Social Percolation and the Influence of Mass Media

Mass media shift the percolative phase transition observed in the marketing model of Solomon and Weisbuch.Comment: 6 pages including 4 figure

Reply to Comment: Quantum Cryptography Based on Orthogonal States?

This is our Reply to Peres' Comment [quant-ph/9509003] to "Quantum Cryptography Based on Orthogonal States" [Phys. Rev. Lett. 75, 1239 (1995)].Comment: 3 pages, LaTex, no figure

Small and Large Scale Granular Statics

Recent experimental results on the static or quasistatic response of granular materials have been interpreted to suggest the inapplicability of the traditional engineering approaches, which are based on elasto-plastic models (which are elliptic in nature). Propagating (hyperbolic) or diffusive (parabolic) models have been proposed to replace the `old' models. Since several recent experiments were performed on small systems, one should not really be surprised that (continuum) elasticity, a macroscopic theory, is not directly applicable, and should be replaced by a grain-scale (``microscopic'') description. Such a description concerns the interparticle forces, while a macroscopic description is given in terms of the stress field. These descriptions are related, but not equivalent, and the distinction is important in interpreting the experimental results. There are indications that at least some large scale properties of granular assemblies can be described by elasticity, although not necessarily its isotropic version. The purely repulsive interparticle forces (in non-cohesive materials) may lead to modifications of the contact network upon the application of external forces, which may strongly affect the anisotropy of the system. This effect is expected to be small (in non-isostatic systems) for small applied forces and for pre-stressed systems (in particular for disordered systems). Otherwise, it may be accounted for using a nonlinear, incrementally elastic model, with stress-history dependent elastic moduli. Although many features of the experiments may be reproduced using models of frictionless particles, results demonstrating the importance of accounting for friction are presented.Comment: 10 pages, 9 figures. Accepted for publication in "Granular Matter" (special issue: 4th Int. Conf. on Conveying and Handling of Particulate Solids, Budapest, Hungary, May 2003). v2: Minor revisions to text and figure

On the Microscopic Foundations of Elasticity

The modeling of the elastic properties of disordered or nanoscale solids requires the foundations of the theory of elasticity to be revisited, as one explores scales at which this theory may no longer hold. The only cases for which microscopically based derivations of elasticity are documented are (nearly) uniformly strained lattices. A microscopic approach to elasticity is proposed. As a first step, microscopically exact expressions for the displacement, strain and stress fields are derived. Conditions under which linear elastic constitutive relations hold are studied theoretically and numerically. It turns out that standard continuum elasticity is not self-evident, and applies only above certain spatial scales, which depend on details of the considered system and boundary conditions. Possible relevance to granular materials is briefly discussed.Comment: 6 pages, 5 figures, LaTeX2e with svjour.cls and svepj.clo, submitted to EPJ E, minor error corrected in v

Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations

This paper is focused on the performance analysis of binary linear block codes (or ensembles) whose transmission takes place over independent and memoryless parallel channels. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived. These bounds are applied to various ensembles of turbo-like codes, focusing especially on repeat-accumulate codes and their recent variations which possess low encoding and decoding complexity and exhibit remarkable performance under iterative decoding. The framework of the second version of the Duman and Salehi (DS2) bounds is generalized to the case of parallel channels, along with the derivation of their optimized tilting measures. The connection between the generalized DS2 and the 1961 Gallager bounds, addressed by Divsalar and by Sason and Shamai for a single channel, is explored in the case of an arbitrary number of independent parallel channels. The generalization of the DS2 bound for parallel channels enables to re-derive specific bounds which were originally derived by Liu et al. as special cases of the Gallager bound. In the asymptotic case where we let the block length tend to infinity, the new bounds are used to obtain improved inner bounds on the attainable channel regions under ML decoding. The tightness of the new bounds for independent parallel channels is exemplified for structured ensembles of turbo-like codes. The improved bounds with their optimized tilting measures show, irrespectively of the block length of the codes, an improvement over the union bound and other previously reported bounds for independent parallel channels; this improvement is especially pronounced for moderate to large block lengths.Comment: Submitted to IEEE Trans. on Information Theory, June 2006 (57 pages, 9 figures

Hardness Amplification of Optimization Problems

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products. We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows: If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27. As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium

Particle displacements in the elastic deformation of amorphous materials: local fluctuations vs. non-affine field

We study the local disorder in the deformation of amorphous materials by decomposing the particle displacements into a continuous, inhomogeneous field and the corresponding fluctuations. We compare these fields to the commonly used non-affine displacements in an elastically deformed 2D Lennard-Jones glass. Unlike the non-affine field, the fluctuations are very localized, and exhibit a much smaller (and system size independent) correlation length, on the order of a particle diameter, supporting the applicability of the notion of local "defects" to such materials. We propose a scalar "noise" field to characterize the fluctuations, as an additional field for extended continuum models, e.g., to describe the localized irreversible events observed during plastic deformation.Comment: Minor corrections to match the published versio