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

Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the spectral sequence considerably expanded. Other minor change

Data compression and regression based on local principal curves.

Frequently the predictor space of a multivariate regression problem of the type y = m(x_1, …, x_p ) + ε is intrinsically one-dimensional, or at least of far lower dimension than p. Usual modeling attempts such as the additive model y = m_1(x_1) + … + m_p (x_p ) + ε, which try to reduce the complexity of the regression problem by making additional structural assumptions, are then inefficient as they ignore the inherent structure of the predictor space and involve complicated model and variable selection stages. In a fundamentally different approach, one may consider first approximating the predictor space by a (usually nonlinear) curve passing through it, and then regressing the response only against the one-dimensional projections onto this curve. This entails the reduction from a p- to a one-dimensional regression problem. As a tool for the compression of the predictor space we apply local principal curves. Taking things on from the results presented in Einbeck et al. (Classification – The Ubiquitous Challenge. Springer, Heidelberg, 2005, pp. 256–263), we show how local principal curves can be parametrized and how the projections are obtained. The regression step can then be carried out using any nonparametric smoother. We illustrate the technique using data from the physical sciences

Pharmacokinetics of recombinant human erythropoietin applied subcutaneously to children with chronic renal failure

The single-dose pharmacokinetics of recombinant human erythropoietin (rHuEPO) given SC was investigated in 20 patients aged 7-20 years at different stages of chronic renal failure. In a pilot study we confirmed the lower bioavailability of the drug in 2 children when given SC compared with the IV route (24% and 43%, respectively). Following administration of 4,000 units/m2, rHuEPO SC effective serum erythropoietin concentrations increased from a mean baseline level (+/- SD) of 23 +/- 13 units/l to a mean peak concentration of 265 +/- 123 units/l, which was reached after 14.3 +/- 9.4 h, followed by a slow decline until baseline values were attained at 72 h. Mean residence time was 30 +/- 9 h and mean elimination half-time 14.3 +/- 7 h. The single-dose kinetics of SC rHuEPO in children with different degrees of renal failure are comparable to those in adult patients. Possibly, the higher efficacy of SC rHuEPO in patients with renal anaemia compared with IV rHuEPO is related to its prolonged action

Two-Dimensional General Rate Model of Liquid Chromatography Incorporating Finite Rates of Adsorption−Desorption Kinetics and Core−Shell Particles

A two-dimensional general rate model of liquid chromatography incorporating slow rates of adsorption–desorption kinetics, axial and radial dispersions, and core–shell particles is formulated. Radial concentration gradients are generated inside the column by considering different regions of injection at the inlet. Analytical solutions are obtained for a single-component linear model by simultaneously utilizing the Laplace and Hankel transformations for the considered two sets of boundary conditions. These linear solutions are useful for simulating liquid-chromatographic columns with diluted or small-volume samples and those in which radial concentration gradients are significant. To gain further insight into the process, analytical moments are also deduced from the Laplace–Hankel-domain solutions. For situations of concentrated and large-volume samples, which are not solvable analytically, formulation of nonlinear models is necessary. In this study, a semidiscrete, high-resolution, finite-volume scheme is extended to approximate the resulting nonlinear-model equations for multicomponent mixtures. The performance of the column is analyzed by implementing a specified criterion of performance. A few numerical case studies are conducted to inspect the effects of the model parameters on the elution profiles

Coisotropic Branes, Noncommutativity, and the Mirror Correspondence

We study coisotropic A-branes in the sigma model on a four-torus by explicitly constructing examples. We find that morphisms between coisotropic branes can be equated with a fundamental representation of the noncommutatively deformed algebra of functions on the intersection. The noncommutativity parameter is expressed in terms of the bundles on the branes. We conjecture these findings hold in general. To check mirror symmetry, we verify that the dimensions of morphism spaces are equal to the corresponding dimensions of morphisms between mirror objects.Comment: 13 page

Giant phonon anomalies in the pseudo-gap phase of TiOCl

We report infrared and Raman spectroscopy results of the spin-1/2 quantum magnet TiOCl. Giant anomalies are found in the temperature dependence of the phonon spectrum, which hint to unusual coupling of the electronic degrees of freedom to the lattice. These anomalies develop over a broad temperature interval, suggesting the presence of an extended fluctuation regime. This defines a pseudo-gap phase, characterized by a local spin-gap. Below 100 K a dimensionality cross-over leads to a dimerized ground state with a global spin-gap of about 2$\Delta_{spin}\approx$~430 K.Comment: 4 pages, 3 figures, for further information see http://www.peter-lemmens.d

Symplectic cohomology and q-intersection numbers

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of sign