. Ensemble classification analysis utilizing classical machine learning classifiers yielded an accuracy of 90 % when classifying samples according to aflatoxin contamination levels. A functional relationship between the Bhattacharyya distance of noncontaminated samples to contaminated samples and aflatoxin contamination levels was modeled as a first order polynomial with an R2 value of 0.9723.

MATHEMATICAL MODELS OF THE THERMAL PROTECTION COATING OF THE GAS GENERATOR OF THE HYDROGEN STORAGE AND SUPPLY SYSTEM

The article describes the properties of the thermal protection coating of the gas generator of the hydrogen storage and supply system by two transfer functions with Hurwitz polynomials of the third order. Obtaining such transfer functions is based on the solution of the non-stationary heat conduction equation, represented in operator form using the integral Laplace transform, and in which approximating spline functions in the form of second-order polynomials are used. The article gives the solution of the non-stationary thermal conductivity equation, which describes the thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system under the thermal effect of a fire. This solution comes in the operator form for the surface temperature of the heat-protective coating on the side of the gas generator wall. The peculiarity of this decision is the presence of hyperbolic functions of an irrational argument in its composition. The structural and dynamic scheme of the thermodynamic system ‘gas generator wall – heat-protective coating’ is presented, the feature of which is the presence of two entrances. The signal at the scheme’s first input reflects the thermal effect of the fire, and the signal at the second input reflects the thermal state of the gas generator cavity. An equivalent mathematical transition to the description of thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system was carried out. This transition happened due to the use of spline functions, which approximate the hyperbolic functions of the irrational argument and are polynomials of the second order. The article gives a verbal interpretation of the algorithm for determining the transfer functions of the heat-protective coating of the gas generator of the hydrogen storage and supply system and also an example of its implementation. Furthermore, it shows that for the given conditions of functioning of the gas generator heat-protective coating, the relative errors in approximation of hyperbolic functions by second-order polynomials do not exceed 1.7 %, and the average relative error when equivalent replacement of transfer functions does not exceed 3.8 %. Keywords: gas generator, hydrogen storage and supply system, thermal protection coating.

Three-dimensional compact multi-resolution weighted essentially non-oscillatory reconstruction under the Arbitrary Lagrange–Euler framework

A third-order compact multi-resolution weighted essentially non-oscillatory (CMR-WENO) reconstruction method for three-dimensional (3D) hybrid unstructured grids is developed using the Arbitrary Lagrange–Euler framework. The finite volume method is used to discretize the governing equations, and some turbulent and moving boundary problems are simulated. Only one compact center stencil comprising the neighboring cells of each control cell is required to construct the polynomials in the algorithm. As a result, the number of stencils and stencil cells is significantly reduced when compared with the traditional WENO scheme. This simplifies the code and improves the robustness of the algorithm. By ensuring the cell average and first-order derivatives are consistent with that in stencil cells an over-determined system of equations can be used to reconstruct the polynomials. This system can then be solved using the compact least squares method to avoid an ill-conditioned coefficient matrix. Furthermore, a coupled implicit iteration strategy is used to solve for the unknown coefficients, so no extra determination is required for the derivatives of each control cell. The final interpolation function for discontinuities in the flow field is obtained using CMR-WENO to nonlinearly combine polynomials of different orders, which further improves the stability of the algorithm. The CMR-WENO can be implemented on 3D hybrid unstructured grids and can be used to simulate complex problems such as those involving turbulence and moving boundaries. Finally, the algorithm presented here is verified to be third-order accurate and to exhibit good robustness when used on several representative numerical examples.

Stability of the linearization method applied in tricom-partmental polynomial catenary systems of order (α + β)

This paper aims to identify exchange coefficients of a nonlinear polynomial tri-compartmental catenary system of (α + β) order. This is based on two principal procedures. The first procedure presented is related to the recommended solution consisting of introducing an adequate time t∗ &gt; 0 in a way to be defined. That is to say: wait a moment to allow the exchange to settle in the polynomial (α +β) order nonlinear catenary system after injecting the quantity into the main compartment, then measure this compartment with compartment 2, at this time t∗ &gt; 0. In the second procedure, we apply the Taylor formula to linearize the nonlinear system and identify the exchange coefficients. In the end, we will prove that the linearization method is stable

Higher Order Polynomial Transformer for Fine-Grained Freezing of Gait Detection.

Freezing of Gait (FoG) is a common symptom of Parkinson's disease (PD), manifesting as a brief, episodic absence, or marked reduction in walking, despite a patient's intention to move. Clinical assessment of FoG events from manual observations by experts is both time-consuming and highly subjective. Therefore, machine learning-based FoG identification methods would be desirable. In this article, we address this task as a fine-grained human action recognition problem based on vision inputs. A novel deep learning architecture, namely, higher order polynomial transformer (HP-Transformer), is proposed to incorporate pose and appearance feature sequences to formulate fine-grained FoG patterns. In particular, a higher order self-attention mechanism is proposed based on higher order polynomials. To this end, linear, bilinear, and trilinear transformers are formulated in pursuit of discriminative fine-grained representations. These representations are treated as multiple streams and further fused by a cross-order fusion strategy for FoG detection. Comprehensive experiments on a large in-house dataset collected during clinical assessments demonstrate the effectiveness of the proposed method, and an area under the receiver operating characteristic (ROC) curve (AUC) of 0.92 is achieved for detecting FoG.

Advancing Fixed-Source Reactor Analysis Through OpenNode Implementation

OpenNode, a new open-source Fortran code, enables the simulation of fixed-source reactors. Powered by the nodal expansion method (NEM) and seamless Python integration, OpenNode competes with existing nuclear reactor simulation tools. The fixed-source problem plays a crucial role in radioprotection, providing flux calculations and detailed insights into heating effects and dose rates. Users can define and configure reactor models via a JSON input file, specifying critical parameters like geometry, materials, cross-section data, boundary conditions, and fixed sources. OpenNode further empowers users with a Python interface for preprocessing and postprocessing, streamlining result analysis. Distinguishing itself from conventional NEM codes, OpenNode supports three-dimensional (3D) Cartesian geometries, facilitating intricate reactor design simulations. Customization options, including mesh sizes, polynomial orders, and calculation modes, enhance precision and efficiency. In our comprehensive study, we verified OpenNode’s fixed-source mode using the 3D-IAEA reactor, comparing it with the SANM code KOMODO. The results underscore OpenNode’s exceptional accuracy in computing critical parameters, like the effective multiplication factor, power distribution, and nodal flux, within fixed-source reactors. OpenNode stands as a reliable, user-friendly tool poised to advance nuclear reactor simulations.

VEMcomp: a Virtual Elements MATLAB package for bulk-surface PDEs in 2D and 3D

AbstractWe present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulk-surface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order $${k=1}$$ k = 1 on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order $${k=1}$$ k = 1 as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.

Serum anti-Mullerian hormone levels in Turkish girls aged 18 and younger for ovarian reserve determination

Our aim was to show that anti-Mullerian hormone (AMH) may be used as a quantitative marker of ovarian reserve in Turkish girls aged 18 years and younger and establish the reference values for AMH in Turkish girls. This retrospective study included girls between 8-18 years old, without premature ovarian failure or without genetic factors resulting in ovarian dysgenesis. Blood specimens were collected after overnight fasting early in the morning during the early follicular phase. Measurement of serum levels of gonadotropins and AMH was done. Mean serum AMH levels of different age groups and best fitting curve representing AMH percentiles (10th, 25th, 50th, 75th, 90th) were calculated. In total 785 girls with a mean age of 16.16±1.90 years were included, divided into seven age groups. The mean serum AMH level for the total cohort was 5.20±4.19 ng/mL. There was a significant difference between the mean values of AMH in age groups as follows: ≤12 and 17-≤18 (p=0.011). The best fitting curves for AMH percentiles were 4th order polynomial functions. There was a significant correlation between AMH and age and follicle stimulating hormone levels (r=0.148, p<0.001 and r=-0.092, p=0.010). Our results reflect the real-life data for serum AMH values in Turkish girls. Our nomogram may be useful for counseling adolescents about their ovarian reserve and diagnosing other gynecological diseases. A longitudinal study is necessary for improving the predictive value of AMH values in girls aged 18 and younger.

Adaptive polynomial Kalman filter for nonlinear state estimation in modified AR time series with fixed coefficients

AbstractThis article presents a novel approach for adaptive nonlinear state estimation in a modified autoregressive time series with fixed coefficients, leveraging an adaptive polynomial Kalman filter (APKF). The proposed APKF dynamically adjusts the evolving system dynamics by selecting an appropriate autoregressive time‐series model corresponding to the optimal polynomial order, based on the minimum residual error. This dynamic selection enhances the robustness of the state estimation process, ensuring accurate predictions, even in the presence of varying system complexities and noise. The proposed methodology involves predicting the next state using polynomial extrapolation. Extensive simulations were conducted to validate the performance of the APKF, demonstrating its superiority in accurately estimating the true system state compared with traditional Kalman filtering methods. The root‐mean‐square error was evaluated for various combinations of standard deviations of sensor noise and process noise for different sample sizes. On average, the root‐mean‐square error value, which represents the disparity between the true sensor reading and estimate derived from the adaptive Kalman filter, was 35.31% more accurate than that of the traditional Kalman filter. The comparative analysis highlights the efficacy of the APKF, showing significant improvements in state estimation accuracy and noise resilience.

The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element

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