Search Results (12)

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 inner product and vector parts and give the polar forms for lightlike generalized octonions. Furthermore, the matrix representations of the generalized octonions are given and some properties of these representations are achieved. Also, powers and roots of the matrix representations are presented. All calculations in the article are achieved by using MATLAB R2023a and these codes are presented with an illustrative example. Full article

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 consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)$\otimes$SU(2)$\otimes$U(1) symmetry. Full article

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) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example. Full article

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 left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions $\mathbb{O}=\mathbb{C}\oplus {\mathbb{C}}^3$, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space $\mathcal{S}$ of $C{\ell }_{10}$ Majorana spinors is generated by the $C{\ell }_6(\subset C{\ell }_{10})$ volume form, ${\omega }_6={\gamma }_1\cdots {\gamma }_6$, and is left invariant by the Pati–Salam subgroup of $Spin\left(10\right)$, $G_{\mathrm{PS}}=Spin\left(4\right)\times Spin\left(6\right)/{\mathbb{Z}}_2$. While the $Spin\left(10\right)$ invariant volume form ${\omega }_{10}={\gamma }_1\dots {\gamma }_{10}$ of $C{\ell }_{10}$ is known to split $\mathcal{S}$ on a complex basis into left and right chiral (semi)spinors, $\mathcal{P}=\frac{1}{2}(1-i{\omega }_6)$ is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of $G_{\mathrm{PS}}$ that preserves the sterile neutrino (which is identified with the Fock vacuum). The ${\mathbb{Z}}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product $\mathcal{A}\subset \mathcal{P}C{\ell }_{10}\mathcal{P}=C{\ell }_4\otimes C{\ell }_6^0$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part $C{\ell }_4^1$ of the first factor. The fact that the projection of $C{\ell }_{10}$ only involves the even part $C{\ell }_6^0$ of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle. Full article

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 transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform ($\mathbb{O}-$SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed ($\mathbb{O}-$SAFT). Afterwards, we generalize several uncertainty relations for the ($\mathbb{O}-$SAFT) which include Pitt’s inequality, Heisenberg–Weyl inequality, logarithmic uncertainty inequality, Hausdorff–Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform. Full article

Six-dimensional spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati–Salam, $SU\left(5\right)$, and $Spin\left(10\right)$ grand unified theories are found when a single generation is isolated. For spacetime symmetries, $Spin(4,2)$ may be used for conformal symmetry, $Ad{S}_5\to d{S}_4$, or simply broken to $Spin(3,1)$ of a Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $Ad{S}_3$ with various GUTs. An action for three generations of fermions in the Majorana–Weyl spinor $\mathbf{128}$ of $Spin(4,12)$ is found with $Spin\left(3\right)$ flavor symmetry inside $E_{8(-24)}$. The 128 of $Spin(12,4)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{8(-24)}/Spin(12,4)=\left(\mathbb{O}sx\mathbb{O}\right){\mathbb{P}}^2$. Full article

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 measurement, which requires at least three ingredients: object, device and observer, in the non-associative case depends on ordering of operations and is ambiguous. We show that use of octonions as a fundamental algebra, in any measurement, leads to generation of unavoidable 18.6 bit relative entropy of the probability density functions of the active and passive transformations, which correspond to the groups G2 and SO(7), respectively. This algebraical entropy can be used to determine the arrow of time, analogically as thermodynamic entropy does. Full article

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-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with $SU{\left(2\right)}_R$ chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time. Full article

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 information in KGs have gained attention. Existing methods do not well capture the latent dependencies between all components of entities and relations. To address this, we introduce the mathematical theories of octonion, a more expressive generalized form of complex number and quaternion, and propose a translation-based KGC model with octonion (TransO). TransO models entities as octonion coordinate vectors, relations as the combination of octonion component matrices and coordinate vectors, and uses specific grouping calculation rules to interact between entities and relations. In addition, since hyperbolic Poincaré space in non-Euclidean mathematics can represent hierarchical data more accurately and effectively than traditional Euclidean space, we propose a Poincaré-extended TransO model (PTransO). PTransO transforms octonion coordinate vectors into hyperbolic embeddings by exponential mapping, and integrates the Euclidean-based calculations into hyperbolic space by operations such as Möbius addition and hyperbolic distance. The experimental results of link prediction indicate that TransO outperforms other translation-based models on the WN18 benchmark, and PTransO further achieves state-of-the-art performance in low-dimensional space on the well-established WN18RR and FB15k-237 benchmarks. Full article

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 ${\mathbb{Z}}_p$. 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. Full article

Given a 2^N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units e_a , 1 ≤ a ≤ 2^N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {e_a, e_b , e_c} , 1 ≤ a < b < c ≤ 2^N - 1 and e_ae_b = ±e_c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C_N; in particular, C₃ (octonions) is isomorphic to the Pasch (6₂, 4₃) -configuration, C₄ (sedenions) is the famous Desargues (10₃) -configuration, C₅ (32-nions) coincides with the Cayley-Salmon (15₄, 20₃) -configuration found in the well-known Pascal mystic hexagram and C₆ (64-nions) is identical with a particular (21₅, 35₃) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C_N leads to a conjecture that C_N is isomorphic to a combinatorial Grassmannian of type G₂(N + 1). Full article

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 construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature. Full article