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We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.

Wireless Sensor Networks (WSN) offer a flexible low-cost solution to the problem of event monitoring, especially in places with limited accessibility or that represent danger to humans. WSNs are made of resource-constrained wireless devices, which require energy efficient mechanisms, algorithms and protocols. One of these mechanisms is Topology Control (TC) composed of two mechanisms, Topology Construction and Topology Maintenance. This dissertation expands the knowledge of TC in many ways. First, it introduces a comprehensive taxonomy for topology construction and maintenance algorithms for the first time. Second, it includes four new topology construction protocols: A3, A3Lite, A3Cov and A3LiteCov. These protocols reduce the number of active nodes by building a Connected Dominating Set (CDS) and then turning off unnecessary nodes. The A3 and A3-Lite protocols guarantee a connected reduced structure in a very energy efficient manner. The A3Cov and A3LiteCov protocols are extensions of their predecessors that increase the sensing coverage of the network. All these protocols are distributed -they do not require localization information, and present low message and computational complexity. Third, this dissertation also includes and evaluates the performance of four topology maintenance protocols: Recreation (DGTRec), Rotation (SGTRot), Rotation and Recreation (HGTRotRec), and Dynamic Local-DSR (DLDSR). Finally, an event-driven simulation tool named Atarraya was developed for teaching, researching and evaluating topology control protocols, which fills a need in the area of topology control that other simulators cannot. Atarraya was used to implement all the topology construction and maintenance cited, and to evaluate their performance. The results show that A3Lite produces a similar number of active nodes when compared to A3, while spending less energy due to its lower message complexity. A3Cov and A3CovLite show better or similar coverage than the other distributed protocols discussed here, while preserving the connectivity and energy efficiency from A3 and A3Lite. In terms of network lifetime, depending on the scenarios, it is shown that there can be a substantial increase in the network lifetime of 450% when a topology construction method is applied, and of 3200% when both topology construction and maintenance are applied, compared to the case where no topology control is used.

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This thesis is devoted to the study of the effect of Legendrian surgery on contact manifolds. In particular, we study the effect of this surgery on the Reeb dynamics of the contact manifold on which we perform such a surgery along Legendrian links. We obtain an exact sequence of cyclic Legendrian homology for the Legendrian links. Then we present the applications in 3-dimension and higher dimensions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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We investigate quotient-fine nearness frames, showing that they are reflective in the category of strong nearness frames, and that, in those with spatial completion, any near subset is contained in a near grill. We construct two categories, each of which is shown to be equivalent to that of quotient-fine nearness frames. We also consider some subcategories of the category of nearness frames, which are co-hereditary and closed under coproducts. We give due attention to relations between these subcategories. We introduce totally strong nearness frames, whose category we show to be closed under completions. We investigate N-homomorphisms and remote points in the context of totally bounded uniform frames, showing the role played by these uniform N-homomorphisms in the transfer of remote points, and their relationship with C -quotient maps. A further study on grills enables us to establish, among other things, that grills are precisely unions of prime filters. We conclude the thesis by showing that the lattice of all nearnesses on a regular frame is a pseudo-frame, by which we mean a poset pretty much like a frame except for the possible absence of the bottom element. / Mathematical Sciences / Ph.D. (Mathematics)

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Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the M$\ddot{o}$bius strip and Figure 8 Klein bottle have slower changes in curvature, but the classic Klein bottle has a quicker change in curvature.