Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in Euclidean space

We prove $\epsilon$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $\delta$-small compared to the mean curvature. We give the explicit dependence of $\delta$ on $\epsilon$ within the class of uniformly convex hypersurfaces with bounded volume.Comment: 11 pages. Comments, discussions or suggestions are welcom

Isotropic functions revisited

To a smooth and symmetric function $f$ defined on a symmetric open set $\Gamma\subset\mathbb{R}^{n}$ and a real $n$-dimensional vector space $V$ we assign an associated operator function $F$ defined on an open subset $\Omega\subset\mathcal{L}(V)$ of linear transformations of $V$, such that for each inner product $g$ on $V$, on the subspace $\Sigma_{g}(V)\subset\mathcal{L}(V)$ of $g$-selfadjoint operators, $F_{g}=F_{|\Sigma_{g}(V)}$ is the isotropic function associated to $f$, which means that $F_{g}(A)=f(\mathrm{EV}(A))$, where $\mathrm{EV}(A)$ denotes the ordered $n$-tuple of real eigenvalues of $A$. We extend some well known relations between the derivatives of $f$ and each $F_{g}$ to relations between $f$ and $F$. By means of an example we show that well known regularity properties of $F_{g}$ do not carry over to $F$.Comment: 13 pages. Added an example to show that loss of regularity is possible. Extended the bibliography. Comments are welcom

Making English a New Latin

The paper looks at various aspects of the so-called Latin-English analogy and particularly at the ways in which English may share the fate of Latin in ultimately becoming a victim of its own success. A critical factor in the history of Latin was a conceptual split between its native and non-native varieties, which eventually proved instrumental in establishing its reputation as a dead language. The author wishes to argue that current proposals for a codification of English as a Lingua Franca, aimed at providing vast numbers of L2 learners with a pedagogical alternative that does not emulate L1 standards, may be regarded as major steps towards making English a new Latin: creating a similar split between native versus foreigners' English

Gradient estimates for inverse curvature flows in hyperbolic space

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface.Comment: 7 pages. Discussions are welcom

The inverse mean curvature flow in warped cylinders of non-positive radial curvature

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over $S^n$ the flow exists for all times and remains a graph over $S^{n}$. Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal $C^2$ estimates in case that the ambient pinching improves.Comment: 29 pages. Thoroughly checked revised version. Fixed some computational errors and added another small technical assumption to make the main theorem work. Comments and discussions are very welcom

Optimal Asset Taxes in Financial Markets with Aggregate Uncertainty

This paper studies Pareto-optimal risk-sharing arrangements in a private information economy with aggregate uncertainty and ex ante heterogeneous agents. I show how to implement Pareto-optima as equilibria when agents can trade claims to consumption contingent on aggregate shocks in financial markets. The first result is that if aggregate and idiosyncratic shocks are independent, the implementation of optimal allocations does not require any interventions in financial markets. This result can be extended to dynamic settings in the sense that, in this case, only savings need to be distorted, but not trades in financial markets. Second, I characterize optimal trading distortions in financial markets when aggregate and idiosyncratic shocks are not independent. In this case, optimal asset taxes must be higher for those securities that pay out in aggregate states in which consumption is more volatile. For instance, this can provide an efficiency justification for the frequently observed differential tax treatment of different asset classes, such as debt and equity claims.

Two-dimensional optical ring resonators based on radial Bragg resonance

A Bragg-reflection-based ring resonator is proposed and analyzed. Closed-form expressions for the field and dispersion curves for radial Bragg gratings and photonic bandgap crystals are derived. The required gratings exhibit a chirped period and a varying index profile. Small bending radii and strong control over the resonator dispersion are possible by the Bragg confinement. Large free spectral range and low radiation loss are predicted theoretically

Giant Fiber Lasers: A New Paradigm for Secure Key Distribution

We propose and analyze a new concept for secure key distribution based on establishing laser oscillations between the sender and receiver. Compared to quantum mechanics based systems, our scheme allows for significantly higher key-establishing rates and longer ranges. By properly designing the laser structure, it is possible to increase the difficulty of eavesdropping almost arbitrarily, thus making our scheme an intriguing alternative and a complementary technology to quantum key-distribution systems

All-optical gates facilitated by soliton interactions in a multilayered Kerr medium

All-optical soliton logic operations, facilitated by incoherent interactions of multiple spatial solitons with nonlinear interfaces, are proposed and analyzed. A particlelike model, validated by beam propagation simulations, was developed for calculating the soliton trajectories and was employed for the analysis of the soliton-based logic gates