Given the subjective preferences of n roommates in an n-bedroom apartment, one can use Sperner's lemma to find a division of the rent such that each roommate is content with a distinct room. At the given price distribution, no roommate has a strictly stronger preference for a different room. We give a new elementary proof that the subjective preferences of only n − 1 of the roommates actually suffice to achieve this envy-free rent division. Our proof, in particular, yields an algorithm to find such a fair division of rent. The techniques also give generalizations of Sperner's lemma including a new proof of a conjecture of the third author. arXiv:1702.07325v2 [math.CO]
For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice Z d absorbed (or killed) at the boundary. The analysis is based on Poincaré, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.