14,224 research outputs found
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
Revisiting the Impact of Atmospheric Dispersion and Differential Refraction on Widefield Multiobject Spectroscopic Observations. From VLT/VIMOS to Next Generation Instruments
(Abridged) Atmospheric dispersion and field differential refraction impose
severe constraints on widefield MOS observations. Flux reduction and spectral
distortions must be minimised by a careful planning of the observations --
which is especially true for instruments that use slits instead of fibres. This
is the case of VIMOS at the VLT, where MOS observations have been restricted,
since the start of operations, to a narrow two-hour range from the meridian to
minimise slit losses. We revisit in detail the impact of atmospheric effects on
the quality of VIMOS-MOS spectra. We model slit losses across the entire VIMOS
FOV as a function of target declination. We explore two different slit
orientations at the meridian: along the parallactic angle (North-South), and
perpendicular to it (East-West). We show that, for fields culminating at zenith
distances larger than 20 deg, slit losses are minimised with slits oriented
along the parallactic angle at the meridian. The two-hour angle rule holds for
these observations using N-S orientations. Conversely, for fields with zenith
angles smaller than 20 deg at culmination, losses are minimised with slits
oriented perpendicular to the parallactic angle at the meridian. MOS
observations can be effectively extended to plus/minus three hours from the
meridian in these cases. In general, night-long observations of a single field
will benefit from using the E-W orientation. All-sky or service mode
observations, however, require a more elaborate planning that depends on the
target declination, and the hour angle of the observations. We establish
general rules for the alignment of slits in MOS observations that will increase
target observability, enhance the efficiency of operations, and speed up the
completion of programmes -- a particularly relevant aspect for the forthcoming
spectroscopic public surveys with VIMOS.Comment: Accepted to A&A. 11 pages, 15 figures. This paper presents the new
recommendations for optimal slit alignment in VLT/VIMOS observation
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
GEMs and amplitude bounds in the colored Boulatov model
In this paper we construct a methodology for separating the divergencies due
to different topological manifolds dual to Feynman graphs in colored group
field theory. After having introduced the amplitude bounds using propagator
cuts, we show how Graph-Encoded-Manifolds (GEM) techniques can be used in order
to factorize divergencies related to different parts of the dual topologies of
the Feynman graphs in the general case. We show the potential of the formalism
in the case of 3-dimensional solid torii in the colored Boulatov model.Comment: 20 pages; 20 Figures; Style changed, discussion improved and typos
corrected, citations added; These GEMs are not related to "Global Embedding
Minkowskian spacetimes
The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs
This is the second of a series of four papers in which we prove the following
relaxation of the Loebl-Komlos--Sos Conjecture: For every there
exists a number such that for every every -vertex graph
with at least vertices of degree at least
contains each tree of order as a subgraph.
In the first paper of the series, we gave a decomposition of the graph
into several parts of different characteristics; this decomposition might be
viewed as an analogue of a regular partition for sparse graphs. In the present
paper, we find a combinatorial structure inside this decomposition. In the last
two papers, we refine the structure and use it for embedding the tree .Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDM
Measurable versions of Vizing's theorem
We establish two versions of Vizing's theorem for Borel multi-graphs whose
vertex degrees and edge multiplicities are uniformly bounded by respectively
and . The ``approximate'' version states that, for any Borel
probability measure on the edge set and any , we can properly
colour all but -fraction of edges with colours in a
Borel way. The ``measurable'' version, which is our main result, states that
if, additionally, the measure is invariant, then there is a measurable proper
edge colouring of the whole edge set with at most colours
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