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This study is concerned with the development of extended fluid-dynamic models for the prediction of micro-scale flows. When compared to classical fluid descriptions, such models must remain valid on scales where traditional techniques fail. Also, knowing that solution to these equations will be sought by numerical methods, the nature of the extended models must also be such that they are amenable to solution using computational techniques. Moment closures of kinetic theory offer the promise of satisfying both of these requirements. It is shown that the hyperbolic nature of moment equations imbue them with several numerical advantages including an extra order of spacial accuracy for a given reconstuction when compared to the Navier-Stokes equations and a reduced sensitivity to grid irregularities. In addition to this, the expanded set of parameters governed by the moment closures allow them to accurately model many strong non-equilibrium effects that are typical of micro-scale flows. Unfortunately, traditional moment models have suffered from various closure breakdowns, and robust models that offer a treatment for non-equilibrium viscous heat-conducting gas flows have been elusive. To address these issues, a regularized 10-moment closure is first proposed herein based on the maximum-entropy Gaussian moment closure. This mathematically well-behaved model avoids closure breakdown through a strictly hyperbolic treatment for viscous effects and an elliptic formulation that accounts for non-equilibrium thermal diffusion. Moreover, steps toward the development of fully hyperbolic moment closures for the prediction of non-equilibrium viscous gas flow are made via two novel approaches. A thorough study of each of the proposed techniques is made through numerical solution of many classical flow problems.

Normal coloring - an equivalent version of Petersen coloring - is a special proper 5-edge-coloring of cubic graphs. Every edge in a normally colored graph is normal, i.e. it uses together with its four neighbours either only three colors or all five colors. Jaeger conjectured that every bridgeless cubic graph has a normal coloring. This conjecture, if true, imply for example Cycle double cover conjecture. Here we solve a weakened version of Jaeger's problem. We are looking for a proper 5-edge-coloring such that at least a part of the edges is normal. We show a coloring of generalized prisms with two thirds of the edges normal and a coloring of graphs without short cycles with almost half of the edges normal. Then we propose a new approach to normal coloring - chains. We use chains to prove that there cannot be only one single mistake in an almost normally colored graph. We also prove some statements about cuts in a normally colored graph which also follow from nowhere-zero Petersen flow. Finally, we examine a four-cycle in a normally colored graph. 1

Un graphe est un ensemble de noeuds, ensemble de liens reliant des paires de noeuds. Avec la quantité accumulée de données collectées, il existe un intérêt croissant pour la compréhension des structures et du comportement de très grands graphes. Néanmoins, l’augmentation rapide de la taille des grands graphes rend l’étude de tous les graphes de moins en moins efficace. Ainsi, il existe une demande impérieuse pour des méthodes plus efficaces pour étudier de grands graphes sans nécessiter la connaissance de tous les graphes. Une méthode prometteuse pour comprendre le comportement de grands graphes consiste à exploiter des propriétés spécifiques de structures locales, telles que la taille des grappes ou la présence locale d’un motif spécifique, c’est-à-dire un graphe donné (généralement petit). Un exemple classique de la théorie des graphes (cas avérés de la conjecture d'Erdos-Hajnal) est que, si un graphe de grande taille ne contient pas de motif spécifique, il doit alors avoir un ensemble de noeuds liés par paires ou non liés, de taille exponentiellement plus grande que prévue. Cette thèse abordera certains aspects de deux questions fondamentales de la théorie des graphes concernant la présence, en abondance ou à peine, d’un motif donné dans un grand graphe : - Le grand graphe peut-il être partitionné en copies du motif ? - Le grand graphe contient-il une copie du motif ? Nous discuterons de certaines des conjectures les plus connues de la théorie des graphes sur ce sujet: les conjectures de Tutte sur les flots dans les graphes et la conjecture d'Erdos-Hajnal mentionnée ci-dessus, et présenterons des preuves pour plusieurs conjectures connexes - y compris la conjecture de Barát-Thomassen, une conjecture de Haggkvist et Krissell, un cas particulier de la conjecture de Jaeger-Linial-Payan-Tarsi, une conjecture de Berger et al, et une autre d'Albouker et al. / A graph is a set of nodes, together links connecting pairs of nodes. With the accumulating amount of data collected, there is a growing interest in understanding the structures and behavior of very large graphs. Nevertheless, the rapid increasing in size of large graphs makes studying the entire graphs becomes less and less efficient. Thus, there is a compelling demand for more effective methods to study large graphs without requiring the knowledge of the graphs in whole. One promising method to understand the behavior of large graphs is via exploiting specific properties of local structures, such as the size of clusters or the presence locally of some specific pattern, i.e. a given (usually small) graph. A classical example from Graph Theory (proven cases of the Erdos-Hajnal conjecture) is that if a large graph does not contain some specific pattern, then it must have a set of nodes pairwise linked or not linked of size exponentially larger than expected. This thesis will address some aspects of two fundamental questions in Graph Theory about the presence, abundantly or scarcely, of a given pattern in some large graph: - Can the large graph be partitioned into copies of the pattern? - Does the large graph contain any copy of the pattern?We will discuss some of the most well-known conjectures in Graph Theory on this topic: the Tutte's flow conjectures on flows in graphs and the Erdos-Hajnal conjecture mentioned above, and present proofs for several related conjectures -- including the Barát-Thomassen conjecture, a conjecture of Haggkvist and Krissell, a special case of Jaeger-Linial-Payan-Tarsi's conjecture, a conjecture of Berger et al, and another one by Albouker et al.

Regulation and alteration of streams and rivers have a long history and dams provide us with resources. It also degrades the ecosystems connected to them by cutting off transportation routes and changing the flow regime. Riparian ecosystems have evolved to the natural flow regimes, making them sensitive to hydrological alterations. Riparian plant communities are often species rich and have a positive impact on their surroundings, but are often degraded because of flow alteration. Studies predict that riparian zones will lose species and extent because of climate change, making it increasingly important to manage and restore them.  This report examines the potential effects of two scenarios of a planned re-introduction of natural flow variation in the regulated river Juktån in Sweden. Riparian vegetation composition and cover was inventoried in plots with different amount of yearly flooding to assess current relationships between flooding and riparian vegetation. A rating curve was used to determine the relationship between discharge and river stage. GIS-modeling using LiDAR data and the rating curve made it possible to predict the extent of flooding with the two scenarios, and resulting potential changes in vegetation extent and composition. No significant correlation was found between species richness and number of days flooded under current flooding regime. Positive correlation was however found between elevation and species richness, bare soil and flooding, and vascular plant cover and flooding. The three most species-rich vegetation belts are expected to increase substantially in area with both scenarios, suggesting that positive response is likely with the introduction seasonal flow variation.