Stress Propagation and Arching in Static Sandpiles

We present a new approach to the modelling of stress propagation in static granular media, focussing on the conical sandpile constructed from a point source. We view the medium as consisting of cohesionless hard particles held up by static frictional forces; these are subject to microscopic indeterminacy which corresponds macroscopically to the fact that the equations of stress continuity are incomplete -- no strain variable can be defined. We propose that in general the continuity equations should be closed by means of a constitutive relation (or relations) between different components of the (mesoscopically averaged) stress tensor. The primary constitutive relation relates radial and vertical shear and normal stresses (in two dimensions, this is all one needs). We argue that the constitutive relation(s) should be local, and should encode the construction history of the pile: this history determines the organization of the grains at a mesoscopic scale, and thereby the local relationship between stresses. To the accuracy of published experiments, the pattern of stresses beneath a pile shows a scaling between piles of different heights (RSF scaling) which severely limits the form the constitutive relation can take ...Comment: 38 pages, 24 Postscript figures, LATEX, minor misspellings corrected, Journal de Physique I, Ref. Nr. 6.1125, accepte

Quasivariational solutions for first order quasilinear equations with gradient constraint

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable {\em a priori} estimates. We obtain also the existence of stationary solutions, by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results

Introduction to tropical series and wave dynamic on them

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves

Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit toru

Scaling limit of the odometer in divisible sandpiles

In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus

Scaling limit of the odometer in divisible sandpiles

In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems