Wikipedia:Reference desk/Archives/Mathematics/2010 October 10

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October 10

Intersection of 2 dense subspaces of a normed vector space necessarily dense?

As the title says, is the intersection of 2 dense subspaces of a normed vector space necessarily dense? I suspect not, but I'm struggling to come up with an example - I imagine there's one where the intersection is the zero vector only, but please could anyone help me a bit further since this line of thought has got me nowhere? Thanks! 62.3.246.201 (talk) 02:25, 10 October 2010 (UTC)

Consider $\mathbb {Q}$ and $\{r{\sqrt {2}}\ {\big |}\ r\in \mathbb {Q} \}$ as subspaces of the reals. 71.167.238.13 (talk) 02:41, 10 October 2010 (UTC)

Gotcha, thanks! 62.3.246.201 (talk) 11:22, 10 October 2010 (UTC)

Usually, normed vector space are meant to be vector spaces over the field of real or complex numbers. To quote an example with real vector spaces, just think to the Banach space $L^{1}([0,1])$ with the subspace of the continuous functions, and of the simple functions. Both are dense linear subspaces, but their intersection is a line -the constant functions. --pma 19:17, 11 October 2010 (UTC)

Very straightforward probability

I'm kind of embarrassed that I'm stuck on this, but here's my question:

Suppose you have a box with N black marbles and 1 gold marble. If you randomly select n marbles from the box, what's the probaility that you have selected a gold marble?

So I tried to set this up as #number of combinations with a gold marble/# total number of combinations, but I keep getting answers that don't make sense. I don't even know if I should treat the black marbles as distinct or not! Please help! 74.15.136.172 (talk) 04:45, 10 October 2010 (UTC)

Put N+1 white marbles in a box; take n of them up and put on the table beside the box. Now close your eyes while a demon appears and paints all the marbles black except a randomly selected one which he paints gold. What is the probability that the one he selected was one of those you've already taken from the box? –Henning Makholm (talk) 05:08, 10 October 2010 (UTC)

Right, obviously. Thanks a lot! 74.15.136.172 (talk) 15:05, 10 October 2010 (UTC)

the way to do these things is to subtract from 1. what is the probability you will roll a 6 at least once if you roll a die 6 times? Well, the probability that you don't once is 5/6, that you don't twice is (5/6)*(5/6) and that you don't six times is (5/6)^6. That's about 0.334897977, so if that's the probability you don't roll even one six in the six rolls, if you subtract it from one you get the probability that you do. Do the same thing for your problem. 92.224.205.56 (talk) 10:25, 10 October 2010 (UTC)

Makholms way is far easier. Taemyr (talk) 10:30, 10 October 2010 (UTC)

And more relevant - dice-rolling is selection with replacement, removing marbles is without replacement. I do wish that all replies were helpful →81.131.164.39 (talk) 16:11, 10 October 2010 (UTC)

I said do the "same thing" for your problem. If N is 5, and n is 3, you would do 1-(5/6) * (4/5) * (3/4). I thought it would be obvious. you can work out the general formula for N and n in the same way. 92.224.205.56 (talk) 20:43, 10 October 2010 (UTC)

So why not give this example straight away, instead of claiming that a power of a constant fraction is the "same thing" as the product of fractions with numerator and denominator each decreasing by 1? Why claim that "the way to do these things" is a method which arrives at the straightforward (to quote the OP) answer only after tedious algebraic cancellation?→81.131.164.39 (talk) 19:51, 12 October 2010 (UTC)

See Hypergeometric distribution. Bo Jacoby (talk) 17:38, 10 October 2010 (UTC).

See Overkill. Seriously though, knowledge of different distributions isn't supposed to replace probability common sense. -- Meni Rosenfeld (talk) 19:00, 10 October 2010 (UTC)

Right. But the purpose of a reference desk is to provide reference. Bo Jacoby (talk) 04:41, 12 October 2010 (UTC).

A mirror problem

Dear all:

I am working on the following problem:

A mirror has the property that whenever a ray of light emanates from the origin it reflects parallel to the x-axis. Find the equation of the mirror.

I have produced the following illustration:

and got as far as working out the following equations for the three lines:

Incident line: ${\frac {y-y_{c}}{x-x_{c}}}={\frac {0-y_{c}}{0-x_{c}}}\rightarrow {\frac {y-y_{c}}{x-x_{c}}}={\frac {y_{c}}{x_{c}}}$

Normal line: $y-y_{c}=-{\frac {1}{\frac {dy_{c}}{dx_{c}}}}(x-x_{c})\rightarrow y-y_{c}=-{\frac {dx_{c}}{dy_{c}}}(x-x_{c})$

Reflected line: $y=y_{c}$

However, I am stuck with trying to relate the three line equations into a single ODE that captures the relationship between these three lines: namely that the angle between the incident line and the normal line is equal to the angle between the normal line and the reflected line. Could anyone help me out?

Thanks for all your help.

L33th4x0r (talk) 12:46, 10 October 2010 (UTC)

Let $\alpha$ be the angle between the reflected and normal line. The (negative of the) slope of the normal line is $\tan \alpha ={\frac {dx_{c}}{dy_{c}}}$. The slope of the incident line is $\tan(2\alpha )={\frac {-y_{c}}{x_{c}}}$. Now use the trigonometric identity $\tan(2\alpha )={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}$. -- Meni Rosenfeld (talk) 13:03, 10 October 2010 (UTC)

You can solve it using some vector calculus. In particular the Frenet–Serret frame for plane curves. At each point of a regular plane curve you have a unit tangent vector T and a unit normal vector N. Let the plane curve be given by γ(t) = (x(t),y(t)). Then we have

${\mathbf {T} }={\frac {({\dot {x}}(t),{\dot {y}}(t))}{\sqrt {{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}}}}\ {\text{ and }}\ {\mathbf {N} }={\frac {(-{\dot {y}}(t),{\dot {x}}(t))}{\sqrt {{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}}}}.$

The chord joining the origin to the curve is exactly γ(t). This can be written in the form γ = αT + βN where α and β are some numbers. Since T⋅N = 0, T⋅T = N⋅N = 1 we can take the dot products to show that α = γ⋅T and β = γ⋅N, thus γ = (γ⋅T)T + (γ⋅N)N. The light ray reflects in the normal line, so the tangential component is reversed. That means that reflecting γ in the normal line gives ρ = (γ⋅N)N – (γ⋅T)T. The vector ρ lies on the line of the reflected light. For that to be parallel with the x-axis we need ρ⋅(0,1) = 0. Doing the substitutions and taking the denominator gives a differential equation:

${\dot {x}}^{2}y-y{\dot {y}}^{2}-2x{\dot {x}}{\dot {y}}=0\ .$

For the problem we know that the curve's tangent line must he vertical when it crosses the x-axis. So we can assume that y(t) = t, giving:

$t({\dot {x}}^{2}-1)-2x{\dot {x}}=0\ .$

We can solve this differential equation, and we see that for non-zero k:

$x(t)={\frac {1}{2k}}(t^{2}-k^{2}).$

Since y(t) = t you now have your final equation for the curve. You actually have two one-parameter families of curve given for k < 0 and k > 0.

You could replace k by the x-intercept if you liked. When y = 0 you have x = –1⁄2k, i.e. k = –2x. Putting this into the formula the equations become

$x=p-{\frac {y^{2}}{4p}}\ ,$

where p ≠ 0 is the x-intercept. — Fly by Night (talk) 14:48, 10 October 2010 (UTC)

The solution is well-known to be a parabola whose axis is parallel to the x-axis, but some of the ways of showing that above are cumbersome. There are some high-school-geometry-style arguments for it. I'll see if I can put one here later if there's some interest. Michael Hardy (talk) 15:44, 10 October 2010 (UTC)

There are other non-continuous, piecewise smooth solutions which aren't parabolas. — Fly by Night (talk) 17:24, 10 October 2010 (UTC)