Euclidean Division Research Articles

On the stability of parametric polynomials

This work introduces analytical tools for investigating stability regions of parametric polynomials. Our approach extends the classical D-decomposition method to accommodate any number of parameters, whether linearly or non-linearly involved in the coefficients of a polynomial ψ. Through Hermite's stability method, we establish the existence of stability regions within the parameter space, where all roots of ψ reside exclusively in the closed left half complex plane, defining the stability boundary. To identify points within the stability region, we employ the quotient and remainder from the Euclidean division of ψ by specific polynomials. Additionally, we introduce the ‘dynamics correlator’, a polynomial expressing ψ in its assignable roots space, whose zero set generalises the Root Locus. Confluence polynomials and loci, derived from the dynamics correlator, are used to realise stabilisability strategies. Extensive numerical examples demonstrate the application and effectiveness of the proposed methods in stabilising challenging systems and meeting additional performance criteria.

Should I Learn Division Algorithm?: An Investigation of Elementary Students’ Solution Strategies on Division with Remainder (DWR) Problems

The division with remainder (DWR) problems offer significant potential for students to make sense of the division operation. The purpose of the study is to investigate elementary school students' solution strategies for DWR problems. In particular, this study aims to compare the problem-solving strategies in DWR problems employed by second-grade students, who are versed in multiplication, but have not been introduced to division; with those of third and fourth-grade students who are familiar with division but have yet to engage with the interpretation of remainders. This qualitative research obtained data from 144 students in second, third, and fourth-grades in a public primary school. A total of six different DWR problems were presented to the students, including types as remainder divisible, remainder not divisible and remainder as a whole. The findings indicated that the strategies used by students in solving DWR problems differed. While second-grade students prefer strategies such as repetitive addition, repetitive subtraction, grouping, verbal explanation and using models, there is a noticeable tendency to use the division algorithm by fourth-grade students. However, it was noticed that students were unable to interpret the remainder in a meaningful way, especially from the third-grade, when they began to learn the division algorithm. According to the study, rather than moving immediately to the division algorithm, teachers should spend their time helping students understand division through contextual problems and representations.

Euclidean Domain in the Ring Q[\(\sqrt{-43}\)

An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: R \(\to\) \(\mathbb{Z}\)+ which admits proper generalization of the Euclidean division of integers. Every Euclidean domain (ED) is a Principal ideal domain (PID), but not all principal ideals are Euclidean. We provide detailed proof that the quadratic algebraic integer ring Q[\(\sqrt{-43}\)] is not Euclidean domain. We proved that the ring of algebraic integer in the quadratic complex field Q[\(\sqrt{-43}\)] is a principal ideal domain using the developed inequalities and field norm axioms in [1]. We proved that the ring Q[\(\sqrt{-43}\)] fails to have universal side divisors, thus, fails to be Euclidean domain (ED). This article extended the result application of [1] proving that ring R of algebraic integer in complex quadratic fields Q[\(\sqrt{-M}\)] for M = 43 is non-Euclidean PID in an understandable manner. We hope to look into the formation of these rings, thus, non-Euclidean geometries where the practical application will be more useful. E.g., Elliptic curves on finite fields.

Euclidean affine functions and their application to calendar algorithms

AbstractIn everyday life, dates are specified in terms of year, month and day, but this is not how digital devices represent them. Such devices continuously count elapsed days since a certain reference date, usually 1 January 1970. Accordingly, the date exactly one year after this reference is 1 January 1971 and digital devices represent it as 365. Conversions between machine and human formats are, arguably, amongst the most common operations performed by digital devices and constitute the subject of this article. We introduce Euclidean affine functions (EAFs) and study their properties. EAFs are of the form , where , , , and are integers and where denotes the quotient of Euclidean division. We derive algebraic relations and numerical approximations that are important for the efficient evaluation of these expressions in modern CPUs. Since division is a particular case of an EAF (when and ), the optimisations proposed in this article can also be applied to division. The main application presented in this article is the derivation of conversion algorithms for the Gregorian calendar. We will show that they can be implemented substantially more efficiently than is currently the case in widely used C, C++, C#, and Java open source libraries. Gains in speed of a factor of two or more are common. These algorithms have been implemented in GCC, the Linux Kernel and .NET.

Impact of programming on primary mathematics learning

The aim of this study is to investigate whether a programming activity might serve as a learning vehicle for mathematics acquisition in grades four and five. For this purpose, the effects of a programming activity, an essential component of computational thinking, were evaluated on learning outcomes of three mathematical notions: Euclidean division (N = 1,880), additive decomposition (N = 1,763) and fractions (N = 644). Classes were randomly assigned to the programming (with Scratch) and control conditions. Multilevel analyses indicate negative effects (effect size range −0.16 to −0.21) of the programming condition for the three mathematical notions. A potential explanation of these results is the difficulties in the transfer of learning from programming to mathematics.

Periodic solutions of approximated normal forms and simplification of normal transform by using Gröbner basis

The paper proposes a methodology based on the normal form perturbation method and Gröbner basis approach. Spectrum rearrangement allows to exhibit resonant terms associated with periodic behaviors of the system. Normal forms generate compatibility equations when one looks for periodic solutions and define amplitude-frequency conditions. Gröbner generators are defined from these equations and associated ideal is generated in order to simplify results of final normal form using generalized Euclidean division. Normal transform are simplified, and in some cases can be linearized with nonlinearity remaining hidden in amplitude-frequency conditions.

A Note on the Girth of (3, 19)-Regular Tanner’s Quasi-Cyclic LDPC Codes

In this article, we study the cycle structure of (3, 19)-regular Tanner’s quasi-cyclic (QC) LDPC codes with code length $19p$ , where $p$ is a prime and $p\equiv 1~(\bmod ~57)$ , and transform the conditions for the existence of cycles of lengths not more than 10 into polynomial equations in a 57th root of unity of the prime field $\mathbb {F}_{p}$ . By employing the Euclidean division algorithm to check whether these equations have solutions over the prime field $\mathbb {F}_{p}$ , the girth values of (3, 19)-regular Tanner’s QC-LDPC codes of code length $19p$ are determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.

Conhecimento Especializado de um formador de professores de Matemática ao ensinar o Teorema do Algoritmo da Divisão Euclidiana: um foco nos exemplos e explicações

De acordo com algumas perspectivas, qualquer profissional envolvido e responsável pela formação de professores pode ser considerado um formador. Nesse sentido, um matemático que trabalha em cursos de formação inicial é considerado também como formador de professores, e seu conhecimento influencia diretamente essa formação. Neste artigo analisamos a prática de um matemático que atua na formação de professores e discutimos o conhecimento especializado desse formador, considerando a perspectiva do Mathematics Teachers’ Specialized Knowledge. Os resultados obtidos permitem destacar a natureza especializada e o conteúdo do conhecimento do formador, que é matemático e não possui formação específica para assumir o papel de formador.

Blind recognition of binary BCH codes based on Euclidean algorithm

A novel method based on Euclidean algorithm is proposed to solve the problem of blind recognition of binary Bose–Chaudhuri–Hocquenghem (BCH) codes in non-cooperative applications. By carrying out iterative Euclidean divisions on the demodulator output bit-stream, the proposed method can determine the codeword length and generator polynomial of unknown BCH code. The computational complexity is derived as O (n 3 ). Simulation results show the efficiency of the proposed method.

Elliptic curve cryptography arithmetic in terms of one variable polynomial division

This paper develops an approach for point addition and doubling on elliptic curves over finite fields using one variable polynomial arithmetic based on Euclidean division. This approach succeeds in computing these operations on realistic curves over large finite fields due to a striking observation about computing the gcd of two polynomials one which represents the elliptic curve as its roots and the other representing the lines which intersect or are tangent to the curve. This paper provides a verification of correctness of this approach for different finite fields. The resulting algorithm is tested on realistic elliptic curves and is shown to be practically scalable to perform the computations.

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.

The Comparison of Degree of Complementary Dominating Set of Dominating Set using Euclidean Division Algorithm with Divisor 5 for Interval Graph G

The Comparison of Degree of Complementary Dominating Set of Dominating Set using Euclidean Division Algorithm with Divisor 5 for Interval Graph G

A method for recovering Jacobi matrices with mixed spectral data

In this paper we employ the Euclidean division for polynomials to recover uniquely a Jacobi matrix in terms of the mixed spectral data consisting of its partial entries and the information given on its full spectrum together with a subset of eigenvalues of its truncated matrix obtained by deleting the last row and last column, or its rank-one modification matrix modified by adding a constant to the last element. A necessary and sufficient condition is provided for the existence of the inverse problem. A numerical algorithm and a numerical example are given.

Arithmetic of Koblitz Curve <i>Secp256k1</i> Used in Bitcoin Cryptocurrency Based on One Variable Polynomial Division

Bitcoin uses a specific Koblitz curve secp256k1 is a type of elliptic curve defined by the Standards for Efficient Cryptography Group (SECG). This paper develops an approach for arithmetic (point addition and doubling) on secp256k1 Koblitz curve over finite fields using one variable polynomial based on Euclidean division. This approach succeeds in computing these operations due to a striking observation about computing the gcd of two polynomials one which represents the elliptic curve as its roots and the other representing the lines which intersect or are tangent to the curve. The resulting algorithm is tested on realistic secp256k1 Koblitz curve and is shown to be scalable to perform the computations.

Girth Analysis of Tanner’s (3, 17)-Regular QC-LDPC Codes Based on Euclidean Division Algorithm

In this paper, the girth distribution of the Tanner's (3, 17)-regular quasi-cyclic LDPC (QCLDPC) codes with code length 17p is determined, where p is a prime and p ≡ 1 (mod 51). By analyzing their cycle structure, five equivalent types of cycles with length not more than 10 are obtained. The existence of these five types of cycles is transmitted into polynomial equations in a 51st unit root of the prime field Fp. By using the Euclidean division algorithm to check the existence of solutions for such polynomial equations, the girth values of the Tanner's (3, 17)-regular QC-LDPC codes are obtained.

Némethi's division algorithm for zeta-functions of plumbed 3-manifolds

A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbed 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta-function, for the explicit calculation of the polynomial, in particular for the Seiberg--Witten invariant.

Realization of Nonlinear Time-Delay Input–Output Equations

In this letter the problem of transforming a nonlinear multi-input multi-output time-delay control system, described by a set of higher order functional differential equations relating system inputs and outputs, into a set of first order functional differential equations, is studied. The globally linearized system equations are described in terms of polynomials in a delay operator and derivative operator. The Euclidean left division of polynomials is used to compute the basis of a certain free submodule of differential 1-forms. The integrability of the submodule is proved to be necessary and sufficient for the solvability of the problem.

Remarks on the realization of time-varying systems

The realization problem of nonlinear time-varying input–output equations is considered. Differentials of the state coordinates, necessary for realization, are determined by the vector space of differential one-forms, spanned over the field of meromorphic functions. Formulas for computing the basis one-forms are given, based on the Euclidean division of non-commutative polynomials. Moreover, it is shown that in the case of a reducible system, the subspace admits a basis with certain structure, explicitly related to reduced input–output equations.

Hardware Division by Small Integer Constants

This article studies the design of custom circuits for division by a small positive constant. Such circuits can be useful for specific FPGA and ASIC applications. The first problem studied is the Euclidean division of an unsigned integer by a constant, computing a quotient and remainder. Several new solutions are proposed and compared against the state-of-the-art. As the proposed solutions use small look-up tables, they match well with the hardware resources of an FPGA. The article then studies whether the division by the product of two constants is better implemented as two successive dividers or as one atomic divider. It also considers the case when only a quotient or only a remainder is needed. Finally, it addresses the correct rounding of the division of a floating-point number by a small integer constant. All these solutions, and the previous state-of-the-art, are compared in terms of timing, area, and area-timing product. In general, the relevance domains of the various techniques are different on FPGA and on ASIC.

The Brun gcd algorithm in high dimensions is almost always subtractive

We introduce and study an algorithm which computes the gcd of d+1 entries. This is a natural extension of the usual Euclid algorithm, and coincides with it for d=1; it performs Euclidean divisions, between the largest entry and the second largest entry, and then re-orderings. This is the discrete version of a multidimensional continued fraction algorithm due to Brun, in d dimensions. We perform the average-case analysis of this algorithm, and prove that the mean number of steps is linear with respect to the size of the entry. The method is based on the study of the underlying Brun dynamical system, and relies on dynamical analysis. All the constants that arise in the analysis are mathematical constants which are defined via the Brun dynamical system: for instance, the main constant of the analysis involves the entropy of the system, and the other constants of the analysis are related to fine properties of the invariant measure of the system. We are then led to the asymptotic behaviour of the Brun dynamical system, when dimension d becomes large, which is of independent interest. We show that the asymptotic Brun dynamical system almost always involves quotients that are equal to 1, and is then almost always subtractive. This explains the inefficiency of the Brun gcd algorithm, particularly when it is compared in high dimensions to another extension of the Euclid algorithm, proposed by Knuth, and already analyzed by the authors.