Wikipedia:Reference desk/Archives/Mathematics/2013 June 10

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June 10

Why is the angle defined via area instead of arc length?

Both the circular angle and the hyperbolic angle are defined via the area instead of arc length. Why?

My guess is that the area is related to some global property of the space and thus put strong constraint on the definition of the angle. While the arc length is a local property of a curve. For a given arc length, the curve can change arbitarily so there will be not a good definition of angle via the arc length.

Armeria wiki (talk) 01:16, 10 June 2013 (UTC)

As far as I've ever been able to discern, this odd notion of "angle" comes up only when discussing the hyperbolic functions, which are important for all sorts of reasons, but hardly at all for any reasons that directly relate to this notion of angle. The hyperbolic functions are sort of the trig functions rotated 90 degrees in the complex plane, and they show up all over the place. If you want a "reason", the best one that I can think of is that (the simple ones) are solutions to

${\frac {d^{2}y}{dx^{2}}}=y$

, just as the simple trig functions are solutions to

${\frac {d^{2}y}{dx^{2}}}=-y$

. I'm not convinced there's really any one identifiable "reason"; that's just the best one I can come up with. --Trovatore (talk) 04:24, 10 June 2013 (UTC)

Oh, so to finish my chain of reasoning, which I'm not sure was quite obvious, the reason that the hyperbolic angle is defined as it is is just because that's the definition that gives the hyperbolic functions, which are important for other reasons. I've never really liked this way of presenting them — to me it seems more clever than insightful. But maybe there's an insight somewhere that I've just missed. --Trovatore (talk) 04:31, 10 June 2013 (UTC)

A fair hypothesis could be that in terms of area is probably the only direct geometric definition that has been found to work (even though it may regarded more as a geometric accident than an insightful definition), and it can be related back to an almost identical definition in the trigonometric case (only with a different curve), if only for similarity when discussing both. — Quondum 04:53, 10 June 2013 (UTC)

I think that's pretty much what I said, except for the bit about it being a geometric definition. The question is, what's the point of making the definition geometric? I question the value of that, when it comes out as odd-looking as this one. --Trovatore (talk) 05:29, 10 June 2013 (UTC)

The circle x²+y²=1 is parametrized by x=cos(u), y=sin(u) because cos²(u)+sin²(u)=1, and the hyperbola x²−y²=1 is parametrized by x=cosh(u), y=sinh(u) because cosh²(u)−sinh²(u)=1. Without geometry it is hard to understand the names hyperbolic sine and hyperbolic cosine. Bo Jacoby (talk) 06:26, 10 June 2013 (UTC).

That's true, but those are just names. Few of the applications of the hyperbolic functions seem to have much to do with hyperbolas. --Trovatore (talk) 06:57, 10 June 2013 (UTC)

See rapidity. Bo Jacoby (talk) 08:44, 10 June 2013 (UTC).

Exactly. Has nothing to do with hyperbolas. --Trovatore (talk) 19:07, 10 June 2013 (UTC)

I think this is also because area is conserved by affine transformations (up to multiplication by constant) but length is not. --84.228.201.241 (talk) 10:38, 10 June 2013 (UTC)

Both angles are just the arc length. One is with respect to the Euclidean line element $ds^{2}=dx^{2}+dy^{2}$, and the other is with the Lorentzian line element $ds^{2}=dx^{2}-dy^{2}$. (Both are also areas, essentially by Green's theorem.) Sławomir Biały (talk) 11:45, 10 June 2013 (UTC)

Thanks! I checked the Lorentian line element $ds^{2}=dx^{2}-dy^{2}$ which is correct. From the circular angle from arc length of unit circle in 2D, we have the solid angle from a unit sphere in 3D. Do we have similar "hyperbolic angle in 3D" with a hyperbolic surface? Thanks. Armeria wiki (talk) 10:46, 11 June 2013 (UTC)

Yes, there is. This is area in the hyperboloid model of the hyperbolic plane. The hyperboloid $x^{2}+y^{2}-z^{2}=-1,z>0$ carries a line element $ds^{2}=dx^{2}+dy^{2}-dz^{2}$ which induces an area form that can be integrated. This can all be done explicitly in a spherical coordinate system by setting $x=\cos \theta \sinh \phi ,y=\sin \theta \sinh \phi ,z=\cosh \phi$ for $\phi \geq 0$. The hyperbolic area is obtained by integrating $\sinh \phi \,d\phi \,d\theta$ over the region of interest on the hyperboloid. (A similar thing works in any dimension, and on the hyperboloid of one sheet.) Sławomir Biały (talk) 11:16, 11 June 2013 (UTC)

This is certainly a nice way of understanding it. Going back to the original question (why is it defined this way?), I think Trovatore's observation may apply: that it is usually defined for its algebraic and differential utility, but we can see that this geometric interpretation also emerges. So it is merely one way one could define it, but competing definitions are probably more common, being more natural in a non-geometric context. — Quondum 12:14, 11 June 2013 (UTC)

It probably also has to do with the fact that most people just learning about hyperbolic functions would not be comfortable with a Lorentzian notion of arc length, but would be comfortable with the ordinary Euclidean notion of area. Sławomir Biały (talk) 12:35, 11 June 2013 (UTC)

Great! Thanks a lot! I didn't know these things before. By the way, could you please explain a little bit in details how to apply Green's theorem to the case of hyperbolic angle with Lorentzian line element? Thanks. Armeria wiki (talk) 11:49, 11 June 2013 (UTC)

The (ordinary Euclidean) area subtended by a hyperbolic angle is given by the line integral over the arc of the hyperbola $A={\frac {1}{2}}\int x\,dy-y\,dx$ by Green's theorem. Substituting in the parametrization of the hyperbola by hyperbolic angle $x=\cosh t$, $y=\sinh t$, the line integral is just the integral of 1, and so just gives back the hyperbolic angle. Sławomir Biały (talk) 12:27, 11 June 2013 (UTC)

Thanks. But how does integral of area above by Green's theorem relate to the Lorentzian line element $ds^{2}=dx^{2}-dy^{2}$? Armeria wiki (talk) 12:36, 11 June 2013 (UTC)

It enters via the parametrization that I've used, which satisfies $x'=y$ and $y'=x$, so the integrand is just $((x')^{2}-(y')^{2})\,dt$. Sławomir Biały (talk) 12:54, 11 June 2013 (UTC)

Or, to look at it a different way, $x\,dy-y\,dx=ds$ (up to a sign which I must have switched at some point), which is true independently of the parametrization. Sławomir Biały (talk) 13:09, 11 June 2013 (UTC)

What is the difference between $ds=x\,dy-y\,dx$ above and a normal arc length $ds={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx$? Armeria wiki (talk) 13:48, 11 June 2013 (UTC)

They're identically equal on the unit circle. On the unit hyperbola, you have $ds={\sqrt {1-(dy/dx)^{2}}}dx=xdy-ydx$ is the Lorentzian arc length. Sławomir Biały (talk) 14:16, 11 June 2013 (UTC)

Is the indentical equal for the unit circle just a coincidence? For a circle or a hyperbola or some other curve, we can always calculate the normal arc length $ds_{normal}={\sqrt {1+(dy/dx)^{2}}}dx$. Similarly, we can calculate the "arc length" from the Green's theorem $ds_{Green}=x\,dy-y\,dx$ for a curve. My original question is about the relation among angle, area and arc length. It seems that the normal arc length has nothing to do with area in general. While the $ds_{Green}=x\,dy-y\,dx$ defined via Green's theorem is related to the area and can be used to define an angle related a curve. So besides a circle or a hyperbola, for an abitary curve, for example $x^{m}+a*y^{n}=C$, or something as you like, can we use the Green's theorem to define some "arc length" related to the area and some correponding "angle"? Armeria wiki (talk) 14:57, 11 June 2013 (UTC)

I think you're reading too much significance into Green's theorem here. It's all really quite a trivial observation. In general, you have $xdy-ydx=r^{2}d\theta =\rho ^{2}d\phi$ where r is the euclidean distance theta the circular angle rho the Lorentzian distance and phi the hyperbolic angle. Its only when r=1 or rho=1 that this is naturally thought of as a length. Sławomir Biały (talk) 15:22, 11 June 2013 (UTC)

I think for $x^{2}+y^{2}=2$ with $r={\sqrt {2}}$ or $x^{2}-y^{2}=3$ with $\rho ={\sqrt {3}}$, we can still define "Green's" arc length and angle. As an example for arbitary curve with related angle, for $x=1$, we can define "Green's" arc length as $ds_{Green}=dy$ and corresponding angle $\theta _{Green}=y$.Armeria wiki (talk) 15:57, 11 June 2013 (UTC)

Somehow I'm not very satisfied with the argument depending on the parameterization. For instance, for the hyperbola $x^{2}-y^{2}=1$, we certainly can parameterize this curve in a standard way as $x=\cosh t$, $y=\sinh t$. Then we have the Lorentzian arc length segement $ds_{Green}=xdy-ydx={\sqrt {dx^{2}-dy^{2}}}=dt$ with the hyperbolic angle $dt$. But there are many other ways to parameterize the curve, such as $x=\sec u$, $y=\tan u$, with the Lorentzian arc length segement $ds_{Green}=xdy-ydx={\sqrt {dx^{2}-dy^{2}}}=(\sec u)du=d\ln |\sec u+\tan u|$. I think here the paramter $u$ (or $\ln |\sec u+\tan u|$?) can be viewed as an angle, which is different from the normal hyperbolic angle. Another example is ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$. If we parameterize the ellipse with the usual Euclidean circular angle $\theta$ with ${\frac {y}{x}}=\tan \theta$, we have $x={\frac {ab}{\sqrt {a^{2}\tan ^{2}\theta +b^{2}}}}$, $y={\frac {ab\tan \theta }{\sqrt {a^{2}\tan ^{2}\theta +b^{2}}}}$, and the arc length segement $ds_{Green}=xdy-ydx={\frac {a^{2}b^{2}\sec ^{2}\theta }{a^{2}\tan ^{2}\theta +b^{2}}}d\theta =d(ab\arctan({\frac {a}{b}}\tan \theta ))$. We also might try another paremeterization for the ellipse by $x=a\cos \phi$, $y=a\sin \phi$ and $ds_{Green}=xdy-ydx=abd\phi$ or by $x=a\cos {\frac {\phi }{ab}}$, $y=a\sin {\frac {\phi }{ab}}$ and $ds_{Green}=xdy-ydx=d\phi$. In an extreme case, we can all parameterize the curve as $x=x$, $y=f(x)$ and $ds_{Green}=xdy-ydx=(xf^{\prime }(x)-f(x))dx=d(xf(x)-2F(x))$ with $F^{\prime }(x)=f(x)$. Different parameterization will give different value of angle. But I think the definition of angle via the area or via the corresponding arc length from Green's theorem should be indepedent on the parameterization. Armeria wiki (talk) 01:04, 13 June 2013 (UTC)

The parameterization $(\cosh t,\sinh t)$ was chosen because it is the Lorentzian arclength parameterization, so the line integral of $x\,dy-y\,dx$ is just the integral of 1, so returns the hyperbolic angle. You could have used any parametrization of the hyperbola, but you would have gotten an integral that was not so easy to calculate and express in terms of the hyperbolic angle (but it would have been the same value, being the same integral after a change of variables). The general formula is that on the unit hyperbola $ds=y\,dx-x\,dy$. This doesn't require you to work in a particular parameterizationSławomir Biały (talk) 12:11, 13 June 2013 (UTC)

Resolved

 – — Thank you all very much!

Changing a matrix so that a multiplication is preserved

Hi all,

I have a 5x5 matrix that has each column holds a percentage of the money held in a fund. The matrix has the following properties: >Every entry is a decimal between 0 and 1 (just a percentage) >Every column sums to less than 1. So we have $F_{1}$ to $F_{5}$ fixed amounts of the funds. $L_{1}$ to $L_{5}$ fixed amounts (remaining constant throughout) and our matrix:

$\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1,5}\\a_{21}&a_{22}&\cdots &a_{2,5}\\\vdots &\vdots &\ddots &\vdots \\a_{5,1}&a_{5,2}&\cdots &a_{5,5}\end{bmatrix}}.$

With each $0<a_{i,j}<1$. Such that:

${\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1,5}\\a_{21}&a_{22}&\cdots &a_{2,5}\\\vdots &\vdots &\ddots &\vdots \\a_{5,1}&a_{5,2}&\cdots &a_{5,5}\end{bmatrix}}{\begin{bmatrix}F_{1}\\F_{2}\\\vdots \\F_{5}\end{bmatrix}}={\begin{bmatrix}L_{1}\\L_{2}\\\vdots \\L_{5}\end{bmatrix}}$

(So the different proportions of the 5 funds sum to each $L$). My question is this: is it possible to multiply this matrix $A$ by another matrix which has special properties, so that the above condition is preserved in the resulting matrix, as well as the column sums being between 0 and 1? Eventually I want to randomly generarate these "special" matrices then test to see if an output further down the line is maximised.

I'm pretty sure I am overcomplicating things, but thanks for any help given! 80.254.147.164 (talk) 13:49, 10 June 2013 (UTC)

Let me try to simplify this: You have a 5x5 matrix $A$ with entries between 0 and 1. You have a particular vector $f$ with non-negative entries. You want to find a matrix $B$ such that $BAf=Af$, and each column of $BA$ sums to a value between 0 and 1. Sound right?--80.109.106.49 (talk) 15:41, 10 June 2013 (UTC)

It could, more generally, be $XAYf=Af$ (where either X or Y could be the identity matrix, or both in the trivial solution). MChesterMC (talk) 08:20, 11 June 2013 (UTC)

Yes that's right - a way of finding that matrix $B$ (or $X$ and $Y$ in the general case) given the avialable info. I knew I was making it too complcated! 80.254.147.164 (talk) 08:26, 11 June 2013 (UTC)

Actually, your case is easier, since that equation boils down to $Bl=l$, i.e. $l$ is an eigenvector of $B$ with eigenvalue 1. Given that, I'm sure someone better at math than me can give a statement of which properties $B$ must have. MChesterMC (talk) 09:45, 11 June 2013 (UTC)

Just some initial thoughts - since this matrix has eigenvalue 1 the possible set of matricies for $A$ is the set of all of the matricies that preserve the distance and size of $l$. This includes rotations around that vector, skewing that preserves that vector etc 80.254.147.164 (talk) 11:56, 11 June 2013 (UTC)

Well, expanding the eigenvalue equation gives $\sum _{j=1}^{5}b_{ij}L_{j}=L_{i}$, which lets you put (e.g.) $b_{ik}$ in terms of the other $b_{ij}$ (for j!=k). There seems to be free choice (except for one $b_{ij}$ in each row), so we could treat B as a 5x4 core matrix B', with the last row dictated by the values of B'. Any matrix B' should work to give something that satisfies the eigenvalue equation (but not necessarily the sum of columns < 1 constraint). The sum of each column will depend on every value in B (the sum of column i is $\sum _{i=1}^{5}\sum _{k=1}^{5}B_{ik}A_{kj}$), so finding a general form will probably be rather difficult. MChesterMC (talk) 13:51, 11 June 2013 (UTC)

Basic Demonstration Required

$\sum _{k=0}^{p}{(-1)^{k}\ {\frac {n}{n+k}}\ C_{p}^{k}}\ =\ {\frac {p!\ n!}{(p+n)!}}\ =\ {\frac {1}{C_{p+n}^{n}}}\ =\ {\frac {1}{C_{p+n}^{p}}}\quad -$ 79.113.223.107 (talk) 16:18, 10 June 2013 (UTC)

Integrate $\scriptstyle x^{n-1}(1-x)^{p}$ from 0 to 1. This will give the LHS of your formula up to a factor of n. The rest now follows from standard Beta function identities. Sławomir Biały (talk) 16:27, 10 June 2013 (UTC)

I've obtained it by integrating $\scriptstyle \left(1-{\sqrt[{n}]{x}}\right)^{p}$ from 0 to 1. — 79.113.243.176 (talk) 19:36, 10 June 2013 (UTC)

$I=\int _{0}^{1}\left(1-{\sqrt[{n}]{x}}\right)^{p}dx\ ,\qquad t={\sqrt[{n}]{x}}\ ,\qquad x=t^{n}\ ,\qquad dx=d(t^{n})=n\ t^{n-1}\ dt$

$I=n\int _{0}^{1}t^{n-1}(1-t)^{p}\ dt\ =\ n\ \mathrm {B} (n,\ p+1)\ =\ n\ {\frac {(n-1)!\ p!}{(n+p)!}}\ =\ {\frac {n!\ p!}{(n+p)!}}$

79.113.243.176 (talk) 22:43, 10 June 2013 (UTC)

.I guess the real question then would be how do we prove Euler's integral formula for the beta function to be equal to. $\color {blue}{\tfrac {m!\ n!}{(m+n)!}}$ . — 79.113.243.176 (talk) 00:58, 11 June 2013 (UTC)

Consider the integral

$\int _{0}^{1}\exp(xt)\exp \left[(1-x)u\right]dx$

Count Iblis (talk) 13:27, 11 June 2013 (UTC)

It is equal to ${\frac {e^{t}-e^{u}}{t-u}}$ — 79.113.237.30 (talk) 15:24, 11 June 2013 (UTC)

You can then consider this function for complex u and t. The nth derivative w.r.t. t and the mth w.r.t. u in the limit for both u and t to zero will yield the desired integral and you can compute that by expanding the function [exp(t) - exp(u)]/(t-u). If we take |u| > |t|, then we have to expand 1/(t-u) in positive powers of t and negative powers of u. Only the term exp(u)/(u-t) will then contribute. Count Iblis (talk) 16:04, 11 June 2013 (UTC)

I assume u and v are considered independent variables in this scenario, and the umpteenth derivate of our function will have a form similar to:

$\left({\frac {e^{x}}{x}}\right)^{(n)}={\frac {e^{x}}{x^{n+1}}}\sum _{k=0}^{n}{(-1)^{k}(x^{n})^{(k)}}$

And then we'll somehow use this formula, or some version thereof, to write some sort of Taylor series for the function in question... Maybe even replace e^x with its own Taylor series expansion, all in order to... do what exactly ? :-( — 79.113.237.30 (talk) 17:29, 11 June 2013 (UTC)

I need to demonstrate it from scratch, without any recourse to either beta functions or gamma functions. Because that's where I began, and now I'm trying to work it backwards. We may use the properties of combinations and factorials, but not those of their integral extensions to the entire domain of real numbers. That's the whole point of my endeavour. — 79.113.223.107 (talk) 16:41, 10 June 2013 (UTC)

Ok, well I think another way to do it is to express the matrix whose n,k entry is ${\tfrac {n}{n+k}}$ as a Cauchy matrix then compute its inverse from the (known) formula. I haven't actually done this, but I think it (or some variation) will probably work.

Is there no easier way ? :-( — 79.113.223.107 (talk) 17:08, 10 June 2013 (UTC)

There probably is, but it's a strange formula from the combinatorial perspective, since it doesn't just involve whole numbers. One has to devise a way to go from combinatorial functions to their reciprocals. It's natural to attempt that by matrix inversion. There are other methods like Möbius inversion, but you would have to think hard about the combinatorics of the problem. Other than that, it might be worth checking John Riordan's "Combinatorial identities" to see if it appears there. (If so, I should prepare myself to be humbled by a much simpler solution :-) Sławomir Biały (talk) 17:49, 10 June 2013 (UTC)

Isn't there some simple high-school-level proof from mathematical induction that we're missing ? Will it help, for instance, if we multiply both sides by (p+n)! ? — 79.113.243.176 (talk) 19:10, 10 June 2013 (UTC)

For large n, the RHS scales as n^-p but on the LHS, n/(n+k) goes to 1, so the LHS scales as 1. I don't see how this can be true. --Mark viking (talk) 17:40, 10 June 2013 (UTC)

The LHS tends to zero as n goes to infinity by the binomial theorem. Sławomir Biały (talk) 17:49, 10 June 2013 (UTC)

Ah, of course, I missed the cancellation of terms. Sorry about the noise. --Mark viking (talk) 19:06, 10 June 2013 (UTC)

Read this book. Count Iblis (talk) 18:00, 10 June 2013 (UTC)

Browsing it as we speak... (Do you have some specific page or chapter in mind ?) — 79.113.243.176 (talk) 19:14, 10 June 2013 (UTC)

Yes, I am aware that the expression becomes the hypergeometric function ₂F₁ when the sum does not alternate, i.e. when the (-1)^k term is missing. — 79.113.243.176 (talk) 22:51, 10 June 2013 (UTC)

I think you can get at this from finite difference calculus. From Finite difference#n-th difference, the RHS is proportional to the pth-forward difference of the inverse:

$\sum _{k=0}^{p}{(-1)^{k}\ {\frac {n}{n+k}}\ C_{p}^{k}}=-1^{p}n\Delta ^{p}(1/n)$

where the forward difference is respect to n. Then we have $1/n=(n-1)_{-1}$ with $()_{-1}$ the Pochammer symbol. Then

$\Delta ^{p}(n-1)_{-1}=(-1)_{p}(n-1)_{-p-1}=(-1)^{p}p!/\left(n(n+1)...(n+p)\right)$

and the original sum becomes

$\sum _{k=0}^{p}{(-1)^{k}\ {\frac {n}{n+k}}\ C_{p}^{k}}=(-1)^{2p}p!/\left((n+1)...(n+p)\right)=p!n!/(n+p)!$

--Mark viking (talk) 21:40, 10 June 2013 (UTC)

Resolved

On my talk page they asked for something without even differences, Anyway using recursion and the binomial theorem, hope they're acceptable, we can do it as follows

Multiply the left by the divisor on the right to make the result zero, so we want to prove

$\sum _{k=0}^{p}(-1)^{k}{\frac {n}{n+k}}{\frac {p!}{k!\ (p-k)!}}{\frac {(p+n)!}{p!\ n!}}=1$

Recurse on p subtract pth version of the lhs from the next and we should get zero, I'll take it as read that p=0 gives 1.

$\sum _{k=0}^{p+1}(-1)^{k}\left({\frac {n}{n+k}}{\frac {(p+1!}{k!\ (p+1-k)!}}{\frac {(p+1+n)!}{(p+1)!\ n!}}-{\frac {n}{n+k}}{\frac {p!}{k!\ (p-k)!}}{\frac {(p+n)!}{p!\ n!}}\right)$

Simplify

$\sum _{k=0}^{p+1}(-1)^{k}{\frac {n}{(n+k)\ n!\ k!}}\left({\frac {(p+1+n)!}{(p+1-k)!}}-{\frac {(p+n)!}{(p-k)!}}\right)=\sum _{k=0}^{p+1}(-1)^{k}{\frac {n\ (p+n)!}{(n+k)\ n!\ k!\ (p-k)!}}\left({\frac {p+1+n}{p+1-k}}-{\frac {1}{1}}\right)$

Which simplifiy then gives

$\sum _{k=0}^{p+1}(-1)^{k}{\frac {n\ (p+n)!}{n!\ k!\ (p+1-k)!}}={\frac {(p+n)!}{(n-1)!}}\sum _{k=0}^{p+1}(-1)^{k}{\frac {1}{k!\ (p+1-k)!}}$

Making the thing summed look like a binomial expansion we get

${\frac {(p+n)!}{(n-1)!\ (p+1)!}}\sum _{k=0}^{p+1}(-1)^{k}{\frac {(p+1)!}{k!\ (p+1-k)!}}={\frac {(p+n)!}{(n-1)!\ (p+1)!}}(1-1)^{p+1}=0$

Therefore the value of the difference does not change when you increment p and therefore the expression is true for all natural numbers p. Dmcq (talk) 14:48, 13 June 2013 (UTC)

Resolved

 – — Thank you all very, very much !

Vector projection

What's the equation for the vector of the projection of ${\vec {u}}$ onto ${\vec {v}}$? --Melab±1 ☎ 20:23, 10 June 2013 (UTC)

See scalar projection. — 79.113.243.176 (talk) 20:47, 10 June 2013 (UTC)