Order Polynomial Research Articles

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

Екстремальні значення найкращих наближень лінійних комбінацій гармонічних функцій

Extreme problems and their practical applications have been under the scrutiny of mathematicians since ancient times. An important step in the development of extreme problems was made by P. L. Chebyshev, who in the 50s of the 19th century laid the foundations of a section of destructive function theory – the theory of approximation. A significant role of the formation of the theory of approximation of functions was played by Carl Weierstrass’s theorem on the convergence to zero of best approximations by polynomials of a continuous function. As is well known, Weierstrass’s theorem is not constructive – it does not contained estimates of the approach speed. Thanks to the work of D. Jackson, S. N. Bernstein, Vallee-Poussin and others, such estimates began to appear in works on approximation theory. At the same time, at the first stages of the development of the theory of approximation, approximations of individual functions were studied. That beginning of a new period, a dipper study of the deviation values of functions from their approximating polynomials, dates back to the 30s and 40s of the 20th century and is associated with the names of A. M. Kolmogorov, S. M. Nikolsky, J. Favard, N. I. Achieser, M. G. Crane and B. Nagy. Thanks to their works, the main emphasis in the theory of approximations is shifted to the study of the best approximations or other approximation characteristics of functions that have certain differential-difference or smoothness properties. In particular, in 1936, J. Favard calculated the exact values of the best uniform approximations by trigonometric polynomials of order no higher than n – 1 on classes of differentiable 2π-periodic functions, whose r-th (r – natural) derivatives are in a unit sphere of the space of essentially bounded functions. The problem of obtaining exact values of the best approximations in uniform and integral metrics for various functional compacts was in sight of many prominent mathematicians of the XX century. General issues related to the study of the best approximation functional: the existence of a polynomial of the best approximation, its characteristic properties, are destribed in detail in many monographs, in particular, for example, in the book by M. P. Korneichuk [1]. In the 80s and 90s of the XX century, O. I. Stepanets (see, [2, section III]) developed a new approach to the classification of periodic functions, which allowed for a fairly fine classification of extremely wide sets of periodic functions. At the same time, the results obtained for these classes are, on the one hand, general, and on the other hand, they give a number of new, hitherto unknown results that were impossible to obtain on previously known classes. Following the approaches to the requirements of function classification, we can consider a linear combination of function classes of a more complex nature. And then the problem of finding the exact values of the upper bounds of the best joint approximations will be reduced to the problem of the best approximation of this composite class corresponding to convolutions with the composite kernel.

3D Bézier Guarding: Boundary-Conforming Curved Tetrahedral Meshing

We present a method for the generation of higher-order tetrahedral meshes. In contrast to previous methods, the curved tetrahedral elements are guaranteed to be free of degeneracies and inversions while conforming exactly to prescribed piecewise polynomial surfaces, such as domain boundaries or material interfaces. Arbitrary polynomial order is supported. Algorithmically, the polynomial input surfaces are first covered by a single layer of carefully constructed curved elements using a recursive refinement procedure that provably avoids degeneracies and inversions. These tetrahedral elements are designed such that the remaining space is bounded piecewise linearly. In this way, our method effectively reduces the curved meshing problem to the classical problem of linear mesh generation (for the remaining space).

Systematic analysis of non-intrusive polynomial chaos expansion to determine rotary blood pump performance over the entire operating range

This study applies non-intrusive polynomial chaos expansion (NIPCE) surrogate modeling to analyze the performance of a rotary blood pump (RBP) across its operating range. We systematically investigate key parameters, including polynomial order, training data points, and data smoothness, while comparing them to test data. Using a polynomial order of 4 and a minimum of 20 training points, we successfully train a NIPCE model that accurately predicts pressure head and axial force within the specified operating point range ([0–5000] rpm and [0–7] l/min). We also assess the NIPCE model's ability to predict two-dimensional velocity data across the given range and find good overall agreement (mean absolute error = 0.1 m/s) with a test simulation under the same operating condition. Our approach extends current NIPCE modeling of RBPs by considering the entire operating range and providing validation guidelines. While acknowledging computational benefits, we emphasize the challenge of modeling discontinuous data and its relevance to clinically realistic operating points. We offer open access to our raw data and Python code, promoting reproducibility and accessibility within the scientific community. In conclusion, this study advances comprehensive NIPCE modeling of RBP performance and underlines how critically NIPCE parameters and rigorous validation affect results.

On-orbit geometric calibration and preliminary accuracy verification of GaoFen-14 (GF-14) optical two linear-array stereo camera

ABSTRACT The GaoFen-14 (GF-14) satellite is China’s most recent high-resolution earth observation satellite system. It is equipped with a two linear-array stereo camera and is intend for topographic mapping at 1:10,000-scale without ground control points (GCPs). The technical parameters of payloads will change once the satellite enters orbit. As a result, strict on-orbit geometric calibration is necessary. This study performs the on-orbit geometric calibration of the GF-14 stereo camera using ground calibration data. The exterior orientation errors are corrected by a comprehensive bias matrix, internal orientation errors are described by a fifth order polynomial. The results of Yinchuan calibration field show that the planar accuracy is better than 286 m (RMS) before calibration, and improved to 2.11 m (front camera) and 1.51 m (back camera) by external orientation, and further improved to 1.46 m (front camera) and 0.99 m (back camera) after internal calibration. The front intersection accuracy reaches 0.67 m in plane (RMS) and 1.10 m in elevation (RMS), respectively. Checking by multiple global check points (CKPs), the planar and elevation RMS reaches 2.38 m (5.09 m in CE90) and 2.08 m (3.43 m in LE90), respectively.

Predicted effects of image acquisition and analysis conditions on DTMRI tractography-based muscle architecture estimates.

To quantify the effects of the intrinsic signal pattern, image acquisition conditions, and data analysis conditions on diffusion-tensor MRI (DTMRI) tractography-based muscle architecture estimates using a sampling-reconstruction assessment framework. Numerical models of muscles were constructed with realistic architectural properties. DTMRI signals were computed at signal-to-noise ratio (SNR) of 24-96 and common voxel sizes. Fiber tracking was performed, and the results were compared with the known architectural properties. SNR exerted the most significant impact on the outcome. The outcome variables approached asymptotes at SNR ≈ 54. Large in-plane voxel dimensions reduced the similarity between reconstructed fibers and the known architectural properties. Higher order polynomials helped reconstruct fibers with more complicated geometry but overfit noise for less complex geometries. The intrinsic fiber curvature also affected the robustness of polynomial smoothing to SNR. Other conditions, such as the fiber dimensionality, voxel aspect ratio, and slice thickness, did not affect the outcomes. SNR ≥ 54 is recommended for accurate muscle architecture characterization using DTMRI. Averaged across all simulated conditions, the greatest percent errors under SNR = 54 were -5.6% and -4.0% for the pennation angle and fiber-tract length estimates, respectively. For fiber tracts with intermediate intrinsic curvature, the greatest percent error for the curvature estimate was 9.8% for SNR = 54. Smaller in-plane voxel size (≤1.5 mm) is preferred to minimize the estimation error in architectural properties. If necessary, slice thickness may be adjusted within typical ranges to achieve sufficient SNR when slices are aligned near the fiber direction. Third-order polynomial fitting is appropriate for smoothing fiber tracts.

Harmonic Oscillator with a Step and/or a Ramp

We discuss the one-dimensional Schrödinger equation for a harmonic oscillator with a finite step at the origin and/or an external field described by a ramp function. The first half of this paper is a partial review of our recent work. The latter half is devoted to an extension of the problem, i.e., imposing an external linear field on the negative half line. The solvability of the problem via the Hermite polynomials is discussed. We demonstrate that a harmonic oscillator with a step and a ramp can have one eigenstate whose wavefunction is expressed in terms of Hermite polynomials of different orders. Explicit examples are also provided at appropriate places in the text.

Hermite-scan imaging for differentiating glioblastoma from normal brain: Simulations and ex vivo studies for applications in intra-operative tumor identificationa).

Hermite-scan (H-scan) imaging is a tissue characterization technique based on the analysis of raw ultrasound radio frequency (RF) echoes. It matches the RF echoes to Gaussian-weighted Hermite polynomials of various orders to extract information related to scatterer diameter. It provides a color map of large and small scatterers in the red and blue H-scan image channels, respectively. H-scan has been previously reported for characterizing breast, pancreatic, and thyroid tumors. The present work evaluated H-scan imaging to differentiate glioblastoma tumors from normal brain tissue ex vivo. First, we conducted 2-D numerical simulations using the k-wave toolbox to assess the performance of parameters derived from H-scan images of acoustic scatterers (15-150 μm diameters) and concentrations (0.2%-1% w/v). We found that the parameter intensity-weighted percentage of red (IWPR) was sensitive to changes in scatterer diameters independent of concentration. Next, we assessed the feasibility of using the IWPR parameter for differentiating glioblastoma and normal brain tissues (n = 11 samples per group). The IWPR parameter estimates for normal tissue (44.1% ± 1.4%) were significantly different (p < 0.0001) from those for glioblastoma (36.2% ± 0.65%). These findings advance the development of H-scan imaging for potential use in differentiating glioblastoma tumors from normal brain tissue during resection surgery.

