Wikipedia:Reference desk/Archives/Mathematics/2010 February 3

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February 3[edit]

Transcendental Galois extension[edit]

Resolved

Is there such a thing? I'm especially unsure about the possible topology. It cannot be (as I understand) the usual profinite one (since a Galois group may not be compact.) Algebraically, there is a separability for transcendental extensions (due to Mac Lane?) but don't know if it is a good idea to define Galois = (Mac Lane) separable + normal, where "normal" being the usual one. I would be delighted if you know something more. -- Taku (talk) 02:43, 3 February 2010 (UTC)

I've encountered no such notion myself and Wikipedia seems unaware of it, but it doesn't seem impossible. The article Galois extension characterizes a Galois extension as an algebraic extension K/F whose automorphism group has fixed field F; if we drop the condition of being algebraic, the definition still seems sensible on the face of it. For example, take F as a field of characteristic other than 2, and consider the transcendental extension F(X)/F; the automorphism of F(X) that fixes F and maps X to -X has fixed field F, so therefore that extension would be Galois. Plausibly such a notion might be useful. Others? Eric. 131.215.159.171 (talk) 06:31, 3 February 2010 (UTC)
Are you sure about your example? The polynomial p(X)=X^2 is mapped onto itself under the automorphism you mentioned, and thus should lie in the automorphism's fixed field. PST 13:04, 3 February 2010 (UTC)
Indeed. The fixed field in this case is F(x2), and the extension F(x)/F(x2) is algebraic. — Emil J. 13:22, 3 February 2010 (UTC)
You are right; I wasn't thinking carefully. Eric. 131.215.159.171 (talk) 02:18, 5 February 2010 (UTC)
The whole point of Galois extensions is that they satisfy the fundamental theorem of Galois theory, i.e., there is a dual correspondence of intermediate extensions to closed subgroups of the Galois group. If the extension is transcendental, merely requiring that Fix(Aut(K/F)) = F comes nowhere near ensuring this goal for whatever definition of "closed" (thought I can't remember the specific counterexample ATM), and is unlikely to be a very useful property by itself. — Emil J. 13:18, 3 February 2010 (UTC)
Well, of course you could argue that for the purposes of "pure field theory", the existence of a Galois connection should be somehow related to the notion of a "Galois extension", and I agree. But field extensions also occur often in other branches of mathematics, and although an example does not immediately come to mind (I vaguely recall one, and if I remember, I will note it here), perhaps some definition of "Galois extension" for transcendental extensions may be useful in algebraic geometry (for instance). If Taku is looking for an example of how to use the theory of transcendental extensions in Galois theory, one striking example is the fact that if F is a purely transcendental finitely generated extension of \mathbb{Q}, and if E is Galois over F (in the usual sense), there is a Galois extension K of \mathbb{Q}, such that the Galois group of K over \mathbb{Q} is isomorphic to the Galois group of E over F; most standard proofs of this fact involve Hilbert's irreducibility theorem. The result may be of interest if you were looking for connections between "transcendental extensions" and "Galois theory". PST 13:38, 3 February 2010 (UTC)

Well, I wasn't thinking anything fancy. I had a very innocent example like \mathbf{C}/\mathbf{Q}. (It's not a Galois extension since it's not algebraic.) I thought, in application, it makes sense to start with the top field \mathbf{C}; since this way you can apply analytic results (e.g., Lie groups). The problem is that there are transcendental elements when we go down. (Or maybe I'm missing the story completely.) Hence, my somehow rhetorical question. I think we can agree that It would be nice if there is such a thing as a transcendental Galois extension. (I don't know any possible applications to algebraic geometry that PST mentioned.) Of course it is possible to define a extension to be Galois by the closure property: i.e., F^{**} = F where * means taking a Galois group and taking fixed field respectively. As Emil. J pointed out, such a definition is vacuous. (And, if I understand correctly, if we require a Galois group to be profinite and further assume that there is a Galois connection, then it would follow that the extension is algebraic (since the union of finite extensions is algebraic.) -- Taku (talk) 02:23, 4 February 2010 (UTC)

This is the answer to myself. Apparently there is "transcendental Galois theory": a lecture note by J.S. Milne has a section on this. [1] As I noted above, a Galois group is not compact. But then what is it? Do we know? -- Taku (talk) 02:47, 4 February 2010 (UTC)

This book connects moduli spaces and Galois theory. In general, most of the connections between Galois theory and geometry involve the inverse Galois problem (as far as I know). PST 03:48, 4 February 2010 (UTC)

A book like that is probably too advanced for me. But it still sounds very interesting. Thanks. -- Taku (talk) 12:34, 4 February 2010 (UTC)

Smooth maps[edit]

Resolved

I'm working on a problem where I am supposed to give necessary and sufficient conditions for a map, f, from one smooth manifold on the reals to another is a smooth map. With very little work, I showed that this is equivalent to showing f^3 is smooth in the normal calculus sense. Here smooth means C^\infty. Now, do you think my answer should be that the cube of f is smooth on the reals? I mean that is necessary and sufficient, but I don't know if there's more that can be said, like "Every such function would look like ...". I can see that products and sums of smooth functions product smooth functions, so if f is smooth, then f^3 should be smooth. But, the opposite is not true as f(x) = x^{1/3} is not smooth but its cube is. Any thoughts? Thanks. StatisticsMan (talk) 05:03, 3 February 2010 (UTC)

I asked my professor and he said that is all he is looking for. StatisticsMan (talk) 21:02, 3 February 2010 (UTC)

Induction[edit]

Everyone knows how to do induction and recursive definition on well-ordered sets. Is there a generalized notion of well ordering to partially ordered sets so we can do things similar to induction and recursion on them? Money is tight (talk) 07:37, 3 February 2010 (UTC) Nvm I found what I was searching for in Well-founded_induction Money is tight (talk) 07:46, 3 February 2010 (UTC)

I'd say the Zorn lemma. --pma 09:36, 3 February 2010 (UTC)
I don't see that Zorn's lemma answers that question. Certainly on some posets that are not linearly ordered people do inductive arguments. For example when one proves Los's theorem, true of all formulas of first-order logic. "Well-founded" is essential; "well-ordered" is more than what is needed. Michael Hardy (talk) 17:51, 4 February 2010 (UTC)
PS: See well-founded relation. Michael Hardy (talk) 17:52, 4 February 2010 (UTC)
I disagree that "Everyone knows how to do induction and recursive definition on well-ordered sets." Speak for yourself. Edison (talk) 00:22, 6 February 2010 (UTC)

Winning eight games out of ten[edit]

Sometimes I play a series of ten Reversi games online against ten different opponents, selected at random. If I win eight of the ten games, and there are no draws, could I use that to calculate or estimate at what percentile in the Reversi-player ability range I am? Ignoring that its a small sample size. Thanks. 78.146.251.66 (talk) 12:27, 3 February 2010 (UTC)

If on an average you win eight of the ten games you play, and if everything stated in your question is assumed, you should effectively defeat 80% of the population in Reversi. But that implies that your percentile rank is 80. PST 12:58, 3 February 2010 (UTC)
Given that you had a uniform chance of any ranking beforehand the probability density afterwards of where you are from 0 to 1 is as x8(1-x)2 normalized so it all adds up to to 1 which is the Beta distribution with parameters 9 and 3. That expression is x8-2x9+x10. The integral is x9/9-2x10/10+x11/11. Its integral between 0 and 1 is 1/9-2/10+1/11=2/990 so thats what you divide by to get your final result. The limits of the 8th percentile are I believe 0.7 and 0.8 but you might want something else. You work out the integral at the two end figures, subtract and divide by that total between 0 and 1. And that gives your chance of being in that percentile. But I'm afraid I can't do that in my head. Dmcq (talk) 13:42, 3 February 2010 (UTC)
BTW by that article the average (arithmetic mean) of your ranking is 75% and the most likely value (mode (statistics))) is 80%. Dmcq (talk) 13:51, 3 February 2010 (UTC)
No, no, no. All these calculations confuse "percentile of Reversi ability" with "probability to defeat a random opponent".
First, playing ability probably cannot be placed on a one-dimensional scale. It's very possible to have cycles where player A dominates B (defeats him with probability > 0.5), B dominates C and C dominates A. It's possible that the world champion of reversi, which dominates all other powerful players, is so confused by the cluelessness of weaker players that he beats them less often than he should. Thus the player in the 100th percentile would have a relatively low probability to defeat a random opponent.
Even if we ignore this and choose "probability to defeat a random opponent" as our measure for ability, this quantity will only be monotonous with percentile, not identical to it. It's very possible that the best player, at the 100th percentile, is only able to defeat a random opponent 70% of the time.
So we see that the OP's 80% winning record, even if measured for a large sample, does not tell us what the percentile is. It is possible the he is really the world champion, and it is possible that he is on the 60th percentile (I think it can't be lower).
For estimating "probability to defeat a random opponent", Dmcq gives the right ideas, but the prior needn't be uniform. It depends on the distribution of this parameter among all players, and your objective estimates about your own ability. -- Meni Rosenfeld (talk) 15:44, 3 February 2010 (UTC)
Sorry you're quite right, it doesn't give the percentiles at all and one would need a lot more information to do that. Thanks for pointing that out for me, very silly. Dmcq (talk) 17:10, 3 February 2010 (UTC)
Any attempt to assess your relative ranking depends upon some other things that aren't being stated. First and foremost, if you are playing where there are ratings, you should use that as a guide rather than your performance in specific games against players who may be at either end of the spectrum. Generally, stronger players will avoid playing much weaker ones so as to waste little of their time. If you played a truly random selection of opponents, the percentage of games you would win would roughly match up with the percentage of players you are stronger than; but, for example, in chess the player generally considered strongest historically, Gary Kasparov, only had about a 70-30 record, since he only played other chess players in the top hundredth of a percentile through most of his career. [Incidentally, one of the strongest Reversi/Othello players is Imre Leader, the godson of Imre Lakatos (recently mentioned at this desk for his book, Proofs and Refutations).]
One thing said here is absolutely false. A (very) strong player will beat a weak one 100% of the time unless s/he falls asleep during the game. It's not a subject for this desk, but it shouldn't go unchallenged.Julzes (talk) 17:59, 3 February 2010 (UTC)
You misunderstood me. I have no knowledge about Reversi, and I wasn't trying to make a statement about how likely a strong Reversi player is to beat a weak one. I was talking about games in general, using "Reversi" as a placeholder. It requires specific domain knowledge to show that Reversi does not exhibit any of the scenarios I mentioned (if this is so).
For proffesionally played games that do not involve randomness, the probability in question is indeed usually close to 100%; for games that do it is usually less.
"The percentage of games you would win would roughly match up with the percentage of players you are stronger than" is false in general, and a strong statement about Reversi in particular. -- Meni Rosenfeld (talk) 19:40, 3 February 2010 (UTC)
Reversi doesn't use dice or cards. The question was not about games with randomness, but about random selection of opponents. The matchup between proportion of games won and percentile of skill is greater the more highly graded skill-levels are in a game without randomness. In general, one won't enjoy success over opponents one is stronger than 100% of the time, so it is certainly true that the matchup is not perfect. It is close, and, given a large sample of games against truly randomly chosen opponents, I think it's probably the best estimator for someone in the middle ranks (I don't know if this question has been researched). This, however, probably excludes players who only barely know the game. Someone with only a modicum of skill may very well have a hard time getting good results against a rank beginner (somewhat as you said about strong versus weak players). At any rate, one thing that can be argued is that all games exhibit some randomness, to the extent that weak players may choose their moves with no more skill (and sometimes less) than a random-move selector would. And also, as I said, opponent selection cannot possibly be random and the rating systems that are available are a better guide to determining skill level. There is a lack of transitivity in the ranking question as well, and I think this is the point that Mr. Rosenfeld was trying to get across. Such things as variations in styles of play and specific preparation for specific opponents can either make comparisons impossible or yield false results. In some cases comparisons can be made but are not made well with head-to-head results, and in others comparison is effectively impossible. And then there is the question of which game of Reversi or chess one is talking about, from 1 minute per game to postal.Julzes (talk) 21:36, 3 February 2010 (UTC)
Non-transitivity is just a small part of the point I was trying to get across. What I was saying is:
  • Given a game G, assuming for simplicity that draw is impossible, the pool of players is fixed and their ability doesn't change, there is a function f(x,y) which gives the probability that x will beat y in a match where the assignment of sides between them is chosen randomly. This takes into account randomness in both the rules of the game and the choices made by players.
  • The function f is not necessarily simple.
  • A possible simplifying assumption is that ability is on a one-dimensional scale: There is a function g(x) such that g(x) > g(y) \iff \forall z(f(x,z)> f(y,z)). We also assume that g is injective.
  • The percentile ranking of player x is the proportion of players who have a lower g. The probability that player x will beat a player chosen uniformly randomly among all players is \tfrac1n\sum_{y}f(x,y) (I'm assuming n, the total number of players, is large). PST and Dmcq assumed that g exists and that \sum_yI(g(x)>g(y)) = \sum_yf(x,y)\;\!.
  • This doesn't hold in general. It obviously holds if f(x,y)=I(g(x)>g(y), but it can also hold without it. It is also approximately true if a small difference in g leads f to be close to 0\1.
  • If the choice of opponent is not uniform among all players, then obviously the measurements are different.
  • Subtractive rating systems like Elo's assume that there is a function h such that f(x,y)=h(g(x)-g(y)).
  • Any or all of the mentioned assumptions may hold for Reversi, and I take your word for anything you say on the matter.
-- Meni Rosenfeld (talk) 08:44, 4 February 2010 (UTC)
I see what you're saying, and it's basically something I understood despite what I said. I would be close to correct if, say, each decile had total dominance over any decile two or more below it, but this is a rather strong statement. It would require some actual data analysis to answer the OP's question from a statistical rather than mathematical perspective, but your analysis is correct from the mathematical one.Julzes (talk) 17:38, 5 February 2010 (UTC)
The article about the Elo rating system (used in chess and tennis) might be of some help here. 66.127.55.192 (talk) 18:07, 3 February 2010 (UTC)

