Extremal polynomials in stratified groups

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials tre related to a new algebraic characterization of abnormal sub-Riemannian extremals in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations, in both normal and abnormal case

Force sensor for chameleon and Casimir force experiments with parallel-plate configuration

The search for non-Newtonian forces has been pursued following many different paths. Recently it was suggested that hypothetical chameleon interactions, which might explain the mechanisms behind dark energy, could be detected in a high-precision force measurement. In such an experiment, interactions between parallel plates kept at constant separation could be measured as a function of the pressure of an ambient gas, thereby identifying chameleon interactions by their unique inverse dependence on the local mass density. During the past years we have been developing a new kind of setup complying with the high requirements of the proposed experiment. In this article we present the first and most important part of this setup -- the force sensor. We discuss its design, fabrication, and characterization. From the results of the latter we derive limits on chameleon interaction parameters that could be set by the forthcoming experiment. Finally, we describe the opportunity to use the same setup to measure Casimir forces at large surface separations with unprecedented accuracy, thereby potentially giving unambiguous answers to long standing open questions

Effective thermal dynamics following a quantum quench in a spin chain

We study the nonequilibrium dynamics of the Quantum Ising Model following an abrupt quench of the transverse field. We focus on the on-site autocorrelation function of the order parameter, and extract the phase coherence time $\tau^{\phi}_Q$ from its asymptotic behavior. We show that the initial state determines $\tau^{\phi}_Q$ only through an effective temperature set by its energy and the final Hamiltonian. Moreover, we observe that the dependence of $\tau^{\phi}_Q$ on the effective temperature fairly agrees with that obtained in thermal equilibrium as a function of the equilibrium temperature.Comment: 4 pages, 4 figures. Published versio

Hypercompact stellar clusters: morphological renditions and spectro-photometric models

Numerical relativity predicts that the coalescence of a black hole-binary causes the newly formed black hole to recoil, and evidence for such recoils has been found in the gravitational waves observed during the merger of stellar-mass black holes. Recoiling (super)massive black holes are expected to reside in hypercompact stellar clusters (HCSCs). Simulations of galaxy assembly predict that hundreds of HCSCs should be present in the halo of a Milky Way-type galaxy, and a fraction of those around the Milky Way should have magnitudes within the sensitivity limit of existing surveys. However, recoiling black holes and their HCSCs are still waiting to be securely identified. With the goal of enabling searches through recent and forthcoming databases, we improve over existing literature to produce realistic renditions of HCSCs bound to black holes with a mass of 10$^{5}$ M$_{\odot}$. Including the effects of a population of blue stragglers, we simulate their appearance in Pan-STARRS and in forthcoming $Euclid$ images. We also derive broad-band spectra and the corresponding multi-wavelength colours, finding that the great majority of the simulated HCSCs fall on the colour-colour loci defined by stars and galaxies, with their spectra resembling those of giant K-type stars. We discuss the clusters properties, search strategies, and possible interlopers.Comment: Accepted for publication on MNRAS, 17 pages, 7 figure

A Simple Generative Model of Collective Online Behaviour

Human activities increasingly take place in online environments, providing novel opportunities for relating individual behaviours to population-level outcomes. In this paper, we introduce a simple generative model for the collective behaviour of millions of social networking site users who are deciding between different software applications. Our model incorporates two distinct components: one is associated with recent decisions of users, and the other reflects the cumulative popularity of each application. Importantly, although various combinations of the two mechanisms yield long-time behaviour that is consistent with data, the only models that reproduce the observed temporal dynamics are those that strongly emphasize the recent popularity of applications over their cumulative popularity. This demonstrates---even when using purely observational data without experimental design---that temporal data-driven modelling can effectively distinguish between competing microscopic mechanisms, allowing us to uncover new aspects of collective online behaviour.Comment: Updated, with new figures and Supplementary Informatio