Wikipedia:Reference desk/Archives/Mathematics/2011 February 16

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February 16

Modeling High Dimensional Dynamical System

I am looking into questions involving spatial economics and have some practical questions for people with experience in modelling dynamical systems.

In continuous time, when you've got a bunch of nonlinear differential equations strung together, and you model them using, say, RungeKutta45, how many dimensions can you keep adding before the computational rounding-error and noise will take over?

Can a numerical model of a 20 or 30 dimensional system of differential equations be trusted? Is Matlab okay for this or should you be using Fortran or something?

What about a Coupled map lattice? Can these be done with continuous systems using RungeKutta or should you discretize your system first? The example that they give is for a logistic map, which is 1D. What if each point on your coupled map lattice was a five or six (or more) D system? Then a 20*20 lattice would be a system in something like 2000 dimensions. Is this a reasonable thing to attempt? —Preceding unsigned comment added by 130.102.158.15 (talk) 07:41, 16 February 2011 (UTC)

It's less a matter of dimensions than stability (see for example stiff equation), accuracy of the initial conditions, etc. Anyway, 20 or 30 variables is nothing. Runge-Kutta is a rather low tech method and I gather there is much better stuff for systems of that size. I don't know anything about coupled map lattices but I think you can do fine consulting a numerical methods book. I thought Numerical Recipes was pretty good but I have the impression that some experts think it gives bad advice at times, so who knows. 71.141.88.54 (talk) 09:32, 16 February 2011 (UTC)

Hi, could you elaborate on what methods are better than Runge-Kutta for large-sized systems? I was under the impression that Matlab's 'ode45' was as good as it gets. —Preceding unsigned comment added by 118.208.0.88 (talk) 09:36, 16 February 2011 (UTC)

Actually unless you've got unusually difficult or large volumes of problems, I'd just go with whatever is more convenient, so if you're used to ode45 and Matlab, I'd suggest trying it on your problem and if it works out well enough for you, that's great. I'm not any kind of expert at this stuff; what I mostly remember is that Runge-Kutta is quite reliable and simple, and fancier methods like multistep methods and the Bulirsch-Stoer algorithm are more efficient but can be finicky. By that standard I guess RK is the first thing to try. 71.141.88.54 (talk) 17:50, 16 February 2011 (UTC)

Generalisations of the Monty Hall problem

The Monty Hall problem generalises into more than three doors, of which one contains the car and the rest contain goats. After Monty opens one door containing a goat, switching to a different door increases the probability of getting the car by a factor of $1+{\frac {1}{n-2}}$, where n is the number of doors. But how does it generalise into a situation where there's more than one car, and it's enough to get one of them to win? JIP | Talk 09:09, 16 February 2011 (UTC)

Let the total number of doors be n, the number of cars m and the number of doors revealed k. Also let w be the probability of winning when switching, t when not switching and r when choosing one of the $n-k$ hidden doors uniformly randomly. Then $t={\frac {m}{n}}$ and $r={\frac {m}{n-k}}$. But also $r={\frac {1}{n-k}}t+{\frac {n-k-1}{n-k}}w$ and so $w={\frac {m(n-1)}{n(n-k-1)}}$. -- Meni Rosenfeld (talk) 09:31, 16 February 2011 (UTC)

So this means, then, that the factor the probability of winning increases by is ${\frac {w}{t}}$, which is ${\frac {\frac {m(n-1)}{n(n-k-1)}}{\frac {m}{n}}}$, which is ${\frac {m(n-1)}{n(n-k-1)}}\times {\frac {n}{m}}$, which simplifies to ${\frac {n-1}{n-k-1}}$. So the number of cars doesn't matter, it only depends on the number of doors in total and the number of doors revealed. JIP | Talk 10:25, 16 February 2011 (UTC)

Non linear second order ODEs

I am having trouble with my maths work for this week, I am nto asking you to solve this for me, but just how to get started with each one:

the questions are at: http://i56.tinypic.com/11kgkd0.jpg

Any advice at all will be greatly appreciated.

I can see that for :

a) I think i need to reduce it to a first order equation and then use bernoulli's equation to solve it, but I am unsure how to get it first order since no variables are missing so I don't know what substitution must be made

b) I think i just need to use an auxilliary equation, so do I sub in the exponential identities for sin and cos and then try to get the aux equation from there?

c) Short of using the substitution in question and getting y = xp^2 + p^3, I have no idea.

Sorry for not putting the questions here, the link is just an image of the questions.

Thanks a lot guys

--94.193.65.158 (talk) 14:54, 16 February 2011 (UTC)

[This is written after this was archived; hopefully someone will find it helpful.] First, for posterity, I'll rewrite the equations:

1. $y''+{\frac {y'^{2}}{y}}+2{\frac {y'}{x}}=0$
2. $y'''\sin x+3y''\cos x-3y'\sin x-y\cos x=e^{x}$
3. $y=xy'^{2}+y'^{3}$ (hint: write $y'=p$ and solve for x)

(#3 isn't second-order, and #2 is neither nonlinear nor second-order.)

1. This has a very simple solution as well as another more complicated solution; I don't know how you're supposed to know how many to look for.
2. All the coefficients on the left are exponentials with the rate $\pm i$, whereas the right side has rate 1. The product of exponentials has the sum of their rates, so $y,y',y'',y'''$ must all share a pair of rates.
3. This also has a very simple solution and a very complicated solution; I don't know exactly how you're supposed to get them. For the simple one, the only hint that seems useful is that only the left-hand side changes if you add a constant to y. --Tardis (talk) 23:51, 24 February 2011 (UTC)

Simultaneous functions

Given 3 functions of 3 variables where there are 2 constants K:

${\begin{aligned}y&=K_{R}\cdot r+(1-K_{R}-K_{B})\cdot g+K_{B}\cdot b\\p_{B}&={\frac {1}{2}}\cdot {\frac {b-y}{1-K_{B}}}\\p_{R}&={\frac {1}{2}}\cdot {\frac {r-y}{1-K_{R}}}\end{aligned}}$

what are the reverse functions?

${\begin{aligned}b&=f(y,p_{B},p_{R})\\g&=f(y,p_{B},p_{R})\\r&=f(y,p_{B},p_{R})\\\end{aligned}}$

I have tried to solve this as a system of 3 simultaneous equations but get results that seem too complex considering the equations are linear with only 2 constants involved. Help? Cuddlyable3 (talk) 18:28, 16 February 2011 (UTC)

Hard to say whether your results are really too complex or not without seeing them. It you try to compose them with the forward transform symbolically, do they yield the identity transform or not?

(By they way, YCbCr, in case anyone is wondering where the transformation comes from). –Henning Makholm (talk) 03:24, 17 February 2011 (UTC)

Yes this question comes from video color spaces. The YCbCr article quotes conversions in both directions for one numerical case which I have not found to work. Therefore I need to calculate the inverse transform symbolically from first principles. Can you do it? Cuddlyable3 (talk) 07:59, 17 February 2011 (UTC)

I got

${\begin{aligned}b&=y+2p_{B}(1-K_{B})\\g&=y+{\frac {2K_{B}p_{B}(1-K_{B})+2K_{R}p_{R}(1-K_{R})}{K_{B}+K_{R}-1}}\\r&=y+2p_{R}(1-K_{R})\end{aligned}}$

-- Meni Rosenfeld (talk) 10:20, 17 February 2011 (UTC)

Dividing a Fraction by a whole number

Hi, I need to divide 2/3 into 208.50, how would that look if I put that into a calculator, I don't need the answer just the steps how to compute that?

Thanks. —Preceding unsigned comment added by 71.137.240.110 (talk) 20:09, 16 February 2011 (UTC)

I assume that your heading should have been "dividing a number by a fraction", so you start with the number first in your calculator. You need to know that 2/3 means 2 divided by 3, and that multiplication is the inverse of division. Then you can use the simple rule that to divide by a fraction, you can just multiply by its inverse. Ask again if this information is not sufficient. I'm not giving a simple reply in case this is homework. Dbfirs 21:28, 16 February 2011 (UTC)

The keystrokes are:
2 0 8 . 5 X 3 ÷ 2 =
Cuddlyable3 (talk) 08:10, 17 February 2011 (UTC)

Or
2 0 8 . 5 ÷ 2 X 3=
→86.132.235.49 (talk) 16:02, 17 February 2011 (UTC)

... or, if you happen to have an old calculator with "automatic constant", you can press: 2 ÷ 3 ÷÷ 208.5 = (where ÷÷ is interpreted as "divided into"). This probably doesn't work on modern calculators. Dbfirs 09:42, 18 February 2011 (UTC)

Infinite groups

This is an exercise from Koblitz's book "Introduction to Elliptic Curves and Modular Forms" but it's not for a class, as we don't have a class on elliptic curves or modular forms at my school. Even if you haven't heard of a fundamental domain, I explain basically what I'm trying to figure out (which is only part of the problem) after the statement of the problem. Also, the action here should be that $g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in GL_{2}(\mathbb {R} )$ acts on $z\in \mathbb {C}$ by linear fractional transformations, $gz={\frac {az+b}{cz+d}}$.

Statement of Problem: Suppose that $\Gamma _{1}$ and $\Gamma _{2}$ are two subgroups of finite index in $SL_{2}(\mathbb {Z} )$, and $\Gamma _{1}=\alpha \Gamma _{2}\alpha ^{-1}$ for some $\alpha \in GL_{2}(\mathbb {Q} )$. If $F_{2}$ is a fundamental domain for $\Gamma _{2}$, prove that $\alpha F_{2}$ is a fundamental domain for $\Gamma _{1}$.

For starters, I need to show that for any $z$ in the upper half plane, there exists $g_{1}\in \Gamma _{1}$ such that $g_{1}z\in \alpha F_{2}$ and the statement of the problem tells us that for every $z$ in the upper half plane, there exists $g_{2}\in \Gamma _{2}$ such that $g_{2}z\in F_{2}$. Any hints here? I can't come up with anything. I know that $\alpha g_{2}z\in \alpha F_{2}$. But, $\alpha g_{2}$ is not necessarily in $\Gamma _{1}$. It's probably not even in $SL_{2}(\mathbb {Z} )$.

Thanks StatisticsMan (talk) 22:50, 16 February 2011 (UTC)

We have a Fundamental domain article.

Concretely, a hint for you is that "any z" in the conclusion and "every z" in the premise do not need to be the same element.

More generally, it sounds like you're allowing all the extraneous detail to confuse you when in fact what you need to do is purely group theoretic: You have a group G that acts on some set X, and two conjugate subgroups Γ₁ and Γ₂ related by α. Furthermore, you have a subset F of X containing exactly one representative for each orbit in X under Γ₂. You want to show that αF contains exactly one representative for each orbit in X under Γ₁.

If this is to hold for any eligible F, a natural approach is to show that the action of α preserves entire orbits -- that is, if x and y are in the same Γ₂-orbit, then αx and αy are in the same Γ₁-orbit, and vice versa. This can be done without any knowledge about what G or X is, or how the action works, or any topology. –Henning Makholm (talk) 04:00, 17 February 2011 (UTC)