Hierarchical Divergence-Conforming Vector Bases for Pyramid Cells

Divergence-conforming hierarchical vector bases for the pyramid consist of face- and volume-based functions obtained by a simple procedure that uses a new paradigm recently introduced by this author to produce pyramid bases. In order to define the bases’ order, the procedure starts by mapping the pyramids into a cube of a new Cartesian space, which we call the grandparent space, where the basis functions and their divergences take on polynomial form. Then we get the face-based functions of zero polynomial order and the volume-based functions of the first order. Functions of arbitrarily high order are obtained by multiplying the vector functions of the lowest order by independent scalar polynomials of higher order. Our face-based functions conform to those of other differently shaped elements to allow the use of hybrid meshes, while the multiplicative construction technique generates right away the volume-based basis functions. The completeness of the bases is demonstrated and all the basis functions we obtain are suitably normalized; their expression involves orthogonal polynomials which are easy to implement and alleviate the loss of linear independence.

Curved Pseudo Elastic Impedance based on Elastic Attribute Rotation Scheme for Reservoir Petrophysical Property Prediction

Seismic based reservoir characterization commonly attempts to estimate petrophysical properties of a reservoir from a combination of elastic attributes resulted from seismic inversion. Two most wanted main reservoir properties to be estimated are porosity variation and hydrocarbon saturation across the reservoir. A combination of acoustic impedance (AI) and the ratio between P- and S-wave velocities (Vp/Vs ratio) based on rock physics models had been used in the earlier study to form Curved Pseudo Elastic Impedance (CPEI) attribute. However, the approach involved a fourth order polynomial in the logarithmic domain of AI and Vp/Vs ratio. A much simpler approach of combining these elastic properties to form CPEI attributes which is based on attribute rotation scheme is proposed in this study. To illustrate the methodology, we applied to well and seismic data located in the Natuna Basin, Indonesia. Results show that better attributes correlation to porosity and hydrocarbon saturation variation is achieved.

Development of low cost pyranometer IoT based solar irradiance measurement station

This paper presents the design of self-sustainable IoT pyranometer station using low cost BPW34 photodiode. The sustainable power of the station is achieved by a 20W solar panel charges a 7.2Ah lead acid battery through a charge controller. This station uses low cost BPW34 sensors which are connected to a 16-bit ADC module to convert the analog voltage into digital data. This module is connected to an ESP32 microcontroller that connects to Wi-Fi and sends sensor readings to ThingSpeak IoT platform to allow monitoring. This study collected over 36000 data points throughout February 2023 with an average daily data loss of 2.47%. Polynomial regression method was used to determine a best fit curve equation between the BPW34 sensor and the reference sensor. The result shows that the proposed BPW34 sensor achieve the highest error of ±80W/m2 with approximate 3% of daily data. The summary, a second order polynomial equation is found best fit for the proposed low cost BPW34 sensor.

A New Nonlinear One-Dimensional Shape Function applied to Simulate a Steady State Heat Transfer Problem

Generally linear shape functions are considered for two noded elements as coefficients of higher order polynomial cannot be determined from two nodes. In the present work, attempts have been made to fit a non-linear trigonometric shape function for one dimensional two noded elements. As in case of nonlinearity, the dependent variable varies nonlinearly with the independent one, consideration of linear shape function may lead to erroneous results. Here, a one dimensional heat transfer problem is solved analytically, with the help of linear shape functions and with the help of newly developed nonlinear shape functions. Analysis shows that results obtained by considering nonlinear shape functions have quite good match with that by linear shape functions.

An explainable AI model for power plant NOx emission control

In recent years, developing Artificial Intelligence (AI) models for complex system has become a popular research area. There have been several successful AI models for predicting the Selective Non-Catalytic Reduction (SNCR) system in power plants and large boilers. However, all these models are in essence black box models and lack of explainability, which are not able to give new knowledge. In this study, a novel explainable AI (XAI) model that combines the polynomial kernel method with Sparse Identification of Nonlinear Dynamics (SINDy) model is proposed to find the governing equation of SNCR system based on 5-year operation data from a power plant. This proposed model identifies the system's governing equation in a simple polynomial format with polynomial order of 1 and only 1 independent variable among original 68 input variables. In addition, the explainable AI model achieves a considerable accuracy with less than 21 % deviation from base-line models of partial least squares model and artificial neural network model.

