280 research outputs found
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 . It
emerges as the Fourier transform of the autocorrelation measure of .
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
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