Fakultät für Mathematik, Informatik und Statistik - Digitale Hochschulschriften der LMU - Teil 01/02

The Brown–Ravenhall model is used in quantum physics and chemistry to describe relativistic multiparticle systems, particularly atoms and molecules. In this dissertation we analyse some general properties of this model on the mathematically rigorous level. We show that under very general assumptions on the interaction potentials the essential spectrum of multiparticle Brown–Ravenhall operators is the right semiaxis starting from the minimal energy possible for the decompositions of the system into two clusters. This result, usually called HVZ theorem, is the fundamental starting point in the spectral analysis of multiparticle Hamiltonians with decaying potentials.

Suppose now that the particles constituting the system repel each other but are confined by an external field decaying at infinity. In this situation we prove that the eigenfunctions corresponding to the eigenvalues below the essential spectrum decay exponentially.

If some particles of the system are identical, the laws of quantum mechanics often require to reduce the operator to the subspace of functions which transform according to some irreducible representation of the group of permutation

of identical particles. On the other hand, the interactions are often invariant under some rotations and reflections. We prove that both the HVZ theorem and the exponential decay of eigenfunctions hold true for operators reduced to the irreducible representations of the above groups.

Our results are potentially important in further studies of the spectrum and in the scattering theory of Brown–Ravenhall operators.

Visions of the next generation Web such as the "Semantic Web" or the "Web 2.0" have triggered the emergence of a multitude of data formats. These formats have different characteristics as far as the shape of data is concerned (for example tree- vs. graph-shaped). They are accompanied by a puzzlingly large number of query languages each limited to one data format. Thus, a key feature of the Web, namely to make it possible to access anything published by anyone, is compromised.

This thesis is devoted to versatile query languages capable of accessing data in a variety of Web formats. The issue is addressed from three angles: language design, common, yet uniform semantics, and common, yet uniform evaluation. % Thus it is divided in three parts:

First, we consider the query language Xcerpt as an example of the advocated class of versatile Web query languages. Using this concrete exemplar allows us to clarify and discuss the vision of versatility in detail.

Second, a number of query languages, XPath, XQuery, SPARQL, and Xcerpt, are translated into a common intermediary language, CIQLog. This language has a purely logical semantics, which makes it easily amenable to optimizations. As a side effect, this provides the, to the best of our knowledge, first logical semantics for XQuery and SPARQL. It is a very useful tool for understanding the commonalities and differences of the considered languages.

Third, the intermediate logical language is translated into a query algebra, CIQCAG. The core feature of CIQCAG is that it scales from tree- to graph-shaped data and queries without efficiency losses when tree-data and -queries are considered: it is shown that, in these cases, optimal complexities are achieved. CIQCAG is also shown to evaluate each of the aforementioned query languages with a complexity at least as good as the best known evaluation methods so far. For example, navigational XPath is evaluated with space complexity O(q d) and time complexity O(q n) where q is the query size, n the data size, and d the depth of the (tree-shaped) data.

CIQCAG is further shown to provide linear time and space evaluation of tree-shaped queries for a larger class of graph-shaped data than any method previously proposed. This larger class of graph-shaped data, called continuous-image graphs, short CIGs, is introduced for the first time in this thesis. A (directed) graph is a CIG if its nodes can be totally ordered in such a manner that, for this order, the children of any node form a continuous interval.

CIQCAG achieves these properties by employing a novel data structure, called sequence map, that allows an efficient evaluation of tree-shaped queries, or of tree-shaped cores of graph-shaped queries on any graph-shaped data. While being ideally suited to trees and CIGs, the data structure gracefully degrades to unrestricted graphs. It yields a remarkably efficient evaluation on graph-shaped data that only a few edges prevent from being trees or CIGs.

Kryptologie, die Wissenschaft von den Geheimschriften und ihrer Entschlüsselung, erfährt in

der gegenwärtigen Zeit zunehmend praktische Bedeutung. Sie gewährleistet nicht nur Vertraulichkeit,

Integrität und Authentizität beim Nachrichtenaustausch, sondern bestimmt auch die

Sicherheit des elektronischen Geschäftsverkehrs, des Datenschutzes und ermöglicht digitale

Signaturen. Gleichzeitig stützen sich moderne Verfahren der Kryptologie auf die Zahlentheorie,

deren Grundlagen bereits in der Sekundarstufe I bereitgestellt werden. In der Vereinigung von

mathematischen Inhalten, historischen Entwicklungen und aktuellen gesellschaftspolitischen

Aspekten bietet Kryptologie vielfältige Einsatzmöglichkeiten für den Unterricht an allgemeinbildenden

Schulen. Dass kryptologische Inhalte dennoch nicht in den Schulunterricht integriert

sind, ist Anlass dieser Arbeit.

Ausgehend vom Bildungs- und Erziehungsauftrag allgemeinbildender Schulen wird zunächst

die Berechtigung eines Unterrichts in Kryptologie nachgewiesen. Anschließend wird dessen

didaktischer Ort festgelegt, es werden geeignete Lerninhalte ausgewählt und in eine zeitliche

Abfolge gebracht. In einer Unterrichtssequenz werden diese kryptologischen Lerninhalte didaktisch

aufbereitet, auf Verständnisebene der Schüler transferiert und mit methodischen Hinweisen

für Lehrkräfte versehen.

Although the definition of symplectic field theory suggests that one has to count holomorphic curves in cylindrical manifolds equipped with a cylindrical almost complex structure, it is already well-known from Gromov-Witten theory that, due to the presence of multiply-covered curves, we in general cannot achieve transversality for all moduli spaces even for generic choices. In this thesis we treat the transversality problem of symplectic field theory in two important cases. In the first part of this thesis we are concerned with the rational symplectic field theory of Hamiltonian mapping tori, which is also called the Floer case. For this observe that in the general geometric setup for symplectic field theory, the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder and allows us to compute the full contact homology. The second part of this thesis is devoted to the branched covers of trivial cylinders over closed Reeb orbits, which are the trivial examples of punctured holomorphic curves studied in rational symplectic field theory. Since all moduli spaces of trivial curves with virtual dimension one cannot be regular, we use obstruction bundles in order to find compact perturbations making the Cauchy-Riemann operator transversal to the zero section and show that the algebraic count of elements in the resulting regular moduli spaces is zero. Once the analytical foundations of symplectic field theory are established, our result

implies that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration. After introducing additional marked points and differential forms on the target manifold we finally use our result to compute the second page of the corresponding spectral sequence for filtered complexes.

Nuclear magnetic resonance spectroscopy (NMR) is one of the most important methods for measuring the three-dimensional structure of biomolecules. Despite major progress in the NMR methodology, the solution of a protein structure is still a tedious and time-consuming task. The goal of this thesis is to develop bioinformatics methods which may strongly accelerate the NMR process. This work concentrates on a special type of measurements, the so-called chemical shifts. Chemical shifts are routinely measured at the beginning of a structure resolution process. As all data from the laboratory, chemical shifts may be error-prone, which might complicate or even circumvent the use of this data. Therefore, as the first result of the thesis, we present CheckShift, a method which automatically corrects a frequent error in NMR chemical shift data. However, the main goal of this thesis is the extraction of structural information hidden in chemical shifts. SimShift was developed as a first step in this direction. SimShift is the first approach to identify structural similarities between proteins based on chemical shifts. Compared to methods based on the amino acid sequence alone, SimShift shows its strength in detecting distant structural relationships. As a natural further development of the pairwise comparison of proteins, the SimShift algorithm is adapted for database searching. Given a protein, the improved algorithm, named SimShiftDB, searches a database of solved proteins for structurally homologue entries. The search is based only on the amino acid sequence and the associated chemical shifts. The detected similarities are additionally ranked based on calculations of statistical significance. Finally, the Chemical Shift Pipeline, the main result of this work, is presented. By combining automatic chemical shift error correction (CheckShift) and the database search algorithm (SimShiftDB), it is possible to achieve very high quality in 70% to 80% of the similarities identified. Thereby, only about 10% of the predictions are in error.

In dieser Arbeit wir ein pragmatischer Ansatz zur Typisierung, statischen Analyse und Optimierung von Web-Anfragespachen, speziell Xcerpt, untersucht. Pragmatisch ist der Ansatz in dem Sinne, dass dem Benutzer keinerlei Einschränkungen aus Entscheidbarkeits- oder Effizienzgründen auf modellierbare Typen gestellt werden. Effizienz und Entscheidbarkeit werden stattdessen, falls nötig, durch Vergröberungen bei der Typprüfung erkauft.

Eine Typsprache zur Typisierung von Graph-strukturierten Daten im Web wird eingeführt. Modellierbare Graphen sind so genannte gewurzelte Graphen, welche aus einem Spannbaum und Querreferenzen aufgebaut sind. Die Typsprache basiert auf

reguläre Baum Grammatiken, welche um typisierte Referenzen erweitert wurde. Neben wie im Web mit XML üblichen geordneten strukturierten Daten, sind auch ungeordnete Daten, wie etwa in Xcerpt oder RDF üblich, modellierbar. Der dazu verwendete Ansatz---ungeordnete Interpretation Regulärer Ausdrücke---ist neu. Eine operationale Semantik für geordnete wie ungeordnete Typen wird auf Basis spezialisierter Baumautomaten und sog. Counting Constraints (welche wiederum auf presburgerarithmetische Ausdrücke) basieren. Es wird ferner statische Typ-Prüfung und -Inferenz von Xcerpt Anfrage- und Konstrukttermen, wie auch Optimierung von Xcerpt Anfragen auf Basis von Typinformation eingeführt.

In der vorliegenden Arbeit wird eine Technik vorgestellt, die Software bereits aufgrund ihrer Modellbeschreibung in ihrem zeitlichen Verhalten bewertbar macht. Dazu wird eine Technik zur Beschreibung dynamischer Ausführungsmodelle für Zeitverhalten, sogenannte dynamische Performanz-Modelle, eingeführt. Ein Modellierungtechnik für Umgebungen wird beschrieben. Schließlich erlaubt die Definition einer zeitbehafteten Semantik die Bewertung einzelner Bearbeitungsabläufe.

Mittels einer Petrinetzsemantik wird einerseits das Modellverhalten exakt festgelegt und andererseits ein Wohlgeformtheitsbegriff eingeführt, der einer späteren Werkzeugimplementierung dient. Die vorteilhaften technischen Eigenschaften dieses Entwurfes erlauben zudem eine Übersetzung aus UML-Aktivitätsdiagrammen.

Anhand der Bewertung einzelner Abläufe wird das zeitliche Verhalten eines dynamischen Modells analysiert. Diese einzelnen Abläufe, sogenannte Szenarien, werden durch das Werkzeug bewertet. Jeweils ein Tripel aus einem dynamischen Modell, einem Szenario und einer passenden Umgebung läßt sich mit einen Zeitwert bewerten. Dynamischer Modelle können direkt aus UML-Aktivitätsdiagrammen erzeugt werden. Szenarien können graphisch editiert und Last- und Ressourcenmodellen aus Meßwerttabellen generiert werden. Die ermittelten Zeitwerte können dann graphisch aufbereitet und zur Beurteilung der dynamischen Eigenschaften herangezogen werden.

Die hier entwickelten Modelle und eingesetzten formalen Spezifikationstechniken werden in der Werkzeugstudie vollständig gekapselt und müssen vom Werkzeuganwender nicht beherrscht werden. Somit stellt diese Arbeit auch ein Musterbeispiel für den Software-Entwurf unter Einbeziehung formaler Methoden dar.

Two recent technological advances have extended the diversity of domains and social contexts of Human-Computer Interaction: the embedding of computing capabilities into physical hand-held objects, and the emergence of large interactive surfaces, such as tabletops and wall boards. Both interactive surfaces and small computational devices usually allow for direct and space-multiplex input, i.e., for the spatial coincidence of physical action and digital output, in multiple points simultaneously. Such a powerful combination opens novel opportunities for the design of what are considered as hybrid interactions in this work.

This thesis explores the affordances of physical interaction as resources for interface design of such hybrid interactions. The hybrid systems that are elaborated in this work are envisioned to support specific social and physical contexts, such as collaborative cooking in a domestic kitchen, or collaborative creativity in a design process. In particular, different aspects of physicality characteristic of those specific domains are explored, with the aim of promoting skill transfer across domains.

irst, different approaches to the design of space-multiplex, function-specific interfaces are considered and investigated. Such design approaches build on related work on Graspable User Interfaces and extend the design space to direct touch interfaces such as touch-sensitive surfaces, in different sizes and orientations (i.e., tablets, interactive tabletops, and walls).

These approaches are instantiated in the design of several experience prototypes: These are evaluated in different settings to assess the contextual implications of integrating aspects of physicality in the design of the interface. Such implications are observed both at the pragmatic level of interaction (i.e., patterns of users' behaviors on first contact with the interface), as well as on user' subjective response. The results indicate that the context of interaction affects the perception of the affordances of the system, and that some qualities of physicality such as the 3D space of manipulation and relative haptic feedback can affect the feeling of engagement and control. Building on these findings, two controlled studies are conducted to observe more systematically the implications of integrating some of the qualities of physical interaction into the design of hybrid ones.

The results indicate that, despite the fact that several aspects of physical interaction are mimicked in the interface, the interaction with digital media is quite different and seems to reveal existing mental models and expectations resulting from previous experience with the WIMP paradigm on the desktop PC.

In recent years data sets have become increasingly more complex

requiring more flexible instruments for their analysis. Such a flexible

instrument is regression analysis based on a structured additive

predictor which allows an appropriate modelling for different

types of information, e.g.~by using smooth functions for spatial information,

nonlinear functions for continuous covariates or by using effects for the

modelling of cluster--specific heterogeneity.

In this thesis, we review many important effects.

Moreover, we place an emphasis on interaction terms and introduce a

possibility for the simple modelling of a complex interaction between two continuous covariates. \\

Mainly, this thesis is concerned with the topic of variable and

smoothing parameter selection within structured additive regression

models. For this purpose, we introduce an efficient algorithm that

simultaneously selects relevant covariates and the degree of smoothness

for their effects. This algorithm is even capable of handling complex situations with many covariates and observations.

Thereby, the validation of different models is based

on goodness of fit criteria, like e.g.~AIC, BIC or GCV. The methodological

development was strongly motivated by case studies from different areas. As examples, we analyse

two different data sets regarding determinants of undernutrition in India and of rate making for

insurance companies. Furthermore, we examine the performance or our selection

approach in several extensive simulation studies.