73,386 research outputs found
Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications
In this paper, we redesign and simplify an algorithm due to Remy et al. for
the generation of rooted planar trees that satisfies a given partition of
degrees. This new version is now optimal in terms of random bit complexity, up
to a multiplicative constant. We then apply a natural process
"simulate-guess-and-proof" to analyze the height of a random Motzkin in
function of its frequency of unary nodes. When the number of unary nodes
dominates, we prove some unconventional height phenomenon (i.e. outside the
universal square root behaviour.)Comment: 19 page
Gravitational waves from scattering of stellar-mass black holes in galactic nuclei
Stellar mass black holes (BHs) are expected to segregate and form a steep
density cusp around supermassive black holes (SMBHs) in galactic nuclei. We
follow the evolution of a multi-mass system of BHs and stars by numerically
integrating the Fokker-Planck energy diffusion equations for a variety of BH
mass distributions. We find that the BHs "self-segregate'', and that the
rarest, most massive BHs dominate the scattering rate closest to the SMBH (<
0.1 pc). BH--BH binaries form out of gravitational wave emission during BH
encounters. We find that the expected rate of BH coalescence events detectable
by Advanced LIGO is ~1 - 100/yr, depending on the initial mass function of
stars in galactic nuclei and the mass of the most massive BHs. We find that the
actual merger rate is likely ~10 times larger than this due to the intrinsic
scatter of stellar densities in many different galaxies. The BH binaries that
form this way in galactic nuclei have significant eccentricities as they enter
the LIGO band (90% with e > 0.9), and are therefore distinguishable from other
binaries, which circularise before becoming detectable. We also show that
eccentric mergers can be detected to larger distances and greater BH masses
than circular mergers, up to ~ 700 solar masses. Future ground-based
gravitational wave observatories will be able to constrain both the mass
function of BHs and stars in galactic nuclei.Comment: 22 pages, 13 figures. MNRAS accepted, in pres
Discrete curvature approximations and segmentation of polyhedral surfaces
The segmentation of digitized data to divide a free form surface into patches is one of the key steps required to perform a reverse engineering process of an object. To this end, discrete curvature approximations are introduced as the basis of a segmentation process that lead to a decomposition of digitized data into areas that will help the construction of parametric surface patches. The approach proposed relies on the use of a polyhedral representation of the object built from the digitized data input. Then, it is shown how noise reduction, edge swapping techniques and adapted remeshing schemes can participate to different preparation phases to provide a geometry that highlights useful characteristics for the segmentation process. The segmentation process is performed with various approximations of discrete curvatures evaluated on the polyhedron produced during the preparation phases. The segmentation process proposed involves two phases: the identification of characteristic polygonal lines and the identification of polyhedral areas useful for a patch construction process. Discrete curvature criteria are adapted to each phase and the concept of invariant evaluation of curvatures is introduced to generate criteria that are constant over equivalent meshes. A description of the segmentation procedure is provided together with examples of results for free form object surfaces
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