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There's a discussion taking place at this question Mathematical study of Mpemba effect? where the author UwF is arguing that this question isn't clearly off-topic and would like some explanations for its closure. But I think there could be a broader discussion as well about the scope of MO and specifically what sorts of questions that are roughly speaking "applied mathematics" should be admissible, and whether some more words about this should be included at the help center.

I would like to hear from the general community, but perhaps especially from those who identify themselves as "applied mathematicians" (Steve Huntsman for example?). I am cognizant of the fact that many people feel that applied math areas are rather underrepresented at MO. I'm not sure that the MO question that I linked to is a critical test case, but maybe it will help get the ball rolling.

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    $\begingroup$ In my opinion, if a physics question is translated into a proper mathematical question, it is on-topic on MO. However if it is asked as a physics question, it is rather off-topic. The given question on the Mpemba effect simply explains this effect and asks for explanations in terms of mathematical models -- hence it does not provide this translation. Therefore I think it qualifies as a reasonable physics question, but rather not as a mathematical question. $\endgroup$ – Stefan Kohl Jan 7 '14 at 15:32
  • $\begingroup$ I am not translating a physics question into a proper mathematical question, that is true, and I think this rarely happens. Mathematicians and physicists generally ask different questions, even when their discussion is motivated by the same "real world" phenomena or effect. My question is mathematical, the question is whether numerical analysis and simulation methods can deal with the basic equations describing heat flow with convection and evaporation or other fundamental processes that are responsable for the Mpemba, according to the various theories proposed by physicists and chemists. $\endgroup$ – UwF Jan 7 '14 at 15:47
  • $\begingroup$ I added an introductory paragraph on the Mpemba effect to provide a context for my question. If that is too much physics for MO, then I can delete it or replace it by a link to wikipedia. $\endgroup$ – UwF Jan 7 '14 at 15:51
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    $\begingroup$ @UwF: I doubt the first paragraph is the issue. Look at the second paragraph. It seems like your asking for experiments verifying proposed hypotheses. Perhaps you're specifically looking for in silico experiments? (That's what the third paragraph suggests.) Are you looking for theoretical work at all? Maybe? Mostly? The question is not clear about any of this. I don't think classifying the question as math or physics makes sense before those items are clarified. $\endgroup$ – François G. Dorais Jan 7 '14 at 16:04
  • $\begingroup$ I am looking for theoretical work, yes. But I am not sure why you doubt this, can you please explain what you mean by theoretical work? $\endgroup$ – UwF Jan 7 '14 at 16:08
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    $\begingroup$ @UwF: The point of what I wrote is that your question is too vague, which has nothing to do with whether it's physics or not. Once you make your question precise, it will be clear whether it is on-topic for MO. If you leave it to the reader to make the question precise, it is extremely likely that 5 users will do that in a way that makes the question off-topic. $\endgroup$ – François G. Dorais Jan 7 '14 at 16:13
  • $\begingroup$ @ François My question was a simple reference request. I am sorry that I gave some motiviation for my interest in this question and that this is not 100% purely mathematical. But I don't understand why some people here feel that they have to purge MO of anything interdisciplinary "none-pure". I can not agree with such a restricted view of mathematics. $\endgroup$ – UwF Jan 7 '14 at 16:22
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    $\begingroup$ @UwF: Please read my comments again; I explicitly said that motivation isn't the main issue I see with the question. (By the way, in general, talking about "some people" is not a good way to make friends.) $\endgroup$ – François G. Dorais Jan 7 '14 at 16:32
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    $\begingroup$ @UwF Obviously there is no need to be "sorry" (or was that sarcasm?). Possibly it would help to relegate your motivation to a section titled "background", and delineate as clearly as possible what exactly you want to know; François several comments ago was expressing his confusion over what the question is. Also: try not to be argumentative ("I don't understand why some people here feel that they have to purge MO of anything interdisciplinary "none-pure"." -- that's kind of argumentative). $\endgroup$ – Todd Trimble Jan 7 '14 at 16:32
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    $\begingroup$ There are different kinds of applied mathematics. The kind that proves theorems is clearly on-topic for MO. However, there is another kind that focuses on model building, numerical simulation, etc. This is a much fuzzier area. Technical questions about the mathematical tools they use (e.g. about numerical analysis) seem to me to be on-topic as long as they are at a sufficiently high level. But questions that are more directly about how to model some phenomena (like the one here) don't seem to me to be on-topic. The problem is that the issues here are not the mathematics, $\endgroup$ – Andy Putman Jan 7 '14 at 16:44
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    $\begingroup$ (continued) but the physics (or chemistry, or biology, or...) of the thing they are trying to model. While these are certainly interesting and important areas, I simply don't think that they are a good fit for MO. $\endgroup$ – Andy Putman Jan 7 '14 at 16:47
  • $\begingroup$ Francois, Sorry, I didn't mean you by "some people". I mean those that just downvote or vote to close, but don't get involved in the discussion. Todd, Yes, I now see that I should have organized my question in a different way. But I am not happy with the attitude towards applied and/or interdisciplinary subjects that the reaction to my post reflects. I asked for references and added a few lines to explain my interest in the subject. Why do those who don't know any reference and are not interested in the question have to close it? Why not simply ignore it? Does my question do any harm to MO? $\endgroup$ – UwF Jan 7 '14 at 16:55
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    $\begingroup$ @UwF I shouldn't presume to speak for the closers for this particular question, but I think in general a lot of people don't want MO to become a free-for-all. Thus, we have distributed moderation. You don't have to be interested in a question or have a reference at hand in order to feel it's not right for MO. With regard to attitudes toward "applied math", etc. -- this is why I made this post: to bring such things into the open. I would like readers from a broad spectrum to enter this discussion. I do sense that many "applied mathematicians" (faute de mieux) share some of your attitudes. $\endgroup$ – Todd Trimble Jan 7 '14 at 17:33
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    $\begingroup$ @UwF: I think your grieviance towards the MO community is unjustified. I have not seen any bias against applied or interdisciplinary question on MO, provided that the mathematical core of the questions is 1). Well elucidated in the question AND 2). And this core is mathematically interesting in itself. I think your question (in its current form) has failed 1). and hence it is being closed. Usually doing 1). correctly involves the step of modeling the physical phenomenon, which is expected from the person asking the question. I don't see any effort on your part to do this so far. $\endgroup$ – Piyush Grover Jan 7 '14 at 18:14
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    $\begingroup$ Possibly it could be useful to figure out what type of "applied" (for lack of a better words) questions should be asked on MO to consider which of these might be better asked on the Computational Science SE where some of the top users are mathematicians too (also in the 'narrow sense' of mathematcians proving thms and working in academia). Of course that there is another site does not make it off-topic here and question can be on-topic on multiple sites. Still the existance of that site could perhaps play a role in deciding what is and is not for MO. $\endgroup$ – user9072 Jan 7 '14 at 19:04
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I am definitely doing applied math and here my two cents: In applications of mathematics one has to face a lot of questions and issues that are not really mathematical like modeling of real world phenomena, tuning parameters in models or algorithms, software engineering or issues of implementation. Of course there are mathematical theories involved in all these issues but often one just uses basic (but often not simple by any means) tools from these areas rather than doing any research in that respect. For questions from these areas there are better sites in the stackexchange network than MO (e.g. https://scicomp.stackexchange.com/, https://stackoverflow.com/ or https://physics.stackexchange.com/). The reason is not that applied math question are not welcomed here, but that the community on other sites is better suited for these kind of questions.

My personal feeling is that it is hard to formulate a question on applied math well to make it suited for MO. I do not blame anybody for that - it's just that one has to remove all vague things (e.g. coming from underlying model inaccuracies or whatever) from the issue to get a crisp question for which a correct mathematical answer can be expected. However, this often helps to think about the core mathematical problem harder and that is often the first or crucial step towards a solution. It may well happen that something that starts as a question on an application of mathematics turns into a question in which nothing of the application can be seen anymore (and that may be a reason that one sees little questions which are obviously about some application).

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    $\begingroup$ +1 I'm very glad to have your take on this, Dirk. $\endgroup$ – Todd Trimble Jan 7 '14 at 21:57
  • $\begingroup$ Hi Dirk, Having other sites in the stackexchange netwerk that can be relevant to a good research-level question should not be a consideration by us. Also, since MO is, by a good margin, higher research-quality site than others, if you want to get an answer also for an interdiciplinary question, asking on MO can be a good strategy. $\endgroup$ – Gil Kalai Jan 10 '14 at 6:48
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    $\begingroup$ @GilKalai I partly agree. In applied math research one sometimes needs mathematical advice on tools that one wants to use (e.g. Which iterative method will perform best on this huge linear system? What pitfalls are there when I set up finite elements for this coupled system of PDEs?). Well, these question are about research but I have the feeling that I will get better answers on scicomp (in these examples) than here. Also, other users of scicomp may also find these questions/answers a useful resource. So my questions on computational tools would go to scicomp while (cont.) $\endgroup$ – Dirk Jan 10 '14 at 8:41
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    $\begingroup$ for questions on an application that need somehow broader expertise I would also consider MO. $\endgroup$ – Dirk Jan 10 '14 at 8:42
  • $\begingroup$ This recent question is a good counterexample to my point. $\endgroup$ – Dirk Jan 21 '14 at 10:13
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I voted to close.

Mathematics can be used in many other areas. However, the field of applied mathematics is different from the collection of all applications of mathematics.

When you can't figure out what is going on in a chemical system, sometimes the problem is with your understanding of the relevant chemistry instead of the analysis of a mathematical model. If the problem is that we don't understand the physical chemistry, say the phenomenon of supercooling, or the effects of dissolved gases, then this is not an applied math problem, and we can't expect mathematicians to help. We need to ask chemists who study these things. Scientists generally know the basics of mathematical modeling, perhaps better than most mathematicians, so basic modeling issues are not a stumbling block. If the chemical phenomena are understood, but the mathematical model is too unreliable or slow, or we want a general explanation of the results, then it is an applied mathematics problem.

The original question was off-topic because it did not make the question an applied math problem. The competition on the Mpemba effect was run by the Royal Society of Chemists. The arXive paper cited was in the chemical physics section of the arXive and had no mathematics classifier. (I'm skeptical of it, but I can't judge it because I'm not a chemist.) The explanations of the Mpemba effect that I have heard depend on diverse and complicated physical and chemical phenomena. Why was this question asked on MathOverflow instead of chemistry.stackexchange.com or physics.stackexchange.com?

I'm quite skeptical of the suggestion that it's not possible to separate the phenomena experimentally. AFAIK, that's a significant portion of what experimental scientists do. Which separations are possible by clever experimental design, e.g., whether we can control or eliminate convection, is not a mathematical issue.

I welcome actual questions on applied mathematics, and I have argued many times in the past that pure mathematicians have incorrectly closed questions on applied mathematics, that pure mathematicians should welcome the presence of applied mathematics on MathOverflow. The original question just doesn't look like a question on applied mathematics. Perhaps the edit helped, but I don't think it did enough.

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    $\begingroup$ Douglas, I agree with quite a lot of what you wrote, but I'm wary of your bolded sentence "The original question just doesn't look like a question on applied mathematics". I have no opinion on the original question; I just don't think I like that phrase as a general principle. It's too conservative, too reactionary. For instance, one of the major applications of category theory is to computer science. If you saw some question about monads in functional programming, maybe you (or others) wouldn't think it looked like a question on applied mathematics. [continues...] $\endgroup$ – Tom Leinster Jan 9 '14 at 23:35
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    $\begingroup$ [continued] But that's only because of coincidences of history. What we most readily think of as "applied mathematics" is applications of certain parts of mathematics to certain parts of non-mathematics - most traditionally, applications of differential equations to physics. Judging a question according to whether it fits our preconceptions of what applied mathematics "looks like" seems to me to be a recipe for excluding exciting and important developments. $\endgroup$ – Tom Leinster Jan 9 '14 at 23:37
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    $\begingroup$ @Tom Leinster: You don't have to classify all possible future developments. There is room to trust the judgement of people with $3000+$ reputation. If someone posts what appears to be a chemistry problem, whose solution appears to depend on ideas from chemistry and not mathematics, then he or she should explain why this is really an applied math problem despite the appearance. A question which appears to be about chemistry without a good explanation should be closed as off-topic. Saying that science involves mathematical models is not a good explanation. $\endgroup$ – Douglas Zare Jan 9 '14 at 23:49
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    $\begingroup$ The objection raised in this post is perharps reasonable for a vote to close on a certain reading of the original version. For the revised version it seems not to apply at all. The question is a reference request presicely for (applied) mathematical work on an related to certain models for the Mpemba effect or similar problems, namely it asks "Are there any mathematical studies [...] based on the equations proposed to describe or explain the Mpemba effect?" It does not ask for new models, an evaluation of the models, or anything. It asks for a mathemcatical study of some equations. $\endgroup$ – user9072 Jan 10 '14 at 11:13
  • $\begingroup$ There are other objections one could raise, and I do not find this a great question either (although I think it should/could stay) but the one raised here seems besides the point. In addition, why is it asked why it was not asked on physics.SE when the OP links to a question on this asked on physics.SE? This seems like pure rhetoric. Indeed, it seems very clear why it was not asked on physics.SE, namely because it is an applied math question. For example, it seems to me OP would also be interested in math. interesting work inspired by this that is not really pertinent to 'the real world'. $\endgroup$ – user9072 Jan 10 '14 at 11:20
  • $\begingroup$ Douglas: I agree with everything you say in your comment above, although it doesn't seem like a reply to my own comments. $\endgroup$ – Tom Leinster Jan 11 '14 at 21:30
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One issue with cross-disciplinary questions such as this one is that they are often vague. Regardless of what disciplines are involved, such questions have a very slim chance of surviving on MathOverflow because they have a too broad range of interpretations. The odds that 5 voting users interpret the question in a way that they find off-topic are very high, especially since there are already a lot of non-mathematical ideas into the mix! The consequences of this are also very messy since the askers are usually set on their own interpretation and can't see the interpretations that the 5 voters read into their question. In other words, vague cross-disciplinary questions almost inevitably lead to close wars and similar mayhem. Of course, the remedy for this issue is to ask good questions!

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    $\begingroup$ On the other hand, MO is intrinsically not well-suited to some cross-disciplinary questions. Although UwF's question is interesting for its connections to "evaporation, convection, gas dissolved in the water, or interactions on molecular level", it is probably a true statement that the vast majority of people who contribute to MO have next to nothing to say about these issues. One can even assert that (as I just did) without snide asides about "some people"... $\endgroup$ – Lee Mosher Jan 7 '14 at 17:18
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    $\begingroup$ There might be a larger number of people on MO interested in "in silico" experimentation (I love that phrase), and a focussed mathematical question along those lines might work better, as you suggested in a comment. $\endgroup$ – Lee Mosher Jan 7 '14 at 17:19
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I think that questions in applied mathematics and mathematical modeling/understanding of issues from other sciences are suitable for MathOverflow and such questions should be encouraged.

In my opinion UwF's Mpemba effect question is an excellent question of this type.

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  • $\begingroup$ I partly agree. The question in question seems a lot better to me than various other questions of this general type we saw (I would not call it excellent though). I think an unfortuante phrasing at the start gave it an unintended and unfortunate spin (Theo Johnson-Freyd pointed this out on main). $\endgroup$ – user9072 Jan 8 '14 at 12:18
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    $\begingroup$ Thanks, Gil. What constraints, if any, do you think there should be for applied mathematics questions to be suitable for MO? François and Dirk have already listed some considerations which appear rational to me, having to do with a desire to maintain focused questions. I'd add this: my impression is that a successful mathematical modeling often requires a certain extra-mathematical expertise; for example, one might have to acquire detailed knowledge about and visualize exactly what happens when X rays hit DNA and how DNA repairs itself, before developing a viable statistical analysis. (Cont.) $\endgroup$ – Todd Trimble Jan 8 '14 at 18:15
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    $\begingroup$ (Cont.) Should such modeling considerations, far beyond the scope of the usual knowledge of mathematicians, be allowed to enter discussions? My feeling is that such matters would prove rather distracting, as well as difficult for the community to evaluate, and that we get into slippery slope territory. So personally I'd want to pretty much constrain discussions to their specifically mathematical cores. (Feel free to add to your answer to elucidate your beliefs on such matters.) $\endgroup$ – Todd Trimble Jan 8 '14 at 18:19
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    $\begingroup$ Hi Todd, The constraints for questions in applied mathematics (and questions on applications of mathematics in other areas) are just like the constraints in pure mathematics: The question should be of interest to research-level mathematicians, of sufficient academic level, and satisfy our other requirements. $\endgroup$ – Gil Kalai Jan 9 '14 at 17:19
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In my observation it is very hard to predict which "applied" questions are well-received on MO and which are not (here by "applied" I mean question that in some rather direct way touch upon the real world, say, very roughly 'Mathematical explanation/model of [something].'

For example I do not really understand what should make the current question (especially in revised form) much (or at all) less suitable than say The human body's random number generator or The "Dzhanibekov effect" - an exercise in mechanics or fiction? Explain mathematically a video from a space station

And, there are some other questions that in my opinion are a lot worse than the current one (and the two linked to) that were treated rather better than this question.

This contribution is mainly meant to solicit explanation from anybody that does see significant difference between these questions in order to get a better understanding of others criteria to evaluate such questions.

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Note that there is an active proposal for an applied math/mathematical modeling stackexchange site that has garnered many followers but few interesting questions:http://area51.stackexchange.com/proposals/57490/mathematical-modeling

You can consider directing people there.

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