Recent content by cfp

Yes, you are correct about pure imaginary exponents spanning a greater space. I'm only really interested in what's needed to span the "square integrable functions on [0,∞) with exponential tails" though, for which real parts must be negative. Your first claim might be correct, but the proof...

Hi, I have a function on [0,\infty) which is represented as: \sum_{ \stackrel{ \Re( \alpha )\in\mathbb{Q}^+ }{ \Im( \alpha )\in\mathbb{Q} } }{\beta_\alpha e^{-\alpha t}} It seems like this must be a basis for the square integrable functions on [0,\infty) with exponential tails. Am I right...

Yeah I know the differential equation doesn't have a solution satisfying the initial condition. The question I was asking in the first message is how you go about minimizing that functional given this.

The full solution is: C_1\,t+C_2\,\frac{Ei(1,-t)\,t^3+(2+t+t^2)\,\exp{t}}{t^2} where Ei(a,z)=z^{(a-1)} \,\Gamma{(1-a,z)}.

I get that the limit of the general solution as t -> 0 is signum(C2)*infinity where C2 is the constant on the messy complex part. Clearly you can't satisfy the initial conditions...

Yeah there are two constants and the second one multiplies something complex (and messy). There's no reason you can't set the second constant to zero to the best of my knowledge. In any case, whatever the values of those constants, it won't be the case that f(0)=1.

Hi, I am trying to minimize: \int_0^\infty{\exp(-t)(t\,f'(t)-f(t))^2\,dt} by choice of f, subject to f(0)=1 and f'(x)>0 for all x. The (real) solution to the Euler-Lagrange differential equation is: f(t)={C_1}t rather unsurprisingly. However, this violates f(0)=1. If...

For what I'm simulating getting analytic solutions isn't really feasible. There will be several thousand springs, many connected to each other, and all free to move in 2D space. My thought was that given interpenetrating springs, there are three ways of correcting it: 1) You do what I...

Hi, I am working on a simulation and I have a problem of similar nature to the following: Consider a horizontal frictionless pipe containing two damped springs with the same diameter as the pipe. Suppose both of the springs are moving horizontally through the pipe, one faster than the...