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I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation $$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$ I ended up with ...

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...

If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...