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The behaviour of electronic circuits is influenced by ageing effects. Modelling the behaviour of circuits is a standard approach for the design of faster, smaller, more reliable and more robust systems. In this thesis, we propose a formalization of robustness that is derived from a failure model, which is based purely on the behavioural specification of a system. For a given specification, simulation can reveal if a system does not comply with a specification, and thus provide a failure model. Ageing usually works against the specified properties, and ageing models can be incorporated to quantify the impact on specification violations, failures and robustness. We study ageing effects in the context of analogue circuits. Here, models must factor in infinitely many circuit states. Ageing effects have a cause and an impact that require models. On both these ends, the circuit state is highly relevant, an must be factored in. For example, static empirical models for ageing effects are not valid in many cases, because the assumed operating states do not agree with the circuit simulation results. This thesis identifies essential properties of ageing effects and we argue that they need to be taken into account for modelling the interrelation of cause and impact. These properties include frequency dependence, monotonicity, memory and relaxation mechanisms as well as control by arbitrary shaped stress levels. Starting from decay processes, we define a class of ageing models that fits these requirements well while remaining arithmetically accessible by means of a simple structure.
Modeling ageing effects in semiconductor circuits becomes more relevant with higher integration and smaller structure sizes. With respect to miniaturization, digital systems are ahead of analogue systems, and similarly ageing models predominantly focus on digital applications. In the digital domain, the signal levels are either on or off or switching in between. Given an ageing model as a physical effect bound to signal levels, ageing models for components and whole systems can be inferred by means of average operation modes and cycle counts. Functional and faithful ageing effect models for analogue components often require a more fine-grained characterization for physical processes. Here, signal levels can take arbitrary values, to begin with. Such fine-grained, physically inspired ageing models do not scale for larger applications and are hard to simulate in reasonable time. To close the gap between physical processes and system level ageing simulation, we propose a data based modelling strategy, according to which measurement data is turned into ageing models for analogue applications. Ageing data is a set of pairs of stress patterns and the corresponding parameter deviations. Assuming additional properties, such as monotonicity or frequency independence, learning algorithm can find a complete model that is consistent with the data set. These ageing effect models decompose into a controlling stress level, an ageing process, and a parameter that depends on the state of this process. Using this representation, we are able to embed a wide range of ageing effects into behavioural models for circuit components. Based on the developed modelling techniques, we introduce a novel model for the BTI effect, an ageing effect that permits relaxation. In the following, a transistor level ageing model for BTI that targets analogue circuits is proposed. Similarly, we demonstrate how ageing data from analogue transistor level circuit models lift to purely behavioural block models. With this, we are the first to present a data based hierarchical ageing modeling scheme. An ageing simulator for circuits or system level models computes long term transients, solutions of a differential equation. Long term transients are often close to quasi-periodic, in some sense repetitive. If the evaluation of ageing models under quasi-periodic conditions can be done efficiently, long term simulation becomes practical. We describe an adaptive two-time simulation algorithm that basically skips periods during simulation, advancing faster on a second time axis. The bottleneck of two-time simulation is the extrapolation through skipped frames. This involves both the evaluation of the ageing models and the consistency of the boundary conditions. We propose a simulator that computes long term transients exploiting the structure of the proposed ageing models. These models permit extrapolation of the ageing state by means of a locally equivalent stress, a sort of average stress level. This level can be computed efficiently and also gives rise to a dynamic step control mechanism. Ageing simulation has a wide range of applications. This thesis vastly improves the applicability of ageing simulation for analogue circuits in terms of modelling and efficiency. An ageing effect model that is a part of a circuit component model accounts for parametric drift that is directly related to the operation mode. For example asymmetric load on a comparator or power-stage may lead to offset drift, which is not an empiric effect. Monitor circuits can report such effects during operation, when they become significant. Simulating the behaviour of these monitors is important during their development. Ageing effects can be compensated using redundant parts, and annealing can revert broken components to functional. We show that such mechanisms can be simulated in place using our models and algorithms. The aim of automatized circuit synthesis is to create a circuit that implements a specification for a certain use case. Ageing simulation can identify candidates that are more reliable. Efficient ageing simulation allows to factor in various operation modes and helps refining the selection. Using long term ageing simulation, we have analysed the fitness of a set of synthesized operational amplifiers with similar properties concerning various use cases. This procedure enables the selection of the most ageing resilient implementation automatically.
We consider a class of nonautonomous nonlinear competitive parabolic systems on bounded radial domains under Neumann or Dirichlet boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then bounded positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of scalar equations with Neumann or Dirichlet boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.
Given an Abelian semi-group (A, +), an A-valued curvature measure is a valuation with values in A-valued measures. If A = R, complete classifications of Hausdorff-continuous translation-invariant SO(n)-invariant valuations and curvature measures were obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in A = Sym^p R^n and A = Sym^2 Λ^q R^n for p, q ≥ 1 with varying assumptions as for their invariance properties.
In the present work, we classify all smooth translation-invariant SO(n)-covariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle S R^n and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-module-valued SO(n)-covariant valuations. Furthermore, a decomposition of the space of smooth translation-
invariant scalar-valued curvature measures as an SO(n)-module is obtained. As a corollary, we construct explicit bases of continuous translation-invariant scalar-valued valuations and smooth translation-invariant scalar-valued curvature measures.
In recent years using symmetry has proven to be a very useful tool to simplify computations in semidefinite programming. This dissertation examines the possibilities of exploiting discrete symmetries in three contexts: In SDP-based relaxations for polynomial optimization, in testing positivity of symmetric polynomials, and combinatorial optimization. In these contexts the thesis provides new ways for exploiting symmetries and thus deeper insight in the paradigms behind the techniques and studies a concrete combinatorial optimization question.
We study the price-setting problem of market makers under perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as the expectation of a true value of the asset given the market makers partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times.
We analyze the price-setting problem by solving a non-standard filtering problem with an endogenous filtration that depends on the bid and ask price process quoted by the market maker. Under some conditions we show existence and uniqueness of the price processes. In a different setting we construct a counterexample to uniqueness. Further, we discuss the behavior of the spread by a convergence result and simulations.
The condensation phase transition and the number of solutions in random graph and hypergraph models
(2016)
This PhD thesis deals with two different types of questions on random graph and random hypergraph structures.
One part is about the proof of the existence and the determination of the location of the condensation phase transition. This transition will be investigated for large values of $k$ in the problem of $k$-colouring random graphs and in the problem of 2-colouring random $k$-uniform hypergraphs, where in the latter case we investigate a more general model with finite inverse temperature.
The other part deals with establishing the limiting distribution of the number of solutions in these structures in density regimes below the condensation threshold.
The problem of unconstrained or constrained optimization occurs in many branches of mathematics and various fields of application. It is, however, an NP-hard problem in general. In this thesis, we examine an approximation approach based on the class of SAGE exponentials, which are nonnegative exponential sums. We examine this SAGE-cone, its geometry, and generalizations. The thesis consists of three main parts:
1. In the first part, we focus purely on the cone of sums of globally nonnegative exponential sums with at most one negative term, the SAGE-cone. We ex- amine the duality theory, extreme rays of the cone, and provide two efficient optimization approaches over the SAGE-cone and its dual.
2. In the second part, we introduce and study the so-called S-cone, which pro- vides a uniform framework for SAGE exponentials and SONC polynomials. In particular, we focus on second-order representations of the S-cone and its dual using extremality results from the first part.
3. In the third and last part of this thesis, we turn towards examining the con- ditional SAGE-cone. We develop a notion of sublinear circuits leading to new duality results and a partial characterization of extremality. In the case of poly- hedral constraint sets, this examination is simplified and allows us to classify sublinear circuits and extremality for some cases completely. For constraint sets with certain conditions such as sets with symmetries, conic, or polyhedral sets, various optimization and representation results from the unconstrained setting can be applied to the constrained case.
A multiple filter test for the detection of rate changes in renewal processes with varying variance
(2014)
The thesis provides novel procedures in the statistical field of change point detection in time series.
Motivated by a variety of neuronal spike train patterns, a broad stochastic point process model is introduced. This model features points in time (change points), where the associated event rate changes. For purposes of change point detection, filtered derivative processes (MOSUM) are studied. Functional limit theorems for the filtered derivative processes are derived. These results are used to support novel procedures for change point detection; in particular, multiple filters (bandwidths) are applied simultaneously in oder to detect change points in different time scales.
Das libor Markt Modell (LMM) ist seit seiner Entwicklung in den Veröffentlichungen von Brace, Gatarek, Musiela (1997), einerseits, und unabhängig von diesen von Miltersen, Sandmann, Sondermann (1997), andererseits, zu dem anerkanntesten Instrument zur Modellierung der Zinsstruktur und der damit verbundenen Preisfindung für relevante Finanzderivate geworden. libor steht dabei für London Inter-Bank Offered Rate, ein täglich in London fixierter Referenz-Zins für kurzfristige Anlagen. Drei- oder sechsmonatige Laufzeiten sind in Verbindung mit dem LMM üblich. Die Forschung zur Verbesserung dieses Modells hat in den letzten Jahren an Zuwachs gewonnen. Beim Versuch den Fehler der Anpassung an die täglich beobachteten Preise von Zinsoptionen wie Caps und Swaptions zu verringern, erhält man in der Folge auch genauere Bewertungen für andere, exotischere, Derivate. Die zugrunde liegende und zentrale Idee des LMM besteht darin, die Forward (Termin) Zinsen direkt als primären (Vektor) Prozess mehrerer libor Sätze zu betrachten und diese simultan zu modellieren, anstatt sie nur herzuleiten aus einem übergeordneten, unendlich dimensionalen Forward Zinsprozess, wie im zeitlich früher entwickelten Heath-Jarrow-Morton Modell. Das überzeugendste Argument für diese Diskretisierung ist, dass die libor Sätze direkt im Markt beobachtbar sind und ihre Volatilitäten auf eine natürliche Weise in Beziehung gebracht werden können zu bereits liquide gehandelten Produkten, eben jenen Caps und Swaptions. Dennoch beinhaltet das Modell eine gravierende Insuffizienz, indem es keine Krümmung der Volatilitätsoberfläche, im Hinblick auf Optionen mit verschiedenen Basiszinsen, abbildet. Wie im einfachen eindimensionalen Black-Scholes Modell prägen sich auch hier die Ungenauigkeiten der Verteilung in fehlenden heavy tails deutlich aus. Smile und Skew Effekte sind erkennbar. Im klassischen liborMarkt Modell wird in Richtung der Basiszinsdimension nur eine affine Struktur erzeugt, welche bestenfalls als Approximation für die erwünschte Oberfläche dienen kann. Die beobachteten Verzerrungen führen naturgemäss zu einer ungenauen Abbildung der Realität und fehlerhaften Reproduktion der Preise in Regionen, die ein wenig entfernt vom Bereich am Geld liegen. Derartig ungewollte Dissonanzen in Gewinn und Verlustzahlen führten z.B. in 1998 zu gravierenden Verlusten im Zinsderivateportfolio der heutigen Royal Bank of Scotland. ...
Die Populationsgenetik beschäftigt sich mit dem Einfluss von zufälliger Reproduktion, Rekombination, Migration, Mutation und Selektion auf die genetische Struktur einer Population.
In dieser Arbeit mit dem englischen Titel "Ancestral lines under mutation and selection" wird das Zusammenspiel von zufälliger Reproduktion, gerichteter Selektion und Zweiwegmutation untersucht.
Dazu betrachten wir eine haploide Population in der jedes Individuum zu jedem Zeitpunkt genau einen von zwei Typen aus S:={0,1} trägt. Dabei sei 1 der neutrale und 0 der selektiv bevorzugte Typ. Im Diffusionslimes sehr großer Populationen modellieren wir den Prozess der Frequenz der Typ-0-Individuen durch eine Wright-Fisher-Diffusion X:=(X_t) mit Mutation und gerichteter Selektion.
Zu jedem Zeitpunkt s gibt es genau ein Individuum, dessen Nachkommen ab einem bestimmten zukünftigen Zeitpunkt t>s die gesamte Population ausmachen werden. Wir nennen dieses Individuum den gemeinsamen Vorfahren zum Zeitpunkt s, da alle Individuen zu allen Zeitpunkten r>t von ihm abstammen. Sei R_{s} dessen Typ zum Zeitpunkt s. Wir nehmen an, dass der Prozess X zum Zeitpunkt 0 im Gleichgewicht ist und definieren die Wahrscheinlichkeit, dass der gemeinsame Vorfahre zum Zeitpunkt 0 Typ 0 hat, durch h(x):= P(R_{0}=0|X_{0}=x). Eine Darstellung von h(x) wurde bereits von Fearnhead (2002) und Taylor (2007) gefunden und dort mit vorwiegend analytischen Methoden bewiesen. In dieser Arbeit entwickeln wir in Kapitel 3 ein neues Teilchenbild, den pruned lookdown ancestral selection graph (pruned LD-ASG), der für sich selbst genommen interessant ist und eine neue probabilistische Interpretation der Darstellung von h(x) liefert.
Durch Erweiterung des Teilchenbildes auf Nachkommenverteilungen mit schweren Tails und mit Hilfe einer Siegmund Dualität gelingt es uns in Kapitel 4 das Resultat für h(x) von klassischen Wright-Fisher-Diffusionen auf Lambda-Wright-Fisher-Diffuison zu erweitern.
Eine Verbindung zwischen Ideen von Taylor (2007), der den gemeinsamen Prozess (X,R) untersucht hat, und einem von Fearnhead (2002) betrachteten Prozess (R,V), der die Entwicklung des Typs R des gemeinsamen Vorfahren in einer Umgebung von V sogenannten virtuellen Linien beschreibt, stellen wir in Kapitel 6 her. Wir bestimmen die gemeinsame Dynamik des Tripels (X,R,V). In Kapitel 7 betrachten wir ein diskretes Bild mit endlicher Populationsgröße N und schlagen dort eine Brücke zu Resultaten von Kluth, Hustedt und Baake (2013).
Des Weiteren entwickeln wir in Kapitel 5 dieser Arbeit einen Algorithmus zur Simulation der Typen einer Stichprobe von m Individuen, die aus einer Wright-Fisher-Population mit Mutation und Selektion im Gleichgewicht gezogen wird. Mittels dieses Algorithmus illustrieren wir die Typenverteilung für verschiedene Parameterwerte und Stichprobengrößen.