Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-Intersecting Periodic Curves in the Plane   Suppose a smooth planar curve $\gamma$ is $2\pi$-periodic in the $x$
direction and the length of one period is $\ell$. It is shown that if $\gamma$
self-intersects, then it has a segment of length $\ell- 2\pi$ on which it
self-intersects and somewhere its curvature is at least $2\pi/(\ell - 2\pi)$.
The proof involves the projection $\Gamma$ of $\gamma$ onto a cylinder. (The
complex relation between $\gamma$ and $\Gamma$ was recently observed
analytically by T. M. Apostol and M. A. Mnatsakanian. When $\gamma$ is in
general position there is a bijection between self-intersection points of
$\gamma$ modulo the periodicity, and self-intersection points of $\Gamma$ with
winding number 0 around the cylinder. However, our proof depends on the
observation that a loop in $\Gamma$ with winding number 1 leads to a
self-intersection point of $\gamma$.

17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th