Advances in Applied Clifford Algebras最新文献

This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.

The aim of this paper is two-fold. First, we obtain various inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Sasakian space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion.

A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by ({{mathcal {N}}}). The rules of addition and multiplication in ({{mathcal {N}}}) are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A basis of (n+1) null vectors, with pairwise inner products equal to plus or minus one half, defines the Clifford geometric algebras ({mathbb {G}}_{1,n}), or ({mathbb {G}}_{n,1}), respectively, and provides a foundation for a new Cayley–Grassman linear algebra, a theory of complete graphs, and other applications in pure and applied areas of science.

As an addition to proper points of the real plane, we introduce a representation of improper points, i.e. points at infinity, in terms of Geometric Algebra for Conics (GAC) and offer possible use of both types of points. More precisely, we present two algorithms fitting a conic to a dataset with a certain number of points lying on the conic precisely, referred to as the waypoints. Furthermore, we consider inclusion of one or two improper waypoints, which leads to the asymptotic directions of the fitted conic. The number of used waypoints may be up to four and we classify all the cases.

This work presents an extension, called coordinate slice extension, of the union of a finite number of axially symmetric s domains according to the fiber bundle theory and a kind of slice regular functions are defined on this coordinate slice extension.

In this paper we introduce an axiomatic approach to the theory of s-numbers in quaternionic analysis. To this end, Pietsch’s approach to s-number theory is adapted to the quaternionic framework, following the works of Colombo and Sabadini on quaternionic spectral analysis. One of the main results of this paper is the uniqueness of s-numbers over quaternionic Hilbert spaces. Moreover, examples are given in the quaternionic framework together with the introduction of nuclear numbers. A consequence of the presented theory is a basis independent definition of the Schatten classes over quaternionic Hilbert and Banach spaces.

In this paper, we give some new versions of the Plemelj–Sochocki formula under weaker condition in real Clifford Analysis which are different from the result in Luo and Du (Adv Appl Clifford Algebras 27:2531-2583, 2017). By the new versions of the Plemelj–Sochocki formula, we can give a different proof of the generalized Plemelj–Sochocki formula for the symmetric difference of boundary values, which is obtained in Luo and Du (2017), the classical Plemelj–Sochocki formula can be also derived.

This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from nth-order d-bar derivatives for (nin {mathbb {N}}), and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials.

The goal of this paper is to elucidate the hermitian structure of the algebra of geometric quaternions ({textbf {H}}) (that is, the even algebra of the geometric algebra of the euclidean 3d space), use it to couch a geometric and dynamical presentation of the q-bit, and to see its bearing on the geometric structure of q-registers (arrangements of finite number of q-bits) and with that to pay a brief revisit to the formal structure of q-computations, with emphasis on the algebra structure of (textbf{H}^{otimes n}). The main underlying theme is the unraveling of the subtle geometric relations between (textbf{H}) and the sphere (S^2) in the 3d euclidean space.

In this paper, we present the general right-sided quaternionic orthogonal 2D-planes split wave-packet transform that combines windowed and wavelet transforms. We derive fundamental properties: Plancherel–Parseval theorems, reconstruction formulas, and orthogonality relations, and we provide characterization range, convolutions, and some estimates. Additionally, we derive component-wise, directional and logarithmic uncertainty principles for the given transform and give a discrete formula on the square-integrable function space.