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Are there research papers which are written on understanding (rather than deriving new facts and theorems) of topics of Mathematics in some new and unique approach not used before?

For example, a paper that would give a new type of proof of a known result and things like this.

If the answer to this question is 'Yes', then how much weightage is given to such a research paper in terms of its acceptance and publication in a reputed journal?

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    I have no idea what it would mean to understand a topic in a new direction. – Tobias Kildetoft Sep 26 '16 at 7:08
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    Do you mean a paper that would give a new type of proof of a known result and things like this? – quid Sep 26 '16 at 8:31
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    I agree that this question is too unclear as written. I think that perhaps @mathlover might be asking about "expository papers." An example of respectable journal that does publish such papers would be the American Mathematical Monthly. – Brian Borchers Sep 26 '16 at 14:40
  • @quid.............yes – Nitin Uniyal Sep 26 '16 at 15:47
  • @Brian Borchers....see my edit. – Nitin Uniyal Sep 26 '16 at 15:55
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There are many papers out there that give a new proof of a previously-existing theorem. You can search for "a new proof of" in mathscinet (or even google) and find lots of good papers in respectable journals.

However, the new proof must provide new insight to be worth publishing. Two proofs can look different but actually be very similar, in the sense that they are relying on the same key ideas. This can be a bit subtle (and even subjective in edge cases), so it's a good idea if possible to consult with an expert in your field before making your findings public.

But if you manage to prove the result in a more elementary way than the original proof, that's a big point in your favor. It's always good to know that a result can be proven without appealing to a powerful theorem. If you go in the opposite direction (e.g. showing that a theorem from the 1950s also follows from Fermat's Last Theorem), you're less likely to impress people.

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To give a concrete example, the hook-formula in combinatorics, has several proofs. There is a bijective proof by C. Crattenthaler, a probabilistic proof by Greene et al, a geometric polytope proof by I. Pak, a proof based on Lagrange identities by A. N. Kirillov, another bijective proof based on hokey players(?) by D. Zeilberger.

And don't get me started on proofs of the Littlewood-Richardson rule.

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Irit Dinurs paper "The PCP Theorem by Gap Amplification" (JACM) essentially reproved a known theorem in a striking and surprising way. The paper won some very prestigious research-awards. Of course, the paper contains many new ideas and techniques introduced for the purpose of reproving the PCP theorem. In particular, the paper contains lemmas and theorems that where not known before, but the main point of the paper is to provide a better understanding of a known theorem.

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A research paper should deliver novelty, so I wouldn't expect research papers that advance understanding of an existing topic, without providing novel material. By comparison, a review paper or a handbook chapter might achieve such an objective. For instance, Very Short Introductions. Some review papers are highly cited, so presumably they are considered valuable.

EDIT. To conclude, research papers advance knowledge, whereas review papers, handbook chapters, etc. advance our understanding of existing knowledge. Thus, by my reasoning, "research papers which are written on understanding" aren't common, but review papers, handbook chapters, etc. "on understanding" are common.

EDIT II. The scope of the question has changed since I provided this answer. I agree that research papers providing new proofs of known results most certainly exist. My answer acknowledges the existence of such papers, because they deliver novelty.

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    @user2768......Cann't we deliver novelty by providing a way to others in understanding of what is already in existence? – Nitin Uniyal Sep 26 '16 at 16:01
  • That depends on how you define "novelty" ;-) More seriously, let's ignore my choice of words and focus on the message I intended to convey: research papers advance knowledge, whereas review papers, handbook chapters, etc. advance our understanding of existing knowledge. Thus, by my reasoning, "research papers which are written on understanding" aren't common, but review papers, handbook chapters, etc. "on understanding" are common. (Note that the question has been edited since my above answer.) – user2768 Sep 27 '16 at 7:44

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