Topology of random -dimensional cubical complexes

We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

Highly efficient self-trapped exciton emission in a one-dimensional face-shared hybrid lead bromide

A new one-dimensional face-shared hybrid lead bromide exhibits highly efficient broadband yellow-light emission with a quantum yield of 16.8%.

The Effect of Perioral Scan and Artificial Skin Markers on the Accuracy of Virtual Dentofacial Integration: Stereophotogrammetry Versus Smartphone Three-Dimensional Face-Scanning

This study evaluated the effects of different matching methods on the accuracy of dentofacial integration in stereophotogrammetry and smartphone face-scanning systems. The integration was done (N = 30) with different matching areas (n = 10), including teeth image only (TO), perioral area without markers (PN) and with markers (PM). The positional accuracy of the integrated models was assessed by measuring the midline linear deviations and incisal line canting between the experimental groups and laser scanner-based reference standards. Kruskal–Wallis and Mann–Whitney U tests were used for statistical analyses (α = 0.05). The PM method exhibited the smallest linear deviations in both systems; while the highest deviations were found in the TO in stereophotogrammetry; and in PN in smartphone. For the incisal line canting; the canting degree was the lowest in the PM method; followed by that in the TO and the PN in both systems. Although stereophotogrammetry generally exhibited higher accuracy than the smartphone; the two systems demonstrated no significant difference when the perioral areas were used for matching. The use of perioral scans with markers enables accurate dentofacial image integration; however; cautions should be given on the accuracy of the perioral image obtained without the use of markers.