Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completeness of the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in
  S^2}$   It is proved that the set $\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}$,
where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is a fixed constant,
$k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, $s\in S$, is total
in $L^2(S)$. Here $S$ is a smooth, closed, connected surface in $\mathbb{R}^3$.

25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat