Canadian Mathematical Society
Canadian Mathematical Society
  location:  Publicationsjournals
Search results

Search: MSC category 11H06 ( Lattices and convex bodies [See also 11P21, 52C05, 52C07] )

  Expand all        Collapse all Results 1 - 4 of 4

1. CJM 2016 (vol 69 pp. 1143)

Sikirić, Mathieu Dutour
The seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called empty if for any $v\in L$ we have $\Vert v - c\Vert \geq r$. Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay polytope} $P=\operatorname{conv}(S(c,r)\cap L)$. A Delaunay polytope is called {\em perfect} if any affine transformation $\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $n\geq 6$ and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension $7$ which allow us to find that there are only two perfect Delaunay polytopes: $3_{21}$ which is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension $7$ and found $11$ types.

Keywords:Delaunay polytope, enumeration, polyhedral methods
Categories:11H06, 11H31

2. CJM 2011 (vol 63 pp. 1220)

Baake, Michael; Scharlau, Rudolf; Zeiner, Peter
Similar Sublattices of Planar Lattices
The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Categories:11H06, 11R11, 52C05, 82D25

3. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
Hecke $L$-Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47

4. CJM 2002 (vol 54 pp. 449)

Akrout, H.
Théorème de Vorono\"\i\ dans les espaces symétriques
On d\'emontre un th\'eor\`eme de Vorono\"\i\ (caract\'erisation des maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non exceptionnels de type non compact. We prove a theorem of Vorono\"\i\ type (characterisation of local maxima of the Hermite invariant) for the lattices parametrized by irreducible nonexceptional symmetric spaces of noncompact type.

Keywords:réseaux, théorème de Vorono\"\i, espaces symétriques
Categories:11H06, 53C35

© Canadian Mathematical Society, 2017 :