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Search Results: 1 - 10 of 3993 matches for " Achill Schürmann "
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Strict Periodic Extreme Lattices
Achill Schürmann
Mathematics , 2012,
Abstract: A lattice is called periodic extreme if it cannot locally be modified to yield a better periodic sphere packing. It is called strict periodic extreme if its sphere packing density is an isolated local optimum among periodic point sets. In this note we show that a lattice is periodic extreme if and only if it is extreme, that is, locally optimal among lattices. Moreover, we show that a lattice is strict periodic extreme if and only if it is extreme and non-floating.
Perfect, strongly eutactic lattices are periodic extreme
Achill Schürmann
Mathematics , 2008, DOI: 10.1016/j.aim.2010.05.002
Abstract: 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 $d=24$.
Exploiting Symmetries in Polyhedral Computations
Achill Schürmann
Mathematics , 2014,
Abstract: In this note we give a short overview on symmetry exploiting techniques in three different branches of polyhedral computations: The representation conversion problem, integer linear programming and lattice point counting. We describe some of the future challenges and sketch some directions of potential developments.
Exploiting Polyhedral Symmetries in Social Choice
Achill Schürmann
Mathematics , 2011, DOI: 10.1007/s00355-012-0667-1
Abstract: 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.
Energy minimization, periodic sets and spherical designs
Renaud Coulangeon,Achill Schürmann
Mathematics , 2010, DOI: 10.1093/imrn/rnr048
Abstract: 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.
Bounds on generalized Frobenius numbers
Lenny Fukshansky,Achill Schürmann
Mathematics , 2010, DOI: 10.1016/j.ejc.2010.11.001
Abstract: 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 $s$.
Bases of minimal vectors in lattices, III
Jacques Martinet,Achill Schürmann
Mathematics , 2011, DOI: 10.1142/S1793042112500303
Abstract: 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 $n\ge 10$.
On classifying Minkowskian sublattices
Wolfgang Keller,Jacques Martinet,Achill Schürmann
Mathematics , 2009, DOI: 10.1090/S0025-5718-2011-02528-7
Abstract: Let $\Lambda$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lambda'$ be a Minkowskian sublattice of $\Lambda$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lambda$. We extend the classification of possible $\Z/d\Z$-codes of the quotients $\Lambda/\Lambda'$ to dimension~$9$, where $d\Z$ is the annihilator of $\Lambda/\Lambda'$.
Exploiting Symmetry in Integer Convex Optimization using Core Points
Katrin Herr,Thomas Rehn,Achill Schürmann
Mathematics , 2012, DOI: 10.1016/j.orl.2013.02.007
Abstract: We consider convex programming problems with integrality constraints that are invariant under a linear symmetry group. To decompose such problems we introduce the new concept of core points, i.e., integral points whose orbit polytopes are lattice-free. For symmetric integer linear programs we describe two algorithms based on this decomposition. Using a characterization of core points for direct products of symmetric groups, we show that prototype implementations can compete with state-of-the-art commercial solvers, and solve an open MIPLIB problem.
On Lattice-Free Orbit Polytopes
Katrin Herr,Thomas Rehn,Achill Schürmann
Mathematics , 2014, DOI: 10.1007/s00454-014-9638-x
Abstract: Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all $2$-homogeneous groups up to degree twelve. For transitive groups that are not $2$-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.
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