arXiv subjects

Achill Schürmann

Publications and source records attributed to Achill Schürmann.

24 records · Page 2Linked to original sources

arXiv2012

Exploiting Polyhedral Symmetries in Social Choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.

math.OC↗

arXiv2011

Bases of minimal vectors in lattices, III

We prove that all Euclidean lattices of dimension $n\le 9$ which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions $n\ge 10$.

math.NT↗

arXiv2011

Bounds on generalized Frobenius numbers

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$.

math.NT↗

arXiv2011

On classifying Minkowskian sublattices

Let $Λ$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $Λ'$ be a Minkowskian sublattice of $Λ$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $Λ$. We extend the classification of possible $\Z/d\Z$-codes of the quotients $Λ/Λ'$ to dimension~$9$, where $d\Z$ is the annihilator of $Λ/Λ'$.

math.NT↗

arXiv2011

Energy minimization, periodic sets and spherical designs

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.

math.MG↗

arXiv2010

Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.

math.MG↗