Infinite-Dimensional Systems (IDSs) which have been made possible by recent advances in mathematical and computational tools can be used to model complex real phenomena. However, due to physical, economic, or stringent non-invasive constraints on real systems, the underlying characteristics for mathematical models in general (and IDSs in particular) are often missing or subject to uncertainty. Therefore, developing efficient estimation techniques to extract missing pieces of information from available measurements is essential. The human brain is an example of IDSs with severe constraints on information collection from controlled experiments and invasive sensors. Investigating the intriguing modeling potential of the brain is, in fact, the main motivation for this work. Here, we will characterize the hemodynamic behavior of the brain using functional magnetic resonance imaging data. In this regard, we propose efficient estimation methods for two classes of IDSs, namely Partial Differential Equations (PDEs) and Fractional Differential Equations (FDEs). This work is divided into two parts. The first part addresses the joint estimation problem of the state, parameters, and input for a coupled second-order hyperbolic PDE and an infinite-dimensional ordinary differential equation using sampled-in-space measurements. Two estimation techniques are proposed: a Kalman-based algorithm that relies on a reduced finite-dimensional model of the IDS, and an infinite-dimensional adaptive estimator whose convergence proof is based on the Lyapunov approach. We study and discuss the identifiability of the unknown variables for both cases. The second part contributes to the development of estimation methods for FDEs where major challenges arise in estimating fractional differentiation orders and non-smooth pointwise inputs. First, we propose a fractional high-order sliding mode observer to jointly estimate the pseudo-state and input of commensurate FDEs. Second, we propose a modulating function-based algorithm for the joint estimation of the parameters and fractional differentiation orders of non-commensurate FDEs. Sufficient conditions ensuring the local convergence of the proposed algorithm are provided. Subsequently, we extend the latter technique to estimate smooth and non-smooth pointwise inputs. The performance of the proposed estimation techniques is illustrated on a neurovascular-hemodynamic response model. However, the formulations are efficiently generic to be applied to a wide set of additional applications.

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The Unruh effect is one of the most startling predictions of quantum field theory. Its interpretation has been controversially discussed, since the first publications of Fulling, Davies and Unruh in the 1970ties. In a recent paper Buchholz and Solveen proposed an application of basic thermodynamic definitions to clarify the meaning of temperature and thermal equilibrium in the Unruh effect. As a result the interpretation of the KMS-parameter as an expression of local temperature has been questioned. The main result of my diploma thesis asserts quasi-equivalence of the disputed KMS states on a subregion of Rindlerspace that infinitely extends in the direction of travel of a uniformly accelerated Rindler-observer. Exploring the consequences of this result, I will present new insights on the asymptotic behaviour of such KMS states and how this fits into the picture drawn by Buchholz and Solveen.

Image of religion in Bioshock: Infinite computer game Master's thesis - Bc.Jiří Kothera Few mainstream computer games have caused such controversy as Bioshock: Infinite (Irrational Games, 2013). The third installment of the Bioshock series is set in the fictional city of Columbia in an alternate history of early twentieth century, which at first glance appears to be a perfect social utopia. After a while, however, the narrative begins to uncover the multilayered problems of society oppressed by a fraction of the white elite and religious fanaticism embodied by the character of Z.H.Comstock, the charismatic leader of the whole community. The popularity of the game and its stable position at the top of the various popularity charts are not only due to the attractive audiovisual processing and complex game mechanics. It is primarily a story that uses (for a mass audience product) an unprecedented amount of religious symbolism - especially Christian, historical references, polysemic story elements and the story based on the concepts of Frontier myth and American exceptionalism. This work deals with the analysis of narrative and religious-social phenomena appearing in the game, especially those that are directly related to the religious and nationalistic topics in the United States.

This dissertation presents some of the recent developments in the modelling of shape spaces. Forming the basis for a quantitative analysis of shapes, this is relevant for many applications involving image recognition and shape classification. All shape spaces discussed in this work arise from the general situation of a Lie group acting isometrically on some Riemannian manifold. The first chapter summarizes the most important results about this general set-up, which are well known in other branches of mathematics. A particular focus is laid on Hamiltonian methods that explore the relation of symmetry and conserved momenta. As a classical example these results are applied to Kendall’s shape space. More recent approaches of continuous shape models are then summarized and put in the same concise framework. In more detail the square root velocity shape representation, recently developed by Srivastava et al., is being discussed. In particular, the phenomenon of unclosed orbits under the action of reparametrization is addressed. This issue is partially resolved by an extended equivalence relation along with a well defined, non-degenerate, metric on the resulting quotient space.

