Advances in Applied Clifford Algebras最新文献

Let

be an exceptional domain of non-tube type and let (mathcal {U}_{nu }) and (mathcal {W}_{nu }) the associated generalized Hua operators. In this paper, we determine the explicit formula of the action of the group ( E_{6(-14)}) on (mathcal {D}_{16}). We characterized the (L^{p})-range, (1le p < infty ) of the generalized Poisson transform on the Shilov boundary of the domain (mathcal {D}_{16}).

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant in the inequality for the quaternion offset linear canonical transform (QOLCT). We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner–Ville distribution associated with QQPFT and obtain the Lieb’s uncertainty and entropy uncertainty principles for these three transforms.

The aim of this paper is to study the properties of the Möbius addition (oplus ) under the action of the gyration operator gyr[a, b], and the relation between ((sigma ,t))-translation defined by the Möbius addition and the generalized Laplace–Beltrami operator (Delta _{sigma ,t} ) in the octonionic space. Despite the challenges posed by the non-associativity and non-commutativity of octonions, Möbius addition still exhibits many significant properties in the octonionic space, such as the left cancellation law and the gyrocommutative law. We introduce a novel approach to computing the Jacobian determinant of Möbius addition. Then, we discover that the gyration operator is closely related to the Jacobian matrix of Möbius addition. Importantly, we determine that the distinction between (aoplus x) and (xoplus a ) is a specific orthogonal matrix factor. Finally, we demonstrate that the ((sigma ,t))-translation is a unitary operator in (L^2 left( {mathbb {B}^8_t,dtau _{sigma ,t} } right) ) and it commutes with the generalized Laplace–Beltrami operator (Delta _{sigma ,t} ) in the octonionic space.

In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.

Let (X=Sp(1,n)/Sp(n)) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as (Sigma ). In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued (L^2)-space on (Sigma ). Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of Sp(1, n) on (L^2(X)).

Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist spaces are constructed by the join operator. For coupled limbs coupled with closed loops in the MCM, the equivalent limbs can be found by solving the analytical expressions of the twist space on each closed loop’s output link. Then, the twist spaces of the coupled limbs can be easily obtained. The twist space of the MCM’s output link is the intersection of all the limb twist spaces, which can be calculated by the meet operator. The proposed method provides a simplified way of analyzing the mobility of MCMs, and three typical MCMs are chosen to validate this method. The analytical mobility of the MCM’s output link can be obtained, and it naturally indicates both the number and the property of the degrees of freedom (DOFs).

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.

This paper investigates the distribution function and nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for ( mathbb{B}mathbb{C})-valued functions, which characterizes valuable insights into the behavior and structure of (mathbb{B}mathbb{C})-valued functions, allowing to analyze their properties and establish connections with other mathematical concepts. Next, the nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions with the hyperbolic norm are studied. By exploring the nonincreasing rearrangement of (mathbb{B}mathbb{C})-valued functions, it is aimed to determine how the hyperbolic norm influences the rearrangement process and its impact on the function’s behavior and properties.

In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces (L^p) and (L^q.) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras.

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for (A_8), (D_8) and (E_8) for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.