252 research outputs found

### A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for
the primary visual cortex of mammals. This model is neurophysiologically
justified. Further developments of this theory lead to efficient algorithms for
image reconstruction, based upon the consideration of an associated
hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or
certain of its improvements) is a left-invariant structure over the group
$SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
$SE(2,N)$, restricting to a finite number of rotations. This apparently very
simple group is in fact quite atypical: it is maximally almost periodic, which
leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this
semi-discrete model, we improve on previous image-reconstruction algorithms and
we develop a pattern-recognition theory that leads also to very efficient
algorithms in practice.Comment: 123 pages, revised versio

### Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models

We consider least energy solutions to the nonlinear equation $-\Delta_g
u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$
which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds
with unbounded sectional geometry. Partial symmetry and existence of least
energy solutions is proved for quite general nonlinearities $f(r,u)$, where $r$
denotes the geodesic distance from the pole of $M$

### On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables

We discuss various features and details of two versions of the Barrett-Crane
spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian
model and second of the SL(2,C)-symmetric Lorentzian version in which all
tetrahedra are space-like. Recently, Livine and Oriti proposed to introduce a
causal structure into the Lorentzian Barrett--Crane model from which one can
construct a path integral that corresponds to the causal (Feynman) propagator.
We show how to obtain convergent integrals for the 10j-symbols and how a
dimensionless constant can be introduced into the model. We propose a `Wick
rotation' which turns the rapidly oscillating complex amplitudes of the Feynman
path integral into positive real and bounded weights. This construction does
not yet have the status of a theorem, but it can be used as an alternative
definition of the propagator and makes the causal model accessible by standard
numerical simulation algorithms. In addition, we identify the local symmetries
of the models and show how their four-simplex amplitudes can be re-expressed in
terms of the ordinary relativistic 10j-symbols. Finally, motivated by possible
numerical simulations, we express the matrix elements that are defined by the
model, in terms of the continuous connection variables and determine the most
general observable in the connection picture. Everything is done on a fixed
two-complex.Comment: 22 pages, LaTeX 2e, 1 figur

### Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity

The Barrett-Crane models of Lorentzian quantum gravity are a family of spin
foam models based on the Lorentz group. We show that for various choices of
edge and face amplitudes, including the Perez-Rovelli normalization, the
amplitude for every triangulated closed 4-manifold is a non-negative real
number. Roughly speaking, this means that if one sums over triangulations,
there is no interference between the different triangulations. We prove
non-negativity by transforming the model into a ``dual variables'' formulation
in which the amplitude for a given triangulation is expressed as an integral
over three copies of hyperbolic space for each tetrahedron. Then we prove that,
expressed in this way, the integrand is non-negative. In addition to implying
that the amplitude is non-negative, the non-negativity of the integrand is
highly significant from the point of view of numerical computations, as it
allows statistical methods such as the Metropolis algorithm to be used for
efficient computation of expectation values of observables.Comment: 13 page

### An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

We prove \emph{optimal} improvements of the Hardy inequality on the
hyperbolic space. Here, optimal means that the resulting operator is critical
in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated
inequality cannot be further improved. Such inequalities arise from more
general, \emph{optimal} ones valid for the operator $P_{\lambda}:=
-\Delta_{\mathbb{H}^N} - \lambda$ where $0 \leq \lambda \leq
\lambda_{1}(\mathbb{H}^N)$ and $\lambda_{1}(\mathbb{H}^N)$ is the bottom of the
$L^2$ spectrum of $-\Delta_{\mathbb{H}^N}$, a problem that had been studied in
[J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator
$P_{\lambda_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on
$\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact
greater generality since there are shown on general Cartan-Hadamard manifolds
under curvature assumptions, possibly depending on the point.
Existence/nonexistence of extremals for the related Hardy-Poincar\'e
inequalities are also proved using concentration-compactness technique and a
Liouville comparison theorem. As applications of our inequalities we obtain an
improved Rellich inequality and we derive a quantitative version of
Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$Comment: Final Versio

### Stanilov-Tsankov-Videv Theory

We survey some recent results concerning Stanilov-Tsankov-Videv theory,
conformal Osserman geometry, and Walker geometry which relate algebraic
properties of the curvature operator to the underlying geometry of the
manifold.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA

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