scholarly journals Geometric Algebra Framework Applied to Circuits with Non-sinusoidal Voltages and Currents

2021 ◽  
Author(s):  
Jan Cieśliński ◽  
Cezary Walczyk
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Geometric Algebra ◽  
First Principles ◽  
Theoretical Physics ◽  
Electrical Circuits ◽  
Algebra Approach ◽  
Physical Components

We apply a well known technique of theoretical physics, known as Geometric Algebra or Clifford algebra, to linear electrical circuits with non-sinusoidal voltages and currents. We rederive from the first principles the Geometric Algebra approach to the apparent power decomposition. The important new point consists in a choice of a natural convenient basis in the Clifford vector space which simplifies considerably the presentation. Thus we are able to derive a number of general results which are missing in the former papers. In particular, a natural correspondence with the Current Physical Components approach is shown.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

scholarly journals Geometric Algebra Framework Applied to Symmetrical Balanced Three-Phase Systems for Sinusoidal and Non-Sinusoidal Voltage Supply

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1259
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco Manuel Arrabal-Campos ◽  
Javier Roldán Roldán Pérez
Keyword(s):  
Geometric Algebra ◽  
Dimensional Euclidean Space ◽  
Active Components ◽  
Electrical Circuits ◽  
Current Harmonics ◽  
Operation Conditions ◽  
Multiple Phase ◽  
Three Phase ◽  
Phase Systems ◽  
Definition Of

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered.

scholarly journals Clifford algebra approach of 3D Ising model

2018 ◽  
Vol 29 (1) ◽  
Author(s):  
Zhidong Zhang ◽  
Osamu Suzuki ◽  
Norman H. March
Keyword(s):  
Ising Model ◽  
Clifford Algebra ◽  
Algebra Approach ◽  
3D Ising Model

Feature preserving multiresolution subdivision and simplification of point clouds: A conformal geometric algebra approach

2017 ◽  
Vol 41 (11) ◽  
pp. 4074-4087 ◽  
Author(s):  
Shuai Yuan ◽  
Shuai Zhu ◽  
Dong-Shuang Li ◽  
Wen Luo ◽  
Zhao-Yuan Yu ◽  
...  
Keyword(s):  
Geometric Algebra ◽  
Point Clouds ◽  
Conformal Geometric Algebra ◽  
Algebra Approach ◽  
Feature Preserving

scholarly journals THE PROCEDURE OF THE SLIP MODELS OF POLYCRYSTALLINE PLASTICITY IN THE GENERAL CASES

1992 ◽  
Vol 14 (2) ◽  
pp. 13-16
Author(s):  
Bui Huu Dan
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Strain State ◽  
Stress And Strain ◽  
Slip Model ◽  
Dimension Vector ◽  
Computation Procedure ◽  
Five Dimension ◽  
The Many

The computation procedure of the slip model of polycrystalline plasticity was extended for the most general cases, when the stress and strain state are expressed in the five-dimension vector space this extent ion is based on the know ledges of Clifford algebra in the many-dimension (more than 3) vector space. The results would be reduced into the old results given in the more simple cases.

Constitutive relations in classical optics in terms of geometric algebra

2015 ◽  
Vol 55 (2) ◽  
Author(s):  
Adolfas Dargys
Keyword(s):  
Wave Propagation ◽  
Electromagnetic Wave ◽  
Vector Space ◽  
Closed System ◽  
Geometric Algebra ◽  
Electromagnetic Wave Propagation ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Basis Vectors

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric.

scholarly journals The birth of E 8 out of the spinors of the icosahedron

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150504 ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Clifford Algebra ◽  
Root System ◽  
Automorphism Groups ◽  
Root Systems ◽  
Double Cover ◽  
Point Of View ◽  
Three Dimensions ◽  
Theoretical Physics ◽  
Superstring Theory ◽  
3D Space

E 8 is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional (8D) space very different from the space we inhabit; for instance, the Lie group E 8 features heavily in 10D superstring theory. Contrary to that point of view, here we show that the E 8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry. The 240 roots of E 8 arise in the 8D Clifford algebra of 3D space as a double cover of the 120 elements of the icosahedral group, generated by the root system H 3 . As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.

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