22,266 research outputs found
On the lattice programming gap of the group problems
We show that computing the lattice programming gap of the group problems is
NP-hard when the dimension is a part of input. We also obtain lower and upper
bounds for the gap in terms of the cost vector and the determinant of the
lattice
Computing symmetry groups of polyhedra
Knowing the symmetries of a polyhedron can be very useful for the analysis of
its structure as well as for practical polyhedral computations. In this note,
we study symmetry groups preserving the linear, projective and combinatorial
structure of a polyhedron. In each case we give algorithmic methods to compute
the corresponding group and discuss some practical experiences. For practical
purposes the linear symmetry group is the most important, as its computation
can be directly translated into a graph automorphism problem. We indicate how
to compute integral subgroups of the linear symmetry group that are used for
instance in integer linear programming.Comment: 20 pages, 1 figure; containing a corrected and improved revisio
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
Constraints on Flavored 2d CFT Partition Functions
We study the implications of modular invariance on 2d CFT partition functions
with abelian or non-abelian currents when chemical potentials for the charges
are turned on, i.e. when the partition functions are "flavored". We begin with
a new proof of the transformation law for the modular transformation of such
partition functions. Then we proceed to apply modular bootstrap techniques to
constrain the spectrum of charged states in the theory. We improve previous
upper bounds on the state with the greatest "mass-to-charge" ratio in such
theories, as well as upper bounds on the weight of the lightest charged state
and the charge of the weakest charged state in the theory. We apply the
extremal functional method to theories that saturate such bounds, and in
several cases we find the resulting prediction for the occupation numbers are
precisely integers. Because such theories sometimes do not saturate a bound on
the full space of states but do saturate a bound in the neutral sector of
states, we find that adding flavor allows the extremal functional method to
solve for some partition functions that would not be accessible to it
otherwise.Comment: 45 pages, 16 Figures v3: typos corrected, expanded appendix on
numeric implementatio
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