We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.

Laboratory soil columns, 0.3 m diameter $ times$ 0.7 m long, and two computer simulation models, LEACHM-N and NTRM, were used to investigate nitrate-nitrogen ($ rm NO sb{3 sp{-}}$-N) leaching in a sandy loam soil. The following treatments were studied: no-till (NT), reduced tillage (RT), and conventional tillage (CT) practices, with residue (R) and without residue (NR). Nitrogen fertilizer was applied at a rate of 180 kg/ha in granular form (experiment I), and in solution form a year later (experiment II). In both experiments, water was applied 5 times over 3 to 4 weeks duration, with each application lasting for 30 minutes. Each column received an average of 24 mm water in experiment I and 32 mm in experiment II. Soil moisture contents were measured and water for $ rm NO sb{3 sp{-}}$-N concentration determination sampled at 0.1, 0.2, 0.4, and 0.6 m depths, following each water application. / In each experiment I, higher nitrate-nitrogen concentrations ($ lbrack rm NO sb{3 sp{-}}$-N)), occurred at the 0.1 and 0.2 m soil layers in RT and CT treatments initially, but less leached to lower layers, while more $ rm NO sb{3 sp{-}}$-N leached to lower depths (below 0.4 m) in the NT treatment. In experiment II, more $ rm NO sb{3 sp{-}}$-N leached below 0.4 m in RT and CT than in NT treatments. Conventional tillage exhibited the lowest drainage rates. Tillage and residue effects were significant only at early stages (4 hours or before) at some depths of experiment I ($P<0.05$). Maximum $ lbrack rm NO sb{3 sp{-}}$-N) occurred at 0.4 m depth in all treatments. / LEACHM-N estimated more $ rm NO sb{3 sp{-}}$-N leaching below 0.4 m in RT and CT treatments than in NT treatment. The model performed poorly only immediately after fertilizer application, showing up to 50% deviation from observed data. Although LEACHM-N overpredicted $ lbrack rm NO sb{3 sp{-}}$-N) in the 0.2 m soil layers in all treatments, estimations remained within standard deviations of observed data. NTRM performed well below 0.4 m depths, but often underpredicted $ rm NO sb{3 sp{-}}$-N leaching at shallower depths. / From both the laboratory experiments and mathematical simulations it was concluded that when fertilizer is applied in granular form, no till practice is undesirable because deeper $ rm NO sb{3 sp{-}}$-N leaching (below 0.4 m) occurs. Reduced tillage may be the preferred choice in such a situation. When fertilizer is applied in solution, reduced and conventional tillage practices are undesirable because deeper $ rm NO sb{3 sp{-}}$-N leaching occurred. No till practice may be a better choice in such a case.

No description available.

Percolation theory enters in various areas of research including critical phenomena and phase transitions. Bond-site percolation is a generalization of pure percolation motivated by the fact that bond-site is close to many physical realities. This work relies on a numerical study of percolation in lattices. A lattice is a regular pattern of sites also known as nodes or vertices connected by bonds also known as links or edges. Sites may be occupied or unoccupied, where the concentration ps is the fraction of occupied sites. The quantity pb is the fraction of open bonds. A cluster is a set of occupied sites connected by opened bonds. The bond-site percolation problem is formulated as follows: we consider an infinite lattice whose sites and bonds are at random or correlated and either allowed or forbidden with probabilities ps and pb that any site and any bond are occupied and open respectively. If those probabilities are small, there appears a sprinkling of isolated clusters each consisting of occupied sites connected by open bonds surrounded by numerous unoccupied sites. As the probabilities increase, reaching critical values above which there is an infinitely large cluster, then percolation is taking place. This means that one can cross the entire lattice by going successively from one occupied site connected by a opened bond to a neighbouring occupied site. The sudden onset of a spanning cluster happens at particular values of ps and pb, called the critical concentrations. Quantities related to cluster configuration (mean cluster and correlation length) and individual cluster structure (size and gyration radius of clusters ) are determined and compared for different models. In our studies, the Monte Carlo approach is applied while some authors used series expansion and renormalization group methods. The contribution of this work is the application of models in which the probability of opening a bond depends on the occupancy of sites. Compared with models in which probabilities of opening bonds are uncorrelated with the occupancy of sites, in the suppressed bond-site percolation, the higher site occupancy is needed to reach percolation. The approach of suppressed bond-site percolation is extended by considering direction of percolation along bonds (directed suppressed bond-site percolation). Fundamental results for models of suppressed bond-site percolation and directed suppressed bond-site percolation are the numerical determination of phase boundary between the percolating and non-percolating regions. Also, it appears that the spanning cluster around critical concentration is independent on models. This is an intrinsic property of a system. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2007.

