Wikipedia:Reference desk/Archives/Mathematics/2009 January 14

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January 14

Sphere specialists?

I was just wondering whether there are mathematicians specializing only in studying spheres. There are so many important properties of spheres (the well-known Hopf fibration, sphere bundles, parallelizable spheres, how many differentiable structures exist on a given sphere etc...) that I wouldn't be surprised. Any comments are appreciated! Thanks! —Preceding unsigned comment added by 129.143.15.142 (talk) 12:44, 14 January 2009 (UTC)

This sounds like the kind of thing you might see in fiction, like the show Numb3rs. 67.150.254.75 (talk) 13:12, 14 January 2009 (UTC)

The second post should be deleted in my opinion. It seems irrelevant to the question. --Point-set topologist (talk) 13:39, 14 January 2009 (UTC)

Could you not go on about deleting posts please. Personally I get annoyed with administrators saying something and then trying to cover up. I don't think it should be encouraged. Take it up on the talk page if you must. As to the question Hatcher in his book on Algebraic Tpology seems to have made quite a study of the homotopys of the n-sphere. Dmcq (talk) 13:58, 14 January 2009 (UTC)

Who is the administrator here (I am certainly not one) and who is trying to cover something up? I just said that the second post was clearly posted by someone who did not understand the question (i.e a non-mathematician) so the post is irrelevant. Therefore, as promised, I requested someone's opinion on the matter (note that I certainly don't intend on deleting the OP's post). PST (Point-set topologist)

Talk page. Not here. Dmcq (talk) 14:55, 14 January 2009 (UTC)

You might also like Smale's paradox, though why it is called that rather than sphere eversion is beyond me.Dmcq (talk) 14:02, 14 January 2009 (UTC)

Yes, there are mathematicians that specialise in spheres, my tutor is one. I don't really understand what he does, it's differential geometry which I've not studied, but it's something to do with minimal surfaces on higher dimensional spheres (looking at his list of papers it seems he's working his way up the dimensions, I think he's on S⁷ now). --Tango (talk) 16:09, 14 January 2009 (UTC)

$\mathbb {S^{7}}$ is probably the most interesting. PST

Thanks for your help everyone! —Preceding unsigned comment added by 129.143.15.142 (talk) 11:33, 15 January 2009 (UTC)

Modeling using overlapping probability distributions

First off, this isn't for anything serious - I just got to thinking about it, and am curious as to how one would go about doing it. If you wanted to model, say, rainfall, in a semi-realistic way, one way you could go about doing so is to take the past 20+ years of daily precipitation amounts and create a probability distribution from them. To simulate a period of time, then, you would just sample from that probability distribution to find what each day's rainfall would be. However, that doesn't quite work, as dry days tend to cluster together, as do wet days. So instead of just creating a distribution for just a single day, you create a set of probability distributions for 1,2,3 ... days. How would someone go about choosing a series of daily precipitation amounts such that the series follows all of the different probability distributions for each of the cumulative totals? For the single distribution case, I would integrate my normalized distribution, and use a uniform random number generator and select the precipitation amount which matches the running total of area under the probability distribution. However, I'm not able to wrap my head around how to do it for the multiple simultaneous distribution case. -- 99.154.0.155 (talk) 14:55, 14 January 2009 (UTC)

What people try and do is choose as few numbers as possible and as simple a statistical model as possible that gets the distribution as accurately as needed. Beyond that you might as well just use the original data and pick a date at random. You really have to give up trying to do things exactly if you want to model things. The next question is how accurate is good enough, for people wanting funding it is a bit more accurate than the last published model :) There is a nice article about weather forecasting here. Dmcq (talk) 15:17, 14 January 2009 (UTC)

Sounds like you're describing a Markov chain; the basic information of "on how many days did it rain" is redundant given the information about "what patterns of rain/no-rain are seen for every run of three consecutive days", so you just develop a single model with however much memory and go with it. --Tardis (talk) 18:18, 14 January 2009 (UTC)

A Markov chain does seem to be an adequate description. However, I didn't see anything in the article about how one would go about actually calculating the probability of transition for this case. To clarify, I'm not actually interested in weather forcasting, I'm just using it as a conceptual model for this general type of problem. Also, it's not just the binary rain/no-rain patterns, but the cumulative amount of precipitation (though if you imposed a resolution cut off of e.g. 1mm/0.1 in, you'd probably be able to treat the problem as a multi-valued discrete valued one, rather than a continuous valued one). How would one go about calculating the transition probabilities such that a sufficiently long chain would match a given set of probability distributions for 1 day, 2 day, 3 day ... cumulative totals? -- 99.154.0.155 (talk) 02:37, 15 January 2009 (UTC)

Ah — I somehow missed that you wanted a distribution over rainfall amounts. I believe that you can still do much the same thing: calculate (by some sort of interpolation on the data) a joint probability density function for the amounts of rainfall $R_{1},R_{2},\dots ,R_{N}$ on N consecutive days (counting even overlapping runs of N days in the data). Then, at day i, draw $r_{i}$ from the restricted distribution $f(r_{i-N+1},r_{i-N+2},\dots ,r_{i-1},R_{i})$ of one variable. --Tardis (talk) 01:29, 16 January 2009 (UTC)  

It might lead to better results if one tried to model the whole weather together, instead of just rainfall. – b_jonas 16:52, 17 January 2009 (UTC)

definition for geometry situation

If I could draw, it would be very easy.

Imagine a circle, draw a square in that circle so that the circumference is cut or "scribed" into four equal lengths. What is the straight line called that squares off that part of the circumference. In this case, there are four straight lines of equal length, however in general, the lines could be uneven in length and the shape inside is randomKilipaka (talk) 22:11, 14 January 2009 (UTC)

I think you want the word chord. Algebraist 22:14, 14 January 2009 (UTC)

Seems that the question, "In this case, there are four straight lines of equal length...", is asking about the "special" chord: ${\sqrt {r^{2}+r^{2}}}$ but I don't know if it has a "special" name. hydnjo talk 04:00, 15 January 2009 (UTC)

I interpreted that last sentence as meaning the OP was asking about the general case and just giving the special case as an easier to describe example. Either way, I don't know of a name for the special case, either. --Tango (talk) 05:19, 15 January 2009 (UTC)