Polynomials Of Degree Research Articles

Multiple-Vehicle Cooperative Autonomous Parking Trajectory Planning Using Connected Fifth Degree Polynomials and Genetic Algorithm Optimization

Multiple-Vehicle Cooperative Autonomous Parking Trajectory Planning Using Connected Fifth Degree Polynomials and Genetic Algorithm Optimization

Mass-moisture exchange characteristics of carrots

Some thermodynamic characteristics of carrots were calculated and analyzed based on known experimental data - the binding energy of water with the material and the specific isothermal moisture capacity. An innovative nomogram is presented, which can be successfully used for easy and quick determination of the binding energy of water. An empirical equation (third degree polynomial) for the specific isothermal moisture capacity is derived. All results in the paper will be used to design dryers and drying processes for this product.

Development of mathematical optimisation models for predicting the structural properties of rice husk ash (RHA) concrete using Osadebe second degree polynomials

Development of mathematical optimisation models for predicting the structural properties of rice husk ash (RHA) concrete using Osadebe second degree polynomials

Lévy Adaptive B-spline Regression via Overcomplete Systems

The estimation of functions with varying degrees of smoothness is a challenging problem in the nonparametric function estimation. In this paper, we propose the LABS (L\'{e}vy Adaptive B-Spline regression) model, an extension of the LARK models, for the estimation of functions with varying degrees of smoothness. LABS model is a LARK with B-spline bases as generating kernels. The B-spline basis consists of piecewise k degree polynomials with k-1 continuous derivatives and can express systematically functions with varying degrees of smoothness. By changing the orders of the B-spline basis, LABS can systematically adapt the smoothness of functions, i.e., jump discontinuities, sharp peaks, etc. Results of simulation studies and real data examples support that this model catches not only smooth areas but also jumps and sharp peaks of functions. The proposed model also has the best performance in almost all examples. Finally, we provide theoretical results that the mean function for the LABS model belongs to the certain Besov spaces based on the orders of the B-spline basis and that the prior of the model has the full support on the Besov spaces.

Development of mathematical optimisation models for predicting the structural properties of rice husk ash (RHA) concrete using Osadebe second degree polynomials

Development of mathematical optimisation models for predicting the structural properties of rice husk ash (RHA) concrete using Osadebe second degree polynomials

Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives

<abstract><p>In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $ \alpha_i\in(0, 1) $, $ i = 1, 2, \cdots, n $). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $ O(1) $ storage and $ O(N_T) $ computational complexity, where $ N_T $ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $ O\left(\left(\Delta t\right)^{2}+N^{-m}\right) $, where $ \Delta t $, $ N $, and $ m $ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.</p></abstract>

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system.

Efficient Approximations for Matrix-Based Rényi's Entropy on Sequential Data.

The matrix-based Rényi's entropy (MBRE) has recently been introduced as a substitute for the original Rényi's entropy that could be directly obtained from data samples, avoiding the expensive intermediate step of density estimation. Despite its remarkable success in a broad of information-related tasks, the computational cost of MBRE, however, becomes a bottleneck for large-scale applications. The challenge, when facing sequential data, is further amplified due to the requirement of large-scale eigenvalue decomposition on multiple dense kernel matrices constructed by sliding windows in the region of interest, resulting in O(mn3) overall time complexity, where m and n denote the number and the size of windows, respectively. To overcome this issue, we adopt the static MBRE estimator together with a variance reduction criterion to develop randomized approximations for the target entropy, leading to high accuracy with substantially lower query complexity by utilizing the historical estimation results. Specifically, assuming that the changes of adjacent sliding windows are bounded by β << 1 , which is a trivial case in domains, e.g., time-series analysis, we lower the complexity by a factor of √{β} . Polynomial approximation techniques are further adopted to support arbitrary α orders. In general, our algorithms achieve O(mn2√{β}st) total computational complexity, where s, t << n denote the number of vector queries and the polynomial degrees, respectively. Theoretical upper and lower bounds are established in terms of the convergence rate for both s and t , and large-scale experiments on both simulation and real-world data are conducted to validate the effectiveness of our algorithms. The results show that our methods achieve promising speedup with only a trivial loss in performance.

On the Zeros of Apolar Polynomials

The classical notion of apolarity is defined for two complex polynomials of same degree. The main property of two apolar polynomials, f(z) and g(z) was given by Grace’s theorem which states that “every circular domain containing all the zeros of f(z) contains at least one zero of g(z) and vice-versa”. A. Aziz [1] dropped the condition that f(z) and g(z) are of the same degree in case the circular domain is a disk. In this paper, we extend the result of A. Aziz to every kind of circular domain and hence an extension of Grace’s theorem for two arbitrary polynomials is proved. This also allows us to generalise the results of Walsh, Szego, Takagi, Aziz and several other results about apolar polynomials.

Substantiation of the results of uneven leveling of heaps on the dryer conveyor

The article presents the results of finding the optimal combination of the selected factors having the minimum value of the indicator of uneven leveling of the pile on the dryer conveyor, an experiment was conducted, including 8 experiments, constituting a semi-replica of a complete factorial experiment of the type 2859+3, and a program of steep ascent along the response surface was performed. The experiment was carried out three times in a randomized order. In addition, an experiment planning matrix was built using a well-known technique. Based on the results of the experiment, regression coefficients were calculated and their statistical evaluation was carried out. To verify the adequacy of the presented experimental results, the calculated value of the Fisher criterion was determined by a polynomial of the first degree and compared with tabular values.

FACTORIZATION OF DEGREE k-ALMOST SIMPLE POLYNOMIALS

In this work, we introduce the notion of k-almost prime polynomials formed by the product over a Galois field of an arbitrary characteristic p of exactly k irreducible polynomials. The study aims to develop an effective algorithm for factorizing the degree of such composite polynomials providing a minimum of computational complexity. The proposed algorithm is based on the solution of a system of two equations functionally related to the a priori unknown degree of the components of the compound polynomial. One of these equations reflects that the degree of the composite polynomial equals the sum of degrees of elements of this polynomial. The second equation is based on the so-called cycle period of the compound polynomial, which coincides with the least common multiple of the unknown degrees of the components of the compound polynomial. It is found that the expansion of the cycle period contains all the degrees of the factorizable compound polynomial. The computational volume reduction is achieved by switching from the linear scale of the multiplicative group of subtractions generated by the cycle period of the compound polynomial to the logarithmic scale.

Geometric properties of the lattice of polynomials with integer coefficients

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let \(r\), \(n\) be positive integers with \(n \ge 6r\). Let \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) be the space of polynomials of degree at most \(n\) on \([0,1]\) with integer coefficients such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\) and let \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) from \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) in \(L_2(0,1)\). We give rather precise quantitative estimations for successive minima of \(\boldsymbol{P}_n^\mathbb{Z}\) in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\).

On permutation polynomials modulo 7n

A polynomial that permutes the elements of a finite ring R is known as a permutation polynomial. The conditions for the coefficients of a polynomial of degree d to be a permutation polynomial modulo pn, are known for p = 2, 3, 5. In this paper, we obtain the necessary and sufficient conditions on the coefficients of a polynomial modulo 7n to be a permutation polynomial.

Numerical calculation the fracture toughness of square section beam under three-point bending test

A universal equation is proposed to determine specific fracture energy based on numerical calculations by finite element methods in the ANSYS system referring to the three-point bending of a square beam arranged edgewise on supports. Numerical dependences of the specimen compliance on the crack length were represented as the 5th and 6th degree polynomials on the relative value of Δl/h, where Δl is the crack length and h is the half length of the square diagonal. To calculate the specific fracture energy, we used the derivative of specimen compliance by the crack length. The analysis has shown that finite value of specific fracture energy at Δl→0 is obtained when the decomposition term at Δl/h is equal to 0. A satisfactory agreement between the specific fracture energy values for the 5th and 6th degree decompositions is observed only at Δl/h ≈ 1. As the values of Δl decrease, the difference becomes increasingly significant. The averaging of coefficients at corresponding Δl/h degrees of two considered decompositions allows us to find a power function that weakly depends on the Δl/h ratio and leads to a consistent dependence of the specific fracture energy on the crack length. Due to the performed calculations, a universal equation is obtained to determine the specific fracture energy according to the test data of square beams by three-point bending in a wide range of geometric dimensions.

Design and dynamical behavior of a fourth order family of iterative methods for solving nonlinear equations

&lt;abstract&gt;&lt;p&gt;In this paper, a new fourth-order family of iterative schemes for solving nonlinear equations has been proposed. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. For studying the stability of this class, the rational function resulting from applying the iterative expression to a low degree polynomial was analyzed. The dynamics of this rational function allowed us to better understand the performance of the iterative methods of the class. In addition, the critical points have been calculated and the parameter spaces and dynamical planes have been presented, in order to determine the regions with stable and unstable behavior. Finally, some parameter values within and outside the stability region were chosen. The performance of these methods in the numerical section have confirmed not only the theoretical order of convergence, but also their stability. Therefore, the robustness and wideness of the attraction basins have been deduced from these numerical tests, as well as comparisons with other existing methods of the same order of convergence.&lt;/p&gt;&lt;/abstract&gt;

Chebyshev polynomial approximation in CNN for zero-knowledge encrypted data analysis

Integrating Deep Learning (DL) techniques in Convolutional Neural Networks (CNNs) with encrypted data analysis is an emerging field for enhancing data privacy and security. A significant challenge in this domain is the incompatibility of standard non-linear Activation Functions (AF) like Rectified Linear Unit (ReLU) and Hyperbolic Tangent (tanh) with Zero-Knowledge (ZK) encrypted data, which impacts computational efficiency and data privacy. Addressing this, our paper introduces the novel application of Chebyshev Polynomial Approximation (CPA) to adapt these AF to process encrypted data effectively. Utilizing the MNIST dataset, this paper conducted experiments with LeNet and various configurations of AlexNet, extending the range of the ReLU and tanH functions to optimize CPA. Our results reveal an optimal polynomial degree (α), with α = 10 for ReLU and between α = 10 and α = 15 for tanH, beyond which the benefits in accuracy plateau. This finding is crucial for ensuring the accuracy and efficiency of CNNs in processing encrypted data. This recommended study demonstrates that while the accuracy slightly decreases for plaintext data and more significantly for ciphertext data, the overall effectiveness of CPA in CNNs is maintained. This advancement enables CNNs to process encrypted data while preserving privacy and marks a significant step in developing privacy-preserving Machine Learning (ML) and encrypted data analysis.

Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

&lt;abstract&gt;&lt;p&gt;We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.&lt;/p&gt;&lt;/abstract&gt;

Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes

&lt;abstract&gt;&lt;p&gt;Two numerical schemes are proposed and investigated for the Yang–Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang–Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $ L^2 $-norm of the potential and electrical fields in $ \mathcal O(h^{k+1}) $ (provided that the time step is of that order), where $ k $ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.&lt;/p&gt;&lt;/abstract&gt;

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

PREDICTION OF SKINCARE SALES TURNOVER USING THE SUPPORT VECTOR METHOD AT THE WIDYA MSGLOW SIDOARJO COMPANY

Every entrepreneur will certainly follow technological developments in the business world. MsGlow is one of theskincare businesses. The skincare business is one of the businesses that must compete with rapid and complexchanges, and this very competitive makes business people have to think of strategies for business continuity in orderto compete and also survive. One way that can be done is to utilize existing sales data. The importance of fast andprecise operational data processing, information system facilities can be an alternative to solving problems in dataprocessing, minimizing errors and accelerating the data processing process. As the number of sales transactionsincreases, there will be a buildup of data that has not been processed optimally. With the above problems, aforecasting system was created that can forecast skincare sales turnover using the Support Vector Machine (SVM)method. In this study, turnover in several areas will be forecasted. The kernel function variations used in SupportVector Machine (SVM) are RBF, Linear, and, Polynomial Degree 2. The results obtained from this research trialshow that the overall forecasting model is good. The accuracy of the three areas obtained with the RBF kernel has arelatively good MAPE. In the accuracy test to predict skincare sales turnover, the three areas got a fairly goodaccuracy value of 94.46%. In the Sidoarjo area, it is predicted that there will be a lot of decrease in turnover in 2023-2024.