413 research outputs found
Decomposition Theorems for Automorphism Groups of Trees
Motivated by the Bruhat and Cartan decompositions of general linear groups
over local fields, double cosets of the group of label preserving automorphisms
of a label-regular tree over the fixator of an end of the tree and over maximal
compact open subgroups are enumerated. This enumeration is used to show that
every continuous homomorphism from the automorphism group of a label-regular
tree has closed range
Rethinking tipping points in spatial ecosystems
The theory of alternative stable states and tipping points has garnered a lot
of attention in the last decades. It predicts potential critical transitions
from one ecosystem state to a completely different state under increasing
environmental stress. However, typically ecosystem models that predict tipping
do not resolve space explicitly. As ecosystems are inherently spatial, it is
important to understand the effects of incorporating spatial processes in
models, and how those insights translate to the real world. Moreover, spatial
ecosystem structures, such as vegetation patterns, are important in the
prediction of ecosystem response in the face of environmental change. Models
and observations from real savanna ecosystems and drylands have suggested that
they may exhibit both tipping behavior as well as spatial pattern formation.
Hence, in this paper, we use mathematical models of humid savannas and drylands
to illustrate several pattern formation phenomena that may arise when
incorporating spatial dynamics in models that exhibit tipping without resolving
space. We argue that such mechanisms challenge the notion of large scale
critical transitions in response to global change and reveal a more resilient
nature of spatial ecosystems
Axial perturbations of general spherically symmetric spacetimes
The aim of this paper is to present a governing equation for first order
axial metric perturbations of general, not necessarily static, spherically
symmetric spacetimes. Under the non-restrictive assumption of axisymmetric
perturbations, the governing equation is shown to be a two-dimensional wave
equation where the wave function serves as a twist potential for the
axisymmetry generating Killing vector. This wave equation can be written in a
form which is formally a very simple generalization of the Regge-Wheeler
equation governing the axial perturbations of a Schwarzschild black hole, but
in general the equation is accompanied by a source term related to matter
perturbations. The case of a viscous fluid is studied in particular detail.Comment: 16 pages, no figures, minor correction
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