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Say you have an idea and you can't find any paper or book that explicitly states that idea. You don't know for sure if you're right in assuming that you came up first with that idea so you decide to publish it. What are the consequences if you're wrong and the idea has already been published. Do the reviewers point that out for you before the paper gets published? Is your career over because of that?

To be more specific, I have an idea for constructing iterative formulas for solving systems of nonlinear equations for any convergence rate. I also have an elegant proof for that. The problem is that I have found just one article that explicitly states that idea just for functions of one variable, but doesn't expand on it.( just one paragraph and then continues with something else)

BinaryBurst
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    If you're lucky, the reviewers will point out the earlier publication. If you're unlucky, the reviewers will not know about the earlier publication, your paper will get published, and later, when (or if) someone points out the earlier publication, you will feel lousy about it. This happened to me twice; in both cases, I published (in the same journal as my paper) a one-page note acknowledging the prior publication of the result. My career did not end. – Andreas Blass Jul 13 '17 at 16:40
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    Ideas are a dime a dozen. Executing on the idea is the hard part. – Jon Custer Jul 13 '17 at 16:40
  • Possible duplicate of Is it plagiarism if someone else came up with something before you did? – Wrzlprmft Jul 13 '17 at 17:42
  • @Wrzlprmft, I don't really think this is a dup of the post you mentioned, though they are similar. The one asks if it's plagiarism. This one, while using the term plagiarism as well, is really asking what happens if you try to publish something without realizing it's been done. I think I will try to edit the subject. – Fred Douglis Jul 13 '17 at 19:05
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    If it was an honest mistake, it probably won't be a big deal. That said, some authors react worse than others. – Ray Jul 13 '17 at 21:30
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    @JonCuster Kind of a strange conclusion, but I guess it depends on what "idea" means. What does 'executing an idea' mean in mathematics? I'd say that, at least in mathematics, ideas are worth their weight in gold. – Jair Taylor Jul 13 '17 at 23:24
  • @BinaryBurst Are you saying you have an $O(1)$ algorithm for solving systems of nonlinear equations? I mean... you have to at least read in the equations first, right? – Jair Taylor Jul 13 '17 at 23:26
  • No not $O(1)$. It's a method to construct the iterative formula up to the desired convergence rate then you use that formula to iterate until you get a desired number of exact digits. – BinaryBurst Jul 14 '17 at 06:47

2 Answers2

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Ignorance is not plagiarism. If you write that "To the best of my knowledge" the algorithm is new, you imply that you researched the matter duly (I would suggest to consult some experts if you are kind of an outsider to the field) and found no prior publication. If such a publication exists, it will not be plagiarism. It may be embarrassing, but not a career killer by any means (see first comment to your question).

However, if you know that a suggestion exists for the one-dimensional case, you should state that: "This algorithm has been previously suggested for the one-dimensional case by XXXX." You can also qualify this statement to better highlight what your work adds: "...but no proof of convergence was offered." or "Here I generalize the algorithm to n dimensions and provide a proof of convergence."

Stefano
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    We didn't know that a result was already proven (this was in olden times when print journals took years to publish). After the paper was accepted we put a "Note added in proof" mentioning the other paper. We had some citations as "and independently by". (Our proof was different and only a part of the paper.) – Keith McClary Jul 14 '17 at 05:14
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It is the duty of reviewers (in fact, part of THE main duty is to see if the paper makes a original contribution). If you have made an honest effort to find previous work, talked to your colleagues and/or advisor ( if applicable), your job is done. Plus most good ideas are rediscovered ALL THE TIME, in different subfields, different context etc. However, don't be this guy: Rediscovery of calculus in 1994: what should have happened to that paper?

mystupid_acct
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