@0celo7 You should attribute that. I know that many, many people will know the source but this is a physics themed place rather than eitehr CS or numeric methods.
if I have an aerodynamic force model, solved with ode45 in Matlab, and I notice that one of my state variables settles into some final value, how could I explore the stability of this state variable? Do I have to linearize the lift and drag models around such solutions, or is there another process to analyze stability of just one particular state variable?
hmm ... @dmckee, how do I manually perturb it? change the initial value of it to some nonzero value? (sorry, I don't have experience with numerical methods / analysis. This may be beyond me, but I feel it's worth a try anyway.)
that is, do you mean to change the initial conditions a bit? @dmckee thanks,
@D.Hutchinson Change the final value of the parameter you are interested in and re-start the evolver. But if you got there with a minimizer then it is likely to be stable.
Ah, I see, @dmckee. How do I change the final value, though? For instance, I code up the equations, I call ode45, I see all of my vectors of solutions. I see one vector of solutions gives values such as 5.44, 5.38, 5.24, 5.23, 5.2222, 5.211, 5.211, 5.211, 5.211, 5.211, 5.211, 5.211, 5.211...5.211... How can I perturb the value of 5.211?
also, @dmckee, the other state variables are changing in time, it's just this one particular state variable that seems to have settled into a final value, say, 5.211 thanks,
I have no idea at all how your code works. You need to (a) stop the evolver, (b) tweak the value of the parameter you care about and (c) restart the evolver. The details of how that is done depend on how your model is coded.
How to show that the area $A$ of the region
$$\{(x,y)\in\mathbb R^2 : |x|\le y\le e^{-x^2}\}$$
is such that
$$0 < A = 2\int_{0}^{\beta}{(e^{-t^2}-t)dt} < 1$$
in which $\beta>0$ satisfies $e^{-\beta^2}=\beta$
@dmckee, regarding manually perturbing the final value of the parameter to check stability: if I write some even function that stops the solver when an equilibrium solution is reached, then I change the initial condition of that state variable, and then run the model again, do you think that that is the same as perturbing the final value?
@D.Hutchinson That really tests a different thing. That is, does the problem have multiple local optima that can be reached from a small area around your initial conditions.
But if you are arriving at a condition where that one parameter is fixed and the system is not stable then it probably won't come to the same fixed point from different initial conditions.
@dmckee, and so, my method would work anyway, you're saying? In the meantime, I changed the initial conditions of the variable a few times just now and the solutions for this state variable still settle into the same final value. I'm trying to write code that would automatically perturb the state variable, at some time, t > a, where I know when the solution has settled to a final value. But I don't quite have it correct:
ode45(@(t,y) MyFun(t,y,var1,var2,@theta, ...) Then in MyFun, var3 = theta(t) and also,
function var3 = theta(t) if t < 10 var3 = 0; else var3 = 1; end end
@dmckee, the problem with that code (gotten from an old mathworks Q&A discussion) is that t < 10 seems to be only the total time, t, that I run the model for ...
so if I run the model for time scale [0 30], the initial condition on theta doesn't switch from 0 to 1, like I want it to do ...
it'll just use initial condition var3 = 1, since the total time, t was greater than 10 ... @dmckee
I don't see how $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ means $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ where h is a small perturbation and second order terms are dropped
In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways.
If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then
(
A
+
U
B
V
)
−
1
=
A
−
1
−
A...
how nice it is to have public bathroom everywhere.
taking a shower is important for researchers; that's why the only places which I find to have public bathrooms are universities and research institutes.
So... I have been doing some human ethics training module because the uni system failed to realise I have already done them last semester and think I am not doing them. Interestingly, I got the exact same batches of questions wrong. I must be very biased on the way I think
If you want to hide the fact that another moderator also thought your branding of your rooms was deceptive and therefore changed your room names and topics, I have not the slightest inclination to aid you in doing so.
How to show that the area $A$ of the region
$$\{(x,y)\in\mathbb R^2 : |x|\le y\le e^{-x^2}\}$$
is such that
$$0 < A = 2\int_{0}^{\beta}{(e^{-t^2}-t)dt} < 1$$
in which $\beta>0$ satisfies $e^{-\beta^2}=\beta$
