Advances in Applied Clifford Algebras最新文献

In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function (f(x)=e^{textbf{i}lambda x}). For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal and tangential derivative, we solve an associated Neumann-type boundary value problem.

Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an arbitrary subspace Gabor frame, we give a parametric expression of its Gabor duals of type I and type II, and characterize the uniqueness Gabor duals of type I and type II. And as an application, given a Gabor frame for the whole space (L^{2}({mathbb {R}}^{2},,{mathbb {H}})), we give a parametric expression of its all Gabor duals, and derive its unique Gabor dual of type II. Some examples are also provided.

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are (*)-quantum spaces for the quantum orthogonal group (mathcal {O}(SO_q(3))). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of (SO_q(3)). The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules (mathcal {E}_n) of rank 1 and of degree computed to be an even integer (-2n). The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra ({mathcal {U}_{q^{1/2}}(sl_2)}) which is dual to (mathcal {O}(SO_q(3))).

In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds of higher codimension and review some recent striking results, which demonstrate that such higher codimension models may possess symmetries of arbitrarily high jet degree. This disproves the long held belief that the fundamental 2-jet determination results from Chern–Moser theory extend to this case. As a second case, we consider hypersurfaces with singular Levi form at a point, which are of finite multitype. This leads to the study of holomorphically nondegenerate polynomial models. We outline several results on their symmetry algebras including a characterization of models admitting nonlinear symmetries. In the third part we consider the class of structures with everywhere singular Levi forms that has received the most attention recently, namely everywhere 2-nondegenerate structures. We present a computation of their Catlin multitype and results on symmetry algebras of their weighted homogeneous (w.r.t. multitype) models.

Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure.

This work studies the Clifford algebra approach to the density matrix. We discuss elementary examples of pure and mixed states by writing the density matrix as an element of the Clifford algebra of the three-dimensional space (Cl_3). We also revisit the phenomenon of Larmor precession within the framework of Clifford algebra. Additionally, we discuss the geometrical interpretation of the so-called Clifford Density Element (CDE) for pure states in analogy to the Bloch sphere of conventional quantum theory. Finally, we discuss the dynamics of the CDE, which obeys an algebraic form of the Liouville von–Neumann equation.

This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group ({mathbb {S}}), i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when G is of exponent (le 2). In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor results.

The aim of this paper is to analyze and prove different facts related with bicomplex Möbius transformations. Various algebraic and geometric results were obtained, using the decomposition of the bicomplex set as: ({{mathbb {B}}}{{mathbb {C}}}= {{mathbb {D}}}+ textbf{i}{{mathbb {D}}}), and there were used actively both, hyperbolic and bicomplex, geometric objects. The basics of bicomplex Lobachevsky’s geometry are given.

The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann operator, which is motivated by the non-constant coefficients differential operator mentioned above. Then, A Borel–Pompeiu formula for this slice Cauchy–Riemann operator is discovered, which leads to a Cauchy integral formula for slice regular functions. A Plemelj integral formula for the slice Cauchy–Riemann operator is introduced, which gives rise to results on slice regular extension at the end.

From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.