In this thesis, we present a lattice theoretical approach to the field of approximations. Given a pair consisting of a kernel system and a closure system on an underlying lattice, one receives a lattice of approximations. We describe the theory of these lattices of approximations. Furthermore, we put a special focus on the case of concept lattices. As it turns out, approximation of formal concepts can be interpreted as traces, which are preconcepts in a subcontext.:Preface 1. Preliminaries 2. Approximations in Complete Lattices 3. Concept Approximations 4. Rough Sets List of Symbols Index Bibliography / In der vorliegenden Arbeit beschreiben wir einen verbandstheoretischen Zugang zum Thema Approximieren. Ausgehend von einem Kern- und einem Hüllensystem auf einem vollständigen Verband erhält man einen Approximationsverband. Wir beschreiben die Theorie dieser Approximationsverbände. Des Weiteren liegt dabei ein Hauptaugenmerk auf dem Fall zugrundeliegender Begriffsverbände. Wie sich nämlich herausstellt, lassen sich Approximationen formaler Begriffe als Spuren auffassen, welche diese in einem vorgegebenen Teilkontext hinterlassen.:Preface 1. Preliminaries 2. Approximations in Complete Lattices 3. Concept Approximations 4. Rough Sets List of Symbols Index Bibliography

Master of Music / Department of Music / Frank C. Tracz / This document was written after examination, research, score analysis, and lesson planning in preparation for the Graduate Conducting Recital of Travis M. Johnson. This recital was held on Monday, March 8, 2010 at the Cheney High School Auditorium at 6:00 pm. A philosophy of Music Education and criteria for quality literature selection is followed by the theoretical and historical analysis of four works; Flourish for Wind Band by Ralph Vaughn Williams, Air for Band by Frank Erickson, An American Elegy by Frank Ticheli, and Rough Riders by Karl King arranged by James Swearingen. Lesson plans, rehearsal aids, warm up exercises, and student assignments are included in the examination of this process.

Master of Science / Department of Agronomy / Walter H. Fick / Rough-leaved dogwood (Cornus Drummondii Meyer) is one of the most invasive woody plants in rangeland of Kansas. Reduced prescribed burning due to drought and urban encroachment probably has contributed to its spread. Herbicides are commonly recommended for control of rough-leaved dogwood, but minimal data exists for recommendation development. Ten herbicide treatments were applied in June during late flowering at two locations in northeast Kansas in 2005 and 2006. Each treatment was replicated three times in a randomized complete block design with individual plot sizes of 3 x 3 m. Herbicides were applied with hand sprayers in 1017 L/ha solution. Visual evaluations of defoliation were made about 1 and 12 months after treatment (MAT) and mortality was estimated about 1 year after treatment (YAT). Defoliation 1 MAT varied among herbicides with significant location by year and herbicide by year interactions. Treatments providing greater than 70% defoliation 12 MAT both years were triclopyr + 2,4-D (1.12 + 1.06 kg ae/ha), triclopyr + fluroxypyr (1.83 + 0.61 kg ae/ha), picloram + fluroxypyr (0.41 + 0.41 and 0.82 + 0.82 kg ae/ha), and picloram + 2,4-D + triclopyr (0.66 + 2.44 + 2.44 kg ae/ha). Rough-leaved dogwood is difficult to control with a single herbicide application, but treatments exist that will substantially reduce stands. Tebuthiuron pellets (Spike 20P) are another control measure recommended for rough-leaved dogwood. Treatments of 4.4 kg ai/ha (3/4 oz per 100 square feet) tebuthiuron pellets were applied in December 2004. A visual estimate of control indicated tebuthiuron reduced dogwood cover by 65% compared to a 3% decrease on untreated plots. Dogwood density was reduced by 2.2 stems/m2 (P<0.08). Total woody plant cover increased on untreated plots by 6.2 percentage units, but was decreased by 20.9 percentage units on tebuthiuron treated plots. Other woody plants decreased in both treated and untreated plots. Shading by a large elm tree likely caused variation between replications including increases in cover and density of rough-leaved dogwood on treated plots. Tebuthiuron is a photosynthetic inhibitor that often is not effective on shaded plants. Tebuthiuron pellets applied at 4.4 kg ai/ha was an effective control option for unshaded rough-leaved dogwood.

This thesis contains three main results. The first result states that, outside a slim set associated with a Gaussian process with long time memory, paths can be canonically enhanced to geometric rough paths. This allows us to apply the powerful Universal Limit Theorem in rough path theory to study the quasi-sure properties of the solutions of stochastic differential equations driven by Gaussian processes. The key idea is to use a norm, invented by B. Hambly and T.Lyons, which dominates the p-variation distance and the fact that the roughness of a Gaussian sample path is evenly distributed over time. The second result is the almost-sure uniqueness of the signatures of SLE kappa curves for kappa less than or equal to 4. We prove this by first expressing the Fourier transform of the winding angle of the SLE curve in terms of its signature. This formula also gives us a relation between the expected signature and the n-point functions studied in the SLE and Statistical Physics literature. It is important that the Chordal SLE measure in D is supported on simple curves from -1 to 1 for kappa between 0 and 4, and hence the image of the curve determines the curve up to reparametrisation. The third result is a formula for the expected signature of Gaussian processes generated by strictly regular kernels. The idea is to approximate the expected signature of this class of processes by the expected signature of their piecewise linear approximations. This reduces the problem to computing the moments of Gaussian random variables, which can be done using Wick’s formula.

We construct a mathematical model of an order driven market where traders can submit limit orders and market orders to buy and sell securities. We adapt the notion of no free lunch of Harrison and Kreps and Jouini and Kallal to our setting and we prove a no-arbitrage theorem for the model of the order driven market. Furthermore, we compute signatures of order books of different financial markets. Signatures, i.e. the full sequence of definite iterated integrals of a path, are one of the fundamental elements of the theory of rough paths. The theory of rough paths provides a framework to describe the evolution of dynamical systems that are driven by rough signals, including rough paths based on Brownian motion and fractional Brownian motion (see the work of Lyons). We show how we can obtain the solution of a polynomial differential equation and its (truncated) signature from the signature of the driving signal and the initial value. We also present and analyse an ODE method for the numerical solution of rough differential equations. We derive error estimates and we prove that it achieves the same rate of convergence as the corresponding higher order Euler schemes studied by Davie and Friz and Victoir. At the same time, it enhances stability. The method has been implemented for the case of polynomial vector fields as part of the CoRoPa software package which is available at http://coropa.sourceforge.net. We describe both the algorithm and the implementation and we show by giving examples how it can be used to compute the pathwise solution of stochastic rough differential equations driven by Brownian rough paths and fractional Brownian rough paths.

Les surfaces et interfaces rugueuses sont rencontrées dans de nombreuses situations en mécanique et physique des solides. En particulier, une surface ou interface considérée comme lisse à une échelle donnée se révèle souvent rugueuse à autre échelle plus petite. Ce travail étudie les interfaces planes et courbées dont la rugosité peut être raisonnablement décrite comme des ondulations périodiques. Il a pour objectif de modéliser ces interfaces dans des composites et de déterminer leurs effets sur les propriétés effectives élastiques et conductrices des composites concernés. L'approche élaborée pour atteindre cet objectif consiste d'abord à utiliser l'analyse asymptotique pour modéliser une zone d'interface rugueuse comme une interphase hétérogène uniquement suivant son épaisseur et ensuite à faire appel à des schémas micromécaniques pour quantifier les influences de cette interphase sur les propriétés effectives. Ce travail considère trois types de composites dans lesquels de s interfaces périodiquement ondulées sont présentes : composites stratifiés, fibreux et à inclusions. Les résultats obtenus pour ces composites contribuent au développement de la micromécanique et apportent des solutions à des problèmes d'intérêt pratique rencontrés en physique et mécanique des matériaux hétérogènes / Rough surfaces and interfaces are encountered in many situations in mechanics and physics of solids. In particular, a surface or interface considered smooth at a given scale turns out often to be rough at another smaller scale. This work studies the flat and curved interfaces whose roughness can be reasonably described as periodic undulations. It aims to model these interfaces in composites and to determine their effects on the effective elastic and conductive properties of the composites in question. The approach elaborated to achieve this objective consists first in using asymptotic analysis to model a zone of rough interface as an interphase being heterogeneous only along its thickness direction and then in resorting to some micromechanical schemes to quantify the influences of the interphase on the effective properties. This work considers three types of composites in which periodically corrugated interfaces are present: laminated, fibrous and particulate composites. The results obtained for these composites contribute to the development of micromechanics and provide solutions to problems of practical interest encountered in physics and mechanics of heterogeneous materials

