An extension which is relatively twofold mixing but not threefold mixing

We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold

Notes on Austin's multiple ergodic theorem

The purpose of this note is to present my understanding of Tim Austin's proof of the multiple ergodic theorem for commuting transformations, emphasizing on the use of joinings, extensions and factors. The existence of a sated extension, which is a key argument in the proof, is presented in a general context

Notes on Polyphylia Harris with a description of a new species (Coleoptera: Scarabaeidae: Melolonthinae)

Polyphylla aeolus La Rue, new species, is described from the Kelso Sand Dunes, Mojave Desert, San Bernardino County, California, U.S.A. Illustrations of dorsal habitus, significant morphological details, and adult genitalic forms are provided. A description of the type locality including geographical and ecological parameters is presented. Taxonomic problems within the genus and limitations of recently proposed methods of species identification are discussed. A modified key is provided to distinguish the new species. The heretofore undescribed females of Polyphylla anteronivea Hardy, P. mescalerensis Young, P. nubila Van Dyke, and P. pottsorum Hardy, are described. The larval host of P. erratica Hardy is reported, and the adult female is redescribed from pristine specimens. A dorsal habitus illustration of each female is provided. Bionomic and distributional data are presented for Polyphylla avittata Hardy, P. cavifrons LeConte, P. hirsuta Van Dyke, P. monahansensisHardy, P. petiti Guerin, P. stellata Young, andP. squamiventris Cazier

2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process indexed by Z, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the so-called "3-dot system", but in the context of stationary random fields indexed by ZxZ. In this work, we first present an attempt to adapt Ledrappier's construction to the one-dimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the sigma-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes

Zero Krengel Entropy does not kill Poisson Entropy

We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann-Kakutani odometer), but whose associated Poisson suspension has positive entropy

Averaging along Uniform Random Integers

Motivated by giving a meaning to "The probability that a random integer has initial digit d", we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The URI-density, obtained by averaging along the elements of E, and the local URI-density, which we get by considering the k-th element of E and letting k go to infinity. We prove that the elements of E satisfy Benford's law, both in the sense of URI-density and in the sense of local URI-density. Moreover, if b_1 and b_2 are two multiplicatively independent integers, then the mantissae of a natural number in base b_1 and in base b_2 are independent. Connections of URI-density and local URI-density with other well-known notions of densities are established: Both are stronger than the natural density, and URI-density is equivalent to log-density. We also give a stochastic interpretation, in terms of URI-set, of the H_\infty-density

Around King's Rank-One theorems: Flows and Z^n-actions

We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of rank-one flows. In the case of rank-one Z^n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors

Silicon-on-insulator photonic crystal miniature devices with slow light enhanced third-order nonlinearities

The effects of the slow-down factor on third-order nonlinear effects in silicon-on-insulator photonic crystal channel waveguides were investigated. In the slow light regime, with a group index equal to 99, these nonlinear effects are enhanced but the enhancement produced depends on the input peak power level. Simulations indicate the possibility of soliton-like propagation of 1 ps pulses at an input peak power level of 50 mW inside such a photonic crystal waveguide. The increase in the induced phase shift produced by lower group velocities can be used to decrease the size and power requirements needed to operate devices such as optical switches, logic gates, and wavelength translators