Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable. For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs. As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids. / eSSENCE

The self-power map is the function from the set of natural numbers to itself which sends the number $n$ to $n^n$. Motivated by applications to cryptography, we consider the image of this map modulo a prime $p$. We study the question of how large $x$ must be so that $n^n \equiv a \bmod p$ has a solution with $1 \le n \le x$, for every residue class $a$ modulo $p$. While $n^n \bmod p$ is not uniformly distributed, it does appear to behave in certain ways as a random function. We give a heuristic argument to show that the expected $x$ is approximately ${p^2\log \phi(p-1)/\phi(p-1)}$, using the coupon collector problem as a model. Rigorously, we prove the bound $x <p^{2-\alpha}$ for sufficiently large $p$ and a fixed constant $\alpha > 0$ independent of $p$, using a counting argument and exponential sum bounds. Additionally, we prove nontrivial bounds on the number of solutions of $n^n \equiv a \bmod p$ for a fixed residue class $a$ when $1 \le n \le x$, extending the known bounds when $1 \le n \le p-1$.

The self-power map is the function from the set of natural numbers to itself which sends the number $n$ to $n^n$. Motivated by applications to cryptography, we consider the image of this map modulo a prime $p$. We study the question of how large $x$ must be so that $n^n \equiv a \bmod p$ has a solution with $1 \le n \le x$, for every residue class $a$ modulo $p$. While $n^n \bmod p$ is not uniformly distributed, it does appear to behave in certain ways as a random function. We give a heuristic argument to show that the expected $x$ is approximately ${p^2\log \phi(p-1)/\phi(p-1)}$, using the coupon collector problem as a model. Rigorously, we prove the bound $x <p^{2-\alpha}$ for sufficiently large $p$ and a fixed constant $\alpha > 0$ independent of $p$, using a counting argument and exponential sum bounds. Additionally, we prove nontrivial bounds on the number of solutions of $n^n \equiv a \bmod p$ for a fixed residue class $a$ when $1 \le n \le x$, extending the known bounds when $1 \le n \le p-1$.

Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán. / This mémoire contains some results about numerical integration. They are related to the famous quadrature formula of K. F. Gauss. A very interesting generalization of the formula of Gauss was obtained by P.Turán. It is contained in a paper that was published in 1948, only a few years after the second world war. Due to adverse circunstances he was in at the time, the author (Turán) left many details for the reader to fill in. Otherwise, the article of Turán inspired a multitude of research, and his formula has been extended in many ways and several papers have been written on this subject. However, there is no single book or paper where one can nd a clear and comprehensive account of the basic results pertaining to Turán's formula. Thus, I would like my Master's mémoire to contain enough details that can enlighten the reader and present an exposition of much that has been done on this subject. Here is how we have arranged the contents of the mémoire. 1-a. The original formula of Gauss for polynomials - statement along with a proof. 1-b. Turán's point of view - detailed account of the results contained in his paper. 2-a. A formula for trigonometric polynomials analogous to that of Gauss. 2-b. A formula for trigonometric polynomials analogous to that of Turán. 3-a. Two formulae for entire functions of exponential type, analogous to the one of Gauss for polynomials. 3-b. A formula for entire functions of exponential type, analogous to that of Turán. 4-a. Annexe A - Basic facts about Legendre polynomials. 4-b. Annexe B - Polynomial interpolation. 4-c. Annexe C - Basic facts about entire functions of exponential type. 4-d. Annexe D - Paper of P. Turán.

Wireless Sensor Networks (WSNs) have enticed a strong attention in the research community due to the broad range of applications and services they support. WSNs are composed of intelligent sensor nodes that have the capabilities to monitor different types of environmental phenomena or critical activities. Sensor nodes operate under stringent requirements of scarce power resources, limited storage capacities, limited processing capabilities, and hostile environmental surroundings. However, conserving sensor nodes’ power resources is the top priority requirement in the design of a WSN as it has a direct impact on its lifetime. The IEEE 802.15.4 standard defines a set of specifications for both the PHY layer and the MAC sub-layer that abide by the distinguished requirements of WSNs. The standard’s MAC protocol employs an intelligent backoff algorithm, called the Binary Exponent Backoff (BEB), that minimizes the drainage of power in these networks. In this thesis we present an in-depth study of the IEEE 802.15.4 MAC protocol to highlight both its strong and weak aspects. We show that we have enticing opportunities to improve the performance of this protocol in the context of WSNs. We propose three new backoff algorithms, namely, the Standby-BEB (SB-BEB), the Adaptive Backoff Algorithm (ABA), and the Priority-Based BEB (PB-BEB), to replace the standard BEB. The main contribution of the thesis is that it develops a new design concept that drives the design of efficient backoff algorithms for the IEEE 802.15.4-based WSNs. The concept dictates that controlling the algorithms parameters probabilistically has a direct impact on enhancing the backoff algorithm’s performance. We provide detailed discrete-time Markov-based models (for AB-BEB and ABA) and extensive simulation studies (for the three algorithms) to prove the superiority of our new algorithms over the standard BEB.

The subject of this thesis is the maximization of the information divergence from an exponential family on a finite set, a problem first formulated by Nihat Ay. A special case is the maximization of the mutual information or the multiinformation between different parts of a composite system. My thesis contributes mainly to the mathematical aspects of the optimization problem. A reformulation is found that relates the maximization of the information divergence with the maximization of an entropic quantity, defined on the normal space of the exponential family. This reformulation simplifies calculations in concrete cases and gives theoretical insight about the general problem. A second emphasis of the thesis is on examples that demonstrate how the theoretical results can be applied in particular cases. Third, my thesis contain first results on the characterization of exponential families with a small maximum value of the information divergence.

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For a stationary two dimensional random field, we utilize the classical Kolmogorov-Wiener theory to develop prediction methodology which requires minimal assumptions on the dependence structure of the random field. We also provide solutions for several non-standard prediction problems which deals with the "modified past," in which a finite number of observations are added to the past. These non-standard prediction problems are motivated by the network site selection in the environmental and geostatistical applications. Unlike the time series situation, the prediction results for random fields seem to be expressible only in terms of the moving average parameters, and attempts to express them in terms of the autoregressive parameters lead to a new and mysterious projection operator which captures the nature of edge-effects. We put forward an approach for estimating the predictor coefficients by carrying out an extension of the exponential models. Through simulation studies and real data example, we demonstrate the impressive performance of our prediction method. To the best of our knowledge, the proposed method is the first to deliver a unified framework for forecasting random fields both in the time and spectral domain without making a subjective choice of the covariance structure. Finally, we focus on the estimation of the hurst parameter for long range dependence stationary random fields, which draws its motivation from applications in the environmental and atmospheric processes. Current methods for estimation of the Hurst parameter include parametric models like fractional autoregressive integrated moving average models, and semiparametric estimators which are either inefficient or inconsistent. We propose a novel semiparametric estimator based on the fractional exponential spectrum. We develop three data-driven methods which can automatically select the optimal model order for the fractional exponential models. Extensive simulation studies and analysis of Mercer and Hall?s wheat data are used to illustrate the performance of the proposed estimator and model order selection criteria. The results show that our estimator outperforms existing estimators, including the GPH (Geweke and Porter-Hudak) estimator. We show that the proposed estimator is consistent, works for different definitions of long range dependent random fields, is computationally simple and is not susceptible to model misspecification or poor efficiency.

Knowledge of the terms sex and gender has important political, health and equity considerations. This thesis investigates the macrostructural assertions of Connell’s social theory of gender which is fundamentally concerned with demonstrating the relational and hierarchical nature of gender. A major criticism of the theory has been its lack of account of the individual and the ways in which gender is performed in local settings. Therefore, investigation primarily concerns whether Connell’s macrostructural theory is explicable in local social contexts. A theoretical critique and reframing of the theory lead to articulating the interdependency between structural, cultural and individual factors. By explicitly stating Connell’s implicit assertions, what becomes evident is that people’s gendered relations are interrelated with beliefs which are held personally and shared culturally. Specifically, a major theoretical impasse is overcome when recognising that the “ideology of supremacy” of a dominative masculinity is necessarily interdependent with the structural relations of power. / From here I have suggested that there are particular patterns of these structures and beliefs which can be seen in macrostructural terms, but also in local settings. These hypotheses are reframed into social network terms for an empirical investigation of Connell’s theory in local contexts. To determine whether the predicted hypotheses for Connell’s theory occur at greater than chance levels, a particular type of statistical model for social networks, called exponential random graph (p*) models, is employed. Importantly, such models utilize a methodological approach which specifically acknowledges the interdependency of structural, individual and cultural factors, thus aligning Connell’s theory with the method of investigation. / Primarily, Connell’s theory is concerned with differing configurations of masculinity, and for this reason my focus is predominantly on males and their relations with one another. To this end, two general local settings were chosen to explore these issues – secondary schools and all-male elite-level (AFL) sporting teams. Social network models were used to examine the relations between masculinities in six schools and four AFL clubs. Importantly, Connell has suggested that local contexts are likely to differ from one another in the degree to which they support gendered relations of power. Results for schools and clubs vary considerably from one another in the ways in which they provide local level support for Connell’s theory. Significantly though, there are some contexts which do show support for Connell’s theory. That such evidence can be found to endorse specifically defined local-level predictions for Connell’s theory, even when controlling for complex micro-level social structures, and also for other individual level effects, and still produce statistically significant effects supporting these predictions suggests that support is not trivial. There is strong evidence that attitudes towards masculinity can be an important organising principle in the emergence of hierarchy, not universally, but in some contexts. / It can be concluded that gender relations tend to operate in ways predicted by Connell’s theory, though local context is particularly important. The specific findings from local social contexts do have wider implications for Connell’s theory, including how hierarchy in gender can be tied to other structures of power, where femininities fit into the theory, a more precise account of hegemony and an exploration of the impact culture has in local settings.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 105-108). Also available on the Internet.