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It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on…
In every finite mixture of different normal distributions, there will always be exactly one of those distributions that not only is over-represented in the right tail of the mixture, but even completely overwhelms all other subpopulations…
We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup…
The paper develops a general framework for constrained clustering which is based on the close connection of geometric clustering and diagrams. Various new structural and algorithmic results are proved (and known results generalized and…
A.Olevskii and A.Ulanovskii obtained a scale of density results, which correspond to how well an exponential system approximates a uniformly minimal system over a compact set. We extend their result in several directions. First, we show…
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if…
We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles…
We introduce a non-partisan probability distribution on congressional redistricting of North Carolina which emphasizes the equal partition of the population and the compactness of districts. When random districts are drawn and the results…
We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for finding control policies are unlikely to or simply don't have guarantees of finding policies within a constant factor or a…
Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory,…
Politics around the world exhibits increasing polarization, demonstrated in part by rigid voting configurations in institutions like legislatures or courts. A crux of polarization is separation along a unidimensional ideological axis, but…
In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each…
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is…
Demographic noise causes unlimited population growth in a broad class of models which, without noise, would predict a stable finite population. We study this effect on the example of a stochastic birth-death model which includes…
Star-shaped branching patterns of genealogies are common in marine species. High-fecundity marine populations are characterized by low ratios of effective to actual population size, which reflect high variance in reproductive success among…
Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the…
We introduce a simple model of opinion dynamics in which binary-state agents evolve due to the influence of agents in a local neighborhood. In a single update step, a fixed-size group is defined and all agents in the group adopt the state…
This preprint offers a detailed look, both qualitative and quantitative, at districting with respect to recent voting patterns in one state: Pennsylvania. We investigate how much the partisan playing field is tilted by political geography.…
Let \(G\) be a non-discrete, locally compact group with Haar measure \(m\). We prove that there exists a compact set \(K \subset G\) with \(m(K)=0\) such that \(KK^{-1}\) contains a neighborhood of the identity. Moreover, such a set may be…
All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…