# Mathematical Equations in Research Papers and Plagiarism

Good Day - I am writing a review paper of some techniques, which involve mathematical models. The authors utilize some equations to represent their model comprised of different components. Since I need to explain their methodology, its not possible to do it without referring to their equations. So I have to add their equations in the review. I will properly cite every author's work, which makes sense as I am explaining their work.

What could be the issues I might face when it comes to plagiarism in this case?

Following up on my comments, you state that your context is writing a review article on variational approximations. There's no need to cite individual equations. The standard format might be:

"Rasmussen et al 2021 introduce the SuperVar method, which uses a factorised joint p(x, y) = p(x)p(y). Under these assumptions, they show that [ equations ]. They also bound the variance of the SuperVar estimator as [ equations involving a,b,c ] where a,b,c are some things."

Since this is a review article, there's no a priori expectation that any equations/results are your own. So much so that if you were to give any of your own results, I'd point that out, rather than pointing out equations from other papers.

• Yes - At the start of paragraph, I write it like this "In order to tackle the issues pointed above, abc et. al  proposed another algorithm, named ... " ---- after some details, I present their methodology, which basically includes how they setup the priors, posteriors and estimations. Usually, there are around 6-7 equations per paper, but as far as I see it, its necessary to include the equations, because of the nature of papers under discussion. Mar 4, 2021 at 14:10

Plagiarism is representing the work of another as you own. You aren't doing this here if you properly cite the work, so no, it isn't plagiarism. Just make sure that your citation is somehow "attached" to the equations, perhaps by using appropriate phrasing when including them in addition to the citation. "The following, from the paper:...".

Don't confuse simple copying with plagiarism. Even paraphrasing can be plagiarism if it isn't cited.

I'll also assume here that you don't have copyright issues (copying too much from covered works), since there are some exceptions for mathematical work. But copyright and plagiarism are, again, different issues.

I assume that the work of other authors that you are referring to has already been published.

You can add someone else's equation in your work, but it has to be absolutely clear to the reader that this is the original work of someone else. And it must be clear if you chose to modify it a bit, e.g., by using different greek letters, or maybe shorten it a bit for the benefit of your readers.

We all lean heavily on other's work; plagiarism is about making the impression that the work is yours. It can for sure be a bit difficult in the beginning to reference correctly. This two-pager by Neidinger can be a place to start: https://zdaugherty.ccnysites.cuny.edu/teaching/plagiarism.pdf

• Strongly disagree that you need to point out 'using different greek letters'. Theorems/lemmas that originate from other papers will almost always be stated with notation matching the current paper, not the source. There's absolutely no expectation in maths/tcs that a cited equation/result is not modified/shortened, just that the implication wrt the context of the current paper matches. Mar 3, 2021 at 14:22
• @Oxonon -- So as far as I read this line "Theorems/lemmas that originate from other papers will almost always be stated with notation matching the current paper, not the source."----- So for instance, we should notify at the start that in this paper, we use X to represent a dataset, W a weight, and C as a constant.. no matter what happens, we should replace the equations used in all papers with respective letters as defined in our paper.. but still our definition can conflict with others.. for example we all usually denote X as a dataset in our models. Mar 3, 2021 at 14:26
• @Oxonon - Moreover, if we use something that looks odd, for instance a Z to denote a dataset, whereas usually it is taken as hidden variables in our case.. so things can get confusing.. Mar 3, 2021 at 14:28
• Within a formal statement like a theorem or a lemma, you'll likely redefine the key quantities within the statement (matching your own notation). E.g. "For a dataset X of size n and a kernel k ...". Within the main body of your writing, you'd have introduced X as the dataset of size n at the start of your paper, and then when referring to other work you'd state "Authors XYZ show that f(X) scales as n^3", or "The gradients of f show that f(X)=O(n^3) (Authors XYZ 2021)." Mar 3, 2021 at 14:47
• Your notation should be self contained and self consistent --- I should never need to look the work you cite to figure out what Z,X,W and C are. And if you are introducing an equation from other work that applies to your current notation exactly, you state it in your current notation, you don't go ahead and introduce their notation for the sake of that equation, since that'd be confusing/duplicant. You absolutely do not need to notify the reader that you do not use the same notation/wording, the reader has no expectation that it is so. Mar 3, 2021 at 14:50

Since you will properly cite every author's work, you shouldn't face any plagiarism issues. Just make sure to explain that the mathematical models include the equations you present. For instance:

Alice, Bob, and Charlie propose a mathematical model comprising Equations (1)-(3) [ABC21].

You can then go on to explain a bit more about the model and present each of the equations. It is absolutely clear to the reader that each of those equations belongs to Alice, Bob, and Charlie, rather than you.

• Actually, there are around 20+ such models which I am summarizing. So every review starts typically like: Author ABC proposed a methodology based upon Variational Approximation. The factorization is given by the following equation: ---eq. here---- Moreover, the model uses the prior given below, which in turn is a Gaussian with parameters a,b --- eq. here ---- So against a review, I have around 5,6 equations which I copy, but try to tweak these a little based using different alphabets/letters. However not all the time, its possible to change tweak the equations. Mar 3, 2021 at 14:18
• @rockstone435 Ultimately, your exact presentation will depend on your exact requirements. With 20+ models, I expect categorisation into several sections. Each section can open with a brief overview, perhaps explaining any commonalities and tweaks, e.g., mentioning using different alphabets/letters. Thereafter each subsection can introduce a different model. Rather than explicitly referencing Equations (1)-(3), you could simply mention that each subsection contains equations from the original authors, except when you state otherwise. You are then free to simplify/tweak equations (and say so Mar 3, 2021 at 15:13