Mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Mathematical modelling in mathematics education and instruction

This paper aims at giving a concise survey of the present state-of-the-art of mathematical modelling in mathematics education and instruction. It will consist of four parts. In part 1, some basic concepts relevant to the topic will be clarified and, in particular, mathematical modelling will be defined in a broad, comprehensive sense. Part 2 will review arguments for the inclusion of modelling in mathematics teaching at schools and universities, and identify certain schools of thought within mathematics education. Part 3 will describe the role of modelling in present mathematics curricula and in everyday teaching practice. Some obstacles for mathematical modelling in the classroom will be analysed, as well as the opportunities and risks of computer usage. In part 4, selected materials and resources for teaching mathematical modelling, developed in the last few years in America, Australia and Europe, will be presented. The examples will demonstrate many promising directions of development.

Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications

Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.

Regelbasiertes Reasoning auf massiv paralleler Hardware

Eine wesentliche Funktionalität bei der Verwendung semantischer Technologien besteht in dem als Reasoning bezeichneten Prozess des Ableitens von impliziten Fakten aus einer explizit gegebenen Wissensbasis. Der Vorgang des Reasonings stellt vor dem Hintergrund der stetig wachsenden Menge an (semantischen) Informationen zunehmend eine Herausforderung in Bezug auf die notwendigen Ressourcen sowie der Ausführungsgeschwindigkeit dar. Um diesen Herausforderungen zu begegnen, adressiert die vorliegende Arbeit das Reasoning durch eine massive Parallelisierung der zugrunde liegenden Algorithmen und der Einführung von Konzepten für eine ressourceneffiziente Ausführung. Diese Ziele werden unter Berücksichtigung der Verwendung eines regelbasierten Systems verfolgt, dass im Gegensatz zur Implementierung einer festen Semantik die Definition der anzuwendenden Ableitungsregeln während der Laufzeit erlaubt und so eine größere Flexibilität bei der Nutzung des Systems bietet. Ausgehend von einer Betrachtung der Grundlagen des Reasonings und den verwandten Arbeiten aus den Bereichen des parallelen sowie des regelbasierten Reasonings werden zunächst die Funktionsweise von Production Systems sowie die dazu bereits existierenden Ansätze für die Optimierung und im Speziellen der Parallelisierung betrachtet. Production Systems beschreiben die grundlegende Funktionalität der regelbasierten Verarbeitung und sind somit auch die Ausgangsbasis für den RETE-Algorithmus, der zur Erreichung der Zielsetzung der vorliegenden Arbeit parallelisiert und für die Ausführung auf Grafikprozessoren (GPUs) vorbereitet wird. Im Gegensatz zu bestehenden Ansätzen unterscheidet sich die Parallelisierung insbesondere durch die gewählte Granularität, die nicht durch die anzuwendenden Regeln, sondern von den Eingabedaten bestimmt wird und sich damit an der Zielarchitektur orientiert. Aufbauend auf dem Konzept der parallelen Ausführung des RETE-Algorithmus werden Methoden der Partitionierung und Verteilung der Arbeitslast eingeführt, die zusammen mit Konzepten der Datenkomprimierung sowie der Verteilung von Daten zwischen Haupt- und Festplattenspeicher ein Reasoning über Datensätze mit mehreren Milliarden Fakten auf einzelnen Rechnern erlauben. Eine Evaluation der eingeführten Konzepte durch eine prototypische Implementierung zeigt für die adressierten leichtgewichtigen Ontologiesprachen einerseits die Möglichkeit des Reasonings über eine Milliarde Fakten auf einem Laptop, was durch die Reduzierung des Speicherbedarfs um rund 90% ermöglicht wird. Andererseits kann der dabei erzielte Durchsatz mit aktuellen State of the Art Reasonern verglichen werden, die eine Vielzahl an Rechnern in einem Cluster verwenden.