Transversal structures on triangulations: a combinatorial study and straight-line drawings

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change

Three-dimensional magnetic flux rope structure formed by multiple sequential X-line reconnection at the magnetopause

On 14 June 2007, four Time History of Events and Macroscale Interactions during Substorms spacecraft observed a flux transfer event (FTE) on the dayside magnetopause, which has been previously proved to be generated by multiple, sequential X-line reconnection (MSXR) in a 2-D context. This paper reports a further study of the MSXR event to show the 3-D viewpoint based on additional measurements. The 3-D structure of the FTE flux rope across the magnetospheric boundary is obtained on the basis of multipoint measurements taken on both sides of the magnetopause. The flux rope's azimuthally extended section is found to lie approximately on the magnetopause surface and parallel to the X-line direction; while the axis of the magnetospheric branch is essentially along the local unperturbed magnetospheric field lines. In the central region of the flux rope, as distinct from the traditional viewpoint, we find from the electron distributions that two types of magnetic field topology coexist: opened magnetic field lines connecting the magnetosphere and the magnetosheath and closed field lines connecting the Southern and Northern hemispheres. We confirm, therefore, for the first time, the characteristic feature of the 3-D reconnected magnetic flux rope, formed through MSXR, through a determination of the field topology and the plasma distributions within the flux rope. Knowledge of the complex geometry of FTE flux ropes will improve our understanding of solar wind-magnetosphere interaction.Astronomy & AstrophysicsSCI(E)5ARTICLE51904-191111

Quantification of Nematic Cell Polarity in Three-dimensional Tissues

How epithelial cells coordinate their polarity to form functional tissues is an open question in cell biology. Here, we characterize a unique type of polarity found in liver tissue, nematic cell polarity, which is different from vectorial cell polarity in simple, sheet-like epithelia. We propose a conceptual and algorithmic framework to characterize complex patterns of polarity proteins on the surface of a cell in terms of a multipole expansion. To rigorously quantify previously observed tissue-level patterns of nematic cell polarity (Morales-Navarette et al., eLife 8:e44860, 2019), we introduce the concept of co-orientational order parameters, which generalize the known biaxial order parameters of the theory of liquid crystals. Applying these concepts to three-dimensional reconstructions of single cells from high-resolution imaging data of mouse liver tissue, we show that the axes of nematic cell polarity of hepatocytes exhibit local coordination and are aligned with the biaxially anisotropic sinusoidal network for blood transport. Our study characterizes liver tissue as a biological example of a biaxial liquid crystal. The general methodology developed here could be applied to other tissues or in-vitro organoids.Comment: 27 pages, 9 color figure