Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integrable systems and effectivisation of Riemann theorem about domaims
  of the complex plane   Consider a closed analytic curve $\gamma$ in the complex plane and denote by
> $D_+$ and $D_-$ the interior and exterior domains with respect to the curve.
The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem
there exists a function $w(z)=\frac 1r z+\sum_{j=0}^\infty p_j z^{-j}$, mapping
$D_-$ to the exterior of the unit disk $\{w\in C|| w | >1\}$. It is follow from
[arXiv : hep-th /0005259] that this function is described by formula $\log
w=\log z-\partial_{t_0} (\frac 12\partial_{t_0}+\sum\limits_{k\geqslant
1}\frac{z^{-k}}{k} \partial_{t_k})v$, where $v=v(t_0, t_1, \bar t_1, t_2, \bar
t_2,...)$ is a function from the area $t_0$ of $D_+$ and the momemts $t_k$ of
$D_-$. Moreover, this function satisfies the dispersionless Hirota equation for
2D Toda lattice hierarchy.
  Thus for an effectivisation of Riemann theorem it is sufficiently to find a
representation of $v$ in the form of Taylor series $v=\sum N(i_0 | i_1,...,i_k|
\bar i_1,...,\bar i_{\bar k})t_0 t_{i_1},...,t_k \bar t_{\bar i_1},...,\bar
t_{\bar i_{\bar k}}$. The numbers $N(i_0 | i_1,...,i_k | \bar i_1, ..., \bar
i_{\bar k})$ for $i_\alpha, \bar i_\beta\leqslant 2$ is found in [arXiv:
hep-th/0005259]. In this paper we find some recurrence relations that give a
possible to find all $N(i_0\bigl| i_1,...,i_k|\bar i_1,...,\bar i_{\bar k})$.

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The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th