A study of students’ mathematical identity was carried out in February 2009 involving participants from two colleges of education, one in Dublin (Republic of Ireland) and one in Belfast (Northern Ireland). All participants were pre-service primary school teachers in the third year of their B.Ed. programme, having chosen to specialize in mathematics. Data was gathered using a questionnaire (with, mainly, open-ended questions) followed by focus groups, involving the same participants, on each campus. This paper considers how students’ exploration of their mathematical identity led them to deepen their insight into learning and teaching mathematics. Recommendations are made for how the methods used in this research might be beneficial on a larger scale, in different environments.

The progress of blended learning has given rise to the need to prepare quality electronic materials, especially those which use the greatest advantage of an electronic document – its interactivity. This paper presents several types of PDF materials – interactive exercises, tests and games created by LaTeX packages (AcroTeX eDucation Bundle) with a contribution of other supporting instruments (3D graphics, fancytooltips, AcroFLeX). Differential equations, as an important tool of continuous mathematical modeling, have been chosen to demonstrate the still increasing power of PDF documents. This strategy allowed me to introduce innovative approaches in explaining and exercising this part of mathematics at the same time. To create such materials some LaTeX knowledge is needed; nevertheless this article is for all math teachers who are looking for quality interactive materials.

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Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Background: Clonal competition in cancer describes the process in which the progeny of a cell clone supersedes or succumbs to other competing clones due to differences in their functional characteristics, mostly based on subsequently acquired mutations. Even though the patterns of those mutations are well explored in many tumors, the dynamical process of clonal selection is underexposed. Methods: We studied the dynamics of clonal competition in a BcrAbl-induced leukemia using a γ-retroviral vector library encoding the oncogene in conjunction with genetic barcodes. To this end, we studied the growth dynamics of transduced cells on the clonal level both in vitro and in vivo in transplanted mice. Results: While we detected moderate changes in clonal abundancies in vitro, we observed monoclonal leukemias in 6/30 mice after transplantation, which intriguingly were caused by only two different BcrAbl clones. To analyze the success of these clones, we applied a mathematical model of hematopoietic tissue maintenance, which indicated that a differential engraftment capacity of these two dominant clones provides a possible explanation of our observations. These findings were further supported by additional transplantation experiments and increased BcrAbl transcript levels in both clones. Conclusion: Our findings show that clonal competition is not an absolute process based on mutations, but highly dependent on selection mechanisms in a given environmental context.

Background: Clonal competition in cancer describes the process in which the progeny of a cell clone supersedes or succumbs to other competing clones due to differences in their functional characteristics, mostly based on subsequently acquired mutations. Even though the patterns of those mutations are well explored in many tumors, the dynamical process of clonal selection is underexposed. Methods: We studied the dynamics of clonal competition in a BcrAbl-induced leukemia using a γ-retroviral vector library encoding the oncogene in conjunction with genetic barcodes. To this end, we studied the growth dynamics of transduced cells on the clonal level both in vitro and in vivo in transplanted mice. Results: While we detected moderate changes in clonal abundancies in vitro, we observed monoclonal leukemias in 6/30 mice after transplantation, which intriguingly were caused by only two different BcrAbl clones. To analyze the success of these clones, we applied a mathematical model of hematopoietic tissue maintenance, which indicated that a differential engraftment capacity of these two dominant clones provides a possible explanation of our observations. These findings were further supported by additional transplantation experiments and increased BcrAbl transcript levels in both clones. Conclusion: Our findings show that clonal competition is not an absolute process based on mutations, but highly dependent on selection mechanisms in a given environmental context.

Die Arbeit behandelt Methoden zur Analyse bzw. Synthese elektromechanischer Systeme mit endlichem Freiheitsgrad (EMS). Dabei wird von einer einheitlichen mathematischen Modellierung solcher Systeme basierend auf dem Prinzip der virtuellen Arbeit in Lagrange'scher Fassung ausgegangen. Als Analysemethoden für strukturfeste EMS werden neben der numerischen Integration die Bestimmung von Gleichgewichtszuständen und die Herleitung der linearisierten Gleichungen zur Schwingungsanalyse dargelegt. Auf die Analyse von strukturvariablen EMS wird ausführlich eingegangen. Dazu werden Phänomene der Strukturvariabilität domänenunabhängig als unilaterale Bindungen aufgefasst und mittels komplementärer Variablen beschrieben. Die kombinatorische Aufgabe der Strukturfindung wird mittels eines linearen Komplementaritätsproblems gelöst. Die Synthese eines EMS wird als inverses Problem der Dynamik aufgefasst. Bei fester Gesamtkonfiguration führt das auf die nichtlineare dynamische Steuerung solcher Systeme. Dazu wird ein so genannter erweiterter PD-Regler - bestehend aus einer nichtlinearen Vorsteuerung auf Basis der inversen Dynamik des EMS und einer linearen Rückführung des Lage- und Geschwindigkeitsfehlers - entworfen. Die globale asymptotische Stabilität dieses Regelgesetzes wird durch explizite Konstruktion einer Lyapunov-Funktion nachgewiesen. Einige Beispiele zur Anwendung der aufgeführten Analyse- und Synthesemethoden runden die Arbeit ab. The thesis deals with methods for analysis and synthesis of electromechanical systems with finite degrees of freedom (EMS). Starting point is a unified mathematical approach to modelling such systems based on the principle of virtual work in Lagrange's formulation. Numerical integration, determination of equilibrium states and derivation of linearized equations are used as analytical methods for EMS with fixed structure. Electromechanical systems with variable structure are regarded explicitly. Phenomena of structural variability are comprehended as unilateral constraints and described using complementary variables. The combinatorial task of finding a valid structure is solved using a linear complementarity problem. The synthesis of EMS is understood as an inverse task of dynamics. Using a fixed configuration, this approach leads to non-linear dynamic control of such systems. A so-called augmented PD-controller - consisting, on the one hand, of a non-linear feedforward based on inverse dynamics of the EMS and, on the other hand, of a linear feedback using position and velocity errors - is designed. Global asymptotic stability is proven by explicit construction of a Lyapunov-function. Some examples showing the usage of the corresponding analytical and synthetic methods are given.

We prove exponential decay for a fractional power of the Green's function for some correlated Anderson models using the fractional moment method.