Undecidable properties of self-affine sets and multi-tape automata

We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.Comment: 10 pages, v2 includes some corrections to match the published versio

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Heritability of attention problems in children II: longitudinal results from a study of twins age 3 to 12.

this paper we present data of large samples of twin families, with an equal number of girls and boys. The well-known gender difference with boys displaying more OA and AP was observed at each age. Even at the age of 3, boys display more OA problems than girls. Clinical studies have indicated that severe problem behavior can be identified in very young children (see for review, Campbell, 1995; Keenan & Wakschlag, 2000; Shaw, Owens, Giovannelli, & Winslow, 2001) and that the onset of ADHD is during the pre-school period (Barkley, Fisher, Edelbrock, & Smallish, 1990; Table 6 Top part includes percentages of total variances (diagonal) and covariances (off-diagonal) explained by additive genetic, genetic dominance, and unique environmental components based on best fitting models. Percentages for boys and girls are reported below and above diagonal, respectively. Lower part includes correlations calculated for additive genetic, genetic dominance, and unique environmental sources of variance between different ages. Correlations for boys and girls are reported below and above diagonal, respectively Relative proportions of variance and covariance BoysnGirls A% D% E% OA 3 AP 7 AP 10 AP 12 OA 3 AP 7 AP 10 AP 12 OA 3 AP 7 AP 10 AP 12 OA 3 50n41 73 79 75 22n33 17 13 14 28n26 10 8 11 AP 7 59 33n57 50 53 31 39n16 31 28 10 28n27 19 19 AP 10 86 31 41n48 47 6 51 31n25 32 8 18 28n27 21 AP 12 71 24 31 40n54 16 55 45 30n18 13 21 24 30n28 Correlations between different ages BoysnGirls ADE OA 3 AP 7 AP 10 AP 12 OA 3 AP 7 AP 10 AP 12 OA 3 AP 7 AP 10 AP 12 OA 3 1.00 .60 .66 .57 1.00 .30 .16 .20 1.00 .15 .12 .14 AP 7 .57 1.00 .62 .57 .41 1.00 .99 1.00 .15 1.00 .46 .41 AP 10 .68 .56 1.00 .61 .08 .94 1.00 1.00 .11 .42 1.00 .50 AP 12 .49 .42 .53 1.00 .20 .98 .99 1.00 .14 .45 .58 1.00 ..

Coming to terms with heritability

The complex mechanisms of heredity are little appreciated by non-specialists, in some measure, because of misunderstandings that are perpetuated when words used for technical terms have other, more widely understood, folk meanings. When a word has both technical and folk meanings, it is the responsibility of the specialist to avoid promoting confusion by either using extremely cautious and precise language when using the term or, in cases when confusion is inevitable, abandoning the term in favor of one without a widely understood folk meaning. The study of heredity is beset by such confusion, and the term heritability appears to be at the heart of some of the confusion. In this article, I discuss both the technical and folk meanings of heritability and examine the bridge between them. By continuing to use the term heritability, we risk promulgating serious misunderstanding about the workings of heredity, therefore I suggest selectability as an alternative term to avoid such pitfalls.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42804/1/10709_2005_Article_BF02259512.pd