Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The principal eigenfunction of the Dirichlet Laplacian with prescribed
  numbers of critical points on the upper half of a topological torus   We consider the principal eigenvalue problem for the Laplace-Beltrami
operator on the upper half of a topological torus under the Dirichlet boundary
condition. We present a construction of the upper half of a topological torus
that admits the principal eigenfunction having exact numbers of critical
points. Furthermore, we manage to identify the locations of all the critical
points of the principal eigenfunction explicitly.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph