Doctor of Philosophy, The Ohio State University, 1973, Graduate School

Master of Science, The Ohio State University, 2011, Computer Science and Engineering

We present a framework to interactively explore features present in a vector field through three geometric measurements: curl, curvature, and torsion. Measuring each streamline results in a distribution of measurements. The key assertion to our framework is that these distributions reflect characteristics of the streamline effectively. Therefore, from these distributions we may begin to compare each streamline. We perform dimensionality reduction using two methods, principal component analysis and multidimensional scaling. The MDS is performed by calculating the pair wise distance between cumulative distribution functions and the principal component analysis is performed on a set of descriptive vectors built from summary statistics. The statistics in the vector are built from one or more distributions of geometric measurements for every streamline. The idea is to place streamlines in feature space relative to how different they are with respect to each streamline's geometric measurement distribution. If the distributions are the same then the streamlines will occupy the same spot in feature space. Finally, our framework mitigates feature occlusion by using a feature space from which to select streamlines. To explore the vector field, streamline selections of various sizes are made from the feature space. This renders a subset of streamlines ideally revealing a feature of the underlying vector field.

PHD, Kent State University, 2010, College of Arts and Sciences / Department of Computer Science

Geometry Manipulative Authoring Systems (GMAS) have existed for over twenty years. With the advent of Internet, people began to place geometry manipulatives on the Web. However, there has been little research on how to utilize the Web to the greatest extent in the design and implementation of a GMAS. This dissertation presents GeometryEditor that provides not only online manipulative authoring, storing sharing, and publishing, but also models and solutions for various Web orientation topics. Manipulatives can be presented in authoring mode and learning mode. An interoperation model is proposed for the interaction among manipulatives and the enclosing page. An educational page authoring environment based on this model is presented. GeometryEditor is also a package for building Web applications that need mathematical drawing support. Client Web sites can interact with GeoSite via data Web service. In addition to providing basic authoring support that most existing systems have, GeometryEditor has a few unique and very important authoring features such as Synchronized Copy, multiple coordinate systems, event handling on geometric objects and so on. The research gives the theoretical analysis of the relation and behavior in tools, macros and expressions. GeometryEditor is the only system that is able to identify all the acceptable input object types for macros and expressions. GeometryEditor defines XML schema for geometric object types, and Geometry Construction Markup Language (GCML), the XML representation to describe Object Data Generators (ODG). GCML allows users with some basic programming background to provide extension of authoring support to the system. GeometryEditor also provides a reusable implementation layer that can be used as the infrastructure for building other Web-based manipulative authoring systems.

MS, Kent State University, 2023, College of Arts and Sciences / Department of Earth Sciences

Hydraulic heterogeneity in aquifers contributes to non-Fickian transport characteristics, i.e., which cannot be defined by the continuum-scale advection-dispersion equation (ADE). We investigate the role of first-order heterogeneity, i.e., pore geometry's effect on the dispersion phenomenon of porous media. The research questions addressed are; how can we determine dispersion coefficient and dispersivity as a function of pore-scale geometry and various flow rate? Does dispersivity scale with length-scale even at the pore-scale? In this computational study, a series of intra-pore geometries are designed and quantified by a dimensionless pore geometry factor (β), which captures a broad range of pores that likely exists due to diagenetic processes. Navier-Stokes and Advection-Diffusion equations are solved to examine the transport phenomenon via breakthrough curve (BTC) and residence time distribution (RTD). We determine a length-scale when non-Fickian features transition to the Fickian transport regime by sequentially extending the number of pores. Our results indicate that not only is the velocity distribution and its variance (σ2) are dependent on the pore geometry, but its impact is amplified with flow rate. Consequently, the magnitude of non-Fickian becomes significant for complex pore shapes and require a longer length-scale for the Fickian transport. Thus, a larger velocity variance due to the effect of pore geometry and flow rate contributes to a larger dispersion and Dispersity where variations are found to be a function of β and flow rate. We determine various constitutive equations to predict the length-scale needed for Fickian dispersion, the magnitude of non-Fickian features, the Fickian dispersion and dispersivity coefficients as a function of pore geometry factor (β) and velocity variance (σ2) for various flow regimes, bridging the gap between the pore-scale and the continuum-sale.

BA, Oberlin College, 2020, Mathematics

One area of interest in studying plane curves is intersection. Namely, given two plane curves, we are interested in understanding how they intersect. In this paper, we will build the machinery necessary to describe this intersection. Our discussion will include developing algebraic tools, describing how two curves intersect at a given point, and accounting for points at infinity by way of projective space. With all these tools, we will prove Bezout's theorem, a robust description of the intersection between two curves relating the degrees of the defining polynomials to the number of points in the intersection.

Master of Sciences, Case Western Reserve University, 2019, Mathematics

Vector bundles play a prominent role in the study of projective algebraic varieties. Vector bundles can describe facets of the intrinsic geometry of a variety, as well as its relationship to other varieties, especially projective spaces. Additionally, being among the simplest examples of coherent sheaves, they can be manipulated by a wealth of technical machinery. Here we outline the general theory of vector bundles and describe their classification and structure. We also consider some special bundles and general results in low dimensions, especially rank 2 bundles and surfaces, as well as bundles on projective spaces. Finally, we indicate some open problems and current areas of research.

Doctor of Philosophy, The Ohio State University, 2018, Mathematics

The first steps in defining a notion of spherical tropicalization were recently taken by Tassos Vogiannou in his thesis and by Kiumars Kaveh and Christopher Manon in a related paper. Broadly speaking, the classical notion of tropicalization concerns itself with valuations on the function field of a toric variety that are invariant under the action of the torus. Spherical tropicalization is similar, but considers instead spherical G-varieties and G-invariant valuations. The core idea of my dissertation is the construction of the extended tropicalization of a spherical embedding. Vogiannou, Kaveh, and Manon only concern themselves with subvarieties of a spherical homogeneous space G/H. My thesis describes how to tropicalize a spherical embedding by tropicalizing the additional G-orbits of X and adding them to the tropicalization of G/H as limit points. This generalizes work done by Kajiwara and Payne for toric varieties and affords a means for understanding how to tropicalize the compactification of a subvariety of G/H in X. The extended tropicalization construction can be described from three different perspectives. The first uses the polyhedral geometry of the colored fan and the second extends the Grobner theory definition given by Kaveh and Manon. The third method works by embedding the spherical variety in a specially-constructed toric variety, tropicalizing there with the standard theory, and then applying a particular piecewise-projection map. This final perspective introduces a novel means for tropicalizing a homogeneous space that allows us to prove several statements about the structure of a spherical tropicalization by transferring results from the toric world where more is known. We also suggest a definition for the tropicalization of subvarieties of a homogeneous space whose defining equations have coefficients with non-trivial valuation. All the previous theory has been done in the constant coefficient case, i.e. when th (open full item for complete abstract)

Doctor of Philosophy, The Ohio State University, 2018, Mathematics

This work deals with various phenomena relating to complex geometry. We are particularly interested in non-Kahler Hermitian manifolds, and most of the work here was done to try to understand the geometry of these spaces by understanding the torsion. Chapter 1 introduces some background material as well as various equations and inequalities on Hermitian manifolds. We are focused primarily on the inequalities that are useful for the analysis that we do later in the thesis. In particular, we focus on k-Gauduchon complex structures, which were initially defined by Fu, Wang, and Wu. Chapter 2 discusses the spectral geometry of Hermitian manifolds. In particular, we estimate the real eigenvalues of the complex Laplacian from below. In doing so, we prove a theorem on non-self-adjoint drift Laplace operators with bounded drift. This result is of independent interest, apart from its application to complex geometry. The work in this section is largely based on the Li-Yau estimate as well as an ansatz due to Hamel, Nadirashvili and Russ. Chapter 3 considers orthogonal complex structures to a given Riemannian metric. Much of the work in this section is conjectural in nature, but we believe that this is a promising approach to studying Hermitian geometry. We do prove several concrete results as well. In particular, we show how the moduli space of k-Gauduchon orthogonal complex structures is pre-compact.

Doctor of Philosophy, The Ohio State University, 2017, EDU Teaching and Learning

Mathematical classroom discourse has been identified as a key element in students' cognitive development (e.g., Forman, 1996; Lampert & Cobb, 2003; Yackel, Cobb, & Wood, 1991). In constructivist-based, inquiry classrooms, discussions can be an important way for students to use interactions within their social environment to build personal mathematical understanding. This approach is also consistent with sociocultural views on learning. “Academic mathematical Discourse practices can be understood in general as using language and other symbols systems to talk, think, and participate in the practices that lead to literate mathematical Discourse practices that are the `the objective of school learning'” (Moskovich, 2007, p. 28). The iterative process of sending and receiving communications within the mathematics classroom can shape students' learning and mathematical dispositions. Researchers argue that as students write about their thinking, talk about their thinking with others, and respond thoughtfully to others' mathematical ideas, they build understandings of mathematics from these interactions (Empson, 2003; National Council of Teachers of Mathematics, 2001; Yackel, 2002). While researchers agree that mathematical classroom discourse is a crucial element in students' cognitive development, exactly what role discourse plays in regulating students' cognitive development is undefined. Given a cognitive continuum from the conceptual understanding and reasoning students bring to school to the standards or school learning outcomes of mathematics classrooms, learning progressions describe “the successively more sophisticated ways of thinking about an idea that follow one another as students learn” (Wilson & Bertenthal, 2005, p. 48). Battista's research on geometry learning (1992, 2009) and Learning Progression for Geometric Shapes (2007) makes evident that increases in sophistication of conceptualization are marked by and coincide with changes in language. For (open full item for complete abstract)

Doctor of Philosophy, The Ohio State University, 2017, EDU Teaching and Learning

Many researchers have argued that proving is essential to doing and knowing mathematics because it is the basis of mathematical understanding (Cirillo & Herbst, 2012; Hanna & Jahnke, 1996). However, research studies conducted on students' performance with constructing proofs has found that the majority of high school geometry students are unable to construct valid proofs (McCrone & Martin, 2004; Senk, 1985). The present study contributes to this literature and focuses on the formal proofs that use triangle congruence postulates, which students construct in high school geometry. This qualitative study utilizes a psychological constructivist perspective to investigate how students construct and reason about geometric proofs. The following research questions are answered. What are the different ways that students think and reason while attempting formal geometry proofs that use triangle congruence postulates? How can the information gained from answering the first research question be used to start developing a first draft of a learning progression for geometry proofs? The study presents detailed descriptions of the cognitive processes that students use to construct and reason about geometric proofs as well as describes some of the errors and difficulties students exhibit when doing proofs. The data was collected by administering a series of one-on-one semi-structured task-based interviews to six high school geometry students who were asked to complete a series of proof problems. Students were interviewed for four to five one-hour sessions in which they "thought aloud" as they worked on twelve proof problems. All interviews were video recorded and later transcribed. Data analysis methods implemented were the constant comparative method and retrospective analysis. Findings suggest that most of the proofs that students wrote were not formally correct, but that many students wrote proofs that were not reflective of the often sound proof reasoning they stat (open full item for complete abstract)

Doctor of Philosophy, The Ohio State University, 1981, Graduate School

Doctor of Philosophy, The Ohio State University, 1979, Graduate School

Doctor of Philosophy, The Ohio State University, 1972, Graduate School

Doctor of Philosophy, The Ohio State University, 1964, Graduate School

MS, Kent State University, 2016, College of Arts and Sciences / Department of Mathematical Sciences

Here, we prove the theorem due to Hopf that a surface with constant mean curvature that is homeomorphic to a sphere must be a sphere.

Master of Science, The Ohio State University, 2015, Mathematics

Moduli spaces of curves have been central objects for decades in algebraic geometry. This paper reviews a generalization by Hassett in 2003 of the classic moduli problem. Hassett's moduli spaces classify the stable n-pointed curves of given genus g, with weighted data on the marked points. Hassett proved the existence of such coarse moduli spaces as projective schemes. In the first chapters we review the classic moduli problems and provide a sketch of GIT construction of moduli spaces. Then we review the reductions maps between moduli space of weighted pointed stable curves. Next we discuss the chamber decomposition and wall crossing maps among our moduli spaces. The last sections provided an exposition of the application to several birational constructions of moduli spaces. We review Kapranov, Keel, and Losev-Manin's examples, and discuss the realizations of their examples by successive reductions between weighted pointed moduli spaces.

PHD, Kent State University, 2015, College of Arts and Sciences / Department of Mathematical Sciences

I consider the following problems from Tomography. Suppose that the projections (sections) of two given bodies onto (by) every subspace of a fixed dimension are related by a certain condition. Does this imply that the bodies satisfy a similar condition in the ambient space? There are two major parts to this dissertation. The first one is on bodies with directly congruent projections or sections. The second part is about containment of two bodies and relations between their volumes, provided the projections (sections) of the first body can be rotated to be contained in the corresponding projection (section) of the second one.

Master of Sciences, Case Western Reserve University, 0, EMC - Aerospace Engineering

Flame spread over solid fuels with simple geometries has been extensively studied in the past, but few have investigated the effects of complex fuel geometry. This study uses numerical modeling to analyze the flame spread and burning of wavy (corrugated) thin solids and the effect of varying the wave amplitude. Sensitivity to gas phase chemical kinetics is also analyzed. Fire Dynamics Simulator is utilized for modeling. The simulations are two-dimensional Direct Numerical Simulations including finite-rate combustion, first-order pyrolysis, and gray gas radiation. Changing the fuel structure configuration has a significant effect on all stages of flame spread. Corrugated samples exhibit flame shrinkage and break-up into flamelets, behavior not seen for flat samples. Increasing the corrugation amplitude increases the flame growth rate, decreases the burnout rate, and can suppress flamelet propagation after shrinkage. Faster kinetics result in slightly faster growth and more surviving flamelets. These results qualitatively agreement with experiments.

Doctor of Philosophy, The Ohio State University, 2014, Geodetic Science and Surveying

Knowing the position of an object in real-time has tremendous meaning. The most widely used and well-known positioning system is GPS (Global Positioning System), which is now used widely as invisible infrastructure. However, GPS is only available for outdoor uses. GPS signals are not available for most indoor scenarios. Although much research has focused on vision-based indoor positioning, it is still a challenging problem because of limitations in both the vision sensor itself and processing power. This dissertation focuses on real-time 3D positioning of a moving object using multiple static cameras. A real-time, multiple static camera system for object detection, tracking, and 3D positioning that is run on a single laptop computer was designed and implemented. The system successfully shows less than ±5 mm in real-time 3D positioning accuracy at an update rate of 6 Hz to 10 Hz in a room measuring 8×5×2.5 meters. Implementation and experimental analysis has demonstrated that this system can be used for real-time indoor object positioning. In addition, `collinearity condition equations of motion' were derived that represent the geometric relationship between 2D motions and 3D motion. From these equations, a `tracking from geometry' method was developed that combines these collinearity condition equations of motion with an existing tracking method to simultaneously estimate 3D motion as well as 2D motions directly from the stereo camera system. A stereo camera system was built to test the proposed methods. Experiments with real-time image sequences showed that the proposed method provides accurate 3D motion results. The calculated 3D positions were compared with the results from an existing 2D tracking method that uses space intersection. The differences between results of the two methods were less than ±0.01 mm in all X, Y, and Z directions. The advantage of the tracking from geometry method is that this method calculates 2D motions and 3D motion simultaneously, w (open full item for complete abstract)

MS, University of Cincinnati, 2012, Engineering and Applied Science: Aerospace Engineering

Turbomachinery blades are an integral part of air breathing propulsion systems, gas and steam turbines and other energy conversion devices. The blade design is a very important process since it defines component performance. A parametric approach for the blade geometry design has been implemented. A variety of three dimensional blade shapes can be created using only a few basic parameters and limited interaction with a CAD system. Using a general approach for creating the blade geometries makes the process easy and robust for creating 3D blade shapes for various turbomachinery components. The geometry of the blade is defined by a very basic set of geometric and aerodynamic parameters. Parameters such as flow angles, axial chord, thickness to chord ratio and streamline meridional coordinates are defined. The leading edge and trailing edge are defined by curves as part of the input. Using these parameters, a specified number of 2D airfoils are created and are radially stacked on the desired stacking axis. The sweep and lean perturbations of the blade are defined by splines as a function of a few control points. The design tool generates a specified number of 3D blade sections and each section consists of a defined number of coordinates in the cartesian coordinate system. These sections can be lofted in a CAD package to obtain a solid 3D blade model, which has been demonstrated using Unigraphics-NX. Parametric update of the spline points defining the 3D blade sections creates new blade shapes without going back into the CAD interface. This approach for the design is very beneficial as the geometry can be modified quickly and easily as per the needs of the designer at any point of time. Using this tool, blade shapes of a 10 stage compressor similar to the GE/NASA EEE HPC, a 3 stage booster, a reverse engineered GE 1.5 MW wind turbine and a centrifugal compressor based on a NASA design are constructed as examples. The general capabilty of the design tool is demonstrated (open full item for complete abstract)