by Wu Fung Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 92-95). / Acknowledgements --- p.i / Notations --- p.ii / Introduction --- p.1 / Chapter 1 --- Preliminaries --- p.8 / Chapter 1.1 --- Formulations of Harmonic Maps --- p.8 / Chapter 1.2 --- Function Spaces --- p.11 / Chapter 1.3 --- Penalized Equations --- p.13 / Chapter 2 --- Main Lemmas --- p.15 / Chapter 2.1 --- Short Time Existence --- p.16 / Chapter 2.2 --- Energy Inequalities --- p.18 / Chapter 2.3 --- The Monotonicity Inequalities --- p.23 / Chapter 2.4 --- e - Regularity Theorem --- p.30 / Chapter 3 --- The Compact Case --- p.39 / Chapter 3.1 --- Existence and Regularity for dim M = 2 --- p.39 / Chapter 3.2 --- Existence and Regularity for dim M ≥ 2 --- p.49 / Chapter 3.3 --- Blow-up Results --- p.61 / Chapter 3.4 --- Existence of Harmonic maps --- p.69 / Chapter 4 --- The Noncompact Case --- p.74 / Chapter 4.1 --- Heat-flows from Rm --- p.75 / Chapter 4.2 --- Basic Lemmas --- p.77 / Chapter 4.3 --- Nonpositive Curvature Target Manifolds --- p.83 / Chapter 4.4 --- Dirichlet Problem at Infinity --- p.88 / Bibliography --- p.92

Dans cette thèse, qui s’inscrit dans l’étude de la modélisation géométrique via les méthodes formelles, nous proposons un langage graphique à base de règles dédié à la description des opérations topologiques des cartes généralisées. Notre langage est fondé sur la théorie des transformations de graphes. Dans nos règles, les variables permettent d’abstraire les cellules topologiques (sommets, arêtes, faces, volumes, etc.) manipulées dans les opérations topologiques. Nous avons défini des critères syntaxiques sur les règles assurant que les objets obtenus par application des règles satisfont les contraintes de cohérence des cartes généralisées. La conception de ce langage a été motivée par l’étude de la dynamique de l’appareil de Golgi. Il est connu que dans cette organelle, la topologie des compartiments joue un rôle essentiel. Néanmoins, la structure globale de l’appareil de Golgi reste encore méconnue. Plusieurs hypothèses de fonctionnement sont ainsi avancées par les biologistes. Notre langage à base de règles fournit un cadre pour la simulation puis la comparaison de ces différentes hypothèses d’appareil de Golgi. / This PhD thesis is in line with the study of geometric modeling by the means of formal methods. We propose a graphical rule-based language dedicated to the description of the topological operations of the generalized maps. Our language is based on the graphs transformations theory. In our rules, variables allow one to abstract the topological cells (vertices, edges, faces, volumes, etc.) handled in the topological operations. We have defined some syntactic criteria over rules, which ensure that the objects computed by rules application satisfy the consistency constraints of the generalized maps. The conception of this language has been motivated by the study of the Golgi apparatus dynamics. It is known that in this organelle, the topology of the compartments plays a decisive role. Nevertheless, the global Golgi apparatus structure is still unknown. The biologists thus propose several functional hypotheses. Our rule-based language provides a framework to simulate and the to compare these different hypotheses.

Thesis (M.Sc. (Mathematics)) --University of Limpopo, 2007 / The category Loc of locales and continuous maps is dual to the category Frm of frames and frame homomorphisms. Regular subobjects of a locale A are elements of the form Aj = fj : A ! A j j(a) = ag: The subobjects of this form are called sublocales of A. They arise from the lattice OX of open sets of a topological space X in a natural way. The right adjoint of a frame homomorphism maps closed (dually, open) sublocales to closed (dually, open) sublocales. Simple coverings and separated frames are studied and conditions under which they are closed (or open) are those that are related to coequalizers are shown. Under suitable conditions, simple coverings are regular epimorphisms. Extremal epimorphisms and strong epimorphisms in the setting of locales are studied and it is shown that strong epimorphisms compose. In the category Loc of locales and continuous maps, closed surjections are regular epimorphisms at least for those surjections with subfit domains. / National Research Foundation

During the summer and fall of 1982, personnel from the Geophysics Group in the School of Oceanography at Oregon State University conducted an aeromagnetic survey in the northern Oregon Cascades to assess geothermal potential and study the thermal evolution of the Cascade volcanic arc. Total field and low-pass filtered magnetic anomaly maps obtained from the survey data show high amplitude positive and negative anomalies associated with volcanic cones and shallow source bodies along the axis of the High Cascades. Spectral analysis of the aeromagnetic data yielded source depths and depths-to-the-bottom of the magnetic sources. The magnetic source bottom, in the northern Oregon Cascades, is interpreted as the depth to the Curie-point isotherm. The northern Oregon study area shows shallow Curie-point isotherm depths of 5 to 9 km below sea level (BSL) beneath the axis of the High Cascades from the southern boundary (44°N latitude) to near Mt. Wilson (45°N latitude). A smaller region of shallow Curie-point depths of 6 to 9 km BSL lies west of Mt. Wilson (45°N latitude, 122°W longitude). The shallow Curie-point isotherm suggests the emplacement of relatively recent intrusive bodies in the upper crust beneath the axis of the High Cascades and west of Mt. Wilson. A major northeast trending structure observed in magnetic and residual gravity anomalies near Mt. Wilson, is the northernmost. extent of shallow Curie-point depths and high geothermal gradients mapped in the northern Oregon Cascades. This northeast trending structure appears to mark a division between high intrusive activity in localized areas south of Mt. Wilson and intrusive activity confined beneath the major cones north of Mt. Wilson. / Graduation date: 1986

Thesis (Ph. D.)--Texas State University-San Marcos, 2006. / Vita. Appendices: leaves 162-247. Includes bibliographical references (leaves 248-262).

We study the geometric classification of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as defined by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (Ω,B,??).

This work proposes a two-stage method that reconstructs the map of a scene from tagged photographs of that scene. In the first stage, several methods are proposed that transform tag data from the photographs into an intermediary distance matrix. These methods are compared against each other. In the second stage, an approach based on the physical mass-spring system is proposed that transforms the distance matrix into a map. This approach is compared against and outperforms MDS-MAP(P) when given human tagged input photographs. Experiments are carried out on two test datasets, one with 67 tags, and the other with 19. An evaluation method is described and the optimal overall reconstruction generates maps with accuracies of 47% and 66% respectively for the two test datasets, both scoring roughly 40% higher than a random reconstruction. The map reconstruction method is applied to three sample datasets and the resulting maps are qualitatively evaluated.

This work proposes a two-stage method that reconstructs the map of a scene from tagged photographs of that scene. In the first stage, several methods are proposed that transform tag data from the photographs into an intermediary distance matrix. These methods are compared against each other. In the second stage, an approach based on the physical mass-spring system is proposed that transforms the distance matrix into a map. This approach is compared against and outperforms MDS-MAP(P) when given human tagged input photographs. Experiments are carried out on two test datasets, one with 67 tags, and the other with 19. An evaluation method is described and the optimal overall reconstruction generates maps with accuracies of 47% and 66% respectively for the two test datasets, both scoring roughly 40% higher than a random reconstruction. The map reconstruction method is applied to three sample datasets and the resulting maps are qualitatively evaluated.

Given a surface of genus g with fundamental group π, a representation of π into PSL(2,R) is a homomorphism that assigns to each generator of π an element of P SL(2, R). The group P SL(2, R) acts on Hom(π, P SL(2, R)) by conjugation. Define therepresentationspaceRg tobethequotientspaceHom(π,PSL(2,R))\PSL(2,R). Associated to each representation ρ is a number e(ρ) called its Euler class. Goldman showed that the space Rg has components that can be indexed by Euler classes of rep- resentations, and that there is one component for each integer e satisfying |e| ≤ 2g−2. The two maximal components correspond to Teichmu ̈ller space, the space of isotopy classes of hyperbolic structures on a surface. Teichmu ̈ller space is known to be homeomorphic to a ball of dimension 6g − 6. The other components of Rg are not as well understood. The theory of harmonic maps between non-positively curved manifolds has been used to study Teichmu ̈ller space. Given a harmonic map between hyperbolic surfaces, there is an associated quadratic differential on the domain surface called the Hopf differential. Wolf, following Sampson, proved that via the Hopf differential, harmonic maps parametrize Teichmu ̈ller space. This thesis extends his work to the case of branched hyperbolic structures, which correspond to certain elements in non- maximal components of representation space. More precisely, a branched hyperbolic structure is a pair (M, σ|dz|2) where M is a compact surface of genus g and σ|dz|2 is a hyperbolic metric with integral order cone singularities at a finite number of points expressed in terms of a conformal parameter. Fix a base surface (M, σ|dz|2). For each target surface (M, ρ|dw|2) with the same number and orders of cone points as (M,σ|dz|2), there is a unique harmonic map w : (M,σ|dz|2) → (M,ρ|dw|2) homotopic to the identity that fixes the cone points of M pointwise. Thus we may define another map from the space of branched hyperbolic structures with the same number and orders of cone points to the space of meromorphic quadratic differentials on the base surface M. This map, Φ, takes the harmonic map w associated with a metric ρ|dw|2 to the Hopf differential of w. This thesis shows that the map Φ is injective.

Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically.