arXiv subjects
Publications and source records attributed to Daniel Alayon-Solarz.
We show sufficient and necessary conditions, in terms of some partial differential equations with variable coefficients, for a quaternionic function to admit a continuous derivative in a open set in the sense of C. Schwartz.
We introduce the regular product for Cullen-regular quaternionic functions in a manner that does not depend upon a representation in power series but upon another, weaker kind of representation. The special case when the functions are represented as quaternionic power series is studied. We show that the regular ring of quaternionic power series is a subring of the regular associative ring of real-analytical Hyperholomorphic functions.
We show characterizations of the class of Cullen-regular functions in the sense of Gentili-Struppa for any domain $Ω$ in terms of the Fueter operator. We then state a Integral Theorem and discuss how it can be used to define a more general version of Cullen-regularity, that does not require the function to be of class $C^{1}$.
We introduce the class of semiweak Cullen-regular quaternionic functions by interpreting Cullen-regular functions as solutions to an inhomogeneous PDE in terms of the Fueter operator.
We discuss the proof of a certain integral theorem obtained by C. G. Cullen, originally stated on the class of the analytic intrinsic functions on the quaternions. It is shown that this integral theorem is true for a larger class of quaternionic functions.
Removing al least one point from the unit sphere in $ R^{3}$ allows to factorize the angular part of the laplacian with a Cauchy-Riemann type operator. Solutions to this operator define a complex algebra of potential functions. A family of these solutions is shown to be normalizable on the sphere so it is possible to construct associate solutions for every radial solution to the time-independant Schrödinger equation with a radial potential, such that this family of solutions is square integrable in $R^{3}$. While this family of associated solutions are singular on at least one half-plane, they are square-integrable in almost all of $R^{3}$.
We present some classes of functions that are defined on the quaternions as solutions for a linear operator that resembles the Cauchy-Riemann conditions. Unlike the Fueter regular functions; in this case the identity function is a solution and solutions are closed under the quaternionic product.
We show that Fueter's theorem holds for a more general class of quaternionic functions than those constructed by the Fueter's method.
We call attention to the unusual properties that the 4 dimensional solutions for a modified Fueter-Dirac equations satisfy: In a coordinate-free, constant-free and strictly mathematical way it is possible to show that all the solutions for a modified Fueter-Dirac Equation, which are radial symmetric when restricted to the 3-space, have a nice algebraic structure. Locally, these solutions behave naturally in a algebraic way determined by a certain commutative ring related to the quaternions with non-invertibles. This ring has a nontrivial localization. By using left and right versions of the operator we obtain quirality.