Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On symmetric one-dimensional diffusions   The main purpose of this paper is to explore the structure of local and
regular Dirichlet forms associated with symmetric linear diffusions. Let
$(\mathcal{E},\mathcal{F})$ be a regular and local Dirichlet form on
$L^2(I,m)$, where $I$ is an interval and $m$ is a fully supported Radon measure
on $I$. We shall first present a complete representation for
$(\mathcal{E},\mathcal{F})$, which shows that $(\mathcal{E},\mathcal{F})$ lives
on at most countable disjoint `effective' intervals with corresponding scale
function on each interval, and any point outside these intervals is a trap of
the linear diffusion. Furthermore, we shall give a necessary and sufficient
condition for $C_c^\infty(I)$ being a special standard core of
$(\mathcal{E},\mathcal{F})$ and identify the closure of $C_c^\infty(I)$ in
$(\mathcal{E},\mathcal{F})$ when $C_c^\infty(I)$ is contained but not
necessarily dense in $\mathcal{F}$ relative to the $\mathcal{E}_1$-norm. This
paper is partly motivated by a result of [Hamza, 1975], stated in [FOT, Theorem
3.1.6] and provides a different point of view to this theorem. To illustrate
our results, many examples are provided.

20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph