463 research outputs found
Negativity as a distance from a separable state
The computable measure of the mixed-state entanglement, the negativity, is
shown to admit a clear geometrical interpretation, when applied to
Schmidt-correlated (SC) states: the negativity of a SC state equals a distance
of the state from a pertinent separable state. As a consequence, a SC state is
separable if and only if its negativity vanishes. Another remarkable
consequence is that the negativity of a SC can be estimated "at a glance" on
the density matrix. These results are generalized to mixtures of SC states,
which emerge in certain quantum-dynamical settings.Comment: 9 pages, 1 figur
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
Theory of commensurable magnetic structures in holmium
The tendency for the period of the helically ordered moments in holmium to
lock into values which are commensurable with the lattice is studied
theoretically as a function of temperature and magnetic field. The
commensurable effects are derived in the mean-field approximation from
numerical calculations of the free energy of various commensurable structures,
and the results are compared with the extensive experimental evidence collected
during the last ten years on the magnetic structures in holmium. In general the
stability of the different commensurable structures is found to be in accord
with the experiments, except for the tau=5/18 structure observed a few degrees
below T_N in a b-axis field. The trigonal coupling recently detected in holmium
is found to be the interaction required to explain the increased stability of
the tau=1/5 structure around 42 K, and of the tau=1/4 structure around 96 K,
when a field is applied along the c-axis.Comment: REVTEX, 31 pages, 7 postscript figure
Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra
We give a method for computing upper and lower bounds for the volume of a
non-obtuse hyperbolic polyhedron in terms of the combinatorics of the
1-skeleton. We introduce an algorithm that detects the geometric decomposition
of good 3-orbifolds with planar singular locus and underlying manifold the
3-sphere. The volume bounds follow from techniques related to the proof of
Thurston's Orbifold Theorem, Schl\"afli's formula, and previous results of the
author giving volume bounds for right-angled hyperbolic polyhedra.Comment: 36 pages, 19 figure
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
Additional experimental evidence for a solar influence on nuclear decay rates
Additional experimental evidence is presented in support of the recent
hypothesis that a possible solar influence could explain fluctuations observed
in the measured decay rates of some isotopes. These data were obtained during
routine weekly calibrations of an instrument used for radiological safety at
The Ohio State University Research Reactor using Cl-36. The detector system
used was based on a Geiger-Mueller gas detector, which is a robust detector
system with very low susceptibility to environmental changes. A clear annual
variation is evident in the data, with a maximum relative count rate observed
in January/February, and a minimum relative count rate observed in July/August,
for seven successive years from July 2005 to June 2011. This annual variation
is not likely to have arisen from changes in the detector surroundings, as we
show here.Comment: 8 pages, 6 figure
The maximum number of cliques in a graph embedded in a surface
This paper studies the following question: Given a surface and an
integer , what is the maximum number of cliques in an -vertex graph
embeddable in ? We characterise the extremal graphs for this question,
and prove that the answer is between and , where is the maximum integer such that the
complete graph embeds in . For the surfaces ,
, , , ,
and we establish an exact answer
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
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