12,894 research outputs found
On the reconstruction of planar lattice-convex sets from the covariogram
A finite subset of is said to be lattice-convex if is
the intersection of with a convex set. The covariogram of
is the function associating to each u \in
\integer^d the cardinality of . Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets from . We provide a partial
positive answer to this problem by showing that for and under mild extra
assumptions, determines up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.Comment: accepted in Discrete and Computational Geometr
Magic numbers in the discrete tomography of cyclotomic model sets
We report recent progress in the problem of distinguishing convex subsets of
cyclotomic model sets by (discrete parallel) X-rays in prescribed
-directions. It turns out that for any of these model sets
there exists a `magic number' such that any two
convex subsets of can be distinguished by their X-rays in any set
of prescribed -directions. In particular, for
pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible
numbers are in that very order 11, 9, 11 and 13.Comment: 6 pages, 1 figure; based on the results of arXiv:1101.4149 [math.MG];
presented at Aperiodic 2012 (Cairns, Australia
Densest Lattice Packings of 3-Polytopes
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies
we present an efficient algorithm for computing the density of a densest
lattice packing of an arbitrary 3-polytope. As an application we calculate
densest lattice packings of all regular and Archimedean polytopes.Comment: 37 page
Discrete tomography: Magic numbers for -fold symmetry
We consider the problem of distinguishing convex subsets of -cyclotomic
model sets by (discrete parallel) X-rays in prescribed
-directions. In this context, a `magic number' has
the property that any two convex subsets of can be distinguished
by their X-rays in any set of prescribed
-directions. Recent calculations suggest that (with one exception
in the case ) the least possible magic number for -cyclotomic model
sets might just be , where .Comment: 5 pages, 2 figures; new computer calculations based on the results of
arXiv:1101.4149 and arXiv:1211.6318; presented at ICQ 12 (Cracow, Poland
Lower Bounds for Ground States of Condensed Matter Systems
Standard variational methods tend to obtain upper bounds on the ground state
energy of quantum many-body systems. Here we study a complementary method that
determines lower bounds on the ground state energy in a systematic fashion,
scales polynomially in the system size and gives direct access to correlation
functions. This is achieved by relaxing the positivity constraint on the
density matrix and replacing it by positivity constraints on moment matrices,
thus yielding a semi-definite programme. Further, the number of free parameters
in the optimization problem can be reduced dramatically under the assumption of
translational invariance. A novel numerical approach, principally a combination
of a projected gradient algorithm with Dykstra's algorithm, for solving the
optimization problem in a memory-efficient manner is presented and a proof of
convergence for this iterative method is given. Numerical experiments that
determine lower bounds on the ground state energies for the Ising and
Heisenberg Hamiltonians confirm that the approach can be applied to large
systems, especially under the assumption of translational invariance.Comment: 16 pages, 4 figures, replaced with published versio
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
We consider algebraic Delone sets in the Euclidean plane and
address the problem of distinguishing convex subsets of by X-rays
in prescribed -directions, i.e., directions parallel to nonzero
interpoint vectors of . Here, an X-ray in direction of a finite
set gives the number of points in the set on each line parallel to . It is
shown that for any algebraic Delone set there are four prescribed
-directions such that any two convex subsets of can be
distinguished by the corresponding X-rays. We further prove the existence of a
natural number such that any two convex subsets of
can be distinguished by their X-rays in any set of
prescribed -directions. In particular, this
extends a well-known result of Gardner and Gritzmann on the corresponding
problem for planar lattices to nonperiodic cases that are relevant in
quasicrystallography.Comment: 21 pages, 1 figur
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