Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness and Bubbles Analysis for 1/2-harmonic Maps   In this paper we study compactness and quantization properties of sequences
of 1/2-harmonic maps $u_k\colon\R\to {\cal{S}}^{m-1}$ such that $|u_k|_{\dot
H^{1/2}(\R,{\cal{S}}^{m-1})}\le C.$ More precisely we show that there exist a
weak 1/2-harmonic map $u_\infty\colon\R\to {\cal{S}}^{m-1}$, a possible empty
set ${a_1,...,a_\ell}$ in $\R$ such that up to subsequences
$$(|(-\Delta)^{1/4}u_k|^2 \rightharpoonup
|(-\Delta)^{1/4}u_{\infty}|^2)dx+\sum_{i=1}^{\ell}\lambda_i \delta_{a_i}, in
Radon measure,$$ as $k\to +\infty$, with $\lambda_i\ge 0.$
  The convergence of $u_k$ to $u_\infty$ is strong in $\dot
W^{1/2,p}_{loc}(\R\setminus{a_1,...,a_\ell})$, for every $p\ge 1.$ We quantify
the loss of energy in the weak convergence and we show that in the case of
non-constant 1/2-harmonic maps with values in $ {\cal{S}}^2\,$ one has
$\lambda_i=2 \pi n_i$, with $n_i$ a positive integer.

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The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
1
The categories of this paper are: cs
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph