Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inverse problems for heat equation and space-time fractional diffusion
  equation with one measurement   Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim
M\ge 2$, we consider a space--time fractional diffusion equation with an
interior source that is supported on an open subset $V$ of the manifold. The
time-fractional part of the equation is given by the Caputo derivative of order
$\alpha\in(0,1]$, and the space fractional part by $(-\Delta_g)^\beta$, where
$\beta\in(0,1]$ and $\Delta_g$ is the Laplace--Beltrami operator on the
manifold. The case $\alpha=\beta=1$, which corresponds to the standard heat
equation on the manifold, is an important special case. We construct a specific
source such that measuring the evolution of the corresponding solution on $V$
determines the manifold up to a Riemannian isometry.

8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph