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Journal articles on the topic 'Teaching first degree polynomials'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Antunes, Daniele Mendes Caldas, Cristiana Andrade Poffal, Bárbara Denicol do Amaral Rodriguez, and Fabíola Aiub Sperotto. "Analysis of first degree polynomial function teaching strategies to social applied sciences courses." Ciência e Natura 40 (March 27, 2018): 22. http://dx.doi.org/10.5902/2179460x29526.

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2

Barbosa, Edelweis Jose Tavares, and Anna Paula de Avelar Brito Lima. "Praxéologies du professeur : analyse comparative de manuels scolaires dans l'enseignement des équations polynomiales du premier degréTeacher's Praxeologies: Comparative Analysis of the Textbook in Teaching First-Degree Polynomial Equations." Educação Matemática Pesquisa : Revista do Programa de Estudos Pós-Graduados em Educação Matemática 22, no. 4 (2020): 224–31. http://dx.doi.org/10.23925/1983-3156.2020v22i4p224-231.

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RésuméLe but de cet article est d'analyser, de manière comparative, les livres didactiques et les praxéologies mises en place par les enseignants dans leur pratique pédagogique, concernant l'enseignement des équations polynomiales du premier degré. Cette étude est faite dans le cadre de la théorie anthropologique du didactique (TAD) proposée par Yves Chevallard et ses collaborateurs (1999, 2002, 2009, 2010). La méthodologie est basée sur une approche ethnographique qualitative, dans laquelle les organisations mathématiques et didactiques de trois enseignants sont analysées en les comparant à c

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3

Nascimento, Rosimiro Araujo do, and Marli Teresinha Quartieri. "Investigação Matemática: possibilidade para o ensino de função do 1º grau." Jornal Internacional de Estudos em Educação Matemática 13, no. 2 (2020): 133–44. http://dx.doi.org/10.17921/2176-5634.2020v13n2p133-144.

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ResumoNão são raros os professores de Matemática que identificam dificuldades nos estudantes, iniciantes do Ensino Médio, em compreenderem a função polinomial do 1º grau. Assim, o presente artigo disserta sobre uma das atividades investigativas, que compõe uma dissertação de mestrado, desenvolvida com 26 alunos dispostos em grupos colaborativos, envolvendo a referida função. A pesquisa, na modalidade qualitativa, foi apoiada por uma observação participante na qual questionou-se: como os estudantes do 1º ano do curso Técnico em Agropecuária Integrado ao Ensino Médio possibilitam o ensino e apre

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4

Kousidis, Stavros, and Andreas Wiemers. "On the first fall degree of summation polynomials." Journal of Mathematical Cryptology 13, no. 3-4 (2019): 229–37. http://dx.doi.org/10.1515/jmc-2017-0022.

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Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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5

Cesarano, Clemente, Sandra Pinelas, and Paolo Ricci. "The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree." Symmetry 11, no. 2 (2019): 274. http://dx.doi.org/10.3390/sym11020274.

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New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown.

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6

KHAN, MUHAMMAD YUNAS, SYED IFTIKHAR ALAM, QUTIB ALAM JAN, and Ata-ur Rehman. "FIRST DEGREE HEMORRHOIDS;." Professional Medical Journal 20, no. 04 (2013): 576–80. http://dx.doi.org/10.29309/tpmj/2013.20.04.1086.

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Traditionally the treatment of Ist, 2nd and 3rd degree haemorrhoids is conservative management with fibre rich diet andbetter defecation discipline. If symptoms prevail then there are a wide range of treatment modalities. OBJECTIVE: to evaluate the efficacyof injection Sclerotherapy for First degree haemorrhoids using 5% Phenol in Almond oil. MATERIAL AND METHODS: This case seriesstudy was conducted on 83 patients with first degree piles in Surgical-C unit, Department of Surgery, Khyber Teaching Hospital Peshawarfrom 01/09/2010 till 01/03/2011. After Informed consent, injection Sclerotherapy w

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7

Repka, Marek. "Note on Modular Reduction in Extended Finite Fields and Polynomial Rings for Simple Hardware." Journal of Electrical Engineering 67, no. 1 (2016): 56–60. http://dx.doi.org/10.1515/jee-2016-0008.

