scholarly journals Some Fixed Points Results of Quadratic Functions in Split Quaternions

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Young Chel Kwun ◽  
Mobeen Munir ◽  
Waqas Nazeer ◽  
Shin Min Kang
Keyword(s):  
Fixed Points ◽  
Quadratic Polynomial ◽  
Quadratic Functions ◽  
Split Quaternion ◽  
Quadratic Polynomials ◽  
Split Quaternions

We attempt to find fixed points of a general quadratic polynomial in the algebra of split quaternion. In some cases, we characterize fixed points in terms of the coefficients of these polynomials and also give the cardinality of these points. As a consequence, we give some simple examples to strengthen the infinitude of these points in these cases. We also find the roots of quadratic polynomials as simple consequences.

scholarly journals Fixed Points Results in Algebras of Split Quaternion and Octonion

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 405 ◽  
Author(s):  
Mobeen Munir ◽  
Asim Naseem ◽  
Akhtar Rasool ◽  
Muhammad Saleem ◽  
Shin Kang
Keyword(s):  
Game Theory ◽  
Computer Science ◽  
Fixed Points ◽  
Quadratic Polynomial ◽  
Split Quaternion ◽  
Finite Algebras ◽  
Prime Fields

Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields Z p. Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field.

HOW IS THE DYNAMICS OF KÖNIG ITERATION FUNCTIONS AFFECTED BY THEIR ADDITIONAL FIXED POINTS?

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 327-334 ◽  
Author(s):  
V. DRAKOPOULOS
Keyword(s):  
Fixed Points ◽  
Rational Functions ◽  
Parameter Family ◽  
Quadratic Polynomials ◽  
Iteration Functions ◽  
Extraneous Fixed Points ◽  
Roots Of Equations ◽  
The One ◽  
Cubic Polynomials ◽  
Raphson Method

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

scholarly journals On the Consimilarity of Split Quaternions and Split Quaternion Matrices

2016 ◽  
Vol 24 (3) ◽  
pp. 189-207 ◽  
Author(s):  
Hidayet Hüda Kösal ◽  
Mahmut Akyiğit ◽  
Murat Tosun
Keyword(s):  
Split Quaternion ◽  
Solvability Conditions ◽  
Split Quaternions ◽  
General Solutions ◽  
Quaternion Matrices

AbstractIn this paper, we introduce the concept of consimilarity of split quaternions and split quaternion matrices. In his regard, we examine the solvability conditions and general solutions of the equationsandin split quaternions and split quaternion matrices, respectively. Moreover, coneigenvalue and coneigenvector are defined for split quaternion matrices. Some consequences are also presented.

scholarly journals ON STABLE QUADRATIC POLYNOMIALS

2012 ◽  
Vol 54 (2) ◽  
pp. 359-369 ◽  
Author(s):  
OMRAN AHMADI ◽  
FLORIAN LUCA ◽  
ALINA OSTAFE ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Quadratic Polynomial ◽  
Quadratic Polynomials ◽  
Polynomials Over Finite Fields ◽  
Characteristic 2 ◽  
The Stability ◽  
Almost All

AbstractWe recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

scholarly journals An Algorithm for the Factorization of Split Quaternion Polynomials

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Daniel F. Scharler ◽  
Hans-Peter Schröcker
Keyword(s):  
Hyperbolic Plane ◽  
Dual Quaternion ◽  
Real Polynomial ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Dual Quaternions ◽  
Factorization Theory ◽  
Real Polynomials ◽  
Theory Of Motion ◽  
Univariate Polynomials

AbstractWe present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

scholarly journals ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

2020 ◽  
pp. 423
Author(s):  
Muhammed Talat Sariaydin
Keyword(s):  
Mathematical Model ◽  
Minkowski Space ◽  
Bäcklund Transformations ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Backlund Transformations ◽  
Null Curves ◽  
Basic Concepts ◽  
The Mathematical Model ◽  
End Results

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.

scholarly journals The GCD property and irreduciable quadratic polynomials

1986 ◽  
Vol 9 (4) ◽  
pp. 749-752 ◽  
Author(s):  
Saroj Malik ◽  
Joe L. Mott ◽  
Muhammad Zafrullah
Keyword(s):  
Dedekind Domain ◽  
Quadratic Polynomial ◽  
Prime Element ◽  
Prüfer Domain ◽  
Quadratic Polynomials ◽  
Bezout Domain ◽  
Krull Domain

The proof of the following theorem is presented: IfDis, respectively, a Krull domain, a Dedekind domain, or a Prüfer domain, thenDis correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial inD[X]is a prime element.

scholarly journals Quaternionic Serret-Frenet Frames for Fuzzy Split Quaternion Numbers

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Cansel Yormaz ◽  
Simge Simsek ◽  
Serife Naz Elmas
Keyword(s):  
Natural Extension ◽  
Frenet Frame ◽  
Real Numbers ◽  
Split Quaternion ◽  
Geometric Properties ◽  
Split Quaternions ◽  
Fuzzy Real Numbers

We build the concept of fuzzy split quaternion numbers of a natural extension of fuzzy real numbers in this study. Then, we give some differential geometric properties of this fuzzy quaternion. Moreover, we construct the Frenet frame for fuzzy split quaternions. We investigate Serret-Frenet derivation formulas by using fuzzy quaternion numbers.

Sign in / Sign up

Export Citation Format

Share Document