A combinatorial construction for products of linear transformations over a finite field

AbstractKovacs (J. Combin. Theory, Ser. A 45 (1987), 290–299) has derived an expression for the number of ordered k-tuples, (Ak, …, A1), of n × n matrices over GF(q) whose product Ak … Al has prescribed rank. We give a combinatorial construction for this result

Towards the geometry of double Hurwitz numbers

AbstractDouble Hurwitz numbers count branched covers of CP1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture.In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m=1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves Mg. For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points

The algebra of adjacency patterns: Rees matrix semigroups with reversion

We establish a surprisingly close relationship between universal Horn classes of directed graphs and varieties generated by so-called adjacency semigroups which are Rees matrix semigroups over the trivial group with the unary operation of reversion. In particular, the lattice of subvarieties of the variety generated by adjacency semigroups that are regular unary semigroups is essentially the same as the lattice of universal Horn classes of reflexive directed graphs. A number of examples follow, including a limit variety of regular unary semigroups and finite unary semigroups with NP-hard variety membership problems.Comment: 30 pages, 9 figure

Dual dopamine D2 receptor and β2-adrenoceptor agonists for the treatment of chronic obstructive pulmonary disease: the pre-clinical rationale

AbstractThis paper describes the rationale for the development of dual dopamine D2-receptor and β2-adrenoceptor agonists as potential treatments for the symptoms of chronic obstructive pulmonary disease (COPD). The putative involvement of pulmonary sensory afferent nerves in mediating the key COPD symptoms of breathlessness, cough and excess sputum production is outlined and the hypothesis that activation of D2-receptors on such nerves would modulate their activity is developed. This premise was tested, in a range of animal models, using the first of a novel class of dual dopamine D2-receptor and β2-adrenoceptor agonists, sibenadet HCI (Viozan™, AR-C68397AA). In the course of these studies it was demonstrated that sibenadet, through activation of D2-receptors, inhibited discharge of rapidly adapting receptors and was effective in reducing reflex-induced tachypnoea, mucus production and cough in the dog. Sibenadet, through its activation of β2-adrenoceptors, was also shown to be an effective bronchodilator with a prolonged duration of action following topical administration to the lungs. These studies also indicated that sibenadet had a wide therapeutic ratio with respect to expected undesirable side-effects such as emesis and cardiovascular disturbances. These results provided a compelling rationale for the initiation of a clinical development programme with sibenadet for the treatment of COPD

Matter wave solitons at finite temperatures

We consider the dynamics of a dark soliton in an elongated harmonically trapped Bose-Einstein condensate. A central question concerns the behavior at finite temperatures, where dissipation arises due to the presence of a thermal cloud. We study this problem using coupled Gross-Pitaevskii and $N$-body simulations, which include the mean field coupling between the condensate and thermal cloud. We find that the soliton decays relatively quickly even at very low temperatures, with the decay rate increasing with rising temperature.Comment: 6 pages, 2 figures, submitted to the Proceedings of QFS '0

Multilateral inversion of A_r, C_r and D_r basic hypergeometric series

In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combining these new multidimensional matrix inverses with A_r and D_r extensions of Jackson's 8-phi-7 summation theorem three balanced very-well-poised 8-psi-8 summation theorems associated with the root systems A_r and C_r are derived.Comment: 24 page

Understanding national variations in reusable packaging: commercial drivers, regulatory factors, and provisioning systems

Given the global environmental impacts associated with single-use plastics, alternatives are being considered, including the reuse of product packaging, which aim to retain the functionality of products while reducing waste and broader environmental impacts. Drawing on the Multi-Level Perspective approach, we analyse three national examples of reuse in Denmark, Germany, and Mexico. While the success of reuse initiatives is often attributed to heightened levels of environmental consciousness, we argue that understanding national variations in levels of reuse requires greater attention to the commercial drivers, regulatory factors, and provisioning systems in which they arise. This argument is developed through case studies of the beverage sector in the three national contexts – specifically beer, mineral water, and soft drinks – as this sector is often regarded as a successful illustration of business-to-consumer reuse systems. To conclude, we consider what the implications might be for nations such as the UK where reuse levels remain comparatively low

A deficit of ultraluminous X-ray sources in luminous infrared galaxies

We present results from a Chandra study of ultraluminous X-ray sources (ULXs) in a sample of 17 nearby (DL < 60 Mpc) luminous infrared galaxies (LIRGs), selected to have star formation rates (SFRs) in excess of 7 M⊙ yr−1 and low foreground Galactic column densities (NH ≲ 5 × 1020 cm−2). A total of 53 ULXs were detected and we confirm that this is a complete catalogue of ULXs for the LIRG sample. We examine the evolution of ULX spectra with luminosity in these galaxies by stacking the spectra of individual objects in three luminosity bins, finding a distinct change in spectral index at luminosity ∼2 × 1039 erg s−1. This may be a change in spectrum as 10 M⊙ black holes transit from an ∼ Eddington to a super-Eddington accretion regime, and is supported by a plausible detection of partially ionized absorption imprinted on the spectrum of the luminous ULX (LX ≈ 5 × 1039 erg s−1) CXOU J024238.9-000055 in NGC 1068, consistent with the highly ionized massive wind that we would expect to see driven by a super-Eddington accretion flow. This sample shows a large deficit in the number of ULXs detected per unit SFR (0.2 versus 2 ULXs, per M⊙ yr−1) compared to the detection rate in nearby (DL < 14.5 Mpc) normal star-forming galaxies. This deficit also manifests itself as a lower differential X-ray luminosity function normalization for the LIRG sample than for samples of other star-forming galaxies. We show that it is unlikely that this deficit is a purely observational effect. Part of this deficit might be attributable to the high metallicity of the LIRGs impeding the production efficiency of ULXs and/or a lag between the star formation starting and the production of ULXs; however, we argue that the evidence – including very low NULX/LFIR, and an even lower ULX incidence in the central regions of the LIRGs – shows that the main culprit for this deficit is likely to be the high column of gas and dust in these galaxies, that fuels the high SFR but also acts to obscure many ULXs from our view

Electrodynamics with radiation reaction

The self force of electrodynamics is derived from a scalar field. The resulting equation of motion is free of all of the problems that plague the Lorentz Abraham Dirac equation. The age-old problem of a particle in a constant field is solved and the solution has intuitive appeal.Comment: 5 page