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

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

Generating vortex rings in Bose-Einstein condensates in the line-source approximation

We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the rotational of the superfluid velocity field is different from zero only on a circumference of given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.Comment: Accepted for publication in Phys. Rev.

Assessment of endocytic traffic and ocrl function in the developing zebrafish neuroepithelium

Endocytosis allows cells to internalise a wide range of molecules from their environment and to maintain their plasma membrane composition. It is vital during development and for maintenance of tissue homeostasis. The ability to visualise endocytosis in vivo requires suitable assays to monitor the process. Here, we describe imaging-based assays to visualise endocytosis in the neuroepithelium of living zebrafish embryos. Injection of fluorescent tracers into the brain ventricles followed by live imaging was used to study fluid-phase or receptor-mediated endocytosis, for which we used receptor-associated protein (RAP, encoded by Lrpap1) as a ligand for low-density lipoprotein receptor-related protein (LRP) receptors. Using dual-colour imaging combined with expression of endocytic markers, it is possible to track the progression of endocytosed tracers and to monitor trafficking dynamics. Using these assays, we reveal a role for the Lowe syndrome protein Ocrl in endocytic trafficking within the neuroepithelium. We also found that the RAP-binding receptor Lrp2 (encoded by lrp2a) appears to contribute only partially to neuroepithelial RAP endocytosis. Altogether, our results provide a basis to track endocytosis within the neuroepithelium in vivo and support a role for Ocrl in this process