Hyperbolic Quaternions Research Articles

Copper ratio obtained by generalizing the Fibonacci sequence

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

NestE: Modeling Nested Relational Structures for Knowledge Graph Reasoning

Reasoning with knowledge graphs (KGs) has primarily focused on triple-shaped facts. Recent advancements have been explored to enhance the semantics of these facts by incorporating more potent representations, such as hyper-relational facts. However, these approaches are limited to atomic facts, which describe a single piece of information. This paper extends beyond atomic facts and delves into nested facts, represented by quoted triples where subjects and objects are triples themselves (e.g., ((BarackObama, holds_position, President), succeed_by, (DonaldTrump, holds_position, President))). These nested facts enable the expression of complex semantics like situations over time and logical patterns} over entities and relations. In response, we introduce NestE, a novel KG embedding approach that captures the semantics of both atomic and nested factual knowledge. NestE represents each atomic fact as a 1*3 matrix, and each nested relation is modeled as a 3*3 matrix that rotates the 1*3 atomic fact matrix through matrix multiplication. Each element of the matrix is represented as a complex number in the generalized 4D hypercomplex space, including (spherical) quaternions, hyperbolic quaternions, and split-quaternions. Through thorough analysis, we demonstrate the embedding's efficacy in capturing diverse logical patterns over nested facts, surpassing the confines of first-order logic-like expressions. Our experimental results showcase NestE's significant performance gains over current baselines in triple prediction and conditional link prediction. The code and pre-trained models are open available at https://github.com/xiongbo010/NestE.

Generalization of the k-Leonardo sequence and their hyperbolic quaternions

In this study, we define the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences, and some terms of these sequences are given. Then, we obtain the generating functions, summation formulas, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, we examine the relations between the terms of the k-Leonardo, k-Leonardo-Lucas, Modified k-Leonardo, Leonardo, Leonardo-Lucas, Modified Leonardo, Francois, Fibonacci, and Lucas sequences. Moreover, we associate the terms of these sequences with matrices. Furthermore, we present on the application of these sequences to hyperbolic quaternions. For these quaternions, we give many properties such as Binet formulas. Finally, the terms of the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences are associated with their hyperbolic quaternion values.

On some Combinatorial Properties of Generalized Commutative Pell and Pell-Lucas Quaternions

Generalized commutative quaternions generalize elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper, we study generalized commutative Pell quaternions and generalized commutative Pell-Lucas quaternions. We present some properties of these numbers and relations between them.

Generalized commutative quaternion polynomials of the Fibonacci type

Generalized commutative quaternions is a number system which generalizes elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper we introduce and study generalized commutative quaternion polynomials of the Fibonacci type.

A generalization of complex, dual and hyperbolic quaternions: Hybrid quaternions

Hybrid numbers are a new non-commutative number system which is a generalization of the complex (i2 = ?1), dual (?2 = 0), and hyperbolic numbers (h2 = 1). In this article, firstly we define a new quaternion system called hybrid quaternions by taking the coefficients of real quaternions as hybrid numbers. This new quaternion system is a combination of complex quaternions (biquaternions), hyperbolic (perplex) quaternions, and dual quaternions, and it can be viewed as a generalization of these quaternion systems. Then, we present the basic properties of hybrid quaternions including fundamental operations, conjugates, inner product, vector product, and norm. Finally, we give a schematic representation of numbers and quaternions.

On the symmetrical second order hyperbolic quaternions sequences

On the symmetrical second order hyperbolic quaternions sequences

The polar form of hyperbolic commutative quaternions

The polar form of hyperbolic commutative quaternions

The Real Representation of Canonical Hyperbolic Quaternion Matrices and Its Applications

In this paper, we construct the real representation matrix of canonical hyperbolic quaternion matrices and give some properties in detail. Then, by means of the real representation, we study linear equations, the inverse and the generalized inverse of the canonical hyperbolic quaternion matrix and get some interesting results.

A note on hyperbolic quaternions

In this work, we introduce hyperbolic quaternions and their algebraic properties. Moreover, we express Euler's and De Moivre's formulas for hyperbolic quaternions.

Split-quaternionic Hermitian Clifford analysis

The theory of split-quaternion analysis and split-quaternion geometry is nowadays under full development. In this article, we extend quaternionic Hermitian Clifford analysis to the case of split-quaternions. The split-quaternionic Hermitian Dirac operator is introduced as a self-map of smooth functions defined in domains of with values in the tensor product of the hyperbolic quaternions , the split-quaternions and the Clifford algebra . It is defined by with being the basis of and denoting the twisted Hermitian Dirac operators in the split-quaternionic Clifford algebra whose definition is based on a delicate construction of the split-quaternionic Witt basis. We establish the counterpart of the Cauchy integral formula.

Hyperbolic Quaternion Formulation of Electromagnetism

In this paper, after presenting the hyperbolic quaternion formalism, an alternative representational method is proposed for the formulation of classical and generalized electromagnetism in the case of the existence of magnetic monopoles and massive photons. Maxwell’s equations and relevant field equations are derived in compact, simpler and elegant forms. These equations are compared with their vectorial, complex quaternionic, dual quaternionic and octonionic representations.

Impurity and quaternions in nonrelativistic scattering from a quantum memory

Models of quantum computing rely on transformations of the states of a quantum memory. We study mathematical aspects of a model proposed by Wu in which the memory state is changed via the scattering of incoming particles. This operation causes the memory content to deviate from a pure state, i.e. induces impurity. For nonrelativistic particles scattered from a two-state memory and sufficiently general interaction potentials in (1+1) dimensions, we express impurity in terms of quaternionic commutators. In this context, pure memory states correspond to null hyperbolic quaternions. In the case with point interactions, the scattering process amounts to appropriate rotations of quaternions in the frequency domain. Our work complements previous analyses by Margetis and Myers (2006 J. Phys. A 39 11567).

Circular and hyperbolic quaternions, octonions, and sedenions—Further results

A sedenion, a 16-dimensional hypercomplex number z = a + Σb n i n + Σc n ε n + di 0, where i n are the classical octonions, ε n are corresponding proper square roots of +1, and i 0 = i n ε n , has three conjugates and three norms, which when composed yield one real modulus. The numbers 1 2 (1 ± ε) generate a unique arithmetic containing zero divisors, idempotents, and multivalued identities. All three quaternionic subalgebras of the sedenions can be factored into polar forms involving multilevel exponents, and the geometry of these polar forms is explored. The conic quaternionic subalgebra is shown to have only four square roots of +1 and four square roots of −1.

Circular and hyperbolic quaternions, octonions, and sedenions

The complex number algebra can be extended to 4-, 8-, and 16-dimensional algebras, some of which are noncommutative and nonassociative. Various assumptions about the norm, including multiplicativity, motivate the arithmetic of these algebras. The 16-dimensional algebra contains nonreal square roots of +1, zero divisors, and nilpotents, and possesses a nonreal norm and a positive modulus. For some of these algebras, exponentials and logarithms are calculated, and striking multilevel exponential polar forms are developed.

Higher-dimensional Riemannian geometry and quaternion and octonion spaces

An eight-dimensional Riemannian geometry is shown to be the basis of a nonsymmetric theory of gravitation. A hyperbolic complex structure is imposed and the group structure is GL(8,R)→GL(4,R)⊗GL(4,R)⊇GL(4,R). Octonion and quaternion division algebras are used to represent geometrical quantities and spinors. A Lagrangian is constructed that is related to supersymmetry and supergravity theories. The group structure for a hyperbolic octonion scheme is GL(8,qH)→GL(4,OH)≂GL(4,qH) ⊗GL(4,qH)⊇GL(4,qH), while a simpler scheme based on hyperbolic quaternions is GL(8,C)→GL(4,qH)≂GL(4,C)⊗GL(4,C)⊇GL(4,C).