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Abstract: Although powers of the Young–Jucys–Murphy elements , , in the symmetric group acting on do not lie in the center of the group algebra of , we show that transitive powers, namely the sum of the contributions from elements that act transitively on are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the center of , and show that they have a polynomiality property. These centrality and polynomiality properties have seemingly unrelated consequences. First, they answer a question raised by Pak [I. Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ary trees, Discrete Math. 204 (1999) 329–335] about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan [J. Irving, A. Rattan, Minimal factorizations of permutations into star transpositions, Discrete Math., in press, math.CO/0610640]; and thirdly, we relate the polynomiality to an existing polynomiality result for a class of double Hurwitz numbers associated with branched covers of the sphere, which therefore suggests that there may be an ELSV-type formula (see [T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001) 297–327]) associated with the star factorization numbers. [Copyright &y& Elsevier]

Abstract: The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a solution to the Plücker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond. [Copyright &y& Elsevier]

Abstract: The existence of a non-negative integer-valued invariant, called the Map–Jack invariant, for rooted maps has been conjectured by Goulden and Jackson [I.P. Goulden, D.M. Jackson, Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996) 873–892]. It has the defining property that, if it is marked by an indeterminate b, then the generating series for rooted maps in locally orientable surfaces is a particular series, involving the Jack symmetric function , that specialises to the generating series for rooted maps in orientable surfaces for (and clearly to all surfaces for ). We propose a candidate Map–Jack invariant η that is determined by depth first search and by the local topological effect of deleting edges from maps, and give results that support its candidacy. In particular, we prove its correctness up to face partition. We also show how the algorithm for determining η may be associated with the Jack symmetric functions. This is achieved by means of the Laplace–Beltrami operator. [Copyright &y& Elsevier]

Abstract: Double Hurwitz numbers count branched covers of 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 (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 . 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. [Copyright &y& Elsevier]

We show that the matrix of chromatic joins, that is associated with the revised Birkhoff–Lewis equations, can be expressed completely in terms of functions defined on a generalization of the Temperley–Lieb algebra. We give a self-contained account of the aspects of the Temperley–Lieb algebra that are essential to this context since these are not easily obtainable in this form. Of interest in the theory of these equations are recursions for the inverse of the matrix of chromatic joins. We show that the approach given here is a natural one which provides clear insight into the investigation of the properties of the inverse, and we give an instance of a recursion. It is hoped that these techniques will be of further value in the study of the revised Birkhoff–Lewis equations. [Copyright &y& Elsevier]

Derives an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points. Combinatiorial topology; Reimann surfaces; Corresponding expressions for surfaces of higher genera; Generalization in terms of a symmetric polynomial; System of linear partial differential equations giving the generating series.

Abstract: Mitochondria play a central role in both the physiological and pathophysiological regulation of cell survival/death. Increasing evidence places mitochondrial dysfunction at the center of many neuropathological conditions. The present study investigates the extent of mitochondrial dysfunction in cortical, hippocampal and cerebellar tissues in a rat model of hypoxia–ischemia (HI). We hypothesized that; mitochondrial dysfunction in situ may be prevented by treatment with clomethiazole (CMZ), a GABAA receptor agonist. Assessment of mitochondrial FAD-linked respiration at both 1- and 3-day post-HI revealed a marked decrease in activity from ipsilateral cortical and hippocampal regions (P<0.001). In addition, small changes were seen in contralateral cortical and hippocampal tissues as well as in the cerebellum at 3-days (P<0.05). Assessment of the mitochondrial electron transport chain (complexes I–V), and mitochondrial markers of integrity (citrate synthase) and oxidative stress (aconitase) confirmed mitochondrial impairment in ipsilateral regions following HI. Complexes I, II–III, V and citrate synthase were also impaired in contralateral regions and cerebellum 3-days post-HI. Treatment with CMZ (414 mg/kg/day via minipumps) provided marked protection to all aspects of neuronal tissue assessed. Circulating cytokine (interleukin [IL]-1α, IL-1β, tumor necrosis factor [TNF]-α, granulocyte macrophage colony-stimulating factor [GM-CSF], IL-4 and IL-10) levels were also assessed in these animals 3-days post-HI. Plasma IL-1α, IL-1β, TNF-α and GM-CSF levels were significantly increased post-HI. Treatment with CMZ ameliorated the increases in IL-1α, IL-1β, TNF-α and GM-CSF levels while increasing plasma IL-4 and IL-10 levels. This study provides evidence of the extent of mitochondrial damage following an HI-insult. In addition, we have shown that protection afforded by CMZ extends to preservation of mitochondrial function and integrity via anti-inflammatory mediated pathways. [Copyright &y& Elsevier]

It is technically demanding to measure the energetic cost of animal flight. Each of the previously available techniques has some disadvantage as well advantages. We compared measurements of the energetic cost of flight in a wind tunnel by four European starlings Sturnus vulgaris made using three independent techniques: heat transfer modelling, doubly labelled water (DLW) and mask respirometry. We based our heat transfer model on thermal images of the surface temperature of the birds and air flow past the body and wings calculated from wing beat kinematics. Metabolic power was not sensitive to uncertainty in the value of efficiency when estimated from heat transfer modelling. A change in the assumed value of whole animal efficiency from 0.19 to 0.07 (the range of estimates in previous studies) only altered metabolic power predicted from heat transfer modelling by 13%. The same change in the assumed value of efficiency would cause a 2.7-fold change in metabolic power if it were predicted from mechanical power. Metabolic power did not differ significantly between measurements made using the three techniques when we assumed an efficiency in the range 0.11-0.19, although the DLW results appeared to form a U-shaped power-speed curve while the heat transfer model and respirometry results increased linearly with speed. This is the first time that techniques for determining metabolic power have been compared using data from the same birds flying under the same conditions. Our data provide reassurance that all the techniques produce similar results and suggest that heat transfer modelling may be a useful method for estimating metabolic rate. [ABSTRACT FROM AUTHOR]

Examines the factors that affect the metabolic power, mechanical power and efficiency during wind tunnel flight by the European starling Sturnus vulgaris. Estimated potential flight ranges of migratory birds; Mechanical components of the power output during flight; Types of stress associated with abnormal sensory input.

Investigates the silver nitrate compresses (SN), silver sulphadiazine cream (SSD), and silver nitrate cream on protecting burns from infections. Comparison on the effect of SN and SSD treatment on patients; Existence of Pseudomonas aeruginosa on SN compresses treatment; Dominance of sulfonamide-resistant Gram-negative bacilli on SSD cream treatment.