Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight''s prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.

In this article, based on Dissection-Motion-Operations, DMO (decomposing a figure into several pieces and composing the resulting pieces into a new figure of equal area), a set of visual representations (models) of mathematical concepts will be introduced. The visual models are producible through manipulation and computer GSP/Cabri software. They are based on the van Hiele’s Levels (van Hiele, 1989) of Thought Development; in particular, Level 2 (Informal Deductive Reasoning) and level 3 (Deductive Reasoning). The basic theme for these models has been visual learning and understanding through manipulatives and computer representations of mathematical concepts vs. rote learning and memorization. The three geometric transformations or motions: Translation, Rotation, Reflection and their possible combinations were used; they are illustrated in several texts. As well, a set of three commonly used dissections or decompositions (Eves, 1972) of objects was utilized.

In this article, based on Dissection-Motion-Operations, DMO (decomposing a figure into several pieces and composing the resulting pieces into a new figure of equal area), a set of visual representations (models) of mathematical concepts will be introduced. The visual models are producible through manipulation and computer GSP/Cabri software. They are based on the van Hiele’s Levels (van Hiele, 1989) of Thought Development; in particular, Level 2 (Informal Deductive Reasoning) and level 3 (Deductive Reasoning). The basic theme for these models has been visual learning and understanding through manipulatives and computer representations of mathematical concepts vs. rote learning and memorization. The three geometric transformations or motions: Translation, Rotation, Reflection and their possible combinations were used; they are illustrated in several texts. As well, a set of three commonly used dissections or decompositions (Eves, 1972) of objects was utilized.

Mit dem Begriff Wellenleiterquantenelektrodynamik (WQED) wird gemeinhin die Physik des quantisierten und in eindimensionalen Wellenleitern geführten Lichtes in Wechselwirkung mit einzelnen Emittern bezeichnet. In dieser Arbeit untersuche ich Effekte der WQED für einzelne Dreiniveausysteme (3NS) bzw. Paare von Zweiniveausystemen (2NS), die in den Wellenleiter eingebettet sind. Hierzu bediene ich mich hauptsächlich numerischer Methoden und betrachte die Modellsysteme im Rahmen der Drehwellennäherung. Ich untersuche die Dynamik der Streuung einzelner Photonen an einzelnen, in den Wellenleiter eingebetteten 3NS. Dabei analysiere ich den Einfluss dunkler bzw. nahezu dunkler Zustände der 3NS auf die Streuung und zeige, wie sich mit Hilfe stationärer elektrischer Treibfelder gezielt auf die Streuung einwirken lässt. Ich quantifiziere Verschränkung zwischen dem Lichtfeld im Wellenleiter und den Emittern mit Hilfe der Schmidt-Zerlegung und untersuche den Einfluss der Form der Einhüllenden eines Einzelphotonpulses auf die Ausbeute der Verschränkungserzeugung bei der Streuung des Photons an einem einzelnen Lambda-System im Wellenleiter. Hier zeigt sich, dass die Breite der Einhüllenden im k-Raum und die Emissionszeiten der beiden Übergänge des 3NS die maßgeblichen Parameter darstellen. Abschließend ergründe ich die Emissionsdynamik zweier im Abstand L in den Wellenleiter eingebetteter 2NS. Diese Dynamik wird insbesondere durch kavitätsartige und polaritonische Zustände des Systems aus Wellenleiter und Emitter ausschlaggebend beeinflusst. Bei der kollektiven Emission der 2NS treten - abhängig vom Abstand L - Sub- bzw. Superradianz auf. Dabei nimmt die Intensität dieser Effekte mit längerem Abstand L zu. Diese Eigenart lässt sich auf die Eindimensionalität des Wellenleiters zurückführen. / The field of waveguide quantum electrodynamics (WQED) deals with the physics of quantised light in one-dimensional (1D) waveguides coupled to single emitters. In this thesis, I investigate WQED effects for single three-level systems (3LS) and pairs of two-level systems (2LS), respectively, which are embedded in the waveguide. To this end, I utilise numerical techniques and consider all model systems within the rotating wave approximation. I investigate the dynamics of single-photon scattering by single, embedded 3LS. In doing so, I analyse the influence of dark and almost-dark states of the 3LS on the scattering dynamics. I also show, how stationary electrical driving fields can control the outcome of the scattering. I quantify entanglement between the waveguide''s light field and single emitters by utilising the Schmidt decomposition. I apply this formalism to a lambda-system embedded in a 1D waveguide and study the generation of entanglement by scattering single-photon pulses with different envelopes on the emitter. I show that this entanglement generation is mainly determined by the photon''s width in k-space and the 3LS''s emission times. Finally, I explore the emission dynamics of a pair of 2LS embedded by a distance L into the waveguide. These dynamics are primarily governed by bound states in the continuum and by polaritonic atom-photon bound-states. For collective emission processes of the two 2LS, sub- and superradiance appear and depend strongly on the 2LS''s distance: the effects increase for larger L. This is an exclusive property of the 1D nature of the waveguide.

Diese Arbeit leistet Beiträge im Bereich der deskriptiven Komplexitätstheorie. Zunächst beschäftigen wir uns mit der ungelösten Frage, ob es eine Logik gibt, welche die Klasse der Polynomialzeit-Eigenschaften (PTIME) charakterisiert. Wir betrachten Graphklassen, die unter induzierten Teilgraphen abgeschlossen sind. Auf solchen Graphklassen lässt sich die 1976 von Gallai eingeführte modulare Zerlegung anwenden. Graphen, die durch modulare Zerlegung nicht zerlegbar sind, heißen prim. Wir stellen ein neues Werkzeug vor: das Modulare Zerlegungstheorem. Es reduziert (definierbare) Kanonisierung einer Graphklasse C auf (definierbare) Kanonisierung der Klasse aller primen Graphen aus C, die mit binären Relationen auf einer linear geordneten Menge gefärbt sind. Mit Hilfe des Modularen Zerlegungstheorems zeigen wir, dass Fixpunktlogik mit Zählen (FP+C) PTIME auf der Klasse aller Permutationsgraphen und auf der Klasse aller chordalen Komparabilitätsgraphen charakterisiert. Wir beweisen zudem, dass modulare Zerlegungsbäume in Symmetrisch-Transitive-Hüllen-Logik mit Zählen (STC+C) definierbar und damit in logarithmischem Platz berechenbar sind. Weiterhin definieren wir eine neue Logik für die Komplexitätsklasse Logarithmischer Platz (LOGSPACE). Wir erweitern die Logik erster Stufe mit Zählen um einen Operator, der eine in logarithmischem Platz berechenbare Form der Rekursion erlaubt. Die resultierende Logik LREC ist ausdrucksstärker als die Deterministisch-Transitive-Hüllen-Logik mit Zählen (DTC+C) und echt in FP+C enthalten. Wir zeigen, dass LREC LOGSPACE auf gerichteten Bäumen charakterisiert. Zudem betrachten wir eine Erweiterung LREC= von LREC, die sich gegenüber LREC durch bessere Abschlusseigenschaften auszeichnet und im Gegensatz zu LREC ausdrucksstärker als die Symmetrisch-Transitive-Hüllen-Logik (STC) ist. Wir beweisen, dass LREC= LOGSPACE sowohl auf der Klasse der Intervallgraphen als auch auf der Klasse der chordalen klauenfreien Graphen charakterisiert. / This theses is making contributions to the field of descriptive complexity theory. First, we look at the main open problem in this area: the question of whether there exists a logic that captures polynomial time (PTIME). We consider classes of graphs that are closed under taking induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed-point logic with counting (FP+C) captures PTIME on the class of permutation graphs and the class of chordal comparability graphs. We also prove that the modular decomposition tree is definable in symmetric transitive closure logic with counting (STC+C), and therefore, computable in logarithmic space. Further, we introduce a new logic for the complexity class logarithmic space (LOGSPACE). We extend first-order logic with counting by a new operator that allows it to formalize a limited form of recursion which can be evaluated in logarithmic space. We prove that the resulting logic LREC is strictly more expressive than deterministic transitive closure logic with counting (DTC+C) and that it is strictly contained in FP+C. We show that LREC captures LOGSPACE on the class of directed trees. We also study an extension LREC= of LREC that has nicer closure properties and that, unlike LREC, is more expressive than symmetric transitive closure logic (STC). We prove that LREC= captures LOGSPACE on the class of interval graphs and on the class of chordal claw-free graphs.

Subject of study: Cilia and flagella are slender appendages of eukaryotic cells. They are actively bending structures and display regular bending waves. These active flagellar bending waves drive fluid flows on cell surfaces like in the case of the ciliated trachea or propels single cell micro-swimmers like sperm or alga. Objective: The axoneme is the evolutionarily conserved mechanical apparatus within cilia and flagella. It is comprised of a cylindrical arrangement of microtubule doublets, which are the elastic elements and dyneins, which are the force generating elements in the axonemal structure. Dyneins collectively bend the axoneme and must be specifically regulated to generate symmetric and highly asymmetric waveforms. In this thesis, I address the question of the molecular origin of the asymmetric waveform and test different theoretical descriptions for motor regulation. Approach: I introduce the isolated and reactivated Chlamydomonas axoneme as an experimental model for the symmetric and asymmetric flagellar beat. This system allows to study the beat in a controlled and cell free environment. I use high-speed microscopy to record shapes with high spatial and temporal resolution. Through image analysis and shape parameterization I extract a minimal set of parameters that describe the flagellar waveform. Using Chlamydomonas, I make use of different structural and motor mutants to study their effect on the shape in different reactivation conditions. Although the isolated axoneme system has many advantages compared to the cell-bound flagellum, to my knowledge, it has not been characterized yet. Results: I present a shape parameterization of the asymmetric beat using Fourier decomposition methods and find, that the asymmetric waveform can be understood as a sinusoidal beat around a circular arc. This reveals the similarities of the two different beat types: the symmetric and the asymmetric beat. I investigate the origin of the beat-asymmetry and find evidence for a specific dynein motor to be responsible for the asymmetry. I furthermore find experimental evidence for a strong sliding restriction at the basal end of the axoneme, which is important to establish a static bend. In collaboration with P. Sartori and F. Jülicher, I compare theoretical descriptions of different motor control mechanisms and find that a curvature controlled motor-regulation mechanism describes the experimental data best. We furthermore find, that in the dynamic case an additional sliding restriction at the base is unnecessary. By comparing the waveforms of intact cells and isolated reactivated axonemes, I reveal the effect of hydrodynamic loading, and the influence of boundary conditions on the shape of the beating flagella.

