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Journal articles on the topic 'Octonians'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

Goodaire, Edgar G., and Guilherme Leal. "Associative subalgebras of the octonians." Journal of Algebra and Its Applications 15, no. 10 (2016): 1650188. http://dx.doi.org/10.1142/s0219498816501887.

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In the mid 1970s, Michel Racine classified the maximal subalgebras of an octonian algebra. In this paper, we classify the maximal associative subalgebras. It turns out that there are four, up to isomorphism, all of dimension [Formula: see text]. In final sections, we apply our findings to investigate the groups that sit inside the Moufang loop of invertible elements of the split octonians and also to show that a well-known inequality of Jørgensen holds in a new context.

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2

Kalmbach HE, Gudrun. "Octonians and the EM-Compass." Acta Scientific Women's Health 3, no. 3 (2021): 57–60. http://dx.doi.org/10.31080/aswh.2020.03.0190.

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3

Kalmbach HE, Gudrun. "Octonians and the EM-Compass." Acta Scientific Women's Health 3, no. 3 (2021): 157–60. http://dx.doi.org/10.31080/aswh.2021.03.0190.

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4

Hashimoto, Hideya. "Oriented 6-dimensional submanifolds in the octonians, III." International Journal of Mathematics and Mathematical Sciences 18, no. 1 (1995): 111–20. http://dx.doi.org/10.1155/s0161171295000159.

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In this paper, we classify 6-dimensional almost Hermitian submanifolds in the octonians𝕆according to the classification introduced by A. Gray and L. Hervella. We give new examples of quasi-Käthler and∗-Einstein submanifolds in𝕆. Also, we prove that a 6-dimensional weakly∗-Einstein Hermitian submanifold in𝕆is totally geodesic.

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5

Kalmbach H.E., Gudrun. "Projective Gravity." International Journal of Contemporary Research and Review 9, no. 03 (2018): 20181–83. http://dx.doi.org/10.15520/ijcrr/2018/9/03/466.

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In [1] and [3] it was pointed out that octonians can replace an infinite dimensional Hilbert space and psi-waves descriptions concerning the states of deuteron which are finite in number. It is then clear that gravity needs projective and projection geometry to be described in a unified way with the three other basic forces of physics.

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6

Cerda-Morales, Gamaliel. "The Third Order Jacobsthal Octonions: Some Combinatorial Properties." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (2018): 57–72. http://dx.doi.org/10.2478/auom-2018-0033.

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AbstractVarious families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many di erent ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the third order Jacobsthal and third order Jacobsthal-Lucas numbers. We introduce the third order Jacobsthal octonions and the third order Jacobsthal-Lucas octonions and give some of their properties. We derive the relations between third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions.

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7

Gudrun Kalmbach HE. "Atomic kernels as waves and catastrophes." International Journal of Biological and Pharmaceutical Sciences Archive 1, no. 2 (2021): 062–67. http://dx.doi.org/10.30574/ijbpsa.2021.1.2.0020.

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The presentation of atomic kerrnels as particles requires for the physics duality principle that they get a wave description. This is due to presenting the SU (3) GellMann matrix space by octonians which are obtained by doubling the spacetime quaternions. Their multiplication table is different from the SU (3) matrices. The third presentation of this space is a complex 4-dimensional space where the real spacetime coordinates of a 4-dimensional Euclidean Hilbert space R4 are extended to C4. For getting from Deuteron Cooper pairs NP of a neutron and proton atomic kernels AK, the wave package superpositions for AK need the mass defect of neutrons where kg is changed to inner speeds or interaction energies. For kg octonians have a GF measuring base triple as Gleason operator. Using a projective geometrical norming, C4 is normed to CP³, a projective 3-dimensional space. Its cell C³ has spacetime coordinates C², extended by an Einstein energy plane z3 = (m,f), m mass, f = 1/∆t frequency where mass can be transformed into f by using mc² = hf. If C³ is presented as a real space R6, it can be real projective normed to a real projective space P5 for the field presentation of AK. As field the NP‘s have then a common group speed for AK wave packages superpositions with which AK moves in spacetime C² and also a presentation as a Ψ wave. As probability distribution where they can be energetically found in space serves Ψ* Ψ.

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8

CHANYAL, B. C., P. S. BISHT, and O. P. S. NEGI. "OCTONION AND CONSERVATION LAWS FOR DYONS." International Journal of Modern Physics A 28, no. 26 (2013): 1350125. http://dx.doi.org/10.1142/s0217751x1350125x.

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Starting with the usual definitions of octonions and split octonions in terms of Zorn vector matrix realization, we have made an attempt to write the continuity equation and other wave equations of dyons in split octonions. Accordingly, we have investigated the work energy theorem or "Poynting Theorem," Maxwell stress tensor and Lorentz invariant for generalized fields of dyons in split octonion electrodynamics. Our theory of dyons in split octonion formulations is discussed in term of simple and compact notations. This theory reproduces the dynamic of electric (magnetic) in the absence of magnetic (electric) charges.

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9

Rathore, Arun, and Bhupesh Chanyal. "A Representation of an Octonionic Interaction of Color Quarks with the Application of Feynman Diagram." American Journal of Modern Physics 14, no. 2 (2025): 44–51. https://doi.org/10.11648/j.ajmp.20251402.11.

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The SU(3) flavour symmetry for quarks and antiquarks has been demonstrated via the complexified octonion space, where the six complex octonion operators are essentially identical to the SL(3,C) group generators. It has been developed an extensive analysis of the quark flavour theory in the context of complex-octonion space by analyzing the connection between octonions and the SU(3) group. Therefore, it is argued that the extended theory of quark flavors, which preserves the property of non-commutativity, is the complexified variant of octonions. This theoretical model may be further extended to the SU(3) color symmetry, which is regarded as an exact symmetry. In this work, to gain a complete understanding of quark color theory in the framework of complex octonionic space, we have derived the relationship between octonions and the &lt;i&gt;SU(3)c&lt;/i&gt; color group. It has been studied that only eight possibilities of paired gluons are available to provide colorless states of hadrons in order to represent theoretically the octonion glueballs. With the help of Feynman diagrams, we examined the octonionic interaction of color quarks (such as quark-quark, quark anti-quark, and anti-quarks anti-quarks interactions). For the interactions, we have obtained the complex octonion algebraic form of the interaction term, propagator, vertex factor, and color factor. Most importantly, we have examined the conditions for valid and invalid interactions for the complex-octonion formalism.

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10

Vieira, Renata Passos Machado, Milena Carolina dos Santos Mangueira, Francisco Régis Vieira Alves, and Paula Maria Machado Cruz Catarino. "The Generalization of Gaussians and Leonardo’s Octonions." Annales Mathematicae Silesianae 37, no. 1 (2023): 117–37. http://dx.doi.org/10.2478/amsil-2023-0004.

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Abstract In order to explore the Leonardo sequence, the process of complex-ification of this sequence is carried out in this work. With this, the Gaussian and octonion numbers of the Leonardo sequence are presented. Also, the recurrence, generating function, Binet’s formula, and matrix form of Leonardo’s Gaussian and octonion numbers are defined. The development of the Gaussian numbers is performed from the insertion of the imaginary component i in the one-dimensional recurrence of the sequence. Regarding the octonions, the terms of the Leonardo sequence are presented in eight dimensions. Furthermore, the generalizations and inherent properties of Leonardo’s Gaussians and octonions are presented.

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Sürekçi, Arzu Cihan, and Mehmet Ali Güngör. "A Study on Commutative Elliptic Octonion Matrices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 30, no. 1 (2022): 151–69. http://dx.doi.org/10.2478/auom-2022-0009.

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Abstract In this study, firstly notions of similarity and consimilarity are given for commutative elliptic octonion matrices. Then the Kalman-Yakubovich s-conjugate equation is solved for the first conjugate of commutative elliptic octonions. Also, the notions of eigenvalue and eigenvector are studied for commutative elliptic octonion matrices. In this regard, the fundamental theorem of algebra and Gershgorin’s Theorem are proved for commutative elliptic octonion matrices. Finally, some examples related to our theorems are provided.

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12

Weng, Zi-Hua. "Forces in the complex octonion curved space." International Journal of Geometric Methods in Modern Physics 13, no. 06 (2016): 1650076. http://dx.doi.org/10.1142/s0219887816500766.

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The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

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13

Gudrun Kalmbach HE. "Centers and dimensional evolution." International Journal of Scholarly Research in Chemistry and Pharmacy 1, no. 1 (2022): 024–29. http://dx.doi.org/10.56781/ijsrcp.2022.1.1.0041.

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From astronomy, the author cites (from astronomy questions and answers by Michael Lam 2015): if a black hole is rotating, then it will be shaped as an oblate spheroid, slightly larger around the equator than in the direction of the poles. However, the equations of general relativity tell us that rather than having one radius, the location of the event horizon, there are two important radii, the spherical event horizon on the inside, and the oblate spheroidal exterior surface. The region in between the two is called the ergosphere, where particles cannot remain at rest and objects can still escape the black hole. Such a black hole looks like this: … The artists figure is not repeated. There are different subsystems of a black hole mentioned and general relativity is quoted. It is assumed that from a decaying black holes collision or explosion the astronomers universe evolved. In this article a biological Feigenbaum evolution is presented with geometries and symmetries added. Octonians are needed for this presentation.

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14

Kim, Ji-Eun. "Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field." Axioms 13, no. 5 (2024): 291. http://dx.doi.org/10.3390/axioms13050291.

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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 treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.

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15

Gudrun, Kalmbach HE. "Interactions, space presentations, blocks and cross products." GSC Advanced Research and Reviews 06, no. 02 (2021): 061–73. https://doi.org/10.5281/zenodo.4607838.

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Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex  4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.

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16

Cariow, Aleksandr, and Janusz P. Paplinski. "A Parallel Algorithm for Dividing Octonions." Algorithms 14, no. 11 (2021): 309. http://dx.doi.org/10.3390/a14110309.

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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 implementing this operation. In the synthesis of the discussed algorithm, the matrix representation of this operation was used, which allows us to present the division of octonions by means of a vector–matrix product. Taking into account a specific structure of the matrix multiplicand allows for reducing the number of real multiplications necessary for the execution of the octonion division procedure.

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17

Chen, Ray-Ming. "Computational Characterization of the Multiplication Operation of Octonions via Algebraic Approaches." Mathematics 12, no. 8 (2024): 1262. http://dx.doi.org/10.3390/math12081262.

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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 cumbersome. In addition, its internal relation could not be directly visualized via the expression per se. In this article, we study the internal structures of the indexes between imaginary unit octonions. It is then revealed by various copies of isomorphic structures for the multiplication. We isolate one copy and define a multiplicative structure on this. By doing so, we could keep track of all relations between indexes and the signs for cyclic permutations. The final form of our device is expressed in the form of a series of determinants, which shall offer some direct intuition about octonion multiplication and facilitate the further computational aspect of applications.

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18

Weng, Zi-Hua. "Precessional angular velocity and field strength in the complex octonion space." International Journal of Geometric Methods in Modern Physics 17, no. 07 (2020): 2050091. http://dx.doi.org/10.1142/s0219887820500917.

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The paper aims to apply the octonions to explore the precessional angular velocities of several particles in the electromagnetic and gravitational fields. Some scholars utilize the octonions to research the electromagnetic and gravitational fields. One formula can be derived from the octonion torque, calculating the precessional angular velocity generated by the gyroscopic torque. When the octonion force is equal to zero, it is able to deduce the force equilibrium equation and precession equilibrium equation and so forth. From the force equilibrium equation, one can infer the angular velocity of revolution for the particles. Meanwhile, from the precession equilibrium equation, it is capable of ascertaining the precessional angular velocity induced by the torque derivative, including the angular velocity of Larmor precession. Especially, some ingredients of torque derivative are in direct proportion to the field strengths. The study reveals that the precessional angular velocity induced by the torque derivative is independent of that generated by the torque. The precessional angular velocity, induced by the torque derivative, is relevant to the torque derivative and the spatial dimension of precessional velocity. It will be of great benefit to understanding further the precessional angular velocity of the spin angular momentum.

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19

Weng, Zi-Hua. "Basic postulates of some coordinate transformations within material media." AIP Advances 12, no. 12 (2022): 125016. http://dx.doi.org/10.1063/5.0132194.

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This paper aims to explore the physical quantities of several invariants, including the basic postulates of some types of crucial coordinate transformations, conservation laws, and continuity equations, in the electromagnetic and gravitational fields. Maxwell first utilized quaternions to describe the electromagnetic theory. Subsequent scholars make use of octonions to study the physical properties of electromagnetic and gravitational fields simultaneously, including the octonion field strength, field source, angular momentum, torque, and force. When an octonion coordinate system transforms rotationally, the scalar part of one octonion will remain unchanged, although the vector part of the octonion may alter. In the octonion space [Formula: see text], some invariants can be derived from this octonion property. A part of these invariants can be selected as the basic postulates of Galilean transformation or Lorentz transformation. Similarly, several invariants can be derived from the octonion property in the transformed octonion space [Formula: see text], and the invariants can be chosen as the basic postulates of a few new types of coordinate transformations. Furthermore, the combination of invariants in the octonion spaces can be applied as the basic postulates of some new coordinate transformations, relevant to the norm of physical quantities. Through the analysis, it is easy to find that each conserved quantity has its preconditions from the perspective of octonion spaces. This is helpful to deepen the further understanding of the physical properties of conservation laws and other invariants.

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20

Anco, Stephen C., Philic Lam, and Thomas Wolf. "A search for integrable evolution equations with Lax pairs over the octonions." Journal of Physics: Conference Series 2912, no. 1 (2024): 012045. https://doi.org/10.1088/1742-6596/2912/1/012045.

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Abstract Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV scaling weights. A condition for linear differential operators to be a Lax pair over octonions is formulated and solved for the unknown coefficients in the polynomials.

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21

Serôdio, Rogério, Patricia Beites, and José Vitória. "Eigenvalues of matrices related to the octonions." 4open 2 (2019): 16. http://dx.doi.org/10.1051/fopen/2019014.

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A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.

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22

Forrester, Peter J. "Octonions in random matrix theory." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (2017): 20160800. http://dx.doi.org/10.1098/rspa.2016.0800.

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The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N =2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N =3. We then proceed to consider the matrix structure X † X , when X has random octonion entries. Analytic results are obtained from N =2, but are observed to break down in the 3×3 case.

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23

Kişi, Ömer, Selim Çetin, and Mehmet Gürdal. "Generalized Statistical Convergence via Modulus Function in Octonion Valued b-Metric Spaces." Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi 20, no. 1 (2025): 103–25. https://doi.org/10.29233/sdufeffd.1643711.

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The key ideas of summability theory have been the subject of extensive investigation in recent years in a variety of metric space extensions. Octonion-valued metric spaces are based on modifying the triangle inequality of a semi-metric space by multiplying one side of the inequality by a scalar b. This new generalisation of metric spaces is very interesting since octonions are not even a ring since they do not have the associative property of multiplication and the spaces do not satisfy the standard triangle inequality. We are prompted by this to study the notions of strong I-Cesàro summability, I-statistical convergence, I-lacunary statistical convergence, and similar notions that respect the modulus function in octonion valued b-metric spaces, an extension of metric spaces. We also examine the connections among these ideas.

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24

Ibrayev, А. Т. "METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION." PHYSICO-MATHEMATICAL SERIES 6, no. 334 (2020): 5–12. http://dx.doi.org/10.32014/2020.2518-1726.91.

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In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

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Gogberashvili, Merab, and Otari Sakhelashvili. "Geometrical Applications of Split Octonions." Advances in Mathematical Physics 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/196708.

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It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie groupG2. This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how theG2algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.

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26

Weng, Zi-Hua. "Contrastive analysis of two energy gradients in the ultra-strong magnetic fields." International Journal of Modern Physics A 33, no. 35 (2018): 1850212. http://dx.doi.org/10.1142/s0217751x18502123.

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The paper aims to apply the complex-octonions to explore the variable gravitational mass and energy gradient of several particles in the external ultra-strong magnetic fields. J. C. Maxwell was the first to introduce the algebra of quaternions to study the physical properties of electromagnetic fields. Some scholars follow up this method in the field theories. Nowadays, they employ the complex-octonions to analyze simultaneously the physical quantities of electromagnetic and gravitational fields, including the field potential, field strength, field source, linear momentum, angular momentum, torque, and force. When the octonion force is equal to zero, it is able to deduce eight independent equilibrium equations, especially the force equilibrium equation, precessional equilibrium equation, mass continuity equation, and current continuity equation. In the force equilibrium equation, the gravitational mass is variable. The gravitational mass is the sum of the inertial mass and a few tiny terms. These tiny terms will be varied with not only the fluctuation of field strength and of potential energy, but also the spatial dimension of velocity. The study reveals that it is comparatively untoward to attempt to measure directly the variation of these tiny terms of gravitational mass in the ultra-strong magnetic field. However it is not such difficult to measure the energy gradient relevant to the variation of these tiny terms of gravitational mass. In the complex-octonion space, the gravitational mass is a sort of variable physical quantity, rather than an intrinsic property of any physical object. And this inference is accordant with the academic thought of “the mass is not an intrinsic property any more” in the unified electroweak theory.

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Weng, Zi-Hua. "Spin angular momentum of proton spin puzzle in complex octonion spaces." International Journal of Geometric Methods in Modern Physics 14, no. 07 (2017): 1750102. http://dx.doi.org/10.1142/s021988781750102x.

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The paper focuses on considering some special precessional motions as the spin motions, separating the octonion angular momentum of a proton into six components, elucidating the proton angular momentum in the proton spin puzzle, especially the proton spin, decomposition, quarks and gluons, and polarization and so forth. Maxwell was the first to use the quaternions to study the electromagnetic fields. Subsequently the complex octonions are utilized to depict the electromagnetic field, gravitational field, and quantum mechanics and so forth. In the complex octonion space, the precessional equilibrium equation infers the angular velocity of precession. The external electromagnetic strength may induce a new precessional motion, generating a new term of angular momentum, even if the orbital angular momentum is zero. This new term of angular momentum can be regarded as the spin angular momentum, and its angular velocity of precession is different from the angular velocity of revolution. The study reveals that the angular momentum of the proton must be separated into more components than ever before. In the proton spin puzzle, the orbital angular momentum and magnetic dipole moment are independent of each other, and they should be measured and calculated respectively.

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CASTRO, CARLOS. "ON OCTONIONIC GRAVITY, EXCEPTIONAL JORDAN STRINGS AND NONASSOCIATIVE TERNARY GAUGE FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 09, no. 03 (2012): 1250021. http://dx.doi.org/10.1142/s0219887812500211.

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Novel nonassociative octonionic ternary gauge field theories are proposed based on a ternary bracket. This paves the way to the many physical applications of exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A real-valued analog of the Einstein–Hilbert Lagrangian [Formula: see text] involving sums of all the possible contractions of the Ricci tensors plus their octonionic-complex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also re-examined. Finally, we construct (to our knowledge) generalized novel octonionic string and p-brane actions and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of 7,8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, black-holes in string theory and quantum information theory.

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Yamaç Akbıyık, Seda. "On Matrices of Generalized Octonions (Cayley Numbers)." Symmetry 16, no. 12 (2024): 1567. http://dx.doi.org/10.3390/sym16121567.

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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.

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Akbıyık, Mücahit. "On Powers and Roots of Split Octonions." Journal of Mathematics 2023 (January 25, 2023): 1–25. http://dx.doi.org/10.1155/2023/4500295.

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In this article, we obtain the polar forms for two types of split octonions. We calculate De Moivre’s formulas for all polar forms of split octonions. Thus, we give the n t h − powers and roots of split octonions and the matrix representation of split octonions. In addition, we present an illustrative example with Matlab codes.

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31

UYSAL, MINE, MUNESH KUMARI, BAHAR KULOĞLU, KALIKA PRASAD, and ENGIN ÖZKAN. "On The Hyperbolic k-Mersenne And k-Mersenne-Lucas Octonions." Kragujevac Journal of Mathematics 49, no. 5 (2024): 765–79. http://dx.doi.org/10.46793/kgjmat2505.765u.

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In this paper, we introduce the hyperbolic k-Mersenne and k-Mersenne-Lucas octonions and investigate their algebraic properties. We give Binet’s formula and present several interrelations and some well-known identities such as Catalan identity, d’Ocagne identity, Vajda identity, generating functions, etc. of these octonions in closed form. Furthermore, we investigate the relations between hyperbolic k-Mersenne octonions and hyperbolic k-Mersenne-Lucas octonions.

