The Brin-Thompson groups sV are of type F_\infty

We prove that the Brin-Thompson groups sV, also called higher dimensional Thompson's groups, are of type F_\infty for all natural numbers s. This result was previously shown for s up to 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is easier to analyze.Comment: Final version, in Pacific J. Math., 10 pages, 4 figure

Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models

We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold within an error that is exponentially small with respect to the small parameter measuring time scale separation. Second, we apply this result to the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. The traffic jams are waves that travel slowly against the direction of traffic. Equation-free analysis enables us to investigate the behavior of the microscopic traffic model on a macroscopic level. The standard deviation of cars' headways is chosen as the macroscopic measure of the underlying dynamics such that traveling wave solutions correspond to equilibria on the macroscopic level in the equation-free setup. The collapse of the traffic jam to the free flow then corresponds to a saddle-node bifurcation of this macroscopic equilibrium. We continue this bifurcation in two parameters using equation-free analysis.Comment: 35 page

Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems

This is the author accepted manuscript. The final version is available from SIAM via the DOI in this record.41 pages of supplementary material available at https://doi.org/10.6084/m9.figshare.6166421A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): first, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time-scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems. More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp. The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.J. Sieber’s research was supported by funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement number 643073, by the EPSRC Centre for Predictive Modelling in Healthcare (Grant Number EP/N014391/1) and by the EPSRC Fellowship EP/N023544/1. C. Marschler and J. Starke would like to thank Civilingeniør Frederik Christiansens Almennyttige Fond for financial support. J. Starke would also like to thank the Villum Fonden (VKR-Centre of Excellence Ocean Life), the Technical University of Denmark and Queen Mary University of London for financial support

Use Your Strategic Entrepreneurs to Build Your Strategic Partnerships

Internationalisation through strategic partnerships is a goal for many higher education institutions and their upper-level management teams. Yet for institutional objectives to truly flourish, they should get the most out of the various skills that different actors bring to be table. This piece explores the interesting role that can be played by resourceful academic staff in materialising institutional, and individual, aims