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AbstractModular reduction in extended finite fields and polynomial rings is presented, which once implemented works for any random reduction polynomial without changes of the hardware. It is possible to reduce polynomials of whatever degree. Based on the principal defined, two example RTL architectures are designed, and some useful features are noted furthermore. The first architecture is sequential and reduce whatever degree polynomials, taking 2 cycles per term. The second one is Parallel and designed for reduction of polynomials of 2(t−1) degree at most, taking 1 cycle for the whole reducti

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8

Foster, Peter J. "Teaching Engineering at First Degree Level in the UK." Teaching in Higher Education 4, no. 2 (1999): 235–52. http://dx.doi.org/10.1080/1356251990040206.

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9

ROBERTS, DAVID P. "NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5A." International Journal of Number Theory 07, no. 02 (2011): 289–322. http://dx.doi.org/10.1142/s1793042111004113.

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We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±pA for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL2(5)5.10, and field discriminant 569. A closely related polynomial has degree 120, Galois group of the form SL2(5)5.20, and field discriminant 5311. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 54.

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10

WEINGARTNER, ANDREAS. "On the degrees of polynomial divisors over finite fields." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 3 (2016): 469–87. http://dx.doi.org/10.1017/s030500411600044x.

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AbstractWe show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n, is given by cqn−1 + O(n−2). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m. To that end, we first derive an improved estimate for the proportion of polynomials of degree n, all of whose non-constant divisors have degree greater than m. In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permu

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11

De Terán, Fernando, Carla Hernando, and Javier Pérez. "Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis." Electronic Journal of Linear Algebra 37, no. 37 (2021): 35–71. http://dx.doi.org/10.13001/ela.2021.5473.

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In the framework of Polynomial Eigenvalue Problems (PEPs), most of the matrix polynomials arising in applications are structured polynomials (namely, (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve PEPs is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as companion linearizations. It is well known, however, that it is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomial

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12

Gao, Wei, Muhammad Younas, Adeel Farooq, Abaid Virk, and Waqas Nazeer. "Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers." Mathematics 6, no. 10 (2018): 214. http://dx.doi.org/10.3390/math6100214.

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Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly (ethylene amido amine) dendrimers.

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Santos, Rosipléia Souza, and Kisnney Emiliano de Almeida. "ATIVIDADES PARA O ENSINO FUNDAMENTAL E MÉDIO ENVOLVENDO RELAÇÕES DE GIRARD E ARITMÉTICA." Ciência e Natura 37 (August 7, 2015): 439. http://dx.doi.org/10.5902/2179460x14812.

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http://dx.doi.org/10.5902/2179460X14812The objective of this work is to build a didatic proposal for teaching methods to fing the integer or racional roots of polynomials of degree greater than two by using properties of integers learned on the studies of Arithmetic and the relations between coeficients and roots of polynomials known as Relations of Girard, with minimal application of formulas. Thus the student is given the opportunity to exercise simultaneously and dynamically the basics of algebra and arithmetic present in basic education.

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14

Pio Jude, Thayamathy, Elango Panchadcharam, and Koneswaran Masilamani. "Computing Zagreb and Randić Indices of PEG-Cored Dendrimers Used for Drug and Gene Delivery." Journal of Chemistry 2020 (October 14, 2020): 1–7. http://dx.doi.org/10.1155/2020/2821510.

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Zagreb and Randić indices are the most commonly used degree-based topological indices in the study of drug design and development. In molecular topology, M-polynomials are also used to calculate the degree-based topological indices of chemical structures. In this paper, we derive the M-polynomials for the PEG-cored PAMAM, carbosilane, and poly (lysine) dendrimers and calculate their first, second, and second modified Zagreb indices and the Randić index.

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15

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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16

Hussain, Zafar, Mobeen Munir, Muhammad Bilal, Alam Ameer, Shazia Rafique, and Shin Min Kang. "Computational Analysis of new Degree-based descriptors of oxide networks." Open Chemistry 17, no. 1 (2019): 177–82. http://dx.doi.org/10.1515/chem-2019-0023.

