Stabbing Pairwise Intersecting Disks by Five Points

Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points

Routing in Polygonal Domains

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex. Then, we must be able to route a data packet between any two vertices p and q of Pwhere each step must use only the label of the target node q and the routing table of the current node. For any fixed eps > 0, we pre ent a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps. The labels have O(log n) bits, and the routing tables are of size O((eps^{-1} + h) log n). The preprocessing time is O(n^2 log n + hn^2 + eps^{-1}hn). It can be improved to O(n 2 + eps^{-1}n) for simple polygons

Calcitonin substitution in calcitonin deficiency reduces particle-induced osteolysis

<p>Abstract</p> <p>Background</p> <p>Periprosthetic osteolysis is a major cause of aseptic loosening in joint arthroplasty. This study investigates the impact of CT (calcitonin) deficiency and CT substitution under in-vivo circumstances on particle-induced osteolysis in <it>Calca </it>-/- mice.</p> <p>Methods</p> <p>We used the murine calvarial osteolysis model based on ultra-high molecular weight polyethylene (UHMWPE) particles in 10 C57BL/6J wild-type (WT) mice and twenty <it>Calca </it>-/- mice. The mice were divided into six groups: WT without UHMWPE particles (Group 1), WT with UHMWPE particles (Group 2), <it>Calca </it>-/- mice without UHMWPE particles (Group 3), <it>Calca </it>-/- mice with UHMWPE particles (Group 4), <it>Calca </it>-/- mice without UHMWPE particles and calcitonin substitution (Group 5), and <it>Calca </it>-/- mice with UHMWPE particle implantation and calcitonin substitution (Group 6). Analytes were extracted from serum and urine. Bone resorption was measured by bone histomorphometry. The number of osteoclasts was determined by counting the tartrate-resistant acid phosphatase (TRACP) + cells.</p> <p>Results</p> <p>Bone resorption was significantly increased in <it>Calca </it>-/- mice compared with their corresponding WT. The eroded surface in <it>Calca </it>-/- mice with particle implantation was reduced by 20.6% after CT substitution. Osteoclast numbers were significantly increased in <it>Calca </it>-/- mice after particle implantation. Serum OPG (osteoprotegerin) increased significantly after CT substitution.</p> <p>Conclusions</p> <p>As anticipated, <it>Calca </it>-/- mice show extensive osteolysis compared with wild-type mice, and CT substitution reduces particle-induced osteolysis.</p

Routing and Stabbing

ROUTING. Let G be a simple, connected, undirected graph. We consider routing with preprocessing in G. In a preprocessing step, each vertex of G receives a label and a routing table. Then, we must be able to route a packet between any two vertices s and t of G, where each step may use only the label of the target node t, the routing table of the current node and the packet header. This problem has been studied extensively for general graphs, where efficient routing schemes with polylogarithmic routing tables have turned out to be impossible. Thus, special graph classes are of interest. Let P be an x-monotone orthogonal polygon with n vertices. We call P a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points p and q in P are co-visible if and only if the (axis-parallel) rectangle spanned by p and q completely lies in P. In the r-visibility graph Vis(P) of P, we connect two vertices of P with a unit weighted edge if and only if they are co-visible. We present a routing scheme for visibility graphs of simple and of double histograms that have label size log n and table size O(log n deg(v)) for each vertex v of P, where deg(v) is the degree of v in Vis(P). In simple histograms we can route along a shortest path and need no additional header, whereas in double histograms we need headers of size log n and we can route on a path that has twice the length of an optimal path. The preprocessing time is in both cases O(m), where m is the number of edges in Vis(P). Let V be a set of n sites in the plane. The unit disk graph DG(V) of V is the graph with vertex set V where two sites v and w are adjacent if and only if their Euclidean distance is at most 1. The edge weights correspond to the Euclidean distance of its endpoints. Moreover, we use D to denote the diameter of DG(V). We show that for any given eps>0, we can construct a routing scheme for DG(V) that achieves stretch 1+eps, has label size O(eps^(-1)log D log^3n/loglog n), table size eps^(-O(eps^(-2)))log^3n(1+log D/loglog n) and the header needs at most O(log^2n/loglog n) bits. The preprocessing time is O(eps^(-1)n^2 log^2 n). STABBING. Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. This provides a simple---albeit slightly weaker---algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.ROUTEN. Wir betrachten Routen mit Vorverarbeitung in einem Graphen G. Während der Vorverarbeitung von G erhält jeder Knoten ein Label und eine Routingtabelle. Danach müssen wir in der Lage sein ein Datenpaket zwischen je zwei Knoten s und t von G zu routen, wobei jeder Schritt lediglich das Label von t, die Routingtabelle von s und den Header des Pakets benutzen darf. Das Routingproblem wurde bereits ausführlich für allgemeine Graphen erforscht. Es hat sich herausgestellt, dass kompakte (polylogarithmisch große Routingtabellen) und effiziente Routingschemata für allgemeine Graphen nicht existieren. Sei P ein x-monotones orthogonales Polygon mit n Ecken. Wir bezeichnen P als einfaches Histogramm, wenn der obere Teil des Randes eine einzelne Strecke ist. P ist ein doppeltes Histogramm, wenn es eine horizontale Sehne gibt, welche vom linken zum rechten Rand geht. Zwei Punkte p und q sind co-sichtbar genau dann, wenn das achsenparallele Rechteck, aufgespannt von p und q, komplett in P liegt. Im r-Sichtbarkeitsgraphen Vis(P) gibt es eine Kante zwischen je zwei co-sichtbaren Ecken von P. Wir präsentieren ein Routingschema für Sichtbarkeitsgraphen von einfachen und doppelten Histogrammen, welches Labelgröße log n und Routingtabellengröße O(log n deg(v)) für jede Ecke v von P erreicht, wobei deg(v) der Grad von v in Vis(P) ist. In einfachen Histogrammen können wir entlang eines kürzesten Weges routen und benötigen keinen zusätzlichen Header. Für doppelte Histogramme benötigen wir einen Header der Größe log n und erreichen Stretch 2. In beiden Fällen ist die Vorverarbeitungszeit asymptotisch zur Anzahl der Kanten von Vis(P). Sei V eine Menge von n Punkten in der Ebene. Der Einheitskreisgraph DG(V) ist ein Graph mit Knotenmenge V und einer Kante zwischen je zwei Knoten v und w, wenn ihr Euklidischer Abstand höchstens 1 ist. Die Kantengewichte sind die Euklidischen Abstände. Sei außerdem D der Durchmesser des Graphen. Wir konstruieren für jedes eps>0 ein Routingschema für DG(V). Das Schema erreicht Stretch 1+eps, Labelgröße O(eps^(-1)log Dlog^3n/loglog n), Routingtabellengröße eps^(-O(eps^(-2)))log^3n(1+log D/loglog n) und Headergröße O(log^2n/loglog n). Die Vorverarbeitungszeit beträgt O(eps^(-1)n^2log^2 n). PIERCEN. Sei D eine Menge von n sich paarweise schneidenden Kreisscheiben in der Ebene. Eine Punktmenge P pierct D genau dann, wenn jede Kreisscheibe aus D mindestens einen Punkt aus P enthält. Wir präsentieren einen deterministischen Algorithmus, der O(n) Zeit benötigt um 5 Punkte zu finden, die D piercen. Damit liefern wir eine einache, wenn auch etwas schwächere, algorithmische Version des klassischen Ergebnisses von Danzer, wonach eine solche Menge D immer von 4 Punkten gepierct werden kann. Außerdem geben wir ein einfaches Beispiel mit 13 sich paarweise schneidenden Kreisscheiben an, welches nicht von 3 Punkten gepierct werden kann

Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries

The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n / logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau

Time-resolved temperature profile measurements in the exhaust of a single sector gas turbine combustor at realistic operating conditions

Records of the time-varying temperature profile at flight relevant operating conditions are acquired at the exit of a combustion chamber fitted with a staged, lean-burn fuel injector using high-speed laser induced fluorescence (LIF) at a sample rate of 10 kHz. Temperatures are estimated from the concentration dependent fluorescence of the hydroxyl (OH) radical under the assumption of local equilibrium. Beyond the time-series analysis, the acquired data is correlated with simultaneously acquired OH chemiluminescence sampled in the primary zone near the fuel injector. These analyses reveal a strong influence from the precessing vortex core, originating in the primary zone, on oscillations in the temperature profiles measured at the exit of the combustor