Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a Duality between Metrics and $\Sigma$-Proximities   : In studies of discrete structures, functions are frequently used that
express proximity, but are not metrics. We consider a class of such functions
that is characterized by a normalization condition and an inequality that plays
the same role as the triangle inequality does for metrics. We show that the
introduced functions, named $\Sigma$-proximities, are in a definite sense dual
to metrics: there exists a natural one-to-one correspondence between metrics
and $\Sigma$-proximities defined on the same finite set; in contrast to
metrics, $\Sigma$-proximities measure {\it comparative} proximity; the closer
the objects, the greater the $\Sigma$-proximity; diagonal entries of the
$\Sigma$-proximity matrix characterize the ``centrality'' of elements. The
results are extended to arbitrary infinite sets.

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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th