Wikipedia:Reference desk/Archives/Mathematics/2009 March 12

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 12

transcendental numbers and Khinchin's constant

In the biography of Gelfond, it states that before he proved his theorem few explicit transcendental numbers were known. Did not Liouville (constructively) prove the existence of infinitely many using certain decimal expansions?

In the description of Khinchine's constant, it reads "it has not been proven for any specific real number whose full continued fraction representation is not known". Should this not read "known"? For how can the determination be made for a specific number if the digits a0, a1, ... are unknown?

Thanks for any clarification —Preceding unsigned comment added by Aliotra (talk • contribs) 03:10, 12 March 2009 (UTC)

Yes, Liouville proved the existence of transcendental numbers (in fact of continuum-many of them). However, Liouville's proof only works for a class of numbers specifically invented for the purpose, so perhaps what is meant is that few naturally-occurring numbers were known to be transcendental. Algebraist 03:25, 12 March 2009 (UTC)

I take your point. The article at Mactutor about "accessible" real numbers appears not unrelated to this issue. "Naturally occurring" could be interpreted as "accessible", perhaps? I think the numbers Liouville used would be defined as accessible, though.

Looking through the history of the page on Khinchin's constant, I noted that someone else had suggested that the part which I questioned above, needed improvement (and I think you agreed). Is there an expert on number theory (perhaps the original author) to whom I can re-direct this question?142.27.68.72 (talk) 16:53, 16 March 2009 (UTC) 142.27.68.72 (talk) 16:51, 16 March 2009 (UTC)142.27.68.72 (talk) 16:55, 16 March 2009 (UTC)

Sorry about the unsigned post; I've created a user page. Aliotra (talk) 17:59, 14 March 2009 (UTC)

The best place to get expert advice on mathematical issues is at WT:WPM. My phrase 'naturally occurring' was not intended to be mathematically precise, and I'm not sure it can be made so. I would be interested to see an attempt to do so. If by 'accessible' you mean something like 'definable', though, then this doesn't do the job. For any sensible value of 'definable', every naturally-occurring real is definable, but Liouville's original number is definable, though certainly not naturally-occurring. Algebraist 17:01, 16 March 2009 (UTC)

Square root of a limit = limit of square root?

If g_n is a sequence of functions in L^2 and $\int _{0}^{1}|g_{n}|^{2}\to \int _{0}^{1}|g|^{2}$ is it also true that $\left(\int _{0}^{1}|g_{n}|^{2}\right)^{1/2}\to \left(\int _{0}^{1}|g|^{2}\right)^{1/2}$ I sure hope so because I need it to be true. I can also assume g_n and g are continuous if that matters. Thanks StatisticsMan (talk) 04:01, 12 March 2009 (UTC)

The square root of a limit of nonnegative real numbers equals the limit of the square roots, yes. Black Carrot (talk) 06:20, 12 March 2009 (UTC)

And the reason for that is that the square root function is continuous. — Emil J. 11:19, 12 March 2009 (UTC)

I wish that all assertions I make are true (in mathematics) but fortunately this is not the case (otherwise there wouldn't be much fun). --PST 09:25, 12 March 2009 (UTC)

Alright, thanks a lot. StatisticsMan (talk) 12:12, 12 March 2009 (UTC)

"Ecological" study etymology

In statistics, why is a study that deals with only aggregates called an ecological study when it doesn't necessarily relate to ecology? NeonMerlin 04:44, 12 March 2009 (UTC)

The word "ecology" here is being used similarly, but not indentically to the word "population" (see statistical population). In aggregate studies, statisticians work with a large number of aggregates of individuals: the word population would be inappropriate to describe each individual aggregate, because (1) the statistician is not sampling individuals from each aggregate, and (2) the statistician is sampling aggregates from the collection of all aggregates. In this context "population", being used technically (i.e., meaning "that which samples are taken from"), refers to the collection of all aggregates. So some word other than "population" is needed to describe each aggregate of individuals.

The word "ecology" is suggestive because individuals within a given aggregate tend to be highly correlated in the relevant dimensions, and form very messy, far-from-Gaussian distributions (or at least, the statisticians are not interested in the properties of the distributions within an aggregrate, so that messy distributions are expected and not problematic): for example, the example at ecological fallacy#Origin of concept could have a bimodal (at least) distribution of literacy rates within each state, for immigrants and non-immigrants. Although "ecology" is far from the perfect word to describe this, it seems to fit reasonably well. I'd guess that the usage of the word "ecology" arose because this concept arose first in biological studies (this is a guess, I really don't know), and was later generalized to other situations.

I'd welcome discussion from others. Eric. 131.215.158.184 (talk) 08:51, 12 March 2009 (UTC)

Calculation probability for draws

In the aftermath of last nights Champions League matches, I was wondering what the probability for a couple draws for the quarter-finals would be. So there are 4 English teams, 2 Spanish, 1 Portuguese, 1 German. So what's the probability of the four English teams draw each other. No English team draw another. Two of the English teams draw each other. Two Spanish teams draw each other. And the big one, the four English teams draw each other while the two Spanish teams draw each other. (this might perhaps be trivial math though I can't figure out the right formula for calculating) chandler · 09:37, 12 March 2009 (UTC)

If the english teams are ABCD, the spanish EF and the others GH, we can only get your last criteria if the draw is one of: AB CD EF GH, AC BD EF GH, AD BC EF GH. So that is 3 in 7*5*3 (A v One of 7; first alpha left v remaining one of 5; first alpha left v remaining one of 3; the last two play each other.), so 3 in 105 or 0.0286. -- SGBailey (talk) 10:24, 12 March 2009 (UTC)

The article Multiset#Polynomial_notation may be relevant. (?). Bo Jacoby (talk) 19:49, 12 March 2009 (UTC).

bound degree of a region in a planar graph

Hello. Suppose we are given a planar representation of a graph G, and for each of its regions we define the bound degree to be the number of edges enclosing it. Suppose also that it is given that the bound degree of each region is even. I want to show that the graph is bipartite. Obviously as a bipartite graph has no odd cycles, so what I want is to show that cycles involving edges encompassing more then 1 region are also even. How can I do that? Thanks--Shahab (talk) 12:39, 12 March 2009 (UTC)

This is a standard homework exercise, and I hope that is not your reason for asking. Add together the lengths of all the faces inside the cycle. McKay (talk) 21:34, 12 March 2009 (UTC)

No this isn't homework. I am a self learner. Anyway your hint was all I needed in this case. Thanks--Shahab (talk) 06:27, 13 March 2009 (UTC)

What is this probability distribution?

I've recently graphed the following probability distribution:

What is it? It seems to have a fat tail, but that doesn't concern me much; its the oddly mis-shapen nose at the top, and the log-linear sides that have my interest. linas (talk) 14:38, 12 March 2009 (UTC)

Compute the cumulants. Bo Jacoby (talk) 19:57, 12 March 2009 (UTC).

When I saw the title of this section and the graph, I assumed you meant the graph is that of the probability density. But you've labeled it "mutual information". Did you mean that the "mutual information" is itself a random variable? If log ƒ = a|x| then ƒ = e ^−a|x| and you've got a Laplace distribution. That gives you the log-linear sides, but not the funny shape at the top. Michael Hardy (talk) 20:00, 12 March 2009 (UTC)

Extract the constant term from a polynomial, with Maple9

Hi, excuse this naive question. I'm doing a computation with Maple9. As a result I have a huge trigonometric polynomial, and I want to extract the constant term. As a mathematician, I would just integrate over [-pi, pi], but this can not be the right answer. How can I just make it find the constant term? Thanks --131.114.72.215 (talk) 15:31, 12 March 2009 (UTC)

Integration from −π to π will kill all trigonometric terms except the constant term a which will contribute 2πa. So the constant term equals the integral divided by 2π. Is this answering your question? (I don't know Maple9). Bo Jacoby (talk) 20:04, 12 March 2009 (UTC).

Evaluate it at 0? 76.195.10.34 (talk) 21:55, 12 March 2009 (UTC)

That won't work. A trigonometric polynomial generally involves cosine terms that do not vanish when evaluated at 0. Michael Hardy (talk) 22:13, 12 March 2009 (UTC)

A (weird) solution, for the trigonometric polynomial T in x is:

subs(x=0, subs(cos=sin, T));

this transforms all "cos" in "sin" and then evaluates the result at 0, as suggested by 76. This seems much faster than integration (but I guess there is a better way). --pma (talk) 22:33, 12 March 2009 (UTC)

It's hard to give a useful answer without more information. It all depends on what form the expression is in. If you just want to pick out the terms in a sum F that have no sin or cos in them, remove(has,F,{sin,cos}) will do it. However, be careful because if there is only one term remove will operate on the operands of that term (this is a problem in Maple that is endlessly annoying). McKay (talk) 23:24, 12 March 2009 (UTC)