Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp bounds for eigenvalues of triangles   We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle
in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the
perimeter and $A$ is the area of this triangle. We show that the \mbox{constant
9} is optimal and that the optimal constant for the lower bound of the same
form is 16. This gives a positive answer to a conjecture made by P. Freitas.

16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol