Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive definite collections of disks   Let $Q(z,w)=-\prod_{k=1}^n [(z-a_k)(\bar{w}-\bar{a}_k)-R_k^2]$. M. Putinar
and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\leq i,j\leq
n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We
extend this result on symmetric collections of discs with overlapping. More
precisely, we show that in the case when the nodes $a_j$ are situated at the
vertices of a regular $n$-gon inscribed in the unit circle and $\forall i:
R_i\equiv R$, the matrix $Q(a_i,a_j)$ is positive definite if and only if
$R<\rho_n$, where $z=2\rho_n^2-1$ is the smallest $\ne-1$ zero of the Jacobi
polynomial $\mathcal{P}^{n-2\nu,-1}_\nu(z)$, $\nu=[n/2]$.

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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: stat