189 research outputs found
Reheating-volume measure for random-walk inflation
The recently proposed "reheating-volume" (RV) measure promises to solve the
long-standing problem of extracting probabilistic predictions from cosmological
"multiverse" scenarios involving eternal inflation. I give a detailed
description of the new measure and its applications to generic models of
eternal inflation of random-walk type. For those models I derive a general
formula for RV-regulated probability distributions that is suitable for
numerical computations. I show that the results of the RV cutoff in random-walk
type models are always gauge-invariant and independent of the initial
conditions at the beginning of inflation. In a toy model where equal-time
cutoffs lead to the "youngness paradox," the RV cutoff yields unbiased results
that are distinct from previously proposed measures.Comment: Figure 1 updated, version accepted for publication in Phys.Rev.
The Eastwood-Singer gauge in Einstein spaces
Electrodynamics in curved spacetime can be studied in the Eastwood--Singer
gauge, which has the advantage of respecting the invariance under conformal
rescalings of the Maxwell equations. Such a construction is here studied in
Einstein spaces, for which the Ricci tensor is proportional to the metric. The
classical field equations for the potential are then equivalent to first
solving a scalar wave equation with cosmological constant, and then solving a
vector wave equation where the inhomogeneous term is obtained from the gradient
of the solution of the scalar wave equation. The Eastwood--Singer condition
leads to a field equation on the potential which is preserved under gauge
transformations provided that the scalar function therein obeys a fourth-order
equation where the highest-order term is the wave operator composed with
itself. The second-order scalar equation is here solved in de Sitter spacetime,
and also the fourth-order equation in a particular case, and these solutions
are found to admit an exponential decay at large time provided that
square-integrability for positive time is required. Last, the vector wave
equation in the Eastwood-Singer gauge is solved explicitly when the potential
is taken to depend only on the time variable.Comment: 13 pages. Section 6, with new original calculations, has been added,
and the presentation has been improve
Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints
The Einstein constraint equations have been the subject of study for more
than fifty years. The introduction of the conformal method in the 1970's as a
parameterization of initial data for the Einstein equations led to increased
interest in the development of a complete solution theory for the constraints,
with the theory for constant mean curvature (CMC) spatial slices and closed
manifolds completely developed by 1995. The first general non-CMC existence
result was establish by Holst et al. in 2008, with extensions to rough data by
Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC
theory remains mostly open; moreover, recent work of Maxwell on specific
symmetry models sheds light on fundamental non-uniqueness problems with the
conformal method as a parameterization in non-CMC settings. In parallel with
these mathematical developments, computational physicists have uncovered
surprising behavior in numerical solutions to the extended conformal thin
sandwich formulation of the Einstein constraints. In particular, numerical
evidence suggests the existence of multiple solutions with a quadratic fold,
and a recent analysis of a simplified model supports this conclusion. In this
article, we examine this apparent bifurcation phenomena in a methodical way,
using modern techniques in bifurcation theory and in numerical homotopy
methods. We first review the evidence for the presence of bifurcation in the
Hamiltonian constraint in the time-symmetric case. We give a brief introduction
to the mathematical framework for analyzing bifurcation phenomena, and then
develop the main ideas behind the construction of numerical homotopy, or
path-following, methods in the analysis of bifurcation phenomena. We then apply
the continuation software package AUTO to this problem, and verify the presence
of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material
on physical implication
Time-Changed Fast Mean-Reverting Stochastic Volatility Models
We introduce a class of randomly time-changed fast mean-reverting stochastic
volatility models and, using spectral theory and singular perturbation
techniques, we derive an approximation for the prices of European options in
this setting. Three examples of random time-changes are provided and the
implied volatility surfaces induced by these time-changes are examined as a
function of the model parameters. Three key features of our framework are that
we are able to incorporate jumps into the price process of the underlying
asset, allow for the leverage effect, and accommodate multiple factors of
volatility, which operate on different time-scales
Quantum statistical properties of the radiation field in a cavity with a movable mirror
A quantum system composed of a cavity radiation field interacting with a
movable mirror is considered and quantum statistical properties of the field
are studied. Such a system can serve in principle as an idealized meter for
detection of a weak classical force coupled to the mirror which is modelled by
a quantum harmonic oscillator. It is shown that the standard quantum limit on
the measurement of the mirror position arises naturally from the properties of
the system during its dynamical evolution. However, the force detection
sensitivity of the system falls short of the corresponding standard quantum
limit. We also study the effect of the nonlinear interaction between the moving
mirror and the radiation pressure on the quadrature fluctuations of the
initially coherent cavity field.Comment: REVTeX, 9 pages, 5 figures. More info on
http://www.ligo.caltech.edu/~cbrif/science.htm
Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes
We construct a class of solutions to the Cauchy problem of the Klein-Gordon
equation on any standard static spacetime. Specifically, we have constructed
solutions to the Cauchy problem based on any self-adjoint extension (satisfying
a technical condition: "acceptability") of (some variant of) the
Laplace-Beltrami operator defined on test functions in an -space of the
static hypersurface. The proof of the existence of this construction completes
and extends work originally done by Wald. Further results include the
uniqueness of these solutions, their support properties, the construction of
the space of solutions and the energy and symplectic form on this space, an
analysis of certain symmetries on the space of solutions and of various
examples of this method, including the construction of a non-bounded below
acceptable self-adjoint extension generating the dynamics
A Sparse Reformulation of the Green's Function Formalism Allows Efficient Simulations of Morphological Neuron Models
We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale as O (n) with the number n of inputs locations, contrary to the previously reported O (n(2)) scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.Peer reviewedFinal Accepted Versio
The Generalized Lyapunov Theorem and its Application to Quantum Channels
We give a simple and physically intuitive necessary and sufficient condition
for a map acting on a compact metric space to be mixing (i.e. infinitely many
applications of the map transfer any input into a fixed convergency point).
This is a generalization of the "Lyapunov direct method". First we prove this
theorem in topological spaces and for arbitrary continuous maps. Finally we
apply our theorem to maps which are relevant in Open Quantum Systems and
Quantum Information, namely Quantum Channels. In this context we also discuss
the relations between mixing and ergodicity (i.e. the property that there exist
only a single input state which is left invariant by a single application of
the map) showing that the two are equivalent when the invariant point of the
ergodic map is pure.Comment: 13 pages, 3 figure
Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film
It has been experimentally observed that weakly conducting suspended films of
smectic liquid crystals undergo electroconvection when subjected to a large
enough potential difference. The resulting counter-rotating vortices form a
very simple convection pattern and exhibit a variety of interesting nonlinear
effects. The linear stability problem for this system has recently been solved.
The convection mechanism, which involves charge separation at the free surfaces
of the film, is applicable to any sufficiently two-dimensional fluid. In this
paper, we derive an amplitude equation which describes the weakly nonlinear
regime, by starting from the basic electrohydrodynamic equations. This regime
has been the subject of several recent experimental studies. The lowest order
amplitude equation we derive is of the Ginzburg-Landau form, and describes a
forward bifurcation as is observed experimentally. The coefficients of the
amplitude equation are calculated and compared with the values independently
deduced from the linear stability calculation.Comment: 26 pages, 2 included eps figures, submitted to Phys Rev E. For more
information, see http://mobydick.physics.utoronto.c
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
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