Francis Edward Su

RENTAL HARMONY: SPERNER'S LEMMA IN FAIR DIVISION

My friend’s dilemma was a practical question that mathematics could answer, both elegantly and constructively. He and his housemates were moving to a house with rooms of various sizes and features, and were having trouble deciding who should get which room and for what part of the total rent. He asked, “Do you think there’s always a way to partition the rent so that each person will prefer a different room?” As we shall see, with mild assumptions, the answer is yes. This rent-partitioning problem is really a kind of fair-division question. It can be viewed as a generalization of the age-old cake-cutting problem, in which one seeks to divide a cake fairly among several people, and the chore-division problem, posed by Martin Gardner in [6, p. 124], in which one seeks to fairly divide an undesirable entity, such as a list of chores. Lately, there has been much interest in fair division (see, for example, the recent books [3] and [11]), and each of the related problems has been treated before (see [1], [4], [10]). We wish to explain a powerful approach to fair-division questions that unifies these problems and provides new methods for achieving approximate envy-free divisions, in which each person feels she received the “best” share. This approach was carried out by Forest Simmons [13] for cake-cutting and depends on a simple combinatorial result known as Sperner’s lemma. We show that the Sperner’s lemma approach can be adapted to treat chore division and rent-partitioning as well, and it generalizes easily to any number of players. From a pedagogical perspective, this approach provides a nice, elementary demonstration of how ideas from many pure disciplines—combinatorics, topology, and analysis—can combine to address a real-world problem. Better yet, the proofs can be converted into constructive fair-division procedures.

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 believe the portions are equally split. 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.

The Shapley value of phylogenetic trees

Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley’s original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.

A polytopal generalization of Sperner's lemma

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar. 32 (1996), 71-74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,...., vn. Label the vertices of T by 1,2,..., n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n - d full dimensional simplices of T, each labelled with d + 1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in Alekseyevskaya (Discrete Math. 157 (1996), 15-37) and Billera et al. (J. Combin. Theory Ser. B 57 (1993), 258-268).

BORSUK-ULAM IMPLIES BROUWER: A DIRECT CONSTRUCTION

The Borsuk-Ulam theorem and the Brouwer fixed point theorem are well-known theorems of topology with a very similar flavor. Both are non-constructive existence results with somewhat surprising conclusions. Most topology textbooks that cover these theorems (e.g., [4], [5], [6]) do not mention the two are related—although, in fact, the Borsuk-Ulam theorem implies the Brouwer Fixed Point Theorem. The theorems themselves are often proved using the machinery of algebraic topology or the concept of degree of a map. That one theorem implies the other can therefore be established once one understands this machinery, but this requires background. Moreover, such proofs tend to be indirect, relying on the equivalence of these existence theorems with corresponding non-existence theorems. For instance, Dugundji and Granas [3] show that the Borsuk-Ulam theorem is equivalent to the statement that no antipode-preserving, continuous map f : S → S can be homotopic to a constant map. From this one can see that the Brouwer fixed point theorem is a special case, because it can be shown equivalent to the statement that the identity map id : S → S (which is antipode-preserving) is not homotopic to a constant map. However, such an indirect approach is not really necessary, and perhaps a more direct proof would give insight as to how the two theorems are related. The purpose of this note is to provide a completely elementary proof that the Borsuk-Ulam theorem implies the Brouwer theorem by a direct construction, in which the existence of antipodal points in one theorem yields the asserted fixed point in the other.

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a proof of Ky Fans combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fans lemma to allow for triangulations of Sn that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tuckers lemma that holds for a more general class of triangulations than the usual version.

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Abstract 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.

Convergence of Random Walks on the Circle Generated by an Irrational Rotation

Fix α ∈ [0, 1). Consider the random walk on the circle S1 which proceeds by repeatedly rotating points forward or backward, with probability 1 2 , by an angle 2πα. This paper analyzes the rate of convergence of this walk to the uniform distribution under “discrepancy” distance. The rate depends on the continued fraction properties of the number ξ = 2α. We obtain bounds for rates when ξ is any irrational, and a sharp rate when ξ is a quadratic irrational. In that case the discrepancy falls as k− 1 2 (up to constant factors), where k is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of ξ which allows for tighter bounds on terms which appear in the Erdős-Turan inequality.

Splitting Fields and Periods of Fibonacci Sequences Modulo Primes

Summary We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

Counting on Continued Fractions

have the same numerator. These fractions simplify to 1 a3993 and 103993 respectively. _T3_102 355 In this paper, we provide a combinatorial interpretation for the numerators and denominators of continued fractions which makes this reversal phenomenon easy to see. Through the use of counting arguments, we illustrate how this and other important identities involving continued fractions can be easily visualized, derived, and remembered. We begin by defining some basic terminology. Given an infinite sequence of integers ao ? 0, a, ? 1, a2 ? 1,. let [ao, a n.a,] denote the finite continued fraction