Instruct: Identify the main and secondary category of Arxiv papers based on the titles
Query: Analogs of Jacobian conditions for subrings
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int