Galen's Avoiding Distress provides an opportunity for scholars to qualify Galen's philosophical eclecticism because his ἄσκησις to avoid distress intersects theory and practice. My thesis carefully analyzes the theoretical framework behind Galen's claim that he “trained his φαντασἰαι for the loss of all his possessions” as well as the specific practices that constitute this training regimen. I trace the concept of φαντασἰα back to the first philosophical discussions in Plato's Theaetetus-Sophist structure and Aristotle's De anima to answer the questions “What are the φαντασἰαι that he talks about?” and “How do they participate in cognition?” I analyze Galen's On the doctrines of Hippocrates and Plato, Affections and Errors, and Thrasybulus to identify Galen's specific practices and relate them to what Galen thinks is the purpose of all humans. My inquiry allows me to argue that while Galen uses his imagination to condition himself not to fear the atrocities of Commodus he subordinates emotional tranquility and practices that promote it to the greater goal of doing good deeds for others.

In this thesis, motivated by modelling polymers, the topological entanglement complexity of systems of two self-avoiding polygons (2SAPs), stretched polygons and systems of self-avoiding walks (SSAWs) in a tubular sublattice of Z3 are investigated. In particular, knotting and linking probabilities are used to measure a polygonfs selfentanglement and its entanglement with other polygons respectively. For the case of 2SAPs, it is established that the homological linking probability goes to one at least as fast as 1-O(n^(-1/2)) and that the topological linking probability goes to one exponentially rapidly as n, the size of the 2SAP, goes to infinity. For the case of stretched polygons, used to model ring polymers under the influence of an external force f, it is shown that, no matter the strength or direction of the external force, the knotting probability goes to one exponentially as n, the size of the polygon, goes to infinity. Associating a two-component link to each stretched polygon, it is also proved that the topological linking probability goes to unity exponentially fast as n→∞. Furthermore, a set of entangled chains confined to a tube is modelled by a system of self- and mutually avoiding walks (SSAW). It is shown that there exists a positive number γ such that the probability that an SSAW of size n has entanglement complexity (EC), as defined in this thesis, greater than γn approaches one exponentially as n→∞. It is also established that EC of an SSAW is bounded above by a linear function of its size. Using a transfer-matrix approach, the asymptotic form of the free energy for the SSAW model is also obtained and the average edge-density for span m SSAWs is proved to approach a constant as m→∞. Hence, it is shown that EC is a ggoodh measure of entanglement complexity for dense polymer systems modelled by SSAWs, in particular, because EC increases linearly with system size, as the size of the system goes to infinity.

In this thesis, motivated by modelling polymers, the topological entanglement complexity of systems of two self-avoiding polygons (2SAPs), stretched polygons and systems of self-avoiding walks (SSAWs) in a tubular sublattice of Z3 are investigated. In particular, knotting and linking probabilities are used to measure a polygonfs selfentanglement and its entanglement with other polygons respectively. For the case of 2SAPs, it is established that the homological linking probability goes to one at least as fast as 1-O(n^(-1/2)) and that the topological linking probability goes to one exponentially rapidly as n, the size of the 2SAP, goes to infinity. For the case of stretched polygons, used to model ring polymers under the influence of an external force f, it is shown that, no matter the strength or direction of the external force, the knotting probability goes to one exponentially as n, the size of the polygon, goes to infinity. Associating a two-component link to each stretched polygon, it is also proved that the topological linking probability goes to unity exponentially fast as n→∞. Furthermore, a set of entangled chains confined to a tube is modelled by a system of self- and mutually avoiding walks (SSAW). It is shown that there exists a positive number γ such that the probability that an SSAW of size n has entanglement complexity (EC), as defined in this thesis, greater than γn approaches one exponentially as n→∞. It is also established that EC of an SSAW is bounded above by a linear function of its size. Using a transfer-matrix approach, the asymptotic form of the free energy for the SSAW model is also obtained and the average edge-density for span m SSAWs is proved to approach a constant as m→∞. Hence, it is shown that EC is a ggoodh measure of entanglement complexity for dense polymer systems modelled by SSAWs, in particular, because EC increases linearly with system size, as the size of the system goes to infinity.

We study the collapse transition of a lattice based protein model including an explicit coarse-grained model of a solvent. This model accounts for explicit hydrophobic interactions, and it is studied by Monte Carlo simulation. The protein is modelled as self-avoiding random walk with nearest neighbor interactions on a two dimensional lattice. Without the solvent, universal quantities of the chain around the collapse transition temperature are well known. Hydrophobicity is then modelled through a lattice of solvent molecules in which each molecule can have Q states depending of an orientation variable. Only one state is energetically favored, when two neighboring solvent molecules are both in the same state of orientation. The monomers are placed in interstitial position of the solvent lattice, and are only allowed to occupy sites surrounded by solvent cells of the same orientation. The potential of mean force between two interstitial solute molecules is calculated, showing a solvent mediated attraction typical of hydrophobic interactions. We then show that this potential increases with the energy of hydrogen bond formation as it appears in the model, while its characteristic range decreases. More importantly, we show that the chain embedded in the solvent undergoes a collapse transition, with the temperature of the transition being shifted relative to that of the chain in isolation. We calculate several critical exponents near the collapse transition, and we observe that their values are not conserved in presence of the explicit solvent.

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Book Summary: When White people of faith act in a particular way, their motivations are almost always attributed to their religious orientation. Yet when religious people of color act in a particular way, their motivations are usually attributed to their racial positioning. Religion Is Raced makes the case that religion in America has generally been understood in ways that center White Christian experiences of religion, and argues that all religion must be acknowledged as a raced phenomenon. When we overlook the role race plays in religious belief and action, and how religion in turn spurs public and political action, we lose sight of a key way in which race influences religiously-based claims-making in the public sphere. With contributions exploring a variety of religious traditions, from Buddhism and Islam to Judaism and Protestantism, as well as pieces on atheists and humanists, Religion Is Raced brings discussions about the racialized nature of religion from the margins of scholarly and religious debate to the center. The volume offers a new model for thinking about religion that emphasizes how racial dynamics interact with religious identity, and how we can in turn better understand the roles religion—and Whiteness—play in politics and public life, especially in the United States. It includes clear recommendations for researchers, including pollsters, on how to better recognize moving forward that religion is a raced phenomenon.

This work presents a numerical investigation of self-avoiding walks (SAWs) on percolation clusters, a canonical model for polymers in disordered media. A new algorithm has been developed allowing exact enumeration of over ten thousand steps. This is an increase of several orders of magnitude compared to previously existing enumeration methods, which allow for barely more than forty steps. Such an increase is achieved by exploiting the fractal structure of critical percolation clusters: they are hierarchically organized into a tree of loosely connected nested regions in which the walks segments are enumerated separately. After the enumeration process, a region is \"decimated\" and behaves in the following effectively as a single point. Since this method only works efficiently near the percolation threshold, a chain-growth Monte Carlo algorithm has also been used. Main focus of the investigations was the asymptotic scaling behavior of the average end-to-end distance as function of the number of steps on critical clusters in different dimensions. Thanks the highly efficient new method, existing estimates of the scaling exponents could be improved substantially. Also investigated were the number of possible chain conformation and the average entropy, which were found to follow an unusual scaling behavior. For concentrations above the percolation threshold the exponent describing the growth of the end-to-end distance turned out to differ from that on regular lattices, defying the prediction of the accepted theory. Finally, SAWs with short range attractions on percolation clusters are discussed. Here, it emerged that there seems to be no temperature-driven collapse transition as the asymptotic scaling behavior of the end-to-end distance even at zero temperature is the same as for athermal SAWs. / Die vorliegenden Arbeit präsentiert eine numerische Studie von selbstvermeidenden Zufallswegen (SAWs) auf Perkolationsclustern, ein kanonisches Modell für Polymere in stark ungeordneten Medien. Hierfür wurde ein neuer Algorithmus entwickelt, welcher es ermöglicht SAWs von mehr als zehntausend Schritten exakt auszuzählen. Dies bedeutet eine Steigerung von mehreren Größenordnungen gegenüber der zuvor existierenden Methode, welche kaum mehr als vierzig Schritte zulässt. Solch eine Steigerung wird erreicht, indem die fraktale Struktur der Perkolationscluster geziehlt ausgenutzt wird: Die Cluster werden hierarchisch in lose verbundene Gebiete unterteilt, innerhalb welcher Wegstücke separat ausgezählt werden können. Nach dem Auszählen wird ein Gebiet \"dezimiert\" und verhält sich während der Behandlung größerer Gebiete effektiv wie ein Gitterpunkt. Da diese neue Methode nur nahe der Perkolationsschwelle funktioniert, wurde zum Erzielen der Ergebnisse zudem ein Kettenwachstums-Monte-Carlo-Algorithmus (PERM) eingesetzt. Untersucht wurde zunächst das asymptotische Skalenverhalten des Abstands der beiden Kettenenden als Funktion der Schrittzahl auf kritischen Clustern in verschiedenen Dimensionen. Dank der neuen hochperformanten Methode konnten die bisherigen Schätzer für den dies beschreibenden Exponenten signifikant verbessert werden. Neben dem Abstand wurde zudem die Anzahl der möglichen Konformationen und die mittlere Entropie angeschaut, für welche ein ungewöhnliches Skalenverhalten gefunden wurde. Für Konzentrationen oberhalb der Perkolationsschwelle wurde festgestellt, dass der Exponent, welcher das Wachstum des Endabstands beschreibt, nicht dem für freie SAWs entspricht, was nach gängiger Lehrmeinung der Fall sein sollte. Schlussendlich wurden SAWs mit Anziehung zwischen benachbarten Monomeren untersucht. Hier zeigte sich, dass es auf kritischen Perkolationsclustern keinen Phasenübergang zu geben scheint, an welchem die Ketten kollabieren, sondern dass das Skalenverhalten des Endabstands selbst am absoluten Nullpunkt der Temperatur unverändert ist.

