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In this review, the formation, evolution, and decay of the large-scale structure of the Universe is discussed in the context of observational data, numerical simulations, and the Cosmological Standard Model (CSM). Problems concerning…
A tree-based network on a set $X$ of $n$ leaves is said to be universal if any rooted binary phylogenetic tree on $X$ can be its base tree. Francis and Steel showed that there is a universal tree-based network on $X$ in the case of $n=3$,…
Multiple-merger coalescents, e.g. $\Lambda$-$n$-coalescents, have been proposed as models of the genealogy of $n$ sampled individuals for a range of populations whose genealogical structures are not captured well by Kingman's…
Many biological studies involve inferring the evolutionary history of a sample of individuals from a large population and interpreting the reconstructed tree. Such an ascertained tree typically represents only a small part of a…
In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been…
Mutation rate variation across loci is well known to cause difficulties, notably identifiability issues, in the reconstruction of evolutionary trees from molecular sequences. Here we introduce a new approach for estimating general…
Models of random walks are considered in which walkers are born at one location and die at all other locations with uniform death rate. Steady-state distributions of random walkers exhibit dimensionally dependent critical behavior as a…
We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices,…
We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal…
The correlation functions of the multi-arc complex matrix model are shown to be universal for any finite number of arcs. The universality classes are characterized by the support of the eigenvalue density and are conjectured to fall into…
We consider random recursive trees that are grown via community modulated schemes that involve random attachment or degree based attachment. The aim of this paper is to derive general techniques based on continuous time embedding to study…
Generalized Polya urn models can describe the dynamics of finite populations of interacting genotypes. Three basic questions these models can address are: Under what conditions does a population exhibit growth? On the event of growth, at…
Directed percolation is one of the generic universality classes for dynamic processes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model, with a parameter $r$…
We perform extensive numerical simulations of different versions of the sandpile model. We find that previous claims about universality classes are unfounded, since the method previously employed to analyze the data suffered a systematic…
The multi-species coalescent provides an elegant theoretical framework for estimating species trees and species demographics from genetic markers. Practical applications of the multi-species coalescent model are, however, limited by the…
Molecular phenotypes are important links between genomic information and organismic functions, fitness, and evolution. Complex phenotypes, which are also called quantitative traits, often depend on multiple genomic loci. Their evolution…
We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive $\varepsilon_1$ and $\varepsilon_2$ such that every tree that is neither a path nor a star has inducibility at…
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the…
Ecological and evolutionary processes show various population dynamics depending on internal interactions and environmental changes. While crucial in predicting biological processes, discovering general relations for such nonlinear dynamics…
Considering a random binary tree with $n$ labelled leaves, we use a pruning procedure on this tree in order to construct a $\beta(3/2,1/2)$-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning…