Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Nonlocal Transport Equation Describing Roots of Polynomials Under
  Differentiation   Let $p_n$ be a polynomial of degree $n$ having all its roots on the real line
distributed according to a smooth function $u(0,x)$. One could wonder how the
distribution of roots behaves under iterated differentation of the function,
i.e. how the density of roots of $p_n^{(k)}$ evolves. We derive a nonlinear
transport equation with nonlocal flux $$ u_t + \frac{1}{\pi}\left( \arctan{
\left( \frac{Hu}{ u}\right)} \right)_x = 0,$$ where $H$ is the Hilbert
transform. This equation has three very different compactly supported
solutions: (1) the arcsine distribution $u(t,x) = (1-x^2)^{-1/2}
\chi_{(-1,1)}$, (2) the family of semicircle distributions $$ u(t,x) =
\frac{2}{\pi} \sqrt{(T-t) - x^2}$$ and (3) a family of solutions contained in
the Marchenko-Pastur law.

3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg