This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.

In this thesis we study two of the exceptional projetive planes P2(CO) and P2(HO). These are the homogenous spaces E6/S1 _C4 Spin(10) and E7=S3 _C2 Spin(12). These spaces both have natural actions by the compact Lie groups F4 and S1 x E6 respectively. The method that we will use to study these spaces is via the decompositions associated to these actions. In particular we will describe the homotopy type of P2(CO) in terms of the octonionic projective plane P2(O) and spaces associated to P2(O). We use this to compute the cohomology of P2(CO). Finally, we give a description of certain orbits of the action on P2(HO).

We present in this thesis some results about sphere graphs of 3-manifolds. If we denote as Mg the connected sum of g copies of S2 x S1, the sphere graph of Mg, denoted as S(Mg), is the graph whose vertices are isotopy classes of essential spheres in Mg, where two vertices are adjacent if the spheres they represent can be realised disjointly. Sphere graphs have turned out to be an important tool in the study of outer automorphisms groups of free groups. The thesis is mainly focused on two projects. As a first project, we develop a tool in the study of sphere graphs, via analysing the intersections of two collections of spheres in the 3-manifold Mg. Elaborating on Hatcher's work and on his definition of normal form for spheres ([15]), we define a standard form for two embedded sphere systems (i.e collections of disjoint spheres) in Mg. We show that such a standard form exists for any couple of maximal sphere systems in Mg, and is unique up to homeomorphisms of Mg inducing the identity on the fundamental group. Our proof uses combinatorial and topological methods. We basically show that most of the information about two embedded maximal sphere systems in Mg is contained in a 2-dimensional CW complex, which we call the square complex associated to the two sphere systems. The second project concerns the connections between arc graphs of surfaces and sphere graphs of 3-manifolds. If S is a compact orientable surface whose fundamental group is the free group Fg, then there is a natural injective map i from the arc graph of the surface S to the sphere graph of the 3-manifold Mg. It has been proved ([12]) that this map is an isometric embedding. We prove, using topological methods, that the map i admits a coarsely defined Lipschitz left inverse.

This thesis will examine Relative Local Cohomology. First we extend many well known theorems about Local Cohomology of finitely generated modules with respect to an ideal of a commutative noetherian rings so that they hold for non-finitely generated modules with respect to certain ideals of non-commutative non-noetherian rings. Then we show how similar results hold for Relative Local Cohomology. In particular we provide a relative version of the Local Duality Theorem. We then examine the links between Relative Homological Algebra and the concept of Structure Theorems and give a bound on the Castelnuovo-Mumford Regularity of rings of invariants based on the Cech Complex.

We introduce open quasitoric manifolds and their functorial properties, including complex bundle maps of their stable tangent bundles, and relate these new spaces to the standard constructions of toric topology: quasitoric manifolds, moment angle manifolds and polyhedral products. We extend the domain of these constructions to countably infinite simplicial complexes, clarifying and generalising constructions of Davis and Januszkiewicz. In particular we describe final spaces in the categories of open quasitoric manifolds and quasitoric spaces, as well as in the categories of characteristic pairs and dicharacteristic pairs. We show how quasitoric manifolds can be constructed smoothly as pullbacks of the final spaces QT(n) for n >= 1, and how stably complex structure also arises this way. We calculate the integral cohomology of quasitoric spaces over Cohen-Macaulay simplicial complexes, including the final spaces QT(n) as a special case. We describe a procedure for calculating the Chern numbers of a quasitoric manifold M and, relating this to our cohomology calculations, show how it may be interpreted in terms of the simplicial homology of H(n), the simplicial complex underlying QT(n).

Apart from the introductory chapter, this thesis is divided into six parts. Chapter two provides a basic discussion of the authorship of the Liu-zi. This part will incorporate and comment previous opinions, and then compare and contrast the Liu-zi and the Wenxin Diaolong. It begins by exploring the opinion that the Liu-zi was written by an Easten Jin scholar. However, this paper will show that the text was written by Liu Zhou of the Beiqi Period(550-577) during the Southern and the Northern Dynasty. Chapter three analyzes the reference sources of the Liu-zi through parallel readings of the Liu-zi the Huainanzi and the Extant Version of the Wenzi. In addition to study parallel passages found in these titles and to show this extensive reworking, this part expounds on typical examples of the different modifications on the Liu-zi by comparing with the Huainanzi and the Extant version of the Wenzi. The Liu-zi tends to use materials from these two earlier texts to present new interpretation of the books. Furthermore, there will be a discussion of a scholastic idea on the timing of the authorship as well as the textual history of the Huainazi and the Extant version of the Wenzi. Chapter four investigates the absorption and abandonment of the theories of Other Scholars in the Liu-zi. By using such connection, this part aims at examining the relationships between the Liu-zi and the other texts to show the differences and inheritance of the Liu-zi among and from those previous texts of Other Scholars. Chapter five inspects opinions of previous scholars on the development of the Eclectics. The chapter further argues with the general misread that the Eclectics is merely Wang-lao Taoism. Then it shows the importance of the Eclectics in the Hanji and determines the academic status of the Liu-zi in the Eclectics, that is, in the history of Chinese Textual Bibliography. Chapter six provides a basic discussion between the Liu-zi and major extant writings of Metaphysics in Wei-Jin Dynasties. This is to look into a special feature, the mutual dependence of Confucianism and Daoism, of the Liu-zi, thus confirming the academic status of the book in Chinese tradition. By providing textual comparison, this aims to provide the relationship and inheritance between the Liu-zi and other major extant writings of Metaphysics in Wei-Jin Dynasties for further research. Chapter seven is a recapitulation of the main points in this thesis. / For other works compiled in the Pre-Han and Han Dynasties which are also parallel to the Liu-zi, please refer to appendix1. / The Liu-zi was traditionally regarded as a work of the Eclectics during the Wei-Jin and the Six Dynasties. Previous discussions of the Liu-zi focus merely on its authorship, but as the issue is intricate, this problem is unresolved even in current researches. Other than the authorship, the general picture, especially the Philological aspect of the book is only rarely explored. This thesis aims to carry out an in-depth investigation of the Liu-zi, including its authorship and a comparison with other related texts, such as the Lushi Chunqiu and the Huainanzi. By doing so, I want to highlight the features and value of the Liu-zi, and demonstrate how the Liu-zi is influenced and adapted from the above two books. / 梁德華. / Adviser: Che Wah Ho. / Source: Dissertation Abstracts International, Volume: 73-03, Section: A, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 290-318). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in Chinese and English. / Liang Dehua.

Anthony J. Silva. / Originally presented as author's thesis (M.S.--Massachusetts Institute of Technology), 1986. / Bibliography: p. 105-106. / Supported in part by the Advanced Research Projects Agency monitored by ONR under contract no. N00014-81-K-0742 Supported in part by the National Science Foundation under grant ECS-8407285

