Browsing by Subject "Geometry"
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Item Behavior of steel tub girders with modified cross-sectional geometry(2019-07-17) Wang, Yang, Ph. D.; Helwig, Todd Aaron, 1965-; Engelhardt, Michael D.; Clayton, Patricia M; Williamson, Eric B; Liechti, Kenneth MSteel trapezoidal box girders, also referred to as steel tub girders, have been an attractive design option for long-span horizontally curved highway bridges. The completed composite bridge system presents an aesthetic appeal profile as well as large torsional stiffness. However, during construction, the open U-shaped steel girder section is relatively flexible in torsion and requires extensive bracing. A recent application on straight bridge system in Waco, Texas showed potential wider utilization of tub girders for span length normally reserved for precast concrete beams. Current design and fabrication practices have several relatively inefficient aspects regarding the section geometry and bracing details. These details include the restrictions on the slope ratio of the webs and the top flange-web attachment. Due to the use of empirical equation for live load distribution factor, the slopes of tub girder webs are restricted to be no greater than 1 Horizontal: 4 Vertical in current AASHTO specification. Significant economy can be achieved by utilizing flatter webs. While keeping the width of bottom flange constant, the use of flatter webs increases the tributary width of individual girder. This leads to potential reduction of the required number of girder lines to support traffic live loads and considerable savings on fabrication time and cost. Additionally, the webs of the tub girder are usually attached at the mid-width of the top flanges. This leaves limited flange width to connect the top lateral braces directly with the flange. Therefore, large gusset plates are frequently used to provide sufficient space for the connection. However, the use of gusset plates leads to poor load transfer and unnecessary fabrication cost. If the top flanges are allowed to offset toward the inside of the box, more flange width would be available for simple bolted bracing connection without using gusset plates. Since these proposed details are not currently permitted by AASHTO Specification, a TxDOT-sponsored research project has been conducted at UT Austin to demonstrate the impact of these improved tub girder details using large-scale experimental study and finite element analyses. This dissertation presents part of the results of this research project to provide better understanding of tub girder behavior as well as design recommendations to improve the efficiency and economy of the steel tub girder systemItem Children's mathematical understandings of tessellations : a cognitive and aesthetic synthesis(2011-12) Eberle, Robert Scott; Carmona Domínguez, Guadalupe de la Paz; Berland, Leema; Empson, Susan; Sinclair, Nathalie; Starbird, Michael; Stroup, WalterTessellations have a rich mathematical structure and are especially appropriate as a context for teaching geometry in the middle grades. Few studies have researched how children conceptualize and learn tessellations in spite of their international use in educational contexts. This exploratory study looks at how fourth grade students conceptualize tessellations before instruction. The analysis is done from a Piagetian, cognitive viewpoint and from an aesthetic viewpoint. It is argued that the aesthetic viewpoint is crucial and foundational to children's mathematical understanding, just as it is for mathematicians. A series of clinical interviews was conducted with six fourth grade children. The results identified common themes of children's understanding, strategies, reasoning, and aesthetic criteria for tessellations. Children's ontology varied between object and process conceptions of tessellations. Children struggled especially with the infinite space of mathematical tessellations. Children's aesthetics, including symmetry, influenced their choices in creating tessellations and are shown to have played a cognitive role in children's mathematical exploration of tessellation structures. Mathematics influences students' aesthetic appreciation of tessellations and, more importantly, aesthetics drives the study of the mathematical structure of tessellations. Children's aesthetic criteria were the same as mathematicians', but with much different emphases. Other results are discussed, including the mathematical content elicited by the tasks, the influence of the tools used to create tessellations, the children's epistemology of their tessellations, and the role symmetry played in giving children confidence. Recommendations for future research and possible implications for curriculum and instruction are noted.Item Conics and geometry(2010-08) Johnson, William Isaac; Daniels, Mark L.; Armendariz, Efraim P.Conics and Geometry is a report that focuses on the development of new approaches in mathematics by breaking from the accepted norm of the time. The conics themselves have their beginning in this manner. The author uses three ancient problems in geometry to illustrate this trend. Doubling the cube, squaring the circle, and trisecting an angle have intrigued mathematicians for centuries. The author shows various approaches at solving these three problems: Hippias’ Quadratrix to trisect an angle and square the circle, Pappus’ hyperbola to trisect an angle, and Little and Harris’ simultaneous solution to all three problems. After presenting these approaches, the focus turns to the conic sections in the non-Euclidean geometry known as Taxicab geometry.Item Course summary of geometry and topology(2010-08) Craig, Tara Theresa; Armendáriz, Efraim P.; Daniels, Mark L.The foundation of Luecke’s course M: 396 Geometry and Topology is that collaboration amongst mathematicians and biologists caused tremendous gains in DNA research. The field of topology has led to significant strides in understanding of the topological properties of the genetic molecule DNA. Through the integration of biological phenomena and knowledge of topology and Euclidean geometry, biologists can describe and quantize enzyme mechanisms and therefore determine enzyme mechanisms causing the changes. Understanding mathematical applications in contexts outside of mathematics on any level helps to explain why mathematics is a core content area in primary and secondary education. Requiring secondary educators to take such a course could result in mathematics taught with real world application on the secondary level as well as on the graduate level, as shown in Luecke’s course.Item Developing a qualitative geometry from the conceptions of young children(2010-05) Greenstein, Steven Baron; Stroup, Walter M.; Empson, Susan B.; Carmona, Guadalupe; Petrosino, Anthony; Starbird, MichaelMore than half a century ago, Piaget concluded from an investigation of children’s representational thinking about the nature of space that the development of children’s representational thought is topological before it is Euclidean. This conclusion, commonly referred to as the “topological primacy thesis,” has essentially been rejected. By giving emphasis to the ideas that develop rather than the order in which they develop, this work set out to develop a new form of non-metric geometry from young children’s early and intuitive topological, or at least non-metric, ideas. I conducted an eighteen-week teaching experiment with two children, ages six and seven. I developed a new dynamic geometry environment called Configure that I used in tandem with clinical interviews in each of the episodes of the experiment to elicit these children’s non-metric conceptions and subsequently support their development. I found that these children developed significant and authentic forms of geometric reasoning. It is these findings, which I refer to as qualitative geometry, that have implications for the teaching of geometry and for research into students’ mathematical reasoning.Item The diagrammatic specification and automatic generation of geometry subroutines(2010-05) Li, Yulin, Ph. D.; Novak, Gordon S.; Porter, Bruce; Boyer, Robert; Gouda, Mohamed; Kant, ElaineProgramming has advanced a great deal since the appearance of the stored-program architecture. Through the successive generations of machine codes, assembly languages, high-level languages, and object-oriented languages, the drive has been toward program descriptions that express more meaning in a shorter space. This trend continues today with domain-specific languages. However, conventional languages rely on a textual formalism (commands, statements, lines of code) to capture the programmer's intent, which, regardless of its level of abstraction, imposes inevitable overheads. Before successful programming activities can take place, the syntax has to be mastered, names and keywords memorized, the library routines mastered, etc. Existing visual programming languages avoid some of these overheads, but do not release the programmer from the task of specifying the program logic, which consumes the main portion of programming time and also is the major source of difficult bugs. Our work aims to minimize the demands a formalism imposes on the programmer of geometric subroutines other than what is inherent in the problem itself. Our approach frees the programmer from syntactic constraints and generates logically correct programs automatically from program descriptions in the form of diagrams. To write a program, the programmer simply draws a few diagrams to depict the problem context and specifies all the necessary parameters through menu operations. Diagrams are succinct, easy to learn, and intuitive to use. They are much easier to modify than code, and they help the user visualize and analyze the problem, in addition to providing information to the computer. Furthermore, diagrams describe a situation rather than a task and thus are reusable for different tasks—in general, a single diagram can generate many programs. For these reasons, we have chosen diagrams as the main specification mechanism. In addition, we leverage the power of automatic inference to reason about diagrams and generic components—the building blocks of our programs—and discover the logic for assembling these components into correct programs. To facilitate inference, symbolic facts encode entities present in the diagrams, their spatial relationships, and the preconditions and effects of reusable components. We have developed a reference implementation and tested it on a number of real-world examples to demonstrate the feasibility and efficacy of our approach.Item The double-elliptic case of the Lie-Riemann-Helmholtz-Hilbert problem of the foundations of geometry(1925) Lubben, Renke Gustav; Not availableItem The effect of temperature and terrace geometry on carbonate precipitation rate in an experimental setting(2012-08) Reid, Ellen Elizabeth; Kim, WonsuckThrough flume experiments we demonstrate the calcite precipitation process seen at geothermal hot springs in the lab setting. A series of four experiments were run, varying temperature and terrace ridge height while all other experimental parameters, including initial substrate slope, spring water discharge, and CO₂ input were kept constant. The goal of the experiments was to measure the temperature and terrace height control quantitatively in terms of the amount of overall travertine aggradation, aggradation rate changes in time and downstream direction, as well as to observe the effect of these parameters on processes occurring during precipitation. Using the final deposit thickness measured manually at the end of each experiment and elevation data obtained from a laser topographic profiler, I conclude that high temperature and small terrace heights favor increased precipitation of travertine. However, the amount of precipitation also depends on location within a terrace pond. Flow velocity increases as it approaches a terrace lip, resulting in enhanced precipitation and greater thicknesses in the downstream direction through increased CO₂ degassing, a process called downstream coarsening.Item Enhancing the model of coarse-grained basin floor fans : characteristic trends within lobes and lobe complexes of the Jurassic Los Molles Fm., Neuquén Basin, Argentina(2019-05) Giacomone, Gabriel; Steel, R. J.; Olariu, CornelBasin floor fans can contain a wide range of grain sizes, though the most frequently described fans have tended to be the fine-grained sandy and muddier ones. A coarse-grained category of fan has been broadly described previously, but it lacks a depositional and facies model. The Mid-Jurassic deep-water marine deposits of Los Molles Fm. in Neuquén Basin, Argentina is an example of this type of fan and it is well exposed in outcrop. A detailed characterization of the basin-floor fans of La Jardinera is used to build a coarse-grained fan model. We made use of a high-resolution satellite image, drone imagery and 4000 m of logs with detailed measurements to build isopach and net/gross (NG) maps that with facies analysis allowed reconstruction of the fan and its lobe complexes (LC1-5). In addition, grain size, facies and bed thickness trends were used to refine the interpretation at a lobe scale within unit LC3. Lithofacies, NG ratios and sandstone body geometry helped define six facies associations; hemipelagic deposits, lobe fringe, off-axis lobes, on-axis lobes, distributary channels and debris flows. The facies associations build lobes (<10 m thick) and these are grouped into lobe complexes (~20-40 m thick). The studied five lobe complexes (LC1-5) are separated by fine-grained intervals (~ 4 m thick in average). The fan shows paleoflow trends towards the northwest at the bottom gradually changing to northeast at the top. The lobe complexes stack forward and backstep gently, with no major switches; they aggrade and shift laterally in an autocyclic manner, following topographic lows left by previous deposits. The maps at lobe complex scale show an overall elongated morphology and serrated geometries downdip, a normal response of focused sediment dispersal associated with channels and high-density turbidity currents. Detailed study of lobes 3 and 4 in LC3 show that proximal to lobe axis beds are thicker (>40 cm), grain size is greater (medium sand to granules) and main facies are conglomerates and structureless sandstones. Off axis, beds are thinner (<40 cm), grain size ranges from fine to medium sand and there is an increase on normally graded and laminated sands. These trends are associated with the confinement and density of the flow. From lobe axis to off-axis, channelized elements disappear and the facies vary from high density to low density turbidites. The present work shows a coarse-grained basin floor fan system that differs from previous models; having a distinct elongated morphology, finger-like geometries and changes in facies associated with channelized features and variations on the type of flow from the axis to the fringes.Item Fingerprinting of Si surface bonds using non-resonant optical second-harmonic generation(2018-08-21) Loumakos, Loucas K.; Downer, Michael Coffin; Demkov, Alexander A; Sitz, Greg O; Ekerdt, John G; Fink, ManfredModern electronic device structures require monitoring and control of surface structure at the atomic level during epitaxial growth. We demonstrate an optical fingerprinting technique that isolates, identifies and monitors individual types of bonds (e.g. step-edge rebonds, terrace dimers) and their chemical activity on a single-domain, vicinal Si(001) surface in ultra-high vacuum. The method uses optical second-harmonic generation (SHG) at a single wavelength, but at multiple incidence angles and polarizations (MAP) hence we call it SHG-MAP. SHG-MAP identifies bonds via the unique dependence of their SHG response on azimuthal sample rotation. Using a simplified bond hyper-polarizability model (SBHM), we developed an automated two-step algorithm for identifying all opportunities for isolating a certain bond type geometrically without multi-parameter fitting: firstly, the full parameter space is used to create a 4-D model of the expected macroscopic SHG radiation and secondly a search is preformed to isolate unique bond group contributions. We demonstrate SHG-MAP by monitoring adsorption of atomic hydrogen and chemical etching of rebonded r-D [subscript B] steps on clean vicinal Si(001) in ultra high vacuum.Item Halsted's Lobatschewsky's Geometry(University of Texas at Austin, 1891-05-01) Lobatschewsky, Nicholaus; Halsted, George BruceItem Infinitesimal symmetries of Dixmier-Douady gerbes(2012-08) Collier, Braxton Livingston; Freed, Daniel S.; Allcock, Daniel; Ben-Zvi, David; Keel, Sean; Meinrenken, EckhardThis thesis introduces the infinitesimal symmetries of Dixmier-Douady gerbes over smooth manifolds. The collection of these symmetries are the counterpart for gerbes of the Lie algebra of circle invariant vector fields on principal circle bundles, and are intimately related to connective structures and curvings. We prove that these symmetries possess a Lie 2-algebra structure, and relate them to equivariant gerbes via a "differentiation functor". We also explain the relationship between the infinitesimal symmetries of gerbes and other mathematical structures including Courant algebroids and the String Lie 2-algebra.Item Integrating topology into the standard high school geometry curriculum(2012-08) Kiker, William George; Odell, E. (Edward); Daniels, MarkThis report conveys some of the modern investigations surrounding the use of topology in a contextual setting. Topics discussed include applications of topology relating to the modeling of biological structures and common objects like sunshades, elementary knot theory, and the connection between the fields of topology and algebra. A brief overview and discussion of the incorporation of elementary topology into the standard Geometry curriculum of secondary schools is also examined.Item Mathematics vocabulary knowledge of eighth-grade students(2019-06-19) Unal, Zehra Emine; Powell, Sarah RannellsThe purpose of this study was to develop a mathematics vocabulary measure for eighth-grade students and to determine the relationships among general vocabulary knowledge, mathematics vocabulary knowledge, and mathematics computation. Students (n=34) took three tests in the following order: (1) mathematics vocabulary, (2) WRAT Computation, and (3) GMRT Vocabulary. Mathematics vocabulary results revealed that the mathematics vocabulary test was highly reliable. Based on students’ scores in all tests, the correlation between mathematics vocabulary knowledge and general vocabulary knowledge as well as the relationship between mathematics vocabulary and mathematics computation were strong. However, there was no significant association between mathematics computation and general vocabulary knowledge. Mathematics vocabulary knowledge was a mediator between the two.Item Morphodynamics and geometry of channels, turbidites and bedforms(2011-12) Peyret, Aymeric-Pierre Bernard; Mohrig, David; Kocurek, Gary; Kim, Wonsuck; Lake, Larry W.; Fulthorpe, CraigThe evolution of landscapes and seascapes in time is the result of the constant interaction between flows and topography. Flows change topography, which in turn change the flow. This feedback causes evolution processes to be highly non-linear and complex. When full analytical derivations of the co-evolution of topography and flow are not possible without oversimplifications, as is the case in river bends, recent large topographical datasets and modern computers allow for correlations between horizontal (planview) and cross-sectional geometry of channels. Numerical analysis in the Mississippi and Trinity rivers indicate that the type of correlation between river radius of curvature and bankfull channel width depends on the migration behavior of the river. In other cases, channel topography may only have a second-order effect on its own evolution, as is the case for fully depositional turbidity currents, and the evolution of aeolian field topography may only be a function of this topography. I show that in these situations, changes