Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O(nlogn) algorithm that checks whether a given bichromatic set has this property.Comment: 31 pages, 24 figure

On the number of fully packed loop configurations with a fixed associated matching

We show that the number of fully packed loop configurations corresponding to a matching with $m$ nested arches is polynomial in $m$ if $m$ is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11 (2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio

Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental Algorithms, 200

Difference frequency generation by quasi-phase matching in periodically intermixed semiconductor superlattice waveguides

Wavelength conversion by difference frequency generation is demonstrated in domain-disordered quasi-phase-matched waveguides. The waveguide structure consisted of a GaAs/AlGaAs superlattice core that was periodically intermixed by ion implantation. For quasi-phase-matching periods of 3.0–3.8 μm, degeneracy pump wavelengths were found by second-harmonic generation experiments for fundamental wavelengths between 1520 and 1620 nm in both type-I and type-II configurations. In the difference frequency generation experiments, output powers up to 8.7 nW were generated for the type-I phase matching interaction and 1.9 nW for the type-II interaction. The conversion bandwidth was measured to be over 100 nm covering the C, L, and U optical communications bands, which agrees with predictions

The Theory of Assortative Matching Based on Costly Signals

We study two-sided markets with a finite numbers of agents on each side, and with two-sided incomplete information. Agents are matched assortatively on the basis of costly signals. A main goal is to identify conditions under which the potential increase in expected output due to assortative matching (relative to random matching) is completely offset by the costs of signalling. We also study how the signalling activity and welfare on each side of the market change when we vary the number of agents and the distribution of their attributes, thereby displaying effects that are particular to small markets. Finally, we look at the continuous version of our two-sided market model and establish the connections to the finite version. Technically, the paper is based on the very elegant theory about stochastic ordering of (normalized) spacings and other linear combinations of order statistics from distributions with monotone failure rates, pioneered by R. Barlow and F. Proschan (1966, 1975) in the framework of reliability theory

Trace formulae for graph Laplacians with applications to recovering matching conditions

Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type. In either case, an infinite series of trace formulae which link together two different graph Laplacians provided that their spectra coincide is derived. Applications are given to the problem of reconstructing matching conditions for a graph Laplacian based on its spectrum

Homotopy Type of the Boolean Complex of a Coxeter System

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic