On the multiplicity of the hyperelliptic integrals

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ a polynomial 1-form in the variables $x$ and $y$. We provide an upper bound on the multiplicity of $I(t)$, away from the critical values of $H$. Namely: $ord\ I(t) \leq n-1+\frac{n(n-1)}{2}$if$\deg \omega <\deg H=n+1$. The reasoning goes as follows: we consider the analytic curve parameterized by the integrals along$\delta(t)$of the$n$``Petrov'' forms of$H$(polynomial 1-forms that freely generate the module of relative cohomology of$H$), and interpret the multiplicity of$I(t)$as the order of contact of$\gamma(t)$and a linear hyperplane of$\textbf C^ n$. Using the Picard-Fuchs system satisfied by$\gamma(t)$, we establish an algebraic identity involving the wronskian determinant of the integrals of the original form$\omega$along a basis of the homology of the generic fiber of$H$. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of$I(t)$. Still, in some cases, related to the geometry at infinity of the curves$\{H=t\} \subseteq \textbf C^2$, the wronskian occurs to be zero identically. In this alternative we show how to adapt the argument to a system of smaller rank, and get a nontrivial wronskian. For a form$\omega$of arbitrary degree, we are led to estimating the order of contact between$\gamma(t)$and a suitable algebraic hypersurface in$\textbf C^{n+1}$. We observe that$ord I(t)$grows like an affine function with respect to$\deg \omega$.Comment: 18 page

Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio

Knowledge of and attitudes to sports drinks of adolescents living in South Wales, UK

Background: The UK sports drinks market has a turnover in excess of £200M. Adolescents consume 15.6% of total energy as free sugars, much higher than the recommended 5%. Sugar sweetened beverages, including sports drinks, account for 30% of total free sugar intake for those aged 11-18 years. Objective: To investigate children’s knowledge and attitudes surrounding sports drinks. Method: 183 self-complete questionnaires were distributed to four schools in South Wales. Children aged 12 - 14 were recruited to take part. Questions focussed on knowledge of who sports drinks are aimed at; the role of sports drinks in physical activity and the possible detrimental effects to oral health. Recognition of brand logo and sports ambassadors and the relationship of knowledge to respondent’s consumption of sports drinks were assessed. Results: There was an 87% (160) response rate. 89.4% (143) claimed to drink sports drinks. 45.9% thought that sports drinks were aimed at everyone; approximately a third (50) viewed teenagers as the target group. Over 2/3rds recognised the brand logos, yet less than a 1/3rd could identify brand ambassadors. About half were aware that dental erosion may result from consumption and approximately 2/3rds knew that they were linked to dental caries and energy provision. Despite this the majority claimed to drink them. As previously reported most of those drinking sports drinks did so because of the taste. Conclusion: Whilst most of the respondents had some understanding of the detrimental effects on health the majority of them were drinking them regularly despite this knowledge. Work is therefore needed at a macro level, with soft drink manufacturers, to consider marketing and reformulation of products for adolescent consumers who appear to enjoy them

Tame Functions with strongly isolated singularities at infinity: a tame version of a Parusinski's Theorem

Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity.Comment: 20 page

First principles simulations of direct coexistence of solid and liquid aluminium

First principles calculations based on density functional theory, with generalised gradient corrections and ultrasoft pseudopotentials, have been used to simulate solid and liquid aluminium in direct coexistence at zero pressure. Simulations have been carried out on systems containing up to 1000 atoms for 15 ps. The points on the melting curve extracted from these simulations are in very good agreement with previous calculations, which employed the same electronic structure method but used an approach based on the explicit calculation of free energies [L. Vo\v{c}adlo and D. Alf\`e, Phys. Rev. B, {\bf 65}, 214105 (2002).]Comment: To appear in Phys. Rev.

Direct calculation of the hard-sphere crystal/melt interfacial free energy

We present a direct calculation by molecular-dynamics computer simulation of the crystal/melt interfacial free energy, $\gamma$, for a system of hard spheres of diameter $\sigma$. The calculation is performed by thermodynamic integration along a reversible path defined by cleaving, using specially constructed movable hard-sphere walls, separate bulk crystal and fluid systems, which are then merged to form an interface. We find the interfacial free energy to be slightly anisotropic with $\gamma$ = 0.62$\pm 0.01$, 0.64$\pm 0.01$ and 0.58$\pm 0.01 k_BT/\sigma^2$ for the (100), (110) and (111) fcc crystal/fluid interfaces, respectively. These values are consistent with earlier density functional calculations and recent experiments measuring the crystal nucleation rates from colloidal fluids of polystyrene spheres that have been interpreted [Marr and Gast, Langmuir {\bf 10}, 1348 (1994)] to give an estimate of $\gamma$ for the hard-sphere system of $0.55 \pm 0.02 k_BT/\sigma^2$, slightly lower than the directly determined value reported here.Comment: 4 pages, 4 figures, submitted to Physical Review Letter

Reduced selenium concentrations and glutathione peroxidase activity in preeclamptic pregnancies

Preeclampsia is a pregnancy-specific condition affecting 2-7% of women and a leading cause of perinatal and maternal morbidity and mortality. Preeclampsia may also predispose the fetus to increased risks of adult cardiovascular disease. Selenium, acting through the selenoprotein glutathione peroxidases, has critical roles in regulating antioxidant status. Recent reports implicate poor maternal selenium status as a nutritional factor predisposing the mother to preeclampsia but the fetus and placenta have not been studied in tandem. Measurement of selenium concentrations, expression and activity levels of glutathione peroxidase and markers of oxidative stress were performed on maternal and umbilical venous blood samples or the placenta from 27 normal pregnant, 25 preeclamptic and 22 healthy age-matched non-pregnant women. The results of this study revealed highly significant reductions in serum selenium concentrations and plasma glutathione peroxidase activity in pregnancy per se compared to non-pregnant controls. Moreover, these levels were further decreased in the preeclamptic mothers and babies compared to normal pregnancies. Umbilical venous selenium was particularly low (42.1±11.8 and 29.0 ± 9.9 mug/L; mean ±s.d.; P<0.05). Both mother and baby had significantly increased levels of markers for oxidative stress in the preeclamptic group. The placental glutathione peroxidase activity and immunohistochemical staining were also reduced in the preeclampsia placentae. Oxidative stress associated with preeclampsia may be a consequence of reduced antioxidant defence pathways specifically involving glutathione peroxidases, perhaps linked to reduced selenium availability. Reduced glutathione peroxidases could be associated with increased generation of toxic lipid peroxides contributing to the endothelial dysfunction and hypertension of preeclampsia

Simulations of Two-Dimensional Melting on the Surface of a Sphere

We have simulated a system of classical particles confined on the surface of a sphere interacting with a repulsive $r^{-12}$ potential. The same system simulated on a plane with periodic boundary conditions has van der Waals loops in pressure-density plots which are usually interpreted as evidence for a first order melting transition, but on the sphere such loops are absent. We also investigated the structure factor and from the width of the first peak as a function of density we can show that the growth of the correlation length is consistent with KTHNY theory. This suggests that simulations of two dimensional melting phenomena are best performed on the surface of a sphere.Comment: 4 eps figure