Advances in Applied Clifford Algebras最新文献

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers (left{ mathbb {D}_{t}right} _{tin mathbb {R}}) are constructed as sub-structures of scaled hypercomplex numbers (left{ mathbb {H}_{t}right} _{tin mathbb {R}}) under the scales (or, the moments) of the set (mathbb {R}) of real numbers. We show that if (t<0), then the classical free probability theory covers our free probability on (left{ mathbb {D}_{t}right} _{t<0}); if (t>0), then our free probability on (left{ mathbb {D}_{t}right} _{t>0}) is represented by the free probability over the classical hyperbolic numbers (mathcal {D}=mathbb {D}_{1}); and if (t=0), then the free probability on (mathbb {D}_{0}) is actually over the dual numbers (textbf{D}=mathbb {D}_{0}). Since the usual free probability theory is over (mathbb {C}), we here concentrate on establishing our free probability theory on (mathcal {D}), or that on (textbf{D}). Our approaches are motivated by the Speicher’s combinatorial free probability. As applications, the (mathbb {D}_{t})-free-probabilistic versions of semicircular elements and circular elements are considered.

We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Cayley Heisenberg group. As a consequence we deduce some qualitative uncertainty principles associated with this transform.

This paper investigates the Lorentz invariance of the multidimensional Dirac–Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac–Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1, n) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action.

In this paper, by utilizing the complex adjoint matrices, we transform the generalized right eigenvalue problem of the split quaternion matrix pencil into an equivalent generalized complex eigenvalue problem. This transformation enables us to propose an effective algebraic method for solving generalized eigenvalues and their corresponding eigenvectors. Additionally, we investigate the corresponding generalized right least squares eigenvalue problem for the split quaternion matrix pencil, providing a comprehensive framework for these types of problems. Secondly, we define the standard generalized right eigenvalues for the split quaternion matrix pencil. We rigorously prove that a split quaternion matrix pencil of order n has exactly n standard generalized right eigenvalues, all of which are complex numbers. Thirdly, we introduce the Rayleigh quotient for the split quaternion matrix pencil and study its fundamental properties. The definition and analysis of the Rayleigh quotient contribute to the theoretical understanding and potential applications of generalized split quaternion eigenvalue problems.

In this paper, we establish and study various properties of the q-numerical range and the q-numerical radius for right linear bounded operators on a right quaternionic Hilbert space.

Conformal manifolds (M_lambda ) are open subsets of (mathbb {R}^n) endowed with the metric

where (lambda ) is called the conformal function. We show that there exists the (alpha )-Dirac operator (D_alpha ), with (alpha in mathbb {R}), acting on functions valued by the Clifford algebra on (M_lambda ). The operator behaves similarly to the usual Euclidean Dirac operator. We develop (alpha )-dependent potential theory for (Delta _alpha ) on conformal manifolds, prove refined Poincaré lemmata, and establish Helmholtz-type decompositions for multivector fields.

We prove an analog of the quaternionic Borel–Pompeiu formula in the sense of proportional fractional (psi )-Cauchy–Riemann operators via Riemann–Liouville derivative with respect to another function.

In the article a class of (mathbb {H})-valued monogenic fractional power functions defined in the reduced quaternions and depending on the parameters (pin mathbb {N}_{0}) and real (lambda > -1) is constructed. These functions are an extension of the well-known class of orthogonal Appell polynomials, which is included as a special case. For the monogenic fractional powers essential properties, i.e. monogenicity, a generalized Appell property and a two-step recurrence formula, are proved and their corresponding Kelvin transforms in terms of a corresponding anti-monogenic fractional power function are given.

In this paper we use the power of the outer exponential (Lambda ^B) of a bivector B to see the so-called invariant decomposition from a different perspective. This is deeply connected with the eigenvalues for the adjoint action of B,  a fact that allows a version of the Cayley–Hamilton theorem which factorises the classical theorem (both the matrix version and the geometric algebra version).

Here I present the clifford package for working with Clifford algebras in the R programming language. Algebras of arbitrary dimension and signature can be manipulated, and a range of different multiplication operators is provided. The algebra is described and package idiom is given; it obeys disordR discipline. A case-study of conformal algebra is presented. The package is available on CRAN and development versions are hosted at github.