The Theory of Difference Potentials in the Three-Dimensional Case

The method of difference potentials can be used to solve discrete elliptic boundary value problems, where all derivatives are approximated by finite differences. Considering the classical potential theory, an integral equation on the boundary will be investigated, which is solved approximately by the help of a quadrature formula. The advantage of the discrete method consists in the establishment of a linear equation system on the boundary, which can be immediately solved on the computer. The described method of difference potentials is based on the discrete Laplace equation in the three-dimensional case. In the first step the integral representation of the discrete fundamental solution is presented and the convergence behaviour with respect to the continuous fundamental solution is discussed. Because the method can be used to solve boundary value problems in interior as well as in exterior domains, it is necessary to explain some geometrical aspects in relation with the discrete domain and the double-layer boundary. A discrete analogue of the integral representation for functions in will be presented. The main result consists in splitting the difference potential on the boundary into a discrete single- and double-layer potential, respectively. The discrete potentials are used to establish and solve a linear equation system on the boundary. The actual form of this equation systems and the conditions for solvability are presented for Dirichlet and Neumann problems in interior as well as in exterior domain

Fundamental Solutions for Partial Difference Operators and the Solution of Discrete Boundary Value Problems by the Help of Difference Potentials

Im Mittelpunkt der Dissertation steht die Theorie der Differenzenpotentiale, die eng mit der klassischen Potentialtheorie verbunden ist. Vorgestellt wird eine Methode zur Lösung von Randwertproblemen, die nicht auf der Diskretisierung einer Randintegralgleichung beruht, sondern von der Übertragung des Problems in ein Differenzenrandwertproblem ausgeht. Das diskrete Randwertproblem wird mit Hilfe einer Randreduktionsmethode auf eine Randoperatorgleichung transformiert, die detaillierter zu untersuchen ist. Voraussetzung für den Aufbau der Theorie ist die Existenz diskreter Fundamentallösungen. Die Definition der Differenzenpotentiale wird von Ryabenkij übernommen. Seine Herangehensweise führt jedoch zu überbestimmten linearen Gleichungssystemen auf dem Rand. Durch die Aufspaltung des Randpotentials in ein diskretes Einfach- und Doppelschichtpotential wird diese Schwierigkeit in der Dissertation überwunden. Bewiesen werden Eindeutigkeits- und Lösbarkeitsaussagen für Differenzenrandwertprobleme. Das onvergenzverhalten der diskreten Potentiale wird im Kapitel 3 untersucht. Im Kapitel 4 werden numerische Resultate vorgestellt.The theses are based on the theory of difference potentials, which are closely related to the classical potential theory. A method for solving boundary value problems is presented, that does not start from the discretization of a boundary integral equation. In the first step the original problem is replaced by a discrete boundary value problem. By the help of a boundary reduction method the discrete problem is transformed into a boundary operator equation, which is to study in more detail. An important assumption for the theory of difference potentials is the existence of discrete fundamental solutions. The definition of the difference potentials is taken from Ryabenkij. His approach leads to overdetermined linear equation systems on the boundary. By splitting the boundary potential into a discrete single-layer and double-layer potential these problems are solved in the theses. Uniqueness and existence theorems are proved for discrete boundary value problems. The convergence of the discrete potentials is investigated in chapter 3. In chapter 4 numerical results are presented

Diskret holomorphe Funktionen und deren Bedeutung bei der Lösung von Differenzengleichungen

Auf der Grundlage diskreter Cauchy-Riemann Operatoren werden diskret holomorphe Funktionen definiert und detailliert studiert. Darauf aufbauend wird die Lösung von Differenzengleichungen mit Hilfe der diskret holomorphen Funktionen beschrieben

Finite Difference Approximations of the Cauchy-Rieman Operators and the Solution of Discrete Stokes and Navier-Stokes Problems in the Plane

We give a summary of our results based on discrete Cauchy-Riemann operators in the plane. These operators are defined in a way that the factorization of the real Laplacian into two adjoint Cauchy-Riemann operators is possible. This property is similar to the continuous case and can especially be used for calculating the discrete fundamental solution of our finite difference operators. Based on the discrete fundamental solution we define a discrete operator that is right inverse to the discrete Cauchy-Riemann operator. In relation with this operator and an operator on the boundary we are able to prove a discrete version of the Borel-Pompeiu formula. In the second part we present a possibility to solve discrete Stokes and Navier-Stokes problems. The concept is based on the orthogonal decomposition of the space l2 into the space of discrete holomorphic functions and its orthogonal complement. By introducing the orthoprojectors P+h and Q+h we can prove the existence and uniqueness of the solution of discrete Stokes problems. In addition we state a problem that is equivalent to the discrete Navier-Stokes problem and can be used in an iteration procedure to describe the solution of this problem. For a special case of the Navier-Stokes equations we are able to calculate discrete potential and stream functions. The adapted model includes important algebraical properties and can immediately be used for numerical calculations. A numerical example is presented at the end of the article

THE RELATIONSHIP BETWEEN LINEAR ELASTICITY THEORY AND COMPLEX FUNCTION THEORY STUDIED ON THE BASIS OF FINITE DIFFERENCES

It is well-known that the solution of the fundamental equations of linear elasticity for a homogeneous isotropic material in plane stress and strain state cases can be equivalently reduced to the solution of a biharmonic equation. The discrete version of the Theorem of Goursat is used to describe the solution of the discrete biharmonic equation by the help of two discrete holomorphic functions. In order to obtain a Taylor expansion of discrete holomorphic functions we introduce a basis of discrete polynomials which fulfill the so-called Appell property with respect to the discrete adjoint Cauchy-Riemann operator. All these steps are very important in the field of fracture mechanics, where stress and displacement fields in the neighborhood of singularities caused by cracks and notches have to be calculated with high accuracy. Using the sum representation of holomorphic functions it seems possible to reproduce the order of singularity and to determine important mechanical characteristics

Optimale Trassenführung: Diskretisierung - Splineapproximation - Variationsmethoden

Ausgehend von mathematischen Überlegungen haben wir einfache Modellansätze zur Bearbeitung des folgenden Optimierungsproblems erarbeitet und numerische Tests durchgeführt: Eine Landkarte wird in Quadrate unterteilt, wobei jedes Quadrat mit einem Faktor zu bewerten ist. Dieser Wichtungsfaktor sei klein, wenn das Gebiet problemlos passierbar ist und entsprechend groß, wenn es sich um ein Naturschutz-gebiet, einen See oder ein schwer befahrbares Gebiet handelt. Gesucht wird nach einer günstigen Verbindung vom Punkt A zum Punkt B, wobei die durch den Wichtungsfaktor gegebenen landschaftlichen Besonderheiten zu berücksichtigen sind. Wir formulieren das Problem zunächst als Variationsproblem. Eine notwendige Bedingung, der die Lösungsfunktion genügen muß, ist die Euler-Lagrangesche Differentialgleichung. Mit Hilfe der Hamiltonschen Funktion ist es möglich, diese Differentialgleichung in kanonischer Form zu schreiben. Durch Vereinfachung des Modelles gelingt es, das System der kanonischen Gleichungen so zu konkretisieren, daß es als Ausgangspunkt für numerische Untersuchungen betrachtet werden kann. Dazu verwandeln wir die ursprüngliche Landschaft in eine >Berglandschaft<, wobei hohe Berge schwer passierbare Gebiete charakterisieren. Das einfachste Modell ist ein einzelner Berg, der mit Hilfe der Dichtefunktion einer zweidimensionalen Normalverteilung erzeugt wird. Zusätzlich haben wir Berechnungen an zwei sich überlagernden Bergen sowie einer Schlucht durchgeführt

Diskret holomorphe Funktionen und deren Bedeutung bei der Lösung von Differenzengleichungen

Auf der Grundlage diskreter Cauchy-Riemann Operatoren werden diskret holomorphe Funktionen definiert und detailliert studiert. Darauf aufbauend wird die Lösung von Differenzengleichungen mit Hilfe der diskret holomorphen Funktionen beschrieben

Action comprehension: deriving spatial and functional relations.

A perceived action can be understood only when information about the action carried out and the objects used are taken into account. It was investigated how spatial and functional information contributes to establishing these relations. Participants observed static frames showing a hand wielding an instrument and a potential target object of the action. The 2 elements could either match or mismatch, spatially or functionally. Participants were required to judge only 1 of the 2 relations while ignoring the other. Both irrelevant spatial and functional mismatches affected judgments of the relevant relation. Moreover, the functional relation provided a context for the judgment of the spatial relation but not vice versa. The results are discussed in respect to recent accounts of action understanding

Inducible IL-7 Hyperexpression Influences Lymphocyte Homeostasis and Function and Increases Allograft Rejection

The IL-7/IL-7R pathway is essential for lymphocyte development and disturbances in the pathway can lead to immune deficiency or T cell mediated destruction. Here, the effect of transient hyperexpression of IL-7 was investigated on immune regulation and allograft rejection under immunosuppression. An experimental in vivo immunosuppressive mouse model of IL-7 hyperexpression was developed using transgenic mice (C57BL/6 background) carrying a tetracycline inducible IL-7 expression cassette, which allowed the temporally controlled induction of IL-7 hyperexpression by Dexamethasone and Doxycycline treatment. Upon induction of IL-7, the B220+ c-kit+ Pro/Pre-B I compartment in the bone marrow increased as compared to control mice in a serum IL-7 concentration-correlated manner. IL-7 hyperexpression also preferentially increased the population size of memory CD8+ T cells in secondary lymphoid organs, and reduced the proportion of CD4+Foxp3+ T regulatory cells. Of relevance to disease, conventional CD4+ T cells from an IL-7-rich milieu escaped T regulatory cell-mediated suppression in vitro and in a model of autoimmune diabetes in vivo. These findings were validated using an IL-7/anti-IL7 complex treatment mouse model to create an IL-7 rich environment. To study the effect of IL-7 on islet graft survival in a mismatched allograft model, BALB/c mice were rendered diabetic by streptozotocin und transplanted with IL-7-inducible or control islets from C57BL/6 mice. As expected, Dexamethasone and Doxycycline treatment prolonged graft median survival as compared to the untreated control group in this transplantation mouse model. However, upon induction of local IL-7 hyperexpression in the transplanted islets, graft survival time was decreased and this was accompanied by an increased CD4+ and CD8+ T cell infiltration in the islets. Altogether, the findings show that transient elevations of IL-7 can impair immune regulation and lead to graft loss also under immune suppression