Recent content by wil3

These were both great answers, I agree that using this trick is sketchy in a proof but it seems to work for my application. Thank you!

I am trying to solve the integral \int_{-\infty}^\infty H(x) \delta(x) dx Where H(x) is a unit step and d(x) is a standard Dirac delta. Mathematica chokes on this, but I'm pretty sure that the value is \int_{-\infty}^\infty H(x) \delta(x) dx = \dfrac12 \left(H(0^+) + H(0^-) \right) =...

Typo in question: \omega should not depend on x. I want to solve for \omega , hence why I am calling this an eigenvalue problem. I'm not sure if getting w will also give a and b, like in a standard linear system. \partial_x indicates a partial derivative, which is standard notation in...

Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem: \begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu +...

We mean something like "The concavity of f at x_0 is the value of the second derivative of f at x_0."

no worries, I'm still alive. Three years of physics later, I feel much better about this distinction, but thanks for clarifying.

OH! So I correctly quoted the paper, but the paper itself was incorrect. The author definitely meant tan()-- he switched between two equations. The previous equations had cosh(.) and sinh(.), so I didn't catch the error: Check out 7b in this paper if you're curious where this is from...

Hello! So I was reading a paper in which I came across the following: k = \pi + \pi\ell \tanh{(k)} \approx \pi\ell where "l" is very small. What on Earth is the origin of this approximation? I'm sure it's very simple, but I can't seem to derive it from the angle-sum and small angle...

Can someone concisely clarify the distinction between an ergodic process and a stationary one? Specifically, can anyone provide examples of processes that are ergodic but not stationary or vice-versa? You don't need to provide the definitions; I know what the words mean. But it seems to me...

Nah, I remember this came up in class on a problem of this exact sort. I seem to remember there being a \Gamma*\gamma term going on somewhere. But thanks for the analysis-- I'm sure there is a parallel with the Doppler Effect.

Hello. Let's say that I am in a frame in which I see a rocket traveling at v. This rocket then fires a projectile forward with velocity u in its rest frame. I can find u' easily enough using the Einstein velocity addition formula. However, I recall seeing a version of the formula that uses...

ah, I see. Thanks very much!

That actually very much does clear things up... I had never learned about the equivalency of, for example, spaces of 2x2 tensors and spaces of 1x4 tensors. Where could I find an example of a relationship that establishes the isomorphism? (ie, is there any sort of tensor A would make your...

That idea sounds promising, but I'm afraid I don't quite understand what you mean yet. Would you mind elucidating things a little? I'm new to tensors, and so there could be a critical spatial relationship that I am missing out on here.

Hello. I keep on encountering the need to find the Tensor or Kronecker product of two vectors. Based on the definition, If I found the product of two 2D vectors, I would get a 4-dimensional vector. Some authors claim this is the correct interpretation. However the dyadic product, which many...