Beltrami Equations on Rossi Spheres

Round 1

Reviewer 1 Report

This is a review for the Manuscript Number: Mathematics -1567373- "Beltrami equations on Rossi spheres"

In this paper .The authors studied Beltrami equations ?̅ ?(?)=?(.,?)??(?) on ??, where ?? , |?| < ?, are the Rossi operators (i.e. ?? spans the globally non embeddable ?? structure ?(?) on ?? discovered by ?. Rossi, [15]), are derived such that to describe quasiconformal mappings ?:??→ ? ⊂ ?? from the Rossi sphere (??,?(?)) . The authors Using the Greiner-Kohn-Stein solution (cf. [8]) to the Lewy equation, and the Bargmann representations of the Heisenberg group. The authors solved the Beltrami equations for Sobolev type solutions ?? such that ??−?∈???,?(??,?) with ??∞(??,?(?)).

 

Comments and Suggestions for Authors:

   The topic of this paper is interesting, and the results seem to be correct and interesting. Therefore, it can be published subject to the following minor modifications:

1.  The original papers should be much better presented.
2.  The abstract should be improved by mentioning the major contributions.
3.  English is good but should be checked for some grammar.
4. What is the novelty of the proposed work? Please list the motivation and contribution in the introduction section.
5.  It takes into account the formulation and writing of equations in a more coordinated manner
6.  I suggest write the conclusion and future work.
7.  Coordination of some equations requires reconsideration as well as taking into account punctuation marks. Some equations are not formatted in writing. We hope that this will be taken into account in the final version.
8.  As this paper can be interesting for the researchers in many field of applied mathematic. So, the authors can improve the introduction by the recent developments in this field. Also, the reference section can strengthen by including some of the following papers:
9. Dynamic Hardy-type inequalities with non-conjugate parameters, Alexandria Engineering Journal, 2020 ,1-10.
10. Hilbert Type Inequalities For Time Scale Nabla Calculus, Advances in Difference Equations, 2020, 619,1-21.
11. Some Dynamic Hilbert-Type Inequalities For Two Variables on Time Scales, Journal of Inequalities and Applications, 2021, 31,1-21
12. Well-posedness of Stochastic modified Kawahara equation.

 

Recommendation: After my careful reading, the topic discussed is novel, interesting, significant and important for the related areas. Scientific sound and the problems in the paper is solved in a successful way. Therefore, I recommend the publication of this paper in '' Mathematics " after these minor modifications. So, I want to read the revised version of paper after publishing.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Dear Editor,

 

The new referee report you requested is attached containing the questions and answers you want,

 

Respects.

 

New Review Report(MDPI-M -Manuscript ID Number mathematics-1567373)

 

Article title:         Beltrami equations on Rossi spheres  

 

Possible questions and answers about the article:

 

1. What is the main question addressed by the research?

 

                In the article, the  illustrates, necessary properties, definitions and theorems in finding variationality features of a special “the globally non embeddable CR- spans” structure modelling are characterized Beltrami model equations(based Rossi operators).

 

2. Do you consider the topic original or relevant to the field? Does it address a specific gap in the field?

 

                This topic fills the specific gap in characterizing the original “applied mathematics, Bargmann representation approach, optimization problems” model issue in applied science fields (with the Rossi sphere).

 

3. What does it add to the subject area compared with other published material?

 

                The compared with other published material,there are adding “in determining special approach field, sufficient conditions for a special equation and its contents to be the modelling with special CR manifold structures on the connected Beltrami approach theory” the subject area(under quasiconformal map).

 

4. What specific improvements should the authors consider regarding the methodology? What further controls should be considered?

 

                Authors can improve by developing new aspects of a special approach structure on connecting with the Beltrami equyation to the approach theory equations of Fourier transforms regarding the Methodology, The authors can provide a control mechanism by making new different aspects from the phenomenon of developing special approach features(in high-dimensional Heisenberg groups and models).

 

5. Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?

 

                The results appear consistent with the evidence and arguments.

 

The arguments presented adequately address the main question.

 

6. Are the references appropriate?

 

                In the article, references appear to be relevant to the study subject.

 

7. Please (if any) include any additional comments on the tables and figures.

 

                model mechanisms formed from special Fefferman metric features on approach theory and special Bargmann forms are developed and shown in terms of article images (under the Tanaka-Webster connection).

 

report:In the article, it is seen that  the special " in applied mathematic field method(the theory of applied mathematics and science)( applied theoretic method)" featured definitions and calculations are made (the special Beltrami forms) for some special applied approach form equations (with Lewy operatör).

 

findings: In the article, the special applied Beltrami approach form equations are seen (the special calculations) with the help of some special forms (under quasiconformal map).

 

strengths: In the article, there are proficiency-enhancing conditions for the special applied approach forms.

 

weaknesses: In the article, it may be necessary to develop some extra new conditions for the special applied approach forms of this type.

 

any minor issue: Although there are some minor weaknesses in terms of mathematical language, the article can be considered sufficient.

 

result: The article is suitable for publication for the journal.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

See the attached file

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Beltrami equations on 3-sphere are derived such that to describe quasiconformal mappings from the Rossi sphere, in this paper . using the Greiner-Kohn-Stein solution to the Lewy equation, and the Bargmann  representations of the Heisenberg group, the authors solve the Beltrami equations for Sobolev type solutions. This result is interesting and important,  so I think this paper can be accepted to be published in the journal mathematics.

Author Response

See attachment.

Author Response File: Author Response.pdf