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Rademacher-Carlitz polynomials

Matthias Beck, Florian Kohl (2014)

Acta Arithmetica

We introduce and study the Rademacher-Carlitz polynomial R ( u , v , s , t , a , b ) : = ∑ k = ⌈ s ⌉ ⌈ s ⌉ + b - 1 u ⌊ ( k a + t ) ⌋ / b v k where a , b ∈ ℤ > 0 , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum r t ( a , b ) : = ∑ k = 0 b - 1 ( ( ( k a + t ) / b ) ) ( ( k / b ) ) , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...

Remplissage de l’espace euclidien par des complexes polyédriques d’orientation imposée et de rotondité uniforme

Vincent Feuvrier (2012)

Bulletin de la Société Mathématique de France

Nous donnons une méthode de construction de complexes polyédriques dans ℝ n permettant de relier entre elles des grilles dyadiques d’orientations différentes tout en s’assurant que les polyèdres utilisés ne soient pas trop plats, y compris leurs sous-faces de toutes dimensions. Pour cela, après avoir rappelé quelques définitions et propriétés simples des polyèdres euclidiens compacts et des complexes, on se dote d’un outil qui permet de remplir de polyèdres n -dimensionnels un ouvert en forme de tube...

Riemann sums over polytopes

Victor Guillemin, Shlomo Sternberg (2007)

Annales de l’institut Fourier

It is well-known that the N -th Riemann sum of a compactly supported function on the real line converges to the Riemann integral at a much faster rate than the standard O ( 1 / N ) rate of convergence if the sum is over the lattice, Z / N . In this paper we prove an n-dimensional version of this result for Riemann sums over polytopes.

Rigidity and flexibility of virtual polytopes

G. Panina (2003)

Open Mathematics

All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.

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