A strong law of large numbers for branching processes: almost sure spine events

We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.Comment: 6 page

Tarsal Tunnel Syndrome – a narrative review of the literature

International audienceThe effect of the mean effective stress on the elastic properties of unbound granular materials is a well-known experimental result. Power laws between the mean effective stress and the shear modulus G vhmax of three natural sands are established for isotropic stress paths using bender elements. Triaxial test results reveal that such power laws are also suitable for contracting deviatoric stress paths whereas it is no longer the case for dilating deviatoric stress paths. Fabric changes during shearing are therefore highlighted. These observations seem to be a typical feature of the behavior of granular materials

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.Comment: 56 pages, 1 figur

Investigating GNSS multipath effects induced by co-located Radar Corner Reflectors

Abstract Radar Corner Reflectors (CR) are increasingly used as reference targets for land surface deformation measurements with the Interferometric Synthetic Aperture Radar (InSAR) technique. When co-located with ground-based Global Navigation Satellite Systems (GNSS) infrastructure, InSAR observations at CR can be used to integrate relative measurements of surface deformation into absolute reference frames defined by GNSS. However, CR are also a potential source of GNSS multipath effects and may therefore have a detrimental effect on the GNSS observations. In this study, we compare daily GNSS coordinate time series and 30-second signal-to-noise ratio (SNR) observations for periods before and after CR deployment at a GNSS site. We find that neither the site coordinates nor the SNR values are significantly affected by the CR deployment, with average changes being within 0.1 mm for site coordinates and within 1 % for SNR values. Furthermore, we generate empirical site models by spatially stacking GNSS observation residuals to visualise and compare the spatial pattern in the surroundings of GNSS sites. The resulting stacking maps indicate oscillating patterns at elevation angles above 60 degrees which can be attributed to the CR deployed at the analysed sites. The effect depends on the GNSS antenna used at a site with the magnitude of multipath patterns being around three times smaller for a high-quality choke ring antenna compared to a ground plane antenna without choke rings. In general, the CR-induced multipath is small compared to multipath effects at other GNSS sites located in a different environment (e. g. mounted on a building)