Cette thèse s'inscrit dans l'étude mathématique de la percolation, qui regroupe une famille de modèles présentant une transition de phase. Des avancées majeures au cours des quinze dernières années, notamment l'invention du SLE et la preuve de l'invariance conforme de la percolation de Bernoulli critique, nous permettent aujourd'hui d'avoir une image très complète de la percolation de Bernoulli sur le réseau triangulaire. Cependant, de nombreuses questions demeurent ouvertes, et ont motivé notre travail.La première d'entre elle est l'universalité de la percolation plane, qui affirme que les propriétés macroscopiques de la percolation plane critique ne devraient pas dépendre du réseau sous-jacent à sa définition. Nous montrons, dans le cadre de la percolation Divide and Color, un résultat qui va dans le sens de cette universalité et identifions, dans ce contexte, des phénomènes macroscopiques indépendants du réseau microscopique. Une version plus faible d'universalité est donnée par la théorie de Russo-Seymour-Welsh (RSW), et sa validité est connue pour la percolation de Bernoulli (sans dépendance) sur les réseaux plans suffisamment symétriques. Nous étudions de nouveaux arguments de type RSW pour des modèles de percolation avec dépendance. La deuxième question que nous avons abordée est celle de l'absence d'une composante connexe ouverte infinie au point critique, une question importante du point de vue physique, puisqu'elle traduit la continuité de la transition de phase. Dans deux travaux en collaboration avec Hugo Duminil-Copin et Vladas Sidoravicius, nous montrons que la transition de phase est continue pour la percolation de Bernoulli sur le graphe Z^2x{0,...,k}, et pour la percolation FK sur le réseau carré avec paramètre q inférieur ou égal à 4. Enfin, la dernière question qui nous a guidés est la localité du point critique : la donnée des boules de grands rayons d'un graphe suffit-elle à identifier avec une bonne précision la valeur du point critique? Dans un travail en collaboration avec Sébastien Martineau, nous répondons de manière affirmative à cette question dans le cadre des graphes de Cayley de groupes abéliens.

The density of states of elastic waves in a porous amorphous “mesoglass” has been measured in the strong-scattering regime. Samples were constructed by sintering glass beads percolated on a random lattice. This structure was investigated via x-ray tomography, and fractal behaviour was observed with fractal dimension D = 2.6. Using sufficiently small samples, the individual modes of vibration could be resolved and counted in the Fourier transform of each transmitted ultrasonic pulse. A statistical treatment of the data, designed to account for the possibility of missing modes, was developed, yielding a robust method for measuring the density of states. In the strong-scattering regime, the data are in good agreement with a simple model based on mode conservation, though the density of states significantly exceeds the predictions of the Debye approximation at low frequencies. At intermediate frequencies, an average density of states of 47.1 ± 0.3 MHz⁻¹ mm⁻³ was found, with a frequency dependence of f^(0.01 ± 0.04).

The density of states of elastic waves in a porous amorphous “mesoglass” has been measured in the strong-scattering regime. Samples were constructed by sintering glass beads percolated on a random lattice. This structure was investigated via x-ray tomography, and fractal behaviour was observed with fractal dimension D = 2.6. Using sufficiently small samples, the individual modes of vibration could be resolved and counted in the Fourier transform of each transmitted ultrasonic pulse. A statistical treatment of the data, designed to account for the possibility of missing modes, was developed, yielding a robust method for measuring the density of states. In the strong-scattering regime, the data are in good agreement with a simple model based on mode conservation, though the density of states significantly exceeds the predictions of the Debye approximation at low frequencies. At intermediate frequencies, an average density of states of 47.1 ± 0.3 MHz⁻¹ mm⁻³ was found, with a frequency dependence of f^(0.01 ± 0.04).

The final Chapter concerns a problem of partitioning graphs subject to certain restrictions. We prove that several subproblems are NP-complete.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 6 uppsatser.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.