Recent content by Mute

Would the proofs be at all accessible to another reader if you didn't fill in the gaps in explanations of the logic and explaining this or that step? A paper that no one understands is not much better (in terms of communicating research) than no paper at all. If your role is making that...

Note also that you write the indefinite integral as $$\int dx~x\cos(bx) = \frac{bx\sin(bx) + \cos(bx) - 1}{b^2} + C'.$$ You can do because the factor I introduced, ##-1/b^2##, is just a constant which could be re-absorbed into C'. If you now take the limit as ##b \rightarrow 0##, you will find...

g(z) is the initial condition; integrating G(z;z') against f(z') should give the solution in this case, although... Is z the only variable in this DE? How can the initial condition u(0) = g(z) depend on z?

That really depends on you. I took a 3 hour differential equations class in the evening once, and it was fine, but that's me. Any three hour lecture can be tough just because they're so long, but that doesn't mean you won't learn anything. You also usually get a short break halfway through. Try...

Thermodynamic laws such as ##PV = nRT## are only valid in a statistical sense, when there are a large number of particles in the system (because fluctuations in the pressure, volume or temperature are small when there are a large number of particles). In the situations you describe, there are...

If ##x## is the distance from the leftmost edge of the box to the movable wall, then the volume of the gas should just be ##V = Ax##, where A is the surface area of the movable wall. Then, the formula you are given would be ##S \sim N \ln(x T^{3/2})##. If that's the equation the problem gives...

This sounds like the most reasonable interpretation to me. The others that I talked to had the same intuition that ##\dot{p}## was in some sense the more fundamental quantity, but I agree that it probably encapsulates more than just the "force" on the particle, hence why it doesn't boil down to...

The information about the wall's position is contained in the volume in the equation for the entropy. That is, the volume of the gas will be the surface area of the wall times a linear length which contains the variable x. Your attachment doesn't specify what x is measured relative to, so I...

I edited the post to also present the discussion in terms of Fourier transform solution methods, which may make the issue a bit clearer. I hope that also helps!

What's going on, basically, is an abuse of notation in which ##\frac{1}{1-\frac{\partial}{\partial t}}## is a shorthand for the inverse operator ##L^{-1}= \left(1-\frac{\partial}{\partial t}\right)^{-1}##, where ##L = 1 - \frac{\partial}{\partial t}##. Under certain conditions, you can write...

The wikipedia article on Euler-Maclaurin series specifies that the summed function, let's call it f(x), must be "a smooth (meaning: sufficiently often differentiable), analytic function of exponential type ##< 2\pi## defined for all real numbers x in the interval" which the sum covers. This...

Without more information it's hard to say for sure. It could be a coordinate system, as dipole suggests; I could imagine it might be a diagram to help illustrate a scattering problem. He could also just be writing things for a photo op.

Yes, I did my undergrad at a school in Canada that you've probably never heard of, and I was accepted to UIUC. It's no guarantee that you will get in, but it's not out of the question. It helps if you have research experience and good letters of recommendation.

The idea is that you would have a sum over energy states indexed by some discrete label, say ##i##. Your sum is then $$S = \sum_{i=0}^{N} n(E_i),$$ where ##E_i## is the energy of the ith state and ##N## is the maximum number of states (could be infinite). Now, for a large number of finely...

A suuuuuper rough heuristic is that when taking limits, ##\Delta t \rightarrow 0## and ##\Delta B_t \rightarrow 0##, is that ##(\Delta t)^2 \ll \Delta t^2##, and will thus not contribute (leading to ##dt dt = 0##), but ##\Delta B_t## is of order ##\sqrt{\Delta t}##. Hence, ##(\Delta B_t)^2 \sim...