distance of a hyperplane from the origin and the norm[edit]

My question is this: Show that the norm ||f|| of a bounded linear functional f (non-zero) on a normed space X can be interpreted geometrically as the reciprocal of the distance D = inf{||x||: f(x)=1} of the hyperplane H = {x : f(x)=1} from the origin. Firstly what is the meaning of hyperplane? The book I am reading doesn't define hyperplane, it defines hyperplane parallel to a subspace Y as an element of X/Y. Secondly, how should I prove the result? Thanks.-Shahab (talk) 13:30, 3 February 2010 (UTC)

A hyperplane is an affine subspace of codimension 1. In other words, it's a set of the form {x : f(x)=a} for some nonzero linear functional f and scalar a. What do you mean by "recipient"? Algebraist 15:24, 3 February 2010 (UTC)
"Recipient" should perhaps be "reciprocal" ? Gandalf61 (talk) 15:29, 3 February 2010 (UTC)
Yes, that was a mistake. Corrected now. So how do I proceed? Also I believe can think of H as x/f(x)+N where N=Null space of f and x is a fixed element in X-N. But in general how do I interpret a hyperspace as an element of X/Y (what element and what is the subspace Y.) Thanks-Shahab (talk) 17:10, 3 February 2010 (UTC)
Y = {x: f(x) = 0} = your N, and H is literally an element of X/Y if the latter is defined in the obvious way as a set of equivalence classes, it's not necessary to "interpret" it. Anyway, all this talk about hyperplanes and X/Y is just a red herring. The result that ||f|| = 1/inf{||x||: f(x) = 1} = sup{1/||x||: f(x) = 1} follows fairly trivially from the definition of ||f|| = sup{|f(x)|: ||x|| = 1}, just show that the two suprema are taken over the same set (well, except for 0). — Emil J. 17:26, 3 February 2010 (UTC)

Discrete Mathematics: Do quantifier orders matter?[edit]

There's a question in my discrete mathematics book that asks to write in English:

 \forall x \exists y P(x,y)

and then asks us to write:

 \exists y \forall x P(x,y)

My question is: are these equivalent?

Thanks for the help! Sebsile, an alternate account of Saebjorn 16:17, 3 February 2010 (UTC)

Try letting your quantifiers range over people, and letting P(x,y) mean "x loves y". Algebraist 16:19, 3 February 2010 (UTC)
If, unlike xkcd, you don't want to mix math and romance, try P(x,y)\equiv x<y. -- Meni Rosenfeld (talk) 16:25, 3 February 2010 (UTC)
What about this (a kind of variation on Algebraist's example): "for any x there exists a y that can screw x" vs "there exists a y that can screw any x". pma 17:09, 3 February 2010 (UTC)

No they're not equivalent. In the first case the statement is true even if the y-value is different for different x-values. In the second case the statement is not true unless the same y-value works regardless of what the x-value is.

This is precisely the difference between pointwise convergence and uniform convergence. Michael Hardy (talk) 19:52, 3 February 2010 (UTC)

Is there still a distinction between  \exists y :\forall x P(x,y) and  \exists y \forall x :P(x,y)? It's a long time since I used these symbols. Dbfirs 22:37, 3 February 2010 (UTC)
Those colons, as far as I'm aware, mean absolutely nothing. So the difference persists. Algebraist 23:32, 3 February 2010 (UTC)
the colons are old-schoolish symbols for 'such that'. they are kind of unnecessary, so I think modern usage tends to drop them. but to answer the question in english (rather than mathese), the difference would be between for every X there exists (some) Y where P(x,y) as opposed to there exists a (particular) Y for all X where P(x,y). in the first case y can be different for different x's; in the second it's the same y for all x's. --Ludwigs2 00:08, 4 February 2010 (UTC)

There's no difference between the expressions with colons in different places. There is a difference between the expressions with the quantifiers in a different order. Michael Hardy (talk) 17:48, 4 February 2010 (UTC)

OK, I accept the modern view, but, long, long ago, the placing of the colon was used to make the meaning clearer, so  \exists y :\forall x P(x,y) is unambiguous whereas I can see a possible misunderstanding of  \exists y \forall x :P(x,y) (Perhaps others can't, in which case, please ignore this comment.) Dbfirs 09:48, 5 February 2010 (UTC)

Homoeomeric curves[edit]

I don't really need help with this, just thought it would be interesting. The ancient Greeks studied homoeomeric curves, that is curves for which any part can be made to coincide with any other part. Or in more modern language, a connected 1-manifold embedded in Euclidean space so that its symmetry group under isometries of Euclidean space is transitive. Geminus showed there are only three homoeomeric curves (in 3-space): the line, the circle, and the circular helix. It appears though that there is a fourth type of curve in 4-space and in general there are n types in n-space. It appears that there are three types of homoeomeric surfaces in 3-space: the plane, the sphere, and the circular cylinder. It seems natural to ask, can the homoeomeric m-submanifolds of Euclidean n-space be classified?--RDBury (talk) 16:19, 3 February 2010 (UTC)

