Some properties of the norm in a division quaternion algebra

2021 ◽  
Author(s):  
Cristina Flaut ◽  
Diana Savin
Keyword(s):  
Quaternion Algebra

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

On Fibonacci spinors

2020 ◽  
Vol 17 (04) ◽  
pp. 2050065
Author(s):  
Tülay Eri̇şi̇r ◽  
Mehmet Ali̇ Güngör
Keyword(s):  
Quaternion Algebra ◽  
Quaternion Matrix ◽  
Spinor Structure

Spinors are used in physics quite extensively. Basically, the forms of use include Dirac four-spinors, Pauli three-spinors and quaternions. Quaternions in mathematics are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Fibonacci spinors using the Fibonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini formulas which are given for some series of numbers in mathematics for Fibonacci spinors.

scholarly journals Rank one sheaves over quaternion algebras on Enriques surfaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fabian Reede
Keyword(s):  
Quaternion Algebra ◽  
Double Cover ◽  
Complex Numbers ◽  
K3 Surface ◽  
Enriques Surface ◽  
Quaternion Algebras ◽  
Enriques Surfaces ◽  
Rank One ◽  
Torsion Free

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

New Set of Invariant Quaternion Krawtchouk Moments for Color Image Representation and Recognition

2021 ◽  
pp. 2250037
Author(s):  
Gaber Hassan ◽  
Khalid M. Hosny ◽  
R. M. Farouk ◽  
Ahmed M. Alzohairy
Keyword(s):  
Quaternion Algebra ◽  
Color Image ◽  
Image Representation ◽  
Color Images ◽  
Color Channel ◽  
Krawtchouk Polynomials ◽  
Krawtchouk Moments ◽  
Similarity Transformations ◽  
The Stability ◽  
Discrete Moments

One of the most often used techniques to represent color images is quaternion algebra. This study introduces the quaternion Krawtchouk moments, QKrMs, as a new set of moments to represent color images. Krawtchouk moments (KrMs) represent one type of discrete moments. QKrMs use traditional Krawtchouk moments of each color channel to describe color images. This new set of moments is defined by using orthogonal polynomials called the Krawtchouk polynomials. The stability against the translation, rotation, and scaling transformations for QKrMs is discussed. The performance of the proposed QKrMs is evaluated against other discrete quaternion moments for image reconstruction capability, toughness against various types of noise, invariance to similarity transformations, color face image recognition, and CPU elapsed times.

scholarly journals A Method of Robot Base Frame Calibration by Using Dual Quaternion Algebra

IEEE Access ◽  
2018 ◽  
Vol 6 ◽  
pp. 74865-74873
Author(s):  
Gang Wang ◽  
Xiaoping Liu ◽  
Song Han
Keyword(s):  
Quaternion Algebra ◽  
Dual Quaternion

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

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