A solution to the multidimensional additive homological equation

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L_∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L_∞([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0,...

New FPT Algorithms for Finding the Temporal Hybridization Number for Sets of Phylogenetic Trees

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5 ^k· n· m). We also present the concept...

New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees

We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of t binary trees with n leaves each with a running time of O(5^k*n*m) where k is the minimum temporal hybridization number. We also...

Using the slice rank for finding upper bounds on the size of cap sets

The cap set problem consists of finding the maximum size cap sets, i.e. sets without a 3-term arithmetic progression in F₃. In this thesis several known results on the behavior of this number as n → ∞ are presented. In particular we discuss a reformulation by Terence Tao and Will Sawin of a proof found by Dion Gijswijt and Jordan Ellenberg. It...