Self-shrinkers with a rotational symmetry

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $\Sigma^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\mathbb{R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE. We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution $\Sigma^n$ is either a hyperplane $\mathbb{R}^{n}$, the round cylinder $\mathbb{R}\times S^{n-1}$ of radius $\sqrt{2(n-1)}$, the round sphere $S^n$ of radius $\sqrt{2n}$, or is diffeomorphic to an $S^1\times S^{n-1}$ (i.e. a "doughnut" as in [Ang], which when $n=2$ is a torus). In particular for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.Comment: Trans. Amer. Math. Soc. (2011), to appear; 23 pages, 1 figur

Unraveling nonclassicality in the optomechanical instability

Conditional dynamics due to continuous optical measurements has successfully been applied for state reconstruction and feedback cooling in optomechanical systems. In this article, we show that the same measurement techniques can be used to unravel nonclassicality in optomechanical limit cycles. In contrast to unconditional dynamics, our approach gives rise to nonclassical limit cycles even in the sideband-unresolved regime, where the cavity decay rate exceeds the mechanical frequency. We predict a significant reduction of the mechanical amplitude fluctuations for realistic experimental parameters.Comment: 8 pages, 5 figures, equivalent to published versio

Penalized estimation in large-scale generalized linear array models

Large-scale generalized linear array models (GLAMs) can be challenging to fit. Computation and storage of its tensor product design matrix can be impossible due to time and memory constraints, and previously considered design matrix free algorithms do not scale well with the dimension of the parameter vector. A new design matrix free algorithm is proposed for computing the penalized maximum likelihood estimate for GLAMs, which, in particular, handles nondifferentiable penalty functions. The proposed algorithm is implemented and available via the R package \verb+glamlasso+. It combines several ideas -- previously considered separately -- to obtain sparse estimates while at the same time efficiently exploiting the GLAM structure. In this paper the convergence of the algorithm is treated and the performance of its implementation is investigated and compared to that of \verb+glmnet+ on simulated as well as real data. It is shown that the computation time fo

Mean curvature self-shrinkers of high genus: Non-compact examples

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in $\mathbb{R}^3$ over a $2\pi/(g+1)$-periodic graph on an equator of the unit sphere $\mathbb{S}^2\subseteq\mathbb{R}^3$, with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted H\"older spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew. Mat

Numerical Simulation of Heat Transport in Dispersed Gas-Liquid Two-Phase Flow using a Front Tracking Approach

In this paper a simulation model is presented for the Direct Numerical Simulation (DNS) of heat transport in dispersed gas-liquid two-phase flow using the Front Tracking (FT) approach. Our model extends the FT model developed by van Sint Annaland et al. (2006) to non-isothermal conditions. In FT an unstructured dynamic mesh is used to represent and track the interface explicitly by a number of interconnected marker points. The Lagrangian representation of the interface avoids the necessity to reconstruct the interface from the local distribution of the fractions of the phases and, moreover, allows a direct and accurate calculation of the surface tension force circumventing the (problematic) computation of the interface curvature. The extended model is applied to predict the heat exchange rate between the liquid and a hot wall kept at a fixed temperature. It is found that the wall-to-liquid heat transfer coefficient exhibits a maximum in the vicinity of the bubble that can be attributed to the locally decreased thickness of the thermal boundary layer