Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

We solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitz\'eica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitz\'eica equation by an analogue of the Miura transformation for the (modified) Korteweg-de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ``fluid sheets'' of constant Gaussian curvature, namely the plane, sphere and pseudosphere.

Методика анализа объемных дефектов по цифровой модели рельефа поверхности

We developed a technique for revealing and analyzing volumetric surface defects based on geomorphometric modeling, in particular, an analysis of models and maps of some morphometric variables (minimum curvature, maximum curvature, mean curvature, Gaussian curvature, unsphericity, etc.), derived from digital elevation models of a surface. The technique allows one to reveal areas of individual volume defects (cracks, film delaminations, shape deviations, etc.), to determine shape and size of both the defects themselves and adjucent modified areas, as well as to study patterns of their distribution. The technique effectiveness is exempified by defects on silicon–glass and silicon–silicon wafer assemblies, as well as a cracked Ni–W film. The technique can be promising for quality control of manufacturing and diagnostics of damages of various items, in particular, microelectronic products.

Gaussian curvature dilutes the nuclear lamina, favoring nuclear rupture, especially at high strain rate

Nuclear rupture has long been associated with deficits or defects in lamins, with recent results also indicating a role for actomyosin stress, but key physical determinants of rupture remain unclear. Here, lamin-B stably interacts with the nuclear membrane at sites of low Gaussian curvature yet dilutes at high-curvature to favor rupture, whereas lamin-A depletes similarly but only at high strain-rates. Live cell imaging of lamin-B1 gene-edited cancer cells is complemented by fixed-cell imaging of ruptured nuclei in: iPS-derived cells from progeria patients, cells within beating chick embryo hearts, and cancer cells that develop multiple ruptures in migrating through small pores. Dilution and curvature-dependent rupture fit a parsimonious model of a stiff filament that detaches from a curved surface, suggesting an elastic-type response of lamin-B, but rupture is also modestly suppressed by inhibiting myosin-II and by hypotonic stress, which slow the strain rates. Lamin-A dilution and nuclear rupture likelihood indeed increase above a threshold rate of pulling into small pipettes, suggesting a viscoplastic coupling to the envelope for protection against nuclear rupture.

CurD: A Tool for Diffusion Analyses on Curved Membranes

Curved membranes are abundant and functionally relevant in living matter, yet they have eluded computational studies due to methodological limitations. With multiple approaches available for setting up simulations on such curved membranes, there is a growing need for efficient and versatile tools to analyze their outcomes. Here, we present CurD, a freely available tool for analyzing the diffusion of membrane constituents along curved surfaces. The tool efficiently uses the Vertex-oriented Triangle Propagation algorithm to compute geodesic distances significantly faster than conventional implementations of path-finding algorithms, while providing a friendly command-line interface. With CurD, we resolve the effects of curvature and lipid packing densities on the diffusion of lipids on two curved membranes---one with only mean curvature and another with both Mean and Gaussian curvature. We find that Gaussian curvature plays a surprisingly small role, whereas mean curvature or packing of lipid headgroups largely dictates their mobility.