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Register a new userSplines on the Alfeld split of a simplex and type A root systems
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Authors
Hal Schenck
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Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing the Clough-Tocher split of a triangle. A formula for the dimension of the spline space C^r_k(AS(n)) was conjectured recently by Foucart-Sorokina. We prove that the graded module of C^r-splines on the cone over AS(n) is isomorphic to the module D^{r+1}(A_n) of multiderivations on the type A_n Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.
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We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de~Rham complexes, and smoother finite element differential forms.
A family ${A_{0},ldots,A_{d}}$ of $k$-element subsets of $[n]={1,2,ldots,n}$ is called a simplex-cluster if $A_{0}capcdotscap A_{d}=varnothing$, $|A_{0}cupcdotscup A_{d}|le2k$, and the intersection of any $d$ of the sets in ${A_{0},ldots,A_{d}}$ is nonempty. In 2006, Keevash and Mubayi conjectured that for any $d+1le klefrac{d}{d+1}n$, the largest family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster is the family of all $k$-subsets that contain a given element. We prove the conjecture for all $kgezeta n$ for an arbitrarily small $zeta>0$, provided that $nge n_{0}(zeta,d)$. We call a family ${A_{0},ldots,A_{d}}$ of $k$-element subsets of $[n]$ a $(d,k,s)$-cluster if $A_{0}capcdotscap A_{d}=varnothing$ and $|A_{0}cupcdotscup A_{d}|le s$. We also show that for any $zeta nle klefrac{d}{d+1}n$ the largest family of $k$-element subsets of $[n]$ that does not contain a $(d,k,(frac{d+1}{d}+zeta)k)$-cluster is again the family of all $k$-subsets that contain a given element, provided that $nge n_{0}(zeta,d)$. Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.
In this paper we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.
This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the ${ mu }$-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Tits {e}chelonnage root system $Sigma_0$, the Knop root system $widetilde{Sigma}_0$, and the Macdonald root system $Sigma_1$, in terms of Galois actions on the absolute roots $Phi$; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis.
This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.
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