Octonion Algebra Multiplication Tables

2007 ◽  
Keyword(s):  
Octonion Algebra ◽  
Multiplication Tables

scholarly journals An octonion algebra originating in combinatorics

2010 ◽  
Vol 138 (12) ◽  
pp. 4187-4187
Author(s):  
Dragomir Ž. Đoković ◽  
Kaiming Zhao
Keyword(s):  
Octonion Algebra

Latin squares from multiplication tables

2021 ◽  
Author(s):  
Michał Dębski ◽  
Jarosław Grytczuk
Keyword(s):  
Latin Squares ◽  
Multiplication Tables

scholarly journals Trace dynamics and division algebras: towards quantum gravity and unification

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Lie Group ◽  
Planck Scale ◽  
The Self ◽  
Division Algebras ◽  
Quantum Field ◽  
Octonion Algebra ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Massless Spin ◽  
Correct Understanding

AbstractWe have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G2, embedded in the exceptional Lie group F4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

scholarly journals On the Complexity of Some Problems on Groups Input as Multiplication Tables

2001 ◽  
Vol 63 (2) ◽  
pp. 186-200 ◽  
Author(s):  
David Mix Barrington ◽  
Peter Kadau ◽  
Klaus-Jörn Lange ◽  
Pierre McKenzie
Keyword(s):  
Multiplication Tables

On the structure of the split octonion algebra

1990 ◽  
Vol 105 (1) ◽  
pp. 31-41 ◽  
Author(s):  
P. L. Nash
Keyword(s):  
Octonion Algebra

Arithmetical Advice

2021 ◽  
pp. 1-6
Author(s):  
Trevor Davis Lipscombe
Keyword(s):  
High Speed ◽  
Multiplication Tables ◽  
Speed Up

This chapter presents advice on how to avoid simple mistakes when performing mental calculations at high speed. It includes a method to speed up the rate at which you recite your multiplication tables. This can save fractions of a second, which, in an exam with many such multiplications, can be crucial. It urges neat handwriting, and shows the superfluity of zeros at the end, or decimal points in the middle of a number, provided you make estimates before calculating an answer. It presents a quick look at factors, which can slash seconds from the time it take to multiply and divide, and introduces the art of shunting.

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