1,095 research outputs found
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
Full text linkWe consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
Get PDFWe prove convergence results for expanding curvature flows in the Euclidean
and hyperbolic space. The flow speeds have the form , where and
is a positive, strictly monotone and 1-homogeneous curvature function. In
particular this class includes the mean curvature . We prove that a
certain initial pinching condition is preserved and the properly rescaled
hypersurfaces converge smoothly to the unit sphere. We show that an example due
to Andrews-McCoy-Zheng can be used to construct strictly convex initial
hypersurfaces, for which the inverse mean curvature flow to the power
loses convexity, justifying the necessity to impose a certain pinching
condition on the initial hypersurface.Comment: 18 pages. We included an example for the loss of convexity and
pinching. In the third version we dropped the concavity assumption on F.
Comments are welcom
On a Localized Riemannian Penrose Inequality
Full text linkConsider a compact, orientable, three dimensional Riemannian manifold with
boundary with nonnegative scalar curvature. Suppose its boundary is the
disjoint union of two pieces: the horizon boundary and the outer boundary,
where the horizon boundary consists of the unique closed minimal surfaces in
the manifold and the outer boundary is metrically a round sphere. We obtain an
inequality relating the area of the horizon boundary to the area and the total
mean curvature of the outer boundary. Such a manifold may be thought as a
region, surrounding the outermost apparent horizons of black holes, in a
time-symmetric slice of a space-time in the context of general relativity. The
inequality we establish has close ties with the Riemannian Penrose Inequality,
proved by Huisken and Ilmanen, and by Bray.Comment: 16 page
The time evolution of marginally trapped surfaces
Full text linkIn previous work we have shown the existence of a dynamical horizon or
marginally trapped tube (MOTT) containing a given strictly stable marginally
outer trapped surface (MOTS). In this paper we show some results on the global
behavior of MOTTs assuming the null energy condition. In particular we show
that MOTSs persist in the sense that every Cauchy surface in the future of a
given Cauchy surface containing a MOTS also must contain a MOTS. We describe a
situation where the evolving outermost MOTS must jump during the coalescence of
two seperate MOTSs. We furthermore characterize the behavior of MOTSs in the
case that the principal eigenvalue vanishes under a genericity assumption. This
leads to a regularity result for the tube of outermost MOTSs under the
genericity assumption. This tube is then smooth up to finitely many jump times.
Finally we discuss the relation of MOTSs to singularities of a space-time.Comment: 21 pages. This revision corrects some typos and contains more
detailed proofs than the original versio
Differential Geometry of Quantum States, Observables and Evolution
Full text linkThe geometrical description of Quantum Mechanics is reviewed and proposed as
an alternative picture to the standard ones. The basic notions of observables,
states, evolution and composition of systems are analised from this
perspective, the relevant geometrical structures and their associated algebraic
properties are highlighted, and the Qubit example is thoroughly discussed.Comment: 20 pages, comments are welcome
Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"
Full text linkThe main goal of this work consists in showing that the analytic solutions
for a class of characteristic problems for the Einstein vacuum equations have
an existence region larger than the one provided by the Cauchy-Kowalevski
theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove
this result we first describe a geometric way of writing the vacuum Einstein
equations for the characteristic problems we are considering, in a gauge
characterized by the introduction of a double null cone foliation of the
spacetime. Then we prove that the existence region for the analytic solutions
can be extended to a larger region which depends only on the validity of the
apriori estimates for the Weyl equations, associated to the "Bel-Robinson
norms". In particular if the initial data are sufficiently small we show that
the analytic solution is global. Before showing how to extend the existence
region we describe the same result in the case of the Burger equation, which,
even if much simpler, nevertheless requires analogous logical steps required
for the general proof. Due to length of this work, in this paper we mainly
concentrate on the definition of the gauge we use and on writing in a
"geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski
theorem is adapted to the characteristic problem for the Einstein equations and
we describe how the existence region can be extended in the case of the Burger
equation. Finally we describe the structure of the extension proof in the case
of the Einstein equations. The technical parts of this last result is the
content of a second paper.Comment: 68 page
The Dirac operator on untrapped surfaces
Full text linkWe establish a sharp extrinsic lower bound for the first eigenvalue of the
Dirac operator of an untrapped surface in initial data sets without apparent
horizon in terms of the norm of its mean curvature vector. The equality case
leads to rigidity results for the constraint equations with spherical boundary
as well as uniqueness results for constant mean curvature surfaces in Minkowski
space.Comment: 16 page
Time reversal in thermoacoustic tomography - an error estimate
Full text linkThe time reversal method in thermoacoustic tomography is used for
approximating the initial pressure inside a biological object using
measurements of the pressure wave made on a surface surrounding the object.
This article presents error estimates for the time reversal method in the cases
of variable, non-trapping sound speeds.Comment: 16 pages, 6 figures, expanded "Remarks and Conclusions" section,
added one figure, added reference
Stimulus-dependent maximum entropy models of neural population codes
Get PDFNeural populations encode information about their stimulus in a collective
fashion, by joint activity patterns of spiking and silence. A full account of
this mapping from stimulus to neural activity is given by the conditional
probability distribution over neural codewords given the sensory input. To be
able to infer a model for this distribution from large-scale neural recordings,
we introduce a stimulus-dependent maximum entropy (SDME) model---a minimal
extension of the canonical linear-nonlinear model of a single neuron, to a
pairwise-coupled neural population. The model is able to capture the
single-cell response properties as well as the correlations in neural spiking
due to shared stimulus and due to effective neuron-to-neuron connections. Here
we show that in a population of 100 retinal ganglion cells in the salamander
retina responding to temporal white-noise stimuli, dependencies between cells
play an important encoding role. As a result, the SDME model gives a more
accurate account of single cell responses and in particular outperforms
uncoupled models in reproducing the distributions of codewords emitted in
response to a stimulus. We show how the SDME model, in conjunction with static
maximum entropy models of population vocabulary, can be used to estimate
information-theoretic quantities like surprise and information transmission in
a neural population.Comment: 11 pages, 7 figure
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