Lower bounds for the simplexity of the n-cube

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{\frac {n-1} 2}$.Comment: 10 page

Extremal properties for dissections of convex 3-polytopes

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.Comment: 19 page

Lectures on 0/1-polytopes

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e.g. using {\tt polymake}). However, any intuition that is derived from the analysis of examples in ``low dimensions'' will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a {\tt polymake} database.Comment: 45 pages, many figures; to appear in "Polytopes - Combinatorics and Computation" (G. Kalai and G.M. Ziegler, eds.), DMV Seminars Series, Birkh"auser Base

Minimal simplicial dissections and triangulations of convex 3-polytopes

SIGLEAvailable from TIB Hannover: F02B140 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes

This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P . . Can the minimal number of tetrahedra in a triangulation be decreased if one allows the use of interior points of P as vertices? . Can a dissection of P use fewer tetrahedra than a triangulation? . Does the size of a minimal triangulation depend on the geometric realization of P? The main result of this paper is that all these questions have an affirmative answer. Even stronger, the gaps of size produced by allowing interior vertices or by using dissections may be linear in the number of points

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Studies of several tetrahedralization problems

The main purpose of decomposing an object into simpler components is to simplify a problem involving the complex object into a number of subproblems having simpler components. In particular, a tetrahedralization is a partition of the input domain in R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations have applications in the finite element method, mesh generation, computer graphics, and robotics. This thesis investigates four problems in tetrahedralizations and triangulations. The first problem is on the computational complexity of tetrahedralization detections. We present an O(nm log n) algorithm to determine whether a set of line segments .C is the edge set of a tetrahedralization, where m is the number of segments and n is the number of endpoints in .C. We show that it is NP-complete to decide whether .C contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations. After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized in O(n²) time. The third problem is on the tetrahedralization of two nested convex polyhedra. We give a method to tetrahedralize the region between two nested convex polyhedra into a linear number of tetrahedra without introducing Steiner points. This result answers an open problem raised by Bern [16]. The fourth problem is on the lower bound for β-skeletons belonging to minimum weight triangulations. We prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value, the β-skeleton of a point set may not always be a subgraph of the minimum weight triangulation of this point set. This result settles Keil's conjecture [62]

Über ausgewählte numerische Zugänge zu Zellgewebe

Different numerical approaches and algorithms arising in the context of modelling of cellular tissue evolution are discussed in this thesis. Being suited in particular to off-lattice agent-based models, the numerical tool of three-dimensional weighted kinetic and dynamic Delaunay triangulations is introduced and discussed for its applicability to adjacency detection. As there exists no implementation of a code that incorporates all necessary features for tissue modelling, algorithms for incremental insertion or deletion of points in Delaunay triangulations and the restoration of the Delaunay property for triangulations of moving point sets are introduced. In addition, the numerical solution of reaction-diffusion equations and their connection to agent-based cell tissue simulations is discussed. In order to demonstrate the applicability of the numerical algorithms, biological problems are studied for different model systems: For multicellular tumour spheroids, the weighted Delaunay triangulation provides a great advantage for adjacency detection, but due to the large cell numbers the model used for the cell-cell interaction has to be simplified to allow for a numerical solution. The agent-based model reproduces macroscopic experimental signatures, but some parameters cannot be fixed with the data available. A much simpler, but in key properties analogous, continuum model based on reaction-diffusion equations is likewise capable of reproducing the experimental data. Both modelling approaches make differing predictions on non-quantified experimental signatures. In the case of the epidermis, a smaller system is considered which enables a more complete treatment of the equations of motion. In particular, a control mechanism of cell proliferation is analysed. Simple assumptions suffice to explain the flow equilibrium observed in the epidermis. In addition, the effect of adhesion on the survival chances of cancerous cells is studied. For some regions in parameter space, stochastic effects may completely alter the outcome. The findings stress the need of establishing a defined experimental model to fix the unknown model parameters and to rule out further models.Diese Arbeit behandelt verschiedene numerische Verfahren zur Modellierung der Entwicklung von Zellgewebe. Das numerische Hilfsmittel der dreidimensionalen gewichteten, kinetischen und dynamischen, Delaunay-Triangulierung, welches insbesondere für gitterfreie agentenbasierte Modelle geeignet ist, wird eingeführt und auf seine Anwendbarkeit in der Nachbarschaftserkennung diskutiert. Da keine numerische Implementierung existiert, welche alle notwendigen Eigenschaften für die Gewebemodellierung beinhaltet, werden Algorithmen für das inkrementelle Einfügen und Löschen von Punkten in Delaunay-Triangulierungen und das Wiederherstellen der Delaunay-Eigenschaft für Mengen sich bewegender Punkte eingeführt. Weiterhin wird die numerische Lösung von Reaktions-Diffusions-Gleichungen und ihre Verbindung zu agentenbasierten Zellgewebesimulationen diskutiert. Um die Anwendbarkeit der numerischen Algorithmen zu demonstrieren, werden für verschiedene Modellsysteme biologische Probleme studiert: Für multizelluläre Tumorsphäroide stellt die gewichtete Delaunay-Triangulierung einen großen Vorteil für die Nachbarschaftserkennung dar, jedoch muss wegen der großen Zellzahlen das Modell für die Zell-Zell-Wechselwirkung deutlich vereinfacht werden, um eine numerische Lösung zu erlauben. Das agentenbasierte Modell reproduziert makroskopische experimentelle Signaturen, jedoch können nicht alle Parameter mit den verfügbaren Daten bestimmt werden. Ein deutlich einfacheres, aber in Schlüsseleigenschaften analoges Kontinuumsmodell, welches auf Reaktions-Diffusions-Gleichungen basiert, kann gleichfalls die experimentellen Daten reproduzieren. Beide Modellansätze machen jedoch verschiedene Aussagen über nicht-quantifizierte experimentelle Signaturen. Im Falle der Epidermis wird ein kleineres System behandelt, was eine vollständigere Behandlung der Bewegungsgleichungen ermöglicht. Insbesondere wird ein Kontrollmechanismus der Zellproliferation analysiert. Einfache Annahmen reichen aus, um das Fließgleichgewicht zu erklären, welches in der Epidermis beobachtet wird. Zusätzlich wird der Effekt der Adhäsion auf die Überlebenschancen von Krebszellen studiert. Für einige Regionen im Parameterraum können stochastische Effekte den Ausgang komplett verändern. Die Resultate unterstreichen die Notwendigkeit der Etablierung eines definierten experimentellen Modellsystems, um unbekannte Modellparameter zu fixieren und Modelle zu falsifizieren