Wikipedia:Reference desk/Archives/Mathematics/2011 December 2

= December 2 =

Why does the dim function act much like the log function?
(For instance, dim F^n over F is n) — Preceding unsigned comment added by Imanuel.sygal (talk • contribs) 12:25, 2 December 2011 (UTC)
 * How does the dimension of a direct product of two vector spaces compare with the dimensions of the original spaces? (Compare with a well-known log rule.) You can then consider iterated direct products of a vector space with itself, and see another similarity. Icthyos (talk) 14:22, 2 December 2011 (UTC)
 * It isn't really that similar. First, the well known log rule above works out exactly the same if you replace direct product with direct sum in the finite case, which is what we are talking about; this does not hold for logs. Second, while everything, again finite case, you are taking dim of is iso to an F^n, it is not the case that each such object is an F^n; with numbers, log x = n let's you know x, with dim, it doesn't. Finally, dim is interesting as an invariant, if dim happened to be defined so dim F^n = n + 1, it wouldn't really matter that much; however, the function log' x = 1 + log x is not nearly as useful since you are using it for the purpose of calculation. Having said that, were you asking the deeper question of why dim F^n should be n? 71.195.84.120 (talk) 18:48, 7 December 2011 (UTC)
 * For that last point: meaning as in why dimension should be so defined, not as in how to prove that it actually is n.71.195.84.120 (talk) 21:09, 8 December 2011 (UTC)

Riddle about politicians
I have a logical riddle about politicians.

We know that: 1) one always lies, 2) one always says the truth, and 3) one sometimes lies and sometimes tells the truth ?


 * A says that B is a liar.
 * B says that no, he only lies occasionnally.
 * C says that he knows B and knows that he never lies.

Question : Which one is A, B and C? You have to associate letters and numbers. 76.67.157.35 (talk) 18:47, 2 December 2011 (UTC)
 * B
 * A
 * C


 * If you want an explanation from me, you'll have to convince me it isn't homework. — Arthur Rubin  (talk) 19:45, 2 December 2011 (UTC)


 * That seems rather backwards. We usually tell people asking HW questions how to do the prob, but don't provide the answer. StuRat (talk) 19:47, 2 December 2011 (UTC)
 * Perhaps. In this instance, to solve the problem, you need to identify the one who tells the truth.  — Arthur Rubin  (talk) 20:50, 2 December 2011 (UTC)


 * Or, identify one that is certainly not always telling the truth. -- k a i n a w &trade; 20:51, 2 December 2011 (UTC)
 * There are only six possible answers, you could easily solve this by brute force. Widener (talk) 23:49, 2 December 2011 (UTC)

Wave measured at random intervals
Is there a standard way of getting a Fourier series for a waveform where I have measured the value at random spots throughout the wave? Basically the value is measured a number of times at intervals which are longer than a period and rather random within a period but the times and values are measured well. The period is known and accurate, the wave may change a bit unfortunately between periods but overall the shape will always be roughly the same. Dmcq (talk) 19:42, 2 December 2011 (UTC)
 * N measurements will provide you with N equations to determine N coefficients in a trigonometric polynomial. Bo Jacoby (talk) 22:36, 2 December 2011 (UTC).
 * This may be illuminating--78.150.226.102 (talk) 12:02, 3 December 2011 (UTC)


 * Thanks very much. Yes the problem is quite similar to that except in fact quite slow and I can't get as many samples as I'd like. I think I'll have to look into it a bit more deeply myself. I don't want to just assume the first n harmonics and fit exactly as I think that might lead to bad artefacts, I think I'll try optimising something like the power required instead. Dmcq (talk) 12:58, 3 December 2011 (UTC)
 * Yes, fitting a trigonometric polynomial on irregular samples might be ill-conditioned, unlike the case of regular samples which leads to the very well-conditioned Discrete Fourier transform. If that's the case, you can use m measurements to determine only n < m coefficients in a trigonometric polynomial, using Linear least squares (mathematics). Make sure you look at the first figure of the article before you think that linear least squares can only fit lines. 98.248.42.252 (talk) 16:20, 3 December 2011 (UTC)