Wikipedia:Reference desk/Archives/Mathematics/2011 July 11

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July 11

Palm–Khintchine theorem

I'm not a mathematician. I thought that when you added a lot of any kind of probability distribution together, such as for example the Poisson, then the result (with sufficiently large numbers) approximated to a Gaussian distribution.

Please could someone explain in layman's language why the Palm–Khintchine theorem gives a different result - a Poisson distribution rather than the Guassian. 92.24.187.78 (talk) 14:27, 11 July 2011 (UTC)

As Poisson distributions also approximate Gaussian distributions, the results are not necessarily incompatible. Bo Jacoby (talk) 15:25, 11 July 2011 (UTC).

I'm not familiar with the theorem, but note that it talks about Poisson processes, not Poisson distributions. They're closely related but they're different things. If you look at the distribution of number of events in a given interval it will converge to Poisson on one hand, to normal on another, and as Bo says this is compatible.

Note also that the central limit theorem has limitations. Though I don't think this is the case here, it could have been that the Palm–Khintchine theorem talked about a situation where it didn't apply. -- Meni Rosenfeld (talk) 16:43, 11 July 2011 (UTC)

A Poisson distribution is discrete, a Gaussian distribution is continuous. If you have a Poisson distribution whose mean is large (say, 1000 or more), it will be very well modeled by a Gaussian distribution. However, if the mean is small (say, less than 10), a Gaussian is not a good fit. My understanding is that the theorem applies in a regime where you can sample time intervals that give a low number of expected events. Looie496 (talk) 21:39, 11 July 2011 (UTC)

One should speak of adding random variables, not of adding distributions. And you didn't mention independence or any other related hypothesis. That can be weakened, but it can't just be dropped. Also, there is usually a hypothesis that the variance is finite. Although that can be modified, it also cannot just be dropped.

Wikipedia's article titled Palm–Khintchine theorem is quite opaque in its current state, but I don't think it's about adding a large number of independent random variables. Michael Hardy (talk) 01:30, 12 July 2011 (UTC)

4-dimensional geometry

What are the names for the two new directions in 4d geometry? --134.10.114.233 (talk) 16:58, 11 July 2011 (UTC)

I've never heard of direction names for higher dimensions. The closest thing I can think of is to specify coordinates like (x,y,z,w) and then say "the positive x direction" or "the negative w direction". Rckrone (talk) 17:50, 11 July 2011 (UTC)

I guess you could use "before" and "after" (i.e. time as the 4th dimension), but that's by no means standard and would probably just confuse people. Rckrone (talk) 20:54, 11 July 2011 (UTC)

If one of the dimensions is time-like, then the axis variable set is usually {x, y, z, t} instead of {x, y, z, w} unless otherwise specified. 99.24.223.58 (talk) 21:06, 11 July 2011 (UTC)

In and out. – b_jonas 21:23, 11 July 2011 (UTC)

I guess you're looking for names that are analogues of up–down, left–right and forwards–backwards. The turth is that there are no names, and you can label them howsoever you choose. It depends on the applications and on the area of study. As has been mention, some people use wxyz–space, some people use xyzt–space. If you're graphing age, weight, height and shoe size then you might like to label the axes the a–, w–, h– and s–axes, and have awhs–space. The reason there are no names is because the conventional names are borrowed from our everyday experience. But outside of space–time, we have no intuition. A lot of mathematics is done in n–dimensional space, where the dimension of space is a variable itself. For example, see this section of our article on the hypersphere. A hypersphere is like a circle in the plane or a sphere in 3–space. It gives formulas for volume and surface area, where the dimension is a variable itself. Notice that n could, if you wanted, be 1,000,000,000. — Fly by Night (talk) 22:44, 11 July 2011 (UTC)

Charles Howard Hinton called the direction ana and kata. Personally I agree with the in and out. People with good feel for 3d can develop a 4d intuition. However many people don't even have a good feel for 3d, ask them what shape you get if you push the corner of a cube into some modelling clay and they're liable to get it wrong. Dmcq (talk) 22:57, 11 July 2011 (UTC)

That's a nice question. It depends on the physical dimensions of the modelling clay. You can get something homotopy equivalent to a circle or to a point depending on the mutual dimensions of the modelling clay and the cube. In terms of isotopy it'll be a solid torus or a solid sphere. It all depends if the cube pierces the modelling clay. — Fly by Night (talk) 23:19, 11 July 2011 (UTC)

I just meant indenting the clay so you got a pyramid shaped hole and asking how many sides the pyramid has. Many people will say four. Nothing complicated like that if you push it right through you get six sides. Dmcq (talk) 11:35, 12 July 2011 (UTC)

To see why there aren't any "the" names, ask yourself what are "the" names for the ordinary three-dimensional directions. To simplify the discussion, let's forget about up-down, and suppose those are locally well-defined. What are the other ones? Are they left/right, front/back, which is relative to your direction of movement? Are they north/south, east/west, depending on your position on Earth? Are they port/starboard, fore/aft, depending on how your ship is pointing? Upstream/downstream, river's left/river's right, on a river? Uptown/downtown/crosstown, in Manhattan? I could go on.

The ana/kata silliness is even worse than that, because Hinton AFAIK provided no way of specifying which one was ana and which one was kata.

It reminds me of the word globbered, which can

... according to the Ultra-Complete Maximegalon Dictionary of Every Language Ever, mean the noise made by the Lord Sanvavlwag of Hollop on discovering that he has forgotten his wife's birthday for the second year running. Since there was only ever one Lord Sanvavlwag of Hollop and he never married, the word is used only in a negative or speculative sense, and there is an ever-increasing body of opinion that holds that the Ultra-Complete Maximegalon Dictionary is not worth the fleet of trucks that it takes to cart its microstored edition around in.

(That's from Douglas Adams, of course.) --Trovatore (talk) 23:36, 11 July 2011 (UTC)

I tend to use compass directions and directions to the nearest towns even when inside a building. My wife uses direction relative to the way she's looking. Dmcq (talk) 11:41, 12 July 2011 (UTC)

If I told my ex that a place was north of Geary Boulevard, she'd say, "That's away from Golden Gate Park, right?" And I'd wonder: how can you be more sure of the Park, which cannot be seen from most of Geary, than of north? —Tamfang (talk) 23:26, 15 July 2011 (UTC)

Line tangent to a set of circles

Cone as envelope of circles from Sound barrier.

I using a graphing calculator to make something that looks like a mach cone. The circles that are tangent to this mach cone have the form of $(x-2r)^{2}+y^{2}=r^{2}$. To form the mach cone, it would be in the form of $x=|y|n$ where n is whatever number that would be used to stretch $x=|y|$ so that it just touches each circle. What is n in expressed using trigonometric functions? --Melab±1 ☎ 23:35, 11 July 2011 (UTC)

I'm not sure what a "mach cone" is. We don't seem to have an article on it, and I couldn't find anything on Google. If you're looking for something that's tangent to all of those circles then you need to consider the envelope. Let's define a one-parameter family of function $F:\mathbb {R} _{x,y}^{2}\times \mathbb {R} _{r}\to \mathbb {R}$ given by $F((x,y),r):=(x-2r)^{2}+y^{2}-r^{2}.$ We consider the zero level set of this family of functions. For a fixed r ≠ 0 we get the equation of a circle; if r = 0 we get a point – a "circle" of radius zero. As r varies, we get a one-parameter family of circles. Thus, r is the parameter of the family. The envelope of this family of circles is given by solving $F=\partial F/\partial r=0$ with respect to x and y. If we differentiate with respect to $r$ then we get $\partial F/\partial r=6r-4x$ and so $\partial F/\partial r=0\iff x=3r/2.$ Putting that into $F$ gives $F((3r/2,y),r)=y^{2}-3r^{2}/4.$ It follows that $F=\partial F/\partial r=0\iff (x,y)=(3r/2,\pm r{\sqrt {3}}/2).$ For a fixed r ≠ 0 this gives a pair of points, the so-called envelope points. (Informally, these are the points where a circle and the next infinitely close circle in the family intersect). This gives a pair of straight lines through the origin with equations $x\pm y{\sqrt {3}}=0.$ I hope that that's what you were asking about. If not, then my apologies. — Fly by Night (talk) 01:52, 12 July 2011 (UTC)

I think the "Mach cone" might be related to the Mach number, there a picture from Sound barrier which seems to fit.--Salix (talk): 06:49, 12 July 2011 (UTC)

After I had posted this I played around with my graphing calculator and I stumbled upon the answer in the form I was looking for: $x=|y|n$ where $n=1/\tan(\arccos({\sqrt {3}}/2))$. Now if I were too make the original equation $(x-mr)^{2}+y^{2}=r^{2}$, how would I determine $n$? --Melab±1 ☎ 11:48, 12 July 2011 (UTC)

$1/\tan(\arccos({\sqrt {3}}/2))$ is just ${\sqrt {3}}$ as in Fly by Night's reply. Use the method he described to find the result for a general m. -- Meni Rosenfeld (talk) 12:19, 12 July 2011 (UTC)

Well, now I'm looking for how to determine $n$ given $m$, which I determined to be equal to $1/tan(arccos(sqrt{1-1/m^{2}}))$. --Melab±1 ☎ 12:47, 12 July 2011 (UTC)

This is true, but it is equal to simply ${\sqrt {m^{2}-1}}$. -- Meni Rosenfeld (talk) 15:12, 12 July 2011 (UTC)