This thesis argues that in order to demonstrate the possibility and sensibility of Italian philosopher Gianni Vattimo's 'weak religious belief', it should be understood as the becoming uncertain of traditional, metaphysical (strong) belief. The difference between weak belief and strong belief can thereby be understood not as two distinct modes of belief, but as an event of weakening in the history of belief that has yet to be realized by those who believe with the support of metaphysical certainty. Since Vattimo aligns metaphysics with violence, and since he aligns traditional belief with metaphysics, to demonstrate and defend the possibility of Vattimo’s weak belief amounts to the reduction of violence in the world. However, the possibility and validity of weak belief has been called into question by thinkers such as Richard Rorty. In light of a review of the arguments and counter-arguments between Rorty and Vattimo, I argue that it is possible to distinguish weak belief from strong belief as long as this remains a weak distinction.

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0,1}^d where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the 'saturation number' of Q_m in Q_d. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Q_d of bounded average degree such that G does not contain a copy of Q_m but, for every G' such that G ⊊ G' ⊆ Q_d, the graph G' contains a copy of Q_m. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2^{(1-ε)k} subsets of [n] which does not contain a chain of length k+1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0,1) such that the smallest such collection is of cardinality 2^{(1+o(1))^ck} for all k. In Chapter 4, we obtain an exact expression for the 'weak saturation number' of Q_m in Q_d. That is, we determine the minimum number of edges in a spanning subgraph G of Q_d such that the edges of E(Q_d)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Q_m. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of 'axis aligned' copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A₀ of 'infected' vertices in a graph G and, at each step, a 'healthy' vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A₀ percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Q_d has cardinality at least (1+o(1))(d choose r-1)/r. This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r=3 (the minimum cardinality in the case r=2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset 𝒱(q,n) consisting of all subspaces of 𝔽_qⁿordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed 'container method,' to obtain an upper bound on the number of antichains in 𝒱(q,n) and a bound on the size of the largest antichain in a p-random subset of 𝒱(q,n) which holds with high probability for p in a certain range. In Chapter 8, we construct a 'finitely forcible graphon' W for which there exists a sequence (ε_i)^∞_i=1 tending to zero such that, for all i ≥ 1, every weak ε_i-regular partition of W has at least exp(ε_i^-2/2^{5log∗ε_i^-2}) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p,q with p/q ❘≥ 2, a circular (p,q)-colouring of a graph G is a mapping V(G) → ℤ_p such that any two adjacent vertices are mapped to elements of ℤ_p at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p,q)-colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a p,q-colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4.

Ma thèse de doctorat se concentre principalement sur le comportement en temps long des processus de Markov, les inégalités fonctionnelles et les techniques relatives. Plus spécifiquement, Je vais présenter les taux de convergence sous-exponentielle explicites des processus de Markov dans deux approches : la méthode Meyn-Tweedie et l’hypocoercivité (faible). Le document se divise en trois parties. Dans la première partie, Je vais présenter quelques résultats importants et des connaissances connexes. D’abord, un aperçu de mon domaine de recherche sera donné. La convergence exponentielle (ou sous-exponentielle) des chaînes de Markov et des processus de Markov (à temps continu) est un sujet d’actualité dans la théorie des probabilité. La méthode traditionnelle développée et popularisée par Meyn-Tweedie est largement utilisée pour ce problème. Dans la plupart des résultats, le taux de convergence n’est pas explicite, et certains d’entre eux seront brièvement présentés. De plus, la fonction de Lyapunov est cruciale dans l’approche Meyn-Tweedie, et elle est aussi liée à certaines inégalités fonctionnelles (par exemple, inégalité de Poincaré). Cette relation entre fonction de Lyapounov et inégalités fonctionnelles sera donnée avec les résultats au sens L2. En outre, pour l’exemple de l’équation cinétique de Fokker-Planck, un résultat de convergence exponentielle explicite de la solution sera introduite à la manière de Villani : l’hypocoercivité. Ces contenus sont les fondements de mon travail, et mon but est d’étudier la décroissance sous-exponentielle. La deuxième partie, fait l’objet d’un article écrit en coopération avec d’autres sur les taux de convergence sous-exponentielle explicites des processus de Markov à temps continu. Comme nous le savons, les résultats sur les taux de convergence explicites ont été donnés pour le cas exponentiel. Nous les étendons au cas sous-exponentielle par l’approche Meyn-Tweedie. La clé de la preuve est l’estimation du temps de passage dans un ensemble ”petite”, obtenue par Douc, Fort et Guillin, mais pour laquelle nous donnons une preuve plus simple. Nous utilisons aussi la construction du couplage et donnons une ergodicité sous exponentielle explicite. Enfin, nous donnons quelques applications numériques. Dans la dernière partie, mon second article traite de l’équation cinétique de Fokker-Planck. Je prolonge l’hypocoercivité à l’hypocoercivité faible qui correspond à inégalité de Poincaré faible. Grâce à cette extension, on peut obtenir le taux de convergence explicite de la solution, dans des cas sous-exponentiels. La convergence est au sens H1 et au sens L2. A la fin de ce document, j’étudie le cas de l’entropie relative comme Villani, et j’obtiens la convergence au sens de l’entropie. Enfin, Je donne deux exemples pour les potentiels qui impliquent l’inégalité de Poincaré faible ou l’inégalité de Sobolev logarithmique faible pour la mesure invariante. / My Ph.D dissertation mainly focuses on long time behavior of Markov processes, functional inequalities and related techniques. More specifically, I will present the computable sub-exponential convergence rate of the Markov process in two approaches : Meyn-Tweedie’s method and (weak) hypocoercivity. The paper consists of three parts. In the first part, I will introduce some important results and related knowledge. Firstly, overviews of my research field are given. Exponential (or subexponential) convergence of Markov chains and (continuous time) Markov processes is a hot issue in probability. The traditional method - Meyn-Tweedie’s approach is widely applied for this problem. Most of the results about convergence rate is not explicit, and some of them will be introduced briefly. In addition,Lyapunov function is crucial in Meyn-Tweendie’s aproach, and it is also related to some functional inequalities (for example, Poincar´e inequality). The relationship of them will be given with results in L2 sense. Furthermore, as a example of kinetic Fokker-Planck equation, a computable result of exponential convergence of the solution of it will be introduced in Villani’ way - hypocoercivity. These contents are foundations of my work, and my destination is to study the sub-exponential decay. In the second part, it is my article cooperated with others about subexponential convergence rate of continuous time Markov processes. As we all know, the explicit results of convergence rate is about the exponential case. We extend them to sub-exponential case in Meyn-Tweedie’s approach. The key of the proof is the estimation of the hitting time to small set which was got by Douc, Fort and Guillin, for which we also propose an alternative simpler proof. We also use coupling construction as others and give a quantitative sub-exponential ergodicity. At last, we give some calculations for examples. In the last part, my second article deal with the kinetic Fokker-Planck equation. I extend the hypocoercivity to weak hypocoercivity which correspond to weak Poincar´e inequality. Through the extension, one can get the computable rate of convergence of the solution, which is also sub-exponential case. The convergence is in H1 sense and in L2 sense. In the end of this paper, I study the relative entropy case as C.Villani, and get convergence in entropy. Finally, I give two examples for potentials that implies weak Poincar´e inequality or weak logarithmic Sobolve inequality for invarient measure.

In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation, sometimes referred to as the intrinsic mass transport principle. The so-called law of a finite graph is an example of a unimodular measure. We say that a measure is sustained by a countable graph if the set of rooted connected components of the graph has full measure. We demonstrate several new results involving sustained unimodular measures, and provide thorough arguments for known ones. In particular, we give a criterion for unimodularity on connected graphs, deduce that connected graphs sustain at most one unimodular measure, and prove that unimodular measures sustained by disconnected graphs are convex combinations. Furthermore, we discuss weak limits of laws of finite graphs, and construct counterexamples to seemingly reasonable conjectures.

Intelligence, having espionage as its roots, is sometimes misunderstood due to its secret nature. It is due to this that intelligence as a vital component in a state, could be misused by the political regime through less democratic practices that infringe on human rights and the rule of law. The quest of this study (which is not classified, to make the research findings available to both practitioners and scholars of intelligence studies), is to attempt to contribute to the theory and understanding of intelligence studies as a sub-field within the political science in describing, explaining and analysing the intelligence regime. The primary aim is to provide a contextualised and systematic overview of the South African intelligence regime within the framework of the democratisation process in South Africa with specific reference to the period from 1994 to 2014. This study analyses the intelligence regime in South Africa by specifically assessing the role, function and purpose of intelligence through history and within the context of the South African political regime. Therefore, the research problem examined in this study is: given the history and development of South Africa as a democracy, what is the role and function of the intelligence regime during the period 1994 to 2014 – specifically to determine whether intelligence practices were more or less democratic. The theoretical framework formed the basis from which the concepts of state, political regime types and form and degree of government, was explored. It furthermore provided for a comparison of democratic and non-democratic intelligence practices. The study locates intelligence as a reflection of the political regime through the simultaneous typology of both the regime and intelligence, thereby to enable the identification of more or less democratic practices. The notion of a hybrid regime, as having elements of both democratic and non-democratic regimes, presented a fundamental shift in the perception of South Africa‘s democratisation process towards an attempt to reach the goal of being a consolidated democracy. This study draw the conclusion that the role, functions and mandate of an intelligence service within a democracy, should firstly focus on providing the policy-maker with intelligence to be able to make policies; secondly on the identification of threats or potential threats to national security of the state and lastly to protect the constitution. This would ultimately enable the intelligence regime in South Africa to employ more democratic practices which could assist in reaching the goal of democratic consolidation. / MA (Political Studies), North-West University, Potchefstroom Campus, 2014

Intelligence, having espionage as its roots, is sometimes misunderstood due to its secret nature. It is due to this that intelligence as a vital component in a state, could be misused by the political regime through less democratic practices that infringe on human rights and the rule of law. The quest of this study (which is not classified, to make the research findings available to both practitioners and scholars of intelligence studies), is to attempt to contribute to the theory and understanding of intelligence studies as a sub-field within the political science in describing, explaining and analysing the intelligence regime. The primary aim is to provide a contextualised and systematic overview of the South African intelligence regime within the framework of the democratisation process in South Africa with specific reference to the period from 1994 to 2014. This study analyses the intelligence regime in South Africa by specifically assessing the role, function and purpose of intelligence through history and within the context of the South African political regime. Therefore, the research problem examined in this study is: given the history and development of South Africa as a democracy, what is the role and function of the intelligence regime during the period 1994 to 2014 – specifically to determine whether intelligence practices were more or less democratic. The theoretical framework formed the basis from which the concepts of state, political regime types and form and degree of government, was explored. It furthermore provided for a comparison of democratic and non-democratic intelligence practices. The study locates intelligence as a reflection of the political regime through the simultaneous typology of both the regime and intelligence, thereby to enable the identification of more or less democratic practices. The notion of a hybrid regime, as having elements of both democratic and non-democratic regimes, presented a fundamental shift in the perception of South Africa‘s democratisation process towards an attempt to reach the goal of being a consolidated democracy. This study draw the conclusion that the role, functions and mandate of an intelligence service within a democracy, should firstly focus on providing the policy-maker with intelligence to be able to make policies; secondly on the identification of threats or potential threats to national security of the state and lastly to protect the constitution. This would ultimately enable the intelligence regime in South Africa to employ more democratic practices which could assist in reaching the goal of democratic consolidation. / MA (Political Studies), North-West University, Potchefstroom Campus, 2014

This thesis develops the methodology for solving initial boundary value problems with the use of summation-by-parts discretizations. The combination of high orders of accuracy and a systematic approach to construct provably stable boundary and interface procedures makes this methodology especially suitable for scientific computations with high demands on efficiency and robustness. Most classes of high order methods can be applied in a way that satisfies a summation-by-parts rule. These include, but are not limited to, finite difference, spectral and nodal discontinuous Galerkin methods. In the first part of this thesis, the summation-by-parts methodology is extended to the time domain, enabling fully discrete formulations with superior stability properties. The resulting time discretization technique is closely related to fully implicit Runge-Kutta methods, and may alternatively be formulated as either a global method or as a family of multi-stage methods. Both first and second order derivatives in time are considered. In the latter case also including mixed initial and boundary conditions (i.e. conditions involving derivatives in both space and time). The second part of the thesis deals with summation-by-parts discretizations on multi-block and hybrid meshes. A new formulation of general multi-block couplings in several dimensions is presented and analyzed. It collects all multi-block, multi-element and  hybrid summation-by-parts schemes into a single compact framework. The new framework includes a generalized description of non-conforming interfaces based on so called summation-by-parts preserving interpolation operators, for which a new theoretical accuracy result is presented.

An experimental method was further developed to quantify the anisotropy of multi-site fatigue crack initiation behaviors in high strength Al alloys by four-point bend fatigue testing under stress control. In this method, fatigue crack initiation sites (fatigue weak-links, FWLs) were measured on the sample surface at different cyclic stress levels. The FWL density in an alloy could be best described using a three-parameter Weibull function of stress, though other types of sigmoidal functions might also be used to quantify the relationship between FWL density and stress. The strength distribution of the FWLs was derived from the Weibull function determined by fitting the FWLs vs. stress curve experimentally obtained. As materials properties, the FWL density and strength distribution could be used to evaluate the fatigue crack nucleation behaviors of engineering alloys quantitatively and the alloy quality in terms of FWL density and strength distribution. In this work, the effects of environment, types of microstructural heterogeneities and loading direction on FWLs were all studied in detail in AA7075-T651, AA2026-T3511, and A713 Al alloys, etc. It was also found that FWLs should be quantified as a Weibull-type function of strain instead of stress, when the applied maximum cyclic stress exceeded the yield strength of the tested alloys. In this work, four-point bend fatigue tests were conducted on the L-T (Rolling-Transverse), L-S (Rolling-Short transverse) and T-S planes of an AA7075-T651 alloy plate, respectively, at room temperature, 20 Hz, R=0.1, in air. The FWL populations, measured on these surfaces, were a Weibull-type function of the applied maximum cyclic stress, from which FWL density and strength distribution could be determined. The alloy showed a significant anisotropy of FWLs with the weak-link density being 11 mm-2, 15 mm-2 and 4 mm-2 on the L-T, L-S and T-S planes, respectively. Fatigue cracks were predominantly initiated at Fe-containing particles on the L-T and L-S planes, but only at Si-bearing particles on the T-S plane, profoundly demonstrating that the pre-fractured Fe-containing particles were responsible for crack initiation on the L-T and L-S planes, since the pre-fracture of these particles due to extensive deformation in the L direction during the prior rolling operation could only promote crack initiation when the sample was cyclically stressed in the L direction on both the L-T and L-S planes. The fatigue strengths of the L-T, L-S and T-S planes of the AA7075 alloy were measured to be 243.6, 273.0 and 280.6 MPa, respectively. The differences in grain and particle structures between these planes were responsible for the anisotropy of fatigue strength and FWLs on these planes. Three types of fatigue cracks from particles, type-I: the micro-cracks in the particles could not propagate into the matrix, i.e., type-II: the micro-cracks were fully arrested soon after they propagated into the matrix, and type-III: the micro-cracks became long cracks, were observed in the AA7075-T651 alloy after fatigue testing. By cross-sectioning these three-types of particles using Focused Ion Beam (FIB), it was found that the thickness of the particles was the dominant factor controlling fatigue crack initiation at the particles, namely, the thicker a pre-fractured Fe-containing particle, the easier it became a type-III crack on the L-T and L-S planes. On the T-S plane, there were only types-I and III Si-bearing particles at which crack were initiated. The type-I particles were less than 6.5 μm in thickness and type-III particles were thicker than 8.3 μm. Cross-sectioning of these particles using FIB revealed that these particles all contained gas pores which promoted crack initiation at the particles because of higher stress concentration at the pores in connection with the particles. It was also found that fatigue cracks did not always follow the any specific crystallographic planes within each grain, based on the Electron Backscatter Diffraction (EBSD) measurement. Also, the grain orientation did not show a strong influence on crack initiation at particles which were located within the grains. The topography measurements with an Atomic Force Microscope (AFM) revealed that Fe-containing particles were protruded on the mechanically polished surface, while the Si-bearing particles were intruded on the surface, which was consistent with hardness measurements showing that Si-bearing particles were softer (4.030.92 GPa) than Fe-containing ones (8.9 0.87 GPa) in the alloy. To verify the 3-D effects of the pre-fractured particles on fatigue crack initiation in high strength Al alloys, rectangular micro-notches of three different types of dimensions were fabricated using FIB in the selected grains on the T-S planes of both AA2024-T351 and AA7075-T651 Al alloys, to mimic the three types of pre-fractured particles found in these alloys. Fatigue testing on these samples with the micro-notches verified that the wider and deeper the micro-notches, the easier fatigue cracks could be initiated at the notches. In the AA2024-T351 samples, cracks preferred to propagate along the {111} slip plane with the smallest twist angle and relatively large Schmid factor. These experimental data obtained in this work could pave a way to building a 3-D quantitative model for quantification of fatigue crack initiation behaviors by taking into account the driving force and resistance to short crack growth at the particles in the surface of these alloys.

This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BC%7D%5E1" />-smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

This dissertation focuses on a recent but growing movement in Argentina, state-sponsored formal institutions of popular music education. The musics taught in these schools – tango, jazz, and Argentine folk idioms – have historically been excluded from the country’s formal music education systems. Recent moves to standardize and legitimize these musics in this new institutional context raise questions of canon formation, pedagogical praxis, aesthetics and musical meaning that have implications far beyond the classrooms where they are implemented. I examine two of these schools based in and around the capital city of Buenos Aires: the Escuela de Música Popular de Avellaneda, and the Tango and Folklore department of the Conservatorio Superior de Música “Manuel de Falla.” I adopt an ethnographic approach that considers broad structural and policy issues of power distribution, state intervention, and cultural nationalism. I also examine how these structures play out in discourse and practice within and beyond the classroom, shaped by and in turn shaping students’ and teachers’ aesthetics, politics, and subject positions. I then analyze the output of several musical groups composed of current students and recent graduates of these programs, exploring the notion of an emerging institutional aesthetic and the extent to which these institutions act as homogenizing influences or engender creative divergence. Finally, applying Pierre Bourdieu’s concept of a field of cultural production, I question the extent to which this new “música popular” is truly popular, ultimately arguing that it occupies a sort of third space between mass culture and high culture, replicating some avant-garde assumptions about the role of art as anti-commercial, yet simultaneously embracing a symbolic economy that valorizes populist and subaltern identities and ideologies. / text