On a Theorem in Hypercomplex Numbers

1906 ◽  
Vol 26 (1) ◽  
pp. 48-50 ◽  
Author(s):  
J. H. Maclagan-Wedderburn
Keyword(s):  
Quaternion Algebra ◽  
Hypercomplex Numbers

Scheffers in the Mathematische Annalen, vol. xxxix., pp. 364–74, enunciates the following theorem:—If A is an algebra containing the quaternion algebra B as a subalgebra, and if A and B have the same modulus, A can be expressed in the form B C = A = C B, where C is a subalgebra of A every element of which is commutative with every element of B: in other words, if i1, i2, i3, i4 is a basis of B, it is possible to find an algebra C with the basis e1, e2, … ec, such that each of its elements is commutative with every element of B, and such that the elements eris(r = 1, 2, … c, S = 1, … 4) form a basis of A; and if a is the order of A, then a = 4c.

scholarly journals Quaternion Collaborative Filtering for Recommendation

2019 ◽  
Author(s):  
Shuai Zhang ◽  
Lina Yao ◽  
Lucas Vinh Tran ◽  
Aston Zhang ◽  
Yi Tay
Keyword(s):  
Collaborative Filtering ◽  
Quaternion Algebra ◽  
Wide Spectrum ◽  
Representation Learning ◽  
Real Space ◽  
Inner Product ◽  
Hypercomplex Numbers ◽  
Learning Capability ◽  
Inductive Bias ◽  
Real World Datasets

This paper proposes Quaternion Collaborative Filtering (QCF), a novel representation learning method for recommendation. Our proposed QCF relies on and exploits computation with Quaternion algebra, benefiting from the expressiveness and rich representation learning capability of Hamilton products. Quaternion representations, based on hypercomplex numbers, enable rich inter-latent dependencies between imaginary components. This encourages intricate relations to be captured when learning user-item interactions, serving as a strong inductive bias  as compared with the real-space inner product. All in all, we conduct extensive experiments on six real-world datasets, demonstrating the effectiveness of Quaternion algebra in recommender systems. The results exhibit that QCF outperforms a wide spectrum of strong neural baselines on all datasets. Ablative experiments confirm the effectiveness of Hamilton-based composition over multi-embedding composition in real space. 

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.

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