Wikipedia:Reference desk/Archives/Mathematics/2008 May 4

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May 4

standard deviation

 Done

I have a mean of 39.917 and a standard deviation of 27.888 for 24 samples. Isn't that quite a high SD for the mean and a very small number of samples? 71.100.6.147 (talk) 01:51, 4 May 2008 (UTC)

Found Accuracy and precision. 71.100.6.147 (talk) 04:24, 4 May 2008 (UTC)

Norm Equivalence

Let $(\Omega ,F,\mu )$ be a measurable space. Let $L_{K}^{1}(\mu )$ be the quotient space of all $\mu$ integrable functions where two functions are equal (they are in the same equivalence class) if they are equal almost everywhere and the value of the integral can be any real or complex number (i.e., K=R or C). We have the usual norm where
$||f||_{1}=\int _{\Omega }|f|d\mu$
and we define another norm on the same space as
$N(f)=\sup _{A\in F}|\int _{A}fd\mu |$ where $F$ is a sigma algebra on $\Omega$. Now the question is how to show that these two norms are equivalent. If our space was finite dimensional, it would be a very easy proof but our space is not finite dimensional. So I have to find two constants a and b such that
$a||f||_{1}\leq N(f)\leq b||f||_{2}$ for all f in our space.
Now one direction is really easy. I already got it as
$|\int _{A}f|\leq \int _{A}|f|\leq \int _{\Omega }|f|$ so my constant b is one. The question is how to do the other inequality and what is a. This is one of the many questions that I came across while studying for the final. So any tips or hints will be appreciated. Thanks
A Real Kaiser (talk) 03:03, 4 May 2008 (UTC)

If you can show $N(f)=\max\{\int f^{+},\int f^{-}\}$, the work is practically done.Nm420 (talk) 03:53, 4 May 2008 (UTC)

Err, at least when $K=\mathbb {R}$.Nm420 (talk) 03:58, 4 May 2008 (UTC)

So, how can this equivalence be shown for the complex numbers? Does anyone else have any other ideas which might work for both the real and complex numbers?A Real Kaiser (talk) 23:02, 5 May 2008 (UTC)

The idea is that N is the "largest (in magnitude) area under the curve", at least when integrating over the real line. The geometrically intuitive solution to this is that N is either the area under the positive part of the function, or the area under (over?) the negative part of the function. Obviously this needs to be proven rigorously, but you can use your geometric intuition to lead the proof. When integrating over the complex plane, you have f=g+ih, with g and h measurable F and integrable μ, and the norm N is the supremum of a set whose elements are the sum of two squares (namely, the real and imaginary "areas under the curve"). If you can maximize each of those squares, by falling back on the case over the real line, you have found N.Nm420 (talk) 13:46, 6 May 2008 (UTC)

Elliptic curve group addition formula

The article on elliptic curves gives both a geometric and an algebraic definition of addition on an elliptic curve group. How can we prove that these definitions are equivalent? That is, can we derive the algebraic formula from the geometry? --BrainInAVat (talk) 18:46, 4 May 2008 (UTC)

Yes. Note that s is the slope of the line between P and Q. Since R also lies on this line, we know that P, Q and R all lie on the line

$y=sx+c$

P, Q and R also all lie on the curve

$y^{2}=x^{3}-px-q$

so x_P, x_Q and x_R are solutions of

$(sx+c)^{2}=x^{3}-px-q$

Re-arranging this gives us a cubic in x:

$x^{3}-s^{2}x^{2}-(p+2sc)x-(q+c^{2})=0$

The sum of the solutions to this cubic is s² (because -s² is the co-efficient of x²). Therefore

$x_{P}+x_{Q}+x_{r}=s^{2}$

and the equation for y_R then follows from the fact that the slope of the line PR is s. Gandalf61 (talk) 20:57, 4 May 2008 (UTC)