Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Composing generic linearly perturbed mappings and immersions/injections   Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$).
Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$
mapping, respectively. Generally, the composition $F\circ f$ does not
necessarily yield a mapping transverse to a given subfiber-bundle of
$J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any
$\mathcal{A}^1$-invariant fiber, we show that composing generic linearly
perturbed mappings of $F$ and the given immersion $f$ yields a mapping
transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given
fiber. Moreover, we show a specialized transversality theorem on crossings of
compositions of generic linearly perturbed mappings of a given mapping $F:U\to
\mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of
the two main theorems are given.

6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci