Peter Hegarty

Some explicit constructions of sets with more sums than differences

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the n-dimensional binary hypercube, for some n, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concernis with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node v^0 to the all-ones node v^1 tends respectively to 0, 1 and 1, as n tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of v^0 is set to some α = α_n ∈ [0, 1]. We prove that there is a very sharp threshold at α_n = (ln n)/n for the existence of accessible paths from v^0 to v^1 . As a corollary we prove significant concentration, for α below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from v^0 to v^1 existing tends to 1 provided the drift parameter θ = θ_n satisfies n(θ_n) → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.

A Quadratic Lower Bound for the Convergence Rate in the One-Dimensional Hegselmann---Krause Bounded Confidence Dynamics

Let

A Cauchy-Davenport type result for arbitrary regular graphs

f_{k}(n)

On a conjecture of Zimmerman about group automorphisms

fk(n) be the maximum number of time steps taken to reach equilibrium by a system of

The Hegselmann-Krause dynamics on the circle converge

n