MSI Weekly Bulletin - Week starting Monday 26 February, 2007

Compression can be rather addictive. It has an obvious "figure of merit" -- the compressed size of some standard data-set, and it is natural to strive to beat your competitors on this measure. Biological sequences are hard to compress; the more compression the better but there is usually limited value in fighting over the third significant digit. Other properties of a compression method can be just as important or more important. This talk gives some reasons why it is challenging, interesting and useful to compress biological sequences. It also presents two simple models for compressing biological sequences (a possible sub-addiction in compression is to complicated models, but simple is often good); we get good results for DNA and for protein. BIO: http://www.csse.monash.edu.au/~lloyd/

We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

We study Hausdorff dimension for fractals constructed by $\beta$-transformation and $\beta$-expansions. $\beta$-transformations and $\beta$-expansions were first introduced by Renyi (1957) and further explored by Parry (1961). Given $\beta >1$, the $\beta$-transformation $T_\beta$ on [0,1) is defined by $T_\beta x= \beta x (mod 1)$. The $\beta$-expansion of $x\in [0,1)$ is determined by $T_\beta$. This introduces a symbolic dynamical system with one-sided shift map, the $\beta$-shift. The theory of iterated function systems (IFS) developed by Hutchinson, Barnsley, et al is fundamental in analyzing and constructing fractals. Many important fractals such as Sierpinski triangle and Cantor middle-third set appear as attractors of certain IFS's. We will define $\beta$-attractor for an IFS as a compact subset, determined by the $\beta$-shift. In the case that all maps of the IFS are similitudes with the same contractive ratio $0 < r < 1$ and with a disjoint condition the Hausdorff dimension of the $\beta$-attractor is given by $\log(\beta)/(-\log r)$. We will apply this result to compute the Hausdorff dimension for Cantor-type sets constructed by $\beta$-expansions with $\beta >2$. This is joint work with John Hutchinson.