1,048 research outputs found
Some homogenization and corrector results for nonlinear monotone operators
This paper deals with the limit behaviour of the solutions of quasi-linear
equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon
_h},Du_h\right)\right)=f_h on with Dirichlet boundary conditions.
The sequence tends to and the map is
periodic in , monotone in and satisfies suitable continuity
conditions. It is proved that weakly in , where is the solution of a homogenized problem \
-\limfunc{div}(b(x,Du))=f on . We also prove some corrector results,
i.e. we find such that in
Correctors for some nonlinear monotone operators
In this paper we study homogenization of quasi-linear partial differential
equations of the form -\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right)
\right) =f_h on with Dirichlet boundary conditions. Here the
sequence tends to as
and the map is periodic in monotone in
and satisfies suitable continuity conditions. We prove that
weakly in as where
is the solution of a homogenized problem of the form -\mbox{div}\left(
b\left( x,Du\right) \right) =f on We also derive an explicit
expression for the homogenized operator and prove some corrector results,
i.e. we find such that in
An intermediate value theorem in ordered Banach spaces
We consider a monotone increasing operator in an ordered Banach space having
and as a strong super- and subsolution, respectively. In contrast
with the well studied case , we suppose that . Under the
assumption that the order cone is normal and minihedral, we prove the existence
of a fixed point located in the ordered interval $[u_-,u_+].
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
On approximate solutions of semilinear evolution equations
A general framework is presented to discuss the approximate solutions of an
evolution equation in a Banach space, with a linear part generating a semigroup
and a sufficiently smooth nonlinear part. A theorem is presented, allowing to
infer from an approximate solution the existence of an exact solution.
According to this theorem, the interval of existence of the exact solution and
the distance of the latter from the approximate solution can be evaluated
solving a one-dimensional "control" integral equation, where the unknown gives
a bound on the previous distance as a function of time. For example, the
control equation can be applied to the approximation methods based on the
reduction of the evolution equation to finite-dimensional manifolds: among
them, the Galerkin method is discussed in detail. To illustrate this framework,
the nonlinear heat equation is considered. In this case the control equation is
used to evaluate the error of the Galerkin approximation; depending on the
initial datum, this approach either grants global existence of the solution or
gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened
version; the proof of Prop. 3.4. has been simplified
Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
We prove the most general theorem about spectral stability of multi-site
breathers in the discrete Klein-Gordon equation with a small coupling constant.
In the anti-continuum limit, multi-site breathers represent excited
oscillations at different sites of the lattice separated by a number of "holes"
(sites at rest). The theorem describes how the stability or instability of a
multi-site breather depends on the phase difference and distance between the
excited oscillators. Previously, only multi-site breathers with adjacent
excited sites were considered within the first-order perturbation theory. We
show that the stability of multi-site breathers with one-site holes change for
large-amplitude oscillations in soft nonlinear potentials. We also discover and
study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site
breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure
Detailed Spectroscopic and Photometric Analysis of DQ White Dwarfs
We present an analysis of spectroscopic and photometric data for cool DQ
white dwarfs based on improved model atmosphere calculations. In particular, we
revise the atmospheric parameters of the trigonometric parallax sample of
Bergeron et al.(2001), and discuss the astrophysical implications on the
temperature scale and mean mass, as well as the chemical evolution of these
stars. We also analyze 40 new DQ stars discovered in the first data release of
the Sloan Digital Sky Survey.Comment: 6 pages,3 figures, 14th European Workshop on White Dwarfs, ASP
Conference Series, in pres
Ellipsometric measurements by use of photon pairs generated by spontaneous parametric down-conversion
We present a novel interferometric technique for performing ellipsometric
measurements. This technique relies on the use of a non-classical optical
source, namely, polarization-entangled twin photons generated by spontaneous
parametric down-conversion from a nonlinear crystal, in conjunction with a
coincidence-detection scheme. Ellipsometric measurements acquired with this
scheme are absolute; i.e., they do not require source and detector calibration.Comment: 10 pages, accepted for publication in Optics Letter
Adaptive plasticity in the healthy reading network investigated through combined neurostimulation and neuroimaging
The reading network in the human brain comprises several regions, including the left inferior frontal cortex (IFC), ventral occipito-temporal cortex (vOTC) and dorsal temporo-parietal cortex (TPC). The left TPC is crucial for phonological decoding, i.e., for learning and retaining sound-letter mappings. Here, we tested the causal contribution of this area for reading with repetitive transcranial magnetic stimulation (rTMS) and explored the response of the reading network using functional magnetic resonance imaging (fMRI). 28 healthy adult readers overtly read simple and complex words and pseudowords during fMRI after effective or sham TMS over the left TPC. Behaviorally, effective stimulation slowed pseudoword reading. A multivariate pattern analysis showed a shift in activity patterns in the left IFC for pseudoword reading after effective relative to sham TMS. Furthermore, active TMS led to increased effective connectivity from the left vOTC to the left TPC, specifically for pseudoword processing. The observed changes in task-related activity and connectivity suggest compensatory reorganization in the reading network following TMS-induced disruption of the left TPC. Our findings provide first evidence for a causal role of the left TPC for overt pseudoword reading and emphasize the relevance of functional interactions in the healthy reading network for successful pseudoword processing
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