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Rademacher-Carlitz polynomials

Matthias Beck, Florian Kohl (2014)

Acta Arithmetica

We introduce and study the Rademacher-Carlitz polynomial $R (u, v, s, t, a, b) : = \sum_{k = ⌈ s ⌉}^{⌈ s ⌉ + b - 1} u^{⌊ (k a + t) ⌋} / b_{v^{k}}$ where $a, b \in ℤ_{> 0}$ , 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 $r_{t} (a, b) : = \sum_{k = 0}^{b - 1} (((k a + t) / b)) ((k / b))$ , 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...

Radonpartitionen und konvexe Polyeder.

Jürgen Eckhoff (1975)

Journal für die reine und angewandte Mathematik

Random polytopes and the volume-product of symmetric convex bodies.

Shlomo Reisner (1985)

Mathematica Scandinavica

Random Projections of Regular Simplices.

R. Schneider, F. Affentranger (1992)

Discrete & computational geometry

Rapport sur la géométrie $S L_{2} 𝐙$ par morceaux

Peter Greenberg (1991/1992)

Séminaire de théorie spectrale et géométrie

Realizations of regular polytopes.

Peter McMullen (1989)

Aequationes mathematicae

Reciprocal domains and Cohen-Macaulay $d$ -complexes in $ℝ^{d}$ .

Miller, Ezra, Reiner, Victor (2005)

The Electronic Journal of Combinatorics [electronic only]

Recognizing graph theoretic properties with polynomial ideals.

De Loera, Jesús A., Hillar, Christopher J., Malkin, Peter N., Omar, Mohamed (2010)

The Electronic Journal of Combinatorics [electronic only]

Recognizing the topology of the space of closed convex subsets of a Banach space

Taras Banakh, Ivan Hetman, Katsuro Sakai (2013)

Studia Mathematica

Redundance of vertices of the cube relatively to its minimal ellipsoid.

Sarges, Olaf (1996)

Beiträge zur Algebra und Geometrie

Regular triangles and isoclinic triangles in the Grassmann $G_{2} (ℝ^{N})$ .

Masala, G. (1999)

Rendiconti del Seminario Matematico

Rektifizierbare vierdimensionale Simplices.

Johann Berger (1978)

Elemente der Mathematik

Relations among rhombic, Platonic and Archimedean solids

Izidor Hafner, Tomislav Zitko (2002)

Visual Mathematics

Remarks on missing faces and generalized lower bounds on face numbers.

Nevo, Eran (2009)

The Electronic Journal of Combinatorics [electronic only]

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...

Résumé de recherches sur la symétrie des polyèdres non eulériens.

Camille Jordan (1866)

Journal für die reine und angewandte Mathematik

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.

Rigidity and the Alexandrov-Fenchel Inequality.

P. Filliman (1992)

Monatshefte für Mathematik

Rigidity and the lower bound Theorem l.

G. Kalai (1987)

Inventiones mathematicae

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