34,562 research outputs found
Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions
We present an algorithm for enumerating exactly the number of Hamiltonian
chains on regular lattices in low dimensions. By definition, these are sets of
k disjoint paths whose union visits each lattice vertex exactly once. The
well-known Hamiltonian circuits and walks appear as the special cases k=0 and
k=1 respectively. In two dimensions, we enumerate chains on L x L square
lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results
for three dimensions are also given. Using our data we extract several
quantities of physical interest
Path Integral Methods in the Su-Schrieffer-Heeger Polaron Problem
I propose a path integral description of the Su-Schrieffer-Heeger
Hamiltonian, both in one and two dimensions, after mapping the real space model
onto the time scale. While the lattice degrees of freedom are classical
functions of time and are integrated out exactly, the electron particle paths
are treated quantum mechanically. The method accounts for the variable range of
the electronic hopping processes. The free energy of the system and its
temperature derivatives are computed by summing at any over the ensemble of
relevant particle paths which mainly contribute to the total partition
function. In the low regime, the {\it heat capacity over T} ratio shows un
upturn peculiar to a glass-like behavior. This feature is more sizeable in the
square lattice than in the linear chain as the overall hopping potential
contribution to the total action is larger in higher dimensionality. The
effects of the electron-phonon anharmonic interactions on the phonon subsystem
are studied by the path integral cumulant expansion method.Comment: to appear in "Polarons in Advanced Materials" ed. A.S. Alexandrov
(Canopus Books, 2007
Analysing Flow Free with Pairs of Dots In Triangular Graphs
In the puzzle game Flow Free, the player is given a n x n grid with a number of colored point pairings. In order to solve the puzzle, the player must draw a path connecting each pair of points so that the following conditions are met: each pair of dots is connected by a path, each square of the grid is crossed by a path, and no paths intersect. Based on these puzzles, this project examines pairs of points in triangular grid graphs obtained by hexagons for which Hamiltonian paths exist in order to identify which point configurations have solutions. We show that n ≥ 5, any pairs of endpoints admit a Hamiltonian path as they do not surround a corner. This is a solution when n=2 fails when n=3 or 4
Optimizing Quantum Adiabatic Algorithm
In quantum adiabatic algorithm, as the adiabatic parameter changes
slowly from zero to one with finite rate, a transition to excited states
inevitably occurs and this induces an intrinsic computational error. We show
that this computational error depends not only on the total computation time
but also on the time derivatives of the adiabatic parameter at the
beginning and the end of evolution. Previous work (Phys. Rev. A \textbf{82},
052305) also suggested this result. With six typical paths, we systematically
demonstrate how to optimally design an adiabatic path to reduce the
computational errors. Our method has a clear physical picture and also explains
the pattern of computational error. In this paper we focus on quantum adiabatic
search algorithm although our results are general.Comment: 8 pages, 9 figure
Hamiltonian cycles in faulty random geometric networks
In this paper we analyze the Hamiltonian properties of
faulty random networks.
This consideration is of interest when considering wireless
broadcast networks.
A random geometric network is a graph whose vertices
correspond to points
uniformly and independently distributed in the unit square,
and whose edges
connect any pair of vertices if their distance is below some
specified bound.
A faulty random geometric network is a random geometric
network whose vertices
or edges fail at random. Algorithms to find Hamiltonian
cycles in faulty random
geometric networks are presented.Postprint (published version
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
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