Lax matrices for Yang-Baxter maps

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff

The number of ramified coverings of the sphere by the double torus, and a general form for higher genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section

Toroidal Imploding Detonation Wave Initiator for Pulse Detonation Engines

Imploding toroidal detonation waves were used to initiate detonations in propane–air and ethylene–air mixtures inside of a tube. The imploding wave was generated by an initiator consisting of an array of channels filled with acetylene–oxygen gas and ignited with a single spark. The initiator was designed as a low-drag initiator tube for use with pulse detonation engines. To detonate hydrocarbon–air mixtures, the initiator was overfilled so that some acetylene oxygen spilled into the tube. The overfill amount required to detonate propane air was less than 2% of the volume of the 1-m-long, 76-mm-diam tube. The energy necessary to create an implosion strong enough to detonate propane–air mixtures was estimated to be 13% more than that used by a typical initiator tube, although the initiator was also estimated to use less oxygen. Images and pressure traces show a regular, repeatable imploding wave that generates focal pressures in excess of 6 times the Chapman–Jouguet pressure.Atheoretical analysis of the imploding toroidal wave performed using Whitham’s method was found to agree well with experimental data and showed that, unlike imploding cylindrical and spherical geometries, imploding toroids initially experience a period of diffraction before wave focusing occurs. A nonreacting numerical simulation was used to assist in the interpretation of the experimental data

Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross-Pitaevskii equation

Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schroedinger type, which play an important role in the theory of Bose-Einstein condensation. Recent results of Aschenbacher et. al. [AFGST] demonstrate, for a class of 3- dimensional models, that for large boson number (squared L^2 norm), N, the ground state does not have the symmetry properties as the ground state at small N. We present a detailed global study of the symmetry breaking bifurcation for a 1-dimensional model Gross-Pitaevskii equation, in which the external potential (boson trap) is an attractive double-well, consisting of two attractive Dirac delta functions concentrated at distinct points. Using dynamical systems methods, we present a geometric analysis of the symmetry breaking bifurcation of an asymmetric ground state and the exchange of dynamical stability from the symmetric branch to the asymmetric branch at the bifurcation point.Comment: 22 pages, 7 figure

A proof of a conjecture for the number of ramified coverings of the sphere by the torus

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the torus, with elementary branch points and prescribed ramification type over infinity. This proves a conjecture of Goulden, Jackson and Vainshtein for the explicit number of such coverings.Comment: 10 page

Improving efficiency in radio surveys for gravitational lenses

Many lens surveys have hitherto used observations of large samples of background sources to select the small minority which are multiply imaged by lensing galaxies along the line of sight. Recently surveys such as SLACS and OLS have improved the efficiency of surveys by pre-selecting double-redshift systems from SDSS. We explore other ways to improve survey efficiency by optimum use of astrometric and morphological information in existing large-scale optical and radio surveys. The method exploits the small position differences between FIRST radio positions of lensed images and the SDSS lens galaxy positions, together with the marginal resolution of some larger gravitational lens systems by the FIRST beam. We present results of a small pilot study with the VLA and MERLIN, and discuss the desirable criteria for future surveys.Comment: Accepted by MNRAS. 9 pages, 5 figure

Transitive factorizations of permutations and geometry

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th birthda