(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space
Round 1
Reviewer 1 Report
I read the manuscript with some difficulty. So, I believe the text must be profoundly revised. The goal and originality of the work are not clear enough. Therefore, I cannot recommend the manuscript for publication in Axioms in the present form.
Author Response
Comments and Suggestions for Authors
I read the manuscript with some difficulty. So, I believe the text must be profoundly revised. The goal and originality of the work are not clear enough. Therefore, I cannot recommend the manuscript for publication in Axioms in the present form.
Author's Reply to the Review Report (Reviewer 1)
We revised the paper.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The last paragraph before Proposition 2 (page 9, 121-124). The sentence is difficult to read, I suggest adding a comma after "..surfaces, ". Also, at the end the word "is" seems to be redundant.
Then some of the formulas can be improved 76-83:
I would be convenient to denote $\rho = (u^2+v^2)^{-1}$ and rewrite the following formulas, e.g.:
$$x=-\rho^{3} / 15\cdot (3u^{11}-...)$$
The paper would seem more compact.
Lastly, the formulas after the rows 52, 53, 54 can be put in one row, adding "and" between the formulas.
Author Response
Comments and Suggestions for Authors
Author's Reply to the Review Report (Reviewer 2)
- The last paragraph before Proposition 2 (page 9, 121-124). The sentence is difficult to read, I suggest adding a comma after "..surfaces, ". Also, at the end the word "is" seems to be redundant.
Calculating the class for the irreducible algebraic surface equation $\hat{Q}_{7}(a,b,c)=0$ of $\mathfrak{S}_{7}\left( u,v\right) ,$ marked with "$\ast $" in Table 2, was rejected (i.e. "out of memory") by Maple 17 on a laptop Pentium Core i5-4310M 2.00 GHz, 4 GB RAM, with the time given in CPU seconds.
- Then some of the formulas can be improved 76-83: I would be convenient to denote $\rho = (u^2+v^2)^{-1}$ and rewrite the following formulas, e.g.: $$x=-\rho^{3} / 15\cdot (3u^{11}-...)$$
Since the real part of the third part of integral in (\ref{EuclideanWei}) is $2u$, then $z=2u$ for the following all pairs $x$ and $y$. We obtain the following parametric equations $\mathfrak{S}_{m}\left( u,v\right) $ for integers $4\leq m\leq 7$, respectively,
- Lastly, the formulas after the rows 52, 53, 54 can be put in one row, adding "and" between the formulas.
We did it.
Author Response File: Author Response.pdf
Reviewer 3 Report
General Comments:
The paper considers real minimal surfaces family by using Weierstrass data, then computes the irreducible algebraic surfaces of the family in 3-dimensional Euclidean space. The topic is important in mathematics, with possible applications in physics and computer science, especially in computer security. The proposed ideas are interesting; however, the presentation lacks more details.
Specific Comments:
- For the paper to be useful for a wider community, please clarify in brief the meaning of main terms like algebraic, irreducible (over what field?), elimination, etc.
- Line 84: Please clarify why z=2u.
- Please justify the generalizations in Line 103.
- Please explain Proposition2. Is this a conjecture?
- Table 1: It would be more useful to express the time complexity in terms of the surface parameters (e.g., the surface degree) rather than the computer actual time.
Author Response
Comments and Suggestions for Authors
Author's Reply to the Review Report (Reviewer 3)
Comments and Suggestions for Authors
General Comments:
The paper considers real minimal surfaces family by using Weierstrass data, then computes the irreducible algebraic surfaces of the family in 3-dimensional Euclidean space. The topic is important in mathematics, with possible applications in physics and computer science, especially in computer security. The proposed ideas are interesting; however, the presentation lacks more details.
Specific Comments:
1. For the paper to be useful for a wider community, please clarify in brief the meaning of main terms like algebraic, irreducible (over what field?), elimination, etc.
We did it in Section 3.
2. Line 84: Please clarify why z=2u.
Since the real part of the third part of integral in (\ref{EuclideanWei}) is $2u$, then $z=2u$ for the following all pairs $x$ and $y$. We obtain the following parametric equations $\mathfrak{S}_{m}\left( u,v\right) $ for integers $4\leq m\leq 7$, respectively,
3. Please justify the generalizations in Line 103.
Considering above equations, for odd and even numbers $♦m$, we get the following:
We add “Corollary 2” for this.
4. Please explain Proposition 2. Is this a conjecture?
We replace “Proposition 2” with “Conjecture 1”.
5. Table 1: It would be more useful to express the time complexity in terms of the surface parameters (e.g., the surface degree) rather than the computer actual time.
We write “degree” in Table 1, “class” in Table 2.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The revised version has addressed most of the comments carefully. Regarding time complexity in Comment 5, I meant a relationship between the surface parameters and time complexity in terms of the number of mathematical operations. However, this could be a complicated task that can be handled in future works. At present, the paper is useful and deserves publication.
