Approximation of mean curvature flow with generic singularities by smooth flows with surgery

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.Comment: 33 pages. Changes made to introduction and exposition for clarity. Comments welcome

Mean Curvature Flow from Conical Singularities

We prove Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities and resolves a particular case of Ilmanen's strict genus reduction conjecture. Precisely, we prove that the level set flow of a smooth hypersurface $M^n\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.Comment: 22 pages, 3 figures. Comments welcome

Fine analysis of mean curvature flow through singularities

We provide a short survey on the history of the mean curvature flow and the theory of flow through singularities. We establish the existence of a smooth flow with surgery approximating weak mean curvature flows with only spherical and neck pinch singularities, thereby dropping the standard global 2-convexity assumption. This makes use of the resolution of the mean-convex neighbourhood conjecture of Choi–Haslhofer–Hershkovits, and Choi–Haslhofer–Hershkovits–White and a barrier argument for flows with surgery. We conclude our discussion of surgery by utilising the surgery flow, in combination with results of Choi–Chodosh–Mantoulidis–Schulze for generic flows, to increase the known entropy bound for the Schoenflies conjecture in R4. We then consider mean curvature flow of compact hypersurfaces through conical singularities. We demonstrate a uniqueness theorem for flows with tangent flows modelled on the flow generated by a smooth, stable expander with a linearly growing Jacobi field. Moreover, we demonstrate the forward tangent flow at the conical singularity of the outer-most Brakke flows are modelled on the outer-most expanders of the cone, when said expanders are smooth. Combined with work of Ilmanen–White, this demonstrates genus drop for the outer-most flows through such singularities, answering a conjecture of Ilmanen. Finally, we deduce the following dichotomy in dimensions 2 ≤ n ≤ 6: The flow from a compact hypersurface with isolated conical singularity fattens if and only if the flow from the model cone fattens

Approximation of mean curvature flow with generic singularities by smooth flows with surgery

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture

Internet of Things Device with Controllable Lighting Elements

Aspects of the present disclosure are directed to a device with controllable lighting elements or other output elements such as audio, vibratory, gaseous/liquid, and/or olfactory output elements. In particular, the interactive device of the present disclosure can include a structure fashioned in the shape of some fauna or flora (e.g., a tree) that can selectively operate the output elements (e.g., illuminate the lighting elements) based on collected sensor data regarding the surrounding environment. The structure can also respond to specific user interactions and/or receive predetermined control signals that control the output elements

Towards a Bourdieusian analysis of the social composition of the UK film and television workforce

This is an Open Access article is distributed under the terms of the Creative Commons Attribution-NonCommercial 3.0 License (http://www.creativecommons.org/licenses/by-nc/3.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access page(http://www.uk.sagepub.com/aboutus/openaccess.htm).The social composition of the workforce of the UK film and television industries does not reflect the diversity of the population and the industries have been described as white, male and middle class. While the lack of specific demographic representation in employment (for example gender or ethnicity) has been highlighted by both industry and academic commentators, its broader social composition has rarely been addressed by research. This article draws on the work of Bourdieu, particularly the concepts of field, habitus and capitals, to explore perceptions of the barriers to entry into these industries and the way in which individuals negotiate these by drawing on the various capitals to which they have access.Peer reviewe

Onset dynamics of type A botulinum neurotoxin-induced paralysis

Experimental studies have demonstrated that botulinum neurotoxin serotype A (BoNT/A) causes flaccid paralysis by a multi-step mechanism. Following its binding to specific receptors at peripheral cholinergic nerve endings, BoNT/A is internalized by receptor-mediated endocytosis. Subsequently its zinc-dependent catalytic domain translocates into the neuroplasm where it cleaves a vesicle-docking protein, SNAP-25, to block neurally evoked cholinergic neurotransmission. We tested the hypothesis that mathematical models having a minimal number of reactions and reactants can simulate published data concerning the onset of paralysis of skeletal muscles induced by BoNT/A at the isolated rat neuromuscular junction (NMJ) and in other systems. Experimental data from several laboratories were simulated with two different models that were represented by sets of coupled, first-order differential equations. In this study, the 3-step sequential model developed by Simpson (J Pharmacol Exp Ther 212:16–21,1980) was used to estimate upper limits of the times during which anti-toxins and other impermeable inhibitors of BoNT/A can exert an effect. The experimentally determined binding reaction rate was verified to be consistent with published estimates for the rate constants for BoNT/A binding to and dissociating from its receptors. Because this 3-step model was not designed to reproduce temporal changes in paralysis with different toxin concentrations, a new BoNT/A species and rate (kS) were added at the beginning of the reaction sequence to create a 4-step scheme. This unbound initial species is transformed at a rate determined by kS to a free species that is capable of binding. By systematically adjusting the values of kS, the 4-step model simulated the rapid decline in NMJ function (kS ≥0.01), the less rapid onset of paralysis in mice following i.m. injections (kS = 0.001), and the slow onset of the therapeutic effects of BoNT/A (kS < 0.001) in man. This minimal modeling approach was not only verified by simulating experimental results, it helped to quantitatively define the time available for an inhibitor to have some effect (tinhib) and the relation between this time and the rate of paralysis onset. The 4-step model predicted that as the rate of paralysis becomes slower, the estimated upper limits of (tinhib) for impermeable inhibitors become longer. More generally, this modeling approach may be useful in studying the kinetics of other toxins or viruses that invade host cells by similar mechanisms, e.g., receptor-mediated endocytosis

Genetics of asthma: a molecular biologist perspective

Asthma belongs to the category of classical allergic diseases which generally arise due to IgE mediated hypersensitivity to environmental triggers. Since its prevalence is very high in developed or urbanized societies it is also referred to as "disease of civilizations". Due to its increased prevalence among related individuals, it was understood quite long back that it is a genetic disorder. Well designed epidemiological studies reinforced these views. The advent of modern biological technology saw further refinements in our understanding of genetics of asthma and led to the realization that asthma is not a disorder with simple Mendelian mode of inheritance but a multifactorial disorder of the airways brought about by complex interaction between genetic and environmental factors. Current asthma research has witnessed evidences that are compelling researchers to redefine asthma altogether. Although no consensus exists among workers regarding its definition, it seems obvious that several pathologies, all affecting the airways, have been clubbed into one common category called asthma. Needless to say, genetic studies have led from the front in bringing about these transformations. Genomics, molecular biology, immunology and other interrelated disciplines have unearthed data that has changed the way we think about asthma now. In this review, we center our discussions on genetic basis of asthma; the molecular mechanisms involved in its pathogenesis. Taking cue from the existing data we would briefly ponder over the future directions that should improve our understanding of asthma pathogenesis