7,113 research outputs found
Minimal surfaces in the Heisenberg group
We investigate the minimal surface problem in the three dimensional
Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric.
Using a particular surface measure, we characterize minimal surfaces in terms
of a sub-elliptic partial differential equation and prove an existence result
for the Plateau problem in this setting. Further, we provide a link between our
minimal surfaces and Riemannian constant mean curvature surfaces in H equipped
with different Riemannian metrics approximating the Carnot-Caratheodory metric.
We generate a large library of examples of minimal surfaces and use these to
show that the solution to the Dirichlet problem need not be unique. Moreover,
we show that the minimal surfaces we construct are in fact X-minimal surfaces
in the sense of Garofalo and Nhieu.Comment: 26 pages, 12 figure
On complex singularities of the 2D Euler equation at short times
We present a study of complex singularities of a two-parameter family of
solutions for the two-dimensional Euler equation with periodic boundary
conditions and initial conditions F(p) cos p z + F(q) cos q z in the short-time
asymptotic regime. As has been shown numerically in W. Pauls et al., Physica D
219, 40-59 (2006), the type of the singularities depends on the angle between
the modes p and q. Here we show for the two particular cases of the angle going
to zero and to pi that the type of the singularities can be determined very
accurately, being characterised by the values 5/2 and 3 respectively. In these
two cases we are also able to determine the subdominant corrections.
Furthermore, we find that the geometry of the singularities in these two cases
is completely different, the singular manifold being located "over" different
points in the real domain.Comment: 12 pages, 7 figure
A notion of rectifiability modeled on Carnot groups
We introduce a notion of rectifiability modeled on Carnot groups. Precisely,
we say that a subset E of a Carnot group M and N is a subgroup of M, we say E
is N-rectifiable if it is the Lipschitz image of a positive measure subset of
N. First, we discuss the implications of N-rectifiability, where N is a Carnot
group (not merely a subgroup of a Carnot group), which include
N-approximability and the existence of approximate tangent cones isometric to N
almost everywhere in E. Second, we prove that, under a stronger condition
concerning the existence of approximate tangent cones isomorphic to N almost
everywhere in a set E, that E is N-rectifiable. Third, we investigate the
rectifiability properties of level sets of C^1_N functions, where N is a Carnot
group. We show that for almost every real number t and almost every
noncharacteristic point x in a level set of f, there exists a subgroup T_x of H
and r >0 so that f^{-1}(t) intersected with B_H(x,r) is T_x-approximable at x
and an approximate tangent cone isomorphic to T_x at x.Comment: 27 page
Hierarchical Pancaking: Why the Zel'dovich Approximation Describes Coherent Large-Scale Structure in N-Body Simulations of Gravitational Clustering
To explain the rich structure of voids, clusters, sheets, and filaments
apparent in the Universe, we present evidence for the convergence of the two
classic approaches to gravitational clustering, the ``pancake'' and
``hierarchical'' pictures. We compare these two models by looking at agreement
between individual structures -- the ``pancakes'' which are characteristic of
the Zel'dovich Approximation (ZA) and also appear in hierarchical N-body
simulations. We find that we can predict the orientation and position of N-body
simulation objects rather well, with decreasing accuracy for increasing
large- (small scale) power in the initial conditions. We examined an N-body
simulation with initial power spectrum , and found that a
modified version of ZA based on the smoothed initial potential worked well in
this extreme hierarchical case, implying that even here very low-amplitude long
waves dominate over local clumps (although we can see the beginning of the
breakdown expected for ). In this case the correlation length of the
initial potential is extremely small initially, but grows considerably as the
simulation evolves. We show that the nonlinear gravitational potential strongly
resembles the smoothed initial potential. This explains why ZA with smoothed
initial conditions reproduces large-scale structure so well, and probably why
our Universe has a coherent large-scale structure.Comment: 17 pages of uuencoded postscript. There are 8 figures which are too
large to post here. To receive the uuencoded figures by email (or hard copies
by regular mail), please send email to: [email protected]. This is
a revision of a paper posted earlier now in press at MNRA
Spillover modes in multiplex games: double-edged effects on cooperation, and their coevolution
In recent years, there has been growing interest in studying games on
multiplex networks that account for interactions across linked social contexts.
However, little is known about how potential cross-context interference, or
spillover, of individual behavioural strategy impact overall cooperation. We
consider three plausible spillover modes, quantifying and comparing their
effects on the evolution of cooperation. In our model, social interactions take
place on two network layers: one represents repeated interactions with close
neighbours in a lattice, the other represents one-shot interactions with random
individuals across the same population. Spillover can occur during the social
learning process with accidental cross-layer strategy transfer, or during
social interactions with errors in implementation due to contextual
interference. Our analytical results, using extended pair approximation, are in
good agreement with extensive simulations. We find double-edged effects of
spillover on cooperation: increasing the intensity of spillover can promote
cooperation provided cooperation is favoured in one layer, but too much
spillover is detrimental. We also discover a bistability phenomenon of
cooperation: spillover hinders or promotes cooperation depending on initial
frequencies of cooperation in each layer. Furthermore, comparing strategy
combinations that emerge in each spillover mode provides a good indication of
their co-evolutionary dynamics with cooperation. Our results make testable
predictions that inspire future research, and sheds light on human cooperation
across social domains and their interference with one another
Coevolution of Cooperation and Partner Rewiring Range in Spatial Social Networks
In recent years, there has been growing interest in the study of
coevolutionary games on networks. Despite much progress, little attention has
been paid to spatially embedded networks, where the underlying geographic
distance, rather than the graph distance, is an important and relevant aspect
of the partner rewiring process. It thus remains largely unclear how individual
partner rewiring range preference, local vs. global, emerges and affects
cooperation. Here we explicitly address this issue using a coevolutionary model
of cooperation and partner rewiring range preference in spatially embedded
social networks. In contrast to local rewiring, global rewiring has no distance
restriction but incurs a one-time cost upon establishing any long range link.
We find that under a wide range of model parameters, global partner switching
preference can coevolve with cooperation. Moreover, the resulting partner
network is highly degree-heterogeneous with small average shortest path length
while maintaining high clustering, thereby possessing small-world properties.
We also discover an optimum availability of reputation information for the
emergence of global cooperators, who form distant partnerships at a cost to
themselves. From the coevolutionary perspective, our work may help explain the
ubiquity of small-world topologies arising alongside cooperation in the real
world
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