2,250 research outputs found
Wavelength selection beyond Turing
Spatial patterns arising spontaneously due to internal processes are
ubiquitous in nature, varying from regular patterns of dryland vegetation to
complex structures of bacterial colonies. Many of these patterns can be
explained in the context of a Turing instability, where patterns emerge due to
two locally interacting components that diffuse with different speeds in the
medium. Turing patterns are multistable, such that many different patterns with
different wavelengths are possible for the same set of parameters, but in a
given region typically only one such wavelength is dominant. In the Turing
instability region, random initial conditions will mostly lead to a wavelength
that is similar to that of the leading eigenvector that arises from the linear
stability analysis, but when venturing beyond, little is known about the
pattern that will emerge. Using dryland vegetation as a case study, we use
different models of drylands ecosystems to study the wavelength pattern that is
selected in various scenarios beyond the Turing instability region, focusing
the phenomena of localized states and repeated local disturbances
Grothendieck's theorem on non-abelian H^2 and local-global principles
A theorem of Grothendieck asserts that over a perfect field k of
cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic
groups are trivial. The purpose of this paper is to establish a formally real
generalization of this theorem. The generalization -- to the context of perfect
fields of virtual cohomological dimension one -- takes the form of a
local-global principle for the H^2-sets with respect to the orderings of the
field. This principle asserts in particular that an element in H^2 is neutral
precisely when it is neutral in the real closure with respect to every ordering
in a dense subset of the real spectrum of k. Our techniques provide a new proof
of Grothendieck's original theorem. An application to homogeneous spaces over k
is also given.Comment: 22 pages, AMS-TeX; accepted for publication by the Journal of the AM
Trees and Markov convexity
We show that an infinite weighted tree admits a bi-Lipschitz embedding into
Hilbert space if and only if it does not contain arbitrarily large complete
binary trees with uniformly bounded distortion. We also introduce a new metric
invariant called Markov convexity, and show how it can be used to compute the
Euclidean distortion of any metric tree up to universal factors
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