5,112 research outputs found
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 is the local canonical, respectively,
grand-canonical energy to create a critical droplet and 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
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