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Journal articles on the topic 'High-degree polynomials'

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Published: 30 May 2022

Last updated: 31 July 2025

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1

Skrzipek, Michael-Ralf. "Evaluation of High Degree Polynomials." Results in Mathematics 53, no. 3-4 (2009): 427–34. http://dx.doi.org/10.1007/s00025-008-0354-9.

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2

Sitton, G. A., C. S. Burrus, J. W. Fox, and S. Treitel. "Factoring very-high-degree polynomials." IEEE Signal Processing Magazine 20, no. 6 (2003): 27–42. http://dx.doi.org/10.1109/msp.2003.1253552.

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3

Dégot, Jérôme. "Sendov conjecture for high degree polynomials." Proceedings of the American Mathematical Society 142, no. 4 (2014): 1337–49. http://dx.doi.org/10.1090/s0002-9939-2014-11888-0.

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4

Tao, Terence. "Sendov’s conjecture for sufficiently-high-degree polynomials." Acta Mathematica 229, no. 2 (2022): 347–92. http://dx.doi.org/10.4310/acta.2022.v229.n2.a3.

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Beletsky, Anatoly. "An Effective Algorithm for the Synthesis of Irreducible Polynomials over a Galois Fields of Arbitrary Characteristics." WSEAS TRANSACTIONS ON MATHEMATICS 20 (September 30, 2021): 508–19. http://dx.doi.org/10.37394/23206.2021.20.54.

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The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary charact

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Shi, Jun, Jieqing Tan, and Li Zhang. "A Simple Method for Constructing Symmetric Subdivision Schemes with High-Degree Polynomial Reproduction." Symmetry 15, no. 12 (2023): 2202. http://dx.doi.org/10.3390/sym15122202.

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In this paper, we present an efficient method for constructing symmetric subdivision schemes reproducing high-degree polynomials, without solving a system of linear equations. Original symmetric subdivision and its deduced subdivisions have similar increasing characteristics to the family of pseudo-splines from the polynomial reproduction point of view. Several examples are given to illustrate the efficiency of the method.

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Chiang, D. "Factorization of high degree polynomials-a numerical solution." IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 43, no. 9 (1996): 792–94. http://dx.doi.org/10.1109/81.536750.

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Treitel, S., G. A. Sitton, J. W. Fox, and C. S. Burrus. "Factoring high-degree polynomials with applications to geophysics." Leading Edge 25, no. 10 (2006): 1216–19. http://dx.doi.org/10.1190/1.2360607.

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Schrade, Emanuel, Johannes Hanika, and Carsten Dachsbacher. "Sparse high-degree polynomials for wide-angle lenses." Computer Graphics Forum 35, no. 4 (2016): 89–97. http://dx.doi.org/10.1111/cgf.12952.

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Garcia, R. D. M., and C. E. Siewert. "On computing the Chandrasekhar polynomials in high order and high degree." Journal of Quantitative Spectroscopy and Radiative Transfer 43, no. 3 (1990): 201–5. http://dx.doi.org/10.1016/0022-4073(90)90052-8.

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Karakus, Dogan. "FINDING THE BEST-FIT POLYNOMIAL APPROXIMATION IN EVALUATING DRILL DATA: THE APPLICATION OF A GENERALIZED INVERSE MATRIX / POSZUKIWANIE NAJLEPSZEJ ZGODNOŚCI W PRZYBLIŻENIU WIELOMIANOWYM WYKORZYSTANEJ DO OCENY DANYCH Z ODWIERTÓW – ZASTOSOWANIE UOGÓLNIONEJ MACIERZY ODWROTNEJ." Archives of Mining Sciences 58, no. 4 (2013): 1289–300. http://dx.doi.org/10.2478/amsc-2013-0089.

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Abstract In mining, various estimation models are used to accurately assess the size and the grade distribution of an ore body. The estimation of the positional properties of unknown regions using random samples with known positional properties was first performed using polynomial approximations. Although the emergence of computer technologies and statistical evaluation of random variables after the 1950s rendered the polynomial approximations less important, theoretically the best surface passing through the random variables can be expressed as a polynomial approximation. In geoscience studie

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Klimach, Harald G., Jens Zudrop, and Sabine P. Roller. "Generation of high order geometry representations in Octree meshes." PeerJ Computer Science 1 (November 25, 2015): e35. http://dx.doi.org/10.7717/peerj-cs.35.

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We propose a robust method to convert triangulated surface data into polynomial volume data. Such polynomial representations are required for high-order partial differential solvers, as low-order surface representations would diminish the accuracy of their solution. Our proposed method deploys a first order spatial bisection algorithm to find robustly an approximation of given geometries. The resulting voxelization is then used to generate Legendre polynomials of arbitrary degree. By embedding the locally defined polynomials in cubical elements of a coarser mesh, this method can reliably appro

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Rhin, G., and J. M. Sac-Épée. "New Methods Providing High Degree Polynomials with Small Mahler Measure." Experimental Mathematics 12, no. 4 (2003): 457–61. http://dx.doi.org/10.1080/10586458.2003.10504513.

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14

Shiffman, Bernard, and Steve Zelditch. "Random polynomials of high degree and Levy concentration of measure." Asian Journal of Mathematics 7, no. 4 (2003): 627–46. http://dx.doi.org/10.4310/ajm.2003.v7.n4.a11.

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15

Lasserre, Jean B., and Tim Netzer. "SOS approximations of nonnegative polynomials via simple high degree perturbations." Mathematische Zeitschrift 256, no. 1 (2006): 99–112. http://dx.doi.org/10.1007/s00209-006-0061-8.

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16

Nie, Long, Shaowen Yao, and Jing Liu. "High-Precision Leveled Homomorphic Encryption for Rational Numbers." Mathematics 11, no. 2 (2023): 348. http://dx.doi.org/10.3390/math11020348.

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In most homomorphic encryption schemes based on RLWE, native plaintexts are represented as polynomials in a ring Zt[x]/xN+1, where t is a plaintext modulus and xN+1 is a cyclotomic polynomial with a degree power of two. An encoding scheme should be used to transform some natural data types (such as integers and rational numbers) into polynomials in the ring. After homomorphic computations on the polynomial aare finished, the decoding procedure is invoked to obtain the results. We employ the Hensel code for encoding rational numbers and construct a high-precision leveled homomorphic encryption

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17

Kabluchko, Zakhar. "Lee–Yang zeroes of the Curie–Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow." Annales Henri Lebesgue 8 (May 20, 2025): 1–34. https://doi.org/10.5802/ahl.227.

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The backward heat flow on the real line started from the initial condition z n results in the classical n th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large n limit. Similarly, the backward heat flow with the periodic initial condition (sinθ 2) n leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie–Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the n th unitary Hermite polynomial to the expected characterist

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Mirkov, Nikola, Nicola Fabiano, Dušan Nikezić, Vuk Stojiljković, and Milica Ilić. "Symmetries of Bernstein Polynomial Differentiation Matrices and Applications to Initial Value Problems." Symmetry 17, no. 1 (2024): 47. https://doi.org/10.3390/sym17010047.

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In this study, we discuss the symmetries underlying Bernstein polynomial differentiation matrices, as they are used in the collocation method approach to approximate solutions of initial and boundary value problems. The symmetries are brought into connection with those of the Chebyshev pseudospectral method (Chebyshev polynomial differentiation matrices). The treatment discussed here enables a faster and more accurate generation of differentiation matrices. The results are applied in numerical solutions of several initial value problems for the partial differential equation of convection–diffu

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Abd-Elhameed, W. M. "New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems." Scientific World Journal 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/456501.

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This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of

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Chang, Feng Cheng. "Polynomial GCD Derived through Monic Polynomial Subtractions." ISRN Applied Mathematics 2011 (June 13, 2011): 1–7. http://dx.doi.org/10.5402/2011/714102.

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A simple and efficient algorithm for finding the GCD of a pair of univariate polynomials is derived. The process requires only successive subtraction of two monic polynomials. Amazingly, this approach gives the desired GCD for a pair of test polynomials of very high degree, such as and its derivative.

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Turbal, Yurii, Ganna Shlikhta, Mariana Turbal, and Bogdan Turbal. "The polynomial forecasts improvement based on the algorithm of optimal polynomial degree selecting." Eastern-European Journal of Enterprise Technologies 5, no. 4 (125) (2023): 34–42. http://dx.doi.org/10.15587/1729-4061.2023.289292.

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The object of research in the paper is extrapolation problems based on interpolation polynomials. Polynomial-based prediction methods are well known. However, the problem is that such methods often give very large errors in practice. The permissible error of extrapolation even by one grid step is not ensured by the high accuracy of interpolation using polynomials. The paper proposes an algorithm that allows to significantly improve polynomial forecasts by optimizing the procedure for choosing the power of the polynomial, on the basis of which the forecast is built. The algorithm is based on th

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22

Wu, Ruifeng. "Bivariate multiquadric quasi-interpolation operators of Lidstone type." AIMS Mathematics 8, no. 9 (2023): 20914–32. http://dx.doi.org/10.3934/math.20231065.

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<abstract><p>In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate q

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23

Bondarenko, Valeriy Peter. "Formation of Stiffness Matrices of Plane Elements Based on High-Order Degree Polynomials." Journal of Architectural and Engineering Research 1, no. 1 (2021): 13–17. http://dx.doi.org/10.54338/27382656-2021.1-3.

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In this paper, the method for forming the stiffness matrix of a plane element in a plane stress-strain state using functions approximating the displacements of the points of the element in the form of high-order degree polynomials is considered.

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24

Ye, Jun, Xianlin Zhou, Zheng Xu, and Yong Ding. "Verifiable Outsourcing of High-Degree Polynomials and its Application in Keyword Search." Intelligent Automation and Soft Computing 24, no. 1 (2018): 41–46. http://dx.doi.org/10.1080/10798587.2016.1267239.

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25

Polverino, Olga, Giovanni Zini, and Ferdinando Zullo. "On certain linearized polynomials with high degree and kernel of small dimension." Journal of Pure and Applied Algebra 225, no. 2 (2021): 106491. http://dx.doi.org/10.1016/j.jpaa.2020.106491.

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26

Pepin, A., S. S. Beauchemin, S. Léger, and N. Beaudoin. "A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials." Canadian Mathematical Bulletin 63, no. 3 (2019): 655–69. http://dx.doi.org/10.4153/s0008439519000742.

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AbstractEffective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree

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27

Chitambar, Eric, Carl Miller, and Yaoyun Shi. "Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states." Quantum Information and Computation 11, no. 9&10 (2011): 813–19. http://dx.doi.org/10.26421/qic11.9-10-6.

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In this brief report, we consider the equivalence between two sets of $m+1$ bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree $m$ matrix polynomials are unitarily equivalent; i.e. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices $U$ and $V$.

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Sastre, Pedro Galán del, and Rodolfo Bermejo. "A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems." Communications in Computational Physics 9, no. 4 (2011): 1020–39. http://dx.doi.org/10.4208/cicp.041209.160910a.

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AbstractWe perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method. The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom, however, for the same level of accuracy both methods are about the same in terms of CPU time. For polynomials of degree larger than 2, Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU ti

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Ustimenko, Vasyl. "On the Families of Stable Multivariate Transformations of Large Order and Their Cryptographical Applications." Tatra Mountains Mathematical Publications 70, no. 1 (2017): 107–17. http://dx.doi.org/10.1515/tmmp-2017-0021.

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Abstract Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of with n increasing order can be used in the key exchange protocols and El Gamal multivariate cryptosystems related to them. We suggest to use high degree of noncommutativity of affine Cremona group and modify multivariate El Gamal algorithm via conjugations of two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. Recent results on the existence of families of stable transformations of pres

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30

Khan, Waseem A., and M. A. Pathan. "On generalized Lagrange–Hermite–Bernoulli and related polynomials." Acta et Commentationes Universitatis Tartuensis de Mathematica 23, no. 2 (2020): 211–24. http://dx.doi.org/10.12697/acutm.2019.23.19.

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We introduce a new class of generalized polynomials, ascribed to the family of Hermite, Lagrange, Bernoulli, Miller–Lee, and Laguerre polynomials and of their associated forms. These polynomials can be expressed in the form of generating functions, which allow a high degree of exibility for the formulation of the relevant theory. We develop a point of view based on generating relations, exploited in the past, to study some aspects of the theory of special functions. We propose a fairly general analysis allowing a transparent link between different forms of special polynomials.

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Dehesa, Jesús S., and Nahual Sobrino. "Complexity-like properties and parameter asymptotics of Lq -norms of Laguerre and Gegenbauer polynomials." Journal of Physics A: Mathematical and Theoretical 54, no. 49 (2021): 495001. http://dx.doi.org/10.1088/1751-8121/ac3320.

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Abstract The main monotonic statistical complexity-like measures of the Rakhmanov’s probability density associated to the hypergeometric orthogonal polynomials (HOPs) in a real continuous variable, each of them quantifying two configurational facets of spreading, are examined in this work beyond the Cramér–Rao one. The Fisher–Shannon and LMC López-Ruiz–Mancini–Calvet (LMC) complexity measures, which have two entropic components, are analytically expressed in terms of the degree and the orthogonality weight-function’s parameter(s) of the polynomials. The degree and parameter asymptotics of thes

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32

Zanardi, M. C., H. K. Kuga, N. R. da Silveira, and L. Morgan. "High Order and Degree Geopotential and Derivatives Computation Based on the Clenshaw Summation." Applied Mechanics and Materials 706 (December 2014): 191–205. http://dx.doi.org/10.4028/www.scientific.net/amm.706.191.

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An artificial satellite subject to the attraction of the Earth is disturbed due to nonsphericaldistribution and non-symmetrical Earth mass. This uneven distribution of mass is expressed by the socalledspherical harmonic coefficients of the Earth potential. For a faster computation, the accelerationderived from the potential is obtained by a series expansion in terms of these harmonics, the fullynormalized Legendre polynomials and their derivatives, and several recursions associated with thelongitude, geocentric latitude and altitude of the center of mass of the satellite. This paper analyzesth

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Rafiq, Naila, Mudassir Shams, Nazir Ahmad Mir, and Yaé Ulrich Gaba. "A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems." Mathematical Problems in Engineering 2021 (December 22, 2021): 1–22. http://dx.doi.org/10.1155/2021/9826693.

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A highly efficient two-step simultaneous iterative computer method is established here for solving polynomial equations. A suitable special type of correction helps us to achieve a very high computational efficiency as compared to the existing methods so far in the literature. Analysis of simultaneous scheme proves that its convergence order is 14. Residual graphs are also provided to demonstrate the efficiency and performance of the newly constructed simultaneous computer method in comparison with the methods given in the literature. In the end, some engineering problems and some higher degre

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LI, Huixian, Fulei WANG, Chun SHEN, Shiyuan LIU, and Liaojun PANG. "Small interval interpolation fitting bootstrapping method based on residue number system." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 42, no. 5 (2024): 969–78. https://doi.org/10.1051/jnwpu/20244250969.

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Aiming at the problem that the bootstrapping time of approximate homomorphic encryption scheme is too long, a small interval interpolation fitting method based on residue system is proposed. In this paper, the sinusoidal function by using interpolating and fitting method between the multiple cells to avoid the increase in bootstrapping time or decrease in calculation accuracy caused by the high degree of fitting polynomial is calculated. And the efficiency of modular multiplication and modular inversion in the calculation process is improved by combining the residual system. Lagrange interpola

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35

Filimonov, Valery, Olga Lovtskaya, and Alexander Zinoviev. "CALCULATION OF DISSOLVED POLLUTANTS MASS FLOW ACCORDING TO THE DATA OF THEIR CONCENTRATION SPATIAL DISTRIBUTION IN THE SITES OF SMALL PLAINT RIVERS." Eurasian Journal of Mathematical and Computer Applications 11, no. 4 (2023): 14–28. http://dx.doi.org/10.32523/2306-6172-2023-11-4-14-28.

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The paper presents the general solution of the stationary advection-diffusion equa- tion in the integral form for establishing the relationship between the concentration and the total mass flow rate of the pollutant on the plot of the catchment area. The resulting equation was used to solve the inverse problem of estimating the pollutant mass flow rate based on the arbitrary test functions. The unknown dependence of the total mass flow rate on the coordi- nate was represented as a polynomial. The coefficients of polynomials were determined by the least squares method. It was found that the inc

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36

Roelse, Peter. "Factoring high-degree polynomials over $\mathbf F_2$ with Niederreiter's algorithm on the IBM SP2." Mathematics of Computation 68, no. 226 (1999): 869–81. http://dx.doi.org/10.1090/s0025-5718-99-01008-x.

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37

Dunavant, D. A. "Efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube." International Journal for Numerical Methods in Engineering 23, no. 3 (1986): 397–407. http://dx.doi.org/10.1002/nme.1620230306.

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38

Cicuttin, Matteo, Alexandre Ern, and Simon Lemaire. "A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems." Computational Methods in Applied Mathematics 19, no. 4 (2019): 723–48. http://dx.doi.org/10.1515/cmam-2018-0013.

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AbstractWe devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first co

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ARTEMYEV, V. S., and A. S. MAKSIMOV. "IMPLEMENTATION OF SIMOYU METHOD FOR MODELING OF TRANSIENTS OF CONTROL OBJECT." Computational nanotechnology 11, no. 3 (2024): 43–51. http://dx.doi.org/10.33693/2313-223x-2024-11-3-43-51.

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In this paper transients in the control system are investigated on the basis of experimental data. The construction of the transfer function of the control object using Simoyu method is realized by means of Python language. The model of the control system of the object, selection of the regulator and its settings are implemented using SimInTech modeling environment. Within the framework of the conducted research, methodological approaches to the formation of transfer functions of control objects represented in the form of polynomial expressions of various degrees of complexity, starting with p

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Zulkefli, Nor Atirah Izzah, Yeak Su Hoe, and Nor Afifah Hanim Zulkefli. "Fuzzy Predator-Prey Systems by Extended Runge-Kutta Method with Polynomial Interpolation Technique." Semarak International Journal of Machine Learning 1, no. 1 (2024): 11–19. http://dx.doi.org/10.37934/sijml.1.1.1119.

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The growing significance of uncertainty quantification in mathematical modeling of physical phenomena is evident, with fuzzy sets emerging as an alternative approach. This paper focuses on exploring a numerical method for addressing fuzzy differential equations in two-dimensional problems like fuzzy predator-prey systems. The numerical solution employed the extended Runge-Kutta fourth-order method. To overcome challenges related to polynomial fuzzy differential equations, a combination of the polynomial interpolation technique and the extended Runge-Kutta fourth-order method was applied, mitig

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Zulkefli, Nor Atirah Izzah, Yeak Su Hoe, and Nor Afifah Hanim Zulkefli. "Fuzzy Predator-Prey Systems by Extended Runge-Kutta Method with Polynomial Interpolation Technique." Semarak International Journal of Machine Learning 1, no. 1 (2025): 11–19. https://doi.org/10.37934/sijml.1.1.1119a.

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The growing significance of uncertainty quantification in mathematical modeling of physical phenomena is evident, with fuzzy sets emerging as an alternative approach. This paper focuses on exploring a numerical method for addressing fuzzy differential equations in two-dimensional problems like fuzzy predator-prey systems. The numerical solution employed the extended Runge-Kutta fourth-order method. To overcome challenges related to polynomial fuzzy differential equations, a combination of the polynomial interpolation technique and the extended Runge-Kutta fourth-order method was applied, mitig

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Lu, Donghang, Albert Yu, Aniket Kate, and Hemanta Maji. "Polymath: Low-Latency MPC via Secure Polynomial Evaluations and Its Applications." Proceedings on Privacy Enhancing Technologies 2022, no. 1 (2021): 396–416. http://dx.doi.org/10.2478/popets-2022-0020.

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Abstract While the practicality of secure multi-party computation (MPC) has been extensively analyzed and improved over the past decade, we are hitting the limits of efficiency with the traditional approaches of representing the computed functionalities as generic arithmetic or Boolean circuits. This work follows the design principle of identifying and constructing fast and provably-secure MPC protocols to evaluate useful high-level algebraic abstractions; thus, improving the efficiency of all applications relying on them. We present Polymath, a constant-round secure computation protocol suite

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Karpinski, Marek, László Mérai, and Igor E. Shparlinski. "Identity testing and interpolation from high powers of polynomials of large degree over finite fields." Journal of Complexity 49 (December 2018): 74–84. http://dx.doi.org/10.1016/j.jco.2018.07.006.

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Bazarov, Laziz, Kamila Jurayeva, Ivan Bedritskiy, and Mirkomil Mirasadov. "APPROXIMATION OF MAGNETIZATION CURVES OF ELECTRICAL STEEL." Вестник КазАТК 121, no. 2 (2022): 274–81. http://dx.doi.org/10.52167/1609-1817-2022-121-2-274-281.

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It is made a comparison of functions approximating the main magnetization curve of electrical steels, selected according to two main criteria: obtaining the simplest expressions in an analytical study and achieving a minimum calculation error in quantitative calculations of devices with ferromagnetic components. The cases are considered when ferromagnetic elements in devices operate at alternating current and at high values of magnetic induction, that is, they are in a mode close to saturation, while the phenomena of hysteresis are usually neglected, and the main magnetization curve is used as

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Xie, Dawei, Haining Yang, Jing Qin, and Jixin Ma. "Privacy-Preserving and Publicly Verifiable Protocol for Outsourcing Polynomials Evaluation to a Malicious Cloud." International Journal of Digital Crime and Forensics 11, no. 4 (2019): 14–27. http://dx.doi.org/10.4018/ijdcf.2019100102.

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As cloud computing provides affordable and scalable computational resources, delegating heavy computing tasks to the cloud service providers is appealing to individuals and companies. Among different types of specific computations, the polynomial evaluation is an important one due to its wide usage in engineering and scientific fields. Cloud service providers may not be trusted, thus, the validity and the privacy of such computation should be guaranteed. In this article, the authors present a protocol for publicly verifiable delegations of high degree polynomials. Compared with the existing so

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Shoukralla, E. S., and M. A. Markos. "The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind." Asian-European Journal of Mathematics 13, no. 01 (2018): 2050030. http://dx.doi.org/10.1142/s1793557120500308.

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This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynom

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Bonaldi, Francesco, Daniele A. Di Pietro, Giuseppe Geymonat, and Françoise Krasucki. "A Hybrid High-Order method for Kirchhoff–Love plate bending problems." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (2018): 393–421. http://dx.doi.org/10.1051/m2an/2017065.

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We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are nov

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Ibrahim, Anas, Alexander Chefranov, Nagham Hamad, Yousef-Awwad Daraghmi, Ahmad Al-Khasawneh, and Joel J. P. C. Rodrigues. "NTRU-Like Random Congruential Public-Key Cryptosystem for Wireless Sensor Networks." Sensors 20, no. 16 (2020): 4632. http://dx.doi.org/10.3390/s20164632.

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Wireless sensor networks (WSNs) are the core of the Internet of Things and require cryptographic protection. Cryptographic methods for WSN should be fast and consume low power as these networks rely on battery-powered devices and microcontrollers. NTRU, the fastest and secure public key cryptosystem, uses high degree, N, polynomials and is susceptible to the lattice basis reduction attack (LBRA). Congruential public key cryptosystem (CPKC), proposed by the NTRU authors, works on integers modulo q and is easily attackable by LBRA since it uses small numbers for the sake of the correct decryptio

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Dwivedi, Yogendra Swaroop, Rishav Singh, Anuj K. Sharma, Ajay Kumar Sharma, and Carlos Marques. "Intervention of machine learning and explainable artificial intelligence based polynomial transformation approach for improved measurement of oil-water emulsion using FBG sensor." Physica Scripta 100, no. 7 (2025): 076005. https://doi.org/10.1088/1402-4896/addef5.

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Abstract This study focuses on the application of polynomial transformations in combination with machine learning (ML) and explainable artificial intelligence (XAI) techniques to analyze tilted fiber Bragg grating (TFBG) sensor data for oil–water emulsion stability. The dataset consisting of experimental TFBG spectra (wavelength range: 1250–1650 nm) included parameters such as revolutions per minute (RPM) of the rotator, surfactant concentration (Cs), and area (indicating emulsion stability). To enhance the feature space and enable detailed analysis of parameter interactions, polynomial transf

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Farouki, Rida T., and Jeffrey A. Strom. "Computing the roots of sparse high–degree polynomials that arise from the study of random simplicial complexes." Numerical Algorithms 83, no. 4 (2019): 1653–70. http://dx.doi.org/10.1007/s11075-019-00745-3.

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