Development of novel catalysts for the photocatalytic hydrogen formation

Für die zukünftige Umwandlung von Sonnenlicht in chemische Energie wurden schrittweise intramolekular arbeitende Systeme für eine artifizielle Photosynthese aufgebaut. Dabei wurde eine Anzahl von neuen Liganden und den resultierenden Charge-Transfer-Chromophoren (Ru(bpy)3-artig) bzw. katalyseaktiven Komplexen (Pt, Pd) hergestellt. Diese wurden zu heteronuklearen Diaden zusammengefügt, welche aus kovalent verbundenen Funktionsuntereinheiten bestehen. Insbesondere wurden Chromophor und Katalysezentrum so miteinander verbrückt, dass ihre individuellen Eigenschaften weiterbestehen, ihre Einzelfunktionen aber im Zusammenspiel als supramolekularer Photokatalysator fungieren. Somit wird es möglich unter Lichteinstrahlung die Reduktion von Protonen zu Wasserstoff an diesen Katalysatoren durchzuführen. Als Brückeneinheit wurden dafür oligodentate Liganden hergestellt und verwendet, die über Bipyridine und N-heterozyklische Carbene (NHC) die aktiven Metallzentren koordinieren können. Eine Anzahl solcher heterobimetallischen Komplexe, deren Ausgangsverbindungen sowie geeignete Referenzverbindungen wurden hergestellt, charakterisiert und miteinander verglichen. Neben strukturellen, photophysikalischen und elektrochemischen Analysen wurden zusätzlich erfolgreich Katalyseexperimente mit den erzeugten Photokatalysatoren zur lichtgetriebenen Wasserstoffentwicklung aus Wasser durchgeführt. Ebenso wurden repräsentative Kontrollexperimente und zusätzliche Analysen mittels dynamischer Lichtstreuung ausgeführt, um ein tiefergehendes Verständnis von den Prozessen während der Katalyse zu erhalten. Durch Vergleich der Eigenschaften der hergestellten Katalysatoren mit ausgewählten Referenzverbindungen konnten zudem weitere Aussagen über Struktur-Eigenschafts-Beziehungen der Systeme abgeleitet werden

Linear transformation distance for bichromatic matchings

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching} is a plane geometric perfect matching on $P$ such that each edge has one red endpoint and one blue endpoint. Two $BR$-matchings are compatible if their union is also plane. The \emph{transformation graph of $BR$-matchings} contains one node for each $BR$-matching and an edge joining two such nodes if and only if the corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is always connected, but its diameter remained an open question. In this paper we provide an alternative proof for the connectivity of the transformation graph and prove an upper bound of $2n$ for its diameter, which is asymptotically tight

Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $\Omega(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).Comment: This work was presented at the 26th Canadian Conference on Computational Geometry (CCCG 2014), Halifax, Nova Scotia, Canada, 2014. The journal version appeared in Information Processing Letters, 124 (2017), 35--4

An Improved Lower Bound on the Minimum Number of Triangulations

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: (1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. (2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull. (3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture

Packing Short Plane Spanning Trees in Complete Geometric Graphs

Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two trees. Then we show a constant factor approximation for a distributed model in which each point can compute its adjacencies using only local information. This second approach may create cycles, but maintains planarity