Characterizations of distributions using log odds rate

In this paper, we examine the relationships between log odds rate and various reliability measures such as hazard rate and reversed hazard rate in the context of repairable systems. We also prove characterization theorems for some families of distributions viz. Burr, Pearson and log exponential models. We discuss the properties and applications of log odds rate in weighted models. Further we extend the concept to the bivariate set up and study its properties.

Quantile based entropy function in past lifetime

Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distributions and studied its relationship with residual entropy function. In the present paper, we introduce a quantile version of the entropy function in past lifetime and study its properties. Unlike the measure of uncertainty given in Di Crescenzo and Longobardi (2002) the proposed measure uniquely determines the underlying probability distribution. The measure is used to study two nonparametric classes of distributions. We prove characterizations theorems for some well known quantile lifetime distributions

Optimal Sub-Pixel arrangements and coding for ultra-high resolution three-dimensional OLED displays

Information display technology is a rapidly growing research and development field. Using state-of-the-art technology, optical resolution can be increased dramatically by organic light-emitting diode - since the light emitting layer is very thin, under 100nm. The main question is what pixel size is achievable technologically? The next generation of display will considers three-dimensional image display. In 2D , one is considering vertical and horizontal resolutions. In 3D or holographic images, there is another dimension – depth. The major requirement is the high resolution horizontal dimension in order to sustain the third dimension using special lenticular glass or barrier masks, separate views for each eye. The high-resolution 3D display offers hundreds of more different views of objects or landscape. OLEDs have potential to be a key technology for information displays in the future. The display technology presented in this work promises to bring into use bright colour 3D flat panel displays in a unique way. Unlike the conventional TFT matrix, OLED displays have constant brightness and colour, independent from the viewing angle i.e. the observer's position in front of the screen. A sandwich (just 0.1 micron thick) of organic thin films between two conductors makes an OLE Display device. These special materials are named electroluminescent organic semi-conductors (or organic photoconductors (OPC )). When electrical current is applied, a bright light is emitted (electrophosphorescence) from the formed Organic Light-Emitting Diode. Usually for OLED an ITO layer is used as a transparent electrode. Such types of displays were the first for volume manufacture and only a few products are available in the market at present. The key challenges that OLED technology faces in the application areas are: producing high-quality white light achieving low manufacturing costs increasing efficiency and lifetime at high brightness. Looking towards the future, by combining OLED with specially constructed surface lenses and proper image management software it will be possible to achieve 3D images.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Development and applications of density matrix functional theory of generalized Hubbard Models

In dieser Doktorarbeit wird eine akkurate Methode zur Bestimmung von Grundzustandseigenschaften stark korrelierter Elektronen im Rahmen von Gittermodellen entwickelt und angewandt. In der Dichtematrix-Funktional-Theorie (LDFT, vom englischen lattice density functional theory) ist die Ein-Teilchen-Dichtematrix γ die fundamentale Variable. Auf der Basis eines verallgemeinerten Hohenberg-Kohn-Theorems ergibt sich die Grundzustandsenergie Egs[γgs] = min° E[γ] durch die Minimierung des Energiefunktionals E[γ] bezüglich aller physikalischer bzw. repräsentativer γ. Das Energiefunktional kann in zwei Beiträge aufgeteilt werden: Das Funktional der kinetischen Energie T[γ], dessen lineare Abhängigkeit von γ genau bekannt ist, und das Funktional der Korrelationsenergie W[γ], dessen Abhängigkeit von γ nicht explizit bekannt ist. Das Auffinden präziser Näherungen für W[γ] stellt die tatsächliche Herausforderung dieser These dar. Einem Teil dieser Arbeit liegen vorausgegangene Studien zu Grunde, in denen eine Näherung des Funktionals W[γ] für das Hubbardmodell, basierend auf Skalierungshypothesen und exakten analytischen Ergebnissen für das Dimer, hergeleitet wird. Jedoch ist dieser Ansatz begrenzt auf spin-unabhängige und homogene Systeme. Um den Anwendungsbereich von LDFT zu erweitern, entwickeln wir drei verschiedene Ansätze zur Herleitung von W[γ], die das Studium von Systemen mit gebrochener Symmetrie ermöglichen. Zuerst wird das bisherige Skalierungsfunktional erweitert auf Systeme mit Ladungstransfer. Eine systematische Untersuchung der Abhängigkeit des Funktionals W[γ] von der Ladungsverteilung ergibt ähnliche Skalierungseigenschaften wie für den homogenen Fall. Daraufhin wird eine Erweiterung auf das Hubbardmodell auf bipartiten Gittern hergeleitet und an sowohl endlichen als auch unendlichen Systemen mit repulsiver und attraktiver Wechselwirkung angewandt. Die hohe Genauigkeit dieses Funktionals wird aufgezeigt. Es erweist sich jedoch als schwierig, diesen Ansatz auf komplexere Systeme zu übertragen, da bei der Berechnung von W[γ] das System als ganzes betrachtet wird. Um dieses Problem zu bewältigen, leiten wir eine weitere Näherung basierend auf lokalen Skalierungseigenschaften her. Dieses Funktional ist lokal bezüglich der Gitterplätze formuliert und ist daher anwendbar auf jede Art von geordneten oder ungeordneten Hamiltonoperatoren mit lokalen Wechselwirkungen. Als Anwendungen untersuchen wir den Metall-Isolator-Übergang sowohl im ionischen Hubbardmodell in einer und zwei Dimensionen als auch in eindimensionalen Hubbardketten mit nächsten und übernächsten Nachbarn. Schließlich entwickeln wir ein numerisches Verfahren zur Berechnung von W[γ], basierend auf exakten Diagonalisierungen eines effektiven Vielteilchen-Hamilton-Operators, welcher einen von einem effektiven Medium umgebenen Cluster beschreibt. Dieser effektive Hamiltonoperator hängt von der Dichtematrix γ ab und erlaubt die Herleitung von Näherungen an W[γ], dessen Qualität sich systematisch mit steigender Clustergröße verbessert. Die Formulierung ist spinabhängig und ermöglicht eine direkte Verallgemeinerung auf korrelierte Systeme mit mehreren Orbitalen, wie zum Beispiel auf den spd-Hamilton-Operator. Darüber hinaus berücksichtigt sie die Effekte kurzreichweitiger Ladungs- und Spinfluktuationen in dem Funktional. Für das Hubbardmodell wird die Genauigkeit der Methode durch Vergleich mit Bethe-Ansatz-Resultaten (1D) und Quanten-Monte-Carlo-Simulationen (2D) veranschaulicht. Zum Abschluss wird ein Ausblick auf relevante zukünftige Entwicklungen dieser Theorie gegeben.

Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications

Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.

Adsorption kinetic investigations of phthalocyanine derivatives self assembled monolayers (SAMs) on gold: Temperature influence on the SAM formation process and quality

The ordered nano-structured surfaces, like self-assembled monolayers (SAMs) are of a great scientific interest, due to the low cost, simplicity, and versatility of this method. SAMs found numerous of applications in molecular electronics, biochemistry and optical devices. Phthalocyanine (Pc) complexes are of particular interest for the SAM preparation. These molecules exhibit fascinating physical properties and are chemically and thermally stable. Moreover their complex structure is advantageous for the fabrication of switchable surfaces. In this work the adsorption process of Pcs derivatives, namely, subphthalocyanines (SubPcB) and terbium (2TbPc) sandwich complexes on gold has been investigated. The influence of the molecular concentration, chain length of peripheral groups, and temperature on the film formation process has been examined using a number of techniques. The SAMs formation process has been followed in situ and in real time by means of second harmonic generation (SHG) and surface plasmon resonance (SPR) spectroscopy. To investigate the quality of the SAMs prepared at different temperatures atomic force microscopy (AFM) and X-Ray photoelectron spectroscopy (XPS)measurements were performed. Valuable information about SubPcB and 2TbPc adsorbtion process has been obtained in the frame of this work. The kinetic data, obtained with SHG and SPR, shows the best conformance with the first order Langmuir kinetic model. Comparing SHG and SPR results, it has been found, that the film formation occurs faster than the formation of chemical bonds. Such, the maximum amount of molecules on the surface is reached after 6 min for SubPcB and 30 min for 2TbPc. However, at this time the amount of formed chemicals bonds is only 10% and 40% for SubPcB and 2TbPc, respectively. The most intriguing result, among others, was obtained at T = 2 °C, where the formation of the less dense SAMs have been detected with SHG.However, analyzing XPS and AFM data, it has been revealed, that there is the same amount of molecules on the surface at both temperature T = 2 °C, and T = 21 °C, but the amount of formed chemicals bond is different. At T = 2 °C molecules form aggregates, therefore many of available anchor groups stay unattached.

Taming Chaotic Circuits

Control algorithms that exploit chaotic behavior can vastly improve the performance of many practical and useful systems. The program Perfect Moment is built around a collection of such techniques. It autonomously explores a dynamical system's behavior, using rules embodying theorems and definitions from nonlinear dynamics to zero in on interesting and useful parameter ranges and state-space regions. It then constructs a reference trajectory based on that information and causes the system to follow it. This program and its results are illustrated with several examples, among them the phase-locked loop, where sections of chaotic attractors are used to increase the capture range of the circuit.

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