Büchi, Elgot and Trakhtenbrot provided a seminal connection between monadic second-order logic and finite automata for both finite and infinite words. This BET- Theorem has been extended by Lautemann, Schwentick and Thérien to context-free languages by introducing a monadic second-order logic with an additional existentially quantified second-order variable. This new variable models the stack of pushdown au- tomata. A fundamental study by Cohen and Gold extended the context-free languages to infinite words. Our first main result is a second-order logic in the sense of Lautemann, Schwentick and Thérien with the same expressive power as ω-context-free languages. For our argument, we investigate Greibach normal forms of ω-context-free grammars as well as a new type of Büchi pushdown automata, the simple pushdown automata. Simple pushdown automata do not use e-transitions and can change the stack only by at most one symbol. We show that simple pushdown automata of infinite words suffice to accept all ω-context-free languages. This enables us to use Büchi-type results recently developed for infinite nested words. In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Weighted context-free languages of finite words trace back already to Chomsky and Schützenberger. Their work has been extended to infinite words by Ésik and Kuich. As in the theory of formal grammars, these weighted ω-context-free languages, or ω-algebraic series, can be represented as solutions of mixed ω-algebraic systems of equations and by weighted ω-pushdown automata. In our second main result, we show that (mixed) ω-algebraic systems can be trans- formed into Greibach normal form. We then investigate simple pushdown automata in the weighted setting. Here, we give our third main result. We prove that weighted simple pushdown automata of finite words recognize all weighted context-free languages, i.e., generate all algebraic series. Then, we show that weighted simple ω-pushdown automata generate all ω-algebraic series. This latter result uses the former result together with the Greibach normal form that we developed for ω-algebraic systems. As a fourth main result, we prove that for weighted simple ω-pushdown automata, Büchi-acceptance and Muller-acceptance are expressively equivalent. In our fifth main result, we derive a Nivat-like theorem for weighted simple ω- pushdown automata. This theorem states that the behaviors of our automata are precisely the projections of very simple ω-series restricted to ω-context-free languages. The last result, our sixth main result, is a weighted logic with the same expressive power as weighted simple ω-pushdown automata. To prove the equivalence, we use a similar result for weighted nested ω-word automata and apply our present result of expressive equivalence of Muller and Büchi acceptance.

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Masters of Commerce / If share price fluctuations follow a simple random walk then it implies that forecasting models based on historical growth rates have little ability to forecast acceptable share price movements over a certain period. The simple random walk description of share price dynamics is obtained when a large number of investors have equal probability to buy or sell based on their own opinion. This simple random walk description of the stock market is in essence the Efficient Market Hypothesis, EMT. EMT is the central concept around which financial modelling is based which includes the Black-Scholes model and other important theoretical underpinnings of capital market theory like mean-variance portfolio selection, arbitrage pricing theory (APT), security market line and capital asset pricing model (CAPM). These theories, which postulates that risk can be reduced to zero sets the foundation for option pricing and is a key component in financial software packages used for pricing and forecasting in the financial industry. The model used by Black and Scholes and other models mentioned above are Gaussian, i.e. they exhibit a random nature. This Gaussian property and the existence of expected returns and continuous time paths (also Gaussian properties) allow the use of stochastic calculus to solve complex Black- Scholes models. However, if the markets are not Gaussian then the idea that risk can be. (educed to zero can lead to a misleading and potentially disastrous sense of security on the financial markets. This study project test the null hypothesis - share prices on the JSE follow a random walk - by means of graphical techniques such as symmetry plots and Quantile-Quantile plots to analyse the test distributions. In both graphical techniques evidence for the rejection of normality was found. Evidenceleading to the rejection of the hypothesis was also found through nonparametric or distribution free methods at a 1% level of significance for Anderson-Darling and Runs test.

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