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AbstractOxide networks have diverse applications in the polymer and pharmaceutical industries. Polynomials and degree-based topological indices have tendencies to correlate properties of molecular graphs. In this article, we formulate the closed forms of Zagreb and forgotten polynomials and topological indices such as Hyper-Zagreb index, first and second multiple Zagreb indices, forgotten index, Albert index, Bell index, IRM(G) of oxide networks. We also compute the F-index of complement of oxide networks, F-coindex of G and F-coindex of complement of oxide networks. We put graphical analysis

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Sharma, P. L., M. Rehan, and S. Sharma. "Counting irreducible polynomials over GF(3) with first and third coefficients given." Asian-European Journal of Mathematics 08, no. 01 (2015): 1550015. http://dx.doi.org/10.1142/s1793557115500151.

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We obtain some formulae to find the approximate values of the number of irreducible polynomials of degree n > 2 over GF(3) in which the coefficients of xn-1 and xn-3 are prescribed. We compare and find that our estimates are close to the exact values.

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18

ALIABADI, MOHSEN. "A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA." Bulletin of the Australian Mathematical Society 97, no. 3 (2018): 382–85. http://dx.doi.org/10.1017/s0004972718000035.

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The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

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19

AlQudah, Mohammad, and Maalee AlMheidat. "Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations." Symmetry 10, no. 12 (2018): 692. http://dx.doi.org/10.3390/sym10120692.

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Approximating continuous functions by polynomials is vital to scientific computing and numerous numerical techniques. On the other hand, polynomials can be characterized in several ways using different bases, where every form of basis has its advantages and power. By a proper choice of basis, several problems will be removed; for instance, stability and efficiency can be improved, and numerous complications can be resolved. In this paper, we provide an explicit formula of the generalized shifted Chebyshev Koornwinder’s type polynomial of the first kind, T r * ( K 0 , K 1 ) ( x ) , using the Be

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20

Chèze, Guillaume. "Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time." Journal of Complexity 27, no. 2 (2011): 246–62. http://dx.doi.org/10.1016/j.jco.2010.10.004.

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21

Kiltz, Eike, and Arne Winterhof. "On the interpolation of bivariate polynomials related to the Diffie-Hellman mapping." Bulletin of the Australian Mathematical Society 69, no. 2 (2004): 305–15. http://dx.doi.org/10.1017/s0004972700036042.

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We obtain lower bounds on degree and weight of bivariate polynomials representing the Diffie-Hellman mapping for finite fields and the Diffie-Hellman mapping for elliptic curves over finite fields. This complements and improves several earlier results. We also consider some closely related bivariate mappings called P-Diffie-Hellman mappings introduced by the first author. We show that the existence of a low degree polynomial representing a P-Diffie-Hellman mapping would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on weig

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22

Dittmann, Philip. "Irreducibility of polynomials over global fields is diophantine." Compositio Mathematica 154, no. 4 (2018): 761–72. http://dx.doi.org/10.1112/s0010437x17007977.

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Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have a root in $K$. This strengthens a result by Colliot-Thélène and Van Geel [Compositio Math. 151 (2015), 1965–1980] stating that the set of non-$n$th powers in a number field $K$ is diophantine. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order defini

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23

Abreu-Blaya, Ricardo, Sergio Bermudo, José M. Rodríguez, and Eva Tourís. "Topological Indices and f-Polynomials on Some Graph Products." Symmetry 13, no. 2 (2021): 292. http://dx.doi.org/10.3390/sym13020292.

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We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices.

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24

Fieker, Claus, and Jürgen Klüners. "Computation of Galois groups of rational polynomials." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 141–58. http://dx.doi.org/10.1112/s1461157013000302.

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AbstractComputational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$, a $G$-relative $H$-invariant, that is a multivariate polynomial $F$ that is $H$-invariant, but not $G$-invariant. While generic, theoretical methods are known to find such $F$, in general they yield impractical answers. We give a general method for computing invar

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Cesarano, Clemente, Pierpaolo Natalini, and Paolo Ricci. "Pseudo-Lucas Functions of Fractional Degree and Applications." Axioms 10, no. 2 (2021): 51. http://dx.doi.org/10.3390/axioms10020051.

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In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the

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26

Bowron, P. "Teaching Communication Skills in Electrical Engineering Degree Courses." International Journal of Electrical Engineering & Education 27, no. 1 (1990): 3–12. http://dx.doi.org/10.1177/002072099002700101.

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A first-year communication skills course is described which involves the engineering faculty as well as specialist lecturers. It is closely linked to the personal tutor system and exercises are carefully scheduled into a busy timetable. The extent to which the course satisfies EA2 requirements is assessed.

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27

QUESNE, C. "RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM." International Journal of Modern Physics A 26, no. 32 (2011): 5337–47. http://dx.doi.org/10.1142/s0217751x11054942.

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A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invari

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28

Chaudhry, Faryal, Iqra Shoukat, Deeba Afzal, Choonkil Park, Murat Cancan, and Mohammad Reza Farahani. "M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide." Journal of Chemistry 2021 (February 24, 2021): 1–12. http://dx.doi.org/10.1155/2021/6679819.

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Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index and inverse Randić index, and the aug

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ČAP, A., A. R. GOVER, and M. HAMMERL. "NORMAL BGG SOLUTIONS AND POLYNOMIALS." International Journal of Mathematics 23, no. 11 (2012): 1250117. http://dx.doi.org/10.1142/s0129167x12501170.

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First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of lo

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Hussain, Zafar, Mobeen Munir, Shazia Rafique, and Shin Min Kang. "Topological Characterizations and Index-Analysis of New Degree-Based Descriptors of Honeycomb Networks." Symmetry 10, no. 10 (2018): 478. http://dx.doi.org/10.3390/sym10100478.

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Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network. In the present article, new topological characterizations of honeycomb networks are given in the form of degree-based descriptors. In particular, we compute Zagreb and Forgotten polynomials and some topological indices such as the hyper-Zagreb index, first and second multiple Zagreb indices and the Forgotten index, F. We, for the first time, deter

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31

Lara, Teodoro, Nelson Merentes, Roy Quintero, and Edgar Rosales. "On the characterization of Jensen m-convex polynomials." Moroccan Journal of Pure and Applied Analysis 3, no. 2 (2017): 140–48. http://dx.doi.org/10.1515/mjpaa-2017-0012.

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AbstractThe main objective of this research is to characterize all the real polynomial functions of degree less than the fourth which are Jensen m-convex on the set of non-negative real numbers. In the first section, it is established for that class of functions what conditions must satisfy a particular polynomial in order to be starshaped on the same set. Finally, both kinds of results are combined in order to find examples of either Jensen m-convex functions which are not starshaped or viceversa.

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Liu, Shunyi. "Generalized Permanental Polynomials of Graphs." Symmetry 11, no. 2 (2019): 242. http://dx.doi.org/10.3390/sym11020242.

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The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I − ( A ( G ) − μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most

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Stoimenow, A. "On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials." Canadian Journal of Mathematics 59, no. 2 (2007): 418–48. http://dx.doi.org/10.4153/cjm-2007-018-0.

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AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomial Q contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from Q is an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree d ≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of alge

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Zhou, Xiaoqing, Mustafa Habib, Tariq Javeed Zia, Asim Naseem, Anila Hanif, and Ansheng Ye. "Topological invariants for the line graphs of some classes of graphs." Open Chemistry 17, no. 1 (2019): 1483–90. http://dx.doi.org/10.1515/chem-2019-0154.

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AbstractGraph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomial

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Ernst, Thomas. "-Pascal and -Wronskian Matrices with Implications to -Appell Polynomials." Journal of Discrete Mathematics 2013 (March 20, 2013): 1–10. http://dx.doi.org/10.1155/2013/450481.

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We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector o

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Dalto, Jader Otavio, and Verônica Bender Haydu. "Stimulus equivalence for teaching mathematical first-degree functions in the elementary school." Perspectivas em Análise do Comportamento 6, no. 2 (2015): 132–46. http://dx.doi.org/10.18761/pac.2015.022.

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Noguchi, Junjiro. "A remark to a division algorithm in the proof of Oka's first coherence theorem." International Journal of Mathematics 26, no. 04 (2015): 1540005. http://dx.doi.org/10.1142/s0129167x15400054.

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The problem is the locally finite generation of a relation sheaf ℛ(τ1,…,τq) in 𝒪Cn. After τj reduced to Weierstrass' polynomials in zn, it is the key for applying an induction on n to show that elements of ℛ(τ1,…,τq) are expressed as a finite linear sum of zn-polynomial-like elements of degree at most p = max j deg zn τj over 𝒪Cn. In that proof one is used to use a division by τj of the maximum degree, deg zn τj = p (Oka (1948); Cartan (1950); Hörmander (1966); Narasimhan (1966); Nishino (1996), etc.) Here we shall confirm that the division above works by making use of τk of the minimum degree

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Kačala, Viliam, and Csaba Török. "Optimal Approximation of Biquartic Polynomials by Bicubic Splines." EPJ Web of Conferences 173 (2018): 03012. http://dx.doi.org/10.1051/epjconf/201817303012.

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Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the

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Anderson, C. H., and J. Prasad. "On the non-existence of some interpolatory polynomials." International Journal of Mathematics and Mathematical Sciences 9, no. 4 (1986): 753–56. http://dx.doi.org/10.1155/s016117128600090x.

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Here we prove that ifxk,k=1,2,…,n+2are the zeros of(1−x2)Tn(x)whereTn(x)is the Tchebycheff polynomial of first kind of degreen,αj,βj,j=1,2,…,n+2andγj,j=1,2,…,n+1are any real numbers there does not exist a unique polynomialQ3n+3(x)of degree≤3n+3satisfying the conditions:Q3n+3(xj)=αj,Q3n+3(xj)=βj,j=1,2,…,n+2andQ‴3n+3(xj)=γj,j=2,3,…,n+1. Similar result is also obtained by choosing the roots of(1−x2)Pn(x)as the nodes of interpolation wherePn(x)is the Legendre polynomial of degreen.

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Doha, E. H., and Y. H. Youssri. "On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain." Nonlinear Engineering 8, no. 1 (2019): 318–27. http://dx.doi.org/10.1515/nleng-2018-0073.

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Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree

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41

DUDEK, MIROSŁAW R., and TADEUSZ NADZIEJA. "MOLECULAR DYNAMICS SIMULATIONS THROUGH SYMBOLIC PROGRAMMING." International Journal of Modern Physics C 16, no. 03 (2005): 413–25. http://dx.doi.org/10.1142/s0129183105007212.

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We present an algorithm for the Cauchy problem for the ordinary differential equations which is based on symbolic programming. It is complementary to numerical methods, which use the finite differences. The algorithm consists of two steps. First, it yields a set of polynomials of a given degree n in some formal parameter λ. They represent an analytic form of the truncated solution of the system of differential equations under consideration and the larger value of n the larger interval in which this approximation satisfies the chosen accuracy criterion. We put λ = 1 afterall. The second step is

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Binding, P. A., and P. J. Browne. "Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 2 (2000): 239–47. http://dx.doi.org/10.1017/s0308210500000135.

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Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

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Gerrard, Bill, and Keith Macrosson. "Teaching Technology and Business." Industry and Higher Education 11, no. 6 (1997): 378–82. http://dx.doi.org/10.1177/095042229701100608.

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Research conducted 25 years ago in the UK established at that time that businesses required able, enthusiastic and broadly educated people who were willing to master quickly a firm's techniques and procedures and to advance to a position of responsibility and leadership. The BSc Honours degree in Technology and Business Studies (TBS) at the University of Strathclyde was designed to meet that requirement by providing students with an understanding of the language of technology and the language of business, and to link these two to provide a fluency in the communication skills used by managers o

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Lee, Hong-You, and Charles F. Reinholtz. "Inverse Kinematics of Serial-Chain Manipulators." Journal of Mechanical Design 118, no. 3 (1996): 396–404. http://dx.doi.org/10.1115/1.2826899.

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This paper proposes a unified method for the complete solution of the inverse kinematics problem of serial-chain manipulators. This method reduces the inverse kinematics problem for any 6 degree-of-freedom serial-chain manipulator to a single univariate polynomial of minimum degree from the fewest possible closure equations. It is shown that the univariate polynomials of 16th degree for the 6R, 5R-P and 4R-C manipulators with general geometry can be derived from 14, 10 and 6 closure equations, respectively, while the 8th and 4th degree polynomials for all the 4R-2P, 3R-P-C, 2R-2C, 3R-E and 3R-

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DING, JINTAI, and TIMOTHY HODGES. "CRYPTANALYSIS OF AN IMPLEMENTATION SCHEME OF THE TAMED TRANSFORMATION METHOD CRYPTOSYSTEM." Journal of Algebra and Its Applications 03, no. 03 (2004): 273–82. http://dx.doi.org/10.1142/s0219498804000861.

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A Tamed Transformation Method (TTM) cryptosystem was proposed by T. T. Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 233 computations on the finite field with 28 elements. The cipher of the TTM systems are degree 2 polynomial maps derived from composition of invertible maps of either total degree 2 or linear maps which can be easily calculated and can be easily inverted. To ensure the system to be of degree two, the key construction of the implementation schemes of the TTM systems is a multivariable

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Deundyak, Vladimir Mikhailovich, and Anastasiya Vladimirovna Knutova. "Integrability of Systems of the First and Second Degree Polynomials of Several Variables Over Simple Galois Fields." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 2 (2016): 41–46. http://dx.doi.org/10.18522/0321-3005-2016-2-41-46.

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47

Ha, Junsoo. "Smooth Polynomial Solutions to a Ternary Additive Equation." Canadian Journal of Mathematics 70, no. 1 (2018): 117–41. http://dx.doi.org/10.4153/cjm-2017-023-x.

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AbstractLet Fq[T] be the ring of polynomials over the finite field of q elements and Y a large integer. We say a polynomial in Fq[T] is Y-smooth if all of its irreducible factors are of degree at most Y. We show that a ternary additive equation a + b = c over Y-smooth polynomials has many solutions. As an application, if S is the set of first s primes in Fq[T] and s is large, we prove that the S-unit equation u + v = 1 has at least exp solutions.

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BASU, SOHAM. "STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING." Bulletin of the Australian Mathematical Society 104, no. 2 (2021): 249–55. http://dx.doi.org/10.1017/s0004972720001434.

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AbstractWithout resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

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Stojanovic, Nikola, and Negovan Stamenkovic. "Lowpass filters approximation based on the Jacobi polynomials." Facta universitatis - series: Electronics and Energetics 30, no. 3 (2017): 351–62. http://dx.doi.org/10.2298/fuee1703351s.

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A case study related to the design the analog lowpass filter using a set of orthogonal Jacobi polynomials, having four parameters to vary, is considered. The Jacobi polynomial has been modified in order to be used as a filter approximating function. The obtained magnitude response is more general than the response of the classical ultra-spherical filter, due to one additional parameter available in orthogonal Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripple, smaller group delay variation or sharper cutoff slope. Two

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Ahmad, Tashfeen. "Improving political science degree programmes in the twenty-first century." Review of Economics and Political Science 5, no. 3 (2019): 231–47. http://dx.doi.org/10.1108/reps-02-2019-0023.

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Purpose This paper aims to examine the actions lecturers, universities and their administrators can take in improving and making political science undergraduate degrees more relevant in the twenty-first century. This paper will reflect on specific measures undertaken by institutions globally to equip students with unique skills to enhance the value and relevance of their programmes in the context of an increased technologically driven environment. Design/methodology/approach This paper uses a qualitative approach to the review of the literature with implications for practice in examining how u

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