Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bases and Structure Constants of Generalized Splines with Integer
  Coefficients on Cycles   An integer generalized spline is a set of vertex labels on an edge-labeled
graph that satisfy the condition that if two vertices are joined by an edge,
the vertex labels are congruent modulo the edge label. Foundational work on
these objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas
from geometry/topology (equivariant cohomology rings) and algebra (algebraic
splines) to develop the notion of generalized splines. Gilbert, Polster, and
Tymoczko prove that the ring of splines on a graph can be decomposed in terms
of splines on its subgraphs (in particular, on trees and cycles), and then
fully analyze splines on trees. Following Handschy-Melnick-Reinders and Rose,
we analyze splines on cycles, in our case integer generalized splines. The
primary goal of this paper is to establish two new bases for the module of
integer generalized splines on cycles: the triangulation basis and the King
basis. Unlike bases in previous work, we are able to characterize each basis
element completely in terms of the edge labels of the underlying cycle. As an
application we explicitly construct the multiplication table for the ring of
integer generalized splines in terms of the King basis.

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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat