This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the i...
The article presents a parallel hardware-oriented algorithm designed to speed up the division of two octonions. The advantage of the proposed algorithm is that the number of real multiplications is halved as compared to the naive method for implement...
A succinct and systematic form of multiplication for any arbitrary pairs of octonions is devised. A typical expression of multiplication for any pair of octonions involves 64 terms, which, from the computational and theoretical aspect, is too cumbers...
This study investigates curves in a 7-dimensional space, represented by spatial generalized octonion-valued functions of a single variable, where the general octonions include real, split, semi, split semi, quasi, split quasi, and para octonions. We...
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection, error correction, data transmission, and data storage. Codes are studied...
This paper introduces definitions of the octonion cross Wigner distribution (OWD) and the octonion ambiguity function, forming a pair of octonion Fourier transforms. The main part is devoted to the study of the basic properties of the OWD. Among them...
Starting from the 2001 Thomas Friedrich’s work on
Knowledge representation learning achieves the automatic completion of knowledge graphs (KGs) by embedding entities into continuous low-dimensional vector space. In knowledge graph completion (KGC) tasks, the inter-dependencies and hierarchical infor...
In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inve...
In this paper, we investigate the geometric realization of
The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transf...
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 octo...
Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) ha...
We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation o...
This study addresses the preassigned-time (PDT) projective lag synchronization of octonion-valued BAM neural networks (OV-BAMNNs) through exponential quantized event-triggered control (ETC). First, an OV-BAMNN model incorporating discontinuous activa...
In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to tr...
In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we conside...
When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct pre-spacetime, pre-quantu...
We report a theoretical derivation of the Cabibbo–Kobayashi–Maskawa (CKM) matrix parameters and the accompanying mixing angles. These results are arrived at from the exceptional Jordan algebra applied to quark states, and from expressing...
In this essay, we immerse into the framework of normed division algebras as a suitable arena to accommodate the standard model of elementary particles, and we explore some applications to cosmology. Remarkably, they permit interesting non-trivial rea...
In this paper, the fixed points of involutions on the moduli space of principal
This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or power...
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras
This paper develops a unified synchronization framework for octonion-valued fractional-order neural networks (FOOVNNs) subject to mixed delays, Lévy disturbances, and topology switching. A fractional sliding surface is constructed by combining...
This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of le...
Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretica...
By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rota...
Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical...
Usually, it is supposed that irreversibility of time appears only in macrophysics. Here, we attempt to introduce the microphysical arrow of time assuming that at a fundamental level nature could be non-associative. Obtaining numerical results of a me...
The
In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement
This article spanned a new, consistent framework for production, archiving, and provision of analysis ready data (ARD) from multi-source and multi-temporal satellite acquisitions and an subsequent image fusion. The core of the image fusion was an ort...
Six-dimensional spinors with
Given a 2N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe t...