Advances in Applied Clifford Algebras最新文献

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.

The theory of generalized partial-slice monogenic functions is considered as a synthesis of the classical Clifford analysis and the theory of slice monogenic functions. In this paper, we introduce a Cauchy integral formula and a Plemelj formula for generalized partial-slice monogenic functions. Further, we study some properties of the Teodorescu transform in this context. A norm estimation for the Teodorescu transform is discussed as well.

In this paper, the two-sided quaternionic Dunkl transform satisfies some uncertainty principles of quaternion algebra. An analog of the Beurling theorem for the two-sided quaternionic Dunkl transform is obtained. As a direct consequence of Beurling’s theorem, other versions of the uncertainty principle, such as Hardy’s, Gelfand–Shilov’s, Cowling–Price’s and Morgan’s theorems are also deduced.

This paper systematically studies Hilbert boundary value problems for monogenic functions on the hyperplane for the solutions being of any integer orders at infinity, where the negative order cases are new even when restricted to the complex plane context. The explicit solution formulas are provided and the solvability conditions are specified. The results are proved using the Clifford symmetric extension method, which reduces Hilbert boundary value problems to Riemann boundary value problems, involving many innovative geometric techniques.

In this paper, Cauchy theorems for solutions to polynomial Dirac equations with (alpha )-weight in superspace are studied using two methods. First, by constructing a new fundamental solution, the first kind of Cauchy theorem is obtained. Then the connection between polynomial Dirac operators with (alpha )-weight and iterative Dirac operators with (alpha )-weight in superspace is obtained. Finally, using this connection, the second kind of Cauchy theorem is obtained.