Minc conjectured, and Brégman proved, a sharp upper bound on the permanent of an n by n 0-1 matrix with given row sums (equivalently, on the number of perfect matchings in a bipartite graph with each partition class having size n and with fixed degree sequence for one of the two classes). Here we present Radhakrishnan’s entropy proof of Brégman’s theorem, and Alon and Friedland’s proof of an an...

In a recent and ongoing work, Baldwin and Klemperer found a connection between tropical geometry and economics. They gave a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the Unimodularity Theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota. We introduce auction theory for a mat...

This paper describes a prototype of a programmable interactive theorem-proving system. The main new feature of this system is that it supports the construction and manipulation of tree-structured proofs that can contain both metavariables and derived rules that are computed by tactic programs. The proof structure encapsulates the top-down reenement process of proof construction typical of most ...

Galvin solved the Dinitz conjecture by proving that bipartite graphs are ∆edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the k...

A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch’s squari...

This paper shows the existence of independent randommatching of a large (continuum) population in both static and dynamic systems, which has been popular in the economics and genetics literatures. We construct a joint agent-probability space, and randomized mutation, partial matching, and match-induced type-changing functions that satisfy appropriate independence conditions. The proofs are achi...

We construct a bipartite graph having nodes for jobs and machines. For a given job j and machine i, there is an edge e = (j, i) iff Tij > 0. We say a node is tight if the corresponding row or column in the matrix t is tight. We want to compute the matching that matches all the tight nodes. The remaining part of the analysis is to show that in such a graph, there always exists a matching which m...

We derive a simple accuracy matching strategy for inexact Gauss Newton methods and apply it to the numerical solution of boundary value problems of ordinary diierential equations by collocation. The matching strategy is based on an aane contravariant convergence theorem, i.e., the characteristic constants are invariant under aane transformations of the domain. The inexact Gauss Newton method is...

In this paper, we present a partial instantiation theorem prover (INSTANT) which handles sentences in first-order logic in clausal or non-clausal form. INSTANT uses a variant of the GSAT algorithm for determining the satisfiability of a propositional sentence to increase its speed. The algorithm used in INSTANT can be parallelized with good speedup to improve performance. INSTANT is designed fo...

Motivated by Gauss’s first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a...