Wikipedia:Reference desk/Archives/Mathematics/2011 February 3

= February 3 =

IPv4 vs IPv6 for dumdums?
Okay, will someone explain in simple-ish terms what the differences between IPv4 and IPv6 are, why they're incompatible, what switching over entails, what the format is for IPv6 addresses, etc.? I know how to use a computer and the Internet, so don't worry about that. I know what an IP address is, and that we've almost completely exhausted all the IPv4 ones. The IPv6 article kind of assumes you know about DNS and have technical knowledge of IPv4... --- c y m r u . l a s s  (talk me, stalk me) 05:17, 3 February 2011 (UTC)
 * What it means: the end of the world as we know it. IPV4 addresses look like four octets separated by dots, like 123.21.12.34.  IPV6 has 32-bit hex numbers separated by colons, like 2001:0db8:85a3:0000:0000:8a2e:0370:7334, but there are some shortcuts for compressing out blocks of zeros in them, etc.  See IPv6 address.  The basic issues are comparable to what would happen if the phone company suddenly had to start giving out phone numbers with 4x as many digits as the old ones.   Everybody's phones and switches would have to be reprogrammed, etc, stuff will break all over, sort of a Y2K problem but "for real" since almost nobody has actually been getting ready for it.  71.141.88.54 (talk) 06:08, 3 February 2011 (UTC)
 * Huh. Interesting. Is there any way to slowly ease into it, or is it just gonna be a "boom, we're screwed" thing? And aren't some computers already using it? --- c y m r u . l a s s  (talk me, stalk me) 06:41, 3 February 2011 (UTC)
 * See this article by Worthen and Tuna, 'Web running out of addresses,' from the Wall Street Journal on 1 February. EdJohnston (talk) 06:57, 3 February 2011 (UTC)
 * People with existing v4 addresses will be able to keep using them, but it will be very difficult to get new ones (imagining a mob-operated black market in them is only slightly fanciful). But almost nobody has been willing to deal with the hassle of v6 transition since the coming v4 exhaustion hasn't happened yet.  There will be a transitional period of riots, looting teeth gnashing, router reconfiguration confusion, overloaded tech support lines, web site outages, that sort of thing, plus lots of vendors cashing in on opportunities to sell you new crap instead of fixing your old crap.  Lots of old systems will simply fall off of the internet.   Eventually, though, v6 will work on all newer systems.  Added: and yes, v6 has been available for years, with the idea of "start using it now and beat the crunch".  But nobody has been bothering because they all have other stuff to fix "right now" while the v4 network is still humming along.  It's just like putting off homework til the last minute.  71.141.88.54 (talk) 07:46, 3 February 2011 (UTC)
 * And we all know how well that goes... Hmm I'm wondering if it would be a good idea/worth the trouble to switch over. That is, if a lowly random like myself can actually do that :P --- c y m r u . l a s s  (talk me, stalk me) 08:08, 3 February 2011 (UTC)
 * It is mostly your ISP and websites you visit who will have to deal with it. If you're running a very old OS like Windows ME you may have to upgrade, and you may also have a problem if you have an old router.  Otherwise just prepare to deal with crappy service and software patches for a while, as problems get ironed out.  If you're running a recent OS, it should be v6-ready, but various software applications may need upgrades.  Sam Bowne has some very good materials here. 71.141.88.54 (talk) 08:46, 3 February 2011 (UTC)
 * By the way, this question probably should have gone to the computer ref desk rather than the math desk. 71.141.88.54 (talk) 08:47, 3 February 2011 (UTC)

supremum and infimum of a quotient of two series
I need help translating the following into English:

Given two sequences of positive real numbers $$\{ a_n \}_{n \in \mathbb{N}}$$ and $$\{ b_n \}_{n \in \mathbb{N}}$$, we write


 * $$a_n= \mathbb{O}(b_n)$$ if they satisfy $$ \limsup_{n \to \infty} a_n / b_n < \infty $$


 * whereas $$a_n = \Omega (b_n) $$ if they satisfy $$ \liminf_{n \to \infty} a_n / b_n > 0 $$.


