Wikipedia:Reference desk/Archives/Mathematics/2010 December 19

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December 19

Unclear about an equation in an article

Hi there, in Central_limit_theorem#Relation_to_the_law_of_large_numbers, what are $\varphi$, $O$, $\Xi$ and so on? Thanks. It Is Me Here ^{t / c} 11:36, 19 December 2010 (UTC)

I'm not quite clear - do you want to know about the symbols or about their meaning? $\varphi$ (phi) and $\Xi$ (xi) are greek letters, used by mathematicians in moderate distress because they ran out of Latin letters (mathematicians in high distress turned to Hebrew, and eventually just made up new symbols ;-). --Stephan Schulz (talk) 11:43, 19 December 2010 (UTC)

No, no, hehe, I mean: what do these letters represent in the given article section? It Is Me Here ^{t / c} 11:53, 19 December 2010 (UTC)

The part before "Informally, ..." is just an introduction about asymptotics, where f is an arbitrary function and $\phi _{i}$ are also somewhat arbitrary functions that give an expansion of f.

$\xi$ is just the name given to the limit distribution, which the text says must be normal. $\Xi$ is the name given to the limit in the case that the initial distribution does not have a finite mean\variance, which the text says can take various forms but must be stable. -- Meni Rosenfeld (talk) 12:01, 19 December 2010 (UTC)

And the O is as in Big O notation especially the "Infinitesimal asymptotics" section. This is basically saying there are other terms in f which we can ignore when taking the limit.--Salix (talk): 13:25, 19 December 2010 (UTC)

Fourier transform of a multivariate gaussian

I'm trying to compute the Fourier transform of a multivariate Gaussian like

$e^{-{\frac {1}{2}}x^{T}C^{-1}x}$

where C is a matrix.

Does anybody know a reference where I can see this computation?--Pokipsy76 (talk) 13:57, 19 December 2010 (UTC)

I've seen several books where it's done, but none comes immediately to mind. I'll see if I can find one. The Fourier transform is itself a Gaussian function. Michael Hardy (talk) 17:31, 19 December 2010 (UTC)

I'd guess it must be of the form a exp(–½b ξ^TCξ), where a and b are constants involving 2 and π that will depend on your chosen convention for the Fourier transform --Qwfp (talk) 11:26, 20 December 2010 (UTC)

Over the weekend I worked it out for symmetric non-positive-definite matrices C. And the guess above is right. Except that I relied on something that I needed to stop and remember requires proof. That is that the variance of the probability distribution whose density is proportional to

$\exp \left({\frac {-x^{T}C^{-1}x}{2}}\right),$

actually is the matrix C. Qwfp is right except that I used the probabilists' convention where the exponent in the definition of the Fourier transform has a "+". Details later maybe........ Michael Hardy (talk) 19:38, 20 December 2010 (UTC)

Can someone give a simpler explanation of ultraradicals to me?

I'm trying to understand quintics and I'm not understanding the ultraradical function (it's a function, right?). What sort of function is it? Is it differentiable? What's its y=f(x) look like when graphed? Thanks.--70.122.122.203 (talk) 21:39, 19 December 2010 (UTC)

It's the inverse of g(x)=-x⁵-x, so wouldn't the graph of y=f(x) be the graph of x=g(y)? The implicit function theorem implies that f will be differentiable except where g′(y)=0. So it's differentiable on the real line but there are branch points in the complex plane.--RDBury (talk) 03:58, 20 December 2010 (UTC)

But why is a new function required? What makes quintic functions so special that they require a whole new sort of function? What makes the ultraradical function different from a fifth root?--99.179.21.131 (talk) 18:49, 21 December 2010 (UTC)

In serious mathematics it's very common to introduce non-elementary functions, so this isn't that special. The elementary functions are simply too limited to encompass all the wonders mathematics has to offer. For example, you can't express the solutions of $e^{x}=3x$ with only elementary functions, but if you introduce the Lambert W function, you can solve this and similar equations.

If you want to better understand why, specifically, polynomial equations of degree higher than 4 can't be solved with radicals (known as the Abel–Ruffini theorem), you may want to take a look at Galois theory.

A graph of the inverse of $-x^{5}-x$ (I'm following RDBury's lead, I haven't verified this is consistent with our article's notation) can be seen here. -- Meni Rosenfeld (talk) 09:11, 22 December 2010 (UTC)

Ok, the Galois theory article factors $f(x)=x^{5}-x-1$ into $(x^{2}+x+1)(x^{3}+x^{2}+1)$. It also says that it has exactly one real root which cannot be expressed in terms of radicals. The $(x^{2}+x+1)$ term has complex roots, while $(x^{3}+x^{2}+1)$ has one root. I put that into wolfram alpha and it returns the following:

$x=-{\frac {1}{3}}+{\frac {1}{3}}{\sqrt[{3}]{{\frac {25}{2}}-{\frac {3{\sqrt {69}}}{2}}}}+{\frac {1}{3}}{\sqrt[{3}]{{\frac {1}{2}}(25+3{\sqrt {69}})}}$

So... CAN it be expressed in radicals or no?--99.179.21.131 (talk) 04:13, 24 December 2010 (UTC)

$x^{5}-x-1$ factors into $(x^{2}+x+1)(x^{3}+x^{2}+1)$ modulo 2. This is not at all the same as factoring over the reals\complexes, and you can see that over reals

$(x^{2}+x+1)(x^{3}+x^{2}+1)=x^{5}+2x^{4}+2x^{3}+2x^{2}+x+1$.

So the complex roots of $x^{2}+x+1,\ x^{3}+x^{2}+1$ are not roots of $x^{5}-x-1$ - its root cannot be expressed in radicals. -- Meni Rosenfeld (talk) 05:32, 24 December 2010 (UTC)