Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integrable solutions of inhomogeneous refinement type equations on
  intervals   Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set
$\Omega$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function
$\varphi\colon I\times\Omega\to I$ fulfilling some conditions we obtain results
on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement
type equation $$
f(x)=\int_{\Omega}\big|\varphi'_x(x,\omega)\big|f(\varphi(x,\omega))dP(\omega)+g(x).
$$

30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org