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Being a low rep user who is somewhat unfamiliar with the workings of this site, I would appreciate it if I could get some feedback on a question I am considering asking. It would be a reference request question:

"Have cellular automata been applied to solving PDEs on closed curves? If yes, what are some recommended reads on the topic that develop the theory?"

It is motivated by a research problem I am currently struggling with. I am a biomedical engineering grad student though, not mathematics.

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  • $\begingroup$ I know next to nothing about cellular automata or PDE, but I don’t see anything wrong with the question, this sounds like a straightforward reference request. $\endgroup$ – Emil Jeřábek May 6 '14 at 13:22
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    $\begingroup$ I would suggest adding to the body of the question some more details on what you have in mind, as PDEs are a very large subject. You should also be more clear about what you mean by "on closed curves." Also, presumably you did a search on cellular automata and PDE (in google even). Did you find anything? Perhaps you have already read meta.mathoverflow.net/questions/882/… . Expanding on the question with that advice in mind may make it more successful. $\endgroup$ – j.c. May 6 '14 at 13:51
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    $\begingroup$ As an academic question it is reasonable; as a successful question for MO, it should be more specific. Revealing more detail about your motivation would help, as would narrowing the scope, e.g. hyperbolic equation, Dirichlet problem, even Not a Dirichlet problem. $\endgroup$ – The Masked Avenger May 6 '14 at 16:49
  • $\begingroup$ CAs have a lot of serious/deep research but maybe not exactly in the way you describe. CAs are digital, how can they be applied to continuous curves? $\endgroup$ – vzn May 7 '14 at 20:21
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    $\begingroup$ If you ask the question that way, I would vote to close. It feels like a fishing expedition and not a real question. Are there reasons why you suspect there is such a connection? You should mention these reasons as explicitly as possible. Even better if you know of any literature where cellular automata have been applied to related concepts, something concrete that would indicate that there is some hope for this to be the case. $\endgroup$ – Andrés E. Caicedo May 8 '14 at 6:20
  • $\begingroup$ The following related question was bumped today: mathoverflow.net/questions/87612/… perhaps there's something of use to you in the answers $\endgroup$ – j.c. May 14 '14 at 17:25
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What I am writing here is mostly just a digest of other users' comments above. As it stands, the short

Have cellular automata been applied to solving PDEs on closed curves? If yes, what are some recommended reads on the topic that develop the theory?

it not a good question for MathOverflow. It is currently too broad, and it is unclear whether it admits an answer, and it sounds a bit like a fishing expedition since, as vzn wrote

CAs are digital, how can they be applied to continuous curves?

On the other hand, if you edit the question a bit it may become a good question.

  1. Be more precise: one of the easy ways to address vzn's comment is to restrict your attention to the case of numerically solving PDEs. As numerical methods are by definition digital, such an objection cannot arise. But please note that if you are asking a question about the frontiers of numerical methods, the website Computational Science may have more experts with up-to-date knowledge.

    In addition, what exactly to you mean by closed curves? What sorts of PDEs are you thinking about? What benefits are you envisioning coming from CA?

    For example, if you mean by closed curves the smooth image of $\mathbb{S}^1$ in some manifold, then why are you even thinking about PDEs? Shouldn't the relevant question be one of ordinary differential equations? Do you actually mean closed algebraic curves? Then can your problem be described in the context of differential algebra? Is the word "curves" a typo? Are you thinking about some sort of fractal? or some sort of higher dimensional object?

    Next, what kinds of PDEs are you thinking about? If the PDE admits any sort of maximum principle, then on a closed manifold it cannot admit anything other than the trivial solution.

  2. What do you know? Why are you led to ask this question? Are you faced with a particular PDE that cannot be easily resolved using conventional methods? Does the PDE itself has a good physical/computational interpretation that makes the CA implementation sensible? Do you know of any cases where CA has been applied to study any ultimately continuous phenomena? Are there examples where CAs have been shown to be useful for solving a PDE, not necessarily on a closed curve? Have you heard of this idea from someone in particular and want to ask for clarification?

  3. You wrote that the question is inspired by a research problem you are struggling with. Please share as much about the actual problem you are working on as possible. (I understand there may be some desire to keep some details under wraps: but do understand that as mathematicians, we are trained for abstraction. Perhaps your CA idea is complete hogwash, but it is possible if you undo some of your own abstraction and show us something a bit more raw, we can give you something else, which may not be CA, but which may still be useful.)

    (As an aside, this reminds me of the (possibly apocryphal) story of the engineer who walked into the office of a mathematician and asked to know the general form of the Schwarz-Christoffel mapping onto an arbitrary polygon. The mathematician dutifully wrote down the answer and asked about the reason for the question. The engineer's reply was allegedly: "Well, we really want to know the case for the mapping onto a round disc, but we figured that question is too hard, so we'll just ask you the easy one and approximate by taking $n$ to infinity.")

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    $\begingroup$ Hi Willie. I have a concern with point 3: I have struggled a fair bit coming up with an answerable research problem, as there aren't too many low-hanging fruit left in my area of research. If I describe quite a few details, how do I know someone (more capable than myself) won't simply make it their research project, publishing much earlier than myself? It's very selfish of me to think, but I would like some practical advice on that point in particular. The other two points will fall in place naturally as I write a more detailed question. $\endgroup$ – user89 May 9 '14 at 22:44
  • $\begingroup$ @user89: (a) if it is indeed a low hanging fruit, you are essentially describing security by obscurity and it is something that cannot be counted on. (b) Most people here would be personally embarrassed to try to scoop a student. (c) If you are really worried, don't ask on MO. Ask your advisor, or ask your advisor to recommend some people to ask, or ask members of your local math department. (d) Is this going to be your thesis? If not, most mathematicians are not so great in bio-engineering, so maybe you can try to get a joint paper. $\endgroup$ – Willie Wong May 12 '14 at 7:06
  • $\begingroup$ Some tangentially related discussion to point (a) above over at tea: tea.mathoverflow.net/discussion/1230/… tea.mathoverflow.net/discussion/1498/… $\endgroup$ – Willie Wong May 12 '14 at 7:07
  • $\begingroup$ Thanks for your time kindness. I will take all of this information into account if I decide to write out a more detailed question on MO. $\endgroup$ – user89 May 12 '14 at 23:22

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