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Tesi per compendi de publicacions / The study of thermal convection in rotating spherical geometry is fundamental to explain many geophysical and astrophysical phenomena such as the generation of the magnetic fields, or the differential rotation observed in the atmospheres of the major planets. The difficulties associated with the experimental studies enhance the importance of the three-dimensional numerical simulations, such as those presented in this dissertation. In order to obtain the evolution equations, the Boussinesq approximation is applied to the mass, momentum and energy conservation equations, which are rewritten in terms of toroidal and poloidal potentials. Together with the temperature field, they are expanded in spherical harmonics over the sphere, and in the radial direction a collocation method is used. Semi- implicit schemes, based in backward differentiation formulae (IMEX-BDF), implemented with order and time step control (VSVO), are used for time integration. Applying non-slip boundary conditions with internal heating and very low Prandtl numbers (ratio between the thermal diffusive and the viscous time scales), one of the first exhaustive analysis of the linear stability of the conductive state has been performed. In addition, the existence of preferred polar antisymmetric modes at the onset of convection for high rotation rates has been described. A study of the efficiency of different high order time integration schemes, either with fixed time-step or VSVO, has been carried out. In our own time evolution codes we apply the IMEX-BDF formulae with an explicit treatment of the nonlinear terms of the equations. The use of 'matrix-free' methods allows the implicit treatment of the Coriolis term, and makes the implementation of a step and order control easier. The results show that the use of high order methods, especially those with time-step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions. Finally, at low Prandtl number, and with non-slip boundary conditions, the nonlinear dynamics is deeply explored by means of temporal evolutions. The type of solutions is described, and the nonlinear mean flow properties are studied. Using parameters as close as possible to those of the Earth's outer core, the numerical simulations are compared with laboratory experiments and realistic measurements. / L'estudi de la convecció tèrmica en geometria esfèrica en rotació es fonamental per explicar molts fenòmens geofísics i astrofísics, com la generació de camps magnètics, o la rotació diferencial observada en l'atmosfera dels planetes majors. Les dificultats associades amb els estudis experimentals afavoreixen que les simulacions numèriques tridimensionals, com les que es presenten en aquesta memòria, siguin una eina molt important en aquest camp. Per a l'obtenció de les equacions d'evolució, s'aplica l'aproximació de Boussinesq a les equacions de conservació de la massa, la quantitat de moviment i l'energia, i es reescriuen en funció dels potencials toroidal i poloidal. Els potencials i la temperatura es desenvolupen, sobre l'esfera, en harmònics esfèrics i en la variable radial s'usa col·locació. Per a la integració es fan servir esquemes semi-implícits, que en el nostre cas, estan basats en les fórmules de diferenciació regressiva (IMEX-BDF), que s'han implementat amb control d'ordre i pas (VSVO). En primer lloc, sota condicions de contorn d'adherència, calentament intern i nombre de Prandtl (quocient entre les escales de temps de difusió tèrmica i viscosa) molt baix s'ha realizat un dels primers anàlisis exhaustius de l'estabilitat lineal de l'estat conductiu, gràcies a la millora dels mètodes numèrics emprats. Així mateix, s'ha descrit per primera vegada l'existència de modes polars antisimètrics a l'inici de la convecció amb rotacions elevades. En segon lloc s'ha realizat un estudi de l'eficiència de diferents integradors temporals d'ordre alt, amb pas fix o VSVO. En els nostres propis codis temporals apliquem les fórmules IMEX-BDF amb un tractament explícit dels termes no lineals de les equacions. L'ús de mètodes 'matrix-free' fa rentable el tractament implícit del terme de Coriolis i facilita la implementació d'un control d'ordre i pas temporal adequat. Els resultats mostren que amb ordre elevat, amb o sense control de pas i ordre, s'incrementa l'eficiència de la integració i s'obtenen solucions més acurades. Finalment, amb nombre de Prandtl baix i condicions de contorn d'adherència, s'explora exhaustivament la dinàmica no lineal mitjançant evolucions temporals, tot descrivint el tipus de solucions. També s'estudien les propietats mitjanes de fluxos no lineals. Utilizant paràmetres el més similars possible als del nucli extern de la Terra es comparen els resultats de les simulacions numèriques amb experiments de laboratori i amb medicions de situacions reals.

Η εργασία έχει ως αντικείμενο μελέτης τους δενδρίτες. Στο 1ο κεφάλαιο γίνεται μία ιστορική αναδρομή σε τοπολογικές έννοιες. Στο 2ο κεφάλαιο δίνονται όλες οι απαραίτητες έννοιες για την μελέτη των δεδριτών. Στο 3ο κεφάλαιο ορίζεται ο δενδρίτης και μελετώνται βασικές ιδιότητες αυτού. Στο 4ο κεφάλαιο ορίζεται η έννοια του δένδρου και αποδεικνύουμε ότι κάθε δενδρίτης είναι τοπολογικό όριο ακολουθίας δένδρων. Τέλος στο 5ο κεφάλαιο γίνεται κατασκευή του καθολικοιύ δενδρίτη. / In this thesis we define and study the dendrites. We define the notion of dedrite as a inverse limit of topological sequence of trees. Then we construct the universal dendrite.