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32

Kalauni, Pushpa, and J. C. A. Barata. "Role of division algebra in seven-dimensional gauge theory." Modern Physics Letters A 30, no. 10 (2015): 1550047. http://dx.doi.org/10.1142/s0217732315500479.

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The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

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Baez, John C. "The octonions." Bulletin of the American Mathematical Society 39, no. 02 (2001): 145–206. http://dx.doi.org/10.1090/s0273-0979-01-00934-x.

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Bremner, Murray. "Quantum octonions." Communications in Algebra 27, no. 6 (1999): 2809–31. http://dx.doi.org/10.1080/00927879908826594.

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35

Karataş, Adnan, and Serpil Halici. "Horadam Octonions." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 3 (2017): 97–106. http://dx.doi.org/10.1515/auom-2017-0038.

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Abstract In this paper, first we define Horadam octonions by Horadam sequence which is a generalization of second order recurrence relations. Also, we give some fundamental properties involving the elements of that sequence. Then, we obtain their Binet-like formula, ordinary generating function and Cassini identity.

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36

Tokeşer, Ümit, Tuğba Mert, and Yakup Dündar. "Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions." AIMS Mathematics 7, no. 5 (2022): 8645–53. http://dx.doi.org/10.3934/math.2022483.

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<abstract><p>In this paper, we introduce split dual Fibonacci and split dual Lucas octonions over the algebra $ \widetilde{\widetilde{O}}\left(a, b, c\right) $, where $ a, b $ and $ c $ are real numbers. We obtain Binet formulas for these octonions. Also, we give many identities and Vajda theorems for split dual Fibonacci and split dual Lucas octonions including Catalan's identity, Cassini's identity and d'Ocagne's identity.</p></abstract>

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37

Gudrun Kalmbach HE. "Interactions, space presentations, blocks and cross products." GSC Advanced Research and Reviews 6, no. 2 (2021): 061–73. http://dx.doi.org/10.30574/gscarr.2021.6.2.0012.

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Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex 4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.

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Godase, A. D. "Some new properties of hyperbolic k-Fibonacci and k-Lucas octonions." Notes on Number Theory and Discrete Mathematics 30, no. 1 (2024): 100–110. http://dx.doi.org/10.7546/nntdm.2024.30.1.100-110.

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The aim of this paper is to establish some novel identities for hyperbolic k-Fibonacci octonions and k-Lucas octonions. We prove these properties using the identities of k-Fibonacci and k-Lucas numbers, which we determined previously.

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39

Ahmed, W. E. "Powers of Octonions." Applied Mathematics 12, no. 02 (2021): 75–84. http://dx.doi.org/10.4236/am.2021.122006.

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40

Goddard, P., W. Nahm, D. I. Olive, H. Ruegg, and A. Schwimmer. "Fermions and octonions." Communications in Mathematical Physics 112, no. 3 (1987): 385–408. http://dx.doi.org/10.1007/bf01218483.

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41

Elduque, Alberto. "Gradings on Octonions." Journal of Algebra 207, no. 1 (1998): 342–54. http://dx.doi.org/10.1006/jabr.1998.7474.

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42

Keçilioğlu, Osman, and Ilker Akkus. "The Fibonacci Octonions." Advances in Applied Clifford Algebras 25, no. 1 (2014): 151–58. http://dx.doi.org/10.1007/s00006-014-0468-y.

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Lounesto, Pertti. "Octonions and triality." Advances in Applied Clifford Algebras 11, no. 2 (2001): 191–213. http://dx.doi.org/10.1007/bf03042312.

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GOGBERASHVILI, MERAB. "OCTONIONIC VERSION OF DIRAC EQUATIONS." International Journal of Modern Physics A 21, no. 17 (2006): 3513–23. http://dx.doi.org/10.1142/s0217751x06028436.

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It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that are equivalent to the standard Dirac equations. In our approach the particle mass and electromagnetic potentials are part of an octonionic gradient function together with the space–time derivatives. As distinct from previous attempts to translate the Dirac equations into different number systems here the wave functions are real split octonions and not bi-spinors. To formulate positively defined probability amplitudes four different split octonions (transforming into each other by discrete transformations) are necessary, rather then two complex wave functions which correspond to particles and antiparticles in usual Dirac theory.

