We start with the simplest possible model. There are kn objects for sale and n potential buyers. The valuations vi are drawn independently from a common distribution F (vi) for all potential bidders. We assume that F has a continuous density function f . Suppose now that the objects are allocated according to the following rule: kn buyers with the highest bids receive the object and pay the pri...

The number of disjoint cocircuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint cocircuits; consider, for example, M(Kn) and Un,2n. Also the bicircular matroids B(Kn) have arbitrarily large rank and have no 3 disjoint cocircuits. We prove that for each k and n there exists a constant c such that, if M is a matroid ...

The number of disjoint co-circuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint co-circuits; consider, for example, M(Kn) and Un,2n. Also the bicircular matroids B(Kn) have arbitrarily large rank and have no 3 disjoint co-circuits. We prove that for each k and n there exists a constant c such that, if M is a matro...

In this paper we present a randomized algorithm for the multipacket routing problem on an n×nmesh. The algorithm completes with high probability in at the most kn + o(kn) parallel communication steps, with a queue size of k + o(k). The previous best known algorithm [3] takes 5 4 kn+O( kn f(n) ) steps with a queue size of O(k f(n)) (for any 1 ≤ f(n) ≤ n). The algorithm that we will present is op...

A graph G is 1-homogeneous if certain isomorphisms between similarly embedded induced subgraphs of G extend to automorphisms of G. We show that the only connected composite 1-homogeneous graphs are the cube, and Kn ×K2 and Kn ×Kn with n ≥ 2.

The edge set of Kn cannot be decomposed into edge-disjoint hexagons (or 6-cycles) when n =1= 1 or 9 (mod 12). We discuss adding edges to the edge set of Kn so that the resulting graph can be decomposed into edge-disjoint hexagons. This paper gives the solution to this minimum covering of Kn with hexagons problem.

We give a construction of a uniform covering of 2-paths with 5-paths in Kn for all even n ≥ 6, i.e., we construct a set S of 5-paths in Kn having the property that each 2-path in Kn lies in exactly one 5-path in S for all even n ≥ 6.

In the cerebellar cortex, long-term depression (LTD) of synapses between parallel &bers (PF) and Purkinje neurons can spread to neighboring ones, independently of their activation by PF input. This spread of non-speci&c LTD around the activated synapses resembles how units are a3ected in the neighborhood of the winner in a Kohonen Network (KN). However in a classic KN the weight vectors become ...

We give new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbed extensions. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G ⊕ H of graphs G and H . Our results include the following: • The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity. This ex...

Let be a measure with compact support. Assume that is a Lebesgue point of and that 0 is positive and continuous at : Let fAng be a sequence of positive numbers with limit 1. We show that one can choose n 2 An n ; + An n such that lim n!1 Kn n; n + a ~ Kn( n; n) Kn ( n; n ) = sin a a ; uniformly for a in compact subsets of the plane. Here Kn is the nth reproducing kernel for , and ~ Kn is its no...