Multispecies quantum Hurwitz numbers

The construction of hypergeometric 2D Toda $\tau$-functions as generating functions for quantum Hurwitz numbers is extended here to multispecies families. Both the enumerative geometrical significance of these multispecies quantum Hurwitz numbers as weighted enumerations of branched coverings of the Riemann sphere and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived.Comment: 11 pages.This is the revised version posted March 30, 201

Multispecies Weighted Hurwitz Numbers

The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.Comment: this is substantially enhanced version of arXiv:1410.881

The Immediate Practical Implication of the Houghton Report: Provide Green Open Access Now

Among the many important implications of Houghton et al’s (2009) timely and illuminating JISC analysis of the costs and benefits of providing free online access (“Open Access,” OA) to peer-reviewed scholarly and scientific journal articles one stands out as particularly compelling: It would yield a forty-fold benefit/cost ratio if the world’s peer-reviewed research were all self-archived by its authors so as to make it OA. There are many assumptions and estimates underlying Houghton et al’s modelling and analyses, but they are for the most part very reasonable and even conservative. This makes their strongest practical implication particularly striking: The 40-fold benefit/cost ratio of providing Green OA is an order of magnitude greater than all the other potential combinations of alternatives to the status quo analyzed and compared by Houghton et al. This outcome is all the more significant in light of the fact that self-archiving already rests entirely in the hands of the research community (researchers, their institutions and their funders), whereas OA publishing depends on the publishing community. Perhaps most remarkable is the fact that this outcome emerged from studies that approached the problem primarily from the standpoint of the economics of publication rather than the economics of research

RR--Matrix Construction of Electromagnetic Models for the Painlev\'e Transcendents

The Painlev\'e transcendents P_{\rom{I}}--P_{\rom{V}} and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical $R$--matrix Poisson bracket structure on the dual space \wt{\frak{sl}}_R^*(2) of the loop algebra \wt{\frak{sl}}_R(2). The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on \wt{\frak{sl}}_R^*(2) with a time--dependent family of Poisson maps whose images are $4$--dimensional rational coadjoint orbits in \wt{\frak{sl}}_R^*(2). Each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time--varying electromagnetic field. The Painlev\'e equations follow from reduction of these systems by the Hamiltonian flow generated by a second commuting element in the ring of spectral invariants.Comment: 22 pgs, AMSTeX, preprint CRM-2889 (1994