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 $\infty$-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 S3S^3, SO(3)SO(3), SL(2)SL(2) and lens spaces

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ 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 $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ 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-H2_2 inside carbon nanotube

We perform ab-initio Quantum Monte Carlo simulations of para-hydrogen (pH$_2$) at $T=0$ K confined in carbon nanotubes (CNT) of different radii. The radial density profiles show a strong layering of the pH$_2$ 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 $^3$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$_2$ 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 $L^1$ 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