The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc

Presence of celiac disease epitopes in modern and old hexaploid wheat varieties: wheat breeding may have contributed to increased prevalence of celiac disease

Gluten proteins from wheat can induce celiac disease (CD) in genetically susceptible individuals. Specific gluten peptides can be presented by antigen presenting cells to gluten-sensitive T-cell lymphocytes leading to CD. During the last decades, a significant increase has been observed in the prevalence of CD. This may partly be attributed to an increase in awareness and to improved diagnostic techniques, but increased wheat and gluten consumption is also considered a major cause. To analyze whether wheat breeding contributed to the increase of the prevalence of CD, we have compared the genetic diversity of gluten proteins for the presence of two CD epitopes (Glia-α9 and Glia-α20) in 36 modern European wheat varieties and in 50 landraces representing the wheat varieties grown up to around a century ago. Glia-α9 is a major (immunodominant) epitope that is recognized by the majority of CD patients. The minor Glia-α20 was included as a technical reference. Overall, the presence of the Glia-α9 epitope was higher in the modern varieties, whereas the presence of the Glia-α20 epitope was lower, as compared to the landraces. This suggests that modern wheat breeding practices may have led to an increased exposure to CD epitopes. On the other hand, some modern varieties and landraces have been identified that have relatively low contents of both epitopes. Such selected lines may serve as a start to breed wheat for the introduction of ‘low CD toxic’ as a new breeding trait. Large-scale culture and consumption of such varieties would considerably aid in decreasing the prevalence of CD

Fractal coding of monochrome images

\u3cp\u3eFractal techniques have been used by a number of authors (Barnsley and Sloan, 1988l; Jacquin, 1989) to code monochrome images. In this paper we investigate a version of the technique proposed by Jacquin. A number of extensions are made to his coding scheme, and in particular it is shown that one can considerably weaken the requirement on contractivity leading to a coding scheme which preserves high frequencies much better than that proposed by Jacquin.\u3c/p\u3

Spectral analysis of a family of binary inflation rules

The family of primitive binary substitutions defined by 1 ↦ 0 ↦ 01m with integer m ∈ ℕ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation (m=1), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous