27,930 research outputs found
Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box
In -neighbour bootstrap percolation, vertices (sites) of a graph are
infected, round-by-round, if they have neighbours already infected. Once
infected, they remain infected. An initial set of infected sites is said to
percolate if every site is eventually infected. We determine the maximal
percolation time for -neighbour bootstrap percolation on the hypercube for
all as the dimension goes to infinity up to a logarithmic
factor. Surprisingly, it turns out to be , which is in great
contrast with the value for , which is quadratic in , as established by
Przykucki. Furthermore, we discover a link between this problem and a
generalisation of the well-known Snake-in-the-Box problem.Comment: 14 pages, 1 figure, submitte
A fast Bayesian approach to discrete object detection in astronomical datasets - PowellSnakes I
A new fast Bayesian approach is introduced for the detection of discrete
objects immersed in a diffuse background. This new method, called PowellSnakes,
speeds up traditional Bayesian techniques by: i) replacing the standard form of
the likelihood for the parameters characterizing the discrete objects by an
alternative exact form that is much quicker to evaluate; ii) using a
simultaneous multiple minimization code based on Powell's direction set
algorithm to locate rapidly the local maxima in the posterior; and iii)
deciding whether each located posterior peak corresponds to a real object by
performing a Bayesian model selection using an approximate evidence value based
on a local Gaussian approximation to the peak. The construction of this
Gaussian approximation also provides the covariance matrix of the uncertainties
in the derived parameter values for the object in question. This new approach
provides a speed up in performance by a factor of `hundreds' as compared to
existing Bayesian source extraction methods that use MCMC to explore the
parameter space, such as that presented by Hobson & McLachlan. We illustrate
the capabilities of the method by applying to some simplified toy models.
Furthermore PowellSnakes has the advantage of consistently defining the
threshold for acceptance/rejection based on priors which cannot be said of the
frequentist methods. We present here the first implementation of this technique
(Version-I). Further improvements to this implementation are currently under
investigation and will be published shortly. The application of the method to
realistic simulated Planck observations will be presented in a forthcoming
publication.Comment: 30 pages, 15 figures, revised version with minor changes, accepted
for publication in MNRA
Geometric factors influencing the diet of vertebrate predators in marine and terrestrial environments
Predator–prey relationships are vital to ecosystem function and there is a need for greater predictive
understanding of these interactions. We develop a geometric foraging model predicting minimum
prey size scaling in marine and terrestrial vertebrate predators taking into account habitat
dimensionality and biological traits. Our model predicts positive predator–prey size relationships
on land but negative relationships in the sea. To test the model, we compiled data on diets of 794
predators (mammals, snakes, sharks and rays). Consistent with predictions, both terrestrial endotherm
and ectotherm predators have significantly positive predator–prey size relationships. Marine
predators, however, exhibit greater variation. Some of the largest predators specialise on small
invertebrates while others are large vertebrate specialists. Prey–predator mass ratios were generally
higher for ectothermic than endothermic predators, although dietary patterns were similar.
Model-based simulations of predator–prey relationships were consistent with observed relationships,
suggestin
Determination of the Topology of a Directed Network
We consider strongly-connected directed networks of identical synchronous,
finite-state processors with in- and out-degree uniformly bounded by a network
constant. Via a straightforward extension of Ostrovsky and Wilkerson's
Backwards Communication Algorithm in [OW], we exhibit a protocol which solves
the Global Topology Determination Problem, the problem of having the root
processor map the global topology of a network of unknown size and topology,
with running time O(ND) where N represents the number of processors and D
represents the diameter of the network. A simple counting argument suffices to
show that the Global Topology Determination Problem has time-complexity Omega(N
logN) which makes the protocol presented asymptotically time-optimal for many
large networks.Comment: 9 pages, no figures, accepted to appear in IPDPS 2002 (unable to
attend), (journal version to appear in Information Processing Letters
Condorcet domains of tiling type
A Condorcet domain (CD) is a collection of linear orders on a set of
candidates satisfying the following property: for any choice of preferences of
voters from this collection, a simple majority rule does not yield cycles. We
propose a method of constructing "large" CDs by use of rhombus tiling diagrams
and explain that this method unifies several constructions of CDs known
earlier. Finally, we show that three conjectures on the maximal sizes of those
CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic
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