Properties of triangulations obtained by the longest-edge bisection

The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle of the initial angle. The novelty of the proofs is the use of an hyperbolic metric in a shape space for triangles.

On Barycentric Subdivision

Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that, almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values go to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0, 1] ⊂ ℂ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0, 1/2]. In addition we prove that the largest angle converges to π in probability. Our approach is probabilistic, and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0, 1/2]. The stationary distribution of this limit chain is particularly important in our study.

Analysis of the Intact Surface Layer of Caulobacter crescentus by Cryo-Electron Tomography

ABSTRACT The surface layers (S layers) of those bacteria and archaea that elaborate these crystalline structures have been studied for 40 years. However, most structural analysis has been based on electron microscopy of negatively stained S-layer fragments separated from cells, which can introduce staining artifacts and allow rearrangement of structures prone to self-assemble. We present a quantitative analysis of the structure and organization of the S layer on intact growing cells of the Gram-negative bacterium Caulobacter crescentus using cryo-electron tomography (CET) and statistical image processing. Instead of the expected long-range order, we observed different regions with hexagonally organized subunits exhibiting short-range order and a broad distribution of periodicities. Also, areas of stacked double layers were found, and these increased in extent when the S-layer protein (RsaA) expression level was elevated by addition of multiple rsaA copies. Finally, we combined high-resolution amino acid residue-specific Nanogold labeling and subtomogram averaging of CET volumes to improve our understanding of the correlation between the linear protein sequence and the structure at the 2-nm level of resolution that is presently available. The results support the view that the U-shaped RsaA monomer predicted from negative-stain tomography proceeds from the N terminus at one vertex, corresponding to the axis of 3-fold symmetry, to the C terminus at the opposite vertex, which forms the prominent 6-fold symmetry axis. Such information will help future efforts to analyze subunit interactions and guide selection of internal sites for display of heterologous protein segments.

Delving Deeper: N-Mids

In the study of plane geometry, certain figures and points are given considerable attention. Among these are the centroid of a triangle, and the observation that it lies one-third of the way along the median from its foot to the opposite vertex. If we take the equilateral triangle as a special case, the centroid is that point equidistant from the three points that define the triangle. The point equidistant from two points (the endpoints of a segment) lies halfway from one to the other. The simplicity of these fractions almost tells us to look further. The point equidistant from the vertices of a regular tetrahedron lies one-quarter of the way from the base to the vertex.

Fluctuation theory for the Ehrenfest urn via electric networks

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in the d-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

Fluctuation theory for the Ehrenfest urn via electric networks

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in thed-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

Fluctuation Theory for the Ehrenfest urn

The Ehrenfest urn model withdballs, or alternatively random walk on the unit cube inddimensions, is considered in discrete and continuous time, together with related models. Attention is focused on the fluctuation theory of the model—behaviour on unusual states—and in particular on first passage to the opposite vertex. Applications to statistical mechanics, reliability theory and genetics are surveyed, and some new results are obtained.

Fluctuation Theory for the Ehrenfest urn

The Ehrenfest urn model with d balls, or alternatively random walk on the unit cube in d dimensions, is considered in discrete and continuous time, together with related models. Attention is focused on the fluctuation theory of the model—behaviour on unusual states—and in particular on first passage to the opposite vertex. Applications to statistical mechanics, reliability theory and genetics are surveyed, and some new results are obtained.