On Universal Point Sets for Planar Graphs

A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n>=15. Conversely, we use a computer program to show that there exist universal point sets for all n<=10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7'393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.Comment: Fixed incorrect numbers of universal point sets in the last par

The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

Nonlinear stability of chemotactic clustering with discontinuous advection

We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincar\'e inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results

Nonlinear stability of chemotactic clustering with discontinuous advection

We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results

Measuring the bending rigidity of microbial glucolipid (biosurfactant) bioamphiphile self-assembled structures by neutron spin-echo (NSE): interdigitated vesicles, lamellae and fibers

Bending rigidity, k, is classically measured for lipid membranes to characterize their nanoscale mechanical properties as a function of composition. Widely employed as a comparative tool, it helps understanding the relationship between the lipid's molecular structure and the elastic properties of its corresponding bilayer. Widely measured for phospholipid membranes in the shape of giant unilamellar vesicles (GUVs), bending rigidity is determined here for three self-assembled structures formed by a new biobased glucolipid bioamphiphile, rather associated to the family of glycolipid biosurfactants than phospholipids. In its oleyl form, glucolipid G-C18:1 can assemble into vesicles or crystalline fibers, while in its stearyl form, glucolipid G-C18:0 can assemble into lamellar gels. Neutron spin-echo (NSE) is employed in the q-range between 0.3 nm-1 (21 nm) and 1.5 nm-1 (4.1 nm) with a spin-echo time in the range of up to 500 ns to characterize the bending rigidity of three different structures (Vesicle suspension, Lamellar gel, Fiber gel) solely composed of a single glucolipid. The low (k= 0.30 $\pm$ 0.04 kbT) values found for the Vesicle suspension and high values found for the Lamellar (k= 130 $\pm$ 40 kbT) and Fiber gels (k= 900 $\pm$ 500 kbT) are unusual when compared to most phospholipid membranes. By attempting to quantify for the first time the bending rigidity of self-assembled bioamphiphiles, this work not only contributes to the fundamental understanding of these new molecular systems, but it also opens new perspectives in their integration in the field of soft materials

Uniqueness of stationary states for singular Keller-Segel type models

We consider a generalised Keller–Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy–Littlewood–Sobolev type functional inequality in the radial setting

Regulation and Function of the Caspase-1 in an Inflammatory Microenvironment.

The inflammasome is a complex of proteins that has a critical role in mounting an inflammatory response in reply to a harmful stimulus that compromises the homeostatic state of the tissue. The NLRP3 inflammasome, which is found in a wound-like environment, is comprised of three components: the NLRP3, the adaptor protein ASC and caspase-1. Interestingly, although ASC levels do not fluctuate, caspase-1 levels are elevated in both physiological and pathological conditions. Despite the observation that merely raising caspase-1 levels is sufficient to induce inflammation, the crucial question regarding the mechanism governing its expression is unexplored. We found that, in an inflammatory microenvironment, caspase-1 is regulated by NF-κB. Consistent with this association, the inhibition of caspase-1 activity parallels the effects on wound healing caused by the abrogation of NF-κB activation. Surprisingly, not only does inhibition of the NF-κB/caspase-1 axis disrupt the inflammatory phase of the wound-healing program, but it also impairs the stimulation of cutaneous epithelial stem cells of the proliferative phase. These data provide a mechanistic basis for the complex interplay between different phases of the wound-healing response in which the downstream signaling activity of immune cells can kindle the amplification of local stem cells to advance tissue repair