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38
VECTOR PARTITION FUNCTIONS AND GENERALIZED DAHMENMICCHELLI SPACES
, 805
"... Abstract. This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in [4]. Here we introduce a ..."
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Cited by 6 (1 self)
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Abstract. This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in [4]. Here we introduce a space of functions on a lattice which generalizes the space of quasi–polynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X. This space F(X) contains the partition function PX. We prove a ”localization formula” for any f in F(X). In particular, this implies that the partition function PX is a quasi–polynomial on the sets c − B(X) where c is a big cell and B(X) is the zonotope generated by the vectors in X. 1.
On the evaluation of box splines
"... The first (and for some still the only) multivariate Bspline is what today one would call the simplex spline, since it is derived from a simplex, and in distinction to other polyhedral splines, such as the cone spline and the box spline. The simplex spline was first talked about in 1976. However, i ..."
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Cited by 201 (8 self)
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, it was only after Micchelli [Micchelli, 1980] established recurrence relations for them that the topic of simplex splines and other multivariate Bsplines really took o. Their cousins, the box splines, were thought particularly attractive because their recurrence relations turned out to be very simple indeed
Vector partition function and generalized DahmenMicchelli spaces, Preprint arXiv:0805.2907
, 2008
"... Abstract. This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in [4]. Here we introduce a space o ..."
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Cited by 8 (0 self)
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Abstract. This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in [4]. Here we introduce a space of functions on a lattice which generalizes the space of quasi–polynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X. This space F(X) contains the partition function PX. We prove a ”localization formula” for any f in F(X). In particular, this implies that the partition function PX is a quasi–polynomial on the sets c − B(X) where c is a big cell. 1.
Geometric realizations and duality for DahmenMicchelli modules and De ConciniProcesiVergne modules
, 2013
"... ar ..."
Antibodies Used in This Work
"... lgdrsk83a (this work), lgdSH495 (Oh et al., 2003), Dlrev10 SerRX82 FRT82B (Micchelli et al., ..."
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lgdrsk83a (this work), lgdSH495 (Oh et al., 2003), Dlrev10 SerRX82 FRT82B (Micchelli et al.,
Multivariate Bsplines with (almost) arbitrary knots
, 1993
"... Abstract. The multivariate B–spline scheme, due to Dahmen, Micchelli and Seidel [1] imposes some restrictions on the knots which have to be placed such that certain regions have a nonempty interior. Here it will be shown that these conditions can be almost completely dropped without losing the abil ..."
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Cited by 1 (0 self)
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Abstract. The multivariate B–spline scheme, due to Dahmen, Micchelli and Seidel [1] imposes some restrictions on the knots which have to be placed such that certain regions have a nonempty interior. Here it will be shown that these conditions can be almost completely dropped without losing
A Tutte polynomial for toric arrangements
, 2011
"... We introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletionrestriction recursion and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to ..."
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Cited by 16 (5 self)
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to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete DahmenMicchelli space, while M(x, 1) computes the volume and the number
ON MULTIVARIATE SUBDIVISION SCHEMES WITH NONNEGATIVE FINITE MASKS
"... Abstract. We study the convergence of multivariate subdivision schemes with nonnegative finite masks. Consequently, the convergence problem for the multivariate subdivision schemes with nonnegative finite masks supported on centered zonotopes is solved. Roughly speaking, the subdivision schemes defi ..."
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defined by these masks are always convergent, which gives an answer to a question raised by Cavaretta, Dahmen and Micchelli in 1991. 1.
Quadrature formulae for Fourier coefficients
, 2009
"... We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the FourierTchebycheff coefficients given b ..."
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by Micchelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives.
COMBINATORICS AND GEOMETRY OF POWER IDEALS
, 2009
"... We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We pr ..."
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Cited by 26 (2 self)
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prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to DahmenMicchelli
Results 1  10
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