SciPost Submission Page
Using SimTeEx to simplify polynomial expressions with tensors
by Renato Fonseca
Submission summary
| Authors (as registered SciPost users): | Renato Fonseca |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.14390v2 (pdf) |
| Code repository: | https://www.renatofonseca.net/simteex |
| Code version: | 1.0.2 |
| Code license: | GNU General Public License version 3 |
| Date submitted: | 2025-02-11 12:37 |
| Submitted by: | Fonseca, Renato |
| Submitted to: | SciPost Physics Codebases |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Computations with tensors are ubiquitous in fundamental physics, and so is the usage of Einstein's dummy index convention for the contraction of indices. For instance, TiaUaj is readily recognized as the same as TibUbj, but a computer does not know that T[i,a]U[a,j] is equal to T[i,b]U[b,j]. Furthermore, tensors may have symmetries which can be used to simply expressions: if Uij is antisymmetric, then αTiaUaj+βTibUjb=(α−β)TiaUaj. The fact that tensors can have elaborate symmetries, together with the problem of dummy indices, makes it complicated to simplify polynomial expressions with tensors. In this work I will present an algorithm for doing so, which was implemented in the Mathematica package SimTeEx (Simplify Tensor Expressions). It can handle any kind of tensor symmetry.
Current status:
Reports on this Submission
Strengths
1 - There are some well-explained examples in the algorithm's presentation.
Weaknesses
1 - Limited Novelty of the Approach
2 - Absence of Comparison with Existing Tools and Algorithms
3 - Lack of Clarity, Particularly in the Initial Sections
4 - Informal Tone and Redundancy in the Writing
Report
The paper introduces SimTeEx, a Mathematica package designed to simplify polynomial expressions involving tensors with symmetries. The author claims that the method generalizes to any tensor symmetry and provides examples demonstrating its application.
**Overall evaluation:**
Although the problem statement is important and the ideas are sound, the paper requires significant revisions (see below). Also, the paper does not meet the acceptence critera for Scipost Physics Codebase since benchmarks are missing.
Although this might be possible, I am uncertain if this can be managed to be done in a certain time-frame.
These revisions must at least include
(*) Adding benchmarks with existing software
(*) Proving novelty: by comparing the usability with existing software and showing which instances can only be treated by the new approach.
(*) Relating the mathematical background (especially Section 2-3) to the tools presented in the rest of the paper.
**More details on the weaknesses:**
As noted in the Weaknesses section, several aspects of the paper require substantial revision before it can be considered for acceptance.
1 - Limited Novelty of the Approach
While the problem of simplifying tensor expressions is relevant, it has already been addressed in numerous previous works. Presenting new algorithms for this problem is crucial. However, the author states (for example on pages 2-3 and on many other places of the manuscript) that the presented algorithm shares many similarities with existing packages such as Redberry, ATENSOR, Cadabra, and xPerm.
From the manuscript as currently written, it is unclear in what way the methods presented are novel. The author briefly mentions that certain instances can be handled by SimTeEx that one other tool cannot (e.g., at the end of page 11), but the manuscript lacks a thorough and detailed discussion of the specific types of expressions and symmetries that SimTeEx can handle, which are not addressed by existing tools. An in-depth comparison of the capabilities of SimTeEx and other packages would be helpful, particularly outlining the cases where SimTeEx demonstrates clear advantages.
2 - Comparison with Other Tools with respect to Performance Benchmarks:
There is no direct comparison in the paper regarding the complexity of the current tool relative to others. A comparison with existing tools in terms of performance and computational efficiency is essential. It would be valuable to include benchmarks demonstrating how SimTeEx performs relative to packages like Redberry, ATENSOR, Cadabra, or xPerm, both in terms of time complexity and handling more complex tensor expressions.
This is also in conflict with one of the acceptence criteria for this journal: "Benchmarking tests must be provided."
3 - Lack of Coherence and Structure
The manuscript lacks a consistent, cohesive narrative. For example, the connection between Section 2 (dummy indices), Section 3 (polynomials as a vector space) and the rest of the paper is unclear. It is important for the paper to clearly explain how these sections contribute to the overall methodology and how they relate to the implementation of SimTeEx. Without this explanation, the paper seems disjointed, and the relevance of certain sections remains ambiguous.
4 - Informal Tone and Redundancy in the Writing
In several places, the manuscript could benefit from a more formal and concise writing style to enhance clarity and professionalism. There are also multiple grammatical/typographical errors in the manuscript. For example, in the Introduction:
"I have tried to make the discussion there somewhat self-contained, however textbooks on the matter, such as [10], might still come in handy." (page 2)
This phrasing is quite informal and could be revised to present the core ideas more directly without referring to external sources for introductory material.
"Having said this, the reader unfamiliar with the permutation group and its representations shouldn’t be overly concerned as the algorithm described in this paper does not rely on it, nor does one need to know any of this in order to use the main function of the SimTeEx program (to be introduced later), which puts a tensor expression in canonical form." (page 2)
The sentence is lengthy and could be streamlined for improved readability.
Additionally, there are several grammatical issues and typographical errors, which are noted in the requested changes below. The manuscript also occasionally uses the future tense where a different tense would be more appropriate, for example:
"On this topic, I will note that the user is free to [..]"
A more consistent and precise writing style would strengthen the overall presentation of the paper.
Requested changes
In order to decide whether the paper can be accepted the author needs to add several additional information:
(1) A clear information what can/cannot be done by this algorithm compared to other packages such as Redberry, ATENSOR, Cadabra, and xPerm
(2) A benchmark test, comparing the tool with the other packages
(3) Setting a connection between Section 2, 3 and the rest of the paper.
Some of the grammatical and typographical errors found:
"Likewise, under the assumption that k is also fully symmetry [...]" -> is also fully symmetric
well known -> well-known
"Things becomes more complicated [...]" -> become more complicated
"The user does not need to know what is the representation" -> what the representation is
"In fact the user is free to provide" -> In fact, the user is [...]
"code supporting multi-term symmetries which I am aware of" -> that I am aware of
"One should be aware that the function CanonicalForm when applied to two equivalent expressions," -> CanonicalForm, when
"although it is always be true that" -> "although it is always true that"
"performs the sorting operation which is mentioned at the end of section 4." -> performs the sorting operation mentioned at the end of section 4.
"not to introduce new ones which are unfamiliar to the user." -> ones that are unfamiliar
"On often encounters" -> one
"simple modifications to algorithm presented" -> to the algorithm presented
"describe an algorithm which can take into account" -> that can take into account
Recommendation
Ask for major revision