On Alternating Non-Context-Free Grammars

We study several extensions of the notion of alternation from context-free grammars to context-sensitive and arbitrary phrase-structure grammars. Thereby new grammatical characterizations are obtained for the class of languages that are accepted by alternating pushdown automata.

Familiale Bewegungssozialisation

Familiale Bewegungssozialisation – Zum Einfluss der Herkunftsfamilie auf die Bewegungssozialisation von Grundschulkindern. Die zentrale Fragestellung der Schrift ist, welchen Einfluss die soziale Herkunft auf die Bewegungssozialisation und Bewegungsentwicklung von Kindern im Grundschulalter hat. Die Auswirkungen sozialer Ungleichheit auf die Bewegungssozialisation, insbesondere die motorische Entwicklung, wurden in der Sportwissenschaft noch unzureichend untersucht. Der Arbeit liegt die Sozialisationstheorie von Witte (1994) zugrunde; mit ihrer Hilfe wird versucht die motorische Entwicklung theoriegeleitet zu erklären. Dafür werden Merkmale der Bewegungssozialisation und der motorischen Entwicklung in das theoretische Rahmenkonzept von Witte (1994) eingesetzt. Zu Beginn (Kapitel 1) erklärt die Arbeit den Begriff der sozialen Herkunft. Es werden der soziale Status, die Familienform und der Migrationshintergrund als Bestandteile der sozialen Herkunft definiert. Im weiteren Verlauf (Kapitel 2) wird die Sozialisationsinstanz Familie und die verschiedenen Familienformen vor dem Hintergrund sozialer Ungleichheit dargelegt. Das dritte Kapitel widmet sich dem sozialisationstheoretischen Konzept. Es werden die Sozialisationstheorie von Hurrelmann, die Körper- und Bewegungskarriere von Baur und das theoretische Rahmenkonzept der Sozialisation von Witte erklärt. Kapitel 4 beschreibt den Forschungsstand und Kapitel 5 stellt die Modellbildung und die Herleitung der Hypothesen dar. Die empirische Untersuchung fand an ausgewählten Grundschulen der Stadt Kassel statt. Insgesamt wurden 251 Kinder im Alter von 7-10 Jahren mit dem AST 6-11 untersucht und deren Eltern mit einem eigens entwickelten Fragebogen befragt. Die Daten wurden mit Hilfe von multivariaten Verfahren in Beziehung zueinander gesetzt. Die Ergebnisse zeigen, dass sich Grundschulkinder hinsichtlich ihrer motorischen Entwicklung nicht in Abhängigkeit der sozialen Herkunft unterscheiden. Jedoch ist das Sportklima der Familien sehr stark abhängig von der sozialen Herkunft. Es wird deutlich, dass Kinder aus sozial benachteiligten Familien, aus Ein-Eltern-Familien und mit Migrationshintergrund schlechtere Möglichkeiten haben sich in ihrer Bewegungssozialisation zu entfalten. Die Prüfung des Sozialisationsmodells zeigt, neben der guten Operationalisierbarkeit des Modells, dass die Modellvariable „Orientierung“ (Orientierung der Familie hinsichtlich der Bedeutung von Bewegung und Sport) den größten Einfluss auf die Bewegungssozialisation von Kindern hat.

Root parametrized differential equations

The present thesis is about the inverse problem in differential Galois Theory. Given a differential field, the inverse  problem asks which linear algebraic groups can be realized as differential Galois groups of Picard-Vessiot extensions of this field.   In this thesis we will concentrate on the realization of the classical groups as differential Galois groups. We introduce a method for a very general realization of these groups. This means that we present for the classical groups of Lie rank $l$ explicit linear differential equations where the coefficients are differential polynomials in $l$ differential indeterminates over an algebraically closed field of constants $C$, i.e. our differential ground field is purely differential transcendental over the constants.   For the groups of type $A_l$, $B_l$, $C_l$, $D_l$ and $G_2$ we managed to do these realizations at the same time in terms of Abhyankar's program 'Nice Equations for Nice Groups'. Here the choice of the defining matrix is important. We found out that an educated choice of $l$ negative roots for the parametrization together with the positive simple roots leads to a nice differential equation and at the same time defines a sufficiently general element of the Lie algebra. Unfortunately for the groups of type $F_4$ and $E_6$ the linear differential equations for such elements are of enormous length. Therefore we keep in the case of $F_4$ and $E_6$ the defining matrix differential equation which has also an easy and nice shape.   The basic idea for the realization is the application of an upper and lower bound criterion for the differential Galois group to our parameter equations and to show that both bounds coincide. An upper and lower bound criterion can be found in literature. Here we will only use the upper bound, since for the application of the lower bound criterion an important condition has to be satisfied. If the differential ground field is $C_1$, e.g., $C(z)$ with standard derivation, this condition is automatically satisfied. Since our differential ground field is purely differential transcendental over $C$, we have no information whether this condition holds or not.   The main part of this thesis is the development of an alternative lower bound criterion and its application. We introduce the specialization bound. It states that the differential Galois group of a specialization of the parameter equation is contained in the differential Galois group of the parameter equation. Thus for its application we need a differential equation over $C(z)$ with given differential Galois group. A modification of a result from Mitschi and Singer yields such an equation over $C(z)$ up to differential conjugation, i.e. up to transformation to the required shape. The transformation of their equation to a specialization of our parameter equation is done for each of the above groups in the respective transformation lemma.

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.