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.

On the Benefits of Abstraction in Concurrent Haskell

Heutzutage haben selbst durchschnittliche Computersysteme mehrere unabhängige Recheneinheiten (Kerne). Wird ein rechenintensives Problem in mehrere Teilberechnungen unterteilt, können diese parallel und damit schneller verarbeitet werden. Obwohl die Entwicklung paralleler Programme mittels Abstraktionen vereinfacht werden kann, ist es selbst für Experten anspruchsvoll, effiziente und korrekte Programme zu schreiben. Während traditionelle Programmiersprachen auf einem eher geringen Abstraktionsniveau arbeiten, bieten funktionale Programmiersprachen wie z.B. Haskell, Möglichkeiten zur fortgeschrittenen Abstrahierung. Das Ziel der vorliegenden Dissertation war es, zu untersuchen, wie gut verschiedene Arten der Abstraktion das Programmieren mit Concurrent Haskell unterstützen. Concurrent Haskell ist eine Bibliothek für Haskell, die parallele Programmierung auf Systemen mit gemeinsamem Speicher ermöglicht. Im Mittelpunkt der Dissertation standen zwei Forschungsfragen. Erstens wurden verschiedene Synchronisierungsansätze verglichen, die sich in ihrem Abstraktionsgrad unterscheiden. Zweitens wurde untersucht, wie Abstraktionen verwendet werden können, um die Komplexität der Parallelisierung vor dem Entwickler zu verbergen. Bei dem Vergleich der Synchronisierungsansätze wurden Locks, Compare-and-Swap Operationen und Software Transactional Memory berücksichtigt. Die Ansätze wurden zunächst bezüglich ihrer Eignung für die Synchronisation einer Prioritätenwarteschlange auf Basis von Skiplists untersucht. Anschließend wurden verschiedene Varianten des Taskpool Entwurfsmusters implementiert (globale Taskpools sowie private Taskpools mit und ohne Taskdiebstahl). Zusätzlich wurde für das Entwurfsmuster eine Abstraktionsschicht entwickelt, welche eine einfache Formulierung von Taskpool-basierten Algorithmen erlaubt. Für die Untersuchung der Frage, ob Haskells Abstraktionsmethoden die Komplexität paralleler Programmierung verbergen können, wurden zunächst stencil-basierte Algorithmen betrachtet. Es wurde eine Bibliothek entwickelt, die eine deklarative Beschreibung von stencil-basierten Algorithmen sowie ihre parallele Ausführung erlaubt. Mit Hilfe dieses deklarativen Interfaces wurde die parallele Implementation vollständig vor dem Anwender verborgen. Anschließend wurde eine eingebettete domänenspezifische Sprache (EDSL) für Knoten-basierte Graphalgorithmen sowie eine entsprechende Ausführungsplattform entwickelt. Die Plattform erlaubt die automatische parallele Verarbeitung dieser Algorithmen. Verschiedene Beispiele zeigten, dass die EDSL eine knappe und dennoch verständliche Formulierung von Graphalgorithmen ermöglicht.

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.