An Infinite Number of Ways of Algebraic Factorization of a Number and Radical Solution of Higher Degree Polynomial Equations

2018 ◽  
Vol 3 (1) ◽  
Keyword(s):  
Infinite Number ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Radical Solution

scholarly journals On Complex-Valued Triple Controlled Metric Spaces and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Nabil Mlaiki ◽  
Thabet Abdeljawad ◽  
Wasfi Shatanawi ◽  
Hassen Aydi ◽  
Yaé Ulrich Gaba
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Spaces ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Fredholm Type ◽  
Type Integral ◽  
Type Spaces ◽  
Complex Valued

In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Hafez Tari ◽  
Hai-Jun Su
Keyword(s):  
Complete Solution ◽  
Solution Process ◽  
Polynomial Equations ◽  
Homotopy Methods ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Augmented System ◽  
Polynomial Curve ◽  
Four Bar Linkage ◽  
Classical Homotopy

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve.

Forward displacement analysis of a new 1CCC–5SPS parallel mechanism using Gröbner theory

2009 ◽  
Vol 223 (5) ◽  
pp. 1233-1241 ◽  
Author(s):  
D Gan ◽  
Q Liao ◽  
J S Dai ◽  
S Wei ◽  
L D Seneviratne
Keyword(s):  
Parallel Mechanism ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Geometrical Constraint ◽  
Degree Polynomial ◽  
Forward Displacement ◽  
Constraint Equations ◽  
Moving Platform ◽  
Forward Kinematic ◽  
Forward Displacement Analysis

A new parallel mechanism 1CCC–5SPS which has distance and angle constraints is introduced in this article. Degree of freedom and forward kinematic analysis of this new parallel mechanism are presented, in which four equivalent polynomial equations are obtained from the original six geometrical constraint equations. The Gröbner basis theory is used with the four equations and the problem of forward displacement is reduced to a 40th degree polynomial equation in a single unknown from a constructed 10 × 10 Sylvester's matrix which is small in size, from which 40 different locations of the moving platform can be derived. A numerical example confirms the efficiency of the procedure.

scholarly journals Exact solutions of sixth and fifth degree equations

2021 ◽  
Author(s):  
Mohammed El Amine MONIR
Keyword(s):  
Exact Solutions ◽  
Algebraic Polynomial ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
The Real ◽  
Absolute Method ◽  
Quintic Equation ◽  
Degree Equation

Abstract The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

scholarly journals 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

2020 ◽  
Vol 22 (4) ◽  
pp. 224-231
Author(s):  
Edelweis Jose Tavares Barbosa ◽  
Anna Paula de Avelar Brito Lima
Keyword(s):  
Polynomial Equation ◽  
Classroom Teachers ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Mots Clés ◽  
Anthropological Theory ◽  
The Common ◽  
Analyse Comparative ◽  
Théorie Anthropologique Du Didactique ◽  
Anthropological Theory Of Didactics

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 celles des livres de référence. Les résultats indiquent qu'il existe une certaine conformité entre les praxéologies à enseigner, proposées par les auteurs des manuels scolaires et les praxéologies effectivement enseignées par les professeurs en classe. Les enseignants sont les organisateurs des tâches, des techniques et de la technologie de complexité croissante (FONSECA, 2004) qui sont rendus routinières ou problématiques en classe. La résolution d’une équation polynomiale du premier degré du type ax+b=c a été le point commun des trois professeurs, bien que deux des trois enseignants aient aussi travaillé des équations du type a1x+b1=a2x+b2.Mots-clés : Livres didactiques, Équations polynomiales du premier degré, Théorie Anthropologique du didactique.AbstractThe aim of this article was to analyze, comparatively, praxeologies in didactic books and praxeologies carried out by the teacher, concerning the teaching of polynomial equations of the first degree. This study is done within the framework of the Anthropological Theory of Didactics (ATD), proposed by Yves Chevallard and his collaborators (1999, 2002, 2009, 2010). The methodology consists of a qualitative ethnographic approach, in which the mathematical and didactic organizations of three teachers were compared with those of their reference books. The results indicate that there is some conformity between the praxeologies to be taught, proposed by the authors of the textbooks, and the praxeologies effectively taught by the teachers in the classroom. Teachers are the organizers of tasks, techniques, and technology of increasing complexity (FONSECA, 2004) that are made routine or problematic in the classroom. The resolution of a first-degree polynomial equation of the type ax+b=c was the common point among the three teachers, although two of the three teachers also worked on equations of the type a1x+b1=a2x+b2.Keywords: Didactic books, Polynomial equations of the first degree, Anthropological theory of didactics.

scholarly journals The study on general cubic equations over p-adic fields

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1115-1131
Author(s):  
Mansoor Saburov ◽  
Mohd Ahmad ◽  
Murat Alp
Keyword(s):  
Polynomial Equation ◽  
Rational Numbers ◽  
Lower Degree ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Diophantine Problem ◽  
Cubic Equation ◽  
All Solutions ◽  
General Number

A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field Qp belongs to domains Z*p, Zp \ Z*p, Qp \ Zp, Qp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.

Use of Resultant in the Dimensional Synthesis of Linkage Components: Motion Generation With Prescribed Timing

1992 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Angular Displacement ◽  
Solution Process ◽  
Original System ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Closed Form Solutions ◽  
Resultant Theory

Abstract Resultant theory is applied to derive closed-form solutions for the dimensional synthesis of linkage components for a finite number of precision positions for motion generation with prescribed timing. The solutions are in forms of polynomial equations of the exponential of a single unknown angular displacement. The degree of the derived polynomial depends on the number of links in the linkage component and the number of precision positions to be synthesized for, or the number of compatibility equations. The resultant theory is discussed in detail, and the procedure for the derivation of resultant polynomials is demonstrated. This paper shows that, for the case of two compatibility equations, the solution is a six-degree polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree polynomial. The Bernshtein formula is applied to check the exact number of solutions of the original system of polynomial equations and to verify the validity of the derived resultant polynomials. An algorithm is also proposed for screening out extra solutions which may be generated through the solution process.

TWELVE LIMIT CYCLES IN A CUBIC CASE OF THE 16th HILBERT PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 2191-2205 ◽  
Author(s):  
P. YU ◽  
M. HAN
Keyword(s):  
Limit Cycles ◽  
Multiple Scales ◽  
Hilbert Problem ◽  
Perturbation Technique ◽  
Local Limit ◽  
Polynomial Equations ◽  
Computationally Efficient ◽  
Degree Polynomial ◽  
Planar System ◽  
16Th Hilbert Problem

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

scholarly journals Finding Zeros of Quartic Polynomials by Using Radicals

2021 ◽  
Author(s):  
Sureyya Sahin
Keyword(s):  
Polynomial Equation ◽  
Lower Degree ◽  
Polynomial Equations ◽  
Single Variable ◽  
Degree Polynomial ◽  
Numerical Example ◽  
General Polynomial ◽  
Quartic Polynomial ◽  
Quartic Polynomials

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

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