Recent content by RUber

Does this mean that ##I_{max} = 4\pi n I_0## all the time? If so, then that would be all the coincidence needed.

Good point, Lev. The integral that you want to compute is the one around the branch cut. I think you would agree that the larger contour with the notch cut out encloses a region where the function is analytic, and therefore would have an integral equal to zero. Instead of using Cauchy's thm to...

Give ball A some velocity and mass. Ball B and ball C have the same mass but zero velocity. Assume an elastic collision. Conserving kinetic energy gives something like: ##\left(v^2_A m_A + v^2_Bm_B+v^2_Cm_C\right)_{pre}=\left(v^2_A m_A + v^2_Bm_B+v^2_Cm_C\right)_{post} ## Since all the balls are...

From what I've seen, AI is normally a blend of CE and advanced statistics (i.e. mathematics). I think that math would be the best 2nd degree, since it would be able to complement CE well and is a good springboard into the advanced physics needed to understand quantum computing.

The way you have this written is that -1 is greater than x AND x is greater than 1. Is that even possible? Remember when you negate an AND statement, like -1<x<1 which is read -1 is less than x AND x is less than 1, you will get an OR statement.

You should try to do this. At time = 0, you get ## u(x,0)=v(x,0) + h(x) ## Your cosine transform is just in terms of x, right? And for each n, you have an ODE to solve in terms of a function of t, which should be of the form: ## f'(t) = g(t) + c ## where your functions of x are treated as...

I am not sure that there is a standard, since it likely depends on your application. This is simply the Taylor approximation of the cosine function: ## \cos x = 1 - \frac12 x^2 + \frac{1}{24}x^4 - \frac{1}{720}x^6 ...## Therefore, you can cut off the higher order terms whenever you feel they are...

I believe this is usually handled by letting your C2 by a polynomial and solving for the boundary conditions.

That makes sense.

You will probably want to be careful of your divide by zero errors when x = +/- a.

Oh, I see. I read that information as saying that ##f(0,t) = U_0, \quad f(L,t) = U_1##. In this case, you seem to have mixed boundary conditions then. ##\frac{\partial u}{\partial x} (0,t) = 0, u(L,t) = U_1 ## To use the separation of variables after your last step, you will find an appropriate...

Got it. Since you didn't make that explicit in the proof, I was not following your logic. In that case, the only thing missing are the infinite endpoints. If ##\exists m \in \mathbb{E} \, s.t. \forall x \in E, x \geq m## and likewise if ##\exists M \in \mathbb{E} \, s.t. \forall x \in E, x...

To separate the variables, you assume that your function ##F## is a product of two functions, one dependent only on x, ##X(x)## and one dependent only on t ##T(t)##. If you let ## T(0) = 1##, then your initial conditions will fully describe ##X(x)##. In the case that ## \frac{\partial...

In your write-up, you refer to ##f(a_i) ## and ##f(b_i)##. Since ##f## is only defined in ##E##, wouldn't this mean that all your points ##a_i, b_i \in E## as well? I gather that the function you defined is sort of a 'connect the dots' sort of linear interpolation between disjoint points. What...

@Aows, I have not worked much with the FFCT, but it seems like the method is much like that of other transforms. After you have rewritten the derivatives, you should separate the variables, setting F(x,t) = X(x)T(t) and use standard ODE methods to solve for T(t). What is unclear to me it that...