A diffusion process model of the frequency of a mutation • Reversibility of a 1-dimensional diffusion process • Frequency spectrum and age of a mutation General binary coalescent trees • Combinatorial derivation of the age of a mutation • Ewens' sampling formula, a combinatorial derivation • Coalescent lineage distributions Gene trees and Coalescent trees • DNA sequences and the infinitely-many...

Salt concentration governs nucleic acid hybridization according to the Schildkraut-Lifson (1965) equation. High concentrations of sodium dodecyl sulfate (SDS) are used in some common protocols (Church and Gilbert, 1984; Virca et al., 1990) but the effects of SDS on hybridization stringency have not been reported. We investigated hybridization parameters in solutions containing SDS. With targets...

The aim of these notes is to develop the theory of reductive group schemes, incorporating some simplifications into the methods of [SGA3]. We assume the reader is familiar with the basic structure theory over an algebraically closed field k (as developed in [Bo91], [Hum87], [Spr]), but a review is given in §1 in order to fix our terminology, set everything in the framework of k-schemes (rather ...

The last lecture has been more words than equations. Today we are going to do some paradigmatic calculations. We’ll start with -one at high Reynolds number and one at low Reynolds number –which represent critical ideas for the subject of fluid mechanics. The first calculation is arguably the most important applied-mathematical idea of the twentieth century (at least I would argue it), and it is...

2) Suppose that Γ is a 4-paradoxical group and Γ = A1∪A2∪B1∪B2 is a paradoxical decomposition as defined above. Show that Γ plays ping-pong on itself, where the pingpong players are a := a−1 1 a2 and b := b −1 1 b2 .Deduce that Γ contains a non-abelian free subgroup F2. 3) Define the Tarski number T (Γ) of a group Γ to be the smallest integer N if it exists such that Γ is N -paradoxical. By the...

This report from the International Summer School 'From Genome to Life', held at the Institute d'Etudes Scientifiques de Cargèse in Corsica in July 2002, covers the talks of the invited speakers. The topics of the talks can be broadly grouped into the areas of genome annotation, comparative and evolutionary genomics, functional genomics, proteomics, structural genomics, pharmacogenomics, and org...