For the case of curves in three-space, they can be classified in terms of the Frenet–Serret formulas: one with zero curvature (the line), one with nonzero curvature but zero torsion (the circle) and one with nonzero curvature and torsion (the helix). It seems likely that Jordan's extension to n dimensions will nicely classify the homoeomeric curves in dimension n. Algebraist 16:28, 3 February 2010 (UTC)
I'm not sure there are only 4 curves in 4D. Off the top of my head I can think of

\begin{pmatrix} at \\ bt \\ ct \\ dt \end{pmatrix}
\begin{pmatrix} at \\ bt \\ c \cos t \\ c \sin t \end{pmatrix}
\begin{pmatrix} 0 \\ 0 \\ c \cos t \\ c \sin t \end{pmatrix}
\begin{pmatrix} a \cos t \\ a \sin t \\ c \cos t \\ c \sin t \end{pmatrix}
\begin{pmatrix} a \cos nt \\ a \sin nt \\ c \cos mt \\ c \sin mt \end{pmatrix}
The first three are line, spiral and circle - there's only one spiral as the first two components, ate0 + bte1, are orthogonal to the other two whatever the values of a and b. Obviously a = b = 0 gives a circle, c = 0 a straight line.
The last three (including the circle) are related to simple, isoclinic and double rotations: they come from thinking of the paths of points under those rotations, as such path will map to itself via that rotation and powers of it.
The last gives more than one curve as values of m and n can be chosen to generate different closed curves, much like Lissajous curve, except these are homoeomeric. E.g. m = 1, n = 2 is the simplest one that's different from the fourth/isoclinic one. In theory there are as many as there pairs or relatively prime (m, n), i.e. infinitely many. Then there are a class of non-closed curves when m/n is not rational so the curve loops forever and is dense but never joins up.
So it gets a lot more complex just for curves in four dimensions. I can't even think what will happen for surfaces, or for more general m-manifolds in n-dimensions, which gain far more degrees of freedom in spaces with far more complex transformations.
Actually scrub the fourth one. With a suitable change of basis it's just a circle radius \sqrt{a^2 + c^2}. I think my thinking on the paths generated by the general double rotation still makes sense though.--JohnBlackburnewordsdeeds 20:42, 3 February 2010 (UTC)
One more thought - the set of curves described by the last vector above lie on the ridge of a Duocylinder. This as a surface is homeomorphic to the surface of a torus, or a play-area of computer game like Asteroids. The set of curves is just the set of straight lines/geodesics on this surface, which can perhaps be more easily visualised with the above homeomorphisms.--JohnBlackburnewordsdeeds 15:06, 4 February 2010 (UTC)

Clarification on 'Electric Potential Energy' article[edit]

Hi all,

further to my question on charged spheres as capacitors a few days ago, I was discussing electromagnetism with a friend a few years older than me and he pointed out that, in the derivation of the alternate formula for calculating energy ( Electric Potential Energy - under 'Energy stored in an electrostatic field distribution'), when we derive the |E|2 formula for energy, we throw away a surface integral over a sphere of infinite radius because \phi \to 0 as r \to \infty: however, as the potential tends to 0, it's also true that the surface area of the sphere tends to infinity.

Now I can appreciate the basic concept of what's going on here - as we take the limiting values of phi and the surface area for r \to \infty, phi tends to 0 sufficiently fast so that the integral over the surface tends to 0 despite the surface area becoming arbitrarily large, but what I want to know is why? I've consulted 2 textbooks (and the above article) on this, and the only answer I seem to be able to find is a mumbled 'oh, well, the potential just goes to 0 faster...' without any actual justification. Why can we be certain that the potential drops off sufficiently fast that the surface integral becomes negligible? Can anyone give me a proper answer without just hiding behind the fact that '0 times infinity = 0 in this case'?

I greatly appreciate any help you're able to provide, all I want is a proper justified answer or a decent explanation - many thanks, Otherlobby17 (talk) 22:59, 3 February 2010 (UTC)

Far from all charges (you have to assume the charge density is localized, or at least itself eventually falls off with radius "sufficiently fast"), they act like a single point charge, whose potential goes as r^{-1} and whose field goes as r^{-2}. The area of the bounding surface goes as r^2, so the product of the potential, the field, and the area goes as r^{-1} and vanishes as r grows without bound. (If the total charge is 0, the potential and field will drop off at an even faster rate determined by the precise geometry of the charges.) --Tardis (talk) 01:57, 4 February 2010 (UTC)