20,040 research outputs found
Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions
If is the solution to the stochastic porous media equation in
, modelling the self-organized
criticaity and is the critical state, then it is proved that
\int^\9_0m(\cal O\setminus\cal O^t_0)dt<\9, and
\lim_{t\to\9}\int_{\cal O}|X(t)-X_c|d\xi=\ell<\9,\ \mathbb{P}{-a.s.} Here,
is the Lebesgue measure and is the critical region
and a.e.
. If the stochastic Gaussian perturbation has only finitely many
modes (but is still function-valued), \lim_{t\to\9}\int_K|X(t)-X_c|d\xi=0
exponentially fast for all compact with probability one, if
the noise is sufficiently strong. We also recover that in the deterministic
case
Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
We prove the path-by-path well-posedness of stochastic porous media and fast
diffusion equations driven by linear, multiplicative noise. As a consequence,
we obtain the existence of a random dynamical system. This solves an open
problem raised in [Barbu, R\"ockner; JDE, 2011], [Barbu, R\"ockner; JDE,
2018+], and [Gess, AoP, 2014].Comment: 37 page
Multi-valued, singular stochastic evolution inclusions
We provide an abstract variational existence and uniqueness result for
multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert
spaces with general additive and Wiener multiplicative noise. As examples we
discuss certain singular diffusion equations such as the stochastic 1-Laplacian
evolution (total variation flow) in all space dimensions and the stochastic
singular fast diffusion equation. In case of additive Wiener noise we prove the
existence of a unique weak-* mean ergodic invariant measure.Comment: 39 pages, in press: J. Math. Pures Appl. (2013
Menelaus's theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry
Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid's axiomatic basis of Geometry. It is also known as a type of non-Euclidean Geometry, being in many respects similar to Euclidean Geometry
Smarandache's Pedal Polygon Theorem in the Poincare Disc Model of Hyperbolic Geometry
In this note, we present a proof of the hyperbolic a Smarandache's pedal polygon theorem in the Poincare disc model of hyperbolic geometry
Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space
We consider the stochastic reflection problem associated with a self-adjoint
operator and a cylindrical Wiener process on a convex set with nonempty
interior and regular boundary in a Hilbert space . We prove the
existence and uniqueness of a smooth solution for the corresponding elliptic
infinite-dimensional Kolmogorov equation with Neumann boundary condition on
.Comment: Published in at http://dx.doi.org/10.1214/08-AOP438 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Learning Mixtures of Bernoulli Templates by Two-Round EM with Performance Guarantee
Dasgupta and Shulman showed that a two-round variant of the EM algorithm can
learn mixture of Gaussian distributions with near optimal precision with high
probability if the Gaussian distributions are well separated and if the
dimension is sufficiently high. In this paper, we generalize their theory to
learning mixture of high-dimensional Bernoulli templates. Each template is a
binary vector, and a template generates examples by randomly switching its
binary components independently with a certain probability. In computer vision
applications, a binary vector is a feature map of an image, where each binary
component indicates whether a local feature or structure is present or absent
within a certain cell of the image domain. A Bernoulli template can be
considered as a statistical model for images of objects (or parts of objects)
from the same category. We show that the two-round EM algorithm can learn
mixture of Bernoulli templates with near optimal precision with high
probability, if the Bernoulli templates are sufficiently different and if the
number of features is sufficiently high. We illustrate the theoretical results
by synthetic and real examples.Comment: 27 pages, 8 figure
Hierarchical Object Parsing from Structured Noisy Point Clouds
Object parsing and segmentation from point clouds are challenging tasks
because the relevant data is available only as thin structures along object
boundaries or other features, and is corrupted by large amounts of noise. To
handle this kind of data, flexible shape models are desired that can accurately
follow the object boundaries. Popular models such as Active Shape and Active
Appearance models lack the necessary flexibility for this task, while recent
approaches such as the Recursive Compositional Models make model
simplifications in order to obtain computational guarantees. This paper
investigates a hierarchical Bayesian model of shape and appearance in a
generative setting. The input data is explained by an object parsing layer,
which is a deformation of a hidden PCA shape model with Gaussian prior. The
paper also introduces a novel efficient inference algorithm that uses informed
data-driven proposals to initialize local searches for the hidden variables.
Applied to the problem of object parsing from structured point clouds such as
edge detection images, the proposed approach obtains state of the art parsing
errors on two standard datasets without using any intensity information.Comment: 13 pages, 16 figure
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