Wikipedia:Reference desk/Archives/Mathematics/2013 September 20

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September 20

Linear Programming

Maximize Z=4x+9y, subject to teh constraints x>=0,y>=0 s+5y<=200 2x+3y<=134 — Preceding unsigned comment added by Bikramckc (talk • contribs) 04:07, 20 September 2013 (UTC)

Hello. The reference desk will not do your homework for you, although we can help you if you're stuck - only if you have made an attempt to solve the problem.--Jasper Deng (talk) 04:12, 20 September 2013 (UTC)

I assume that you meant x+5y<=200. You don't say whether you are looking for integer values or any real number as the solution. One way to get a solution is by drawing graphs of the constraints to identify the area of the graph that satisfies all the constraints, then draw a line such as 4x + 9y = 36 (just taking Z=36 as a convenient value) and look at lines parallel to this that just cross the constraint area. You may have been taught other methods, so you should use whatever method you have been taught. Dbfirs 14:55, 20 September 2013 (UTC)

Modelling Dissipative Process

I want to model the dissipative process $d\theta =-a\sin \theta dt+\sigma dW(t)$ where $dW(t)=N(t)dt$ is a normally distributed noise process. I would like to model the evolution of $\cos \theta$, using the runge-kutta method.

Given the stochastic nature of the process is some sort of Ito's lemma style correction neccesarry when using runge-kutta models?

In other words, for the purposes of phrasing the problem as ${\frac {dy}{dt}}=f(y,t)$ where $y=\cos \theta$, so that RK methods can be used, is it correct to simply say

${\frac {dy}{dt}}={\frac {dy}{d\theta }}{\frac {d\theta }{dt}}=a\sin ^{2}\theta -\sigma \sin \theta N(t)=a(1-y^{2})-\sigma {\sqrt {1-y^{2}}}N(t)$

where, when modelling, the value of $N(t)$ is drawn from a unit normal distribution at each time step.

Or is it instead correct to say, following Ito, that this is drift diffusion process with $\mu _{t}=-a\sin \theta$ and ${\sigma }_{t}=\sigma$

${\frac {dy}{dt}}=-a\sin \theta {\frac {dy}{d\theta }}+{\frac {\sigma ^{2}}{2}}{\frac {d^{2}y}{d\theta ^{2}}}+\sigma {\frac {dy}{d\theta }}N(s)=a\sin ^{2}\theta -{\frac {\sigma ^{2}}{2}}\cos \theta -\sigma \sin \theta N(t)=a(1-y^{2})+{\frac {\sigma ^{2}}{2}}y-\sigma {\sqrt {1-y^{2}}}N(t)$

I feel like it should be the latter, as the former clearly has a fixed point at y=1, whereas the starting equation does not have a corresponding one at θ=0, however I am not convinced I full understand what are the circumstances under which Ito's correction is valid. — Preceding unsigned comment added by 144.82.172.150 (talk) 17:33, 20 September 2013 (UTC)

The Ito correction should be included, unless the time steps are so small that they fall well within the correlation time of the noise. But if N(t) is purely white noise (it has arbitrary high frequency components), then N(t) and N(t+delta t) are totally uncorrelated for arbitrary small delta t. Count Iblis (talk) 18:04, 20 September 2013 (UTC)

The article Runge–Kutta method (SDE) has a Markov chain approximation method that looks suitable to your problem. --Mark viking (talk) 18:05, 20 September 2013 (UTC)

Thanks guys. For the record (if anyone ever looks back over this) I found this paper useful [1]. — Preceding unsigned comment added by 109.144.154.56 (talk) 10:33, 22 September 2013 (UTC)