Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riesz external field problems on the hypersphere and optimal point
  separation   We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in
the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field
$Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the
Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential
$\log(1/r)$. Characterization theorems of Frostman-type for the associated
extremal measure, previously obtained by the last two authors, are extended to
the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures
supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed
measure and $d-2 \leq s <d$, our results lead to explicit point-separation
estimates for $(Q,s)$-Fekete points, which are $n$-point configurations
minimizing the Riesz $s$-energy on $\mathbb{S}^d$ with external field $Q$. In
the hyper-singular case $s > d$, the short-range pair-interaction enforces
well-separation even in the presence of more general external fields. As a
further application, we determine the extremal and signed equilibria when the
external field is due to a negative point charge outside a positively charged
isolated sphere. Moreover, we provide a rigorous analysis of the three point
external field problem and numerical results for the four point problem.

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The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga