The analysis of time-delayed dynamics on networks may help to understand many systems from physics, biology, and engineering, such as coupled laser arrays, gene-regulatory networks and complex ecosystems. Beside the complexity due to the network structure, the analysis is further complicated by the presence of the delays. Delay systems are in general infinite dimensional and thus can display complex dynamics as oscillations and chaos. The mathematical difficulties related to the delays hinders the analysis of delay networks. Thus, little is known yet about basic relations between network structure and delay dynamics. It has been shown that networks without delays can be studied efficiently with the generalized modeling approach, which analyzes the stability of an assumed steady state by a direct parametrization of the Jacobian matrix. In this thesis, I demonstrate the extension of the generalized modeling approach to delay networks and analyze networks of delay-coupled delay oscillators, with delayed auto-catalytic growth on the nodes and delayed transport between nodes. For degree-homogeneous networks (DHONs), in which each node has the same number of links, the bifurcation lines that border the stable areas can be calculated analytically, where the topology of the network is described only by the eigenvalues of the adjacency matrix. For undirected networks, the stability pattern in the parameter space of growth and transport delay is governed by two periodic sets of tongues of instability, which depend on the largest positive and the smallest negative eigenvalue. The direct relation between the eigenvalue and the bifurcation lines allows us to predict stability patterns for networks with certain topological properties. Thus, bipartite networks display a characteristic periodicity of tongues. In order to analyze the stability of degree-heterogeneous networks (DHENs), I apply a numerical sampling method based on Cauchy\'s Argument Principle. The stability patterns of these networks resembles the pattern of DHONs, which is governed by the two periodic sets. For networks with sufficiently many links, one set disappears, and the stability of DHENs can be approximates by the stability of a fully-connected network with the same average degree. However, random DHENs tend to be more stable than DHONs, and DHENs with a broad degree-distribution tend to be more stable than DHENs with a narrow distribution. Thus, such networks are more likely to give rise to amplitude death, i.e. the stabilization of an unstable steady state through diffusive coupling. The stability pattern of DHENs can be qualitatively different than the pattern in DHONs. However, for small growth delays, close to the critical delay of the single node system, the bifurcation lines of all DHENs with the same average degree coincide. This, is particularly interesting, because there the stability depends on a global property of the network, which suggests a diverging interaction length. In summary, the extension of generalized modeling to time-delay networks reveals basic relations between the delay dynamics and the topology. The generality of our model should allow to apply these results to a large class of real-world systems.

The subject of the thesis is the investigation of asymptotic properties of solutions of the Cauchy problem for structurally damped sigma-evolution operators with time dependent, monotonous, dissipation term. An appropriate energy for solutions of the sigma-evolution equations is defined and some estimates for energies of higher order are proved. In the scale invariant case the optimality of these estimates is shown. Further, the influence of properties of the time dependent dissipation on L^p-L^q estimates for the energy with p and q bigger or equal to 2 and from the conjugate line is clarified. Also smoothing properties of the operators under consideration are investigated. The connection between the regularity of the data and the regularity of the solution in terms of L^2 based Gevrey spaces is considered. Finally, L^1-L^1-estimates in the special case delta = sigma/2 and decreasing dissipative coefficient. / Thema der vorliegenden Dissertation ist die Untersuchung asymptotischer Eigenschaften von Lösungen des Cauchy Problems für strukturell gedämpfte sigma-Evolutions-Operatoren mit zeitabhängigem, monotonen Dissipationskoeffizienten. Es wird eine geeignete Energie definiert und für diese Abschätzungen, auf für entsprechende Energien höherer Ordnung gezeigt. Darüber hinaus wird der Einfluss des Dissipationskoeffizienten auf L^p-L^q Abschätzungen auf und entfernt von der konjugierten Linie untersucht. Im skaleninvarianten Fall wird die Schärfe der Abschätzungen bewiesen. Weiterhin wird der Zusammenhang zwischen der Regularität der Daten und der der Lösung in Termen von L^2-basierten Gevrey-Räumen untersucht. Schließlich werden L^1-L^1-Abschätzungen für den Spezialfall delta = sigma/2 und monoton fallenden Dissipationskoeffizienten gezeigt.

This book is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. We prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

The present thesis considers the problem of reconstructing a function f that is defined on the d-dimensional unit sphere from its mean values along hyperplane sections. In case of the two-dimensional sphere, these plane sections are circles. In many tomographic applications, however, only limited data is available. Therefore, one is interested in the reconstruction of the function f from its mean values with respect to only some subfamily of all hyperplane sections of the sphere. Compared with the full data case, the limited data problem is more challenging and raises several questions. The first one is the injectivity, i.e., can any function be uniquely reconstructed from the available data? Further issues are the stability of the reconstruction, which is closely connected with a description of the range, as well as the demand for actual inversion methods or algorithms. We provide a detailed coverage and answers of these questions for different families of hyperplane sections of the sphere such as vertical slices, sections with hyperplanes through a common point and also incomplete great circles. Such reconstruction problems arise in various practical applications like Compton camera imaging, magnetic resonance imaging, photoacoustic tomography, Radar imaging or seismic imaging. Furthermore, we apply our findings about spherical means to the cone-beam transform and prove its singular value decomposition. / Die vorliegende Arbeit beschäftigt sich mit dem Problem der Rekonstruktion einer Funktion f, die auf der d-dimensionalen Einheitssphäre definiert ist, anhand ihrer Mittelwerte entlang von Schnitten mit Hyperebenen. Im Fall d=2 sind diese Schnitte genau die Kreise auf der Sphäre. In vielen tomografischen Anwendungen sind aber nur eingeschränkte Daten verfügbar. Deshalb besteht das Interesse an der Rekonstruktion der Funktion f nur anhand der Mittelwerte bestimmter Familien von Hyperebenen-Schnitten der Sphäre. Verglichen mit dem Fall vollständiger Daten birgt dieses Problem mehrere Herausforderungen und Fragen. Die erste ist die Injektivität, also können alle Funktionen anhand der gegebenen Daten eindeutig rekonstruiert werden? Weitere Punkte sind die die Frage nach der Stabilität der Rekonstruktion, welche eng mit einer Beschreibung der Bildmenge verbunden ist, sowie der praktische Bedarf an Rekonstruktionsmethoden und -algorithmen. Diese Arbeit gibt einen detaillierten Überblick und Antworten auf diese Fragen für verschiedene Familien von Hyperebenen-Schnitten, angefangen von vertikalen Schnitten über Schnitte mit Hyperebenen durch einen festen Punkt sowie Kreisbögen. Solche Rekonstruktionsprobleme treten in diversen Anwendungen auf wie der Bildgebung mittels Compton-Kamera, Magnetresonanztomografie, fotoakustischen Tomografie, Radar-Bildgebung sowie der Tomografie seismischer Wellen. Weiterhin nutzen wir unsere Ergebnisse über sphärische Mittelwerte, um eine Singulärwertzerlegung für die Kegelstrahltomografie zu zeigen.

In this work we study several important properties of the 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability. Due to the richness of the weight sequence used to measure smoothness, this scale of function spaces incorporates a wide range of function spaces, of which we mention the spaces with variable smoothness. Within the existing characterizations of these spaces, the characterization via smooth atoms is undoubtedly one of the most used when it comes to obtain new results in varied directions. In this work we make use of such characterization to prove several embedding results, such as Sobolev, Franke and Jawerth embeddings, and also to study traces on hyperplanes. Despite the considerable benefits of resorting on the smooth atomic decomposition, there are still some limitations when one tries to use it in order to prove some specific results, such as pointwise multipliers and diffeomorphisms assertions. The non-smooth atomic characterization proved in this work overcome these problems, due to the weaker conditions of the (non-smooth) atoms. Moreover, it also allows us to give an intrinsic characterization of the 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability on the class of regular domains, in which connected bounded Lipschitz domains are included. / In dieser Arbeit untersuchen wir einige wichtige Eigenschaften der 2-microlokalen Besov und Triebel-Lizorkin Räume mit variabler Integrabilität. Weil die Glattheit hier mit einer reicher Gewichtsfolge gemessen wird, beinhaltet diese Skala von Funktionsräumen eine große Anzahl von Funktionsräumen, von denen wir die Räume mit variabler Glattheit erwähnen. Innerhalb der vorhandenen Charakterisierungen dieser Räume ist die Charakterisierung mit glatten Atomen zweifellos eine der am häufigsten verwendeten, um neue Ergebnisse in verschiedenen Richtungen zu erhalten. In dieser Arbeit verwenden wir eine solche Charakterisierung, um mehrere Einbettungsergebnisse zu bewiesen, wie Sobolev-Einbettungen und Einbettungen vom Franke-Jawerth Typ, und auch Spurresultate zu untersuchen. Trotz der beträchtlichen Vorteile des Rückgriffs auf die glatte Atomaren-Zerlegung gibt es immer noch einige Einschränkungen, wenn man versucht, sie zu verwenden, um einige spezifische Ergebnisse zu beweisen, wie beispielsweise punktweise Multiplikatoren und Diffeomorphismen-Assertionen. Die nichtglatte atomare Charakterisierung, die wir in dieser Arbeit beweisen, überwindet diese Probleme aufgrund der schwächeren Bedingungen von (nichtglatten) Atomen. Außerdem erlaubt es uns, eine Intrinsische Charakterisierung der 2-mikrolokalen Besov- und Triebel-Lizorkin-Räume mit variabler Integrabilität auf regulärer Gebieten zu geben.

The subject of the thesis is the investigation of asymptotic properties of solutions of the Cauchy problem for structurally damped sigma-evolution operators with time dependent, monotonous, dissipation term. An appropriate energy for solutions of the sigma-evolution equations is defined and some estimates for energies of higher order are proved. In the scale invariant case the optimality of these estimates is shown. Further, the influence of properties of the time dependent dissipation on L^p-L^q estimates for the energy with p and q bigger or equal to 2 and from the conjugate line is clarified. Also smoothing properties of the operators under consideration are investigated. The connection between the regularity of the data and the regularity of the solution in terms of L^2 based Gevrey spaces is considered. Finally, L^1-L^1-estimates in the special case delta = sigma/2 and decreasing dissipative coefficient. / Thema der vorliegenden Dissertation ist die Untersuchung asymptotischer Eigenschaften von Lösungen des Cauchy Problems für strukturell gedämpfte sigma-Evolutions-Operatoren mit zeitabhängigem, monotonen Dissipationskoeffizienten. Es wird eine geeignete Energie definiert und für diese Abschätzungen, auf für entsprechende Energien höherer Ordnung gezeigt. Darüber hinaus wird der Einfluss des Dissipationskoeffizienten auf L^p-L^q Abschätzungen auf und entfernt von der konjugierten Linie untersucht. Im skaleninvarianten Fall wird die Schärfe der Abschätzungen bewiesen. Weiterhin wird der Zusammenhang zwischen der Regularität der Daten und der der Lösung in Termen von L^2-basierten Gevrey-Räumen untersucht. Schließlich werden L^1-L^1-Abschätzungen für den Spezialfall delta = sigma/2 und monoton fallenden Dissipationskoeffizienten gezeigt.

The main goal of our thesis is to understand qualitative properties of solutions to the Cauchy problem for the semi-linear wave model with time-dependent speed and dissipation. We greatly benefited from very precise estimates for the corresponding linear problem in order to obtain the global existence (in time) of small data solutions. This reason motivated us to introduce very carefully a complete description for classification of our models: scattering, non-effective, effective, over-damping. We have considered those separately.

The analysis of time-delayed dynamics on networks may help to understand many systems from physics, biology, and engineering, such as coupled laser arrays, gene-regulatory networks and complex ecosystems. Beside the complexity due to the network structure, the analysis is further complicated by the presence of the delays. Delay systems are in general infinite dimensional and thus can display complex dynamics as oscillations and chaos. The mathematical difficulties related to the delays hinders the analysis of delay networks. Thus, little is known yet about basic relations between network structure and delay dynamics. It has been shown that networks without delays can be studied efficiently with the generalized modeling approach, which analyzes the stability of an assumed steady state by a direct parametrization of the Jacobian matrix. In this thesis, I demonstrate the extension of the generalized modeling approach to delay networks and analyze networks of delay-coupled delay oscillators, with delayed auto-catalytic growth on the nodes and delayed transport between nodes. For degree-homogeneous networks (DHONs), in which each node has the same number of links, the bifurcation lines that border the stable areas can be calculated analytically, where the topology of the network is described only by the eigenvalues of the adjacency matrix. For undirected networks, the stability pattern in the parameter space of growth and transport delay is governed by two periodic sets of tongues of instability, which depend on the largest positive and the smallest negative eigenvalue. The direct relation between the eigenvalue and the bifurcation lines allows us to predict stability patterns for networks with certain topological properties. Thus, bipartite networks display a characteristic periodicity of tongues. In order to analyze the stability of degree-heterogeneous networks (DHENs), I apply a numerical sampling method based on Cauchy\'s Argument Principle. The stability patterns of these networks resembles the pattern of DHONs, which is governed by the two periodic sets. For networks with sufficiently many links, one set disappears, and the stability of DHENs can be approximates by the stability of a fully-connected network with the same average degree. However, random DHENs tend to be more stable than DHONs, and DHENs with a broad degree-distribution tend to be more stable than DHENs with a narrow distribution. Thus, such networks are more likely to give rise to amplitude death, i.e. the stabilization of an unstable steady state through diffusive coupling. The stability pattern of DHENs can be qualitatively different than the pattern in DHONs. However, for small growth delays, close to the critical delay of the single node system, the bifurcation lines of all DHENs with the same average degree coincide. This, is particularly interesting, because there the stability depends on a global property of the network, which suggests a diverging interaction length. In summary, the extension of generalized modeling to time-delay networks reveals basic relations between the delay dynamics and the topology. The generality of our model should allow to apply these results to a large class of real-world systems.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.