11,102 research outputs found
Large time behavior for the heat equation on Carnot groups
We first generalize a decomposition of functions on Carnot groups as linear
combinations of the Dirac delta and some of its derivatives, where the weights
are the moments of the function.
We then use the decomposition to describe the large time behavior of
solutions of the hypoelliptic heat equation on Carnot groups. The solution is
decomposed as a weighted sum of the hypoelliptic fundamental kernel and its
derivatives, the coefficients being the moments of the initial datum
A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit
We prove existence and uniqueness of solutions to a transport equation
modelling vehicular traffic in which the velocity field depends non-locally on
the downstream traffic density via a discontinuous anisotropic kernel. The
result is obtained recasting the problem in the space of probability measures
equipped with the -Wasserstein distance. We also show convergence of
solutions of a finite dimensional system, which provide a particle method to
approximate the solutions to the original problem
Invariant Carnot-Caratheodory metrics on , , and lens spaces
In this paper we study the invariant Carnot-Caratheodory metrics on
, and induced by their Cartan decomposition
and by the Killing form. Beside computing explicitly geodesics and conjugate
loci, we compute the cut loci (globally) and we give the expression of the
Carnot-Caratheodory distance as the inverse of an elementary function. We then
prove that the metric given on projects on the so called lens spaces
. Also for lens spaces, we compute the cut loci (globally).
For the cut locus is a maximal circle without one point. In all other
cases the cut locus is a stratified set. To our knowledge, this is the first
explicit computation of the whole cut locus in sub-Riemannian geometry, except
for the trivial case of the Heisenberg group
Superfluid behavior of quasi-1D p-H inside carbon nanotube
We perform ab-initio Quantum Monte Carlo simulations of para-hydrogen
(pH) at K confined in carbon nanotubes (CNT) of different radii. The
radial density profiles show a strong layering of the pH molecules which
grow, with increasing number of molecules, in solid concentric cylindrical
shells and eventually a central column. The central column can be considered an
effective one-dimensional (1D) fluid whose properties are well captured by the
Tomonaga-Luttinger liquid theory. The Luttinger parameter is explicitly
computed and interestingly it shows a non-monotonic behavior with the linear
density similar to what found for pure 1D He. Remarkably, for the central
column in a (10,10) CNT, we found an ample linear density range in which the
Luttinger liquid is (i) superfluid and (ii) stable against a weak disordered
external potential, as the one expected inside realistic pores. This superfluid
behavior could be experimentally revealed in bundles of carbon nanotubes, where
deviations from classical inertial values associated with superfluid density
could be measured via torsional oscillator techniques. In summary, our results
suggest that pH within carbon nanopores could be a practical realization of
the long sought-after, elusive superfluid phase of parahydrogen.Comment: 5 pages, 3 figures accepted as PRB rapi
Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes
Motivated by pedestrian modelling, we study evolution of measures in the
Wasserstein space. In particular, we consider the Cauchy problem for a
transport equation, where the velocity field depends on the measure itself.
We deal with numerical schemes for this problem and prove convergence of a
Lagrangian scheme to the solution, when the discretization parameters approach
zero. We also prove convergence of an Eulerian scheme, under more strict
hypotheses. Both schemes are discretizations of the push-forward formula
defined by the transport equation. As a by-product, we obtain existence and
uniqueness of the solution.
All the results of convergence are proved with respect to the Wasserstein
distance. We also show that spaces are not natural for such equations,
since we lose uniqueness of the solution
Negotiating urban change. Strategies and tactics of patrimonialization in Hackney Wick, East London
The heritage category function of building collective identity, condensing in specific goods,
tangible or intangible, a set of locally significant values and practices, has been often pointed
out to show its use in managing relations between the authorities and local communities
(Simonicca 2015).
This happens nowadays with a particularly conscious approach in urban planning, not only
in the practices and rhetorics of institutions, but also in the quotidian experience of people
who develop their own informal planning. In this discussion, the heritage arguments become
an appropriate tool to handle and shape the change, especially when these processes
have a deep impact in the everyday lives of communities and territories, like in cases of
gentrification
Mean-Field Pontryagin Maximum Principle
International audienceWe derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov-type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables
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