Homogeneity degree of fans

The homogeneity degree of a topological space $X$ is the number of orbits of the action of the homeomorphism group of $X$ on $X$. We initiate a study of dendroids of small homogeneity degree, beginning with fans. We classify all smooth fans of homogeneity degree $3$, and discuss non-smooth fans and prove some results on degree $4$

On the elimination of the sweeping interactions from theories of hydrodynamic turbulence

In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity. It is shown that both the elimination of the sweeping interactions and the derivation of the 4/5-law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity. The quasi-Lagrangian transformation, on the other hand, requires an even stronger homogeneity assumption which is many-time rather than one-time but still weaker than many-time global homogeneity. We argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi-Lagrangian transformation.Comment: v1: submitted to Physica D. v2: major revisions; resubmitted to Physica D. v3: minor revisions requested by referee

Homogeneity of Bilayer Graphene

We present non-linear transport measurements on suspended, current annealed bilayer graphene devices. Using a multi-terminal geometry we demonstrate that devices tend to be inhomogeneous and host two different electronic phases next to each other. Both of these phases show gap-like features of different magnitude in non-linear transport at low charge carrier densities, as already observed in previous studies. Here, we investigate the magnetic field dependence and find that both features grow with increasing field, the smaller one with 0.6 meV/T, the larger one with a 5-10 times higher field dependence. We attribute the larger of the two gaps to an interaction induced broken symmetry state and the smaller one to localization in the more disordered parts of the device.Comment: 9 pages, 4 figures, published versio