Galois groups of Schubert problems via homotopy computation

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let \b denote the inverse temperature and let \L_\b \subset \Z^2 be a square box with periodic boundary conditions such that \lim_{\b\to\infty}|\L_\b|=\infty. We run the dynamics on \L_\b starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large \b, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in \L_\b. Using potential-theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as \b\to\infty. It turns out that this time grows as Ke^{\Gamma\b}/|\L_\b| for Glauber dynamics and K\b e^{\Gamma\b}/|\L_\b| for Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |\L_\b|). The fact that the average nucleation time is inversely proportional to |\L_\b| is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition \L_\b.Comment: 45 pages, 11 figure

Bounds and Estimates for the Response to Correlated Fluctuations in Asymmetric Complex Networks

We study the spreading of correlated fluctuations through networks with asymmetric and weighted coupling. This can be found in many real systems such as renewable power grids. These systems have so far only been studied numerically. By formulating a network adapted linear response theory, we derive an analytic bound for the response. For colored we find that vulnerability patterns noise are linked to the left Laplacian eigenvectors of the overdamped modes. We show for a broad class of tree-like flow networks, that fluctuations are enhanced in the opposite direction of the flow. This novel mechanism explains vulnerability patterns that were observed in realistic simulations of renewable power grids

Empirical study of the influence of social groups in evacuation scenarios

The effects of social groups on pedestrian dynamics, especially in evacuation scenarios, have attracted some interest recently. However, due to the lack of reliable empirical data, most of the studies focussed on modelling aspects. It was shown that social groups can have a considerable effect, e.g. on evacuation times. In order to test the model predictions we have performed laboratory experiments of evacuations with different types and sizes of the social groups. The experiments have been performed with pupils of different ages. Parameters that have been considered are (1) group size, (2) strength of intra-group interactions, and (3) composition of the groups (young adults, children, and mixtures). For all the experiments high-quality trajectories for all participants have been obtained using the PeTrack software. This allows for a detailed analysis of the group effects. One surprising observation is a decrease of the evacuation time with increasing group size.Comment: 8 pages, 4 figures, to be published in Traffic and Granular Flow '15 (Springer, 2016

Aurora Volume 12

College formerly located at Olivet, Illinois and known as Olivet University, 1912-1923; Olivet College, 1923-1939, Olivet Nazarene College, 1940-1986, Olivet Nazarene University, 1986-https://digitalcommons.olivet.edu/arch_yrbks/1014/thumbnail.jp