3,392 research outputs found
Voting for Committees in Agreeable Societies
We examine the following voting situation. A committee of people is to be
formed from a pool of n candidates. The voters selecting the committee will
submit a list of candidates that they would prefer to be on the committee.
We assume that . For a chosen committee, a given voter is said to
be satisfied by that committee if her submitted list of candidates is a
subset of that committee. We examine how popular is the most popular committee.
In particular, we show there is always a committee that satisfies a certain
fraction of the voters and examine what characteristics of the voter data will
increase that fraction.Comment: 11 pages; to appear in Contemporary Mathematic
Borsuk-Ulam Implies Brouwer: A Direct Construction
No abstract provided in this article
Teaching Research: Encouraging Discoveries
What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle-school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me
Review: Cake-Cutting Algorithms: Be Fair if You Can
No abstract provided in this article
Rental Harmony: Sperner\u27s Lemma in Fair Division
No abstract provided in this article
Putnam, Pizza & Problem Solving
Ok, here\u27s a difficult question for you.. How can you get roughly 10% of the student body at your college to get up early on a Saturday and spend six hours working on an incredibly difficult exam for which many will get a score of zero
Lower Bounds for Simplicial Covers and Triangulations of Cubes
We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces
A Constructive Proof of Ky Fan\u27s Generalization of Tucker\u27s Lemma
We present a proof of Ky Fan\u27s combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan\u27s lemma to allow for triangulations of Sn that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker\u27s lemma that holds for a more general class of triangulations than the usual version
Consensus-Halving via Theorems of Borsuk-Ulam and Tucker
In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed
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