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This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The first part of the thesis is devoted to formulating and solving specific polynomial systems that reduce the problem of decoding linear codes to the problem of polynomial system solving. We analyze the existing methods (mainly for the cyclic codes) and propose an original method for arbitrary linear codes that in some sense generalizes the Newton identities method widely known for cyclic codes. We investigate the structure of the underlying ideals and show how one can solve the decoding problem - both the so-called bounded decoding and more general nearest codeword decoding - by finding reduced Gröbner bases of these ideals. The main feature of the method is that unlike usual methods based on Gröbner bases for "finite field" situations, we do not add the so-called field equations. This tremendously simplifies the underlying ideals, thus making feasible working with quite large parameters of codes. Further we address complexity issues, by giving some insight to the Macaulay matrix of the underlying systems. By making a series of assumptions we are able to provide an upper bound for the complexity coefficient of our method. We address also finding the minimum distance and the weight distribution. We provide solid experimental material and comparisons with some of the existing methods in this area. In the second part we deal with the algebraic cryptanalysis of block iterative ciphers. Namely, we analyze the small-scale variants of the Advanced Encryption Standard (AES), which is a widely used modern block cipher. Here a cryptanalyst composes the polynomial systems which solutions should yield a secret key used by communicating parties in a symmetric cryptosystem. We analyze the systems formulated by researchers for the algebraic cryptanalysis, and identify the problem that conventional systems have many auxiliary variables that are not actually needed for the key recovery. Moreover, having many such auxiliary variables, specific to a given plaintext/ciphertext pair, complicates the use of several pairs which is common in cryptanalysis. We thus provide a new system where the auxiliary variables are eliminated via normal form reductions. The resulting system in key-variables only is then solved. We present experimental evidence that such an approach is quite good for small scaled ciphers. We investigate further our approach and employ the so-called meet-in-the-middle principle to see how far one can go in analyzing just 2-3 rounds of scaled ciphers. Additional "tuning techniques" are discussed together with experimental material. Overall, we believe that the material of this part of the thesis makes a step further in algebraic cryptanalysis of block ciphers.
In engineering and science, a multitude of problems exhibit an inherently geometric nature. The computational assessment of such problems requires an adequate representation by means of data structures and processing algorithms. One of the most widely adopted and recognized spatial data structures is the Delaunay triangulation which has its canonical dual in the Voronoi diagram. While the Voronoi diagram provides a simple and elegant framework to model spatial proximity, the core of which is the concept of natural neighbors, the Delaunay triangulation provides robust and efficient access to it. This combination explains the immense popularity of Voronoi- and Delaunay-based methods in all areas of science and engineering. This thesis addresses aspects from a variety of applications that share their affinity to the Voronoi diagram and the natural neighbor concept. First, an idea for the generalization of B-spline surfaces to unstructured knot sets over Voronoi diagrams is investigated. Then, a previously proposed method for \(C^2\) smooth natural neighbor interpolation is backed with concrete guidelines for its implementation. Smooth natural neighbor interpolation is also one of many applications requiring derivatives of the input data. The generation of derivative information in scattered data with the help of natural neighbors is described in detail. In a different setting, the computation of a discrete harmonic function in a point cloud is considered, and an observation is presented that relates natural neighbor coordinates to a continuous dependency between discrete harmonic functions and the coordinates of the point cloud. Attention is then turned to integrating the flexibility and meritable properties of natural neighbor interpolation into a framework that allows the algorithmically transparent and smooth extrapolation of any known natural neighbor interpolant. Finally, essential properties are proved for a recently introduced novel finite element tessellation technique in which a Delaunay triangulation is transformed into a unique polygonal tessellation.
Continuous stochastic control theory has found many applications in optimal investment. However, it lacks some reality, as it is based on the assumption that interventions are costless, which yields optimal strategies where the controller has to intervene at every time instant. This thesis consists of the examination of two types of more realistic control methods with possible applications. In the first chapter, we study the stochastic impulse control of a diffusion process. We suppose that the controller minimizes expected discounted costs accumulating as running and controlling cost, respectively. Each control action causes costs which are bounded from below by some positive constant. This makes a continuous control impossible as it would lead to an immediate ruin of the controller. We give a rigorous development of the relevant theory, where our guideline is to establish verification and convergence results under minimal assumptions, without focusing on the existence of solutions to the corresponding (quasi-)variational inequalities. If the impulse control problem can be characterized or approximated by (quasi-)variational inequalities, it remains to solve these equations. In Section 1.2, we solve the stochastic impulse control problem for a one-dimensional diffusion process with constant coefficients and convex running costs. Further, in Section 1.3, we solve a particular multi-dimensional example, where the uncontrolled process is given by an at least two-dimensional Brownian motion and the cost functions are rotationally symmetric. By symmetry, this problem can be reduced to a one-dimensional problem. In the last section of the first chapter, we suggest a new impulse control problem, where the controller is in addition allowed to invest his initial capital into a market consisting of a money market account and a risky asset. The costs which arise upon controlling the diffusion process and upon trading in this market have to be paid out of the controller's bond holdings. The aim of the controller is to minimize the running costs, caused by the abstract diffusion process, without getting ruined. The second chapter is based on a paper which is joint work with Holger Kraft and Frank Seifried. We analyze the portfolio decision of an investor trading in a market where the economy switches randomly between two possible states, a normal state where trading takes place continuously, and an illiquidity state where trading is not allowed at all. We allow for jumps in the market prices at the beginning and at the end of a trading interruption. Section 2.1 provides an explicit representation of the investor's portfolio dynamics in the illiquidity state in an abstract market consisting of two assets. In Section 2.2 we specify this market model and assume that the investor maximizes expected utility from terminal wealth. We establish convergence results, if the maximal number of liquidity breakdowns goes to infinity. In the Markovian framework of Section 2.3, we provide the corresponding Hamilton-Jacobi-Bellman equations and prove a verification result. We apply these results to study the portfolio problem for a logarithmic investor and an investor with a power utility function, respectively. Further, we extend this model to an economy with three regimes. For instance, the third state could model an additional financial crisis where trading is still possible, but the excess return is lower and the volatility is higher than in the normal state.
Forderungen nach kürzeren Entwicklungszyklen bei gleichzeitig höherer Produktqualität führen in allen Bereichen der Nutzfahrzeugtechnik und insbesondere auch bei Baumaschinen zum verstärkten Einsatz von Simulationssoftware. Um in diesem Sinne Lebensdauerberechnungen durchführen zu können, sind jedoch genaue Kenntnisse über die im Kundeneinsatz auftretenden Betriebslasten und Beanspruchungen erforderlich. Für deren Ermittlung hat der Baumaschinenhersteller VOLVO Construction Equipment einen Mobilbagger umfassend mit Messtechnik ausgestattet, die neben den mechanischen Belastungen an der Arbeitsausrüstung auch wesentliche Kenndaten des Hydrauliksystems und des Fahrantriebs erfasst. Dieser Messbagger wurde bereits bei unterschiedlichen Kunden in Europa eingesetzt. Der Artikel beschreibt die methodische Vorgehensweise zur Verarbeitung der erfassten Daten und zur Generierung von repräsentativen Nutzungsprofilen am Beispiel der mechanischen Belastungen an der Arbeitseinrichtung, die im Wesentlichen vom Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM) erarbeitet wurde.
In the ground vehicle industry it is often an important task to simulate full vehicle models based on the wheel forces and moments, which have been measured during driving over certain roads with a prototype vehicle. The models are described by a system of differential algebraic equations (DAE) or ordinary differential equations (ODE). The goal of the simulation is to derive section forces at certain components for a durability assessment. In contrast to handling simulations, which are performed including more or less complex tyre models, a driver model, and a digital road profile, the models we use here usually do not contain the tyres or a driver model. Instead, the measured wheel forces are used for excitation of the unconstrained model. This can be difficult due to noise in the input data, which leads to an undesired drift of the vehicle model in the simulation.
For the numerical simulation of a mechanical multibody system (MBS), dynamical loads are needed as input data, such as a road profile. With given input quantities, the equations of motion of the system can be integrated. Output quantities for further investigations are calculated from the integration results. In this paper, we consider the corresponding inverse problem: We assume, that a dynamical system and some reference output signals are given. The general task is to derive an input signal, such that the system simulation produces the desired reference output. We present the state-of-the-art method in industrial applications, the iterative learning control method (ILC) and give an application example from automotive industry. Then, we discuss three alternative methods based on optimal control theory for differential algebraic equations (DAEs) and give an overview of their general scheme.
Inspired by Kirchhoff’s kinetic analogy, the special Cosserat theory of rods is formulatedin the language of Lagrangian mechanics. A static rod corresponds to an abstract Lagrangian system where the energy density takes the role of the Lagrangian function. The equilibrium equations are derived from a variational principle. Noether’s theorem relates their first integrals to frame-indifference, isotropy and uniformity. These properties can be formulated in terms of Lie group symmetries. The rotational degrees of freedom, present in the geometrically exact beam theory, are represented in terms of orthonormal director triads. To reduce the number of unknowns, Lagrange multipliers associated with the orthonormality constraints are eliminated using null-space matrices. This is done both in the continuous and in the discrete setting. The discrete equilibrium equations are used to compute discrete rod configurations, where different types of boundary conditions can be handled.
In this paper, we present a viscoelastic rod model that is suitable for fast and sufficiently accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (’stiff ’ dof), bending and torsion (’soft’ dof). For inner dissipation, a consistent damping potential from Antman is chosen. Our discrete model is based on a finite difference discretisation on a staggered grid. The right-hand side function f and the Jacobian ∂f/∂(q, v, t) of the dynamical system q˙ = v, v˙ = f(q, v, t) – after index reduction from three to zero – is free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore cheap to evaluate. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computation times within milliseconds, it is suitable for interactivemanipulation in ’virtual reality’ applications. In contrast to fast common VR rod models, our model reflects the structural mechanics solutions sufficiently correct, as comparison with ABAQUS finite element results shows.
A series of (oligo)phenthiazines, thiazolium salts and sulfonic acid functionalized organic/inorganic hybrid materials were synthesized. The organic groups were covalently bound on the inorganic surface through reactions of organosilane precursors with TEOS or with the silanol groups of material surface. These synthetic methods are called the co-condensation process and the post grafting. The structures and the textural parameters of the generated hybrid materials were characterized by XRD, N2 adsorption-desorption measurements, SEM and TEM. The incorporations of the organic groups were verified by elemental analysis, thermogravimetric analysis, FT-IR, UV-Vis, EPR, CV, as well as by 13C CP-MAS NMR and 29Si CP-MAS NMR spectroscopy. Introduction of various organic groups endow different phsysical, chemical properties to these hybrid materials. The (oligo)phenothiazines provide a group of novel redox acitive hybrid materials with special electronic and optic properties. The thiazolium salts modified materials were applied as heterogenized organo catalysts for the benzoin condensation and the cross-coupling of aldehydes with acylimines to yield a-amido ketones. The sulfonic acid containing materials can not only be used as Broensted acid catalysts, but also can serve as ion exchangable supports for further modifications and applications.
Das Zwei-Komponenten System CiaRH von S. pneumoniae ist an der ß-Laktam-Resistenz, Regulation der genetischen Kompetenz, Lyse und Virulenz beteiligt. Unter den 16 Promotoren, die von diesem System kontrolliert werden, befanden sich die fünf stärksten Promotoren in intergenen Regionen. Hieraus resultierte die Vermutung, dass diese Promotoren die Expression von kleinen nichtkodierenden RNAs steuern könnten. Mittels Northern-Analyse konnte nachgewiesen werden, dass von diesen Promotoren aus kleine RNAs mit einer Größe von 87 bis 151 Basen exprimiert werden. Im Stamm mit deletierten ciaR waren diese RNAs nicht bzw. kaum detektierbar. Die fünf CiaRH-abhängigen kleinen RNAs wurden csRNAs benannt (cia controlled small RNAs). Durch Northern-Analyse an vier verschiedenen Zeitpunkten des Wachstums konnte gezeigt werden, dass die csRNAs in hoher Konzentration sowohl während der exponentiellen als auch in der stationären Wachstumsphase in den Zellen von S. pneumoniae vorhanden sind. Mittels 3´-RACE-Analyse wurde die Länge der csRNAs auf die Base genau bestimmt. csRNA1, csRNA2, csRNA3 und csRNA4 bestehen aus 87 bis 101 Nukleotiden. csRNA5 besitzt dagegen eine Länge aus 151 Basen. Die Promotoren der stromabwärts der kleinen RNAs liegenden Gene wurden mittels 5´-RACE-Analyse kartiert und ihre Expression mittels realtime RT-PCR-Analyse im Wildtyp, im ciaR-Deletionsstamm und einem Stamm mit aktiviertem CiaRH-System untersucht. Hierbei konnte gezeigt werden, dass diese Gene unabhängig von CiaRH exprimiert werden. Um die Beteiligung der csRNAs an den CiaRH-assoziierten Phänotypen aufzuklären, wurden die Gene für alle fünf csRNAs deletiert und mit verschiedenen Antibiotika-Resistenz-Genen ersetzt. Durch Kombination aller Deletionen wurde der Stamm S. pneumoniae RK12345 konstruiert, welcher keine csRNA mehr exprimiert. In Stamm RK12345 konnte eine erniedrigte Transformationseffizienz und eine erhöhte Lyserate im Vergleich zum Wildtyp S. pneumoniae R6 beobachtet werden. Hierdurch wurde gezeigt, dass die csRNAs eine Rolle bei den CiaRH-regulierten Prozessen spielen. Kleine RNAs üben ihren regulatorischen Effekt meist durch Wechselwirkung mit der mRNA ihrer Zielgene aus. Die beiden RNA-Moleküle interagieren hierbei über komplementäre Sequenzbereiche. Durch diese Basenpaarungen kommt es zur Hemmung oder Aktivierung der Translation bzw. zum Abbau des RNA-RNA-Interaktionskomplexes. Um die Ursachen der csRNA-vermittelten phänotypischen Veränderungen bestimmen zu können, wurde die Identifizierung der csRNA-Zielgene angestrebt. Mittels bioinformatischer Analyse wurde eine große Anzahl putativer Zielgene vorhergesagt, wovon nach Anlegung verschiedener Kriterien letztendlich 13 zu den weiteren Untersuchungen eingesetzt wurden. Um diese teilweise uncharakterisierten Gene auf eine posttranskriptionelle Regulation durch die csRNAs untersuchen zu können, wurde ein integratives Translations Probe Plasmid namens pTP3 konstruiert. Dieses Plasmid ermöglicht die Klonierung von 5´-Genfragmenten vor das ´lacZ-Gen, wodurch in frame-Zielgen´-´lacZ-Fusionsproteine entstehen. Die Expression des entstandenen Fusionsgens erfolgt hierbei von einem konstitutiven CiaRH-unabhängigen Promotor, namens PvegT. Die Klonierung und Untersuchung der ß-Galaktosidase- Expression der Zielgen´-´lacZ-Fusionsproteine ergab, dass die klonierten Fragmente der Gene spr0081, comC, spr1645 und cibB durch die csRNAs reguliert werden. Die Untersuchung dieser vier Zielgene bei Expression von den eigenen Promotoren und Intaktheit der entsprechenden mRNAs zeigte, dass letztendlich zwei Gene posttranskriptionell negativ durch die csRNAs reguliert werden. Interessanterweise ist eines dieser Zielgene das comC-Gen, welches für das Vorläuferpeptid des Kompetenz-Phäromons CSP kodiert. Diese Beobachtung könnte eine mögliche Ursache für die csRNA-abhängige veränderte Transformierbarkeit von Stamm RK12345 darstellen. Das zweite Zielgen der csRNAs, spr0081, kodiert für einen bisher uncharakterisierten ABC-Transporter, welcher möglicherweise am Transport eines Kohlenhydrats beteiligt ist. Letztendlich wurde die direkte Interaktion der in vitro produzierten mRNA dieser beiden Zielgene mit den csRNAs durch die Entwicklung und Etablierung einer neuartigen Methode der Bandshift-Analyse untersucht. Hierbei konnte nachgewiesen werden, dass die mRNA von comC mit allen fünf csRNAs Interaktions-komplexe bildet und die mRNA von spr0081 befähigt ist mit vier csRNAs Interaktionskomplexe zu bilden. Schließlich wurde der Effekt einzelner RNAs auf die Expression des ComC´-´lacZ-Fusionsproteins getestet. Hierbei konnte gezeigt werden, dass csRNA1, csRNA2 und csRNA3 einerseits bzw. csRNA4 und csRNA5 andererseits genügen, um den Effekt aller fünf RNAs ausüben zu können. Die stärkste Hemmungsaktivität einer einzelnen csRNA konnte bei csRNA4 festgestellt werden.