Steepest Entropy Ascent Quantum Thermodynamics (SEAQT) is a novel theoretical framework unifying quantum mechanics and thermodynamics. This framework employs an equation of motion governed by the principle of steepest entropy ascent to determine the thermodynamic state evolution of modeled systems. The SEAQT framework has seen applied to multiple systems, including quantum and gas phase systems, in addition to solid-state material phenomena. A precise definition of entropy is crucial for the application of this framework. The SEAQT framework defines entropy in terms of an intrinsic property associated with the energy spectrum of a modeled system, namely degeneracy. The degeneracy, or density of states, is the number of unique system configurations for a given energy level. Calculating this quantity is often difficult, limiting many solid-state material studies to the few systems with analytical expressions which define the degeneracy. However, the use of the Replica Exchange Wang Landau (REWL) algorithm has alleviated these challenges. The REWL algorithm is a non-Markovian MC method capable of estimating the density of the state of any discretely described system. Employing the derived discrete energy spectrum and associated degeneracies, combined with the SEAQT equation of motion, has allowed for the investigation of previously indescribable systems. Detail of the complete methods are provided in this document, and the prediction of system kinetics are presented for capillary dynamics, protein folding, polymer brush conformal evolution, and ion sequestration using polymers. The results from each model are compared against experimental results for the thermodynamic paths of systems under varying system conditions are shown. Use of the combined framework has predicted (i) expected grain growth for ceramic and nanoscale metallic systems, (ii) expected conformal evolution of initial collapse of a simple polymer chain, (ii) equilibrium density profile evolution of a polymer brush, (iv) expected functional participation in sequestration of ions in a polar solvent. / Doctor of Philosophy / In this document, a novel computational method is presented for the modeling of various metallic, ceramic and polymeric materials. The computational framework and methodology presented does not model the mechanical evolution of material systems, instead it focuses on a holistic approach accounting for all possible formations, configurations, and associated energies. The basics of the presented frame have seen significant application to several systems of various length scales, though material applications were limited due to the necessity of applying derived analytical expressions from literature. This work expands the application of the method to arbitrary systems, removing prior limitations to simulate several previously indescribable systems. Significant benefits of the presented methods include rapid calculation of the system evolution under variable initial thermal conditions versus conventional models.

If wilderness refers to those spaces that are unoccupied by humans while architecture is one major way that humans occupy space, the terms seem to be mutually exclusive. However, this thesis argues that wilderness and architecture have a fundamental similarity: they are both ways that humans understand and relate to the world. This thesis looks critically at the notion of wilderness by acknowledging that throughout time and history, humans have understood wilderness in innumerable different ways and, as a result, have treated those spaces that are deemed wilderness in innumerable different ways as well. It acknowledges wilderness as a “profoundly human creation[1]” and from this admission, explores the utility of predisposition as a means by which to direct the built environment’s relationship to the un-built environment. Ultimately, this thesis argues that when humans define wilderness in a particular way and then apply that label to a space, they have made a design decision. It investigates the ways in which architecture can apply the label of wilderness to new and even unlikely types of spaces in order to expand the lens through which we, as humans, value and appreciate this Earth. [1] Cronon, William. 1997. "The Trouble with Wilderness:; Or, Getting Back to the Wrong Nature." In Out of the Woods, 28-50: University of Pittsburgh Press. doi:10.2307/j.ctt7zw9qw.8. http://www.jstor.org/stable/j.ctt7zw9qw.8.

The topological entropy of the subdivision map of a finite subdivision rule restricted to the 1-skeleton of its model subdivision complex, which we call textbf{core entropy}, is examined. We consider core entropy for finite subdivision rules realizing quadratic Misiurewicz polynomials and matings of such polynomials. It is shown that for a non-restrictive class of finite subdivision rules realizing quadratic Misiurewicz polynomials, core entropy equals Thurston's core entropy. We also show that the core entropy of formal and degenerate matings of Misiurewicz polynomials is determined by Thurston's core entropy of the mated polynomials. / Doctor of Philosophy / Imagine taking a programmable calculator, inputting a number, and repeatedly pushing one of the buttons which corresponds to one of the calculator's built-in functions. For example, starting by inputting 0.5 and hitting the "x2" button over and over, or starting with 1.47 and repeatedly pressing the "sin(x)" button. The calculator may eventually return numbers that get closer and closer to a specific value, it may repeatedly cycle through some collection of specific numbers, it may not exhibit a clear pattern at all. It is of interest to understand, in some average sense, when, how often, and in what manner these patterns are exhibited and, in a quantitative fashion, compare how complicated the patterns are for different buttons on the calculator corresponding to different functions. For example, is the "x2" button, in some average sense, more or less "complex", in terms of the patterns exhibited by the above procedure, than the "sin(x)" button? Modeling or simulating physical phenomena such as particle motion or the orbits of collections of celestial bodies often entails the use of computer programs. These computer programs carry out calculations which often involves repeated application of various pre-programmed functions. Repeatedly pushing a button on a calculator can be viewed as a simplified version of what goes on with the calculations that a computer carries out in simulating physical phenomena. Understanding how to compare the patterns exhibited by simple, fundamental collections of functions makes for a good starting point for understanding the models that represent various physical phenomena. This work contributes to this endeavor by investigating a quantity which measures the complexity of some fundamental functions.

Kullback-Leibler relative-entropy or KL-entropy of P with respect to R defined as ∫xlnddPRdP , where P and R are probability measures on a measurable space (X, ), plays a basic role in the definitions of classical information measures. It overcomes a shortcoming of Shannon entropy – discrete case definition of which cannot be extended to nondiscrete case naturally. Further, entropy and other classical information measures can be expressed in terms of KL-entropy and hence properties of their measure-theoretic analogs will follow from those of measure-theoretic KL-entropy. An important theorem in this respect is the Gelfand-Yaglom-Perez (GYP) Theorem which equips KL-entropy with a fundamental definition and can be stated as: measure-theoretic KL-entropy equals the supremum of KL-entropies over all measurable partitions of X . In this thesis we provide the measure-theoretic formulations for ‘generalized’ information measures, and state and prove the corresponding GYP-theorem – the ‘generalizations’ being in the sense of R ´enyi and nonextensive, both of which are explained below. Kolmogorov-Nagumo average or quasilinear mean of a vector x = (x1, . . . , xn) with respect to a pmf p= (p1, . . . , pn)is defined ashxiψ=ψ−1nk=1pkψ(xk), whereψis an arbitrarycontinuous and strictly monotone function. Replacing linear averaging in Shannon entropy with Kolmogorov-Nagumo averages (KN-averages) and further imposing the additivity constraint – a characteristic property of underlying information associated with single event, which is logarithmic – leads to the definition of α-entropy or R ´enyi entropy. This is the first formal well-known generalization of Shannon entropy. Using this recipe of R´enyi’s generalization, one can prepare only two information measures: Shannon and R´enyi entropy. Indeed, using this formalism R´enyi characterized these additive entropies in terms of axioms of KN-averages. On the other hand, if one generalizes the information of a single event in the definition of Shannon entropy, by replacing the logarithm with the so called q-logarithm, which is defined as lnqx =x1− 1 −1 −q , one gets what is known as Tsallis entropy. Tsallis entropy is also a generalization of Shannon entropy but it does not satisfy the additivity property. Instead, it satisfies pseudo-additivity of the form x ⊕qy = x + y + (1 − q)xy, and hence it is also known as nonextensive entropy. One can apply R´enyi’s recipe in the nonextensive case by replacing the linear averaging in Tsallis entropy with KN-averages and thereby imposing the constraint of pseudo-additivity. A natural question that arises is what are the various pseudo-additive information measures that can be prepared with this recipe? We prove that Tsallis entropy is the only one. Here, we mention that one of the important characteristics of this generalized entropy is that while canonical distributions resulting from ‘maximization’ of Shannon entropy are exponential in nature, in the Tsallis case they result in power-law distributions. The concept of maximum entropy (ME), originally from physics, has been promoted to a general principle of inference primarily by the works of Jaynes and (later on) Kullback. This connects information theory and statistical mechanics via the principle: the states of thermodynamic equi- librium are states of maximum entropy, and further connects to statistical inference via select the probability distribution that maximizes the entropy. The two fundamental principles related to the concept of maximum entropy are Jaynes maximum entropy principle, which involves maximizing Shannon entropy and the Kullback minimum entropy principle that involves minimizing relative-entropy, with respect to appropriate moment constraints. Though relative-entropy is not a metric, in cases involving distributions resulting from relative-entropy minimization, one can bring forth certain geometrical formulations. These are reminiscent of squared Euclidean distance and satisfy an analogue of the Pythagoras’ theorem. This property is referred to as Pythagoras’ theorem of relative-entropy minimization or triangle equality and plays a fundamental role in geometrical approaches to statistical estimation theory like information geometry. In this thesis we state and prove the equivalent of Pythagoras’ theorem in the nonextensive formalism. For this purpose we study relative-entropy minimization in detail and present some results. Finally, we demonstrate the use of power-law distributions, resulting from ME-rescriptions of Tsallis entropy, in evolutionary algorithms. This work is motivated by the recently proposed generalized simulated annealing algorithm based on Tsallis statistics. To sum up, in light of their well-known axiomatic and operational justifications, this thesis establishes some results pertaining to the mathematical significance of generalized measures of information. We believe that these results represent an important contribution towards the ongoing research on understanding the phenomina of information. (For formulas pl see the original document) ii

The purpose of this study is to estimate and compare the entropy and redundancy of written English and Swedish. We also investigate and compare the entropy and redundancy of Twitter language. This is done by extracting n consecutive characters called n-grams and calculating their frequencies. No precise values are obtained, due to the amount of text being finite, while the entropy is estimated for text length tending towards infinity. However we do obtain results for n = 1,...,6  and the results show that written Swedish has higher entropy than written English and that the redundancy is lower for Swedish language. When comparing Twitter with the standard languages we find that for Twitter, the entropy is higher and the redundancy is lower.

Estudamos a entropia em várias situações na gravidade, verificando se seu comportamento é holográfico, obedecendo à lei de área de Bekenstein. Inicialmente, usando o método da \"parede de tijolos\", calculamos, em diversos casos, a entropia estatística de um campo escalar num fundo não trivial. Tal fundo é gerado por buracos negros de 4 ou 5 dimensões com cargas. A fórmula da entropia de Bekenstein é geralmente satisfeita, mas algumas correções são discutidas no caso pentadimensional. Este método é também aplicado para soluções tipo buracos negros na gravidade de Lovelock. Resulta que o método de \"parede de tijolos\", apesar de correto para a teoria de Einstein-Hilbert, pode não ser válido em geral, o que também acontece com a lei área. Algumas propriedades concernentes à teoria de cordas, especialmente a tecnologia das D-branas, são revistas naqueles aspectos necessários para este trabalho. Também estudamos e calculamos o limite superior da entropia para a gravidade de Lovelock. Finalmente, verificamos a validade do princípio holográfico num universo de (4 + n) dimensões numa fase inflacionária assimétrica. / We study the entropy for various situations in gravity, checking whether its behavior is holographic, obeying Bekensteins area law. First, using the brick wall method, we compute the statistical entropy of a scalar field in a nontrivial background in different cases. Such a background is generated by four and five dimensional black holes with charges. The Bekenstein entropy formula is generally obeyed, but corrections are discussed in the latter case. This method is applied also to the black hole solutions of the Lovelock gravity. It turns out that the brickwall method, through correct for the Einstein-Hilbert theory, may fail in general. The same happens to the area law. Some properties concerning string theory, especially the D-branes technology, are reviewed while necessary to this work. Furthermore, we study and compute the upper limit of the entropy for the Lovelock gravity. Finally, we check the validity of the holographic principle in a (4+n) dimensional universe in an asymmetric inflationary phase.

We hypothesized that temporal features of EEG are altered in sleep apnea subjects comparedto normal subjects. The initial aim was to develop a measure to discriminate sleep stages innormals. The longer-term goal was to apply these methods to identify differences in EEGactivity in sleep apnea subjects from normals. We analyzed the C3A2 EEG and anelectrooculogram (EOG) recorded from 9 normal adults awake and in rapid eye movement(REM) and non-REM sleep. The EEG signals were filtered to remove EOG contamination. Twomeasures of the irregularity of EEG signals, Sample Entropy (SpEn) and Tsallis Entropy, wereevaluated for their ability to discriminate sleep stages. SpEn changes with sleep state, beinglargest in Wake. Stage 3/4 had the smallest SpEn (0.57??0.11) normalized to Wake values,followed by Stage 2 (0.72??0.09), REM (0.75??0.1) and Stage 1 (0.89??0.05). This pattern wasconsistent in all the polysomnogram records analyzed. Similar pattern was observed in leadO1A2 as well. We conclude that SpEn may be useful as part of a montage for assessing sleepstate. We analyzed data from sleep apnea subjects having obstructive and central apnea eventsand have made some preliminary observations; the SpEn values were more similar across sleepstages and also high correlation with oxygen saturation was observed.

Thesis (MSc.(Physics) - University of Pretoria, 2007. / Summary in English.

Estudamos a entropia em várias situações na gravidade, verificando se seu comportamento é holográfico, obedecendo à lei de área de Bekenstein. Inicialmente, usando o método da \"parede de tijolos\", calculamos, em diversos casos, a entropia estatística de um campo escalar num fundo não trivial. Tal fundo é gerado por buracos negros de 4 ou 5 dimensões com cargas. A fórmula da entropia de Bekenstein é geralmente satisfeita, mas algumas correções são discutidas no caso pentadimensional. Este método é também aplicado para soluções tipo buracos negros na gravidade de Lovelock. Resulta que o método de \"parede de tijolos\", apesar de correto para a teoria de Einstein-Hilbert, pode não ser válido em geral, o que também acontece com a lei área. Algumas propriedades concernentes à teoria de cordas, especialmente a tecnologia das D-branas, são revistas naqueles aspectos necessários para este trabalho. Também estudamos e calculamos o limite superior da entropia para a gravidade de Lovelock. Finalmente, verificamos a validade do princípio holográfico num universo de (4 + n) dimensões numa fase inflacionária assimétrica. / We study the entropy for various situations in gravity, checking whether its behavior is holographic, obeying Bekensteins area law. First, using the brick wall method, we compute the statistical entropy of a scalar field in a nontrivial background in different cases. Such a background is generated by four and five dimensional black holes with charges. The Bekenstein entropy formula is generally obeyed, but corrections are discussed in the latter case. This method is applied also to the black hole solutions of the Lovelock gravity. It turns out that the brickwall method, through correct for the Einstein-Hilbert theory, may fail in general. The same happens to the area law. Some properties concerning string theory, especially the D-branes technology, are reviewed while necessary to this work. Furthermore, we study and compute the upper limit of the entropy for the Lovelock gravity. Finally, we check the validity of the holographic principle in a (4+n) dimensional universe in an asymmetric inflationary phase.

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