Wikipedia:Reference desk/Archives/Mathematics/2013 October 13

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 13

Product of Sines

$\prod _{k=1}^{n-1}\sin {\Big (}{\tfrac {k}{n}}\cdot \pi {\Big )}\ =\ {\frac {n}{2^{n-1}}}\qquad -$ 79.113.215.238 (talk) 12:56, 13 October 2013 (UTC)

Are you asking us or telling us? AndrewWTaylor (talk) 16:21, 13 October 2013 (UTC)

(Problems getting the notation to appear correctly - seems to work only occasionally). AndrewWTaylor (talk) 16:23, 13 October 2013 (UTC)

1. I've seen it yesterday on a math site, and, albeit it looks so deceitfully simple, I've already tried two approaches and failed... 2. Yes, I know, I've been having the same problems for months now... (I think it's due to page size, I'm not sure...) — 79.113.210.178 (talk) 02:30, 14 October 2013 (UTC)

The identity is mentioned in List_of_trigonometric_identities#Identities_without_variables, which references Mathworld (maybe where you saw it?), where the only source given is "personal communication". AndrewWTaylor (talk) 08:23, 14 October 2013 (UTC)

Possibly this identity:
      $\prod _{k=1}^{n}\cos \theta _{k}={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})$
where
      $S=\{1,-1\}^{n}$
(mentioned in List of trigonometric identities#Product-to-sum and sum-to-product identities) has something to do with the one above? --CiaPan (talk) 14:26, 16 October 2013 (UTC)

Question about SAT scores

How rare (or how common) are "perfect" SAT scores ... in any given year, or in general? And, has that trend gone up or down, over time? And how about "perfect" ACT scores? Thanks. Joseph A. Spadaro (talk) 14:03, 13 October 2013 (UTC)

See SAT#Raw scores, scaled scores, and percentiles. Duoduoduo (talk) 16:25, 13 October 2013 (UTC)

Yes, thanks. I had seen that. But, those are merely percentiles. I am not seeking the relative number (i.e., the percentage) of students who achieve a perfect score, but rather the absolute number of students who do (e.g., "In 2011, there were 43 students who achieved a perfect SAT score." ... or some such statement.). Thanks. Joseph A. Spadaro (talk) 17:22, 13 October 2013 (UTC)

That section does say something close to what you want:

The older SAT (before 1995) had a very high ceiling. In any given year, only seven of the million test-takers scored above 1580. A score above 1580 was equivalent to the 99.9995 percentile.

This can't be exactly right, because I doubt that the number was the same in every year as in every other year. But it's probably intended to be a statement of the average number over all the years prior to 1995. Duoduoduo (talk) 18:53, 13 October 2013 (UTC)

This dotcom site says:

Perfect Scores: A perfect score is 2,400 points. Approximately 1,000,000 students take the SAT each year and on average, only 20 of them get a perfect score.

I found this by googling "perfect score SAT". You might want to look at some of the other hits that come up for this search. Duoduoduo (talk) 19:01, 13 October 2013 (UTC)

The number of students getting 36 on the ACT is on the order of several hundred a year out of millions of test-takers, according to the Wikipedia article on the ACT.--Jasper Deng (talk) 19:16, 13 October 2013 (UTC)

Thanks to all. Much appreciated! Joseph A. Spadaro (talk) 02:50, 16 October 2013 (UTC)

Identifying a unitary

Given a complex m by n matrices $X$ and $Y=XU$ for $m\gg n$ such that $U$ is uniquely defined, what is the computationally least intensive way of identifying the n by n unitary $U$. The only approach which occurs to me is svd followed by comparison of the right hand unitaries, however this seems to involve evaluating a lot of redundant information. — Preceding unsigned comment added by 81.155.161.131 (talk) 22:51, 13 October 2013 (UTC)

As a followup question it would also be useful to know about the case where $Y\approx XU$ and I would like to find the matrix $U$ which minimises the error as determined by some defensible measure (preferably the induced norm of $Y-XU$) where the restriction of $U$ to the unitaries remains. Any help greatly appreciated. — Preceding unsigned comment added by 81.155.161.131 (talk) 23:12, 13 October 2013 (UTC)

Any help at all? If someone could even provide some pointers to a useful method or relevant article it would be useful! — Preceding unsigned comment added by 85.210.44.227 (talk) 20:42, 15 October 2013 (UTC)

Sorry this took so long for me to remember such a trivial approach; I've been thinking about this since you posted it. First, note that you do not need $m\gg n$ or even $m=n$: each (independent) row of X gives you $2n$ constraints (from being complex), and a unitary matrix has only $n^{2}$ degrees of freedom despite being complex. So just truncate X to n rows and calculate $U=X_{n}^{-1}Y$. If finding a set of n linearly independent rows is hard, you can try the Moore–Penrose pseudoinverse of the whole matrix: $U=X^{+}Y$. (You can even try the psuedoinverse in the $m<n$ case, but I don't think that it generally finds the unitary solution when that condition is necessary to give uniqueness.) Numerical error may of course give a matrix that is not quite unitary; I suppose you could average ${\tilde {U}}$ and $\left({\tilde {U}}^{-1}\right)^{*}$ or so to project it into that space reasonably well. --Tardis (talk) 13:56, 17 October 2013 (UTC)