Wikipedia:Reference desk/Archives/Mathematics/2010 March 1

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March 1

Critical point of a constant function

I was asked this question at Arabic Wikipedia: Does the constant function have infinite number of critical points? I thought it shouldn't have at all but the definition of the critical point allows this since its derivative is always zero at any point.--Email4mobile (talk) 02:59, 1 March 2010 (UTC)

Technically, yes, every point is critical. --Tango (talk) 03:01, 1 March 2010 (UTC)

... or, more precisely, every point is a stationary point (since the whole function is stationary). Dbfirs 12:42, 1 March 2010 (UTC)

But for a critical point or a stationary point, I think there should be some description as: ... where the function "stops" increasing or decreasing. This is not the same in case of a constant function.--Email4mobile (talk) 03:19, 2 March 2010 (UTC)

That's the motivation behind the definition, certainly, but definitions often behave oddly in corner cases. One could explicitly exclude points around which the function is stationary from the definition, but there really isn't much to be gained by doing so. --Tango (talk) 03:53, 2 March 2010 (UTC)

Sums & products of random numbers.

I have a process that takes two random numbers A and B - each in the range 0..1 with an even distribution (ie every value is equally probable). What is the distribution of (A+B)/2, (AxB) and the fractional part of (A+B)?

I should be able to figure this out - but somehow I have a brain blockage this morning! (Be gentle - I'm not a math geek!)

TIA SteveBaker (talk) 14:38, 1 March 2010 (UTC)

If A and B are independent and uniformly distributed on (0,1), then (A+B)/2 has a triangular distribution, while the fractional part of A+B is again uniformly distributed on (0,1). My cursory search hasn't found a standard name for the distribution of the product, but it's given by Prob(A*B<x)=x-ln(x) for x in (0,1). Algebraist 15:07, 1 March 2010 (UTC)

Many thanks! That's exactly what I needed. SteveBaker (talk) 15:11, 1 March 2010 (UTC)

The last one can't be right, as $\lim _{x\to 0+}x-\ln x=+\infty$.—Emil J. 15:13, 1 March 2010 (UTC)

The correct formula is Pr(A × B < x) = x(1 − ln x) for x in (0,1].—Emil J. 15:25, 1 March 2010 (UTC)

Yea it looks like it was a typo. --pma 17:13, 1 March 2010 (UTC)

And maybe it's worth adding the distribution of max(A,B) which is just Pr(max(A,B)<x)=x² -just because max(A,B)<x is the same as A<x AND B<x, and by the assumed independence the probability of the latter event is Pr(A<x)Pr(B<x)=x²--pma 17:13, 1 March 2010 (UTC)

Mathworld has Uniform Product Distribution for the product of a number of uniform distributions. Dmcq (talk) 18:49, 1 March 2010 (UTC)

Integers into a Line with an Overlap

Consider the map f : Z --> S^1, from the integers to the unit circle, given by f(n) = (cos(n),sin(n)). I'm interested in the image of this map. How will the images of the integers fill up the circle? For example, is it like the irrationals fill up the reals, or like the transcendentals fill up the reals? (This is an open answer, and not a dichotomy between my two examples) •• Fly by Night (talk) 18:21, 1 March 2010 (UTC)

The transcendentals are uncountable, so it definitely won't be like them. --Tango (talk) 18:29, 1 March 2010 (UTC)

And so are the irrationals. In fact, transcendentals and irrationals are homeomorphic. The image of the map is a countable dense subset, so in that respect it is similar to the situation of rationals within reals. However, it is hard to answer the question better without a hint of what kind of "being like" you have in mind.—Emil J. 18:37, 1 March 2010 (UTC)

I was hoping for a tag or a name for the way that the image is distributed about the circle. I mean topologically. •• Fly by Night (talk) 18:47, 1 March 2010 (UTC)

Well, it's dense and countable, as I wrote. These properties determine it uniquely, as far as topology is concerned: for every two countable dense subsets of S¹ there is a homeomorphism of S¹ to itself which transforms one of the sets into the other.—Emil J. 19:27, 1 March 2010 (UTC)

So it's like the rational numbers? •• Fly by Night (talk) 20:27, 1 March 2010 (UTC)

Yes, it's analogous to the rationals in the reals - a countable set that is dense within an uncountable set. --Tango (talk) 20:34, 1 March 2010 (UTC)

pma's reply should have been here. -- Meni Rosenfeld (talk) 08:04, 2 March 2010 (UTC)

Sorry, I misread that as "rationals". Yes, the transcendentals and irrationals are essentially the same (they differ by a countable, and thus null, set, the algebraic irrationals). --Tango (talk) 20:34, 1 March 2010 (UTC)

Very nice. You can make this homeomorpism as close to the identity as you wish, right? I'm quite sure one can even make it an analytic diffeomorphism. --pma 21:21, 1 March 2010 (UTC)

I beg your pardon, but I didn't mention anything about making the identity close to anything (whatever that might mean). What's an analytic diffeomorphism? •• Fly by Night (talk) 22:00, 1 March 2010 (UTC)

I think pma's comment was a (misplaced) reply to Emil J.. An analytic diffeomorphism is a homeomorphism that is analytic (can be expressed as a power series) with respect to some differential structure (see those articles for a more precise/rigorous definition). For S¹, there is an obvious differential structure inherited from the plane. --Tango (talk) 22:13, 1 March 2010 (UTC)

Btw, to be precise you need much more than a differentiable structure to define analytic maps, that is, an analytic structure. And you don't need to derive the (real) analytic structure of S¹ from the plane. It has analytic charts of the form exp(it). Or, think to the quotient R/2πZ.--pma 09:20, 2 March 2010 (UTC)

Was it misplaced? Indentation does have a meaning, and if possible posts should follow the chronological order in the thread.--pma 22:45, 1 March 2010 (UTC)

Your comment was immediately below mine and indented more than it, which would usually mean it was a reply to mine, which it wasn't. So, yes, it was misplaced. --Tango (talk) 03:55, 2 March 2010 (UTC)

Is what Tango says above the common rule? I'm a bit curious to know --pma 07:38, 2 March 2010 (UTC)

Yes, keeping replies adjacent to posts trumps chronological order. Think of it as a tree structure. I've marked the place where your reply should have been. See also Wikipedia:Indentation. -- Meni Rosenfeld (talk) 08:04, 2 March 2010 (UTC)

thank you both! --pma 09:21, 2 March 2010 (UTC)

You're welcome. Wikipedia:Talk page#Indentation is the official guideline on the subject (although it isn't actually tagged as a guideline, for some reason). --Tango (talk) 19:31, 2 March 2010 (UTC)

The spinor map

I've recently been faced with quite a surprising result, and am having difficulty explaining it - I'll try to pose the problem as succinctly as possible.

Call H the standard epimorphism from the unit quaternions, $S^{3}$, to SO(3) (described here). Let $q=(a,b,c,d)\in S^{3}$ and define a homomorphism $\alpha$ from SO(3) to PSL(2,C) via

$\alpha (H(q))={\begin{pmatrix}a+id&-c+ib\\c+ib&a-id\end{pmatrix}}$. Now, let $\sigma$ be the spinor map from PSL(2,C) to the orthochronous, proper Lorentz transformations, L.

It turns out that the top left 3x3 block matrix of $\sigma (\alpha (H(q)))$ is $H(q)$ -- so, in some sense, $\sigma \circ \alpha$ is the identity on SO(3), since there is a monomorphism embedding SO(3) into L.

I was quite surprised by this -- should I be? I just composed $\sigma$ with $\alpha$ because I could, I wasn't expecting this to fall out so neatly, and can't explain why it happens. Is there something trivial that I'm missing? Obviously $\alpha$ maps onto PSU(2), and the spinor map takes this (or, certainly SU(2)) onto the embedding of SO(3) in L, but that doesn't really demystify anything. Any help is appreciated - thanks, Icthyos (talk) 22:28, 1 March 2010 (UTC)

I don't really believe your map $\alpha$ is well-defined. H(q)=H(-q), but unless I'm missing something $\alpha$ does not respect this Tinfoilcat (talk) 11:11, 2 March 2010 (UTC)

Well, $\alpha$ maps into PSL(2,C), so $\alpha (H(q))={\begin{pmatrix}a+id&-c+ib\\c+ib&a-id\end{pmatrix}}=-{\begin{pmatrix}a+id&-c+ib\\c+ib&a-id\end{pmatrix}}={\begin{pmatrix}-a-id&c-ib\\-c-ib&-a+id\end{pmatrix}}=\alpha (H(-q))$ . Icthyos (talk) 12:12, 2 March 2010 (UTC)

I knew I was missing something...Tinfoilcat (talk) 12:38, 2 March 2010 (UTC)

Think of the quaternions as (certain) 2x2 matrices over the complex numbers in the usual way: then the surjection from the group of unit quaternions to SO(3) composed with your map alpha is simply the natural map that sends a matrix in SL(2,C) to its image in PSL(2,C). Remember that the action of quaternions on 3-space by rotations is as follows: the quaternion q acts on a vector v by sending v to $qvq^{-1}$, where v is identified with a quaternion with zero real part. So the image of q in SO(3,R) is this "conjugation rotation". But the spinor map composed with the natural map SU(2) to PSL(2,C) also sends q to the map it induces by conjugation, this time on a certain set of matrices. The 3+1ness is because these matrices are of the form tI + (something independent of t), so conjugation doesn't move tI, but the action on the x,y,z part will be the same as the action on 3-space.Tinfoilcat (talk) 12:57, 2 March 2010 (UTC)

Thanks so much - I can't believe I overlooked the conjugation connection. I've had pretty much every fact you used written out on a side of A4, but just couldn't thread it together. Thanks again, Icthyos (talk) 15:24, 2 March 2010 (UTC)

You're welcome. If you want to look at this from a different point of view, you might recast it as a result that says certain representations of SU(2) are isomorphic... Tinfoilcat (talk) 16:42, 3 March 2010 (UTC)