98 research outputs found
On Chaotic Dynamics in Rational Polygonal Billiards
We discuss the interplay between the piece-line regular and vertex-angle
singular boundary effects, related to integrability and chaotic features in
rational polygonal billiards. The approach to controversial issue of regular
and irregular motion in polygons is taken within the alternative deterministic
and stochastic frameworks. The analysis is developed in terms of the
billiard-wall collision distribution and the particle survival probability,
simulated in closed and weakly open polygons, respectively. In the multi-vertex
polygons, the late-time wall-collision events result in the circular-like
regular periodic trajectories (sliding orbits), which, in the open billiard
case are likely transformed into the surviving collective excitations
(vortices). Having no topological analogy with the regular orbits in the
geometrically corresponding circular billiard, sliding orbits and vortices are
well distinguished in the weakly open polygons via the universal and
non-universal relaxation dynamics.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
On almost hypohamiltonian graphs
A graph is almost hypohamiltonian (a.h.) if is non-hamiltonian, there
exists a vertex in such that is non-hamiltonian, and is
hamiltonian for every vertex in . The second author asked in [J.
Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here
we solve this problem. To this end, we present a specialised algorithm which
generates complete sets of a.h. graphs for various orders. Furthermore, we show
that the smallest cubic a.h. graphs have order 26. We provide a lower bound for
the order of the smallest planar a.h. graph and improve the upper bound for the
order of the smallest planar a.h. graph containing a cubic vertex. We also
determine the smallest planar a.h. graphs of girth 5, both in the general and
cubic case. Finally, we extend a result of Steffen on snarks and improve two
bounds on longest paths and longest cycles in polyhedral graphs due to
Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
String patterns in the doped Hubbard model
Understanding strongly correlated quantum many-body states is one of the most
difficult challenges in modern physics. For example, there remain fundamental
open questions on the phase diagram of the Hubbard model, which describes
strongly correlated electrons in solids. In this work we realize the Hubbard
Hamiltonian and search for specific patterns within the individual images of
many realizations of strongly correlated ultracold fermions in an optical
lattice. Upon doping a cold-atom antiferromagnet we find consistency with
geometric strings, entities that may explain the relationship between hole
motion and spin order, in both pattern-based and conventional observables. Our
results demonstrate the potential for pattern recognition to provide key
insights into cold-atom quantum many-body systems.Comment: 8+28 pages, 5+10 figure
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