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Bilgici, Göksal, and Ahmet Daşdemir. "Some unrestricted Fibonacci and Lucas hyper-complex numbers." Acta et Commentationes Universitatis Tartuensis de Mathematica 24, no. 1 (2020): 37–48. http://dx.doi.org/10.12697/acutm.2020.24.03.

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A number of studies have investigated the Fibonacci quaternions and octonions that include consecutive terms of the Fibonacci sequence. This paper presents a new generalization of Fibonacci quaternions, octonions and sedenions, where non-consecutive Fibonacci numbers are used. We present the Binet formulas, generating functions and some identities for these new types of hyper-complex numbers.

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46

Krasnov, Kirill. "Spin(11, 3), particles, and octonions." Journal of Mathematical Physics 63, no. 3 (2022): 031701. http://dx.doi.org/10.1063/5.0070058.

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The fermionic fields of one generation of the Standard Model (SM), including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with [Formula: see text], where [Formula: see text] are the usual and split octonions, respectively. It is then well known that choosing a unit imaginary octonion [Formula: see text] equips [Formula: see text] with a complex structure J. Similarly, choosing a unit imaginary split octonion [Formula: see text] equips [Formula: see text] with a complex structure [Formula: see text], except that there are now two inequivalent complex structures, one parameterized by a choice of a timelike and the other of a spacelike unit [Formula: see text]. In either case, the identification [Formula: see text] implies that there are two natural commuting complex structures [Formula: see text] on S+. Our main new observation is that the subgroup of Spin(11, 3) that commutes with both [Formula: see text] on S+ is the direct product Spin(6) × Spin(4) × Spin(1, 3) of the Pati–Salam and Lorentz groups, when [Formula: see text] is chosen to be timelike. The splitting of S+ into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S+ into eigenspaces of [Formula: see text] corresponds to splitting of Lorentz Dirac spinors into two different chiralities. This provides an efficient bookkeeping in which particles are identified with components of such an elegant structure as [Formula: see text]. We also study the simplest possible symmetry breaking scenario with the “Higgs” field taking values in the representation that corresponds to three-forms in [Formula: see text]. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM and thus breaks Spin(11, 3) down to the product of the SM, Lorentz, and U(1) B− L groups, with the last one remaining unbroken. This three-form Higgs field also produces the Dirac mass terms for all the particles.

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Bilge, Ayşe Hümeyra, Tekin Dereli, and Şahin Kocak. "A trip around octonions." Journal of Physics: Conference Series 2191, no. 1 (2022): 012006. http://dx.doi.org/10.1088/1742-6596/2191/1/012006.

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Abstract In these expository notes, after a contemplation on the dawn of octonions, we give proofs for the Frobenius theorem and the Hurwitz theorem, we review the basics of Clifford algebras and spin groups, and exemplify the startling role played by the octonions in 7 — and 8 —dimensional phenomena such as the special 3— and 4—forms, the Bonan form, Spin(7) and Spin(8) groups and the mysterious triality.

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Kim, Myeung Su, Yeonjun Kim, Jae-Hyouk Lee, Seungbum Nam, and Kiho Park. "Multiplication of integral octonions." Journal of Algebra and Its Applications 15, no. 08 (2016): 1650144. http://dx.doi.org/10.1142/s0219498816501449.

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The integral subsets of octonions are an analog of integers in real numbers and related to many interesting topics in geometry and physics via [Formula: see text]-lattices. In this paper, we study the properties of the multiplication of the integral subsets of octonions by studying configuration of Fano plane via blocks and operations on them. And we show that the integral subsets are integral indeed by introducing new and elementary methods.

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BOYA, LUIS J., and RUTWIG CAMPOAMOR-STURSBERG. "COMPOSITION ALGEBRAS AND THE TWO FACES OF G2." International Journal of Geometric Methods in Modern Physics 07, no. 03 (2010): 367–78. http://dx.doi.org/10.1142/s0219887810004348.

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We consider composition and division algebras over the real numbers: We note two rôles for the group G2: as automorphism group of the octonions and as the isotropy group of a generic three-form in seven dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.

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Kılıç, Nayil. "On dual Horadam octonions." Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 137–49. http://dx.doi.org/10.7546/nntdm.2019.25.1.137-149.

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