Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang

DOI10.2307/2001125zbMath0718.41017OpenAlexW4243595006WikidataQ56503921 ScholiaQ56503921MaRDI QIDQ5749755

Billera, Louis J.

Publication date: 1988

Full work available at URL: https://doi.org/10.2307/2001125

zbMATH Keywords

nonsimplicial decompositions of a plane region

Mathematics Subject Classification ID

Numerical computation using splines (65D07) Exact enumeration problems, generating functions (05A15) Triangulating manifolds (57Q15) Spline approximation (41A15) Other problems of combinatorial convexity (52A37) Homology with local coefficients, equivariant cohomology (55N25)

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Cites Work

• Multivariate spline spaces
• The algebra of continuous piecewise polynomials
• Bounds on the dimension of spaces of multivariate piecewise polynomials
• The dimension of bivariate spline spaces of smoothness r for degree \(d\geq 4r+1\)
• A matroid on hypergraphs, with applications in scene analysis and geometry
• Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree \(d\geq 4r+1\)
• On Smooth Multivariate Spline Functions
• The Algebraic Geometry of Stresses in Frameworks
• Certain Reflexive Sheaves on P n c and a Problem in Approximation Theory
• An Explicit Basis for $C^1 $ Quartic Bivariate Splines
• A Nodal Basis for C 1 Piecewise Polynomials of Degree n ≥5
• Piecewise polynomials and the finite element method
• Variational methods for the solution of problems of equilibrium and vibrations
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