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.

Changes of traditional farming systems and their effects on land degradation and socio-economic conditions in the Inle Lake region, Myanmar

At many locations in Myanmar, ongoing changes in land use have negative environmental impacts and threaten natural ecosystems at local, regional and national scales. In particular, the watershed area of Inle Lake in eastern Myanmar is strongly affected by the environmental effects of deforestation and soil erosion caused by agricultural intensification and expansion of agricultural land, which are exacerbated by the increasing population pressure and the growing number of tourists. This thesis, therefore, focuses on land use changes in traditional farming systems and their effects on socio-economic and biophysical factors to improve our understanding of sustainable natural resource management of this wetland ecosystem. The main objectives of this research were to: (1) assess the noticeable land transformations in space and time, (2) identify the typical farming systems as well as the divergent livelihood strategies, and finally, (3) estimate soil erosion risk in the different agro-ecological zones surrounding the Inle Lake watershed area. GIS and remote sensing techniques allowed to identify the dynamic land use and land cover changes (LUCC) during the past 40 years based on historical Corona images (1968) and Landsat images (1989, 2000 and 2009). In this study, 12 land cover classes were identified and a supervised classification was used for the Landsat datasets, whereas a visual interpretation approach was conducted for the Corona images. Within the past 40 years, the main landscape transformation processes were deforestation (- 49%), urbanization (+ 203%), agricultural expansion (+ 34%) with a notably increase of floating gardens (+ 390%), land abandonment (+ 167%), and marshlands losses in wetland area (- 83%) and water bodies (- 16%). The main driving forces of LUCC appeared to be high population growth, urbanization and settlements, a lack of sustainable land use and environmental management policies, wide-spread rural poverty, an open market economy and changes in market prices and access. To identify the diverse livelihood strategies in the Inle Lake watershed area and the diversity of income generating activities, household surveys were conducted (total: 301 households) using a stratified random sampling design in three different agro-ecological zones: floating gardens (FG), lowland cultivation (LL) and upland cultivation (UP). A cluster and discriminant analysis revealed that livelihood strategies and socio-economic situations of local communities differed significantly in the different zones. For all three zones, different livelihood strategies were identified which differed mainly in the amount of on-farm and off-farm income, and the level of income diversification. The gross margin for each household from agricultural production in the floating garden, lowland and upland cultivation was US$ 2108, 892 and 619 ha-1 respectively. Among the typical farming systems in these zones, tomato (Lycopersicon esculentum L.) plantation in the floating gardens yielded the highest net benefits, but caused negative environmental impacts given the overuse of inorganic fertilizers and pesticides. The Revised Universal Soil Loss Equation (RUSLE) and spatial analysis within GIS were applied to estimate soil erosion risk in the different agricultural zones and for the main cropping systems of the study region. The results revealed that the average soil losses in year 1989, 2000 and 2009 amounted to 20, 10 and 26 t ha-1, respectively and barren land along the steep slopes had the highest soil erosion risk with 85% of the total soil losses in the study area. Yearly fluctuations were mainly caused by changes in the amount of annual precipitation and the dynamics of LUCC such as deforestation and agriculture extension with inappropriate land use and unsustainable cropping systems. Among the typical cropping systems, upland rainfed rice (Oryza sativa L.) cultivation had the highest rate of soil erosion (20 t ha-1yr-1) followed by sebesten (Cordia dichotoma) and turmeric (Curcuma longa) plantation in the UP zone. This study indicated that the hotspot region of soil erosion risk were upland mountain areas, especially in the western part of the Inle lake. Soil conservation practices are thus urgently needed to control soil erosion and lake sedimentation and to conserve the wetland ecosystem. Most farmers have not yet implemented soil conservation measures to reduce soil erosion impacts such as land degradation, sedimentation and water pollution in Inle Lake, which is partly due to the low economic development and poverty in the region. Key challenges of agriculture in the hilly landscapes can be summarized as follows: fostering the sustainable land use of farming systems for the maintenance of ecosystem services and functions while improving the social and economic well-being of the population, integrated natural resources management policies and increasing the diversification of income opportunities to reduce pressure on forest and natural resources.