Doctor of Philosophy / Department of Mathematics / Diego Maldonado / Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs. Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory. Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].

O presente trabalho tem como objetivo o estudo da execução residual na tutela dos interesses individuais homogêneos e as questões procedimentais que envolvem o instituto no direito processual brasileiro. Foram abordados aspectos relativos aos fundamentos do instituto no direito brasileiro, os requisitos para sua aplicação e o procedimento para a restituição da indenização devida aos indivíduos nos casos em que esta se faz necessária. / This paper aims to study the fluid recovery in the protection of homogeneous individual interests in Brazilian Law and the procedural issues surrounding the institute. The paper approached aspects related to the fundamentals of the institute in Brazilian law, the requirements for their application and the procedure for due compensation for individuals in cases where this is necessary.

Foi estudado experimentalmente o desempenho biológico de um tipo de enxerto ósseo homógeno, processado quimicamente, esterilizado em óxido de etileno e embebido em medula óssea autógena. De cães doadores adulto-jovens foram obtidos blocos cilíndricos de 1,0 x 1,0 cm da epífise distal do fêmur, com auxílio de uma trefina. Os ossos assim obtidos foram clareados, desengordurados, esterilizados em óxido de etileno e mantidos em estoque. Trinta cães adultos jovens foram usados como receptores do enxerto e foram alocados em dois grupos. No grupo I (experimental) os animais foram operados e, criado, transversalmente, com trefina de 1,0 cm de diâmetro externo, um espaço cilíndrico de 1,0 cm de diâmetro por 1,0 cm de altura na epífise distal do fêmur direito, onde foi encaixado o enxerto preparado e que foi previamente embebido em medula óssea do próprio animal retirada por punção óssea na crista ilíaca. Três semanas depois, o mesmo animal foi submetido ao mesmo procedimento cirúrgico no fêmur esquerdo. Os cães deste grupo foram sacrificados seis semanas após a primeira cirurgia. O grupo II constituiu o controle e foi formado por vinte cães adulto-jovens em que, primeiro, foi retirado um cilindro de osso da epífise distal do fêmur esquerdo e, em seguida, no mesmo tempo cirúrgico, foi criado o mesmo espaço no fêmur direito, com uma trefina de um centímetro de diâmetro interno e, então, encaixado o bloco de osso retirado do fêmur do outro lado do mesmo animal. Estes animais foram subdivididos em dois subgrupos de dez cães cada, em relação ao tempo de sacrifício. Em um subgrupo os cães foram sacrificados três semanas após, enquanto que, no outro subgrupo, foram sacrificados seis semanas após a cirurgia de implantação do enxerto. O processo de incorporação do enxerto foi avaliado pela histologia convencional e histologia de fluorescência óssea, pela injeção prévia de tetraciclina. A comparação foi entre os grupos I e II, nos períodos de três e seis semanas pós-implante. As áreas dos enxertos homógenos com 21 dias pós-implante, geralmente mostraram-se visíveis, na maioria das vezes, formadas por trabéculas finas irregulares, sem osteócitos, porém com áreas de neoformação óssea. Já os enxertos autógenos, a área do osso implantado era bem visível, com trabéculas de osso antigo, mais adelgaçadas, em menor número que o osso receptor, porém com intensa deposição de osso neoformado. Para os enxertos homógenos com 42 dias pós-implante, a área do enxerto estava bem definida e integrada ao osso adjacente, composta por trabéculas antigas com predomínio de osso neoformado sobre a superfície. Nos enxertos autógenos com 42 dias pós-implante, as trabéculas tinham orientação comum, com espaço intertrabecular preenchido por tecido conjuntivo denso, em algumas áreas, e por medula óssea madura em outras. O enxerto homógeno processado e esterilizado em óxido de etileno e embebido em medula óssea apresentou boa atividade biológica, embora, com integração mais lenta e menor desempenho em relação ao enxerto autógeno, o que o torna um bom substituto para este último. / In the present investigation the biological performance of the homogenous bone graft chemically processed, sterilized with ethylene oxide and soaked in autogenous bone marrow was investigated. Thirty dogs were assigned into two groups. In group I, ten young adult dogs received a cylindrical block of the above mentioned graft. Previously, the bone graft was immersed in autogenous bone marrow and then inserted in a cylindrical hole, with the same dimensions, created at the distal femoral epiphysis of the right femur. After three weeks the same animal was submitted to the same surgical procedure, but on the left femur. The animal was killed six weeks later. The group II was made up of twenty dogs that received an autogenous graft obtained from the left distal femur and implanted into the right distal femur (same shape, dimensions and technique). From this group, 10 dogs were killed three weeks later and 10 dogs were killed six weeks later. The graft incorporation was evaluated by light microscopy and fluorescence microscopy. Results showed that for 21 day implant in the homogenous group, there was osteogenesis at the periphery of the bone graft, but most of its trabeculae were still inviable. Conversely, for the autogenous grafts newly-formed bone was found on the trabecular surface, even in the inner parts of the implanted bone. At 42 days there was active osteogenesis in homogenous grafts with new bone deposition on the trabeculae but, in two cases, the graft was not incorporated (reabsorbed in one case and sequestred in the other one). The autogenous graft showed newly formed bone arranged in a well mature fashion with new bone marrow filling the intertrabecular gaps. It was concluded that the homogenous graft as prepared herein is a good alternative for autogenous grafts, although with a slower osteogenesis rate an less biological performance.

