The Basis Invariant of Generalized n-Cube Symmetries Group with Odd Degrees

Round 1

Reviewer 1 Report

The authors make a study on the invariant polynomial algebra of the generalized n-cube. Although no essentially new techniques appear to be introduced, the results may be of interest to readers interested in the subject. Nevertheless, I believe it is necessary to revise the article starting from the following points:

The Introduction should be enriched by relating the topic to preceding research. In this sense, current references should be introduced and the article should be placed in the context of current research.

In line 53, a "less or equal" should be replaced by a "less".

Although there is a section called Main Result, no theorem is formulated that captures that result. It is recommended that this section be structured in theorem-demonstration form.

The computations on pages 6 to 10 are too long. The authors should revise them in several ways: (i) eliminate the steps that are elementary, highlighting those that are key; (ii) explain the most important steps, to facilitate reading and make the reasoning used explicit; (iii) structure that part by adding lemmas, if possible.

The Conclusions section only summarizes the results obtained. It would be useful for the authors to explain some consequences or applications of these results.

Authors are also advised to make a stylistic and typographical revision of their article in order to increase the elegance of their presentation. For example, there are paragraphs that are too short that could be unified (e.g., lines 45 to 49) or typographical mistakes (e.g., in the line before equation (6), the parentheses that include the reference [8] are superfluous).

Author Response

Dear reviewer,

We response to your comments and suggestions one by one as follows. Please take a look over.

The authors make a study on the invariant polynomial algebra of the generalized n-cube. Although no essentially new techniques appear to be introduced, the results may be of interest to readers interested in the subject. Nevertheless, I believe it is necessary to revise the article starting from the following points:

The Introduction should be enriched by relating the topic to preceding research. In this sense, current references should be introduced and the article should be placed in the context of current research.

• We enriched the introduction. We noticed literature origin of used results. Some of literature we consult to find another point of view on the theme we concerned. In doing so, we find a literature that are not concerned with. In introduction we mentioned it.
• If it is not enough, please give us suggestions.

In line 53, a "less or equal" should be replaced by a "less".

• We replace it and correct other formulas as much we can find ourselves mistakes.

Although there is a section called Main Result, no theorem is formulated that captures that result. It is recommended that this section be structured in theorem-demonstration form.

• Now you can see Theorem formally. The proof is given after it in the rest of section called Main Result.

The computations on pages 6 to 10 are too long. The authors should revise them in several ways: (i) eliminate the steps that are elementary, highlighting those that are key; (ii) explain the most important steps, to facilitate reading and make the reasoning used explicit; (iii) structure that part by adding lemmas, if possible.

• We shorted the calculation expressions according our opinion. Maybe we can cut it more. Please advise us if more is needed.

The Conclusions section only summarizes the results obtained. It would be useful for the authors to explain some consequences or applications of these results.

• We change the Conclusions with some further investigation possibilities.

Authors are also advised to make a stylistic and typographical revision of their article in order to increase the elegance of their presentation. For example, there are paragraphs that are too short that could be unified (e.g., lines 45 to 49) or typographical mistakes (e.g., in the line before equation (6), the parentheses that include the reference [8] are superfluous).

• In the version revised we try to beautify the look of manuscript aware that this is the first step in attracting the reader.

Our new version is attached below 

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper has significant grammatical and typographical problems.

For example, nearly every line has such a problem. It is clear that not much care has been put into the presentation. In the long calculations they have double equality indicators.

 

Also, they assert a statement to be true for every r, but only seem to 

consider r=2,4,6.

Another sample of bad writing is given in Eq. (5), (6) and (7), where the notation R_{2r} is used twice for seemingly different things. Eq. (5) and (7) seem to be the same thing, but are not the same.

The author's English is so bad that it is difficult to know what they are really doing.

 

Author Response

Dear reviewer,

 

we response to your comments and suggestions one by one as follows. Please take a look over.

This paper has significant grammatical and typographical problems.

• Aware of the problems, we fixed them thoroughly. We sure some remains and ask to warn us.

For example, nearly every line has such a problem. It is clear that not much care has been put into the presentation. In the long calculations they have double equality indicators.

• We admit that we were in a hurry compiling the manuscript together. We thank for your patience reviewing it.

Also, they assert a statement to be true for every r, but only seem to 

consider r=2,4,6.

• We formalize our result into the Theorem 2. After it, we use the rest of section 4. Main result to prove it. After case r=6, we consider general case. According the recommendation from other reviewers, we shortened calculation by avoiding unnecessary parts.

Another sample of bad writing is given in Eq. (5), (6) and (7), where the notation R_{2r} is used twice for seemingly different things. Eq. (5) and (7) seem to be the same thing, but are not the same.

• Thank you to notice the mistake which was being there in front of our eyes obviously. Namely, this is a crucial connection between our intention to picture the idea in 3-dimensional case for general n-cube result idea.
• We sure that mistakes remain, so we will be thankful for any further warning.

Down below our new version is attached. Please, take a look over.

Author Response File: Author Response.pdf

Reviewer 3 Report

There are many grammatical and spelling errors throughout the manuscript. In the abstract, the authors use the word "detaily". Detaily is not an English vocabulary word. Line 10: should be Euclid, not Euklid. Line 27: "hiperplane" is spelled wrong. There are many sentences throughout that are incomplete or not clear. 

The authors mention "There are various methods for constructing
20 basis invariant of this algebra, but we consider that one in our work." Discuss or cite these other "various methods". 

In Equations (2) and (3), you can rewrite them clearer by writing $1\leq i< j \leq n$. 

Line 70: you typed "vertexes given by by the". Should be "vertices", and you typed "by by" twice. 

The authors typed a theorem from their reference (10), but what page in the reference is this theorem located?

Their main result is hidden as a sentence in Line 122. You need to state clearly what your theorem is. Secondly, the authors spent 4 pages doing small case calculations, r = 2, 4, 6. Why did you include this much calculation? Shouldn't r = 2 or r = 4 be enough?

It is difficult to see where the proof begins and where it ends. And why do the authors derive J_{m(r-1)}^* partially, rather than prove that they have found all the invariants? 

This is an incomplete manuscript, and very difficult to read and follow. It appears that the authors put it together without any revision and rushed to submit it. 

Author Response

Dear reviewer,

 

we response to your comments and suggestions one by one as follows. Please take a look over.

There are many grammatical and spelling errors throughout the manuscript. In the abstract, the authors use the word "detaily". Detaily is not an English vocabulary word. Line 10: should be Euclid, not Euklid. Line 27: "hiperplane" is spelled wrong. There are many sentences throughout that are incomplete or not clear. 

• We improve mistakes through all article. But we are sure that some mistakes always survive. If you find another one, please inform us.
• We do our best to write grammatically correct and now we hope manuscript is in the sense of English language. Although every translation in terms of language lacks its original meaning.

The authors mention "There are various methods for constructing
20 basis invariant of this algebra, but we consider that one in our work." Discuss or cite these other "various methods". 

• Searching the literature, we couldn’t find reference about similar investigation. This motivated us to refresh the problem which is possibly forgotten. This is a reason that we explain our calculation in detail.

In Equations (2) and (3), you can rewrite them clearer by writing $1\leq i< j \leq n$. 

• We agree with your suggestion.

Line 70: you typed "vertexes given by by the". Should be "vertices", and you typed "by by" twice. 

• In present version we hope you will not find something similarly again. Nevertheless, the mistakes are always possible and it is hard to find the own fault.

The authors typed a theorem from their reference (10), but what page in the reference is this theorem located?

• The only Theorem we cited from one old result from reference [6]. There the result was given without a proof, so we prove our calculations more detail that is usual.
• We use reference [10] for relating item’s definitions and to find informations about items we use

Their main result is hidden as a sentence in Line 122. You need to state clearly what your theorem is. Secondly, the authors spent 4 pages doing small case calculations, r = 2, 4, 6. Why did you include this much calculation? Shouldn't r = 2 or r = 4 be enough?

• Now we state our theorem. And reduce calculations. For r=6 we give only the last step with discussion.

It is difficult to see where the proof begins and where it ends. And why do the authors derive J_{m(r-1)}^* partially, rather than prove that they have found all the invariants? 

• The whole part of Section 4. Main result after Theorem is the proof. We think it is better was to present it informally. But we can change our point of view.

This is an incomplete manuscript, and very difficult to read and follow. It appears that the authors put it together without any revision and rushed to submit it. 

• Unfortunately, we agree with you. We rush to submit it. So we put it together without serious revision. We thank you for your patience and every omission you pointed out to us. And hope not disappoint you much.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have responded adequately to the comments raised and have significantly improved their work. In the current version, I consider the article to be publishable.

Reviewer 2 Report

OK, so I got the new version.

In my initial review I reported that there were major grammatical changes needed and I find the following:

I look at the abstract, which is four lines long, and I find eight (8) grammatical/typographical problems in those four lines. I find that the major object of study is referenced twice in this abstract, each time using a different name. I also find a statement in their reply to the effect that I should let them know of any other errors that I might find. This is not the job of the referee!! They are responsible for fixing their own errors.

From this I take it that the authors have no interest in revising their manuscript in a serious way, and so I can only recommend that the paper be rejected. 

Reviewer 3 Report

This paper has improved significantly. Pending a minor revision (for example, see lines 153, 156), this manuscript should be published.