7,007 research outputs found
Asymptotic of geometrical navigation on a random set of points of the plane
A navigation on a set of points is a rule for choosing which point to
move to from the present point in order to progress toward a specified target.
We study some navigations in the plane where is a non uniform Poisson point
process (in a finite domain) with intensity going to . We show the
convergence of the traveller path lengths, the number of stages done, and the
geometry of the traveller trajectories, uniformly for all starting points and
targets, for several navigations of geometric nature. Other costs are also
considered. This leads to asymptotic results on the stretch factors of random
Yao-graphs and random -graphs
Almost harmonic spinors
We show that any closed spin manifold not diffeomorphic to the two-sphere
admits a sequence of volume-one-Riemannian metrics for which the smallest
non-zero Dirac eigenvalue tends to zero. As an application, we compare the
Dirac spectrum with the conformal volume.Comment: minor modifications of the published versio
The combinatorics of the colliding bullets problem
The finite colliding bullets problem is the following simple problem:
consider a gun, whose barrel remains in a fixed direction; let be an i.i.d.\ family of random variables with uniform distribution on
; shoot bullets one after another at times , where the
th bullet has speed . When two bullets collide, they both annihilate.
We give the distribution of the number of surviving bullets, and in some
generalisation of this model. While the distribution is relatively simple (and
we found a number of bold claims online), our proof is surprisingly intricate
and mixes combinatorial and geometric arguments; we argue that any rigorous
argument must very likely be rather elaborate.Comment: 29 page
Random recursive triangulations of the disk via fragmentation theory
We introduce and study an infinite random triangulation of the unit disk that
arises as the limit of several recursive models. This triangulation is
generated by throwing chords uniformly at random in the unit disk and keeping
only those chords that do not intersect the previous ones. After throwing
infinitely many chords and taking the closure of the resulting set, one gets a
random compact subset of the unit disk whose complement is a countable union of
triangles. We show that this limiting random set has Hausdorff dimension
, where , and that it can be described as
the geodesic lamination coded by a random continuous function which is
H\"{o}lder continuous with exponent , for every
. We also discuss recursive constructions of triangulations of
the -gon that give rise to the same continuous limit when tends to
infinity.Comment: Published in at http://dx.doi.org/10.1214/10-AOP608 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Anarchy and Neutrino Physics
The neutrino sector of a seesaw-extended Standard Model is investigated under
the anarchy hypothesis. The previously derived probability density functions
for neutrino masses and mixings, which characterize the type I-III seesaw
ensemble of complex random matrices, are used to extract
information on the relevant physical parameters. For and , the
distributions of the light neutrino masses, as well as the mixing angles and
phases, are obtained using numerical integration methods. A systematic
comparison with the much simpler type II seesaw ensemble is also performed to
point out the fundamental differences between the two ensembles. It is found
that the type I-III seesaw ensemble is better suited to accommodate
experimental data. Moreover, the results indicate a strong preference for the
mass splitting associated to normal hierarchy. However, since all permutations
of the singular values are found to be equally probable for a particular mass
splitting, predictions regarding the hierarchy of the mass spectrum remains out
of reach in the framework of anarchy.Comment: 1+22 pages, 8 figures, typos fixed, added referenc
Relaxation approximation of Friedrich's systems under convex constraints
This paper is devoted to present an approximation of a Cauchy problem for
Friedrichs' systems under convex constraints. It is proved the strong
convergence in L^2\_{loc} of a parabolic-relaxed approximation towards the
unique constrained solution
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