Enumeration of pattern avoiding compositions of numbers Abstract The aim of this work was to find some new results for 3-regular compositions, i.e., for those compositions which avoid the set of patterns {121, 212, 11}. Those compositions can be regarded as a generalization of Carlitz composition. Based on the generating function of compositions avoiding the set of patterns {121, 11} and {212, 11} we derive an upper bound for the coefficients of the power series of the generating function of 3-regular compositions. Using the theory of finite automata we derive its lower bound. We develop this result further by defining 3-block compositions. For the generating function of 3-regular compositions we prove a recursive ralation. Besides that we also compute the generating function of compositions avoiding the set of patterns {312, 321} whose parts are in the set [d]. In the last section we prove that the generating function of Carlitz compositions is transcendental.

<p>An Archimedean lattice is a graph of a regular tiling of the plane, such that all corners are equivalent. A tiling is regular if all tiles are regular polygons: equilateral triangles, squares, et cetera. There exist exactly 11 Archimedean lattices. Being planar graphs, the Archimedean lattices have duals, 3 of which are Archimedean, the other 8 are called Laves lattices.</p><p>In the thesis, three measures of connectivity of these 19 graphs are studied: the connective constant for self-avoiding walks, and bond and site percolation critical probabilities. The connective constant measures connectivity by the number of walks in which all visited vertices are unique. The critical probabilities quantify the proportion of edges or vertices that can be removed, so that the produced subgraph has a large connected component.</p><p>A common issue for these measures is that they, although intensely studied by both mathematicians and scientists from other fields, have been calculated only for very few graphs. With the goal of comparing the induced orders of the Archimedean and Laves lattices under the three measures, the thesis gives improved bounds and estimates for many graphs. </p><p>A large part of the thesis focuses on the problem of deciding whether a given graph is a subgraph of another graph. This, surprisingly difficult problem, is considered for the set of Archimedean and Laves lattices, and for the set of matching Archimedean and Laves lattices.</p>

An Archimedean lattice is a graph of a regular tiling of the plane, such that all corners are equivalent. A tiling is regular if all tiles are regular polygons: equilateral triangles, squares, et cetera. There exist exactly 11 Archimedean lattices. Being planar graphs, the Archimedean lattices have duals, 3 of which are Archimedean, the other 8 are called Laves lattices. In the thesis, three measures of connectivity of these 19 graphs are studied: the connective constant for self-avoiding walks, and bond and site percolation critical probabilities. The connective constant measures connectivity by the number of walks in which all visited vertices are unique. The critical probabilities quantify the proportion of edges or vertices that can be removed, so that the produced subgraph has a large connected component. A common issue for these measures is that they, although intensely studied by both mathematicians and scientists from other fields, have been calculated only for very few graphs. With the goal of comparing the induced orders of the Archimedean and Laves lattices under the three measures, the thesis gives improved bounds and estimates for many graphs. A large part of the thesis focuses on the problem of deciding whether a given graph is a subgraph of another graph. This, surprisingly difficult problem, is considered for the set of Archimedean and Laves lattices, and for the set of matching Archimedean and Laves lattices.