La diffusion électromagnétique par des surfaces rugueuses concerne un ensemble très vaste de problèmes actuels en optronique (maîtrise des signatures/cloaking, analyse des signatures laser et infrarouge, imagerie active, localisation de la lumière, imagerie optique haute résolution, modélisation des interactions lumière matière et des signatures optiques, applications photovoltaïques et détecteurs infrarouges, biotechnologie). Les travaux réalisés dans le cadre de la thèse "Contrôle de la lumière par des éléments de surface désordonnés'' consistent à réaliser expérimentalement des surfaces rugueuses dont les propriétés statistiques sont contrôlées. Pour cela, un banc expérimental de photofabrication a été développé, utilisant sles propriétés statistiques des figures de speckle issues d'un faisceau laser mis en forme spatialement.Les surfaces réalisées présentent ainsi des propriétés statistiques qui n'existent pas à l'état naturel (fonction d'autocorrélation non gaussienne). Au delà de ces surfaces photofabriquées, le travaux de cette thèse s'intéressent également aux propriétés de surfaces de silicium fortement rugueuses caractérisées de Black Silicon. / Scattering of electromagnetic waves from rough surfaces is involved in a wide area of research in optronics (cloaking, laser and infrared signature analysis, active imaging, light localisation, high resolution optical imaging, laser-matter interaction and optical signature modelling, photovoltaics, infrared sensors, biotechnologies). Studies performed during the thesis ``Ligth control by random surface elements'' consist in the experimental fabrication of rough surfaces which statistical properties are controled. An experimental setup has been implemented, using the properties of a spatially shaped laser speckle pattern. The photofabricated surfaces show statistical properties that do not exist in nature such as non Gaussian autocorrelation function. Beyond these photofabricated surfaces, we also studied very rough surfaces of semi-conductor known as Black Silicon.

Master of Science / Department of Agronomy / Walt Fick / Prescribed burning has been a common conservation practice on native prairie dating back to the days of pioneer settlement. Advantages include increased forage quality, reduction of undesirable plants, improved wildlife habitat, removal of accumulated dead plant litter and relatively low costs. While spring is the commonly accepted time to burn, little research has been conducted on late-summer and fall burning for specific objectives that include targeting undesirable plant species and measuring potential effects on soil physical properties. The first part of this study was to evaluate the effect that prescribed burning has on population dynamics of sericea lespedeza (Lespedeza cuneata [Dumont] G. Don), rough-leaf dogwood (Cornus drummondii Meyer), and additional woody species. Stem counts and cover estimates were taken from 20, 0.25-m[superscript]2 frames prior to and post-burn. Change in botanical composition, plant density, frequency, and Daubenmire canopy cover estimates were calculated. Sericea lespedeza plant frequency across all clay upland burns decreased 2.27% and increased 4.76% across all loamy/limy upland burns the first growing season post-burn. Dogwood densities increased 3.12 stems m[superscript]-[superscript]2 on spring burns compared to a decrease of 0.30 stems m[superscript]-[superscript]2 on unburned plots the first growing season post-burn. Changes in frequency of other woody species the first growing season post-burn showed significant interactions between burn treatment and ecological site, and between ecological site and year. A significant interaction between burn treatment and ecological site was found on total woody species plant composition changes two growing seasons post-burn for the first year of burn treatments. The secondary part of this study was to evaluate the effect of prescribed burning on soil bulk density and wet-aggregate stability. Soil samples were collected along the same line-transects used for vegetation sampling. Significant differences among mean weight diameters (MWD), percent water-stable aggregates (WSA), and WSA size fractions occurred between burned and unburned soils following burning in the fall of 2011. Monitoring plant and soil response to prescribed burning in different seasons may lead to adjustments being made in management of rangelands where sericea lespedeza, dogwood, and additional woody species occur.

La « présence chinoise » en Afrique est devenue une réalité démographique incontournable, elle a pris de l'ampleur au cours des dernières années ; la diaspora chinoise sur le territoire africain représente pas moins d'un million de personnes, contre moins de 100 000 en 2001. Afin d'appréhender notre objet de recherche, et saisir les mécanismes économiques intrinsèques aux sociétés africaines, nous adopterons une approche « par le bas » car le niveau local semble plus approprié pour comprendre les ressorts de la croissance économique. / The "Chinese presence" in Africa has become an inescapable demographic reality, it has grown in recent years the Chinese Diaspora on the African territory is not less than one million people, against less than 100 000 in 2001. To understand our research object, and capture the economic mechanisms intrinsic to African societies, we adopt an approach "bottom" because the local level seems more appropriate to understand the springs of economic growth.

The present understanding of the structure and dynamics of turbulent boundary layers on aerodynamically smooth walls has been clarified over the last decade or so. However, the dynamics of turbulent boundary layers over rough surfaces is much less well known. Nevertheless, there are many industrial and environmental flow applications that require understanding of the mean velocity and turbulence in the immediate vicinity of the roughness elements.</p> <p>This thesis reports the effects of surface roughness on the flow characteristics in a turbulent boundary layer. Both experimental and numerical investigations are used in the present study. For the experimental study, comprehensive data sets are obtained for two-dimensional zero pressure-gradient turbulent boundary layers on a smooth surface and ten different rough surfaces created from sand paper, perforated sheet, and woven wire mesh. The physical size and geometry of the roughness elements and freestream velocity were chosen to encompass both transitionally rough and fully rough flow regimes. Three different probes, namely, Pitot probe, single hot-wire, and cross hot-film, were used to measure the velocity fields in the turbulent boundary layer. A Pitot probe was used to measure the streamwise mean velocity, while the single hot-wire and cross hot-film probes were used to measure the fluctuating velocity components across the boundary layer. The flow Reynolds number based on momentum thickness, , ranged from 3730 to 13,550. The data reported include mean velocity, streamwise and wall-normal turbulence intensities, Reynolds shear stress, triple correlations, as well as skewness and flatness factors. Different scaling parameters were used to interpret and assess both the smooth- and rough-wall data at different Reynolds numbers, for approximately the same freestream velocity. The appropriateness of the logarithmic law and power law proposed by George and Castillo (1997) to describe the mean velocity in the overlap region was also investigated. The present results were interpreted within the context of the Townsends wall similarity hypothesis. </p> <p>Based on the mean velocity data, a novel correlation that relates the skin friction to the ratio of the displacement and boundary layer thicknesses, which is valid for both smooth- and rough-wall flows, was proposed. In addition, it was also found that the application of a mixed outer scale caused the velocity profile in the outer region to collapse onto the same curve, irrespective of Reynolds numbers and roughness conditions. The present results showed that there is a common region within the overlap region of the mean velocity profile where both the log law and power law are indistinguishable, irrespective of the surface conditions. For the power law formulation, functional relationships between the roughness shift, and the power law coefficient and exponent were developed for the transitionally rough flows. The present results also suggested that the effect of surface roughness on the turbulence field depends to some degree on the specific characteristics of the roughness elements and also the component of the Reynolds stress tensor being considered. </p> <p>In the case of the numerical study, a new wall function formulation based on a power law was proposed for smooth and fully rough wall turbulent pipe flow. The new formulation correctly predicted the friction factors for smooth and fully rough wall turbulent pipe flow. The existing two-layer model realistically predicted the velocity shift on a log-law plot for the fully rough turbulent boundary layer. The two-layer model results also showed the effect of roughness is to enhance the level of turbulence kinetic energy and Reynolds shear stress compared to that on a smooth wall. This enhanced level extends into the outer region of the flow, which appears to be consistent with present and recent experimental results for the boundary layer.