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

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April 12

When (or how do i find out when) the integral of x^P exp(-x^Q) converges over (0, infinity)?

Not much more to explain than what the title says really! I'm revising up on my analysis but I'm uncertain as to how to go about finding out where (if anywhere) in terms of P,Q the integral $\int _{0}^{\infty }x^{P}exp(-x^{Q})\,dx$ converges, where P and Q > 0. If someone can just point me in the direction of a theorem or a wikipage which I could begin to use to approach the problem (or help me begin to approach it if you're feeling generous!), I'd be hugely appreciative - thanks a lot,

Mathmos6 (talk) 04:20, 12 April 2009 (UTC)

First, change variable: $\textstyle x^{Q}=y$. This reduce to the case Q=1 (with another P, that turns out to be > -1 ), which is the Euler integral for the gamma function. So you can actually compute the integral in terms of P & Q. Then, is it clear to you the convergence of the Euler integral (i.e. in your notations, P > -1, Q=1)? --pma (talk) 09:02, 12 April 2009 (UTC)

Since $exp(-x^{Q})\!$ goes to 0 pretty quickly and $x^{P}\!$ is only a polynomial, this integral converges for all positive values of P and Q. In general, determining the convergence of improper integrals has more to do with asymptotics than with integration. The techniques tend to be the same as those for determining the convergence of infinite series. Jim (talk) 16:38, 12 April 2009 (UTC)

Intrinsic equation

Given that the intrinsic equation of a curve is $s=asec^{3}\psi -a$ where a is a non zero constant, I have to show that $y=atan^{3}\psi$. I start with the Whewell equation, namely $y=\int sin\psi ds$. However I get $sin\psi =[1-[{\frac {a}{s+a}}]^{2/3}]^{1/2}$ and from here cannot make any progress. Integrals.com can't integrate that expression so I doubt I'll have any luck with it. Where have I gone wrong? Thanks 92.0.38.75 (talk) 11:04, 12 April 2009 (UTC)

Notice that the intrinsic equation is translation invariant, so you have also to assume (0,0) belongs to the curve, if you want to show $y=a\tan ^{3}\psi$. That said, I do not see anything wrong in your approach, although there are possibly other ways. You have

$x^{\prime }(s)=\cos \psi (s)={\Big (}{\frac {a}{a+s}}{\Big )}^{\frac {1}{3}},$

$y^{\prime }(s)=\sin \psi (s)=\left[1-{\Big (}{\frac {a}{a+s}}{\Big )}^{\frac {2}{3}}\right]^{\frac {1}{2}}.$

Integrate, using $\textstyle x(0)=0$ and $\textstyle y(0)=0$:

$x(s)=\int _{0}^{s}{\Big (}{\frac {a}{a+\sigma }}{\Big )}^{\frac {1}{3}}\,d\sigma ={\frac {3}{2}}a^{\frac {1}{3}}{\Big [}(a+s)^{\frac {2}{3}}-a^{\frac {2}{3}}{\Big ]},$

$y(s)=\int _{0}^{s}\left[1-{\Big (}{\frac {a}{a+\sigma }}{\Big )}^{\frac {2}{3}}\right]^{\frac {1}{2}}\,d\sigma ={\Big [}(a+s)^{\frac {2}{3}}-a^{\frac {2}{3}}{\Big ]}^{\frac {3}{2}}.$

From these you can easily obtain your $\textstyle y=a\tan ^{3}\psi$ and the cartesian equation

$ay^{2}={\frac {8}{27}}x^{3}.$

So, it's old semicubic parabola. (PS: notice the TeX for trigonometric functions and parentheses).--pma (talk) 11:35, 14 April 2009 (UTC)

Invariant sets

Let

$T:X\to X$

and

$X':=\bigcap _{n=0}^{\infty }T^{n}(X)$

is it true that

$T(X')=X'$

? --pokipsy76 (talk) 15:47, 12 April 2009 (UTC)

No, in general you only have $\scriptstyle T(X')\subset X'$. There are simple examples; maybe you can see it: try with X a countable tree. pma (talk) 16:31, 12 April 2009 (UTC)

Which T do you have in mind on the tree?--pokipsy76 (talk) 17:43, 12 April 2009 (UTC)

T is the map going one step up the tree towards the root, as in the example below. Algebraist 20:01, 12 April 2009 (UTC)

