Testwiki:Reference desk/Archives/Mathematics/2010 March 17

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March 17

Plank Time

It says

planck time = $5.39124(27)\times 10^{-44}$

What is the

$27$

?174.3.107.176 (talk) 01:40, 17 March 2010 (UTC)

It's the uncertainty. It means (5.39124 +/- 0.00027) *10^-44. The last two digits (the 24) are +/- the two digits in the brackets. 0.00027*10^-44 is the standard error. --01:47, 17 March 2010 (UTC)

So 5.39127 *10^-44, ± 0.00024?174.3.107.176 (talk) 03:23, 17 March 2010 (UTC)

5.39124 *10^-44, ± 0.00027*10^-44. HOOTmag (talk) 08:22, 17 March 2010 (UTC)

Local connection between measure preserving transformations

Suppose we have two maps F and G which preserve the Lebesgue measure and such that d(F,G)<ε (say in C^k).

Problem: does it exist a continuous family of maps F_t such that

• F_t preserves the Lebesgue measure
• d(F_t,F)<ε
• F₀=F and F₁=G?

--Pokipsy76 (talk) 19:37, 17 March 2010 (UTC)

Character question

Working on a proof from "Introduction to Elliptic Curves and Modular Forms" by Koblitz. It involves characters and Gauss sums, which I have little experience with and I don't know what's going on. It's Prop 17 on P127 if you happen to have the book. The proof is on P128. There is a limited preview on Google Books but it does not include P128 so the proof is not included.

${\chi }_1$ is a primitive Dirichlet character modulo N, so a multiplicative character, and $\xi =e^{2\pi i/N}$ is an additive character. $g=\sum _{j=0}^{N-1}{\chi }_1(j){\xi }^j$ is the Gauss sum, though I don't know if this is important yet. We define a function $f_{{\chi }_1}(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{\chi }_1(n)q^n$, where $q=e^{2\pi iz}$ and the $a_n$ come from a modular form we start out with, $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{q}^n$. So, that's just the background and I am stuck on step 1 of the proof. It's probably not very hard. The claim is

$f_{{\chi }_1}(z)=\sum _{l=0}^{N-1}{\chi }_1(l)\sum _{n=0}^{\mathrm{\infty }}\left(\frac{1}{N}\sum _{\nu =0}^{N-1}{\xi }^{(l-n)\nu }\right)a_n{q}^n$.

I guess they're just rewriting ${\chi }_1(n)$ in some other form??? I have no idea. Thanks for any help. StatisticsMan (talk) 20:50, 17 March 2010 (UTC)

$\frac{1}{N}\sum _{\nu =0}^{N-1}{\xi }^{(l-n)\nu }$ is 0 when l and n are different mod N, and is 1 when they are the same. So that form is just stacking up the terms for each equivalence class mod N. Rckrone (talk) 06:38, 18 March 2010 (UTC)

Lambert W function for a base other than e

y = xn^x

solve for x in terms of y and n?--203.22.23.9 (talk) 21:22, 17 March 2010 (UTC)

Never mind, I've figured it out--203.22.23.9 (talk) 21:23, 17 March 2010 (UTC)