Wikipedia:Reference desk/Archives/Mathematics/2012 August 25#Dirichlet kernel

I got interested in the impact that chance, contingent factors, etc. can have on student grades. I reduced my doubts to limit cases in which either an impossible question is added to an exam, or a giveaway question is added. I was surprised to discover that for good students a free question does not make up for an impossible question.

I then made a more thorough study to try to discover how many questions would needed to reduce the effect of the impossible or give-away question to having no important influence. Following that I devised a way of computing how "luck of the draw" would influence a student's grades if the sample size was very small. The method I developed is a simple application of probability theory, but as I have done it the process involves a great deal of computation.

Working the whole thing out would probably not be as much work as calculating a table of trig functions, but I am hopeful that somebody has already done the work or attacked the problem in some mathematically more efficient way. I have posted an explanation in more detail, and a spreadsheet based on a limited sample size used to evaluate spelling competence for a year's work, i.e., a vocabulary list of several hundred words.

http://www.china-learn.info/MathConcepts.xhtml

Can anybody point me at other work on this level? I am aware that work has been done on the subjects of "reliability" and "validity," but the math seems much more intense, and the preparatory work would be much more arduous. I am looking for materials that could easily be understood by secondary school teachers who may think that ten spelling words on a final exam will give all students fair grades.

Thanks.P0M (talk) 03:51, 25 August 2012 (UTC)

Assuming an 8-character password that is either

1) all lower-case letters (e.g. fjwkdosj, or jefokds)

2) all lower-case letters, but one single digit in one place (e.g. fjds4sds)

which ruile has higher entropy? This is not homework. --80.99.254.208 (talk) 07:24, 25 August 2012 (UTC)

:Looking at the number of possibilities, it's 26⁸ in the first case and 26⁷(10)(8) in the later case, assuming that the digit can be in any position, which means a tad more than 3 times as many possibilities, using a single digit. See if you can figure out entropy from there. StuRat (talk) 08:42, 25 August 2012 (UTC)

Show that $\backslash frac\{1\}\{2\}\backslash int\_0^x\; \backslash left(\backslash frac\{\backslash sin((N+1/2)t)\}\{\backslash sin(t/2)\}-1\backslash right)dt$ is uniformly bounded in $N$ and $x\; \backslash in\; [-\backslash pi,\backslash pi]$ using the fact that .

Obviously the strategy is to manipulate the first expression into $\backslash int\_0^\backslash infty\; \backslash frac\{\backslash sin\; t\}\{t\}\; dt$ or something similar.

So, $\backslash frac\{1\}\{2\}\backslash int\_0^x\; \backslash left(\backslash frac\{\backslash sin(Nt)\backslash cos(t/2)+\backslash cos(Nt)\backslash sin(t/2)\}\{\backslash sin(t/2)\}-1\backslash right)dt$

$=\backslash frac\{1\}\{2\}\backslash int\_0^x\; \backslash left(\backslash frac\{\backslash sin(Nt)\backslash cos(t/2)\}\{\backslash sin(t/2)\}+\backslash cos(Nt)-1\backslash right)dt$

$=\backslash frac\{1\}\{2\}\backslash int\_0^x\; \backslash left(\backslash frac\{\backslash sin(Nt)\backslash cos(t/2)\}\{\backslash sin(t/2)\}+\backslash cos(Nt)-1\backslash right)dt$

Widener (talk) 13:47, 25 August 2012 (UTC)

$\backslash int\_0^x\; (\backslash cos(Nt)-1)\; dt$ is uniformly bounded in N and $x\; \backslash in\; [-\backslash pi,\backslash pi]$ at least. Widener (talk) 13:50, 25 August 2012 (UTC)

:That isn't going to help. The first thing you need to observe is that $\backslash int\_0^\backslash infty\; \backslash sin\; t\; \backslash frac\{dt\}\{t\}\; =\; \backslash int\_0^\backslash infty\; \backslash sin\; (N+1/2)t\; \backslash ,\backslash frac\{dt\}\{t\}$. Sławomir Biały (talk) 14:54, 25 August 2012 (UTC)

::How are they equal? Widener (talk) 15:39, 25 August 2012 (UTC)

:::dt/t is the multiplicative Haar measure for the group $\backslash mathbb\{R\}\_+$ (or just do a substitution). Sławomir Biały (talk) 15:55, 25 August 2012 (UTC)

Am I correct in saying that a surface is a manifold if, at every point on the surface, there exists a coordinate that can locally be written as a function of the others? Widener (talk) 14:03, 25 August 2012 (UTC)

:I assume you mean a smooth manifold and smooth function. If so, the answer is "yes" (by the implicit function theorem). Sławomir Biały (talk) 14:55, 25 August 2012 (UTC)

::I don't understand... the Euclidean plane is a smooth manifold, but its coordinates aren't locally degenerate. --Tango (talk) 15:01, 25 August 2012 (UTC)

:::Here I'm assuming that the question is specifically about surfaces in $\backslash mathbb\{R\}^3$, in which case the Euclidean plane is just $z=0$. (Of course, there are surfaces that cannot be realized as manifolds in $\backslash mathbb\{R\}^3$, but I don't think that's what the question was about.) Sławomir Biały (talk) 15:18, 25 August 2012 (UTC)

::::Ah, I didn't realise we were talking about an embedding of a surface. The question and answer make far more sense now! Thanks. --Tango (talk) 15:20, 25 August 2012 (UTC)