Subject of study: Cilia and flagella are slender appendages of eukaryotic cells. They are actively bending structures and display regular bending waves. These active flagellar bending waves drive fluid flows on cell surfaces like in the case of the ciliated trachea or propels single cell micro-swimmers like sperm or alga. Objective: The axoneme is the evolutionarily conserved mechanical apparatus within cilia and flagella. It is comprised of a cylindrical arrangement of microtubule doublets, which are the elastic elements and dyneins, which are the force generating elements in the axonemal structure. Dyneins collectively bend the axoneme and must be specifically regulated to generate symmetric and highly asymmetric waveforms. In this thesis, I address the question of the molecular origin of the asymmetric waveform and test different theoretical descriptions for motor regulation. Approach: I introduce the isolated and reactivated Chlamydomonas axoneme as an experimental model for the symmetric and asymmetric flagellar beat. This system allows to study the beat in a controlled and cell free environment. I use high-speed microscopy to record shapes with high spatial and temporal resolution. Through image analysis and shape parameterization I extract a minimal set of parameters that describe the flagellar waveform. Using Chlamydomonas, I make use of different structural and motor mutants to study their effect on the shape in different reactivation conditions. Although the isolated axoneme system has many advantages compared to the cell-bound flagellum, to my knowledge, it has not been characterized yet. Results: I present a shape parameterization of the asymmetric beat using Fourier decomposition methods and find, that the asymmetric waveform can be understood as a sinusoidal beat around a circular arc. This reveals the similarities of the two different beat types: the symmetric and the asymmetric beat. I investigate the origin of the beat-asymmetry and find evidence for a specific dynein motor to be responsible for the asymmetry. I furthermore find experimental evidence for a strong sliding restriction at the basal end of the axoneme, which is important to establish a static bend. In collaboration with P. Sartori and F. Jülicher, I compare theoretical descriptions of different motor control mechanisms and find that a curvature controlled motor-regulation mechanism describes the experimental data best. We furthermore find, that in the dynamic case an additional sliding restriction at the base is unnecessary. By comparing the waveforms of intact cells and isolated reactivated axonemes, I reveal the effect of hydrodynamic loading, and the influence of boundary conditions on the shape of the beating flagella.:Contents 1 Introduction. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Biology of Cilia and Flagella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The dimensions of flagellated micro-swimmers . . . . . . . . . . . . . . . . . 4 1.1.2 The symmetric and the asymmetric beat . . . . . . . . .. . . . . . . . . . . . 5 1.1.3 Chlamydomonas reinhardtii as a flagella model . . . . . . . . . . 5 1.2 The axoneme is the internal structure in eukaryotic cilia and flagella . . 6 1.3 Structure and function of microtubules and dyneins . . . . . . . . . . . 9 1.3.1 Microtubules: The structural elements in the axoneme . . . . . . 9 1.3.2 Dyneins: The force generators that drive the axonemal beat . . . 10 1.3.3 The asymmetries in the axoneme and consequences for the beat 17 1.4 Axonemal waveform models and mechanisms: from sliding to bending to beating . . . . . . . . . . . . . . 20 1.5 Geometrical representation and parameterization of the axonemal beat . . . . . . . . . . . . . . . 23 2 Questions addressed in this thesis . . . . . . . . . . . . . . 27 3 Material and Methods . . . . . . . . . . . . . . 29 3.1 Chlamydomonas cells: Axoneme preparation and motility assays . . . . 29 3.1.1 Culturing of Chlamydomonas reinhardtii cells . . . . . . . . . . . 29 3.1.2 Isolation, demembranation and storage of axonemes . . . . . . . 33 3.1.3 Reactivation of axonemes in controlled conditions . . . . . . . . . 35 3.1.4 Axoneme gliding assay using kinesin 1 . . . . . . . . . . . . . . . 36 3.2 Imaging and image processing . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 High-speed imaging of the flagella and axonemes . . . . . . . . . 38 3.2.2 Precise tracking of isolated axonemes and the flagella of cells . . 42 3.2.3 High throughput frequency evaluation of isolated axonemes . . . 47 3.2.4 Beat frequency characterization of the reactivated WT axoneme . . . . . . . . . . . . . . 49 4 Results and Discussion . . . . . . . . . . . . . . 53 4.1 The beat of the axoneme propagates from base to tip . . . . . . . . . . . 53 4.1.1 TEM study reveals no sliding at the base of a bend axoneme . . 53 4.1.2 The direction of wave propagation is directly determined from the reactivation of polarity marked axonemes . . . . . . . . . . 57 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 The asymmetric beat is the superposition of a static semi-circular arc and a sinusoidal beat . . . . . . .. . . . . . . . . . . . . . . . . 61 4.2.1 The waveform is parameterized by Fourier decomposition in time . . . . . . . . . . . . . . 61 4.2.2 The 0th and 1st Fourier modes describe the axonemal waveform . . . . . . . . . . . . . . 65 4.2.3 General dependence of shape parameters on axoneme length and beat frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.4 The isolated axoneme is a good model for the intact flagellum . .. . . . . . . . . . . . . . 71 4.2.5 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 The circular motion is a consequence of the axonemal waveform . . . . . . . . . . . . . . . . . . . 79 4.3.1 Hydrodynamic relations for small amplitude waves explain the relation between swimming and shape of axonemes . . . . 79 4.3.2 The swimming path can be reconstructed using shape information and a hydrodynamic model . . . . . . . . . . . . . . . . 83 4.3.3 Motor mutations alter the direction of rotation of reactivated axonemes. . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.4 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 The molecular origin of the circular mean shape. . . . . . . . . . . . . . 89 4.4.1 Motor Mutations do not abolish the mean shape, a structural mutation does . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.2 The axoneme is straight in absence of ATP but bend at low ATP concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.3 Viscous load decreases the mean curvature . . . . . . . . . . . . 99 4.4.4 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Curvature-dependent dynein activation accounts for the shape of the beat of the isolated axoneme . . . . . . . . . . . . . . . . 103 5 Conclusions and Outlook . . . . . . . . . . . . . . . . 109 5.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Abbreviations . . . . . . . . . . . . . . . . 111 List of figures . . . . . . . . . . . . . . . . 116 List of tables . . . . . . . . . . . . . . . . 118 Bibliography

Die vorliegende Arbeit befasst sich mit der Fragestellung, inwiefern sich gegebene Verfahren zur Approximation von rationalen Krylow-Räumen zur Berechnung von Matrixfunktionen eignen. Als Modellproblem wird dazu eine 2.5D-Formulierung eines Problems aus der Gleichstrom-Geoelektrik mit finiten Elementen formuliert und dann mittels Matrixfunktionen auf rationalen Krylow-Unterräumen gelöst. Ein weiterer Teil beschäftigt sich mit dem Vergleich zweier Verfahren zur Transformation bestehender rationaler Krylow-Räume. Bei beiden Varianten werden die zugrunde liegenden Pole getauscht ohne dass ein explizites Invertieren von Matrizen notwendig ist. In dieser Arbeit werden die über mehrere Publikationen verteilten Grundlagen einheitlich zusammengetragen und fehlende Zusammenhänge ergänzt. Beide Verfahren eignen sich prinzipiell um rationale Krylow-Räume zu approximieren. Dies wird anhand mehrerer Beispiele gezeigt. Anhand des Modellproblems werden Beschränkungen der Methoden verdeutlicht.