Short Remark on (p₁,p₂,p₃)-Complex Numbers

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

Review: Short remark on (p1, p2, p3)-complex numbers

by Wolf-Dieter Richter

I attach my insights

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The article is written in acceptable English language.

Author Response

Dear reviewer, thank you very much for your assessement of my work and your suggestions for improvement.  In the order of the suggestions you gave, I responded as follows:

1. I have included a corresponding overview of the sources mentioned in a section at the very end of Section 4 and another one at the very beginning of Section 5 and marked it green.
2. Such a section has now been added at the end of Section  6 and, in addition, Remark 6 has been significantly expanded in the spirit suggested.
3.  Corresponding changes are marked red in the text.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript by Wolf-Dieter Richter explores the concept of (p1, p2, p3)-complex numbers and their properties. It introduces the (p1, p2, p3)-spherical coordinate transformation and defines (p1, p2, p3)-complex numbers as an algebraic structure. The author employs generalized spheres to define multiplication and introduces a functional crucial in defining the density of the three-dimensional p-generalized Gaussian probability law, building upon Richter's previous work in 2021.

The main result involves deriving the three-complex Cartesian coordinate product of (p1, p2, p3)-complex vectors, demonstrating its coincidence with the three-complex (p1, p2, p3)-spherical coordinate product. The manuscript also provides formulas for calculating the Lebesgue measure of sets using the (p1, p2, p3)-spherical dynamical intersection proportion function.

In this paper, the author introduces the new (p1, p2, p3)-complex structure and corresponding trigonometric Euler-type formulae, explores invariant probability densities, and discusses [p1, p2, p3]-spherical probability densities and their invariance property. The paper introduces a density generating function and exemplifies light-tailed and heavy-tailed distributions. It also discusses generalized uniform distributions on spheres and dynamic measure disintegration, covering dynamic ball numbers. The Appendix includes quotations of functions from the literature useful for constructing alternative generalized complex number systems.

In particular, Section 2 initiates a discussion on the geometric approach and introduces coordinates for describing movements on the manifold S, defining the Lie group (S, G) as the basis for the (p1, p2, p3)-complex structure. It extends well-known formulas for trigonometric functions and the exponential function to the three-dimensional case.

Section 3 delves into invariant probability densities, exploring [p1, p2, p3]-spherical probability densities and their invariance property. The section introduces density generating functions, providing examples of light-tailed and heavy-tailed distributions. The invariance property is highlighted as a basis for testing statistical hypotheses on [p1, p2, p3]-sphericity.

Section 4 discusses generalized uniform distributions on the sphere S and dynamic geometric disintegration of the Lebesgue measure in R^3. It emphasizes adapting the Lebesgue measure to account for dynamic changes in shape and size of spheres defined by (p1, p2, p3)-complex numbers, offering mathematical formulations and theorems.

Section 5 introduces ball number functions for a p-sphere S, generalizing properties of circle numbers. It extends the concept of ? to include spheres with different shapes and dimensions based on parameters \(p1, p2,\) and \(p3\). The dynamic nature of the structure is emphasized, contributing significantly to three-dimensional space distributions.

Section 6 advances the geometric method, interpreting the vector-valued vector product and its generalization to p-generalized three-dimensional complex numbers. It builds upon previous work, emphasizing the role of angles and rotation matrices.

The paper is well-written, offering original insights into a dynamic three-dimensional complex structure with a new (p1, p2, p3)-complex number system. It presents a novel approach to analyzing invariant probability densities and generalized distributions on spheres. I recommend accepting it for publication in the journal Symmetry in its current form.

Author Response

Dear reviewer,

thank you very much for your assessement of my work and in particular the very clear description of its content, which will be very useful to the reader.