Wikipedia:Reference desk/Archives/Mathematics/2010 October 7

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

October 7

More general form (and an image) for sine's continued fraction.

Hi.

I'd like to add a more general notation for the "continued fraction" section of the Sine article.

There are other notation styles listed here: Generalized continued fraction#Notation.

Any suggestions for an image/diagram for this section would be appreciated too.

Thanks —Pengo

binomial coeffcient

Resolved

I want to find out the maximum value of ${\tbinom {n}{k}}$ for a fixed postive integer n, and varying k. Intutively the answer is obvious to me, for even n, it is achieved at k=n/2 and for odd n, it is achieved at k=(n-1)/2. But how do I prove this result. I dont want to differentiate and equate to zero. (Anyway, the binomial coefficient for me as defined as a function of k, where k is a positive integer only.) Thanks. -Shahab (talk) 11:23, 7 October 2010 (UTC)

Just compute the ratio ${\tbinom {n}{k+1}}/{\tbinom {n}{k}}$.—Emil J. 11:43, 7 October 2010 (UTC)

${\tbinom {n}{k}}=n!/(k!(n-k)!)=n(n-1)\dots (n-k+1)/k(k-1)\dots (1)$. Increasing $k$ by 1 adds a $k+1$ in the denominator and a $n-(k+1)+1=n-k$ in the numerator. Since $(n-k)/(k+1)>1$ for $n>2k+1$, the maximum follows from an easy induction. By the way, for odd n, there are actually two answers, $k=(n-1)/2$ or $k=(n+1)/2$. 67.158.43.41 (talk) 11:52, 7 October 2010 (UTC)

Thanks-Shahab (talk) 17:10, 7 October 2010 (UTC)

btw, the proper adjective for describing the up-down behaviour of any row in the Pascal triangle is unimodal --pma 18:38, 7 October 2010 (UTC)

hexagon torus

Hi. If you stitch together opposite sides of a square, you get a torus or a klein bottle or a projective plane. What do you get if you stitch opposite sides of a hexagon together? Do these objects have a name? Robinh (talk) 19:35, 7 October 2010 (UTC)

what about stitching the opposite sides together of an n-gon? (monogon, duogon, triangle, square, pentagon, hexagon, heptagon, octogon, etc) —Preceding unsigned comment added by 85.181.147.249 (talk) 06:00, 8 October 2010 (UTC)

Depending on what direction the stitchings occur in, you will get either a sphere, a connected sum of tori or a connected sum of projective planes. —Preceding unsigned comment added by 203.97.79.114 (talk) 08:13, 8 October 2010 (UTC)

I think the same three things: there are in theory four ways you can do the stitching, with 0, 1, 2, 3 opposite edges stitched in the same direction, so you stay on the same side. These correspond to stitching 0, 1, 1 and 2 opposite edges of a square, giving a Klein bottle in two cases. I think you'd only get a sphere if non-opposite sides were stitched together.--JohnBlackburne^words_deeds 08:59, 8 October 2010 (UTC)

The double torus

You'll find that 203.97.79.114 has the right answer. Take the hexagon and identify opposite edges with the same orientation; like you do with the sqaure to get the torus. Image you walk around the edges anti-clockwise. The first edge will be a and you'll be walking in the direction of a's orientation. The next three edges will be b, c and d; with you travelling in the direction of each edge's orientation. The next edges will be a, b, c and d, but this time you'll be walking against each edge's orientation. This whole walk can be represented by a word:

$abcda^{-1}b^{-1}c^{-1}d^{-1}\,\ ,$

where the inverses mean you're walking against the edge's orientation. In this case we have an orientable, compact surface of genus 2, i.e. a double torus. (The connected sum of two tori) You can get lots of other surface from the same hexagon but with different orientations. — Fly by Night (talk) 09:33, 8 October 2010 (UTC)

Yes, thinking on it the first two cases, zero and one edges stitched in opposite directions, are different to the square. But if only one or zero are stitched the same way it's the same, so the Klein bottle and projective plane, and for not just the hexagon but any even-sided polygon.--JohnBlackburne^words_deeds 10:19, 8 October 2010 (UTC)

I'm not sure about even. If you have a 4n-sided polygon and you identify the edges to give the word

$a_{1}a_{2}\ldots a_{2n}a_{1}^{-1}a_{2}^{-1}\!\!\!\ldots a_{2n}^{-1}\ ,$

then you get the n-torus after you've finished all of the identifications. Not sure what a six sided one would give:

$abca^{-1}b^{-1}d\sim \ ?$

My guess would be a sliced torus, which might not be a torus and just fall apart... Hmmm... — Fly by Night (talk) 10:39, 8 October 2010 (UTC)

I was only thinking of if all, or all but one pair, of the opposite edges are reversed and joined to the opposite edge. If they are all reversed it's like a disc with each point on the edge identified with it's opposite point, so the projective plane. If all but one pair are reversed the reversed edges are like one edge of the square, the non-reversed edges are like the other, so the Klein bottle. In each case this works for not just six sides but any even sided polygon where opposite sides are stitched together.--JohnBlackburne^words_deeds 15:49, 8 October 2010 (UTC)

You can check it using the words. If all the edges are reversed then you'll have

$abcabc=(abc)(abc)=\alpha \alpha \,,$

which is just a projective plane like you say. — Fly by Night (talk) 19:27, 8 October 2010 (UTC)

Quick group theory question

Let p be a prime. If a group has more then p-1 elements of order p, why cant the group be cyclic. I can see that the group cant possibly be Z_p. (Btw sorry about asking so many questions these days. All this isnt homework or anything.)-Shahab (talk) 19:54, 7 October 2010 (UTC)

Well, how many order p elements does Z_n have? Algebraist 23:44, 7 October 2010 (UTC)

p-1? for there is a unique subgroup of order p and it has p-1 generators. Is this correct? But now I have another question. How do I count the number of order m elements in Z_n in general. For example how many elements of order 8 in Z_{8000000}? Can I just say phi(8)? Somehow I am not feeling too sure about this.-Shahab (talk) 04:07, 8 October 2010 (UTC)

Why should Z_n have a unique subgroup of order p (when it has one at all)? What are the order-8 elements of Z_8000000? Algebraist 05:06, 8 October 2010 (UTC)

Suppose G is cyclic. I assume G=Z_n. Now p divides n by hypothesis. Then since Z_n is cyclic it will have a unique subgroup H of order p. All order p elements will be within that cyclic subgroup H as if there was a order p element outside H then H will no longer remain unique. H is cyclic and isomorphic to Z_p and so will have p-1 generators. In other words there are p-1 elements of order p in G, contrary to the hypothesis. So unless G has exactly p-1 elements of order p it cant be cyclic. For Z_8000000 I use a similar logic since 8 divides 8000000. Is this reasoning correct? I am not quite sure because 8000000 is never coming into the picture properly.-Shahab (talk) 08:52, 8 October 2010 (UTC)

The reasoning is correct if m divides n. Otherwise there are no elements of order m.—Emil J. 10:49, 8 October 2010 (UTC)