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-$k$ (small scale) power in the initial conditions. We examined an N-body simulation with initial power spectrum $P(k) \propto k^3$, 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 $k^4$). 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