by Huang Yu. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 145-150). / Introduction --- p.1 / Chapter 1 --- Global Attractors of Semigroups --- p.9 / Introduction --- p.9 / Chapter 1.1 --- Basic Notions --- p.9 / Chapter 1.2 --- Semigroup of Class K --- p.11 / Chapter 1.3 --- Semigroup of Class AK --- p.15 / Chapter 1.4 --- Hausdorff and Fractal Dimensions of Attractors --- p.19 / Chapter 1.4.1 --- Hausdorff and Fractal dimensions --- p.20 / Chapter 1.4.2 --- The Dimensions of Invariant Sets --- p.22 / Chapter 1.4.3 --- An Application to Evolution Equations --- p.35 / Notes --- p.39 / Chapter 2 --- Invariant Manifolds and Inertial Manifolds --- p.40 / Introduction --- p.40 / Chapter 2.1 --- Preliminary --- p.41 / Chapter 2.1.1 --- Notions --- p.41 / Chapter 2.1.2 --- Nemytskii Operator --- p.43 / Chapter 2.1.3 --- Contractions on Embedded Banach Spaces --- p.47 / Chapter 2.2 --- Linear and Nonlinear Integral Equations --- p.49 / Chapter 2.3 --- Invariant Manifolds --- p.55 / Chapter 2.4 --- Inertial Manifolds --- p.59 / Notes --- p.63 / Chapter 3 --- Semilinear Parabolic Variational Inequalities --- p.64 / Introduction --- p.64 / Chapter 3.1 --- Existence Results --- p.66 / Chapter 3.2 --- The Existence of Global Attractors --- p.69 / Chapter 3.3 --- The Weakly Approximating Inertial Manifolds --- p.76 / Chapter 3.4 --- An Application: The Obstacle Problem --- p.87 / Chapter 4 --- Semilinear Wave Equations with Damping and Critical Expo- nent --- p.91 / Introduction --- p.91 / Chapter 4.1 --- Existence Results --- p.93 / Chapter 4.2 --- The Global Attractor for the Problem --- p.96 / Chapter 4.2.1 --- A Proposition on Uniform Decay --- p.98 / Chapter 4.2.2 --- Compactness of the Trajectories of (4.2.7) --- p.102 / Chapter 4.3 --- A Particular Case-Linear Damping --- p.105 / Chapter 4.4 --- Estimate of the Dimensions of the Global Attractor --- p.111 / Chapter 4.4.1 --- The Linearized Equation --- p.114 / Chapter 4.4.2 --- The Hausdorff and Fractal Dimensions of the Attractor --- p.117 / Chapter 5 --- Partially Dissipative Evolution Equations --- p.123 / Introduction --- p.123 / Chapter 5.1 --- Basic Notions --- p.124 / Chapter 5.2 --- Semilinear Parabolic Equations and Systems --- p.128 / Chapter 5.3 --- Semilinera Hyperbolic Equation with Damping --- p.136 / Reference --- p.145

Procedural Content Generation (PCG) and Dynamic Difficulty Adjustment (DDA) have been used separately in games to improve player experience. We explore using PCG and DDA together in a feedback loop to keep a player in the "flow zone." The central tenet of this work is a conjecture about how the shape of the performance versus difficulty curve changes at the boundaries of the flow zone. Based on this conjecture, we have developed an algorithm that detects when the player has left the flow zone and appropriately adjusts the difficulty to bring the gameplay back into flow, even as the skill of the player is changing. We developed a game-independent algorithm, implemented our algorithm for the open-source Infinite Mario Bros (IMB) game and conducted a user study that supports the hypothesis that players will enjoy the game more with DDA - PCG algorithm.

In 1943 R. C. Buck showed that a sequence x is convergent if some regular matrix sums every subsequence of x. Thus, for example, if every subsequence of x is Cesaro summable, then x is actually convergent. Buck's result was quite surprising, since research in summability theory up to that time gave no hint of such a remarkable theorem. The appearance of Buck's result in the Bulletin of the American Mathematical Society (3) created immediate interest and has prompted considerable research which has taken the following directions: (i) to study regular matrix transformations in order to shed light on Buck's theorem, (ii) to extend Buck's theorem, (iii) to obtain analogs of Buck's theorem for sequence spaces other than the space of convergent sequences, and (iv) to obtain analogs of Buck's theorem involving processes other than subsequencing, such as stretching. The purpose of the present paper is to contribute to all facets of the problem, particularly to (i), (iii), and (iv).

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Mathematical modelling, simulation, and optimisation are core methodologies for future developments in engineering, natural, and life sciences. This work aims at applying these mathematical techniques in the field of biological processes with a focus on the wine fermentation process that is chosen as a representative model. In the literature, basic models for the wine fermentation process consist of a system of ordinary differential equations. They model the evolution of the yeast population number as well as the concentrations of assimilable nitrogen, sugar, and ethanol. In this thesis, the concentration of molecular oxygen is also included in order to model the change of the metabolism of the yeast from an aerobic to an anaerobic one. Further, a more sophisticated toxicity function is used. It provides simulation results that match experimental measurements better than a linear toxicity model. Moreover, a further equation for the temperature plays a crucial role in this work as it opens a way to influence the fermentation process in a desired way by changing the temperature of the system via a cooling mechanism. From the view of the wine industry, it is necessary to cope with large scale fermentation vessels, where spatial inhomogeneities of concentrations and temperature are likely to arise. Therefore, a system of reaction-diffusion equations is formulated in this work, which acts as an approximation for a model including computationally very expensive fluid dynamics. In addition to the modelling issues, an optimal control problem for the proposed reaction-diffusion fermentation model with temperature boundary control is presented and analysed. Variational methods are used to prove the existence of unique weak solutions to this non-linear problem. In this framework, it is possible to exploit the Hilbert space structure of state and control spaces to prove the existence of optimal controls. Additionally, first-order necessary optimality conditions are presented. They characterise controls that minimise an objective functional with the purpose to minimise the final sugar concentration. A numerical experiment shows that the final concentration of sugar can be reduced by a suitably chosen temperature control. The second part of this thesis deals with the identification of an unknown function that participates in a dynamical model. For models with ordinary differential equations, where parts of the dynamic cannot be deduced due to the complexity of the underlying phenomena, a minimisation problem is formulated. By minimising the deviations of simulation results and measurements the best possible function from a trial function space is found. The analysis of this function identification problem covers the proof of the differentiability of the function–to–state operator, the existence of minimisers, and the sensitivity analysis by means of the data–to–function mapping. Moreover, the presented function identification method is extended to stochastic differential equations. Here, the objective functional consists of the difference of measured values and the statistical expected value of the stochastic process solving the stochastic differential equation. Using a Fokker-Planck equation that governs the probability density function of the process, the probabilistic problem of simulating a stochastic process is cast to a deterministic partial differential equation. Proofs of unique solvability of the forward equation, the existence of minimisers, and first-order necessary optimality conditions are presented. The application of the function identification framework to the wine fermentation model aims at finding the shape of the toxicity function and is carried out for the deterministic as well as the stochastic case. / Mathematische Modellierung, Simulation und Optimierung sind wichtige Methoden für künftige Entwicklungen in Ingenieurs-, Natur- und Biowissenschaften. Ziel der vorliegende Arbeit ist es diese mathematische Methoden im Bereich von biologischen Prozessen anzuwenden. Dabei wurde die Weingärung als repräsentatives Modell ausgewählt. Erste Modelle der Weingärung, die man in der Literatur findet, bestehen aus gewöhnlichen Differentialgleichungen. Diese modellieren den Verlauf der Populationszahlen der Hefe, sowie die Konzentrationen von verwertbarem Stickstoff, Zucker und Ethanol. In dieser Arbeit wird auch die Konzentration von molekularem Sauerstoff betrachtet um den Wandel des Stoffwechsels der Hefe von aerob zu anaerob zu erfassen. Weiterhin wird eine ausgefeiltere Toxizitätsfunktion benutzt. Diese führt zu Simulationsergebnissen, die im Vergleich zu einem linearen Toxizitätsmodell experimentelle Messungen besser reproduzieren können. Außerdem spielt eine weitere Gleichung für die zeitliche Entwicklung der Temperatur eine wichtige Rolle in dieser Arbeit. Diese eröffnet die Möglichkeit den Gärprozess in einer gewünschten Weise zu beeinflussen, indem man die Temperatur durch einen Kühlmechanismus verändert. Für industrielle Anwendungen muss man sich mit großen Fermentationsgefäßen befassen, in denen räumliche Abweichungen der Konzentrationen und der Temperatur sehr wahrscheinlich sind. Daher ist in dieser Arbeit ein System von Reaktion-Diffusions Gleichungen formuliert, welches eine Approximation an ein Modell mit rechenaufwändiger Strömungsmechanik darstellt. Neben der Modellierung wird in dieser Arbeit ein Optimalsteuerungsproblem für das vorgestellte Gärmodell mit Reaktions-Diffusions Gleichungen und Randkontrolle der Temperatur gezeigt und analysiert. Variationelle Methoden werden benutzt, um die Existenz von eindeutigen schwachen Lösungen von diesem nicht-linearen Modell zu beweisen. Das Ausnutzen der Hilbertraumstruktur von Zustands- und Kontrolraum macht es möglich die Existenz von Optimalsteuerungen zu beweisen. Zusätzlich werden notwendige Optimalitätsbedingungen erster Ordnung vorgestellt. Diese charakterisieren Kontrollen, die das Zielfunktional minimieren. Ein numerisches Experiment zeigt, dass die finale Konzentration des Zuckers durch eine passend ausgewählte Steuerung reduziert werden kann. Der zweite Teil dieser Arbeit beschäftigt sich mit der Identifizierung einer unbekannten Funktion eines dynamischen Modells. Es wird ein Minimierungsproblem für Modelle mit gewöhnlichen Differentialgleichungen, bei denen ein Teil der Dynamik aufgrund der Komplexität der zugrundeliegenden Phänomene nicht hergeleitet werden kann, formuliert. Die bestmögliche Funktion aus einem Testfunktionenraum wird dadurch ausgewählt, dass Abweichungen von Simulationsergebnissen und Messungen minimiert werden. Die Analyse dieses Problems der Funktionenidentifikation beinhaltet den Beweis der Differenzierbarkeit des Funktion–zu–Zustand Operators, die Existenz von Minimierern und die Sensitivitätsanalyse mit Hilfe der Messung–zu–Funktion Abbildung. Weiterhin wird diese Funktionenidentifikationsmethode für stochastische Differentialgleichungen erweitert. Dabei besteht das Zielfunktional aus dem Abstand von Messwerten und dem Erwartungswert des stochastischen Prozesses, der die stochastische Differentialgleichung löst. In dem man die Fokker-Planck Gleichung benutzt wird das wahrscheinlichkeitstheoretische Problem einen stochastischen Prozess zu simulieren in eine deterministische partielle Differentialgleichung überführt. Es werden Beweise für die eindeutige Lösbarkeit der Vorwärtsgleichung, die Existenz von Minimierern und die notwendigen Bedingungen erster Ordnung geführt. Die Anwendung der Funktionenidentifikation auf die Weingärung zielt darauf ab die Form der Toxizitätsfunktion herauszufinden und wird sowohl für den deterministischen als auch für den stochastischen Fall durchgeführt.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 26, 2007) Vita. Includes bibliographical references.

Within the context of the verification of infinite-state systems, 'Regular model checking' is the name of a family of techniques in which states are represented by words or trees, sets of states by finite automata on these objects, and transitions by finite automata operating on pairs of state encodings, i.e. finite-state transducers. In this context, the problem of computing the set of reachable states of a system can be reduced to the one of computing the iterative closure of the finite-state transducer representing its transition relation. This thesis provides several techniques to computing the transitive closure of a finite-state transducer. One of the motivations of the thesis is to show the feasibility and usefulness of this approach through a combination of the necessary theoretical developments, implementation, and experimentation. For systems whose states are encoded by words, the iteration technique proceeds by comparing a finite sequence of successive powers of the transducer, detecting an 'increment' that is added to move from one power to the next, and extrapolating the sequence by allowing arbitrary repetitions of this increment. For systems whose states are represented by trees, the iteration technique proceeds by computing the powers of the transducer and progressively collapsing their states according to an equivalence relation until a fixed point is reached. The proposed iteration techniques can just as well be exploited to compute the closure of a given set of states by repeated applications of the transducer, which has proven to be a very effective way of using the technique. Various examples have been handled completely within the automata-theoretic setting. Another applications of the techniques are the verification of linear temporal properties as well as the computation of the convex hull of a finite set of integer vectors.

We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice $Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)$. In a first part, these processes are characterized as Gibbs states on path spaces of the form $C([0, T],R)Z^{d}$. In a second part, we study the Gibbsian character on $R^{Z}^{d}$ of $v^{t}$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law $v = v^{0}$ is Gibbsian.

We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion / when the initial distribution is Gibbsian.

The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. <br><br> AMS Classifications: 60G15 / 60G60 / 60H10 / 60J60