Haldane linearisation done right: Solving the nonlinear recombination equation the easy way

The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time. This is particularly so if one aims at full generality, thus also including degenerate parameter cases. Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible. This paper shows how to do it, and how to extend the approach to the discrete-time case as well.Comment: 12 pages, 1 figure; some minor update

A note on palindromicity

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.Comment: 12 page

A Glimpse at Mathematical Diffraction Theory

Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure $\omega$. It emerges as the Fourier transform of the autocorrelation measure of $\omega$. The mathematically rigorous approach has produced a number of interesting results in the context of perfect and random systems, some of which are summarized here.Comment: 6 pages; Invited talk at QTS2, Krakow, July 2001; World Scientific proceedings LaTeX styl

Recombination semigroups on measure spaces

The dynamics of recombination in genetics leads to an interesting nonlinear differential equation, which has a natural generalization to a measure valued version. The latter can be solved explicitly under rather general circumstances. It admits a closed formula for the semigroup of nonlinear positive operators that emerges from the forward flow and is, in general, embedded in a multi-parameter semigroup.Comment: 15 page

Colourings of planar quasicrystals

The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and determines the actual colour symmetry groups. Continuing previous work, we present the results of the combinatorial part for planar patterns with n-fold symmetry, where n=7,9,15,16,20,24. This completes the cases with values of n such that Euler's totient function of n is less than or equal to eight.Comment: Talk presented by Max Scheffer at Quasicrystals 2001, Sendai (September 2001). 6 pages, including two colour figure

Single--crossover recombination in discrete time

Modelling the process of recombination leads to a large coupled nonlinear dynamical system. Here, we consider a particular case of recombination in {\em discrete} time, allowing only for {\em single crossovers}. While the analogous dynamics in {\em continuous} time admits a closed solution, this no longer works for discrete time. A more general model (i.e. without the restriction to single crossovers) has been studied before and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake (2003), we obtain further insight into the single-crossover dynamics and the particular difficulties that arise in discrete time. We then transform the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation. Still, the coefficients of the second step must be determined in a recursive manner, but once this is done for a given system, they allow for an explicit solution valid for all times.Comment: J. Math. Biol., in pres

A Note on Shelling

The radial distribution function is a characteristic geometric quantity of a point set in Euclidean space that reflects itself in the corresponding diffraction spectrum and related objects of physical interest. The underlying combinatorial and algebraic structure is well understood for crystals, but less so for non-periodic arrangements such as mathematical quasicrystals or model sets. In this note, we summarise several aspects of central versus averaged shelling, illustrate the difference with explicit examples, and discuss the obstacles that emerge with aperiodic order.Comment: substantially revised and extended, 15 pages, AMS LaTeX, several figures included; see also math.MG/990715