6 votes

Sobolev density of smooth functions which are zero on a measure zero subset

It is not always clear, what it means for a Sobolev function to vanish on a non-open subset $A\subset \Omega$. Suppose that $f\in H^s(\Omega)$, the $L^2$-bases Sobolev space of order $s\in \mathbb R$,...
  • 693
5 votes

One question about a specific first-order differential equation

Following Euler, let us look at solutions of the form $f(z)=e^{sz}$, where $s$ is a complex number. We obtain a transcendental equation $$s=e^{cs},$$ which for every complex $c\neq 0$ has infinitely ...
2 votes

How to analytically solve this ODEs?

(I doubt this system in general is solvable) but a starting point is to consider $S_A = S_B = 0$ and $W_{AB}, W_{BA}$ are constant $$ \frac{dX_A}{dt} = -X_A(1+W_{AB}X_B^n) \\ \frac{dX_B}{dt} = -X_B(1+...
2 votes

One question about a specific first-order differential equation

There is a well-established method to obtain solutions to this equation, indeed all solutions. It is called the Fourier transform. Its application leads to an algebraic functional equation which ...
  • 29
1 vote

How to rigorously prove that this sequence of stochastic processes converges to a deterministic process?

I am guessing in "The particular thing I'm trying to prove is that,..." you are talking about the convergence of discrete generator to continuous one. The natural topology for these ...
  • 1,677

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