Timelike Ricci bounds for low regularity spacetimes by optimal transport

We prove that a globally hyperbolic smooth spacetime endowed with a $\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $\smash{\mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.Comment: 23 pages. Comments welcom

Heat flow aspects of synthetic Ricci bounds in the extended Kato class

This thesis studies heat flows acting on different objects on possibly singular spaces that admit synthetic lower Ricci curvature bounds by constants, functions, or signed measures. Geometric properties of such spaces and probabilistic features of diffusion processes on these are related to functional inequalities for the involved semigroups. Moreover, heat flow methods are used to set up a second order calculus in the general presence of such measure-valued lower Ricci bounds. First, for a given RCD space, we prove the equivalence of the following synthetic characterizations (with respect to a given lower semicontinuous function k) of the "Ricci curvature at every point being bounded from below by k": geodesic semiconvexity of the relative entropy, the evolution variational inequality, Bochner’s inequality, gradient bounds for the functional heat flow, transport estimates, and the pathwise coupling property. Second, on arbitrary weighted Riemannian manifolds, we prove the equivalence of the previous pathwise coupling property with respect to k and pointwise lower boundedness of the Bakry–Émery Ricci tensor by k, only assuming continuity of k. Under an additional exponential integrability condition on k, which holds if k is in the functional Kato class of the weighted manifold, we prove conservativeness and Bismut–Elworthy–Li’s derivative formula. Third, we extend the second order calculus for RCD spaces from Gigli to Dirichlet spaces which are tamed by a signed extended Kato class measure in the sense of Erbar, Rigoni, Sturm and Tamanini. Inter alia, nonsmooth analogues of Hessians, covariant and exterior derivatives, and the Ricci curvature are defined. Employing these objects, in turn, we define heat flows on 1-forms and vector fields and, along with their basic properties, prove domination of the latter by certain semigroups acting on functions. Fourth, again in the setting of RCD spaces, we obtain improved functional inequalities and regularization properties of the heat flow on 1-forms. The spectrum of its generator, the Hodge Laplacian, is studied as well. Finally, we construct a heat kernel for this heat flow and prove Gaussian upper bounds on its pointwise operator norm

Causal convergence conditions through variable timelike Ricci curvature bounds

We describe a nonsmooth notion of globally hyperbolic, regular length metric spacetimes $(\mathrm{M},l)$. It is based on ideas of Kunzinger-S\"amann, but does not require Lipschitz continuity of causal curves. We study geodesics on $\mathrm{M}$ and the space of probability measures over $\mathrm{M}$ in detail. Furthermore, for such a spacetime endowed with a reference measure $\mathfrak{m}$, a lower semicontinuous function $k\colon \mathrm{M} \to \textbf{R}$, and constants $0<p<1$ and $N\geq 1$, we introduce and study the entropic timelike curvature dimension condition $\smash{\mathrm{TCD}_p^e(k,N)}$ with variable Ricci curvature bound $k$. This provides a unified synthetic approach to general relativistic energy conditions, including $\bullet$ the Hawking-Penrose strong energy condition $\mathrm{Ric}\geq 0$, or more generally $\mathrm{Ric}\geq K$ for constant $K\in\textbf{R}$, in all timelike directions, $\bullet$ the weak energy condition $\mathrm{Ric} \geq \mathrm{scal} - \Lambda$ in all timelike directions, and $\bullet$ the null energy condition $\smash{\mathrm{Ric} \geq 0}$ in all null directions. Our approach also allows for the synthetic quantification of asymptotic conditions or integral controls on the timelike Ricci curvature. For example, we give a nonsmooth generalization of a timelike diameter estimate of Frankel-Galloway (and Schneider), and of a Hawking-type singularity theorem which requires only that the negative Ricci curvature have small enough integral relative to the maximal mean curvature of an achronal slice. As further applications, we discuss the stability of our notion and provide timelike geometric inequalities. To obtain sharp constants in the latter, we develop the localization paradigm in the variable $k$ framework.Comment: 109 pages. Comments welcom

Relationships between serum and meconium biomarkers of prenatal tobacco smoke exposure and their association with infant and early childhood growth

Prenatal active and secondhand tobacco smoke exposures remain a prevalent and preventable risk factor for adverse infant and childhood health outcomes. I used a prospective birth cohort of 389 mothers and their infants who were followed from early pregnancy to three years of age to address two specific aims. First, I validated the utility of meconium as a biological matrix to quantify prenatal tobacco smoke exposure. Second, I examined the association between prenatal tobacco smoke exposure and early childhood body mass index (BMI). I validated the utility of meconium tobacco smoke metabolites as biomarkers of prenatal tobacco smoke exposure against self-report and serum cotinine biomarkers of tobacco smoke exposure. I also estimated and compared associations between meconium and serum metabolite concentrations and infant birth weight. Nicotine, cotinine, and trans-3'-hydroxycotinine were detected in the majority of meconium samples (57-80%). Meconium tobacco smoke metabolite concentrations were positively associated with self-report and serum cotinine biomarkers of tobacco smoke exposure. The association between meconium metabolite concentrations and infant birth weight was similar to serum cotinine associations. Meconium is a promising biological matrix to quantify prenatal environmental toxicant exposure; however, meconium tobacco smoke metabolite concentrations did not provide additional information that could be obtained from a single serum cotinine measurement. In the second aim, prenatal tobacco smoke exposures were quantified using maternal self-report and serum cotinine biomarkers. BMI was calculated from weight and height measurements taken at birth, 4 weeks, and 1, 2, and 3 years of age. During pregnancy, 51% of women had cotinine levels consistent with SHS exposure and 10% had cotinine concentrations indicative of active smoking. After adjustment for confounders, both self-report and serum biomarkers of active tobacco smoke exposures were associated with elevated BMI at 2 and 3 years of age. Estimates of association between self-reported SHS exposures and BMI were attenuated towards the null relative to serum cotinine concentration associations. These results suggest that prenatal tobacco smoke exposures may play a role in the development of overweight in early childhood and that self-reported prenatal SHS exposures are non-differentially misclassified, resulting in biased estimates of association with childhood BMI

A Modular Multilevel Matrix Converter for High Speed Drive Applications

The Modular Multilevel Matrix Converter (M3C) performs a direct three-phase AC to AC power conversion and is highly suitable for medium voltage high power drive applications. One area of application are high speed drives such as compressors. However, additional balancing power components which reduce the output power capability of the M3C when input and output frequencies are similar occur. This paper analytically examines the operation behavior and power capability in these operation points in order to assess whether the M3C can generate these additional components without oversizing the converter\u27s components. Subsequently, the theoretical evaluation is verified by a laboratory scaled prototype with a rated power of 15 kW

Solving the TTC 2011 Compiler Optimization Task with metatools

The authors' "metatools" are a collection of tools for generic programming. This includes generating Java sources from mathematically well-founded specifications, as well as the creation of strictly typed document object models for XML encoded texts. In this context, almost every computer-internal structure is treated as a "model", and every computation is a kind of model transformation. This concept differs significantly from "classical model transformation" executed by specialized tools and languages. Therefore it seemed promising to the organizers of the TTC 2011, as well as to the authors, to apply metatools to one of the challenges, namely to the "compiler optimization task". This is a report on the resulting experiences.Comment: In Proceedings TTC 2011, arXiv:1111.440

Quantification of Fluoroquinolone Uptake through the Outer Membrane Channel OmpF of Escherichia coli.

Decreased drug accumulation is a common cause of antibiotic resistance in microorganisms. However, there are few reliable general techniques capable of quantifying drug uptake through bacterial membranes. We present a semiquantitative optofluidic assay for studying the uptake of autofluorescent drug molecules in single liposomes. We studied the effect of the Escherichia coli outer membrane channel OmpF on the accumulation of the fluoroquinolone antibiotic, norfloxacin, in proteoliposomes. Measurements were performed at pH 5 and pH 7, corresponding to two different charge states of norfloxacin that bacteria are likely to encounter in the human gastrointestinal tract. At both pH values, the porins significantly enhance drug permeation across the proteoliposome membranes. At pH 5, where norfloxacin permeability across pure phospholipid membranes is low, the porins increase drug permeability by 50-fold on average. We estimate a flux of about 10 norfloxacin molecules per second per OmpF trimer in the presence of a 1 mM concentration gradient of norfloxacin. We also performed single channel electrophysiology measurements and found that the application of transmembrane voltages causes an electric field driven uptake in addition to concentration driven diffusion. We use our results to propose a physical mechanism for the pH mediated change in bacterial susceptibility to fluoroquinolone antibiotics.This work was supported by a European Research Council (ERC) Grant (261101 Passmembrane) to UFK. JC acknowledges support from an Internal Graduate Studentship, Trinity College, Cambridge, and a Research Studentship from the Cambridge Philosophical Society. SP was supported by the Leverhulme Trust through an Early Career Fellowship. TM acknowledges support from the Konrad-Adenauer Foundation and the German National Merit Foundation. HB, YB and MW are part of the TRANSLOCATION consortium and have received support from the Innovative Medicines Joint Undertaking under grant agreement 115525, the European Union’s seventh framework program (FP7/2007-2013), and European Federation of Pharmaceutical Industries and Associates companies in-kind contribution. We thank Avelino Javer for help with the MATLAB scripts and Catalin Chimerel for helpful discussions.This is the author accepted manuscript. The final version is available from ACS via http://dx.doi.org/10.1021/jacs.5b0896