Wikipedia:Reference desk/Archives/Mathematics/2010 June 22

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June 22

Euclidean Geometry

A friend asked me to prove that the shortest distance bewteen two points is covered by a straight line. I told him that I couldn't, because this was an axiom of Euclidean geometry. Am I right? 173.179.59.66 (talk) 05:54, 22 June 2010 (UTC)

No, actually, that has been used as an axiom or definition (for example, by Archimedes), but not by Euclid. In his 20th proposition in The Elements, Euclid proves that any two sides of a triangle add up to more than the third side. It follows from this that a straight line is shorter than a route made up of a sequence of straight lines in different directions. I don't know whether or not Euclid also proves that a straight line is also shorter than a curved path; I don't think he would have been able to do it rigorously. --Anonymous, 06:41 UTC, June 22, 2010.

Okay, much thanks. 173.179.59.66 (talk) 06:52, 22 June 2010 (UTC)

Modern mathematicians use the calculus of variations to show that straight lines have the shortest path property. This also allows one to show that, for example, the shortest path along a sphere between two points is a great circle. HTH, Robinh (talk) 07:46, 22 June 2010 (UTC)

If you take the length of a smooth arc $\textstyle \alpha :[0,1]\to \mathbb {R} ^{n}$ to be $\textstyle \int _{0}^{1}|\alpha '(t)|dt$, then the inequality is just $\textstyle \int _{0}^{1}|\alpha '(t)|dt\geq \left|\int _{0}^{1}\alpha '(t)dt\right|=|\alpha (1)-\alpha (0)|.$ The analogous inequality follows if you use the more general notion of total variation of the arc $\alpha$ (then no regularity assumption is needed on $\alpha$). --pma 19:35, 22 June 2010 (UTC)

Kepler's First Law (Antiderivative)

In the process of deriving Kepler's First Law, the antiderivative $\int \!{\frac {ldr}{r{\sqrt {2\mu Er^{2}+2\mu Cr-l^{2}}}}}$ needs to be evaluated. By book gives the solution as an arcsine of something, but I've been having trouble reproducing the result. It seems that every math program I try (as well as my own attempts) yield a messy combination of imaginary numbers and logarithms. Can anyone help me solve this antiderivative? —Preceding unsigned comment added by 173.179.59.66 (talk) 10:45, 22 June 2010 (UTC)

I haven't checked where that equation comes from but are you sure about that r on its own in the denominator? That would be where your log comes from. Dmcq (talk) 11:22, 22 June 2010 (UTC)

Are you sure it isn't arcsec? ${\frac {1}{x{\sqrt {x^{2}-1}}}}={\frac {d}{dx}}\mathrm {arcsec} \,x$ 129.67.37.143 (talk) 11:28, 22 June 2010 (UTC)

Sorry my comment about logs, expression is okay. Try differentiating arcsin(a+b/r) and you should get a similar expression that you can change around to get that what you want. Dmcq (talk) 12:31, 22 June 2010 (UTC)

World Cup group stages - how likely is it that drawing of lots would be required?

I've been following the 2010 FIFA World Cup and there's been some discussion about scenarios that might lead to some of the groups being decided by drawing lots. This seems to me to be an unfair way to make the decision, and I'd like to know the prior possibility (i.e. not from where we are now but from the beginning of the tournament) that this might happen.

There are four teams in each group, each playing each of the others once, and the first two qualify for the next round. FIFA uses the following criteria to rank teams in the Group Stage.

1. greatest number of points in all group matches;
2. goal difference in all group matches;
3. greatest number of goals scored in all group matches;
4. greatest number of points in matches between tied teams;
5. goal difference in matches between tied teams;
6. greatest number of goals scored in matches between tied teams;
7. drawing of lots by the FIFA Organising Committee.

I guess the other thing we'd need to know is the general distribution of scores within the group stages, which we can find by looking at the group stage results from the last three world cups in 2006, 2002 and 1998 (before that there were fewer teams, which might possibly affect results - best to eliminate any variables we can).

Anyone care to have a go at working out the likelihood that one or more groups would have to be decided by drawing lots? Presumably you'd do it by running some kind of simulation, but I know my statistical knowledge isn't up to it. --OpenToppedBus - Talk to the driver 12:07, 22 June 2010 (UTC)

If every game in the group resulted in a equal draw - eg all 1-1 or all 0-0 - then whatever happens you get a 4 way tie. You either draw lots or start using fewest yellow/red cards or choose alphabetically or ... -- SGBailey (talk) 13:16, 22 June 2010 (UTC)

Of course, whatever criteria you have there's a possibility of a tie. My concern is that there's too high a possibility of a tie using the current criteria.

But that's only based on my perception. What I'm looking for is someone to say, "actually, based on a normal distribution of results, that'll only happen once every 50 years" - in which case I'd probably think it's fine. Or alternatively, "statistically, you're likely to have to decide something by drawing lots once in every other tournament", in which case I would think FIFA were remiss in not adding an extra criteria based on number of cards, number of corners, number of times teams have hit the woodwork, etc, any of which would be fairer than just picking names out of a hat. --OpenToppedBus - Talk to the driver 13:48, 22 June 2010 (UTC)

To the best of my knowledge its happened once in the history of the World Cup, in 1990 for Group F to be precise. Republic of Ireland and the Netherlands came 2nd and 3rd respectively after a draw, though both qualifed as that was back in the days of 6 groups where the 4 best 3rd placed teams qualified. Its never actually affected if someone qualified before at World Cup level, though may have happened in other competitions. - Chrism would like to hear from you 19:45, 22 June 2010 (UTC)

I disagree that any of those methods would be better than drawing lots. The object in soccer is very clear: score more goals than the other team and you win. If you use other criteria to decide who advances like how many corners a team takes, that adds new incentives. Then it become somewhat of a different game: a contest to score goals and not allow corner kicks. That's not really what the game is supposed to be. As it stands, it's more valuable to score a goal than to prevent the other team from scoring a goal, which seems a little bit questionable to me but whatever. I guess they want to encourage more aggressive play.

As for drawing lots, I don't see what's unfair about it even it's a really crappy way to lose. Also sorry this doesn't really address the question at all. Rckrone (talk) 22:43, 23 June 2010 (UTC)