in topography may be decoupled from details of the flow field and modeled very easily with a good accuracy.Item Pathway through math : educator perspectives on middle school math acceleration(2021-05-06) Heaton, Amy Joann; Riegle-Crumb, CatherineThis thesis reports a study of middle-school mathematics teachers’ attitudes about teaching Geometry in middle school, along with the difference between the factors they think should be used in placing students in the advanced (Geometry) track and what factors are actually considered. Mathematics is a subject which sees significant racialized tracking due to the sequential nature of its course progression coupled with inequitable data measurements and placement methods. While the Common Core State Standards (CCSS) present a standard course progression that does not include Algebra 1 at the middle school level, many school districts continue to include it, and in some cases, Geometry, as options for higher-performing students. In this study, three middle school teachers from two school districts that offered different middle school mathematics course progressions were surveyed, and the responses were then analyzed and coded. Though these teachers had idealized notions of placement tests being the best measure for a student’s mathematical readiness, additional considerations such as equity concerns and parental disagreement contributed to the actual placement of students into advanced course pathways. This thesis discusses implications for equity in middle school math.Item Realizability of tropical lines in the fan tropical plane(2013-08) Haque, Mohammad Moinul; Helm, David, doctor of mathematicsIn this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable.Item RiverML: a harmonized transfer language for river hydraulic models(2014-08) Jackson, Stephen Robert; Maidment, David R.The multitude of data formats for storing river network, geometry, and flow data presents a challenge for the sharing of information both internally between software applications and externally between agencies. An analysis of existing software applications and data models used for one-dimensional hydraulic modelling of river systems was performed. The commonalities and differences between the model inputs were identified in order to determine the necessary characteristics of a common transfer language. A prototype transfer language was developed using Unified Modeling Language (UML) and implemented as an Extensible Markup Language (XML) schema. This prototype is intended to serve as a first step towards developing an international open standard to facilitate the sharing of hydraulic data. This work was performed in cooperation with the Consortium of Universities for the Advancement of Hydrologic Science, Inc. (CUAHSI) and the Open Geospatial Consortium/World Meteorological Organisation Hydrology Domain Working Group.Item The shapes of sacred space : a proposed system of geometry used to lay out and design Maya art and architecture and some implications concerning Maya cosmology(2010-08) Powell, Christopher, 1959-; Stross, Brian; Stuart, David; Freidel, David; Barnhart, Edwin; Wagner, Logan; Barufaldi, JamesThis dissertation explores the fundamental characteristics of a system of geometry and proportion currently used by Maya house builders and shamans to design vernacular architecture in indigenous Maya communities. An extensive examination of Pre-Columbian Maya art and architecture demonstrates how this system of geometry and proportion was also used by the Maya of the Classic and Post-Classic periods. The dissertation concludes with a brief discussion of how Maya geometry was, and is, an expression of Maya cosmology and religion.Item Spatial ability in high school geometry students(2011-08) Brudigam, Kristin Lea; Crawford, Richard H.; Petrosino, Anthony J.; Marshall, JillThe purpose of this study was to observe the differences in high school PreAP Geometry students in regards to spatial ability. The hypothesis states that students who are enrolled in both high school PreAP Geometry and Introduction to Engineering Design have better spatial ability skills than those students who are solely enrolled in PreAP Geometry. Of the 207 students enrolled in geometry at the test school, there was a smaller population (n = 57) simultaneously enrolled in an engineering graphics course at the high school. No direct or special intervention was given to either group of students. Near the end of the academic year, all students were administered the Purdue Visualization of Rotations Test (ROT). Results showed that students enrolled in the engineering design class performed better than those students not enrolled in the course. Furthermore, the males outperformed the females when all students were considered. However, there was not a significant difference among the males, nor was there a difference between males and females enrolled in engineering. Further research is needed to understand these differences and how geometry education plays a role in the development of spatial ability skills.