Multicomplex hyperfunctions

Vajiac, Adrian; Vajiac, Mihaela

doi:10.1080/17476933.2011.603419

Cited by 20 publications

(14 citation statements)

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“…In this section, we begin by a short introduction of the tricomplex numbers space M(3). One may be refer to [1], [5] and [23] for more details on the next properties.…”

Section: Tricomplex Numbersmentioning

confidence: 99%

A study of dynamics of the tricomplex polynomial $$\eta ^p+c$$ η p + c

Parisé

Rochon

2015

Nonlinear Dyn

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In this article, we give the exact interval of the cross section of the so-called Mandelbric set generated by the polynomial z 3 + c where z and c are complex numbers. Following that result, we show that the Mandelbric defined on the hyperbolic numbers D is a square with its center at the origin. Moreover, we define the Multibrot sets generated by a polynomial of the form Q p,c (η) = η p + c ( p ∈ N and p ≥ 2) for tricomplex numbers. More precisely, we prove that the tricomplex Mandelbric has four principal slices instead of eight principal 3D slices that arise for the case of the tricomplex Mandelbrot set. Finally, we prove that one of these four slices is an octahedron.

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“…In this section, we begin by a short introduction of the tricomplex numbers space M(3). One may be refer to [1], [5] and [23] for more details on the next properties.…”

Section: Tricomplex Numbersmentioning

confidence: 99%

A study of dynamics of the tricomplex polynomial $$\eta ^p+c$$ η p + c

Parisé

Rochon

2015

Nonlinear Dyn

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“…It should be noted that a multicomplex number w = z 1 + z 2 i k ∈ C k can be expressed with respect to any unit, and there are many idempotent representations for multicomplex numbers [43,45]. By definition, we have…”

Section: Expressing Multicomplex Numbersmentioning

confidence: 99%

“…One can extend analytic functions [18] to multicomplex variables, although one may also attempt to extend more complicated functions, such as those which are meromorphic. Recent theorems along these lines are given in [45]. For instance, the bicomplex Riemann zeta function was recently extended to a multicomplex Riemann zeta function [36].…”

Section: Introductionmentioning

confidence: 99%

Multicomplex Wave Functions for Linear And Nonlinear Schrödinger Equations

Theaker

Gorder

2016

Adv. Appl. Clifford Algebras

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We consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto C k. We determine the equivalent real-valued system in recursive form, and derive the relevant continuity equations in order to demonstrate that conservation of probability (a hallmark of standard quantum mechanics) holds in the multicomplex generalization. From here, we obtain the real modulus and demonstrate the generalized multicomplex version of Born's formula for the probability densities. We then turn our attention to possible generalizations of the multicomplex Schrödinger equation, such as the case where the scalar potential is replaced with a multicomplex-valued potential, or the case where the potential involves the real modulus of the wave function, resulting in a multicomplex nonlinear Schrödinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrödinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics.

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“…The product of i 1 and i 2 defines a hyperbolic unit j such that j 2 = 1. They are a particular case of the so-called Multicomplex Numbers (denoted M(n)) [15,16] and [23]. In fact, bicomplex numbers…”

Section: Bicomplex Numbersmentioning

confidence: 99%