Wikipedia:Reference desk/Archives/Mathematics/2006 July 21

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All functions = Circular functions?

I read somewhere (I think it was New Scientist) that someone had recently proved that any function can be expressed in terms of circular functions. Did I remember correctly? And if so, what is this called? --Alexs letterbox 02:27, 21 July 2006 (UTC)[reply]

Sure, any "nice" function can be expressed "almost everywhere" as an infinite sum of sinusoids (not sure about the details); that's the idea behind the continuous Fourier transform. If it's really "recently proved", though, then it must be something else. Fourier analysis is old news. —Keenan Pepper 03:07, 21 July 2006 (UTC)[reply]

In fact, any periodic function (not only nice ones) can be defined with an infinite series of sinusoids. I'm not sure if non-periodic functions can be defined everywhere with sinusoids, but they certainly can for some interval (in a similar fasion to a taylor polynomial if nothing else.) But Keenan is right that this has been known for decades, so if the proof was recent. -48v 05:40, 21 July 2006 (UTC)[reply]

that can't be right. Let x be a real number.Let f(x) = 0 if [x - (integar part of x)] is rational, and f(x) = 1 otherwise. That's a periodic function. Theresa Knott | Taste the Korn 07:06, 21 July 2006 (UTC)[reply]

If you include the "almost everywhere" condition it can be expressed as a sum of sinusoids, because it's equal almost everywhere to the constant function f(x)=1, since the rationals are a set of measure zero. -GTBacchus^(talk) 07:17, 21 July 2006 (UTC)[reply]

Even "almost everywhere" isn't going to be enough in general, e.g., if the function is not measurable. (Cj67 14:10, 21 July 2006 (UTC))[reply]

I'm guessing the recollection is not quite correct about "recently proved". Earlier this year Lennart Carleson was awarded the Abel Prize for his work, especially that related to harmonic analysis. A brief discussion can be found here. --KSmrq^T 19:55, 21 July 2006 (UTC)[reply]

That was it, my memory is shocking. Thanks --Alexs letterbox 04:22, 22 July 2006 (UTC)[reply]

Algorithm

What would be the most effective way to compute numbers as defined in the following definition:

For a given integer n (say 15 for example) calculate the numbers Xi...j that (each) have the following properties:
 1. X can be factorised into n (different) prime numbers.
 2. In the Set of all Substrings of X there are exactly n prime numbers.

Thanks --helohe (talk) 10:52, 21 July 2006 (UTC)[reply]

I have no idea how to deal with the second criterion, other than brute force, but the first is fairly straightforward. Given a set of prime numbers, just multiply them together in every combination of fifteen. That'll produce all and only. BTW, remember that there are actually infinite numbers that are the product of some number n of distinct prime factors, since there are infinitely many primes. Black Carrot 20:32, 21 July 2006 (UTC)[reply]

The first part is no problem. The second is the interessting one. helohe (talk) 02:20, 22 July 2006 (UTC)[reply]

Must the prime numbers in the substrings occur exactly once? For example, 17322165316704618370 = 2×5×7×11×13×17×19×23×29×31×41×43×47×53×59 and contains 2, 3, 5, 7, 17, 31, 37, 53, 61, 67, 73, 83, 167, 173 and 461. But 2, 3 and 7 occur multiple times, and in total there are 21 occurrences of primes in the string. If multiple occurrences count, must the primes be distinct? For example, 1715218867220528670 = 2×3×5×7×11×13×17×23×29×31×37×41×43×47×53 and contains 2 (4 times), 5 (3 times), 7 (3 times), 17, 67, 71, 521, together 15 occurrences, but only seven distinct ones. In any case, the best thing is to generate the products and then filter for the substrings condition. --Lambiam ^Talk 11:02, 22 July 2006 (UTC)[reply]

Elementary Statistics

I need help understanding the formulas in College Elementary Statistics. I need a good resource to go to.

Your teacher/professor, fellow students, graduate students, etc. are all good sources. Any particular formulas? I'm guessing your book has lots of them. Emmett5 23:25, 23 July 2006 (UTC)[reply]

Is there anything else as magical as Dimensional Analysis?

I was very impressed when I first found out about Dimensional analysis and Buckingham's pi theorem. As a non-mathematician it seemed magical how you could obtain a formula for something by just knowing the variables involved, without having to use any calculus either.

Is there anything else as useful in mathematics that I as a non-mathematician am unlikely to know about?

Thanks. --81.104.12.23 20:25, 21 July 2006 (UTC)[reply]

What exactly do you mean by useful? For example, the philosopher WVO Quine, justified realism in mathematics (so treating numbers as real things, and not just a construction by humans) by saying things in the mathematical realm are found in nature. So like Fractals in nature, and other advanced mathematical discoveries that are helpful in doing advanced physics and the like.--droptone 04:14, 22 July 2006 (UTC)[reply]

It's a rather difficult question, as I don't know what sort of thing you find useful or magical or whatnot. Nonetheless, I've always thought manifolds were neat as all get out. They're pretty simple to understand for a non-specialist and they come up in some interesting physics. Gauss' Theorema Egregium, mentioned in the same article, is also pretty cool. --George 05:19, 22 July 2006 (UTC)[reply]

Hi, 81.104.12.23, it seems that you and I share a common interest. You would love this book:

Hornung, Hans (2006). Dimensional Analysis: Examples of the Use of Symmetry. Mineola, NY: Dover. ISBN 0-486-44605-0.

The magic behind the magic of DI is the magic of Lie's theory of the symmetry of differential equations (ordinary or partial) or systems of same. For this try

Cantwell, Brian J. (2002). Introduction to Symmetry Analysis. Cambridge: Cambridge University Press. ISBN 0-521-77740-2.

Bluman, G. W.; Kumei, S. (1989). Symmetries and Differential Equations. New York: Springer-Verlag. (there is a more recent book also coauthored by Bluman but I like this one better)

or the books by Peter J. Olver. ---CH 09:25, 23 July 2006 (UTC)[reply]

Oh, you asked "Is there anything else as useful in mathematics that I as a non-mathematician am unlikely to know about?" Well, ask me on another day and I would give another answer, but today I'd probably say: computational algebraic geometry. ---CH 09:27, 23 July 2006 (UTC)[reply]

Since I just said "computational algebraic geometry", I guess that probability self-actualized. I think that means it has taken on the value unity :-/ (A sick pseudoscience joke, sorry!) ---CH 09:29, 23 July 2006 (UTC)[reply]