The goal of this thesis is to show that institutional and procedural characteristics are affecting the final price of the public procurement. In order to be able to compare the tenders among each other, only public procurement of homogeneous goods is analyzed. The presented model attempts to explain a variation in final price per one unit as a function of estimated unit price, market price and characteristic of procurement procedure - type of procedure, number of bidders and use of electronic auction. In case of electricity and gas public procurement final price elasticity with respect to the estimated price tents to be higher than such elasticity with respect to the market price. This result suggests high rigidity in public procurement procedures. We show that such ineffectiveness is reduced by using open procedure, electronic auction or attracting more bidders.

This thesis describes the development and mechanistic studies of carbocyclization reactions of allene-substituted olefins and 1,3-dienes, catalyzed by palladium(0) and palladium(II). These reactions results in the formation of [n,3,0] bicyclic systems (n = 3-5) with high stereoselectivity and in good to excellent yields. The first carbocyclization presented is a novel palladium(0)-catalyzed cyclo- isomerization of allene-substituted olefins. Secondly an efficient aerobic biomimetic system has been developed for a Pd(II)-catalyzed allylic oxidative carbocyclization of allene-substituted olefins. Additionally, during the studies of palladium-catalyzed carbocyclizations of allene-substituted olefins, it was found that in the absence of palladium a mild thermal ene-reaction occurs. In this manner stereodefined, functionalized bicyclic compounds are obtained with good regioselectivity and in high yields. The third and fourth carbocyclization developed are a palladium(II)-catalyzed oxidation and a palladium(0)-catalyzed intramolecular telomerization of allene-substituted 1,3-dienes. A mechanistic study of the palladium(II)-catalyzed oxidation of allene-substituted 1,3-dienes was made, and reaction intermediates could be isolated. The stereochemistry of the reaction intermediates was assigned, and this made it possible to suggest a mechanism for the reaction. The presented mechanism is a trans carbopalladation of the 1,3-diene, where the allene act as the carbon nucleophile. Due to different stereochemical outcomes of the stoichiometric and catalytic reactions, this mechanism could only explain the stoichiometric reaction. Another mechanism for the catalytic reaction was suggested, which rationalizes both the regio- and stereochemistry of the products.

Mitt syfte med denna studie är att bidra med kunskap hur förskollärarna på en åldershomogen förskola arbetar med barns delaktighet och inflytande, hur arbetet skiljer sig för de olika åldersgrupperna och hur förskollärarna kopplar sitt arbete till läroplanen. Jag har använt mig av intervjuer med fyra förskollärare samt en förskolechef på en åldershomogen förskola för att besvara mina frågeställningar. Resultatet av min undersökning visar att förskollärarna lägger stor vikt vid att barnen ska få delaktighet och inflytande över sin vardag på förskolan. Detta gör de genom att synliggöra det enskilda barnet med hjälp av den pedagogiska dokumentationen. Därefter anpassas verksamhetens arbetssätt och miljö efter de behov samt intressen som barnen visat uttryck för. Detta ansåg förskollärarna hade blivit lättare efter att de börjat med åldershomogena barngrupper till skillnad från när man hade arbetat med åldrarna 1-5. / The purpose of this study is to bring further knowledge into how educators in homogeneous pre-schools work with children's participation as well as their ability to influence their daily school life. The study will also identify what the differences are for different age groups and how educators connect their work with the curriculum. The results have been accumulated by interviewing four educators and a principal in a homogeneous pre-school. The results of the study show that the educators believe that it is important for children to participate and have a say in day to day pre-school life. The use of pedagogical documentation is a tool used to help notice every individual child. Through this documentation they are able to adjust their way of working, the education level and the surrounding environment to cater to the children's interests and needs. The results also show that the educators found it easier to include a child's needs and interests when working with children of the same age.

This research examines present day urban Anglican churches in Zambia which are ethnically “homogeneous” in a heavily multi-ethnic environment. I give Attention to the understanding of the biblical, theological, and missiological background and seek to find a way to attract many ethnic groups into the churches. The research notes that the present scenario does not represent a healthy urban church. I argue that a biblical, theological and holistic Gospel proclamation, that is, in word and deeds, will attract other ethnic groups into the urban church and make it truly multi-ethnic and multi-cultural. / Thesis (MA (Missiology))--North-West University, Potchefstroom Campus, 2013.

This research examines present day urban Anglican churches in Zambia which are ethnically “homogeneous” in a heavily multi-ethnic environment. I give Attention to the understanding of the biblical, theological, and missiological background and seek to find a way to attract many ethnic groups into the churches. The research notes that the present scenario does not represent a healthy urban church. I argue that a biblical, theological and holistic Gospel proclamation, that is, in word and deeds, will attract other ethnic groups into the urban church and make it truly multi-ethnic and multi-cultural. / Thesis (MA (Missiology))--North-West University, Potchefstroom Campus, 2013.

Atmosphere-Ocean General Circulation Models (AOGCMs) are the primary tool for modelling global climate change in the future. However, their coarse spatial resolution does not permit direct application for local scale impact studies. Therefore, either dynamical or statistical downscaling techniques are used for translating AOGCM outputs to local scale climatic variables. The main goal of this study was to improve our understanding of the historical and future climate change at local-scale in the Canadian Prairie Provinces (CPPs) of Alberta, Saskatchewan and Manitoba, comprising 47 diverse watersheds. Given the vast nature of the study area and paucity of recorded data, a novel approach for identifying homogeneous regions for regionalization of precipitation characteristics for the CPPs was proposed. This approach incorporated information about predictors ― large-scale atmospheric covariates from the National Center for Environmental Prediction (NCEP) Reanalysis-I, teleconnection indices and geographical site attributes that impact spatial patterns of precipitation in order to delineate homogeneous precipitation regions using a combination of multivariate approaches. This resulted in the delimitation of five homogeneous climatic regions which were validated independently for homogeneity using statistics computed from observations recorded at 120 stations across the CPPs. For multisite multivariate statistical downscaling, an approach based on the Generalized Linear Model (GLM) framework was developed to downscale daily observations of precipitation and minimum and maximum temperatures from 120 sites located across the CPPs. First, the aforementioned predictors and observed daily precipitation and temperature records were used to calibrate GLMs for the 1971–2000 period. Then the calibrated GLMs were used to generate daily sequences of precipitation and temperatures for the 1962–2005 historical (conditioned on NCEP predictors), and future period (2006–2100) using outputs from six CMIP5 (Coupled Model Intercomparison Project Phase-5) AOGCMs corresponding to Representative Concentration Pathway (RCP): RCP2.6, RCP4.5, and RCP8.5 scenarios. The results indicated that the fitted GLMs were able to capture spatiotemporal characteristics of observed climatic fields. According to the downscaled future climate, mean precipitation is projected to increase in summer and decrease in winter while minimum temperature is expected to warm faster than the maximum temperature. Climate extremes are projected to intensify with increased radiative forcing.

O presente trabalho tem como objetivo o estudo da execução residual na tutela dos interesses individuais homogêneos e as questões procedimentais que envolvem o instituto no direito processual brasileiro. Foram abordados aspectos relativos aos fundamentos do instituto no direito brasileiro, os requisitos para sua aplicação e o procedimento para a restituição da indenização devida aos indivíduos nos casos em que esta se faz necessária. / This paper aims to study the fluid recovery in the protection of homogeneous individual interests in Brazilian Law and the procedural issues surrounding the institute. The paper approached aspects related to the fundamentals of the institute in Brazilian law, the requirements for their application and the procedure for due compensation for individuals in cases where this is necessary.