We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

Numerical simulations of Mach shock reflections have shown that the Mach stem can bifurcate as a result of the slip line jetting forward. Numerical simulations were conducted in this study which determined that these bifurcations occur when the Mach number is high, the ramp angle is high, and specific heat ratio is low. It was clarified that the bifurcation is a result of a sufficiently large velocity difference across the slip line which drives the jet. This bifurcation phenomenon has also been observed after triple point collisions in detonation simulations. A triple point reflection was modelled as an inert shock reflecting off a wedge, and the accuracy of the model at early times after reflection indicates that bifurcations in detonations are a result of the shock reflection process. Further investigations revealed that bifurcations likely contribute to the irregular structure observed in certain detonations.

Numerical simulations of Mach shock reflections have shown that the Mach stem can bifurcate as a result of the slip line jetting forward. Numerical simulations were conducted in this study which determined that these bifurcations occur when the Mach number is high, the ramp angle is high, and specific heat ratio is low. It was clarified that the bifurcation is a result of a sufficiently large velocity difference across the slip line which drives the jet. This bifurcation phenomenon has also been observed after triple point collisions in detonation simulations. A triple point reflection was modelled as an inert shock reflecting off a wedge, and the accuracy of the model at early times after reflection indicates that bifurcations in detonations are a result of the shock reflection process. Further investigations revealed that bifurcations likely contribute to the irregular structure observed in certain detonations.

In this thesis, we present analytical analysis of key distribution schemes on wireless sensor networks. Since wireless sensor network is under unreliable environment, many random key pre-distribution based schemes have been developed to enhance security. Most of these schemes need to guarantee the existence of specific properties, such as disjoint secure paths or disjoint secure cliques, to achieve a secure cooperation among nodes. Two of the basic questions are as follows: 1. Under what conditions does a large-scale sensor network contain a certain structure? 2. How can one give a quantitative analysis behave as n grows to the infinity? However, analyzing such a structure or combinatorial problem is complicated in classical wireless network models such as percolation theories or random geometric graphs. Particularly, proofs in geometric setting models often blend stochastic geometric and combinatorial techniques and are more technically challenging. To overcome this problem, an approximative quasi-random graph is employed to eliminate some properties that are difficult to tackle. The most well-known solutions of this kind problems are probably Szemeredi's regularity lemma for embedding. The main difficulty from the fact that the above questions involve extremely small probabilities. These probabilities are too small to estimate by means of classical tools from probability theory, and thus a specific counting methods is inevitable.

Regularity is a shared memory consistency condition that has received considerable attention, notably in connection with quorum-based shared memory. Lamport's original definition of regularity assumed a single-writer model, however, and is not well defined when each shared variable may have multiple writers. In this thesis, we address this need by formally extending the notion of regularity to a multi-writer model. We have shown that the extension is not trivial. While there exist various ways to extend the single-writer definition, the resulting definitions will have different strengths. Specifically, we give several possible definitions of regularity in the presence of multiple writers. We then present a quorum-based algorithm to implement each of the proposed definitions and prove them correct. We study the relationships between these definitions and a number of other well-known consistency conditions, and give a partial order describing the relative strengths of these consistency conditions. Finally, we provide a practical context for our results by studying the correctness of two well-known algorithms for mutual exclusion under each of our proposed consistency conditions.

This dissertation consists of three parts. In the

Regularity is a shared memory consistency condition that has received considerable attention, notably in connection with quorum-based shared memory. Lamport's original definition of regularity assumed a single-writer model, however, and is not well defined when each shared variable may have multiple writers. In this thesis, we address this need by formally extending the notion of regularity to a multi-writer model. We have shown that the extension is not trivial. While there exist various ways to extend the single-writer definition, the resulting definitions will have different strengths. Specifically, we give several possible definitions of regularity in the presence of multiple writers. We then present a quorum-based algorithm to implement each of the proposed definitions and prove them correct. We study the relationships between these definitions and a number of other well-known consistency conditions, and give a partial order describing the relative strengths of these consistency conditions. Finally, we provide a practical context for our results by studying the correctness of two well-known algorithms for mutual exclusion under each of our proposed consistency conditions.

We study three different problems in non linear PDE. The first problem relates to finding equilibrium configurations in periodic media, more precisely, given an Area-Dirichlet functional J, which is periodic under integer translations and given three planes in R[superscript d], we proof there exists at least one minimizer such that it’s positive part, negative part and zero set remain at a uniform bounded distance of each plane. The second and third problem are related to non local diffusion, in the elliptic non symmetric case and parabolic case. In both cases we are interested in proving interior regularity for solutions of the aforementioned equations. / text

College students are known for having poor sleep and irregular sleep schedules, especially during the first year of college. These sleep habits may contribute to poor academic outcomes down the line, as well as increased risk of developing depression and other disorders. The current study aims to look at the degree of change in sleep variability between high school and college and examine its relationship with mood, emotion regulation, and academic performance. The study also aims to explore the relationship between morningness-eveningness tendencies and academic performance, emotion regulation, and sleep variability, reported both at baseline (as perceived by the students) and over 7 days of daily sleep diaries. Additionally, the study is designed to look at day-to-day effects of sleep on mood. Data were obtained from 311 college freshmen (237 females). Participants were 17-19 years old (M=18.4) and freshmen in college. The study took place over one baseline internet-based session and a week of internet-based daily questionnaires. While students had significantly more variable schedules in college than in high school, this change did not correlate with or predict any measures of interest, including sleep quality, grades, and mood. However, overall variability, as well as eveningness, was associated with a number of negative outcomes, including lower GPA, less adaptive emotion regulation strategies, worse mood, and more depression symptoms.

Thesis (M.A. (Human Movement Science))--North-West University, Potchefstroom Campus, 2009.