96 research outputs found
Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions
Analytic solutions of the Teukolsky equation in Kerr geometries are presented
in the form of series of hypergeometric functions and Coulomb wave functions.
Relations between these solutions are established. The solutions provide a very
powerful method not only for examining the general properties of solutions and
physical quantities when they are applied to, but also for numerical
computations. The solutions are given in the expansion of a small parameter
, being the mass of black hole, which corresponds
to Post-Minkowski expansion by and to post-Newtonian expansion when they
are applied to the gravitational radiation from a particle in circular orbit
around a black hole. It is expected that these solutions will become a powerful
weapon to construct the theoretical template towards LIGO and VIRGO projects.Comment: 24 pages, minor modification
Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion
Analytic solutions of the Regge-Wheeler equation are presented in the form of
series of hypergeometric functions and Coulomb wave functions which have
different regions of convergence. Relations between these solutions are
established. The series solutions are given as the Post-Minkowskian expansion
with respect to a parameter , being the mass of
black hole. This expansion corresponds to the post-Newtonian expansion when
they are applied to the gravitational radiation from a particle in circular
orbit around a black hole. These solutions can also be useful for numerical
computations.Comment: 22 page
A measure-on-graph-valued diffusion: a particle system with collisions, and their applications
A diffusion taking value in probability measures on a graph with a vertex set
, , is studied. The masses on each vertices
satisfy the stochastic differential equation of the form on the simplex, where are independent
standard Brownian motions with skew symmetry and is the neighbour of the
vertex . A dual Markov chain on integer partitions to the Markov semigroup
associated with the diffusion is used to show that the support of an extremal
stationary state of the adjoint semigroup is an independent set of the graph.
We also investigate the diffusion with a linear drift, which gives a killing of
the dual Markov chain on a finite integer lattice. The Markov chain is used to
study the unique stationary state of the diffusion, which generalizes the
Dirichlet distribution. Two applications of the diffusions are discussed:
analysis of an algorithm to find an independent set of a graph, and a Bayesian
graph selection based on computation of probability of a sample by using
coupling from the past.Comment: 30 pages, 3 figure
- β¦