Consider a lipid membrane with a free exposed edge. The energy describing this membrane is quadratic in the extrinsic curvature of its geometry; that describing the edge is proportional to its length. In this note we determine the boundary conditions satisfied by the equilibria of the membrane on this edge, exploiting variational principles. The derivation is free of any assumptions on the symmetry of the membrane geometry. With respect to earlier work for axially symmetric configurations, we discover the existence of an additional boundary condition which is identically satisfied in that limit. By considering the balance of the forces operating at the edge, we provide a physical interpretation for the boundary conditions. We end with a discussion of the effect of the addition of a Gaussian rigidity term for the membrane.

[1] The world's largest observed Slow Slip Events (SSE) occurred in 2001-2002 and 2006 in the Guerrero subduction zone, Mexico. Using an improved GPS processing that accounts for time-varying atmospheric phenomena as well as oceanic, atmospheric and hydrologic loading corrections, the 11 year GPS position time series in Guerrero show a noise reduction of ∼50% with respect to previous studies. Thanks to the improved position time series and, in particular, the simultaneous analysis of the three-dimensional GPS observations, we can provide new information about SSEs in the studied area. First, we detect seven nonperiodic anomalous displacements with subcentimeter amplitude, but no quasi-annual anomalies as proposed previously. The displacements seem to occur simultaneously with the observed peaks of non-volcanic tremor activity in the area. Second, we refine the characteristics of the two major SSEs in terms of timing, duration, and cumulative displacements, and highlight the complex surface spatiotemporal evolution of the displacements during these SSEs. In particular, we observe a clear initiation phase for the 2006 SSE as well as ending phases for both large SSEs. The ending phase shows a strong deceleration of the anomalous displacements with respect to the main displacement phase already observed, for the 2001-2002 and 2006 SSEs. The duration of the SSEs increases by 30-40% including the initiation and ending phases. For the 2006 SSE, the main displacement phase also shows spatiotemporal complexity. Our results demonstrate the need for improved three-dimensional GPS processing technique in order to undertake detailed studies of SSEs.

Consider a closed lipid membrane (vesicle), modeled as a two-dimensional surface, described by a geometrical hamiltonian that depends on its extrinsic curvature. The vanishing of its first variation determines the equilibrium configurations for the system. In this paper, we examine the second variation of the hamiltonian about any given equilibrium, using an explicitly surface covariant geometrical approach. We identify the operator which determines the stability of equilibrium configurations.

In this work we show that corrections to the Newtons second law appears if we assume that the phase space has a symplectic structure consistent with the rules of commutation of the noncommutative quantum mechanics. In the central field case we find that the correction term breaks the rotational symmetry. In particular, for the Kepler problem, this term takes the form of a Coriolis force produced by the weak gravitational field far from a rotating massive object.

The aseismic slow slip event of [2001][2002] in Guerrero, Mexico, with an equivalent magnitude M W ∼ 7.5, is the largest silent earthquake (SQ) among many recently recorded by GPS in different subduction zones (i.e. Japan, Alaska, Cascadia, New Zealand). The sub-horizontal and shallow plate interface in Central Mexico is responsible for specific conditions for the ∼100 km long extended transient zone where the SQs develop from ∼80 to ∼190 km inland from the trench. This wide transient zone and relatively large slow slips of 10 to 20 cm displacements on the subduction fault result in noticeable surface displacements of 5-6 cm during the SQs. Continuous GPS stations allow one to trace the propagation of SQs, and to estimate their arrival time, duration and geometric attenuation. These propagation parameters must be accounted in order to locate source of slow slips events and to understand the triggering effect that they have on large subduction earthquakes. We use longbaseline tiltmeter data to define new time limits This modeling shows that the SQ ceased gradually in the central part of the Oaxaca segment of the subduction zone (west of Puerto Angel, PUAN) and then it apparently triggered another SQ in SE Oaxaca (between PUAN and Salina Cruz, SACR). The estimated horizontal velocities for inter-event epochs at each GPS site are used to assess an average interplate coupling in the Central Oaxaca subduction zone.