Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-zero sum Heffter arrays and their applications   In this paper we introduce a new class of partially filled arrays that, as
Heffter arrays, are related to difference families, graph decompositions and
biembeddings. A non-zero sum Heffter array $\mathrm{N}\mathrm{H}(m,n; h,k)$ is
an $m \times n$ p. f. array with entries in $\mathbb{Z}_{2nk+1}$ such that:
each row contains $h$ filled cells and each column contains $k$ filled cells;
for every $x\in \mathbb{Z}_{2nk+1}\setminus\{0\}$, either $x$ or $-x$ appears
in the array; the sum of the elements in every row and column is different from
$0$ (in $\mathbb{Z}_{2nk+1}$). Here first we explain the connections with
relative difference families and with path decompositions of the complete
multipartite graph. Then we present a complete solution for the existence
problem and a constructive complete solution for the square case and for the
rectangular case with no empty cells when the additional, very restrictive,
property of "globally simple" is required. Finally, we show how these arrays
can be used to construct biembeddings of complete graphs.

1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th