Doctor of Philosophy, The Ohio State University, 2022, Mathematics

We explore lattice point counting and the method of stationary phase through the lens of questions about the number of lattice points on and near surfaces with vanishing curvature. Our focus is on spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic Heisenberg norms for α ≥ 2. We accomplish this through a transformative process that takes a number theory question about counting lattice points and translates it into that of an analytical estimation of measure. This process relies on truncating and scaling the n-dimensional integer lattice to produce a fractal-like set. By introducing a measure on this resulting set and using elementary Fourier analysis, the counting problem is transformed into one of bounding an energy integral. This process uses principles of fractal geometry and oscillatory integrals. Primary challenges that arise are the presence of vanishing curvature and uneven dilations. Following a discussion and formal estimate of the curvature of the Heisenberg spheres, we utilize the method of stationary phase to compute a bound on the Fourier transform of their surface measures. Our work is inspired by that of Iosevich and Taylor (2011) and Garg, Nevo, and Taylor (2015). We present an extension of the main result in the former to surfaces with vanishing curvature. Furthermore, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces. Additionally, we present a mini-course on the basics of stationary phase—a quick-start guide to stationary phase in practice. This includes a discussion of the formulation of oscillatory integrals and their solutions with a focus on the impact of geometric properties (e.g. curvature) on the estimates for the decay of the Fourier transform. It further serves as a supplement to [Shakarchi and Stein, Functional Analysis: Chapter (open full item for complete abstract)

Committee: Krystal Taylor (Advisor); Rodica Costin (Committee Member); Barbara Keyfitz (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2005, Engineering : Mechanical Engineering

The focus of the current research is to analyze the structural behavior of a joined wing aircraft based on Sensorcraft configuration. The joined-wing aircraft is an Unmanned Air Vehicle (UAV) and is being developed by the Air Force Research Laboratory (AFRL) for High-Altitude and Long-Endurance (HALE) missions. HALE aircraft typically find applications in telecommunication relay, environmental sensing and military reconnaissance. The Sensorcraft is designed to operate at high altitudes (60,000 ft) with low speed (300- to 400-knot range) and for long durations of time (60 to 80 hours). At these operating conditions, the density, and hence, the Reynolds number, is low. These conditions necessitate a wing operating with high lift and low drag with high-aspect ratio wings. Moreover, the vehicle must be lightweight and strong, and offer high aerodynamic performance and efficiency. The wings are thus slender and flexible and undergo large deflections during flight. The diamond shape of the joined wing has the primary structural advantage of strength as each wing braces the other against lift loads. The AFRL team has developed an in-house Sensorcraft joined-wing model. The University of Cincinnati (UC), along with its partners, AFRL and Ohio State University are working together to study the complete nonlinear aeroelastic behavior of the joined-wing model. A computational fluid dynamics (CFD) analysis has been performed and the aerodynamic loads acting on the model have been determined. While four different structural models are currently analyzed at UC, the current research focuses on creating and analyzing a reinforced shell joined wing model. Unlike the conventional box-wing models, the reinforced shell wing model has the same shape as that of the aerodynamic model. Here two different models have been created and analyzed. In the first model, the surface mesh for the structure is the same as the grid used for CFD analysis, and hence the pressure loads are applied to the (open full item for complete abstract)

Committee: Dr. Urmila Ghia (Advisor) Subjects:

Doctor of Philosophy, The Ohio State University, 2024, Mathematics

In 1976 breakthrough by S.-T. Yau, the Calabi conjecture was solved by showing the existence of solutions to a fully nonlinear elliptic equation, called the complex Monge-Ampere equation. This is a classical PDE written in terms of the determinant of the Hessian matrix of a function, or alternately as the product of eigenvalues of the Hessian. The special geometrical objects constructed by Yau turned out be a crucial piece in many areas of geometry and mathematical physics, mainly in string theory. An entire new field of mathematics was created by this and other works around the same time that focused on solving geometrical problems by nonlinear PDE methods. This is called geometric analysis. Both aspects of this field seem to be equally challenging. On one side there is the geometrical problem involving metrics and curvature, and on the other, there is a difficult problem of getting a priori estimates for a fully nonlinear PDE. My research is mostly focused on the latter problem of obtaining estimates. It should also be mentioned that the geometric properties of the manifold play an important role in this method. A new generation of fully nonlinear PDEs called the (n − 1) Monge-Ampere equation was introduced in the last decade, mostly with the aim of proving Calabi-Yau type theorems on non-Kahler manifolds. My first project studies a vast generalization of this equation, where we consider equations of symmetric functions of eigenvalues of the Hessian, involving p-plurisubharmonic functions and general nonlinear dependence on lower-order terms. We develop a new technique using the rank of the symmetric function to derive estimates for this equation. We consider the parabolic case of this problem to show the long-time existence of solutions and convergence to the elliptic equation up to a constant. The consideration of p-plurisubharmonic functions and other equations involving (p, p) forms in complex geometry led to consider a new class of PDEs (open full item for complete abstract)

Committee: Bo Guan (Advisor); Justin North (Committee Member); Hsian-Hua Tseng (Committee Member); King-Yeung Lam (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2023, Evolution, Ecology and Organismal Biology

The tooth crowns, tooth roots, and mandible of mammals form an integrated apparatus for the acquisition and breakdown of food. The mandible houses the teeth and anchors the muscles of mastication, with its proportions determining the mechanical advantage along the tooth row; the tooth roots anchor the teeth in their sockets and dissipate the stresses of biting and chewing to the bone; the tooth crowns perform direct mechanical breakdown of food items. All three serve vital and distinct functions for a mammal's ecology, and, of particular interest to paleontologists, all three tend to fossilize well, making them useful sources of dietary information for extinct mammals. Here, I analyze the functional signal that can be detected in the morphology of all three features, and where possible I use this dietary signal to infer details about the ecology of a group of extinct mammals known as archaic ungulates. Archaic ungulates are a group of morphologically similar lineages mostly dating to the early Paleogene that were numerous and diverse but which have been difficult to place both phylogenetically and ecologically. Looking at mandible shape of modern mammals, I find that phylogenetic signal tends to overwhelm dietary signal, making dietary inference difficult, but the dietary signal that is present indicates that archaic ungulates diversified in diet early in the Paleocene into niches that likely included specialized herbivory and faunivory. For the tooth roots, I conducted a survey of root number diversity in modern Mammalia, and found several previously unrecognized patterns in root number. I preliminarily establish that root number is not directly linked to tooth dimensions, tooth cusp number, or body size. It is also largely unrelated to diet, meaning that this while this character is far more complex than previously realized, it has no clear utility for dietary inference. For the tooth crowns, I performed a meta-analysis of studies that use one of the most popular (open full item for complete abstract)

Committee: John Hunter (Advisor); Debbie Guatelli-Steinberg (Committee Member); Jill Leonard-Pingel (Committee Member); Jonathan Calede (Committee Member) Subjects: Evolution and Development; Morphology; Paleontology

Bachelor of Science, Wittenberg University, 2020, Math

The objective of this thesis is to understand the systematic analytic treatment of the model presented in Anna Ghazaryan and Vahagn Manukian's journal article, “Coherent Structures in a Population Model for Mussel-Algae Interaction," which concentrates on the formation of mussel beds on soft sediments, like those found on cobble beaches. The study will investigate how the tidal flow of the water is the main structure that creates the mussel-algae interaction observed on soft sediments. With this investigation, the idea of fast-time and slow-time systems is explicated according to Geometric Singular Perturbation Theory, how Invariant Manifold Theory proves the existence of our solutions, the process of non-dimensionalization, and the re-scaling of the model. It will apply concepts found in nonlinear dynamics to discover equilibria and nullclines of the system. Finally, the study will discuss what the findings mean in context of the population model and the implications of tidal flow on other ecological relationships.

Committee: Adam Parker (Advisor); Alyssa Hoofnagle (Committee Member); Jeremiah Williams (Committee Member) Subjects: Applied Mathematics; Aquatic Sciences; Ecology; Mathematics

Master of Mathematical Sciences, The Ohio State University, 2020, Mathematical Sciences

Depression is a mental disorder that causes changes in mood, appetite, and behaviors such as sleep patterns. The most common symptom may be the lack of interest in things that were previously pleasant to the individual. For a long time the cause of this disorder has been questioned by scientists and by the 1960s the serotonin hypothesis of depression arose, postulating that an imbalance in serotonin levels in the brain results in a depressive disorder. To date, depression has been diagnosed subjectively by observing a set of symptoms in patients. This project is a step towards identifying objective measurable biological differences between data from control subjects and depressed subjects. To this aim, time series measurements of in vivo basal serotonin levels in the hippocampus of control mice and mice depressed via the Chronic Mild Stress (CMS) protocol were used for a quantitative analysis. While the basal serotonin level of CMS mice tended to be lower than that of control mice, neither the average basal level nor several other criteria were sufficient to distinguish CMS from control data. A geometric data analysis approach was proposed, related to topological data analysis of functions using the sub-level set filtration. This geometric approach extracts features of the time series that capture dynamic differences between CMS and control data and is sufficient to distinguish this data.

Committee: Janet Best (Advisor); Adriana Dawes (Committee Member) Subjects: Biology; Mathematics

Doctor of Philosophy, The Ohio State University, 2019, Mathematics

In this thesis, we study second order fully nonlinear elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We mainly focus on the elliptic equations with Neumann boundary condition. We derive oscillation bounds for solutions under the assumption of existence of certain C-subsolutions. We use a parabolic approach to derive the solutions of above elliptic equations. We obtain a priori C^2 estimates for parabolic equations with Neumann boundary conditions. We require some geometric assumptions of background manifolds to derive the second order estimates for non-uniformly-elliptic equations. We obtain long-time existence and uniform convergence results, through which we derive the solutions of above elliptic equations. In the appendix we derive a parabolic Harnack inequality for linear uniformly parabolic operators with vanishing Neumann boundary condition (it does not require any curvatures assumption), which is essential for the uniform convergence results. In application, we apply the Neumann problem for the elliptic equations above to study the Calabi-Yau type problem for compact Kahler manifolds with smooth boundary.

Committee: Bo Guan (Advisor); Barbara Keyfitz (Committee Member); King Yeung Lam (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2019, Mathematics

In this thesis we extend techniques from additive combinatorics to the setting of harmonic analysis and geometric measure theory. We focus on studying the distribution of three-term arithmetic progressions (3APs) within the supports of singular measures in Euclidean space. In Chapter 2 we prove a relativized version of Roth's theorem on existence of 3APs for positive measure subsets of pseudorandom measures on R and show that a positive measure of points are the basepoints for three-term arithmetic progressions within these measures' supports. In Chapter 3 we combine Mattila's approach to the Falconer distance conjecture with Green and Tao's arithmetic regularity lemma to show that measures on R^d with sufficiently small Fourier transform as measured by an L^p-norm have supports with an abundance of three-term arithmetic progressions of various step-sizes. In Chapter 4 we develop a novel regularity lemma to show that measures on R^d with sufficiently large dimension, as measured by a gauge function, must either contain non-trivial three-term arithmetic progressions in their supports or else be structured in a specific quantitative manner, which can be qualitatively described as, at infinitely many scales, placing a large amount of mass on at least two distinct cosets of a long arithmetic progression.

Committee: Vitaly Bergelson (Advisor); Alexander Leibman (Committee Member); Krystal Taylor (Committee Member) Subjects: Mathematics

Master of Science, The Ohio State University, 2018, Civil Engineering

The design of residential structures has changed and evolved throughout years of research based on numerical modelling of real-world conditions. Wood design is controlled heavily by member use, baseline material load capacities, and wood species,all of which determine material properties. Extensive work has been done to determine the effects of altering material properties due to environmental stimuli, however, certain types of decay and wood rot have yet to be fully tested and understood. Certain microbial organisms, given the right conditions, can cause irreversible damage to wood structures. A particularly critical mode of failure is premature column collapse which is driven in part by degraded cross sectional and material properties. Using stiffness relationships smartly programmed into matrix form, it is possible to calculate reduced buckling capacities for these degraded members to draw some important conclusions. Two important items are to determine what degrees of wood decay are critical and what column heights are particularly susceptible to this type of premature failure. Finally, specifications and code should be developed further to assist engineers make choices about the criticality of wood decay without the use of personalized software. This can be done using pre-generated aids to help designers make smarter and more cost effective choices.

Committee: Natassia Brenkus PhD (Advisor); Anthony Massari PhD (Committee Member); Nan Hu PhD (Committee Member) Subjects: Civil Engineering

Doctor of Philosophy (PhD), Ohio University, 2018, Biological Sciences (Arts and Sciences)

Crurotarsi are a clade of archosauromorphs ranging in age from the Middle Triassic to Recent that includes two semiaquatic, faunivorous subclades: Crocodylia and the predominantly late-Triassic Phytosauria. Phytosaurs and crocodylians exhibit generally similar overall body morphology, and each exhibits a range of narrow to broad rostral cranium morphotypes. These morphological similarities lead to the commonly adopted hypothesis that the two clades exhibited a number of ecological and behavioral similarities. One such hypothesized ecological similarity is that phytosaurs utilized a range of food item types that was the same as that utilized by extant crocodylians. In particular, phytosaurs possessing slender rostra with a high aspect ratio were previously hypothesized to have been strictly or primarily piscivorous, much like extant crocodylians with slender, high aspect ratio rostra that have been described as piscivorous. However, a review of available literature reporting on direct observations and other dietary data in extant crocodylians revealed that no extant crocodylian taxa are either strictly piscivorous or lacking at least one population that consumes teleosts as a primary food source. Instead of being correlated with consumption of a specific food type, then, rostrum morphology in extant crocodylians appears to be correlated with the relative size of food items that can be consumed by a given individual. Cranium shape, jaw musculature, and biomechanical performance were assessed in both phytosaurs and extant crocodylians to test hypotheses of morphological and functional similarities of the cranium between these two clades. Results of these analyses were interpreted in the context of prey:predator size ratios correlating with rostrum morphology in extant crocodylians in order to better constrain inferred diet ranges and variation among phytosaur taxa. In general, phytosaurs were more similar to crocodylians with very high aspect ratio rostra in most facet (open full item for complete abstract)

Committee: Patrick O'Connor Ph.D. (Advisor); John Cotton Ph.D. (Committee Member); Shawn Kuchta Ph.D. (Committee Member); Susan Williams Ph.D. (Committee Member); Lawrence Witmer Ph.D. (Committee Member) Subjects: Anatomy and Physiology; Animals; Biology; Biomechanics; Ecology; Evolution and Development; Paleoecology; Paleontology

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.

Committee: Fangyang Zheng (Advisor); Bo Guan (Committee Member); King-Yeung Lam (Committee Member); Jean-Francois Lafont (Committee Member); Mario Miranda (Committee Member) Subjects: Mathematics

Master of Science in Engineering (MSEgr), Wright State University, 2014, Mechanical Engineering

An improved spatial statistical approach and probabilistic prediction method for mistuned integrally bladed rotors is proposed and validated with a large population of rotors. Prior work utilized blade-alone principal component analysis to model spatial variation arising from geometric deviations contributing to forced response mistuning amplification. Often, these studies considered a single rotor measured by contact probe coordinate measurement machines to assess the predictive capabilities of spatial statistics through principal component analysis. The validity of the approach has not yet been demonstrated on a large population of mistuned rotors representative of operating fleets, a shortcoming addressed in this work. Furthermore, this work improves the existing predictions by applying principal component methods to sets of airfoil (rotor) measurements, thus effectively capturing blade-to-blade spatial correlations. In conjunction with bootstrap sampling, the method is validated with a set of 40 rotors and quantifies the subset size needed to characterize the population. The work combines a novel statistical representation of rotor geometric mistuning with that of probabilistic techniques to predict the known distribution of forced response amplitudes.

Committee: Joseph C. Slater Ph.D., P.E. (Advisor); Jeffrey M. Brown Ph.D. (Committee Member); J. Mitch Wolff Ph.D. (Committee Member); Ha-Rok Bae Ph.D. (Committee Member) Subjects: Aerospace Engineering; Engineering; Mechanical Engineering

Doctor of Philosophy, The Ohio State University, 2004, Industrial and Systems Engineering

There are usually two concerns for die casting designers, thermal characteristics and fill pattern because they are closely related to casting quality and die life. The traditional way to obtain the results is numerical simulation. However, due to the high computational cost, numerical simulation is not a perfect tool during the early stages of product development. In this study, a quick algorithm to compute the equilibrium temperature of the die and ejection temperature of the part is presented. The equilibrium temperature is defined as the time average temperature over a cycle after the process reaches the quasi steady state. This can help the cycle and die cooling/heating design. A few models to compute the heat released from part are tested and the combined asymptotic and surrogate model is applied. Special attention is paid to heat transfer calculation at the part-die interface and computational efficiency improvement. The algorithm also addresses the modeling of cooling/heat line, spray effects and techniques for die splitting at the parting line. The algorithm has been implemented in the software CastView based on the finite difference method. The previous algorithm used in CastView for fill pattern analysis based on geometric reasoning is redesigned. In this qualitative method, the flow behavior is calculated using the cavity geometric information. Many shortcomings in the old algorithm were fixed and improved. The new algorithm includes considerations which affect the flow behavior, such as flow resistance, more flow angle search and influence within neighborhood. Special attention is paid to computational efficiency improvement. The fill pattern algorithm for die casting process is adapted for slow fill processes including gravity casting and squeeze casting. The dominant term for flow behavior for different process is defined from dimensionless Navier-Stokes equations. Based on this analysis, the fill pattern algorithm for die casting is modified for slow (open full item for complete abstract)

Committee: R. Miller (Advisor) Subjects: Engineering, Industrial

Master of Science (MS), Ohio University, 2000, Mechanical Engineering (Engineering)

This thesis focuses on the forging metal forming process, and newly found techniques of geometric analysis that pertain to this operation. This project is part of an "ongoing development of a design environment that integrates models for materials and processes and allows selection and optimization of materials and manufacturing processes for components such as those used in aircraft structures and engines." The forging of axisymmetric turbine-engine disks similar to those used in aircraft engines is the focus of this project. The objective of this project was to create low-fidelity models that deliver reasonable results in quick, cost-effective manner. Models were created using Matlab® that can greatly aid in the ability to run the simulation and optimization based design of multi-stage manufacturing processes. This project adds to the above-mentioned work, the ability to compute the minimum offset and the mutual volumetric discretization of axisymmetric disk profiles. These programs deal with "a new process design method for controlling microstructures and mechanical properties through the optimization of preform and die shapes." The models presented here will be used to perform the "simulation of a metalforming process similar to those used for the manufacturing of turbine disks." Such a system will "allow the evaluation, with respect to quality, performance, and cost of alternate materials, processes, and process parameters for the affordable manufacturing of reliable components."

Committee: Bhavin Mehta (Advisor) Subjects: Engineering, Mechanical