Learning Spline Models with the EM Algorithm for Shape Recognition

This paper demonstrates how cubic Spline (B-Spline) models can be used to recognize 2-dimension nonrigid handwritten isolated characters. Each handwritten character is represented by a set of nonoverlapping uniformly distributed landmarks. The Spline models are constructed by utilizing cubic order of polynomial to model the shapes under study. The approach is a two-stage process. The first stage is learning, we construct a mixture of spline class parameters to capture the variations in spline coefficients using the apparatus Expectation Maximization algorithm. The second stage is recognition, here we use the Fréchet distance to compute the variations between the spline models and test spline shape for recognition. We test the approach on a set of handwritten Arabic letters

Convolution finite element based digital image correlation for displacement and strain measurements

This work presents a novel global digital image correlation (DIC) method, based on a newly developed convolution finite element (C-FE) approximation. The convolution approximation can rely on the mesh of linear finite elements and enables arbitrarily high order approximations without adding more degrees of freedom. Therefore, the C-FE based DIC can be more accurate than the usual FE based DIC by providing highly smooth and accurate displacement and strain results with the same element size. The detailed formulation and implementation of the method have been discussed in this work. The controlling parameters in the method include polynomial order, patch size, and dilation. A general choice of the parameters and their potential adaptivity have been discussed. The proposed DIC method has been tested by several representative examples, including the DIC challenge 2.0 benchmark problems, with comparison to the usual FE based DIC. C-FE outperformed FE in all the DIC results for the tested examples. This work demonstrates the potential of C-FE and opens a new avenue to enable highly smooth, accurate, and robust DIC analysis for full-field displacement and strain measurements.

Optimization of extraction of phenolic compounds and antioxidants from passion fruit and rambutan seeds using response surface methodology

Passion fruit seed (PS) and rambutan seed (RS) contain sizable amounts of phenolic compounds (PCs) with bioactive compounds including antioxidant activity (AOA). The aim of this study was to optimize the extraction of PCs from PS and RS using response surface methodology (RSM). Preliminary results indicated that the solid to liquid (S-L) ratio of 1:40 (w/v) showed the highest yield (EY) for PS and RS, with values of 11.59 ± 0.08 and 14.55 ± 1.67 g/100 g samples, respectively. The S-L ratio of 1:30 (w/v) was selected as an economic logistic. The predicted response compared to experimental data for EY, EPC, DPPH, ABTS and FRAP of the extracts were 12.41 % (12.68 %), 715 (707), 85 (81), 539 (522) and 107 (160) mg GAE/100 g dry weight for RS and 13.68 % (14.41 %), 3141 (3235), 1327 (1434), 5793 (5564) and 1567 (1509) mg GAE/100 g dry weight for PS, respectively. The experimental data, including EY, EPC, and AOA, were fitted using a 2nd order polynomial equation. The statistical analysis revealed that ethanol concentration and extraction temperature significantly influenced PS (p < 0.001), whereas ethanol concentration primarily affected RS on the EY and EPC. The multiple regression analysis suggested that AOA of PS was significantly affected by linear, quadratic, and interaction of ethanol concentration, extraction time and temperature. Nonetheless, the AOA of RS was only affected by ethanol concentration. The best possible combinations for PS and RS were ethanol concentration 67 and 54 %, extraction temperature 94 and 63 °C, and extraction time 186 and 221 min, respectively, to obtain the maximum EY, extractable phenolic compounds (EPC) and AOA. The composite desirability values for the optimal conditions of PS and RS were 0.74 and 0.68, respectively. Optimal aqueous phenolic extracts from PS and RS could serve as sustainable food-grade ingredients and nutraceutical products.

Analytical solution of the fractional neutron point kinetic equations using the Mittag-Leffler function

A new analytical solution of the fractional point kinetic equations is developed in the present work, which considers one group of delayed neutrons as well as a constant reactivity. One of the main novelties and theoretical contributions of the developed solution is that it does not require approximating the Inverse Laplace transform by numerical means, as other solutions do, and instead it is obtained analytically using the Green and the Mittag-Leffler functions. A set of algorithms and MATLAB codes are developed with the purpose to implement and compute the developed solution in an adequate way, which represents the major computational contribution. Numerical experiments show that the developed solution is very accurate solving short-time dynamics phenomena, being able to reproduce, in a precise way, data that is reported in literature. The development of the proposed analytic solution represents an important contribution to validate numerical methods that are currently used to solve the neutron fractional point kinetic equations. Program SummaryProgram Title: Mittag-Leffler FNPKECPC Library link to program files:https://doi.org/10.17632/v4kfhzw7sf.1Developer's repository link:https://github.com/Cruz-Lopez-Carlos-Antonio/Mittag-Leffler-FNPKE/tree/mainLicensing provisions: Creative Commons by 4.0Programming language: MATLAB (2021a) / Python 3Nature of problem:Fractional Neutron Point Kinetic Equations describe the neutron population in a nuclear reactor, considering the changes on transport and thermohydraulic properties of the system. For such task, a mass balance approach based on fractional and integral derivatives is used, which improves and generalizes the integer case that uses the classical definitions of these mathematical operators. From a physical point of view, this new approach takes into account memory and non-local effects that are omitted in the standard neutron diffusion theory and which are relevant in scenarios where the current neutron density undergoes strongly variations on time. From a mathematical point of view, using fractional derivatives leads to a new and more general type of differential equations of arbitrary order that requires different methods of solution and approximation, and which must include the integer approach as particular case. In such context, the development of analytical solutions is a fundamental problem because it provides a framework to carry out verifications of numerical methods, as well as an insight to perform theoretical comparisons with the classical model. The developed Mittag-Leffler FNPK codes analytically solve the Fractional Neutron Point Kinetic Equations defined in terms of the Caputo's derivatives and subject to classical initial conditions.Solution method:The developed codes are written in MATLAB and Python programming languages, and solve the mentioned fractional differential equation system using a power series expansion as well as the Laplace transform. Such solution considers a single group of precursors of delayed neutrons and it is expressed in terms of the 2-parameter Mittag-Leffler function, being accurately computed considering a small number of terms of the infinite sum. Unlike other approaches, in the present case it is not necessary to use a numerical procedure to find the inverse Laplace transform, because it is analytically found using the Green function. Even when the developed solution is found for a constant reactivity case, the Mittag-Leffler FNPKE codes can extended it to more general scenarios where the reactivity is a function of time or where it depends on the neutron density itself. In these cases, it is necessary to apply a subinterval methodology where the time is divided in small steps, assuming a constant reactivity in each of them. The smaller of the intervals, the more accurate the approximation. With the purpose to overcome the numerical issues related to the computation of the 2-parameter Mittag-Leffler function, the developed codes integrate the most advanced algorithm reported in literature, which is useful also to estimate the high-order derivatives of such function.Additional comments including restrictions and unusual features:The developed codes do not face issues related to arithmetical precision digits, as other solutions do, because it is not expressed in terms of high order polynomials and instead it is written as a linear combination of the Green function. Additionally, several steps can be precomputed without regard the neutron parameters and initial conditions, which represents an important advantage in terms of the computational time.

Equal-order finite element method for the Stokes equations with variable viscosity

In this brief note, we extend the initial investigation proposed in John et al. (2016) to study the Stokes equations with variable viscosity considering a stabilized finite element scheme based on equal order polynomials for approximating the velocity and the pressure. We establish optimal a priori error estimates for this new stabilized scheme, which are further confirmed by numerical tests. Moreover, these tests show a weak viscosity dependence in the error bounds for different orders of magnitude between νmax and νmin.

Bivariate Chromatic Polynomials of Mixed Graphs

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.

Buckling analysis of thin plates with circular cut-outs for sustainable ventilated food packaging design

Thin plates with circular cut-outs are widely used in engineering structures such as bridges, construction, and ventilated corrugated paperboard boxes in food packaging. This study investigates the critical buckling loads of thin elastic plates comprised of a single circular hole at different locations and with different sizes. To determine the critical loads, experimental tests and finite element method simulations are conducted. The vertical location and the diameter of the hole impact the buckling results significantly. The results indicate a significant correlation between the experimental and finite element data. The study develops second and third order polynomial formulas that predict buckling loads. In this study, corrugated paperboard plates were used as samples to gain insights into ventilated food packaging designs, aiming to enhance their buckling strength while reducing packaging material consumption for packaging sustainability purposes. The solution proposed in this study can be applied to predict the buckling of other elastic thin plate structures with a circular hole, relevant to industries such as construction and aerospace.