10,982 research outputs found
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
Renormalization group improvement of the spectrum of Hydrogen-like atoms with massless fermions
We obtain the next-to-next-to-leading-log renormalization group improvement
of the spectrum of Hydrogen-like atoms with massless fermions by using
potential NRQED. These results can also be applied to the computation of the
muonic Hydrogen spectrum where we are able to reproduce some known double logs
at O(m\alpha^6). We compare with other formalisms dealing with log resummation
available in the literature.Comment: 9 pages, LaTeX. Minor changes, note added, final versio
Preparing the bound instance of quantum entanglement
Among the possibly most intriguing aspects of quantum entanglement is that it
comes in "free" and "bound" instances. Bound entangled states require entangled
states in preparation but, once realized, no free entanglement and therefore no
pure maximally entangled pairs can be regained. Their existence hence certifies
an intrinsic irreversibility of entanglement in nature and suggests a
connection with thermodynamics. In this work, we present a first experimental
unconditional preparation and detection of a bound entangled state of light. We
consider continuous-variable entanglement, use convex optimization to identify
regimes rendering its bound character well certifiable, and realize an
experiment that continuously produced a distributed bound entangled state with
an extraordinary and unprecedented significance of more than ten standard
deviations away from both separability and distillability. Our results show
that the approach chosen allows for the efficient and precise preparation of
multimode entangled states of light with various applications in quantum
information, quantum state engineering and high precision metrology.Comment: The final version accounts for a recent comment in Nature Physics
[24] clarifying that a previous claim of having generated bound entanglement
[23] was not supported by the authors' data. We also extended our
introduction and discussion and also added reference
A trivial observation on time reversal in random matrix theory
It is commonly thought that a state-dependent quantity, after being averaged
over a classical ensemble of random Hamiltonians, will always become
independent of the state. We point out that this is in general incorrect: if
the ensemble of Hamiltonians is time reversal invariant, and the quantity
involves the state in higher than bilinear order, then we show that the
quantity is only a constant over the orbits of the invariance group on the
Hilbert space. Examples include fidelity and decoherence in appropriate models.Comment: 7 pages 3 figure
Effective field theories for heavy quarkonium
We review recent theoretical developments in heavy quarkonium physics from
the point of view of Effective Field Theories of QCD. We discuss
Non-Relativistic QCD and concentrate on potential Non-Relativistic QCD. Our
main goal will be to derive QCD Schr\"odinger-like equations that govern the
heavy quarkonium physics in the weak and strong coupling regime. We also
discuss a selected set of applications, which include spectroscopy, inclusive
decays and electromagnetic threshold production.Comment: 162 pages, 30 figures, revised version, references added. Accepted
for publication in Reviews of Modern Physic
Striations in the Taurus molecular cloud: Kelvin-Helmholtz instability or MHD waves?
The origin of striations aligned along the local magnetic field direction in
the translucent envelope of the Taurus molecular cloud is examined with new
observations of 12CO and 13CO J=2-1 emission obtained with the 10~m
submillimeter telescope of the Arizona Radio Observatory. These data identify a
periodic pattern of excess blue and redshifted emission that is responsible for
the striations. For both 12CO and 13CO, spatial variations of the J=2-1 to
J=1-0 line ratio are small and are not spatially correlated with the striation
locations. A medium comprised of unresolved CO emitting substructures (cells)
with a beam area filling factor less than unity at any velocity is required to
explain the average line ratios and brightness temperatures. We propose that
the striations result from the modulation of velocities and the beam filling
factor of the cells as a result of either the Kelvin-Helmholtz instability or
magnetosonic waves propagating through the envelope of the Taurus molecular
cloud. Both processes are likely common features in molecular clouds that are
sub-Alfvenic and may explain low column density, cirrus-like features similarly
aligned with the magnetic field observed throughout the interstellar medium in
far-infrared surveys of dust emission.Comment: 11 pages, 4 figures. Accepted for publication in MNRA
Noisy continuous--opinion dynamics
We study the Deffuant et al. model for continuous--opinion dynamics under the
influence of noise. In the original version of this model, individuals meet in
random pairwise encounters after which they compromise or not depending of a
confidence parameter. Free will is introduced in the form of noisy
perturbations: individuals are given the opportunity to change their opinion,
with a given probability, to a randomly selected opinion inside the whole
opinion space. We derive the master equation of this process. One of the main
effects of noise is to induce an order-disorder transition. In the disordered
state the opinion distribution tends to be uniform, while for the ordered state
a set of well defined opinion groups are formed, although with some opinion
spread inside them. Using a linear stability analysis we can derive approximate
conditions for the transition between opinion groups and the disordered state.
The master equation analysis is compared with direct Monte-Carlo simulations.
We find that the master equation and the Monte-Carlo simulations do not always
agree due to finite-size induced fluctuations that we analyze in some detail
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