Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Construction of $C^2$ cubic splines on arbitrary triangulations   In this paper, we address the problem of constructing $C^2$ cubic spline
functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we
endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The
$C^2$ cubic space on such a refined triangulation has a stable dimension and
optimal approximation power. Moreover, any spline function in such space can be
locally built on each of the macro-triangles independently via Hermite
interpolation. We provide a simplex spline basis for the space of $C^2$ cubics
defined on a single macro-triangle which behaves like a Bernstein/B-spline
basis over the triangle. The basis functions inherit recurrence relations and
differentiation formulas from the simplex spline construction, they form a
nonnegative partition of unity, they admit simple conditions for $C^2$ joins
across the edges of neighboring triangles, and they enjoy a Marsden-like
identity. Also, there is a single control net to facilitate control and early
visualization of a spline function over the macro-triangle. Thanks to these
properties, the complex geometry of the Wang-Shi macro-structure is transparent
to the user. Stable global bases for the full space of $C^2$ cubics on the
Wang-Shi refined triangulation $\mathcal{T}$ are deduced from the local simplex
spline basis by extending the concept of minimal determining sets.

18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
3
The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin