19 research outputs found
Hyperelliptic Theta-Functions and Spectral Methods
A code for the numerical evaluation of hyperelliptic theta-functions is
presented. Characteristic quantities of the underlying Riemann surface such as
its periods are determined with the help of spectral methods. The code is
optimized for solutions of the Ernst equation where the branch points of the
Riemann surface are parameterized by the physical coordinates. An exploration
of the whole parameter space of the solution is thus only possible with an
efficient code. The use of spectral approximations allows for an efficient
calculation of all quantities in the solution with high precision. The case of
almost degenerate Riemann surfaces is addressed. Tests of the numerics using
identities for periods on the Riemann surface and integral identities for the
Ernst potential and its derivatives are performed. It is shown that an accuracy
of the order of machine precision can be achieved. These accurate solutions are
used to provide boundary conditions for a code which solves the axisymmetric
stationary Einstein equations. The resulting solution agrees with the
theta-functional solution to very high precision.Comment: 25 pages, 12 figure
Computation of the Generalized F Distribution
Exact expressions are given for the distribution function of the ratio of a
weighted sum of independent chi-squared variables to a single chi-square
variable, scaled appropriately. This distribution is the generalization of the
classical F distribution to mixtures of chi-squared variables. The distribution
is given in terms of the Lauricella functions. The truncation error bounds are
given in terms of hypergeometric functions. Applications to detecting joint
outliers and Hotelling's misspecified T^2 distribution are given.Comment: Latex, 15 page
Classical and Quantum Transport Through Entropic Barriers Modelled by Hardwall Hyperboloidal Constrictions
We study the quantum transport through entropic barriers induced by hardwall
constrictions of hyperboloidal shape in two and three spatial dimensions. Using
the separability of the Schrodinger equation and the classical equations of
motion for these geometries we study in detail the quantum transmission
probabilities and the associated quantum resonances, and relate them to the
classical phase structures which govern the transport through the
constrictions. These classical phase structures are compared to the analogous
structures which, as has been shown only recently, govern reaction type
dynamics in smooth systems. Although the systems studied in this paper are
special due their separability they can be taken as a guide to study entropic
barriers resulting from constriction geometries that lead to non-separable
dynamics.Comment: 59 pages, 22 EPS figures
Geodesic flow on the ellipsoid with equal semi-axes
The equations for the geodesic flow on the ellipsoid are well known, and were first
solved by Jacobi in 1838 by separating the variables of the Hamilton–Jacobi equation. In
1979 Moser showed that the equations for the geodesic flow on the general ellipsoid with
distinct semi-axes are Liouville-integrable, and described a set of integrals which weren't
known classically. These integrals break down in the case of coinciding semi-axes.
After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid
with distinct semi-axes, the three-dimensional ellipsoid with the two middle semi-axes being
equal, corresponding to a Hamiltonian invariant under rotations, is investigated, using
the tools of singular reduction and invariant theory. The system is Liouville-integrable
and thus the invariant manifolds corresponding to regular points of the energy momentum
map are 3-dimensional tori. An analysis of the critical points of the energy momentum
map gives the bifurcation diagram. The fibres of the critical values of the energy momentum
map are found, and an analysis is carried out of the action variables. The obstruction
to the existence of single valued globally smooth action variables is monodromy. [Continues.
Dynamische Systeme (hybrid meeting)
This workshop continued a biannual series of workshops at Oberwolfach on
dynamical systems that started with a meeting organized by Moser and Zehnder in 1981.
Workshops in this series focus on new results and developments in
dynamical systems and related areas of mathematics, with symplectic geometry playing an important role in recent years in connection with Hamiltonian dynamics. In this year special emphasis was placed on various kinds of spectra (in contact geometry, in Riemannian geometry, in dynamical systems and in symplectic topology) and their applications to dynamics
Gravitating discs around black holes
Fluid discs and tori around black holes are discussed within different
approaches and with the emphasis on the role of disc gravity. First reviewed
are the prospects of investigating the gravitational field of a black
hole--disc system by analytical solutions of stationary, axially symmetric
Einstein's equations. Then, more detailed considerations are focused to middle
and outer parts of extended disc-like configurations where relativistic effects
are small and the Newtonian description is adequate.
Within general relativity, only a static case has been analysed in detail.
Results are often very inspiring, however, simplifying assumptions must be
imposed: ad hoc profiles of the disc density are commonly assumed and the
effects of frame-dragging and completely lacking. Astrophysical discs (e.g.
accretion discs in active galactic nuclei) typically extend far beyond the
relativistic domain and are fairly diluted. However, self-gravity is still
essential for their structure and evolution, as well as for their radiation
emission and the impact on the environment around. For example, a nuclear star
cluster in a galactic centre may bear various imprints of mutual star--disc
interactions, which can be recognised in observational properties, such as the
relation between the central mass and stellar velocity dispersion.Comment: Accepted for publication in CQG; high-resolution figures will be
available from http://www.iop.org/EJ/journal/CQ