Almost graphical hypersurfaces become graphical under mean curvature flow

Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow is graphical inside the cylinder of half the radius. Here we prove a lower bound on this period depending on the Lipschitz-constant of the initial graphical representation. This is used to deal with a mean curvature flow that lies inside a slab and is initially graphical inside a cylinder except for a small set. We show that such a flow will become graphical inside the cylinder of half the radius. The proofs are mainly based on White's regularity theorem.Comment: 33 page

A new version of Brakke's local regularity theorem

Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's local regularity theorem yields that $(\mu_t)$ is actually smooth and graphical inside a smaller ball for times $t\in (C,T-C)$ for some constant $C$. Here we extend this result to times $t\in (C,T)$. The main idea is to prove that a Brakke flow that is initially locally graphical with small gradient will remain graphical for some time. Moreover we use the new local regularity theorem to generalise White's regularity theorem to Brakke flows.Comment: 40 pages, corrected an error in the former Lemma 4.4 (now Lemma 5.4), added a section about curvature estimates (used for the mentioned correction

Supersymmetric QCD corrections to e+e−→tbˉH−e^+e^-\to t\bar{b}H^- and the Bernstein-Tkachov method of loop integration

The discovery of charged Higgs bosons is of particular importance, since their existence is predicted by supersymmetry and they are absent in the Standard Model (SM). If the charged Higgs bosons are too heavy to be produced in pairs at future linear colliders, single production associated with a top and a bottom quark is enhanced in parts of the parameter space. We present the next-to-leading-order calculation in supersymmetric QCD within the minimal supersymmetric SM (MSSM), completing a previous calculation of the SM-QCD corrections. In addition to the usual approach to perform the loop integration analytically, we apply a numerical approach based on the Bernstein-Tkachov theorem. In this framework, we avoid some of the generic problems connected with the analytical method.Comment: 14 pages, 6 figures, accepted for publication in Phys. Rev.

Regularität des Brakke-Flusses

This work is about the regularity of the Brakke flow. A Brakke flow is a generalised version of mean curvature flow, which describes a family of surfaces parameterized by time such that each point at each time is moved with velocity equal to the mean curvature vector of the surface at that point. The central Result is Brakke's local regularity theorem, which considers Brakke flows that lie in a slab. If this slab is narrow enough and also the area ratio in suitable balls is controlled by certain bounds, then in a smaller region it is actually smooth and graphical. Brakke's general regularity theorem says that at a time, where no sudden loss of area occurs, the singular set of a Brakke flow has top-dimensional measure zero. This result is primarily based on the fact, that for almost every point with a tangent space, we can find a small neighbourhood, where the local regularity theorem can be applied. In the last part we consider Brakke flows in a cylinder, for which the starting surface is graphical except for a set S.If S has small enough measure and if the graphical part satisfies certain gradient- and height- bounds, then one can use the local regularity theorem to show,there are two possibilities: (1) After some time there exists a period of time where the flow is smooth and graphical inside a smaller cylinder. (2) At some later time there exists a smaller cylinder which contains no part of the Brakke flow.Diese Arbeit befasst sich mit der Regularität des Brakke Flusses. Bei einem Brakke Fluss handelt es sich um eine Veralgemeinerung des Mittleren Krümmungsflusses, welcher eine Familie von Flächen beschreibt die nach der Zeit parametrisiert sind, wobei sich jeder Punkt der Fläche zu jedem Zeitpunkt mit Geschwindigkeit gleich dem Mittleren Krümmungsvektor an die Fläche in diesem Punkt bewegt. Zentrales Ergebnis ist Brakkes lokales Regularitätstheorem, dabei werden Brakke Flüsse betrachtet die lokal in einer horizontalen Röhre liegen. Ist nun die Röhre schmal genug und sind des weiteren die Flächenquotienten in bestimmten Kugeln durch bestimmte Schranken kontrolliert, so gibt es ein kleineres Gebiet in dem der Brakke Fluss glatt und graphisch ist. Brakkes allgemeines Regularitätstheorem besagt, dass zu einem Zeitpunkt, zu dem kein abrupter Massenverlust auftritt, die singuläre Menge eines Brakke Flusses Top-dimensionales Maß Null hat. Dieses Ergebnis beruht im Wesentlichen darauf, dass es für fast alle Punkte mit einem Tangentialraum eine kleine Umgebung gibt, in der sich das lokale Regularitätstheorem anwenden lässt. Im letzten Teil betrachten wir Brakke Flüsse in einem Zylinder, deren Anfangsfläche graphisch ist mit Ausnahme einer Menge S. Ist das Maß von S klein genug und genügt der graphische Teil bestimmten Gradienten- und Höhen-schranken, so lässt sich mit Hilfe des lokalen Regulartätstheorems zeigen, dass es zwei Möglichkeiten gibt: (1)Etwas später gibt es eine Zeitperiode während der der Brakke Fluss in einem kleineren Zylinder glatt und graphisch ist. (2)Zu einem späteren Zeitpunkt existiert ein kleinerer Zylinder der keinen Teil des Brakke Flusses enthält

Hyperoxia blunts counterregulation during hypoglycaemia in humans: possible role for the carotid bodies?

Chemoreceptors in the carotid bodies sense arterial oxygen tension and regulate respiration. Isolated carotid body glomus cells also sense glucose, and animal studies have shown the carotid bodies play a role in the counterregulatory response to hypoglycaemia. Thus, we hypothesized that glucose infusion rate would be augmented and neuro-hormonal counterregulation blunted during hypoglycaemia when the carotid bodies were desensitized by hyperoxia. Seven healthy adults (four male, three female) underwent two 180 min hyperinsulinaemic (2 mU (kg fat-free mass (FFM))−1 min−1), hypoglycaemic (3.33 mmol l−1) clamps 1 week apart, randomized to either normoxia (arterial () 111 ± 6.3 mmHg) or hyperoxia ( 345 ± 80.6 mmHg) (P < 0.05). Plasma glucose concentrations were similar during normoxia and hyperoxia at baseline (5.52 ± 0.15 vs. 5.55 ± 0.13 μmol ml−1) and during the clamp (3.4 ± 0.05 vs. 3.3 ± 0.05 μmol ml−1). The glucose infusion rate was 44.2 ± 3.5% higher (P < 0.01) during hyperoxia than normoxia at steady state during the clamp (28.2 ± 0.15 vs. 42.7 ± 0.65 μmol (kg FFM)−1 min−1; P < 0.01). Area under the curve values (expressed as percentage normoxia response) for counterregulatory hormones during hypoglycaemia were significantly suppressed by hyperoxia (noradrenaline 50.7 ± 5.2%, adrenaline 62.6 ± 3.3%, cortisol 63.2 ± 2.1%, growth hormone 53.1 ± 2.7%, glucagon 48.6 ± 2.1%, all P < 0.05 vs. normoxia). These data support the idea that the carotid bodies respond to glucose and play a role in the counterregulatory response to hypoglycaemia in humans

Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems

We present the science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems, targeting an evolution in technology, that might lead to impacts and benefits reaching into most areas of society. This roadmap was developed within the framework of the European Graphene Flagship and outlines the main targets and research areas as best understood at the start of this ambitious project. We provide an overview of the key aspects of graphene and related materials (GRMs), ranging from fundamental research challenges to a variety of applications in a large number of sectors, highlighting the steps necessary to take GRMs from a state of raw potential to a point where they might revolutionize multiple industries. We also define an extensive list of acronyms in an effort to standardize the nomenclature in this emerging field