On Elo based prediction models for the FIFA Worldcup 2018

We propose an approach for the analysis and prediction of a football championship. It is based on Poisson regression models that include the Elo points of the teams as covariates and incorporates differences of team-specific effects. These models for the prediction of the FIFA World Cup 2018 are fitted on all football games on neutral ground of the participating teams since 2010. Based on the model estimates for single matches Monte-Carlo simulations are used to estimate probabilities for reaching the different stages in the FIFA World Cup 2018 for all teams. We propose two score functions for ordinal random variables that serve together with the rank probability score for the validation of our models with the results of the FIFA World Cups 2010 and 2014. All models favor Germany as the new FIFA World Champion. All possible courses of the tournament and their probabilities are visualized using a single Sankey diagram.Comment: 22 pages, 7 figure

Shortest known prion protein allele in highly BSE-susceptible lemurs

We describe the shortest prion protein allele known to date. Surprisingly, it is found as a polymorphism exactly in a species (prosimian lemurs) which seems highly susceptible to oral infection with BSE-derived prions. The truncation of the prion protein we found raises several questions. First, is the truncated octarepeat structure we describe, consisting of two octarepeats, still functional in copper binding? A second question is whether this truncation is related to the remarkable oral infectibility of lemurs with BSE-derived prions. And finally, one could argue that this genotype alone might favour development of a prion disease, even in the absence of exogenous infection

Branching Random Walks on Free Products of Groups

We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter $\lambda$ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by $\Phi(\lambda)$. The main result states that the function $\Phi(\lambda)$ has only one point of discontinuity which is at $\lambda_{c}=R$ where $R$ is the radius of convergence of the Green function of the underlying random walk. Furthermore, $\Phi(R)$ is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of $\Phi(R)-\Phi(\lambda)$ as $\lambda \uparrow R$ is classified. In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if $\lambda\leq \tilde\lambda_{c}$ the end boundary of the trace consists only of infinite words and if $\lambda>\tilde\lambda_{c}$ it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.Comment: 39 pages, 4 figures; final version, accepted for publication in the Proceedings of LM

Capacity of the Range of Random Walks on Free Products of Graphs

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive.Comment: 14 pages, 2 figure