Thesis (M.S.)--University of Wisconsin--Madison, 1969. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Dans cette thèse nous proposons et nous mettons en application plusieurs modèles décrivant la croissance et la morphologie du tissu urbain. Le premier de ces modèles est issu de la percolation en gradient (correlée) déjà proposé de la littérature. Le second, inédit, fait appel à un équation différentielle stochastique. Nos modèles sont paramétrables : les paramètres que nous avons choisi d’appliquer sont naturels et tiennent compte de l’accessibilité des sites. Le résultat des simulations est conforme à la réalité du terrain. Par ailleurs, nous étudions la percolation en gradient: nous démontrons , suivant Nolin, que la frontière de cluster principal se situe dans un voisinage de la courbe critique et nous estimons ses longueurs et largeurs. Enfin, nous menons une étude du processus de croissance SLE. Nous calculons (preuve assistée par ordinateur) l’espérance des carrés des modules pour SLE2 and SLE6. Ces résultats sont liés à la conjecture de Bieberbach. / In this thesis we propose and test models that describe the growth and morphology of cities. The first of these models is used from previously developed correlated gradient percolation model. The second model is related to a stochastic differential equation and has never been proposed before. Both models are parameterizable. The parameters we chose in applications are well justified by physical observations: proximily to axes and accessibility of sites. The result is consistent with actual data. We also study the gradient percolation as a mathematical object. We prove, following Nolin’s ideas, that the front of gradient percolation cluster is localised in a neighborhood of the critical curve with width and length depending on density gradient. Finally, we also study SLE growth processes. We calculate (computer assisted demonstration) the expected value of square of moduli for SLE2 and SLE6 related to the Bieberbach conjecture.

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This thesis focuses a mathematical model from statistical mechanics called the Arboreal gas. The Arboreal gas on a graph $G$ is Bernoulli bond percolation on $G$ with the conditioning that there are no ``loops". This model is related to other models such as the random cluster measure. We mainly study the Arboreal gas and a related model on the $d$-ary wired tree which is simply the $d$-ary wired tree with the leaves identified as a single vertex. Our first result is finding a distribution on the infinite $d$-ary tree that is the weak limit in height $n$ of the Arboreal gas on the $d$-ary wired tree of height $n$. We then study a similar model on the infinite $d$-ary wired tree which is Bernoulli bond percolation with the conditioning that there is at most one loop. In this model, we only have a partial result which proves that the ratio of the partition function of the one loop model in the wired tree of height $n$ and the Arboreal gas model in the wired tree of height $n$ goes to $0$ as $n \rightarrow \infty$. This allows us to prove certain key quantities of this model is actually the same as analogues of that quantity in the Arboreal gas on the $d$-ary wired tree, under an additional assumption. / Graduate / 10000-01-01

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In recent years, there have been efforts to develop a design procedure for swale systems which could be applied to permeable soils. This design should consider space limitations or any other site restriction. Five existing swale sites in the Central Florida area were instrumented to determine flow rates and permeability’s. Twenty-two experiments were performed. It is concluded that the use of a double-ring infiltrometer to determine the infiltration rate is possible for existing swale sites. Another conclusion is that published values of permeability are very variable and accurate, nor specific enough to predict actual infiltration rates. However, very conservative values can be used in design. A swale design procedure, including an equation for the length of triangular and trapezoidal swales, was developed.