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June 5

Topology: Density of rational functions and Euclidean Balls

Hi there:

a while back I asked a question on Euclidean balls, [1] - but I wasn't sure I completely followed the solution: specifically I was looking for how to find the 'k' I mentioned in the question for a ball of radius r slightly > 3 (3.001) containing unit balls, such that the number of balls in R^n is an exponent of (1+k) - and also how to find whether the same exists for a ball of radius slightly >2 (2.001) - I'd be very surprised if the latter did exist since it seems a bit pointless asking a question about '3.001' rather than 3 and then dropping down to just over 2: but how can I show it?

Also, what would be the cleanest way to show that the rational functions $\subset C[0,1]:\mathbb {Q} \mapsto \mathbb {Q}$ are dense in C[0,1] under the metric $d(f,g)=max_{x\in [0,1]}|f(x)-g(x)|$ ?

I could use a good explanation for my first question on the balls if anyone's willing to give one (which I'd very much appreciate!) - the second question I could probably do fine if someone just pointed me in the right direction with the best way to show that its completion is C[0,1]: I'm happy to show any space is dense in its completion, the starting off just seems so obvious in a way I can't see where to go with it!

Thanks very much guys!

Spamalert101 (talk) 07:16, 5 June 2009 (UTC)[reply]

The quickest way to answer your second question is to note that it's an immediate consequence of the Stone–Weierstrass theorem. I don't know if you want something lower-technology than that. Algebraist 11:17, 5 June 2009 (UTC)[reply]

I think part of the first question is answered in Wikipedia:Reference desk/Archives/Mathematics/2009_May_21#Spheres_in_an_n-dimensional_sphere The change from a linear to exponential numbers happens at diameter 1+sqrt(2). Dmcq (talk) 12:29, 5 June 2009 (UTC)[reply]

I'm not sure about what you mean by "the rational functions

\subset C[0,1]:\mathbb {Q} \mapsto \mathbb {Q}

"? --pma (talk) 13:26, 5 June 2009 (UTC)[reply]

I thought the OP just meant the normal rational functions mapping the closed interval [0,1] to itself and I don't know why Algebraist deleted his/her/its/their contribution. Dmcq (talk)

In fact my doubt was about the meaning of

\mathbb {Q} \mapsto \mathbb {Q}

. Maybe he wants rationals functions mapping Q into Q, i.e. with rational coefficients? its: you mean he/she could be just a program?? I also had this suspect as he/she has always perfect answers... instead, we humans happen to say silly things sometimes :) --pma (talk) 16:49, 5 June 2009 (UTC)[reply]

If I did in fact have always perfect answers, this would prove my nonmachinehood. Algebraist 17:14, 5 June 2009 (UTC)[reply]

I try not to make assumptions. Statements like 'If I don't go to bed I'll not be able to get up in the morning' are what I aspire to Dmcq (talk) 17:55, 5 June 2009 (UTC)[reply]

In fact the Approximations section of that article states the version the OP wants with rational coefficents. However it doesn't give a proof which is what the OP wanted. Dmcq (talk) 15:22, 5 June 2009 (UTC)[reply]

Which article is that? As far as I can see, none of the articles linked above has an "Approximations" section. Anyway, Stone–Weierstrass theorem#Weierstrass approximation theorem refers to Bernstein polynomial#Approximating continuous functions which is followed by a proof, though it relies on the weak law of large numbers so it is rather an unusual one. — Emil J. 15:41, 5 June 2009 (UTC)[reply]

Sorry I meant Stone–Weierstrass theorem#Applications. Don't know how that became approximations. Dmcq (talk) 07:02, 7 June 2009 (UTC)[reply]

On reflection, I think the OP wants to show that the continuous functions on [0,1] which take rational values at rational points are dense in the whole space. In that case, my original remark stands, since the set in question is obviously dense in its R-linear span, which is dense in the whole space by the theorem. Algebraist 17:14, 5 June 2009 (UTC)[reply]

volume of torus

How is the volume of a torus (donut) calculated? is it the area of the cross section multiplied by the circumfrence at the center of the cross section (assuming it is a circular cross section and the torus itself is circular?) —Preceding unsigned comment added by 65.121.141.34 (talk) 14:10, 5 June 2009 (UTC)[reply]

Yes, Torus#Geometry says so. PrimeHunter (talk) 14:16, 5 June 2009 (UTC)[reply]

And the reason is Pappus's centroid theorem. It thus does not really matter whether the cross section is circular. — Emil J. 14:19, 5 June 2009 (UTC)[reply]