Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symmetrization of a Cauchy-like kernel on curves   Given a curve $\Gamma\subset \mathbb C$ with specified regularity, we
investigate boundedness and positivity for a certain three-point symmetrization
of a Cauchy-like kernel $K_{\Gamma}$ whose definition is dictated by the
geometry and complex function theory of the domains bounded by $\Gamma$. Our
results show that $\mathtt S[\text{Re} K_{\Gamma}]$ and $\mathtt S[\text{Im}
K_{\Gamma}]$ (namely, the symmetrizations of the real and imaginary parts of
$K_{\Gamma}$) behave very differently from their counterparts for the Cauchy
kernel previously studied in the literature. For instance, the quantities
$\mathtt S[\text{Re} K_{\Gamma}](\mathbf z)$ and $\mathtt S[\text{Im}
K_{\Gamma}](\mathbf z)$ can behave like $\frac32c^2(\mathbf z)$ and
$-\frac12c^2(\mathbf z)$, where $\mathbf z$ is any three-tuple of points in
$\Gamma$ and $c(\mathbf z)$ is the Menger curvature of $\mathbf z$. For the
original Cauchy kernel, an iconic result of M. Melnikov gives that the
symmetrized forms of the real and imaginary parts are each equal to
$\frac12c^2(\mathbf z)$ for all three-tuples in $\mathbb C$.

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The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an