14,551 research outputs found
Class numbers of totally real fields and applications to the Weber class number problem
The determination of the class number of totally real fields of large
discriminant is known to be a difficult problem. The Minkowski bound is too
large to be useful, and the root discriminant of the field can be too large to
be treated by Odlyzko's discriminant bounds. We describe a new technique for
determining the class number of such fields, allowing us to attack the class
number problem for a large class of number fields not treatable by previously
known methods. We give an application to Weber's class number problem, which is
the conjecture that all real cyclotomic fields of power of 2 conductor have
class number 1.Comment: Accepted for publication by Acta Arithmetic
Primordial black hole formation in the early universe: critical behaviour and self-similarity
Following on after three previous papers discussing the formation of
primordial black holes during the radiative era of the early universe, we
present here a further investigation of the critical nature of the process
involved, aimed at making contact with some of the basic underlying ideas from
the literature on critical collapse. We focus on the intermediate state, which
we have found appearing in cases with perturbations close to the critical
limit, and examine the connection between this and the similarity solutions
which play a fundamental role in the standard picture of critical collapse. We
have derived a set of self-similar equations for the null-slicing form of the
metric which we are using for our numerical calculations, and have then
compared the results obtained by integrating these with the ones coming from
our simulations for collapse of cosmological perturbations within an expanding
universe. We find that the similarity solution is asymptotically approached in
a region which grows to cover both the contracting matter and part of the
semi-void which forms outside it. Our main interest is in the situation
relevant for primordial black hole formation in the radiative era of the early
universe, where the relation between the pressure and the energy density
can be reasonably approximated by an expression of the form with
. However, we have also looked at other values of , both because
these have been considered in previous literature and also because they can be
helpful for giving further insight into situations relevant for primordial
black hole formation. As in our previous work, we have started our simulations
with initial supra-horizon scale perturbations of a type which could have come
from inflation.Comment: 23 pages, 8 figures, new abstract, submitted to Classical and Quantum
Gravity. This new version of the paper has been completely rewritten with
respect the previous one, with several changes and substantial additional
wor
Measuring the Effects of Artificial Viscosity in SPH Simulations of Rotating Fluid Flows
A commonly cited drawback of SPH is the introduction of spurious shear
viscosity by the artificial viscosity term in situations involving rotation.
Existing approaches for quantifying its effect include approximate analytic
formulae and disc-averaged be- haviour in specific ring-spreading simulations,
based on the kinematic effects produced by the artificial viscosity. These
methods have disadvantages, in that they typically are applicable to a very
small range of physical scenarios, have a large number of simplifying
assumptions, and often are tied to specific SPH formulations which do not
include corrective (e.g., Balsara) or time-dependent viscosity terms. In this
study we have developed a simple, generally applicable and practical technique
for evaluating the local effect of artificial viscosity directly from the
creation of specific entropy for each SPH particle. This local approach is
simple and quick to implement, and it al- lows a detailed characterization of
viscous effects as a function of position. Several advantages of this method
are discussed, including its ease in evaluation, its greater accuracy and its
broad applicability. In order to compare this new method with ex- isting ones,
simple disc flow examples are used. Even in these basic cases, the very roughly
approximate nature of the previous methods is shown. Our local method pro-
vides a detailed description of the effects of the artificial viscosity
throughout the disc, even for extended examples which implement Balsara
corrections. As a further use of this approach, explicit dependencies of the
effective viscosity in terms of SPH and flow parameters are estimated from the
example cases. In an appendix, a method for the initial placement of SPH
particles is discussed which is very effective in reducing numerical
fluctuations.Comment: 15 pages, 9 figures, resubmitted to MNRA
Funnel-flow accretion onto highly magnetized neutron stars and shock generation
In this paper, we initiate a new study of steady funnel-flow accretion onto
strongly magnetized neutron stars, including a full treatment of shock
generation. As a first step, we adopt a simplified model considering the flow
within Newtonian theory and neglecting radiative pressure and cooling. The flow
is taken to start from an accretion disc and then to follow magnetic field
lines, forming a transonic funnel flow onto the magnetic poles. A standing
shock occurs at a certain point in the flow and beyond this material accretes
subsonically onto the star with high pressure and density. We calculate the
location of the standing shock and all other features of the flow within the
assumptions of our model. Applications to observed X-ray pulsars are discussed.Comment: 18 pages, 5 figs, accepted to Progress of Theoretical Physic
Causal Nature and Dynamics of Trapping Horizons in Black Hole Collapse
In calculations of gravitational collapse to form black holes, trapping
horizons (foliated by marginally trapped surfaces) make their first appearance
either within the collapsing matter or where it joins on to a vacuum exterior.
Those which then move outwards with respect to the matter have been proposed
for use in defining black holes, replacing the global concept of an "event
horizon" which has some serious drawbacks for practical applications. We here
present results from a study of the properties of both outgoing and ingoing
trapping horizons, assuming strict spherical symmetry throughout. We have
investigated their causal nature (i.e. whether they are spacelike, timelike or
null), making contact with the Misner-Sharp- Hernandez formalism, which has
often been used for numerical calculations of spherical collapse. We follow two
different approaches, one using a geometrical quantity related to expansions of
null geodesic congruences, and the other using the horizon velocity measured
with respect to the collapsing matter. After an introduction to these concepts,
we then implement them within numerical simulations of stellar collapse,
revisiting pioneering calculations from the 1960s where some features of the
emergence and subsequent behaviour of trapping horizons could already be seen.
Our presentation here is aimed firmly at "real world" applications of interest
to astrophysicists and includes the effects of pressure, which may be important
for the asymptotic behaviour of the ingoing horizon.Comment: 33 pages, 11 figure
Polytropic spheres in Palatini f(R) gravity
We examine static spherically symmetric polytropic spheres in Palatini f(R)
gravity and show that no regular solutions to the field equations exist for
physically relevant cases such as a monatomic isentropic gas or a degenerate
electron gas, thus casting doubt on the validity of Palatini f(R) gravity as an
alternative to General Relativity.Comment: Talk given by EB at the 30th Spanish Relativity Meeting, 10 - 14
September 2007, Tenerife (Spain). Based on arXiv:gr-qc/0703132 and
arXiv:0712.1141 [gr-qc
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