Positivity theorems for solid-angle polynomials
By Matthias Beck, Sinai Robins, Steven V. Sam. 2009
23 citations recorded by the source corpus.
For a lattice polytope P, define A_P(t) as the sum of the solid angles of all the integer points in the dilate tP. Ehrhart and Macdonald proved that A_P(t) is a polynomial in the positive integer variable t. We study the numerator polynomial of the solid-angle series sum_{t >= 0} A_P(t) z^t. In particular, we examine nonnegativity of its coefficients, monotonicity and unimodality questions, and study extremal behavior of the sum of solid angles at vertices of simplices. Some of our results extend to more general valuations.
Subfields: Computational Mechanics, Numerical Analysis
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In Computational Mechanics
- Min Lin
- Yoshua Bengio
- Rahaf Aljundi
- Baptiste Goujaud
- Dougal Maclaurin
- Hugo Larochelle
- Jitendra Malik
- Fabian Pedregosa
In Numerical Analysis
- Yoshua Bengio
- Rahaf Aljundi
- Dougal Maclaurin
- Fabian Pedregosa
- Jitendra Malik
- Gang Luo
- Alain Rakotomamonjy
- Ke Li
Citations in the corpus
- Gradient based sample selection for online continual learning (2019, background): “However, this objective is hard to minimize since the formula of the solid angle is complex, as shown in [11] and [2].”
- Online continual learning with no task boundaries (2019, background): “However, this objective is hard to minimize since the formula of the solid angle is complex, as shown in (Ribando, 2006) and (Beck et al., 2009).”
- Continual Learning in Neural Networks (2019, background): “However, this objective is hard to minimize since the formula of the solid angle is complex, as shown in [123] and [15].”