The fundamental theorem of projective geometry states that a mapping from a projective space to itself whose range has a sufficient number of points in general position is a projective transformation possibly combined with a self-homomorphism of the underlying field. We obtain generalisations of this in many directions, dealing with the case where the mapping is only defined on an open subset of the underlying space, or a subset of positive measure, and dealing with many different spaces over many different rings.

The purpose of this investigation was to describe tenth grade students' views of the purposes of geometric proof within the context of their learning. Classroom observations, the curriculum, assessment tools, journal questions, and a preconceptions questionnaire were used to provide context for the views expressed by students from a single classroom. Eleven classroom episodes selected from the classroom observations were used to describe the instructional context as well as discourse among the students during group work. The episodes provided details about how and when the classroom teacher addressed various purposes of proofs involving geometry concepts throughout two instructional units on coordinate geometry proofs and two-column proofs. The episodes also consisted of student discourse relating to the purposes of geometric proof as students worked on assigned proof problems. The students' views were examined through journal questions given at the beginning of selected days and through a post-instruction questionnaire and individual interviews. There were three main findings of the study. First, several students experienced difficulty in expressing their views of the purposes of geometric proof when asked directly. One-third of the students could only list properties or theorems they encountered during the unit on geometric proof. However, when these students were asked to describe the purpose for each column, all of the students listed both explanation and verification. Second, the students expressed limited views of the purposes of proof, referring mainly to verification. Only a few students mentioned explanation, systematization, and communication. However, students generally referred to at least two purposes of proof (explanation, verification, and communication) when describing the proving process involved in coordinate geometry. Third, the students' views of various purposes of geometric proof were diverse. Recommendations for future research include the examination of students' views of the purposes of geometric proofs for students who use paragraph form and studies to investigate the development of students' views of the purposes of proof as they gain more experience with formal proof writing and other methods of proof. / Graduation date: 2004

The conceptual component of this work is about "reference surfaces'' which are the dual of reference frames often used for shape representation purposes. The theoretical component of this work involves the question of whether one can find a unique (and simple) mapping that aligns two arbitrary perspective views of an opaque textured quadric surface in 3D, given (i) few corresponding points in the two views, or (ii) the outline conic of the surface in one view (only) and few corresponding points in the two views. The practical component of this work is concerned with applying the theoretical results as tools for the task of achieving full correspondence between views of arbitrary objects.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

We de…ne a uniformity on a glued space under uniformly continuous attachment maps. If the component spaces are uniform coverable then the resulting glued space is uniform coverable. We consider examples including the glued uniformity on a …nite dimensional CW complex which is shown to be uniformly coverable. For one dimensional CWcomplexes, the resulting deck group is equivalent to the fundamental group. Other properties of the deck group are explored.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, 1917. / "Private edition, distributed by the University of Chicago Libraries." Includes bibliographical references. Also available on the Internet.

No description available.

The Basal Unit of West Woodbourne Field in Barbados is a 250 m thick succession of finely-interbedded sandstones and mudstones deposited by Paleogene, fine-grained, deep-water systems off the northern South American margin and deformed as sediments were translated to the subduction zone of the Caribbean and Atlantic plates. Closely spaced gamma ray, neutron, density, spontaneous potential, formation microimager and dip meter logs, limited core, and published reports of local outcrops, were used to define three scales of vertical stratigraphic variation within this 1.5 km2 field: (1) decimeters to meters thick log facies; (2) meters to tens of meters thick log successions; and (3) tens to hundred meter thick intervals that are continuous laterally across the field. These variations record changes in sediment supply and depositional energy during progradation and abandonment events varying in scale from local shifts in distributary channels to regional changes in sediment transport along the basin. Well log correlations suggest the Basal Unit comprises a turbidite fan system (250 m thick) trending north to northeast, composed of six, vertically-stacked, distributary channel complexes. Three architectural elements are identified within each distributary channel complex: (1) Major amalgamated channels (30-40 m thick, 150-200 m wide and at least 900 m long) pass down depositional dip into proximal second-order channels that bifurcate basinward (15-20 m thick, symmetric successions); (2) Lobe deposits (20-50 m thick, 400 m wide, and at least 400 m long) are composed of upward-coarsening successions that contain distal second-order channels (1-10 m thick); and (3) Laterally extensive overbank deposits (5-10 m thick), which vertically separate distributary channel-lobe complexes. Reservoir heterogeneities within the Basal Unit are defined by the lateral extent and facies variations across a hierarchy of strata within channel-lobe complexes. Although laterally extensive muddy overbank deposits generally inhibit vertical communication between stacked channel-lobe complexes, in places where high-energy first-order channel sandstones incise underlying muddy overbank deposits, sandstones in subsequent intervals are partially connected. The Basal Unit is bounded on the southwest by a northwest-southeast trending fault that rises 30 degrees towards the northwest to define a structural trap on the northeast side of the field.

In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic differential operators and pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the similarities and differences with the case of smooth symbols on the circle. The second problem is related to the geometry of differential operators and its connection with representations of semi-simple Lie algebras. We show that the semiregular module, naturally associated with a graded semi-simple complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of the diagonal in the square of the flag variety. Namely, we consider the Beilinson-Bernstein localization of the semiregular module and show that it is isomorphic to the D-module obtained by applying the Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the flag variety with support on the diagonal.

In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d, g)-structures.