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

The topic of this thesis is a mathematical treatment of Anderson's orthogonality catastrophe. Named after P.W. Anderson, who studied the phenomenon in the late 1960s, the catastrophe is an intrinsic effect in Fermi gases. In his first work on the topic in [Phys. Rev. Lett. 18:1049--1051], Anderson studied a system of $N$ noninteracting fermions in three space dimensions and found the ground state to be asymptotically orthogonal to the ground state of the same system perturbed by a finite-range scattering potential.

More precisely, let $\Phi_L^N$ be the $N$-body ground state of the fermionic system in a $d$-dimensional box of length $L$,and let $\Psi_L^N$ be the ground state of the corresponding system in the presence of the additional finite-range potential. Then the catastrophe brings about the asymptotic vanishing $$\S_L^N := \<\Phi_L^N, \Psi_L^N\> \sim L^{-\gamma/2}$$ of the overlap $\S_L^N$ of the $N$-body ground states $\Phi_L^N$ and $\Psi_L^N$. The asymptotics is in the thermodynamic limit $L\to\infty$ and $N\to\infty$ with fixed density $N/L^d\to\varrho > 0$.

In [Commun. Math. Phys. 329:979--998], the overlap $\S_L^N$ has been bounded from above with an asymptotic bound of the form $$\abs{\S_L^N}^2 \lesssim L^{-\tilde{\gamma}}$$. The decay exponent $\tilde{\gamma}$ there corresponds to the one of Anderson in [Phys. Rev. Lett. 18:1049--1051]. Another publication by Anderson from the same year, [Phys. Rev. 164:352--359], contains the exact asymptotics with a bigger coefficient $\gamma$.

This thesis features a step towards the exact asymptotics. We prove a bound with a coefficient $\gamma$ that corresponds in a certain sense to the one in [Phys. Rev. 164:352--359], and improves upon the one in [Commun. Math. Phys. 329:979--998]. We use the methods from [Commun. Math. Phys. 329:979--998], but treat every term in a series expansion of $\ln S_L^N$, instead of only the first one. Treating the higher order terms introduces additional arguments since the trace expressions occurring are no longer necessarily nonnegative, which complicates some of the estimates.

The main contents of this thesis will also be published in a forthcoming article co-authored with Martin Gebert, Peter Müller, and Peter Otte.

Prediction problems on high-dimensional molecular data, e.g. the classification of microar-

ray samples into normal and cancer tissues, are complex and ill-posed since the number

of variables usually exceeds the number of observations by orders of magnitude. Recent

research in the area has propagated a variety of new statistical models in order to handle

these new biological datasets. In practice, however, these models are always applied in

combination with preprocessing and variable selection methods as well as model selection

which is mostly performed by cross-validation. Varma and Simon (2006) have used the

term ‘wrapper-algorithm’ for this integration of preprocessing and model selection into the

construction of statistical models. Additionally, they have proposed the method of nested

cross-validation (NCV) as a way of estimating their prediction error which has evolved to

the gold-standard by now.

In the first part, this thesis provides further theoretical and empirical justification for

the usage of NCV in the context of wrapper-algorithms. Moreover, a computationally less

intensive alternative to NCV is proposed which can be motivated in a decision theoretic

framework. The new method can be interpreted as a smoothed variant of NCV and, in

contrast to NCV, guarantees intuitive bounds for the estimation of the prediction error.

The second part focuses on the ranking of wrapper algorithms. Cross-study-validation is

proposed as an alternative concept to the repetition of separated within-study-validations

if several similar prediction problems are available. The concept is demonstrated using

six different wrapper algorithms for survival prediction on censored data on a selection of

eight breast cancer datasets. Additionally, a parametric bootstrap approach for simulating

realistic data from such related prediction problems is described and subsequently applied

to illustrate the concept of cross-study-validation for the ranking of wrapper algorithms.

Eventually, the last part approaches computational aspects of the analyses and simula-

tions performed in the thesis. The preprocessing before the analysis as well as the evaluation

of the prediction models requires the usage of large computing resources. Parallel comput-

ing approaches are illustrated on cluster, cloud and high performance computing resources

using the R programming language. Usage of heterogeneous hardware and processing of

large datasets are covered as well as the implementation of the R-package survHD for

the analysis and evaluation of high-dimensional wrapper algorithms for survival prediction

from censored data.

In heutigen Premiumfahrzeugen kommunizieren bis zu 80 Steuergeräte über bis zu

sechs verschiedene Vernetzungstechnologien. Dabei öffnet sich die

Fahrzeugkommunikation nach außen: Das Fahrzeug kommuniziert auch mit dem

Smartphone des Fahrers und dem Internet.

Für die Kommunikation über verschiedene Anwendungsdomänen im Fahrzeug müssen

heute Gateways eingesetzt werden, die zwischen den nicht-kompatiblen Protokollen

übersetzen. Deswegen geht der Trend auch in der Fahrzeugkommunikation zum

Internet Protocol (IP), das für technologie- und domänenübergreifende Kommunikation entwickelt wurde. Neben dem durchgängigen

Protokoll auf der Vermittlungsschicht ist für die effiziente Entwicklung eines

komplexen, verteilten Systems wie einem Fahrzeug auch eine entsprechende

Kommunikationsmiddleware notwendig.

Die Kommunikation in einem Fahrzeug stellt spezielle Anforderungen an die

Kommunikationsmiddleware. Zum einen werden in Fahrzeugen

unterschiedliche Kommunikationsparadigmen genutzt, beispielsweise signalbasierte

und funktionsbasierte Kommunikation. Zum anderen können sich die

Kommunikationspartner in einem Fahrzeug hinsichtlich ihrer Ressourcen und ihrer

Komplexität erheblich unterscheiden.

Keine existierende IP-basierte Kommunikationsmiddleware erfüllt die in der

vorliegenden Arbeit identifizierten Anforderungen für den Einsatz im Fahrzeug.

Ziel dieser Arbeit ist es daher, eine Kommunikationsmiddleware zu konzipieren,

die für den Einsatz im Fahrzeug geeignet ist.

Die vorgestellte Lösung sieht mehrere interoperable Ausprägungen der

Middleware vor, die den Konflikt zwischen unterschiedlichen funktionalen

Anforderungen einerseits und den sehr heterogenen Kommunikationspartnern

andererseits auflösen.

Ein weiterer elementarer Teil der Lösung ist die Umsetzung der im Fahrzeug

erforderlichen Kommunikationsparadigmen. Das funktionsbasierte Paradigma wird

durch einfache Remote Procedure Calls implementiert. Das signalbasierte

Paradigma wird durch ein darauf aufbauendes Notification-Konzept implementiert.

Somit wird eine stärker am aktuellen Informationsbedarf orientierte

Umsetzung ermöglicht, als dies im heutigen Fahrzeugbordnetz durch das einfache

Verteilen von Daten der Fall ist. Es wird gezeigt, dass sich prinzipiell beide

Kommunikationsparadigmen durch einen einzigen Mechanismus abbilden lassen, der

abhängig von den beteiligten Ausprägungen mit dynamischen oder nur statischen

Daten operiert. Ein skalierbares Marshalling berücksichtigt darüber hinaus

die unterschiedlichen Anforderungen der Anwendungen und die unterschiedliche

Leistungsfähigkeit der beteiligten Steuergeräte. Hiermit wird die Kommunikation zwischen allen Anwendungen im IP-basierten Fahrzeugbordnetz

durchgängig ermöglicht.

Auf dieser Basis wird die Lösung um wichtige Systemdienste erweitert.

Diese Dienste implementieren Funktionen, die nur in der

Kooperation mehrerer Komponenten erbracht werden können oder kapseln allgemeine

Kommunikationsfunktionalität zur einfachen Wiederverwendung.

Zwei für die Anwendung im Fahrzeug wichtige Systemdienste werden

prototypisch dargestellt: Ein Service-Management ermöglicht die Verwaltung von

Diensten in unterschiedlichen Zuständen, ein Security-Management bildet Security-Ziele

auf die bestmögliche Kombination von implementierten Security-Protokollen der

beteiligten Kommunikationspartner

ab. Diese Systemdienste sind selbst skalierbar und lassen sich damit an

das Konzept unterschiedlicher Ausprägungen der Kommunikationsmiddleware

anpassen.

Durch Leistungsmessungen an den im Rahmen dieser Arbeit entstandenen Prototypen

wird gezeigt, dass die konzipierte Kommunikationsmiddleware für den

Einsatz auf eingebetteten Systemen im Fahrzeug geeignet ist. Der

Versuchsaufbau orientiert sich an typischen Anwendungsfällen für die

Fahrzeugkommunikation und verwendet Automotive-qualifizierte, eingebettete

Rechenplattformen. Insbesondere wird nachgewiesen, dass mit dem beschriebenen

Konzept auch

In this thesis, we investigate the question when a non-compact manifold can be quasi-isometric to a leaf in a foliation of a compact manifold. The point of departure is the result of Paul Schweitzer's that every non-compact manifold carries a Riemannian metric so that the resulting Riemannian manifold is not quasi-isometric to a leaf in a codimension one foliation of a compact manifold. We show that the coarse homology of these non-leaves is not finitely generated. This observation motivates the main question of this thesis: Does every leaf in a foliation of a compact manifold have finitely generated coarse homology?

The answer to this question is a double negative: Firstly, we show that there exists a large class of two-dimensional leaves in codimension one foliations that have non-finitely generated coarse homology. Moreover, we improve Schweitzer's construction by showing that every Riemannian metric can be deformed to a codimension one non-leaf without affecting the coarse homology. In particular, we find non-leaves with trivial coarse homology.

In order to answer these questions we develop computational tools for the coarse homology. Furthermore, we show that certain known criteria for manifolds to be a leaf are independent of one another and of the coarse homology.

In recent years, Hilbert's Programme has been resumed within the framework of constructive mathematics. This undertaking has already shown its feasability for a considerable part of commutative algebra. In particular, point-free methods have been playing a primary role, emerging as the appropriate language for expressing the interplay between real and ideal in mathematics.

This dissertation is written within this tradition and has Sambin's notion of formal topology at its core. We start by developing general tools, in order to make this notion more immediate for algebraic application. We revise the Zariski spectrum as an inductively generated basic topology, and we analyse the constructive status of the corresponding principles of spatiality and reducibility. Through a series of examples, we show how the principle of spatiality is recurrent in the mathematical practice.

The tools developed before are applied to specific problems in constructive algebra. In particular, we find an elementary characterization of the notion of codimension for ideals of a commutative ring, by means of which a constructive version of Krull's principal ideal theorem can be stated and proved. We prove a formal version of the projective Eisenbud-Evans-Storch theorem. Finally, guided by the algebraic intuition, we present an application in constructive domain theory, by proving a finite version of Kleene-Kreisel density theorem for non-flat information systems.

Interactive surfaces are a growing trend in many domains. As one possible manifestation of Mark Weiser’s vision of ubiquitous and disappearing computers in everywhere objects, we see touchsensitive screens in many kinds of devices, such as smartphones, tablet computers and interactive tabletops. More advanced concepts of these have been an active research topic for many years. This has also influenced automotive cockpit development: concept cars and recent market releases show integrated touchscreens, growing in size. To meet the increasing information and interaction needs, interactive surfaces offer context-dependent functionality in combination with a direct input paradigm.

However, interfaces in the car need to be operable while driving. Distraction, especially visual distraction from the driving task, can lead to critical situations if the sum of attentional demand emerging from both primary and secondary task overextends the available resources. So far, a touchscreen requires a lot of visual attention since its flat surface does not provide any haptic feedback. There have been approaches to make direct touch interaction accessible while driving for simple tasks. Outside the automotive domain, for example in office environments, concepts for sophisticated handling of large displays have already been introduced. Moreover, technological advances lead to new characteristics for interactive surfaces by enabling arbitrary surface shapes.

In cars, two main characteristics for upcoming interactive surfaces are largeness and shape. On the one hand, spatial extension is not only increasing through larger displays, but also by taking objects in the surrounding into account for interaction. On the other hand, the flatness inherent in current screens can be overcome by upcoming technologies, and interactive surfaces can therefore provide haptically distinguishable surfaces. This thesis describes the systematic exploration of large and shaped interactive surfaces and analyzes their potential for interaction while driving. Therefore, different prototypes for each characteristic have been developed and evaluated in test settings suitable for their maturity level. Those prototypes were used to obtain subjective user feedback and objective data, to investigate effects on driving and glance behavior as well as usability and user experience.

As a contribution, this thesis provides an analysis of the development of interactive surfaces in the car. Two characteristics, largeness and shape, are identified that can improve the interaction compared to conventional touchscreens. The presented studies show that large interactive surfaces can provide new and improved ways of interaction both in driver-only and driver-passenger situations. Furthermore, studies indicate a positive effect on visual distraction when additional static haptic feedback is provided by shaped interactive surfaces. Overall, various, non-exclusively applicable, interaction concepts prove the potential of interactive surfaces for the use in automotive cockpits, which is expected to be beneficial also in further environments where visual attention needs to be focused on additional tasks.

Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model. When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one. However, in the modelling of massively distributed computer systems, even aggregated chains may be still too large for efficient numerical analysis. Recently this problem has been addressed by fluid techniques, where the Markov chain is approximated by a system of ordinary differential equations (ODEs) whose size does not depend on the number of the objects in the model. The technique has been primarily applied in the case of massively replicated sequential processes with small local state space sizes. This thesis devises two different approaches that broaden the scope of applicability of efficient fluid approximations. Fluid lumpability applies in the case where objects are composites of simple objects, and aggregates the potentially massive, naively constructed ODE system into one whose size is independent from the number of composites in the model. Similarly to quasi and near lumpability, we introduce approximate fluid lumpability that covers ODE systems which can be aggregated after a small perturbation in the parameters. The technique of spatial aggregation, instead, applies to models whose objects perform a random walk on a two-dimensional lattice. Specifically, it is shown that the underlying ODE system, whose size is proportional to the number of the regions, converges to a system of partial differential equations of constant size as the number of regions goes to infinity. This allows for an efficient analysis of large-scale mobile models in continuous space like ad hoc networks and multi-agent systems.

The physical picture of particle behaviour that arises from experimental data is that it belongs to one of the following: Either the particle always stays near the origin or the particle comes from infinity, is scattered and escapes to infinity. In the quantum mechanical description these two categories of behaviour are associated with point spectrum and absolutely continuous spectrum respectively. The corresponding spectral measures are point measures or absolutely continuous measures.

According to general results of the measure theory a measure can be decomposed in three parts: a pure point part, an absolutely continuous part and a singular continuous part.

In contrast to the well-known particle behaviour of the other two types of spectrum, the singular continuous spectrum is more difficult to interpret.

For the Schrödinger operator D.B. Pearson constructed an explicit class of potentials that give rise to purely singular continuous spectrum . This example allows the interpretation of the particle behaviour: The particle moves arbitrarily far away from the origin but it feels nevertheless the effect of the potential. Therefore it will recur infinitely often to the vicinity of the origin to run off infinitely often.

The result for the Schröodinger operator leads to the question whether there can be found similar results in relativistic quantum mechanics.

The aim of this paper is to construct for the first time an explicit potential for the Dirac operator that has purely singular continuous spectrum in (-\infty,-1)\cup (1,\infty). The characteristic trait of this potential is that it consists of bumps whose distance is growing rapidly. This allows the particle to depart

from the origin arbitrarily far. But the overall effect of the bumps will always lead the particle back to the origin.