scholarly journals Chirality in Geometric Algebra

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1521
Author(s):  
Michel Petitjean
Keyword(s):  
Geometric Algebra ◽  
Orthogonal Group ◽  
Matrix Representation ◽  
Quadratic Space ◽  
Complex Numbers ◽  
Chiral Algebra

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

scholarly journals Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

2021 ◽  
Author(s):  
Joseph Wilson ◽  
Matt Visser
Keyword(s):  
Efficient Method ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Spin Representation ◽  
Lorentz Transformations ◽  
Rodrigues Formula ◽  
Dimension Formula ◽  
Spacetime Algebra ◽  
Geometric Algebras

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

scholarly journals DOA Estimation of Cylindrical Conformal Array Based on Geometric Algebra

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Minjie Wu ◽  
Xiaofa Zhang ◽  
Jingjian Huang ◽  
Naichang Yuan
Keyword(s):  
Geometric Algebra ◽  
Rotation Angle ◽  
Matrix Representation ◽  
Doa Estimation ◽  
Signal Classification ◽  
Matrix Transformations ◽  
Multiple Signal Classification ◽  
Conformal Array ◽  
Variable Curvature ◽  
Array Structures

Due to the variable curvature of the conformal carrier, the pattern of each element has a different direction. The traditional method of analyzing the conformal array is to use the Euler rotation angle and its matrix representation. However, it is computationally demanding especially for irregular array structures. In this paper, we present a novel algorithm by combining the geometric algebra with Multiple Signal Classification (MUSIC), termed as GA-MUSIC, to solve the direction of arrival (DOA) for cylindrical conformal array. And on this basis, we derive the pattern and array manifold. Compared with the existing algorithms, our proposed one avoids the cumbersome matrix transformations and largely decreases the computational complexity. The simulation results verify the effectiveness of the proposed method.

Widely Linear Estimation with Geometric Algebra

2013 ◽  
pp. 293-308
Author(s):  
Tohru Nitta
Keyword(s):  
Geometric Algebra ◽  
High Dimensional ◽  
Complex Conjugate ◽  
Complex Numbers ◽  
Linear Estimation ◽  
Higher Dimensional ◽  
Dimensional Parameters ◽  
Geometric Algebras ◽  
Widely Linear ◽  
Complex Valued

This chapter reviews the widely linear estimation for complex numbers, quaternions, and geometric algebras (or Clifford algebras) and their application examples. It was proved effective mathematically to add , the complex conjugate number of , as an explanatory variable in estimation of complex-valued data in 1995. Thereafter, the technique has been extended to higher-dimensional algebras. The widely linear estimation improves the accuracy and the efficiency of estimation, then expands the scalability of the estimation framework, and is applicable and useful for many fields including neural computing with high-dimensional parameters.

scholarly journals Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

2019 ◽  
Vol 27 (2) ◽  
pp. 37-65 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdris Ören
Keyword(s):  
Euclidean Space ◽  
Orthogonal Group ◽  
Dimensional Euclidean Space ◽  
Plane Curves ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Special Orthogonal Group ◽  
The Common ◽  
Global Invariants ◽  
The Given

AbstractIn this paper, for the orthogonal group G = O(2) and special orthogonal group G = O+(2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E2 are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.

scholarly journals Dual finite frames for vector spaces over an arbitrary field with applications

2021 ◽  
pp. 1-36
Author(s):  
Patricia Mariela Morillas
Keyword(s):  
Hilbert Spaces ◽  
Matrix Representation ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Dual Frames ◽  
Finite Frames ◽  
Finite Dimensional ◽  
The Matrix ◽  
Normed Vector

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

Some algebraic properties of complex Segre quaternoins

2019 ◽  
Vol 33 ◽  
pp. 158-159
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Matrix Representation ◽  
Complex Numbers ◽  
Polynomial Equations ◽  
Complex Field ◽  
Basic Properties ◽  
Algebraic Properties ◽  
Cauchy Riemann

This paper deals with the basic properties the algebra of Segre quaternions over the field of complex numbers. We study idempotents, ideals, matrix representation and the Peirce decomposition of this algebra. We also investigate the structure of zeros of a polynomial in Segre complex quaternions by reducing it to the system of four polynomial equations in the complex field. In addition, Cauchy-Riemann type conditions are obtained for the differentiability of a function on the complex Segre quaternionic algebra.

Clifford Geometric Algebra

2021 ◽  
Author(s):  
Gennadiy Kondrat'ev
Keyword(s):  
Differential Geometry ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Geometric Interpretation ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Euclidean Manifold ◽  
Clifford Geometric Algebra ◽  
Algebraic Operations

The monograph is devoted to the fundamental aspects of geometric algebra and closely related issues. The category of Clifford algebras is considered as the conjugate category of vector spaces with a quadratic form. Possible constructions in this category and internal algebraic operations of an algebra with a geometric interpretation are studied. An application to the differential geometry of a Euclidean manifold based on a shape tensor is included. We consider products, coproducts and tensor products in the category of associative algebras with application to the decomposition of Clifford algebras into simple components. Spinors are introduced. Methods of matrix representation of the Clifford algebra are studied. It may be of interest to students, postgraduates and specialists in the field of mathematics, physics and cybernetics.

scholarly journals Abelian unipotent subgroups of finite orthogonal groups

1982 ◽  
Vol 32 (2) ◽  
pp. 223-245 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Orthogonal Group ◽  
Quadratic Space ◽  
Orthogonal Groups ◽  
Special Orthogonal Group ◽  
Abelian Subgroups

AbstractIf G is the special orthogonal group O+(V) of a quadratic space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G, extending some results obtained with different methods by Barry (1979).

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