I don't see how it could work. If the tree is a complete binary tree then it seems to me that T(X)=X.--pokipsy76 (talk) 07:34, 13 April 2009 (UTC)

Sorry, should've been clearer: you need to pick the right tree. Algebraist 12:17, 13 April 2009 (UTC)

Or this example with $\scriptstyle X=\mathbb {N}$: define $T(0)=T(1)=0$; $T(2^{n})=1$ for all n>0; $T(x)=x-1$ otherwise. Then $\scriptstyle X'=\{0,\,1\}$ and $\scriptstyle T(X')=\{0\}$. (Uè, paisà!) --pma (talk) 17:11, 12 April 2009 (UTC)

Well if we take $T(x)=x+1$ instead of $T(x)=x-1$ it works. (Uè uè!)--pokipsy76 (talk) 17:41, 12 April 2009 (UTC)

Either works. This is in fact one of the tree examples. Algebraist 20:01, 12 April 2009 (UTC)

Ok, you are right, I didn't see that $T(x)=x-1$ also produces expanding "holes" starting from every $2^{n}-1$.--pokipsy76 (talk) 08:28, 13 April 2009 (UTC)

I'm wandering if there are examples when X is a topological subspace of an euclidean space and T is a continuous function.--pokipsy76 (talk) 18:19, 12 April 2009 (UTC)

Of course there are. Any countable example (such as pma's) can be embedded in N in R. Algebraist 20:01, 12 April 2009 (UTC)

Yes, that was a stupid question and not really what I had in mind... the slightly difficult problem indeed could be to find examples with topological subspaces which are connected and have nonempty interior. For example the intermediate value makes it impossible to extend the above map in R.--pokipsy76 (talk) 07:34, 13 April 2009 (UTC)

You can get a connected example in the plane with a tree again, only this time you put in the edges. You can easily fiddle that to have nonempty interior by fattening out one of the edges. Algebraist 12:17, 13 April 2009 (UTC)

Ok, so what about this "improvement":

Let

$T:X\to X$,

$X_{0}=X$

and for any ordinal number $\alpha$

$X_{\alpha }:=\bigcap _{\beta <\alpha }T(X_{\beta })$

is it true that

$T(X_{\alpha })=X_{\alpha }$

for some $\alpha$ or (at least)

$T(X_{ON})=X_{ON}$

(assuming this makes sense)?--Pokipsy76 (talk) 20:32, 13 April 2009 (UTC)

Yes, this version works. We must have $T(X_{\alpha })=X_{\alpha }$ for some $\alpha$, otherwise $X_{\alpha }$ loses at least one point at every step, and so loses card(ON) many overall, which can't happen since X is a set. Algebraist 21:09, 13 April 2009 (UTC)

And any ordinal of the same cardinality of X may be required in the iteration, before the sequence becomes stationary: in the sense that for any ordinal α with card(α)$\scriptstyle \leq$card(X) there is a map $\scriptstyle T:X\to X$ such that $\scriptstyle T(X_{\alpha })\neq X_{\alpha }$. --pma (talk) 10:01, 14 April 2009 (UTC)

Factorial proof

How cn it be prooved that 0!=1 ? —Preceding unsigned comment added by 201.130.196.160 (talk) 19:14, 12 April 2009 (UTC)

It cannot be proved, it is defined to be that value because it make formulas work. See Factorial#Definition For more info. meshach (talk) 19:24, 12 April 2009 (UTC)

The continuity of the relationship between the gamma function and factorial recquires that 0!=1. The gamma function is an extension of factorial to all real and complex numbers, and is subject to the relationship $(z-1)!=\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt\;.$. If you admit the relationship between gamma and factorial (Which there is proof of on the gamma function article), then evaluation gamma gives proof that 0!=1 since $\Gamma (1)=1$. This is the closest thing to a proof I can offer you. Elocute (talk) 20:25, 12 April 2009 (UTC)

We have $n!=n(n-1)!$, which leads us to $1!=1.0!$. Supposing that 0! is defined, then its value must be 1. Readro (talk) 21:53, 12 April 2009 (UTC)

It is the number of bijections from one empty set to another, which is 1. Strictly speaking it is a convention, but any convention other than 0! = 1 would cause endless trouble. McKay (talk) 00:51, 13 April 2009 (UTC)

Others above have offered quite a bit of techno-babble, but I can show you a very simple derivation by working backwards from, say, 4!

$4!=(4)(3)(2)(1)=24$

$3!={\frac {4!}{4}}={\frac {24}{4}}=6$

$2!={\frac {3!}{3}}={\frac {6}{3}}=2$

$1!={\frac {2!}{2}}={\frac {2}{2}}=1$

$0!={\frac {1!}{1}}={\frac {1}{1}}=1$

Make sense? --69.91.95.139 (talk) 01:55, 13 April 2009 (UTC)

The questioner has specifically requested a proof of the claim that 0! = 1 and yet your "techno-babble" above fails to constitute a proof. Although your argument provides some insight into the claim, it does not constitute a mathematical proof. Furthermore, the other comments above were far more well-reasoned arguments. Please see proof (mathematics) for more details. --PST 04:57, 13 April 2009 (UTC)

Furthermore, User:Readro has provided a well-expressed argument equivalent to yours. Your argument lacks the assumption that 0! is defined, which is necessary to conclude that this entity equals 1. --PST 05:00, 13 April 2009 (UTC)

PST, please use the reference desk for constructive contributions. I have left a note at your talk page. Eric. 131.215.159.99 (talk) 08:32, 13 April 2009 (UTC)

I see nothing wrong in my comment. The previous user seems to have criticized the comments given by the other users when, in fact, he/she is the one to have a flawed argument. And "please use the reference desk for constructive comments" is too general a statement, asserted without sufficient evidence. As I have not contributed anything that is prohibited, and the discussion that you have initiated is not relevant to the question, it is not of my interest to take further part in it. --PST 08:54, 13 April 2009 (UTC)

I apologize for providing Readro's 'proof' more explicitly. In the future I will let the concepts remain written in abstract, potentially confusing symbols rather than written out with numbers. There is no reason, after all, that any argument should be written explicitly for the reader unfamiliar or uncomfortable with the abstract concepts of algebra; that's just silly.

And none of the contributions were formal proofs. You can't prove a convention. You can, however, provide a good argument for it, which is what we've done here. --69.91.95.139 (talk) 11:43, 13 April 2009 (UTC)

The question as to whether your argument is well-designed or not, is not what I am challenging. This is merely my opinion. The point is that you must assume 0! to be defined, before concluding it to be 1. --PST 12:46, 13 April 2009 (UTC)

The other important thing to note is that your comment would have been perfectly OK (it is not of my interest to attack people for their errors) had you not challenged the other arguments. --PST 12:48, 13 April 2009 (UTC)

Then I probably should have said "... which is okay if you're at a high enough level to understand them, but I'm guessing you're not, so...", because that's what I meant. I didn't mean their arguments were bad, just a little too technical for the kind of person who is likely to ask why 0! is 1 (ie, high school algebra level). --69.91.95.139 (talk) 13:35, 13 April 2009 (UTC)

OK. As I said, I am not too concerned about this and nor is anyone else. But in my opinion, you could have phrased your intended comment in a better manner (or just avoiding that comment would not have done any harm). Furthermore, in my understanding, your argument was not logically structured well enough (concluding something equivalent to: f(2) = 2^2 => f(1) = 1^2; and this implication is, of course, not logical), but of course others may see this differently. As there is no longer merit in continuing this argument, noticing that the original poster has left, I no longer see purpose in replying. --PST 03:22, 14 April 2009 (UTC)

Conjugate Diameter?

In A treatise on analytical geometry, conjugate and transverse axes are noted, regarding oblate and prolate spheroids, which agrees with the definition of conjugate and transverse diameters of an ellipse. These appear related to the idea of conjugate points. But, back on pg.107of "A treatise...", the concept of Conjugate Diameters in Ellipseappears to being discussed.

(Diagram of conjugate diameters)

How can that be? Aren't the red and blue lines actually oblique diameters, with the vertical diameter along y being the conjugate diameter, and the horizontal along x, the transverse?
Are there two different meanings of conjugate diameter? ~Kaimbridge~ (talk) 19:46, 12 April 2009 (UTC)

Mandelbrot Set

What are the root and apex points of the Mandelbrot set? 72.197.202.36 (talk) 20:38, 12 April 2009 (UTC)

I believe I have heard 0.25 (the inner point of the cusp of the cardioid) referred to as the root. “Apex point” might refer to −2, which is the point at the very left end. Jim (talk) 01:45, 13 April 2009 (UTC)