Review Reports
by
- Wolf-Dieter Richter
Reviewer 1: Daniel Vasiliu Reviewer 2: Anonymous Reviewer 3: Richard David Gill Reviewer 4: Anonymous
Round 1
Reviewer 1 Report
The paper presents a well written, succinct and, for the most part, self-contained introduction to the concept of lp-complex numbers. The main message is that generalizations of complex numbers have interesting applications such as the extension of many classical results in mathematics or in physics and that lp-complex numbers have significant merit for consideration. The paper discuses in detail the definition and the elementary properties of lp-complex numbers. Among the interesting aspects we distinguish the discussion of invariant properties and a generalization of Euler's formula.
In all, the paper represents a very informative and satisfying read that should be cited whenever the concept of lp-complex numbers is involved.
Author Response
Dear Reviewer,
warm thanks for the words you found for my paper.
Kind regards
Wolf-Dieter Richter
Reviewer 2 Report
While I see that the results generalize complex arithmetic, I could not understand what the purpose of this generalization was. I was looking for some discussion of how using this new binary operation led to some insights or at least simpler proofs for results in ordinary complex analysis. In the end, it just appeared that the paper was checking some algebraic properties of this new binary operator and I could not see if anything new emerged from this process. If this paper is to be published, the authors will have to show how this idea can be applied to some purpose.
Author Response
Dear Reviewer,
thank you for your valuable advice. In a new section at the beginning of the Introduction of the revised manuscript I write now on the use of complex numbers in electrical engineering and hope to stimulate this way ideas of one or another reader for corresponding applications of the new numbers, and I try to do justice to your concern this way.
I'm neither an electrical engineer nor a physicist who knows the most adequate current questions in electrical engineering or quantum physics. Publishing therefore this manuscript with a "Multidisciplinary Publishing Institut" seems to me to be an adapted solution (or at least compromise). I think I can realistically hope that the new complex numbers will now, in the revised form of the manuscript, easier find their useful application and hope very much for your support.
Kind regards
Wolf-Dieter Richter
Reviewer 3 Report
The paper is more or less publishable but English is still very poor. I can understand it well myself, because I am pretty familiar with German, but the author (who is from Germany) clearly is not accustomed to writing in English. The author generates the suggestion hereby that he hasn’t yet communicated his ideas with the international scientific community much. The major criticism to be made against the paper, is that it is presented completely without any practical motivation. So it is just a curiosity. An exercise. Professionally performed, but who would be interested in studying it in detail? I cannot think of an application, either. Surely, the author has tried, but unfortunately, failed. That is not a good signal to send to the reader.
Of course, if the paper were published, and if the paper were attractively written, in particular by showing command of language and nice reporting of relevant literature, then other mathematicians (and maybe even some physicists) might read the paper and might themselves come up with applications.
The paper is about a new kind of number system generalising the complex numbers. Now, as is well known, the complex numbers form a two-dimensional real vector space and are usefully represented in the famous complex plane. Various complex number operations have useful geometric representations in the plane. This property of the space C has already been successfully generalised to higher dimensions. The quaternions form a four dimensional real vector space and moreover turn out to represent rotations in 3D as well as to have enormous meaning in quantum mechanics, through various operations in the complex vector space C^2. The author does know this and makes some side remarks about the topic. After that one can go further and reach the octonions. So we have transitions 1D -> 2D -> 4D -> 8D, and moreover the gaps can't be filled! At each step one loses important properties! Octonion multiplication is no longer even associative and many writers would not like to call the Octonions a number system at all, or not call them an algebra. Nobody talks about the next thing, 16D. At the same time, we have the beautiful Clifford algebras which as real vector spaces have dimensions which are powers of 2 and are related to the number systems I just mentioned. The complex numbers and the quaternions are Clifford algebras. The author writes hopefully "the more properties a system of such numbers is desired to possess, the more dimensions are needed for constructing such a system". But the example of the division algebras of dimensions 1, 2, 4 and 8 only, show that the higher the number of dimensions the less properties one can maintain, and also shows that when one goes higher still, the systems lose just about all interest.
The author seems to be pretty unaware of the huge literature on these topics. The present referee is only an amateur in these matters, but still, he would advise the author at least to study (English) Wikipedia on the topics I have mentioned in this report.
I don't like to be rude but the author does seem to be out of touch with his hoped-for readers. He knows what he knows and thinks that everyone knows the same things by the same name. To give an example: the paper starts by referring to Euler's formula. But Euler, one of the most prolific mathematicians of all times, is responsible for hundreds of famous formulae. The paper nowhere explicitly states what, in the author's opinion, "the" Euler's formula is. I figured out which formula the author meant, after reading through the paper a couple of times, and it is, of course, a very well known result about complex numbers, but in fact, I had never associated it in my mind with the name Euler. OK, so that tells us something about my memory or my education or both, but still, I think it illustrates my point that the author's focus is rather narrow.
I put a lot of yellow marker pen on the manuscript. Mainly I refer to failures of English grammar. I'm sorry I don't have the time right now to write out a proposed replacement text every time. The author has got to find someone in his own circle of friends to help him fix his English. It is not the job of a referee.
Comments for author File: 
Comments.pdf
Author Response
Dear Reviewer,
Thank you for your valuable advice and support. In a new section at the beginning of the Introduction of the revised manuscript I write now on the use of complex numbers in electrical engineering and hope to stimulate this way ideas of one or another reader for corresponding applications of the new numbers, and I try to do justice to your concern this way.
The last four lines of Section 5 are replaced now with three new lines.
The link to Euler's name is now made with reference to the trigonometric representation of complex numbers, both in the abstract and in the text: Section 4: heading and just before Theorem 4.4.
Please find my respond to each of your yellow marks at the same places in the text.
Kind regards
Wolf-Dieter Richter
Reviewer 4 Report
In this manuscript author discuss about $l_p$ complex numbers and some of their properties and further Euler's formula generalization.
I find these results very promising since they are supported by short proves.
Author Response
Dear Reviewer,
many thanks for your assessment.
Kind regards
Wolf-Dieter Richter
Round 2
Reviewer 2 Report
This version is improved but may benefit from some further development. I think the idea of using Clifford algebras is a good one but what would be nice is some analogy with a well-known result that holds or fails (not just the arithmetic properties). I understand that the development of the Euler formula is trying to show this but an application would be even better -- for example, is there some analogy with group characters (the exponential is a character of the additive reals) or more ambitiously, with fourier series?
Author Response
Dear Reviewer,
Thank you for this valuable advice according to which I inserted the new Definitions 4.5 and 4.6 as well as Theorem 4.7 into the manuscript. These items are devoted to analytical instead of algebraic properties and result in the introduction of a character of the real numbers being an analogy to the known exponential one.
Moreover, I cancelled the formerly last line of Section 2 and changed numbering of formulas in a way that all numbered formulas are refered to within the text. The new reference Sasvári is referred to in connection with the consideration on the character at the end of Section 4.
With kind regards
Wolf-Dieter Richter
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
The manuscript introduces classes of Lp -complex numbers with some basic properties and generalization of Euler's formula. Statements and proofs of the theorems are clear. But I'm not convinced of its significance and interests to the readers.
Author Response
Point 1
But I'm not convinced of its significance and interests to the readers.
Response 1
I added a paragraph close to the end of the Introduction where I refer to other types of generalizations and relatives of complex numbers and their applications to physics and electrical engineering hoping this gives more motivation to the reader.
Reviewer 2 Report
The author replaces the classical
trigonometric functions with their lp-counterparts in the trigonometric representation of complex numbers i order to introduce classes of lp-complex numbers. The author also presents some of their fundamental properties. Moreover, by utilizing lp-complex number multiplication, the author generalizes the Euler's formulaand shows that the lp-absolute value of an lp-complex number is remained unchanged. After fixing the notation in section 2, introduces the the geometric vector p-multiplication and the lp-
multiplicative inverse element (Definition 2.1) and the plane of lp-complex numbers and the lp-absolute value (definition 2.2), the author is able to live up to the claims in the abstract. Section 4 is dedicated to the genewralization of Euler's fotmula. The author ends that paper with a discussion which he presents the replacement method's merits and also failure ( when considering a set of numbers which includes the idea of an ordered set).
The author may continue the research and see the other merits of such replacement method.
Author Response
Point 1
The author may continue the research and see the other merits of such replacement method.
Response 1
Thank you very much for encouraging further research on this subject for which I would highly appreciate cooperation with colleagues from physics, electrical engineering and mathematics
Reviewer 3 Report
Dear Author,
The presented considerations are interesting. However, it should be compared at least to the very close structures used in physics. The idea of Clifford modules and works of Chevalley(1954) on the algebraic theory of spinors are related to the topic of the paper.
References
Atiyah, MF, Bott, R, and Shapiro, A (1964). Clifford modules.Topology 3 (Suppl. 1): 3–38.
Brauer, R and Weyl, H (1935). Spinors inndimensions.Amer. J. Math.57: 425–449.
Cartan,É (1938).Théorie des spineurs. Actualités Scientifiques et Industrielles, No. 643 et 701. Paris: Hermann; English transl.: The Theory of Spinors, Paris: Hermann, 1966.
Chevalley, C (1954).The Algebraic Theory of Spinors. New York: Columbia University Press.
Author Response
Point 1
However, it should be compared at least to the very close structures used in physics.
Response 1
Thank you very much for your very constructive and helpfull Comments. I added respective additional paragraph close to the end of the Introduction and inserted the additional references.
Reviewer 4 Report
- The mathematics presented in this work seems correct; no significant mathematical errors were detected. One small issue just after (2.1) is that arctan(y/x) does not distinguish between, say, the first and fourth quadrants.
- The results presented here are relatively minor extensions of commonly known results. While this reviewer is not an expert, it seems that many mathematics could achieve these results without much effort.
- Although no errors where found, please check that the circle-p-products and the circle-2-products were used correctly.
- The English in this manuscript is comprehensible, but not strong. There are a number of places where bits of German sentence structure creep into the English. A few examples:
- ---"This will be the starting point for a generalization of the vector realization of the classical complex structure, here." ("This . . . here" is not acceptable in English, though it may work in German.)
- ---The word "ens" is unusual in English; it does not appear in many dictionaries.
- ---"from a circle . . . within this set": this phrase does not make sense.
- ---"For short" at the beginning of the Discussion should be perhaps "In short"
- ---"some one's" should be "someone's"