Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A solution to the Pompeiu problem   Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the
Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad
\forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain,
the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is
its boundary. Then the comp$ is connected and path connected. By $G$ the group
of all rigid motions of $\R^n$ is denoted. This group consists of all
translations and rotations. A proof of the following theorem is given.
  Theorem 1. {\it Assume that $n=2$, $f\not\equiv 0$, and (*) holds. Then $D$
is a ball.}
  Corollary. {\it If the problem $(\nabla^2+k^2)u=0$ in $D$, $u_N|_S=0$,
$u|_S=const\neq 0$ has a solution, then $D$ is a ball.}
  Here $N$ is the outer unit normal to $S$.

11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph