Independence entropy of \mathbb{Z}^{d}-shift spaces

We introduce a concept of independence entropy for symbolic dynamical systems. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraint defined by the shift space. We show that for a certain class of one-dimensional shift spaces X, the independence entropy coincides with the limiting, as d tends to infinity, topological entropy of the dimensional shift defined by imposing the constraints of X in each of the d cardinal directions. This is of interest because for these shift spaces independence entropy is easy to compute. Thus, while in these cases, the topological entropy of the d-dimensional shift (d≥2) is difficult to compute, the limiting topological entropy is easy to compute. In some cases, we also compute the rate of convergence of the sequence of d-dimensional entropies. This work generalizes earlier work on constrained systems with unconstrained positions.     

The Social Amplifier—Reaction of Human Communities to Emergencies

This paper develops a methodology to aggregate signals in a network regarding some hidden state of the world. We argue that focusing on edges around hubs will under certain circumstances amplify the faint signals disseminating in a network, allowing for more efficient detection of that hidden state. We apply this method to detecting emergencies in mobile phone data, demonstrating that under a broad range of cases and a constraint in how many edges can be observed at a time, focusing on the egocentric networks around key hubs will be more effective than sampling random edges. We support this conclusion analytically, through simulations, and with analysis of a dataset containing the call log data from a major mobile carrier in a European nation.     

Subexponential decay of correlations

We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young’s upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator. Oblatum 14-VII-2001 & 5-VI-2002?Published online: 6 August 2002 Dedicated to B. Weiss on the occasion of his 60th birthday RID="*" ID="*"This research was supported by the Nuffield Foundation, NUF-NAL 00.     

Computer-generated infrared depolarizer using space-variant subwavelength dielectric gratings

We present a novel method for the formation of a complete depolarizer that is based on a polarization-state scrambling procedure over the space domain. Such an element can be achieved by use of cascaded, computer-generated, space-variant subwavelength dielectric gratings. We introduce a theoretical analysis and experimentally demonstrate a depolarizer for infrared radiation at a wavelength of 10.6 microm.     

Breakdown of the internet under intentional attack

We study the tolerance of random networks to intentional attack, whereby a fraction p of the most connected sites is removed. We focus on scale-free networks, having connectivity distribution P(k) approximately k(-alpha), and use percolation theory to study analytically and numerically the critical fraction p(c) needed for the disintegration of the network, as well as the size of the largest connected cluster. We find that even networks with alpha < or = 3, known to be resilient to random removal of sites, are sensitive to intentional attack. We also argue that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, M, as square root of [M], rather than as log (k)M, as expected for random networks away from criticality.     

Thermodynamic formalism for null recurrent potentials

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(?) denote the Gurevic pressure of ? and letL _? be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL _? ^* ν=e ^P(?)ν,L _? h=e ^P(?) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL _? ^k , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.     

Invariant Radon measures for horocycle flows on Abelian covers

We classify the ergodic invariant Radon measures for horocycle flows on ^d–covers of compact Riemannian surfaces of negative curvature, thus proving a conjecture of M. Babillot and F. Ledrappier. An important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary. Dedicated to the memory of M. Babillot     

On standard integral table algebras generated by an element of width 2

The paper studies standard integral table algebras containing an element of width two. It is shown that the table subalgebra generated by such an element is exactly isomorphic to one of the table algebras described in Main Theorems 1.1. and 1.2. The contribution of this author is a part of his Ph.D. thesis at Bar-Ilan University under the supervision of Z. Arad and M. Muzychuk. This author was partially supported by the Israeli Ministry of Absorption.     

Rotating vectorial vortices produced by space-variant subwavelength gratings

A new class of vectorial vortex based on coherent addition of two orthogonal circularly polarized Bessel beams of identical order but with different propagation constants is presented. The transversely space-variant axially symmetric polarization distributions of these vectorial fields rotate as they propagate, while they maintain a propagation-invariant Bessel intensity distribution. These properties were demonstrated by use of discrete space-variant subwavelength gratings for 10.6 microm CO2 laser radiation. The polarization properties were verified by both full space-variant polarization analysis and measurements. Rotating intensity patterns are also demonstrated by transmitting the vectorial vortices through a linear polarizer.