Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Polya's inequality for torsional rigidity and first Dirichlet
  eigenvalue   Let $\Omega$ be an open set in Euclidean space with finite Lebesgue measure
$|\Omega|$. We obtain some properties of the set function $F:\Omega\mapsto
\R^+$ defined by $$ F(\Omega)=\frac{T(\Omega)\lambda_1(\Omega)}{|\Omega|} ,$$
where $T(\Omega)$ and $\lambda_1(\Omega)$ are the torsional rigidity and the
first eigenvalue of the Dirichlet Laplacian respectively. We improve the
classical P\'olya bound $F(\Omega)\le 1,$ and show that $$F(\Omega)\le 1- \nu_m
T(\Omega)|\Omega|^{-1-\frac2m},$$ where $\nu_m$ depends only on $m$. For any
$m=2,3,\dots$ and $\epsilon\in (0,1)$ we construct an open set
$\Omega_{\epsilon}\subset \R^m$ such that $F(\Omega_{\epsilon})\ge 1-\epsilon$.

27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin