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We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z [ x ] any integral polynomial with degree less than 6. The arguments are suitable for building classroom/homework activities in basic algebra courses. [ABSTRACT FROM AUTHOR]

The main object of this present paper is to provide a unified presentation of class of Sinha polynomials which generalizes the well known class of Gegenbauer, Legendre polynomials. In this paper we shall give some basic relations involving the generalized Sinha polynomials and then take up several operational result then we obtain series representation, hypergeometric representation and generating function which are best stated in terms of the generalized polynomial. [ABSTRACT FROM AUTHOR]

The article talks about several theses doctoral degrees in mathematics defended in Spain during 2020. It mentions about several theses such as numerical methods and their teaching; and study of cases in the Ecuadorian university; and limit cycles of small amplitude in polynomial and piecewise polynomial planar vector fields.

Light-field cameras have recently emerged as a powerful tool for one-shot passive 3D shape capture. However, obtaining the shape of glossy objects like metals or plastics remains challenging, since standard Lambertian cues like photo-consistency cannot be easily applied. In this paper, we derive a spatially-varying (SV)BRDF-invariant theory for recovering 3D shape and reflectance from light-field cameras. Our key theoretical insight is a novel analysis of diffuse plus single-lobe SVBRDFs under a light-field setup. We show that, although direct shape recovery is not possible, an equation relating depths and normals can still be derived. Using this equation, we then propose using a polynomial (quadratic) shape prior to resolve the shape ambiguity. Once shape is estimated, we also recover the reflectance. We present extensive synthetic data on the entire MERL BRDF dataset, as well as a number of real examples to validate the theory, where we simultaneously recover shape and BRDFs from a single image taken with a Lytro Illum camera. [ABSTRACT FROM AUTHOR]

Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M2(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial sk is not a Lie polynomial, for k>2. [ABSTRACT FROM AUTHOR]

This article aims to investigate the mathematical knowledge of teachers for the teaching of polynomials in Basic Education. The concept of polynomial was chosen due to its importance within the mathematical knowledge and to its little exploration in research that deals with the teaching and learning of algebraic contents. It is a qualitative interpretive research, which used didactic sequences elaborated by the teachers in their analyzes, which are realized under Ball and colleagues' work. The conclusions of our investigation indicate that the didactic sequences destined to the teaching of the polynomials, elaborated by the teachers, contemplate aspects of a mathematical knowledge for the teaching purely related to procedural aspects and, often, devoid of meanings. It is alerted to the urgency of continuing education, focusing not only on pedagogical knowledge but also on specific mathematical knowledge. [ABSTRACT FROM AUTHOR]

We investigate the strategies used by 64 advanced secondary mathematics students to identify whether a given pair of polynomial representations (graphs, tables, or equations) corresponded to the same function on an assessment of coordinating representations. Participants also completed assessments of domain-related knowledge and background skills. Cluster analysis of strategies by representation pair revealed patterns in the participants' strategy use. Two clusters were identifiable on tasks that required matching equations to graphs and graphs to tables. We identified overlap between these two clusters, suggesting that while the representation pair influenced strategy choice, there was also a general distinction between students who used more and less sophisticated strategies. However, students who used more sophisticated coordination strategies were similar to the others on measures of domain-specific knowledge or background skills. We consider implications for future investigations testing interventions to promote coordinating representations. [ABSTRACT FROM AUTHOR]

When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while performing a task that required connections among graphical and algebraic representations of a polynomial relation. Their reasoning and processes on the task were examined. This revealed more detailed information about the nature of their connections and misconceptions. The study reveals different types of student connections among graphical and algebraic representations. [ABSTRACT FROM AUTHOR]

In this paper, we characterized the balanced bicliques of graphs resulting from the join and the corona of graphs. Furthermore, we established the balanced biclique polynomials of the resulting graphs using combinatorial approach. [ABSTRACT FROM AUTHOR]

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials. [ABSTRACT FROM AUTHOR]

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n = 1, 2,... , k, where fk(1), fk(2),... , fk(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive integers Sk(n) = 1k + 2k + ⋅⋅⋅ + nk, and show that Sk(n) admits the polynomial representations Sk(n) = Pk(n) and Sk(n) = Qk(n) for all n = 1, 2,... , and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for Sk(n) alternative to the well-known formula of Bernoulli. [ABSTRACT FROM AUTHOR]

The paper reports a quasi-experimental study comparing the "traditional flipped classroom" pedagogical design with the "productive failure" (Kapur, 2016) pedagogical design in the flipped classroom for a 2-week curricular unit on polynomials in a Hong Kong Secondary school. Different from the flipped classroom where students are provided video clips with new concepts and associated procedures to review at home before solving problems in class, the "productive failure" pedagogical design in the flipped classroom worked the other way around. Supported by mobile technologies, students explored, discussed and solved problems related to the new concepts first in class even though they might come across failures, followed by consolidating the concepts and associated procedures using video clips at home. The pedagogical design is referred to as "productive failure-based flipped classroom" in this study. The study was carried out in two Grade 7 classes: one with "traditional flipped classroom" and one with the "productive failure-based flipped classroom". Findings show that both classes had significant improvement in procedural knowledge. However, regarding conceptual knowledge, students in the "productive failure" condition performed better than those in traditional flipped classroom. This suggests that the "productive failure-based flipped classroom" pedagogical design may be better able to improve students' problem solving skills. [ABSTRACT FROM AUTHOR]

The article focuses on the complex zeros of cubic polynomial equation, and quartic polynomials.

We study here a pair of sequences of polynomials that arise from a particular iterated mapping on the plane. We show how these sequences come about, and give some of their interesting mathematical properties. [ABSTRACT FROM AUTHOR]

The article presents mathematical theorems related to polynomials with pentagon, square, and cubic curves.

In this paper we consider the teaching of Hermite interpolation. We propose here two nonstandard approaches for exploring Hermite interpolation polynomials in a computer supported environment. As an extension to the standard construction of the interpolation polynomials based on either on the fundamental polynomials or the triangular shaped divided difference table, we first investigate the generalization of the Neville type recursive scheme which may be familiar to the reader or to the student from the chapter about Lagrangian interpolation. Second, we propose an interactive demo tool where by the step-by-step construction of the interpolation polynomial, the interpolation constraints can be considered in an almost arbitrary order. Thus the same interpolating polynomial can be constructed in several different ways. As a by-product, one can also ask an interesting combinatorial problem from the students about the number of compatible orders of the constraints depending on the cardinality of node system. [ABSTRACT FROM AUTHOR]

Different approaches are known for the introduction of the classical Chebyshev polynomials. Although mostly they are introduced in connection with trigonometric identities or defined as orthogonal polynomials or a solution of a particular differential equation, we take extremality as a defining property. We discuss the current computer tools which make the exploration of extremal polynomials possibly more easy and enjoyable. Finally we investigate ways to generalize the notion of classical Chebyshev polynomials as extremal polynomials which are less known. The author used the general computer algebra system Mathematica for the development, but other systems can be also suitbale for the exploration: a part of a GeoGebra book as an illustrative example is also included. [ABSTRACT FROM AUTHOR]

The importance of use educational computer games of Mangahigh platform in the study of mathematics was investigated. Save our dumb planet game and Prodigi quiz were used to discuss first-degree polynomial function. High school students of a state public school in Rio de Janeiro participated of research. Before handle the games, many students had difficulties in understand the concept of function, calculate points to be marked on the Cartesian plane and plot graphics. Students showed easiness and fast adaptation to Mangahigh platform during the lectures with resources. For most, Mangahigh helped to better understand the content. However, statistically, there was no correlation between the traditional learning assessment grades and score performance in the game or quiz. [ABSTRACT FROM AUTHOR]

The article describes an activity program for students in grades 8-10 which asks students to scrutinize and analyze using alternative base systems and polynomial operations.

A Boolean function f on n variables is said to be q- locally correctable if, given a black-box access to a function g which is 'close' to an isomorphism f( x)= f( x,..., x)= f( x,..., x) of f, we can compute f( x) for any $x \in\mathbb{Z}_{2}^{n}$ with good probability using q queries to g. It is known that degree d polynomials are O(2)-locally correctable, and that most k-juntas are O( klog k)-locally correctable, where the closeness parameter, or more precisely the distance between g and f, is required to be exponentially small (in d and k respectively). In this work we relax the requirement for the closeness parameter by allowing the distance between the functions to be a constant. We first investigate the family of juntas, and show that almost every k-junta is O( klog k)-locally correctable for any distance ε<0.001. A similar result is shown for the family of partially symmetric functions, that is functions which are indifferent to any reordering of all but a constant number of their variables. For both families, the algorithms provided here use non-adaptive queries and are applicable to most but not all functions of each family (as it is shown to be impossible to locally correct all of them). Our approach utilizes the measure of symmetric influence introduced in the recent analysis of testing partial symmetry of functions. [ABSTRACT FROM AUTHOR]

The symbolic manipulation capabilities of computer algebra systems, which we refer to as CAS-S, are now becoming instantiated within secondary mathematics textbooks in the United States for the first time. While a number of research studies have examined how teachers use this technology in their classrooms, one of the most important factors in how this technology is used in the classroom is how it is embedded within curricular resources such as textbooks. This study introduces readers to an analytical framework for examining CAS-S within textbooks and presents the results of its application to three secondary U.S. mathematics textbook units involving polynomial functions. The framework consists of two components: application of CAS-S and reflection on CAS-S uses. The analyses identified differences among the three textbook units in pedagogical intent and task connectedness involving CAS-S. The majority of CAS-S tasks were coded as involving low procedural complexity and there were few instances in which the technology was used in the construction of proofs. Textbook developers asked students to reflect on the visible CAS-S result as opposed to the invisible process leading to those results. The implications of these results as well as the potential transformative role of CAS-S are discussed. [ABSTRACT FROM AUTHOR]

We propose a subclass of cyclic Goppa codes given by separable self-reciprocal Goppa polynomials of degree two. We prove that this subclass contains all reversible cyclic codes of length n, n | ( q ± 1), with a generator polynomial g( x), g( α) = 0, i = 0, 1, α = 1, α ∈ GF( q). [ABSTRACT FROM AUTHOR]

This study is about prospective secondary mathematics teachers' understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations-referent preserving versus referent transforming compositions-acting on these quantities. Although multiplicative structures can be modeled by additive structures, they have their own characteristics inherent in their nature. I situate my analysis within a framework of unit coordination with different levels of units supported by a theory of quantitative reasoning and theorems-in-action. Data consist of videotaped qualitative interviews during which prospective mathematics teachers were asked problems on multiplication and factorization of polynomial expressions in x and y. I generated a thematic analysis by undertaking a retrospective analysis, using constant comparison methodology. There was a pattern which showed itself in all my findings. Two student-teachers constantly relied on an additive interpretation of the context, whereas three others were able to distinguish between and when to rely on an additive or a multiplicative interpretation of the context. My results indicate that the identification and coordination of the representational quantities and their units at different categories (multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative reasoning and need to be emphasized in the teaching-learning process. Moreover, representational Cartesian products-in-action at two different levels, indicators of multiplicative thinking, were available to two research participants only. [ABSTRACT FROM AUTHOR]

The article discusses the function of the polynomial in curve fitting and the Lagrange interpolation formula as an important approach to it. The article shows how Lagrange interpolation polynomial can be developed and introduced at precalculus level. According to the author, this is a useful method for numerical analysis for students.

The article presents an evaluation regarding the conception of local extremum by implementing geometry software GeoGebra in compliance with the National Council of Teachers of Mathematics (NCTM) principle and the Common Core State Standards for Mathematics (CCSSM) mathematical practices. It informs that the software aid in constructing a dynamic graph of any polynomial for an instance that of a cubic function in standard form.

The article presents the proofs of irreducibility of cyclotomic polynomial. It presents proof in original and simplified form for prime cyclotomic polynomial due to the Gauss's theorem, a theorem in vector calculus, the Kronecker theorem, a theorem in mathematics and the Einstein's theorem, a geometric theorem. It further presents proofs due to mathematical theorems Dedekind, Landau and Schur for general cyclotomic polynomial.

Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we present several applications of this learning object: study of several transformations in the plane, the problems appearing in numerical analysis concerning roots of complex polynomials with the same modulus or whose multiplicity is greater than 1, the continuity of the roots depending on the coefficients of the polynomial and the study of a classical curve. Interactive learning objects can improve the students' ability, if in addition, they are under suitable guidance. [ABSTRACT FROM AUTHOR]

Copyright of Media4u Magazine is the property of Ing. Jan Chromy, Ph.D. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

It is proved that an integer n ≥ 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Zn, the ring of integers modulo n, such that each element of Zn is a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory. [ABSTRACT FROM AUTHOR]

We give an explicit decoding scheme for the permutation arrays under Hamming distance metric, where the encoding is constructed via a distance-preserving mapping from ternary vectors to permutations ( 3-DPM). [ABSTRACT FROM AUTHOR]

The article examines a cubic polynomials property using geometry and algebra and the calculus derivatives with the aid of GeoGebra interactive geometry application created by Markus Hohenwarter. GeoGebra enables users to come up with applets presented through the Internet. It can be used in translating a cubic polynomial, while showing other polynomials. A direction that can be used by students in proving the connection between conjecture and calculus is also presented.

This article presents different approaches to a problem, dubbed by the author as 'the consecutive pages problem'. The aim of this teaching-oriented article is to promote the teaching of abstract concepts in mathematics, by selecting a challenging amusement problem and then presenting various solutions in such a way that it can engage the attention of a fourth-grade student, a high school senior student, an average college student and scholars. [ABSTRACT FROM AUTHOR]

This article presents several different methods for solving the problem of how to find a certain relation defined in chapter 2. The first method deals with the identities known in the theory of symmetric polynomials as the elements of a certain vector space. The second method is designed around the matrix transformations between symmetric polynomials. The third method is designed around the property of a linear operator and its characteristic polynomial. The fourth method is designed in the area of complex numbers, and introduces the multiplication group of 'complex roots of one'. Significant improvement in the third and fourth method is made by introducing so called 'block method'. It facilitates all calculations by making them much shorter. The article ends with an example showing symmetry and regularity of all procedures. Finally, the article shows how to solve the problem for any degree n of the polynomial, and for any degree k. At the end of the paper solutions for n⩽5 and k ⩽5 are tabulated. [ABSTRACT FROM AUTHOR]

A new class of positive continuous-discrete linear systems with delays in state vector described by the model based on 2D general model is addressed. Necessary and sufficient conditions for the positivity and asymptotic stability of this class of linear systems are established. A procedure for checking the asymptotic stability is proposed. The effectiveness of the procedure is demonstrated on a numerical example. [ABSTRACT FROM AUTHOR]

The article discusses mathematical analysis involving the Polynomial Root Dragging Theorem which qualitatively explains the moving of the polynomial roots. It notes that moving one or more roots of the polynomial will cause every critical point to stay fixed. It adds that the polynomial root motion was considered as the most dynamic problem and yet can be proved useable as standard topics from the undergraduate curriculum.

Copyright of Information Sciences / Informacijos Mokslai is the property of Vilnius University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article describes an activity program for precalculus high school students which teaches the concept of computing simple polynomial derivatives using the power rule by using the example of a cookie.

Subgraphs can results through application of criteria based on matrix which characterize the entire graph. The most important categories of criteria are the ones able to produce connected subgraphs (fragments). Based on theoretical frame on graph theory, the fragmentation algorithm on pair of vertices containing the largest fragments (called MaxF) are exemplified. The counting polynomials are used to enumerate number of all connected substructures and their sizes. For a general class of graphs called dendrimers general formulas giving counting polynomials are obtained and characterized using informational measures. [ABSTRACT FROM AUTHOR]

We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an $${\varepsilon}$$ derivative loss in the angular directions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First we describe a useful procedure for computing the H-polynomials of reductive monoids. Second we use this procedure to compute the H-polynomial of the monoid of n × n matrices in terms of the q-analogues of the rook polynomials of Garsia and Remmel. [ABSTRACT FROM AUTHOR]

The Glaisher rule is an operational identity involving the action of an exponential operator containing the second-order derivatives acting on an exponential function. We use the Crofton and monomiality formalism to derive generalized forms to the multi-dimensional case and show its usefulness in the derivation of old and new forms of generating functions for a wealth of Hermite polynomials families. [ABSTRACT FROM AUTHOR]

A multiplicative version of characteristic transforms for probability measures on × is defined, and continuity theorems in terms of these transforms are proved. [ABSTRACT FROM AUTHOR]

Several authors have defined polynomial sets by means of general Rodrigues formulae in different forms. In this article, an attempt is made to provide an elegant unification of several classes of polynomials. We investigate a general class of polynomials [image omitted] by means of a generalized Rodrigues formula defined as (2) associated with a generalized Mittag-Leffler function [image omitted] , which is recently introduced by Shukla and Prajapati [Shukla, A.K. and Prajapati, J.C., On a generalisation of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Appplications, article in press]. We have also derived several families of generating relations, finite summation formulae for equation (2) by employing operational techniques and bilateral-generating relation. In the end, several special cases have been discussed. [ABSTRACT FROM AUTHOR]

The half-function concept is queried. It is argued that this concept is neither strict nor necessary. Without involving the half-function, the procedure of heuristically constructing complex orthogonal polynomials is presented. [ABSTRACT FROM AUTHOR]

We describe methods for implementing separation algorithms for domino-parity inequalities for the symmetric traveling salesman problem. These inequalities were introduced by Letchford (2000), who showed that the separation problem can be solved in polynomial time when the support graph of the LP solution is planar. In our study we deal with the problem of how to use this algorithm in the general (nonplanar) case, continuing the work of Boyd et al. (2001). Our implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of a 33,810-city instance from the TSPLIB. [ABSTRACT FROM AUTHOR]

A new polynomial neural network (PNN) model for estimating liquid holdup in horizontal two-phase flow is proposed in this article. The PNN evolutionally synthesize network size, connectivity, processing element types, and coefficients for globally optimized structure through training. The framework is established using 330 data sets from different experimental conditions. This self-organizing approach automatically presents internal relationships among data in the polynomial forms, and enhances data approximation and explanation capabilities of resulting data-based learning models. The comparative studies with experimental correlations and artificial neural network applications reveal that the model exhibits significant improvement in the processing structure, and outperforms previous models in overall accuracy across liquid holdup ranges. [ABSTRACT FROM AUTHOR]