Block partitions: an extended view

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a partition of $S$ into $n$ blocks $B_1, \dots , B_n$ with $|b_i - b_j| \le 1$ for every $i, j$. In this paper, we consider a generalization of the problem in higher dimensions

On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by nu(H) the number of vertices of H and by tau(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n > 2 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) - tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively

On the Shape of the Tail of a Two Dimensional Sand Pile

We study the shape of the tail of a heap of granular material. A simple theoretical argument shows that the tail adds a logarithmic correction to the slope given by the angle of repose. This expression is in good agreement with experiments. We present a cellular automaton that contains gravity, dissipation and surface roughness and its simulation also gives the predicted shape.Comment: LaTeX file 4 pages, 4 PS figures, also available at http://pmmh.espci.fr

Filling a silo with a mixture of grains: Friction-induced segregation

We study the filling process of a two-dimensional silo with inelastic particles by simulation of a granular media lattice gas (GMLG) model. We calculate the surface shape and flow profiles for a monodisperse system and we introduce a novel generalization of the GMLG model for a binary mixture of particles of different friction properties where, for the first time, we measure the segregation process on the surface. The results are in good agreement with a recent theory, and we explain the observed small deviations by the nonuniform velocity profile.Comment: 10 pages, 5 figures, to be appear in Europhys. Let

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to permanent chaos, results obtained from the Poincare map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincare surface becomes relevant. However, after introducing a true-time Poincare map, quantities known from the usual Poincare map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincare map.Comment: 4 pages in Revtex with 2 included postscript figures, submitted to Phys. Rev.

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

Fast flowing populations are not well mixed

In evolutionary dynamics, well-mixed populations are almost always associated with all-to-all interactions; mathematical models are based on complete graphs. In most cases, these models do not predict fixation probabilities in groups of individuals mixed by flows. We propose an analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.Comment: 19 pages, 8 figure