Representations of stack triangulations in the plane

Stack triangulations appear as natural objects when defining an increasing family of triangulations by successive additions of vertices. We consider two different probability distributions for such objects. We represent, or "draw" these random stack triangulations in the plane $\R^2$ and study the asymptotic properties of these drawings, viewed as random compact metric spaces. We also look at the occupation measure of the vertices, and show that for these two distributions it converges to some random limit measure.Comment: 29 pages, 13 figure

Potenziale ökologisch wirtschaftender Schulbauernhöfe für Naturschutz und Landschaftspflege

Social or green care farming is becoming a perspective not only for the income of farmers, but has also positive effects on the society. In this investigation organic farms providing space for school classes were investigated by using a questionnaire that was sent to 116 school farms in Germany. 72,4% of those could be analysed. The results give an image of the structure of such farms, but also of their ability to integrate issues like nature conservation and landscape development into their work with the pupils

A natural stochastic extension of the sandpile model on a graph

We introduce a new model of a stochastic sandpile on a graph $G$ containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability $p \in (0,1]$. For $p=1$, this coincides with the standard Abelian sandpile model. In general, for $p\in(0,1)$, the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph $G$. We also define the lacking polynomial $L_G$ as the generating function counting this set according to the number of grains, and show that this polynomial satisfies a recurrence which resembles that of the Tutte polynomial

Combinatorial aspects of sandpile models on wheel and fan graphs

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model's recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with $2n$ vertices, the number of recurrent states with level $n$ is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.Comment: 25 pages, 12 figure

Creep Burst Testing of a Woven Inflatable Module

A woven Vectran inflatable module 88 inches in diameter and 10 feet long was tested at the NASA Johnson Space Center until failure from creep. The module was pressurized pneumatically to an internal pressure of 145 psig, and was held at pressure until burst. The external environment remained at standard atmospheric temperature and pressure. The module burst occurred after 49 minutes at the target pressure. The test article pressure and temperature were monitored, and video footage of the burst was captured at 60 FPS. Photogrammetry was used to obtain strain measurements of some of the webbing. Accelerometers on the test article measured the dynamic response. This paper discusses the test article, test setup, predictions, observations, photogrammetry technique and strain results, structural dynamics methods and quick-look results, and a comparison of the module level creep behavior to the strap level creep behavior

New combinatorial perspectives on MVP parking functions and their outcome map

In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars have the same preferred spot. We study a model introduced by Harris, Kamau, Mori, and Tian in recent work, called the MVP parking problem. In this model, priority is given to the cars arriving later in the sequence. When a car finds its preferred spot occupied by a previous car, it "bumps" that car out of the spot and parks there. The earlier car then has to drive on, and parks in the first available spot it can find. If all cars manage to park through this procedure, we say that the list of preferences is an MVP parking function. We study the outcome map of MVP parking functions, which describes in what order the cars end up. In particular, we link the fibres of the outcome map to certain subgraphs of the inversion graph of the outcome permutation. This allows us to reinterpret and improve bounds from Harris et al. on the fibre sizes. We then focus on a subset of parking functions, called Motzkin parking functions, where every spot is preferred by at most two cars. We generalise results from Harris et al., and exhibit rich connections to Motzkin paths. We also give a closed enumerative formula for the number of MVP parking functions whose outcome is the complete bipartite permutation. Finally, we give a new interpretation of the MVP outcome map in terms of an algorithmic process on recurrent configurations of the Abelian sandpile model.Comment: 33 pages, 25 figures, 6 table

Complete non-ambiguous trees and associated permutations: connections through the Abelian sandpile model

We study a link between complete non-ambiguous trees (CNATs) and permutations exhibited by Daniel Chen and Sebastian Ohlig in recent work. In this, they associate a certain permutation to the leaves of a CNAT, and show that the number of $n$-permutations that are associated with exactly one CNAT is $2^{n-2}$. We connect this to work by the first author and co-authors linking complete non-ambiguous trees and the Abelian sandpile model. This allows us to prove a number of conjectures by Chen and Ohlig on the number of $n$-permutations that are associated with exactly $k$ CNATs for various $k > 1$, via bijective correspondences between such permutations. We also exhibit a new bijection between $(n-1)$-permutations and CNATs whose permutation is the decreasing permutation $n(n-1)\cdots1$. This bijection maps the left-to-right minima of the permutation to dots on the bottom row of the corresponding CNAT, and descents of the permutation to empty rows of the CNAT.Comment: 29 pages, 15 figure

Convergence de cartes et tas de sable

This Thesis studies various problems located at the boundary between Combinatorics and Probability Theory. It is formed of two independent parts. In the first part, we study the asymptotic properties of some families of \maps" (from a non traditional viewpoint). In thesecond part, we introduce and study a natural stochastic extension of the so-called Sandpile Model, which is a dynamic process on a graph. While these parts are independent, they exploit the same thrust, which is the many interactions between Combinatorics and Discrete Probability, with these two areas being of mutual benefit to each other. Chapter 1 is a general introduction to such interactions, and states the main results of this Thesis. Chapter 2 is an introduction to the convergence of random maps. The main contributions of this Thesis can be found in Chapters 3, 4 (for the convergence of maps) and 5 (for the Stochastic Sandpile model).Cette thèse est dédiée à l'étude de divers problèmes se situant à la frontière entre combinatoire et théorie des probabilités. Elle se compose de deux parties indépendantes : la première concerne l'étude asymptotique de certaines familles de \cartes" (en un sens non traditionnel), la seconde concerne l'étude d'une extension stochastique naturelle d'un processus dynamique classique sur un graphe appelé modèle du tas de sable. Même si ces deux parties sont a priori indépendantes, elles exploitent la même idée directrice, à savoir les interactions entre les probabilités et la combinatoire, et comment ces domaines sont amenés à se rendreservice mutuellement. Le Chapitre introductif 1 donne un bref aperçu des interactions possibles entre combinatoire et théorie des probabilités, et annonce les principaux résultats de la thèse. Le Chapitres 2 donne une introduction au domaine de la convergence des cartes. Les contributions principales de cette thèse se situent dans les Chapitres 3, 4 (pour les convergences de cartes) et 5 (pour le modèle stochastique du tas de sable)

Real-Time Observation of Organic Cation Reorientation in Methylammonium Lead Iodide Perovskites.

The introduction of a mobile and polarized organic moiety as a cation in 3D lead-iodide perovskites brings fascinating optoelectronic properties to these materials. The extent and the time scales of the orientational mobility of the organic cation and the molecular mechanism behind its motion remain unclear, with different experimental and computational approaches providing very different qualitative and quantitative description of the molecular dynamics. Here we use ultrafast 2D vibrational spectroscopy of methylammonium (MA) lead iodide to directly resolve the rotation of the organic cations within the MAPbI3 lattice. Our results reveal two characteristic time constants of motion. Using ab initio molecular dynamics simulations, we identify these as a fast (∼300 fs) "wobbling-in-a-cone" motion around the crystal axis and a relatively slow (∼3 ps) jump-like reorientation of the molecular dipole with respect to the iodide lattice. The observed dynamics are essential for understanding the electronic properties of perovskite materials.This work was supported by The Netherlands Organization for Scientific Research (NWO) through the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) research program. A.A.B. also acknowledges a VENI grant from the NWO. A.A.B. is currently a Royal Society University Research Fellow. Z.S. and Z.C. acknowledge the ANR-2011-JS09-004-01-PvCoNano project and the EU Marie Curie Career Integration Grant (303824). A.A.B., Z.S., and Z.C. thank Dutch-French Academy for the support through van Gogh grant.This document is the Accepted Manuscript version of a Published Work that appeared in final form in The Journal of Physical Chemistry Letters, copyright © American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.jpclett.5b0155

(2Z)-2-Fluoro-N-{4-[5-(4-fluoro­phen­yl)-2-methyl­sulfanyl-1H-imidazol-4-yl]-2-pyrid­yl}-3-phenyl­acrylamide

The asymmetric unit of the title compound, C24H18F2N4OS, contains two crystallographically independent mol­ecules, A and B, which are linked into two chains of A and B mol­ecules by inter­molecular N—H⋯O hydrogen bonds. The three-dimensional network is stabilized by π–π inter­actions between the pyridine rings and phenyl rings of different residues, with centroid–centroid distances of 3.793 (1) and 3.666 (2) Å. The mol­ecular conformation is stabilized by intra­molecular N—H⋯F hydrogen bonds (2.15/2.15Å). The imidazole rings make dihedral angles of 39.5 (2)/38.5 (2) and 31.8 (2)/33.2 (2)° with the 4-fluoro­phenyl rings and the pyridine rings, respectively. The methyl group of molecule A is disorderd in a 0.60:0.40 ratio