Advancing Tensor Theories

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the paper under review, the author generalizes tensor theory through the innovative notion of a generalized tensor index, a versatile framework that unifies diverse tensor indices, and explore its transformation properties.

A geometrical interpretation of these generalized tensors is provided by using fractional derivatives.

The present article extends prior conventions by generalizing the tensor concept, introducing the versatile generalized tensor index that encapsulates diverse tensor indices. The transformation of generalized tensors is defined by using fractional derivatives. These extensions have applications to partial differentiation and integration.

The paper under review has a high degree of novelty. The new concepts are interesting.

Author Response

I provide my reply to the comments of the reviewer in the attached manuscript

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper offers a bold and innovative extension of tensor theory. It introduces generalized tensor indices and forges novel connections with category theory via fractional derivatives and abstract algebraic structures. The work is highly ambitious, integrating rigorous mathematical constructs with philosophical depth. However, prior to its acceptance for publication in such a prestigious journal, several improvements are necessary.

1. The notation for generalized indices (such as \( \mathcal{I} = \ell_{k_{-j_i}} \)) and layered transformations (in §3.4–3.5) is extremely intricate. Simplifying the notation or including a glossary would significantly enhance readability.
2.  The figures referred to in §4 (for instance, Fig. 1) are not present, which weakens the geometric interpretations. Incorporating visualizations would clarify abstract concepts like fractional tangent submanifolds.
3.  A practical example is urgently required. For example, how do setorial tensors (§5.1) function in computational scenarios? Which physical systems could potentially benefit from functorial tensors (§5.3)?
4.  Presenting a specific application (such as in gauge theory or machine learning) would firmly establish the utility of these generalizations.
5.  The connection between category theory and tensors (§5.2–5.3) remains rather abstract. A diagrammatic explanation (using, for example, string diagrams or commutative diagrams) could clarify how categorial tensors act as mediators between functors and multilinear maps.
6.  The fractional derivative formalism (§4) heavily relies on Calcagni’s work but lacks a self - contained introduction. A brief review of the geometric implications of fractional calculus would make the paper more self - sufficient (DOI: https://doi.org/10.17654/0974324324024). Additionally, the two - scale fractal derivative could offer an alternative new approach. 

Author Response

I thank the reviewer for their comments. I provide my reply to the comments of the reviewer in the document attached

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Review on the Manuscript: Advancing tensor theories

The manuscript presents a bold and original attempt to generalize tensor theory by introducing “generalized tensor indices,” mapping them to category theory, and exploring fractional differential geometry. It also attempts to define new types of tensors—setorial, categorial, and functorial—using substitution and graphical approaches. The work is highly ambitious, conceptually novel, and ventures into theoretical synthesis of abstract mathematics with philosophical implications.

However, the manuscript suffers from a number of critical issues in presentation, mathematical rigor, clarity, and formatting. While the underlying ideas may be innovative, the current form undermines their potential impact.

1. The term “generic generalised tensor” is introduced early but remains conceptually vague. The mapping T : V → G is presented multiple times, but without clear mathematical structure or motivation.
2. Several concepts are defined and redefined in very similar terms, e.g., generalized tensors in Sections 2.2 and 2.3, or in Theorems 1 through 4. Consolidate similar theorems or explanations. Avoid restating the same ideas unless the new presentation adds clarity or depth.
3. The introduction and “importance criteria” section contains exaggerated or speculative claims about the philosophical and mathematical novelty of the work (e.g., “beautiful mathematics,” “deep mathematics”) without objective evaluation. Let the reader or referee judge the philosophical or aesthetic value based on results and structure. Remove subjective self-praise.
4. The use of functors for substitution is vague and doesn’t align with standard category theory usage.
5. Terms like “middle index,” “generic index,” “triality,” “D-ality,” and “functor of substitution” are introduced without standard definitions and often deviate from accepted mathematical terminology. Align terminology with existing literature when possible, or clearly mark novel definitions. Use footnotes or endnotes to explain novel terms in the context of established theory.
6. The paper uses deeply nested indices (e.g., ℓk...ji) and defines them recursively. However, no concrete structure (e.g., tuples, trees, multi-index sets) is rigorously defined. This makes interpretation and application infeasible.
7. The indexing hierarchy is unnecessarily complex and not grounded in tensor algebra standards.
8. The use of fractional derivatives (e.g., ∂1/2(∂x)1/2) is novel but lacks proper definitions or citations for their interpretation in tensorial settings.
9. The geometrical interpretation of fractional derivatives (Section 4) is speculative and lacks formalism.
10. The generalization of tensor transformation rules (e.g., in Section 3.4) involving fractional powers is nonstandard, and their validity is unclear.
11. The “proofs” in Sections 2.2–2.3 are informal and tautological (e.g., proving that T : V → G holds by defining a substitution SF : R → G).
12. The functor of substitution is treated as a primitive idea but is not rigorously defined.
13. Abstract: “revealing new insights into their structure” → “into its structure” (tense agreement)
14. Page 3: “tensor theory’s intersections with mathematical analysis” → consider rewriting as “intersections of tensor theory with mathematical analysis.”
15. Page 4: “triality as an extension of the concept of duality of vector spaces” → “an extension of vector space duality”
16. Page 4: “This document draws inspiration from modern category theory Definitions” → “definitions” (not capitalized)
17. Page 7: “To prove Theorem 1 , we consider…” → remove the space before the comma.
18. Equation (50): lacks clear motivation and introduces too many variables without context.
19. Section 7 in the table of contents is blank and never appears in the text.
20. Math expressions are inconsistently rendered (sometimes in LaTeX-style code, sometimes written plainly).
21. Some figures (e.g., Figure 1) are referenced but either missing or improperly formatted.
22. Excessive repetition of “we write T : V → G” without incremental clarity.
23. Titles like “Definition of standard tensors” appear multiple times.
24. Many concepts and phrases are repeated verbatim, especially the mapping from vectors to sets and the use of substitution functors.
25. Explicitly define what a “generalized index” is—use set-theoretic or algebraic formalism.
26. Ground all new definitions in standard linear algebra, differential geometry, or category theory when possible.
27. Reduce the depth of indexing and use established multi-index notation when possible.
28. Proofread the document thoroughly to remove typos and improve flow.
29. Use shorter paragraphs, clearer transitions, and better organization (especially in Sections 2 and 3).
30. Include numerical or symbolic examples to illustrate generalized tensors, their transformation, and applications.
31. Provide more intuitive and theoretical motivation for key generalizations.
32. Connect the work more clearly with established literature.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

1. Proofread the document thoroughly to remove typos and improve flow.
2. Use shorter paragraphs, clearer transitions, and better organization (especially in Sections 2 and 3).

Author Response

I thank the reviewer for their comments. I provide my reply to the comments of the reviewer in the document attached

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript proposes a novel generalization of tensor theory through the introduction of generalized tensor indices and attempts to connect this framework with category theory, set theory, and fractional calculus. The author introduces concepts such as generalized tensors, categorial tensors, and functorial tensors, aiming to broaden the scope of traditional tensor analysis and establish connections with modern abstract mathematical theories. Kindly, consider the following comments:

1. The manuscript suffers from numerous grammatical issues, typographical errors, and awkward phrasing throughout. These issues significantly reduce the readability and clarity of the paper, obscuring the underlying mathematical ideas.
Example: “We will forge categories and functors employing generalized tensors, crafting categorial tensors (tensors with categories as elements)” — this sentence could be more clearly and precisely rephrased.

2. Several key definitions (e.g., generalized tensors) lack precise mathematical formalism. These are introduced more through substitutions or descriptive language than through rigorous derivation or formal definitions.

3. Theorems 1 and 2 are stated informally and resemble basic mappings without proper mathematical grounding. I recommend reformulating these theorems using standard mathematical notation and language, accompanied by clearly stated hypotheses and conclusions. In addition, concrete examples illustrating the abstract constructions would significantly enhance comprehension.

4. There is noticeable repetition and conceptual overlap across different sections, especially where both informal and formal statements of similar ideas are repeated. The manuscript would benefit greatly from improved organization, a clearer narrative, and a more logical progression of ideas, which would help reduce redundancy and increase clarity.

5. Concepts such as the “middle index,” “functor of proof,” and “generic generalized tensor” are interesting but lack sufficient motivation and contextualization. Including intuitive explanations, examples, or diagrams would help clarify these notions and make the paper more accessible to a wider mathematical audience.

6. While the manuscript references some classical sources (e.g., Cartan, Leibniz), it lacks adequate engagement with recent literature in tensor theory and category theory. The author is encouraged to strengthen the literature review by comparing the present work with more contemporary research in areas such as higher-category theory, tensor networks, and applied tensor analysis.

Author Response

I thank the reviewer for their comments. I provide my reply to the comments of the reviewer in the document attached

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

This revised version demonstrates significant improvements. In this iteration, the two - scale fractal derivative is presented as an intriguing aspect for future research. Considering its potential to further enrich the concepts you've put forward, I would like to request some relevant citations on this topic. This information would be highly valuable for readers interested in delving deeper into this area. Moreover, it would contribute to strengthening the theoretical basis of your proposed future research direction. I recommend the following sources: 

He, CH; Liu, HW and Liu, C. "A fractal - based approach to the mechanical properties of recycled aggregate concretes", Facta Universitatis, Series: Mechanical Engineering, 22(2)(2024), pp.329 - 342. 

Y.R. Zhang, N. Anjum, D. Tian, A.A. Alsolami. "Fast and accurate population forecasting with two - scale fractal population dynamics and its application to population economics", Fractals, 32(5)(2024), DOI:10.1142/S0218348X24500828. 

Author Response

Round 2 of Comments Reply

 

I would like to thank the referee for their speedy and insightful comments, which have only added to the quality of the final version of the manuscript. Below, I reply to the specific comments of the 2nd reviewer. The rest of the reviewers have agreed with the final version of the manuscript.

 

Reviewer 2:

 

This revised version demonstrates significant improvements. In this iteration, the two - scale fractal derivative is presented as an intriguing aspect for future research. Considering its potential to further enrich the concepts you've put forward, I would like to request some relevant citations on this topic. This information would be highly valuable for readers interested in delving deeper into this area. Moreover, it would contribute to strengthening the theoretical basis of your proposed future research direction. I recommend the following sources: 

He, CH; Liu, HW and Liu, C. "A fractal - based approach to the mechanical properties of recycled aggregate concretes", Facta Universitatis, Series: Mechanical Engineering, 22(2)(2024), pp.329 - 342. 

Y.R. Zhang, N. Anjum, D. Tian, A.A. Alsolami. "Fast and accurate population forecasting with two - scale fractal population dynamics and its application to population economics", Fractals, 32(5)(2024), DOI:10.1142/S0218348X24500828. 

Our reply:

I thank the reviewer for their comment. To address the issue, I added the following phrases to the introduction:

The exploration of two-scale fractal derivatives offers a promising avenue for extending the hierarchical and non-local structures of generalized tensors, particularly in fractional geometry [19,20]. These derivatives, with applications in material science and population dynamics, could further enrich the theoretical and practical implications of our proposed framework.  These interesting aspects are left for a future work.

Thank you very much for your consideration.

Sincerely,

The Author

Reviewer 3 Report

Comments and Suggestions for Authors

I The authors address the comments in a nice way and I am satisfied from their replies. 

Author Response

Round 2 of Comments Reply

 

I would like to thank the referee for their speedy and insightful comments, which have only added to the quality of the final version of the manuscript. Below, I reply to the specific comments of the 2nd reviewer. The rest of the reviewers have agreed with the final version of the manuscript.

 

Reviewer 2:

 

This revised version demonstrates significant improvements. In this iteration, the two - scale fractal derivative is presented as an intriguing aspect for future research. Considering its potential to further enrich the concepts you've put forward, I would like to request some relevant citations on this topic. This information would be highly valuable for readers interested in delving deeper into this area. Moreover, it would contribute to strengthening the theoretical basis of your proposed future research direction. I recommend the following sources: 

He, CH; Liu, HW and Liu, C. "A fractal - based approach to the mechanical properties of recycled aggregate concretes", Facta Universitatis, Series: Mechanical Engineering, 22(2)(2024), pp.329 - 342. 

Y.R. Zhang, N. Anjum, D. Tian, A.A. Alsolami. "Fast and accurate population forecasting with two - scale fractal population dynamics and its application to population economics", Fractals, 32(5)(2024), DOI:10.1142/S0218348X24500828. 

Our reply:

I thank the reviewer for their comment. To address the issue, I added the following phrases to the introduction:

The exploration of two-scale fractal derivatives offers a promising avenue for extending the hierarchical and non-local structures of generalized tensors, particularly in fractional geometry [19,20]. These derivatives, with applications in material science and population dynamics, could further enrich the theoretical and practical implications of our proposed framework.  These interesting aspects are left for a future work.

Thank you very much for your consideration.

Sincerely,

The Author

Reviewer 4 Report

Comments and Suggestions for Authors

Having read through the revised manuscript carefully, I observe that the author addressed the major concerns raised during the initial review, and the paper has been greatly improved. I thus recommend, given the improvements, that the paper can be accepted for publication.

Author Response

Round 2 of Comments Reply

 

I would like to thank all 4 referees for their speedy and insightful comments, which have only added to the quality of the final version of the manuscript. Below, I reply to the specific comments of the 2nd reviewer. The rest of the reviewers have agreed with the final version of the manuscript.

 

Reviewer 2:

 

This revised version demonstrates significant improvements. In this iteration, the two - scale fractal derivative is presented as an intriguing aspect for future research. Considering its potential to further enrich the concepts you've put forward, I would like to request some relevant citations on this topic. This information would be highly valuable for readers interested in delving deeper into this area. Moreover, it would contribute to strengthening the theoretical basis of your proposed future research direction. I recommend the following sources: 

He, CH; Liu, HW and Liu, C. "A fractal - based approach to the mechanical properties of recycled aggregate concretes", Facta Universitatis, Series: Mechanical Engineering, 22(2)(2024), pp.329 - 342. 

Y.R. Zhang, N. Anjum, D. Tian, A.A. Alsolami. "Fast and accurate population forecasting with two - scale fractal population dynamics and its application to population economics", Fractals, 32(5)(2024), DOI:10.1142/S0218348X24500828. 

Our reply:

I thank the reviewer for their comment. To address the issue, I added the following phrases to the introduction:

The exploration of two-scale fractal derivatives offers a promising avenue for extending the hierarchical and non-local structures of generalized tensors, particularly in fractional geometry [19,20]. These derivatives, with applications in material science and population dynamics, could further enrich the theoretical and practical implications of our proposed framework.  These interesting aspects are left for a future work.

Thank you very much for your consideration.

Sincerely,

The Author