Steiner’s formula in the Heisenberg group

Steiner’s tube formula states that the volume of an ϵ-neighborhood of a smooth regular domain in Rn is a polynomial of degree n in the variable ϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.

SrGAP2-Dependent Integration of Membrane Geometry and Slit-Robo-Repulsive Cues Regulates Fibroblast Contact Inhibition of Locomotion

Migrating fibroblasts undergo contact inhibition of locomotion (CIL), a process that was discovered five decades ago and still is not fully understood at the molecular level. We identify the Slit2-Robo4-srGAP2 signaling network as a key regulator of CIL in fibroblasts. CIL involves highly dynamic contact protrusions with a specialized actin cytoskeleton that stochastically explore cell-cell overlaps between colliding fibroblasts. A membrane curvature-sensing F-BAR domain pre-localizes srGAP2 to protruding edges and terminates their extension phase in response to cell collision. A FRET-based biosensor reveals that Rac1 activity is focused in a band at the tip of contact protrusions, in contrast to the broad activation gradient in contact-free protrusions. SrGAP2 specifically controls the duration of Rac1 activity in contact protrusions, but not in contact-free protrusions. We propose that srGAP2 integrates cell edge curvature and Slit-Robo-mediated repulsive cues to fine-tune Rac1 activation dynamics in contact protrusions to spatiotemporally coordinate CIL.

Characterization of OSIRIS NAC filters for the interpretation of multispectral data of comet 67P/Churyumov-Gerasimenko

Context. We interpret multicolor data from OSIRIS NAC for the remote-sensing exploration of comet 67P/Churyumov-Gerasimenko. Aims. We determine the most meaningful definition of color maps for the characterization of surface variegation with filters available on OSIRIS NAC. Methods. We analyzed laboratory spectra of selected minerals and olivine-pyroxene mixtures seen through OSIRIS NAC filters, with spectral methods existing in the literature: reflectance ratios, minimum band wavelength, spectral slopes, band tilt, band curvature, and visible tilt. Results. We emphasize the importance of reflectance ratios and particularly the relation of visible tilt vs. band tilt. This technique provides a reliable diagnostic of the presence of silicates. Color maps constructed by red-green-blue colors defined with the green, orange, red, IR, and Fe2O3 filters let us define regions that may significantly differ in composition.