 * On the other hand $$ a_n = o(b_n)$$ means that $$ \lim_{n \to \infty} a_n / b_n = 0$$


 * and $$a_n = \omega(b_n)$$ means that $$ \lim_{n \to \infty} a_n / b_n = \infty $$

I don't understand. I've read the entries on supremum and infimum but I cant work out what they mean in this particular context... can somebody help break it down in simple terms? —Preceding unsigned comment added by 118.208.23.171 (talk) 06:32, 3 February 2011 (UTC)
 * Fix your typos first. —Preceding unsigned comment added by 99.40.234.78 (talk) 06:53, 3 February 2011 (UTC)


 * Firstly, in case it isn't clear, there are four definitions here. I've edited the formatting to make this clear, and fixed a typo (you had typed $$\infty$$ in place of $$\inf$$). Have you read the entry on Limit superior and limit inferior? Do you have trouble only with the first two definitions, or all four of them? Shreevatsa (talk) 07:04, 3 February 2011 (UTC)


 * Apologies for the typos. My guess now is that there are two series of equal length $$a_n$$ and $$b_n$$, and a new series $$a_n / b_n $$ is created by dividing each element of the first by the corresponding element of the second. The third definition applies in the case that this new series converges to 0 and the fourth definition applies in the case that the new series diverges to infinity. However I am clueless as to the first two definitions, even after reading the article on limit superior and limit inferior.


 * Yes, "equal length" in that both are (countably) infinite. Unfolding the definitions of lim inf and lim sub, and collapsing some nested quantifiers that don't matter in this context, we get:
 * an = O(bn) means that there is some finite number K such that all an/bn are smaller than K.
 * an = &Omega;(bn) is the opposite: there is some K>0 such that all an/bn are larger than K.
 * Beware, incidentally, that this notation is well-established but horribly abuses the equals sign. For a given sequence (bn)n, there are many different sequences that are all O(bn). To make sense of it you should consider each separate instance of O(···) in an equation to mean some unknown series that happens to satisfy the condition. –Henning Makholm (talk) 09:54, 3 February 2011 (UTC)
 * Thanks! —Preceding unsigned comment added by 118.208.23.171 (talk) 10:26, 3 February 2011 (UTC)

Made some further corrections to the notation (LaTeX \limsup, \liminf instead of '\lim \sup' and 'lim \inf'). --CiaPan (talk) 12:54, 3 February 2011 (UTC)

Affine Grassmanian
What's the dimension of the affine Grassmannian AGr(n,k) which consists of all k-dimensional affine subspaces of Rn? In particular, I'm interested in AGr(n,1), i.e. the space of affine one-spaces in Rn, or equivalently the space unorientated lines in Rn. Our article says that, as a homogeneous space, we can identify AGr(n,k) with the quotient
 * $$ \text{AGr}(n,k) = E(n)/\left[E(k) \times O(n-k)\right], $$

where E(n) is the Euclidean group on Rn and O(m) is the orthogonal group on Rm. Doing a quick dimension check, this seems to imply that
 * $$ \dim\left[\text{AGr}(n,k)\right] = (n-k)(n+k-1)/2 \,, $$

but that's not right because AGr(2,1) is diffeomorphic to the Möbius band, and so has dimension two (the lines have equations ax + by = c, which corresponds to (a:b:c), but we must delete (0:0:1) because a = b = 0 doesn't give a valid line. Then the projective plane minus a point is the Möbius band). While the dimension formula gives the dimension to be one. My guess is that AGr(n,1) has dimension 2(n–1). I know that the space of oriented lines in Rn is diffeomorphic to the tangent bundle to the (n–1)-sphere and that that gives a double cover of the space of unoriented lines. — Fly by Night  ( talk )  13:16, 3 February 2011 (UTC)
 * I get (k+1)(n-k) using a heuristic, degrees of freedom approach. It's not rigorous but it agrees with your guess.--RDBury (talk) 20:36, 3 February 2011 (UTC)
 * Yeah, you're exactly right. I misunderstood this section of the article on the orthogonal groups. I think it might need some work that section. In fact, it turns out that
 * $$ \dim \left[ E(m) \right] = m(m+1)/2, \ \mbox{and} \ \dim \left[ O(n) \right] = n(n-1)/2 \,, $$
 * Since AGr(n,k) ≅ E(n)&thinsp;/&thinsp; [&thinsp;E(k) × O(n−k)&thinsp;] it follows that


 * $$ \dim\left[ AGr(n,k) \right] = n(n+1)/2 - \left[k(k+1)/2 + (n-k)(n-k-1)/2 \right] = (k+1)(n-k) \,, $$
 * as you quite rightly said. Thanks for that RDBury; it's appreciated. — Fly by Night  ( talk )  00:51, 4 February 2011 (UTC)

Another way to see it is that an affine subspace of dimension k in Rn can be realized as the intersection of a generic (k+1)-dimensional linear subspace of Rn+1 with the affine hyperplane xn+1=1. This embeds AGr(n,k) as a Zariski open set in Gr(n+1,k+1). So,
 * $$\dim AGr(n,k) = \dim Gr(n+1,k+1) = (k+1)(n-k),$$

which agrees with the other dimension count. Sławomir Biały (talk) 20:34, 4 February 2011 (UTC)


 * I have added this observation to the article.  Sławomir Biały  (talk) 13:30, 5 February 2011 (UTC)


 * Excellent, thanks Sławomir. — Fly by Night  ( talk )  15:36, 5 February 2011 (UTC)

Journal article author affiliation
The article submission guidelines of a mathematics journal says that author affiliations are to be given at the end of the article. Two authors from the same place ( same address) are submitting an article. One of them has another affiliation (permanent address). What is the best way (better than giving the common address of the authors twice) to give the affiliations? 14.139.128.15 (talk) 17:19, 3 February 2011 (UTC)
 * One way to handle it is like this: present author names at start, e.g. Joe Author*, Jane Math*†. At the end, you only need to indicate each affiliation once, e.g. *Affiliation 1, †Affiliation 2. If you want further help, you could name the journal. Also, are you using LaTeX? Many math journals provide authors with templates, .cls files, etc. Lastly, it probably isn't that important for submission. They will not reject you out of hand for some formatting issue. If your work is accepted, then they will work with you to make it right before it is published. SemanticMantis (talk) 20:07, 3 February 2011 (UTC)

Contour integral
My textbook is trying to show that $$\int^{\infty}_{-\infty} \frac{1-\cos(x)}{x^2}\,dx = \pi$$. The author does this by setting up an integral over the complex plane, over a curve that is more-or-less a semi-circle of radius R, centered at the origin, with a diameter along the real axis (looks like an upside-down bowl). For the proof, he uses that $$\left| \frac{1-e^{iz}}{z^2} \right| \le \left| \frac{2}{z^2} \right|$$.

I don't see why this would be true. 74.15.137.130 (talk) 20:49, 3 February 2011 (UTC)
 * The denominators are the same, so we only need to establish the inequality for the numerators. The complex exponential exp(ix) lies on the unit circle of the complex plane (see diagram in linked article). So abs(exp(ix))=1. Is that enough? SemanticMantis (talk) 21:31, 3 February 2011 (UTC)
 * But z is generally complex. At least, the author uses this inequality over the part of the path that is in the complex plane. So wouldn't it have a norm greater than 1? 74.15.137.130 (talk) 23:17, 3 February 2011 (UTC)
 * My mistake, I misread the inequality. What do you know about the norm of z along the path that you're integrating along? This will give you bounds on exp(iz) that will probably do the trick. Try using the Laurent series of the exponential to get some cancellation with the denominators. SemanticMantis (talk) 00:58, 4 February 2011 (UTC)
 * If the path is in the upper half-plane, write z = x + iy, with y ≥ 0. Then |eiz| = |e-y + ix| = e-y ≤ 1. 82.124.101.35 (talk) 03:11, 4 February 2011 (UTC)

Made some minor corrections to LaTeX notation (added \left and \right to vertical bars, so their height corresponds to what they contain). --CiaPan (talk) 07:06, 4 February 2011 (UTC)

Thanks. 74.15.137.130 (talk) 20:46, 4 February 2011 (UTC)