Ph.D. - Mathematics
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Item Algebraic and Combinatorial Applications in Systems and Evolutionary Biology(2024) Curiel, Maize; Gross, Elizabeth; MathematicsItem CONCEPT ANALYSIS IN CATEGORIES(2023) Ferrer, Lance; Pavlovic, Dusko; MathematicsItem Geometric Control of a Quadcopter(University of Hawaii at Manoa, 2023) Gray, Christopher; Chyba, Monique; MathematicsIn recent years, with the rise of affordable commercial grade quadcopters, there has been a lot of research done on modeling the motion of quadcopters, however much of it has been focused on creating robust control schemes for quadcopters. This dissertation works to bridge the gap this has left between motion planning and optimal control, with a focus mainly on two areas. First we work towards creating a model of the equations of motion as realistic as possible and connect the work done in geometric control theory in the past on underwater vehicles, to quadcopters. We also study possible simplifications, a specific control scheme and properties of basic motions. Secondly using an affine control model version of the equations of motion we study the time minimization problem with an emphasis on singular extremals. We observe numerically the possibility for optimal control to have an infinite number of switches in a finite time interval.Item Applications at the Intersection of Hybrid Dynamical Systems and Interaction Networks(University of Hawaii at Manoa, 2023) Carney, Richard Stephen; Chyba, Monique; MathematicsWe consider dynamical systems that exist within a larger network i.e. dynamical systems for which we have a set of local dynamics coupled together via pairwise connections. The particular coupling we have in mind relates to an object called the graph Laplacian. We further employ graph theoretic notions as a way of describing potential changes that the dynamical rules themselves may undergo. Specifically, we describe the notion of a hybrid dynamical system. We focus mainly on two different applications. First, we study protocols based on self-organizing principles, which are used to coordinate the movements of a set of multi-agents. These are meant to model a group of autonomous robotic vehicles operating independently of human control. Implementing a coordinated group of unmanned aerial vehicles, for example, is more efficient for many mission types including tracking, surveillance, and mapping. In this regard, there are numerous reasons why collective control of a group of UAVs is important. Second, inspired by the COVID-19 pandemic, we build a large scale model which accounts for coordination between regions each using travel restrictions in order to mitigate the rapid spread of disease. The model is designed via a family of compartmental SEIR models, and using the formalism of hybrid automata. There is a need for simulations of countries cooperating together, as travel restriction policies were taken by countries without global coordination. It is possible, for instance, that a strategy which appears unfavorable to a region at some point of the pandemic might be best for containing the global spread, or that only by coordinating policies among several regions can a restriction strategy be truly effective.Item Mathematical Epidemiology: Modeling Of Infectious Disease(University of Hawaii at Manoa, 2023) Kunwar, Prateek; Chyba, Monique; MathematicsThe ongoing coronavirus pandemic, caused by severe acute respiratory syndrome coronavirus2 was first identified in December 2019. This dissertation falls into the field of mathematical modelling of infectious diseases. It is composed of two parts: modeling the spread of Covid19, with an application to the Hawaiian archipelago; development of classification tools for the comparison of the evolution of the Covid-19 pandemic at different geographic locations. First we present a generalized discrete deterministic compartmental SEIR model for the spread of Covid-19 which incorporates competing variants of the virus, vaccination, fading of vaccine protection, the possibility of a previously infected individual becoming susceptible and travel restrictions. Using this model on the counties of the State of Hawai‘i , we study the impacts of mitigation measures and the impacts of tourism on the spread of the disease among the local population. Second, we focus on some classification tools. The notion of wave is used to describe the evolution of a pandemic but although the terminology is often used, the current literature does not have a precise definition for it. In this dissertation, we provide a mathematical definition for the notion of a wave and present an algorithm to detect waves from a given set of data. In addition, comparison of the evolution in time of different spread of a disease is also not well addressed in the literature. Here, we introduce topological structures associated to data representing the spread of daily cases or hospital data and define an orientation preserving pseudo-metric on them that can be used to compare the evolution of pandemic between different regions.Item On the Distribution of Complexities of Finite Words, with Connections to Constructive Immunity(University of Hawaii at Manoa, 2023) Birns, Samuel; Kjos-Hanssen, Bjørn; MathematicsThis thesis presents results on complexities of finite and infinite words, where ``complexity" is defined using finite state and Turing machines, respectively. Specifically, the complexity of a word $w$ is defined to be the minimum number of states in a finite state machine needed to uniquely output $w$ among all words of length $\abs{w}$. A theorem counting the number of words of length $n$ and deterministic hidden Markov model complexity $q$ is proven, giving insight into the distribution of deterministic hidden Markov model complexity. In particular, this theorem allows for the computation of deterministic hidden Markov model complexity in polynomial time, an improvement over the exponential-time naive computation. Non-deterministic hidden Markov models of various states are shown to be realized by deterministic hidden Markov models, and the strengths of languages defined by hidden Markov model complexity are investigated. Finally, analogs of these results are expanded to infinite words and to finite-state gambler complexity. When discussing complexities of infinite words in the next topic, $\Sigma^0_1$ and constructively $\Sigma^0_1$ dense sets are introduced. It is shown that these classes are distinct and that they occur in non-$\Delta^0_2$ degrees, high degrees, and c.e. degrees. Then, connections from $\Sigma^0_1$ sets to a problem on the computable dense subsets of $\Q$ is established. Finally, results on effective notions of the complexities of finite words are presented, providing a link between the two topics.Item Mathematical modeling of sensing and feeding by copepods(University of Hawaii at Manoa, 2022) Hachmeister, Julian; Takagi, Daisuke; MathematicsThe ability to feed effectively is a fundamental skill required for survival. Microscopic crustaceans called copepods are a great example of an animal that are very adept at eating. They are tiny creatures, roughly on the order of 1 mm, and so their fluid environment is not like the water that we are accustomed to, but rather, is more like that of honey. To make things even more difficult, they have poor eyesight and are only able to detect the presence of light. Instead of using vision to detect the presence of food, they rely on other sensory mechanics like chemical signals and hydro-mechanical disturbances to remotely detect their prey. Another challenge they are presented with is moving a particle to a desired location for inspection and consumption. Given the high viscosity of their environment, particle transport is difficult since small objects near the boundary of a body will tend to stick to and move with the motion of the body. In this dissertation, we first create a model for three modes of feeding: sinking, swimming and hovering. For each of the three modes, we first create the flow fields by including the antennae, a feature often neglected in previous studies. Then, we measure the magnitude of the disturbance vector induced by a spherical particle located in a plane with sensor locations along the antennae. From this, a detectable volume is generated showing what the model could theoretically detect over a given period of time. What we discover is that sinking may be a preferable mode of feeding if the copepod were surrounded by food. If the copepod is unable to detect anything, then swimming might be best as it would increase the copepod's chances of encountering food. If the copepod came across a dense cluster of food, then positioning underneath the cluster could be the best mode as it would allow the copepod to funnel the food from above to its body. In the second and third chapter, we explore the copepod's ability to transport small particles using the Weis-Fogh fling and clap mechanism. Copepods have been observed performing this motion as early as the 1980's but there have been no mathematical models for the fling and clap in the Stokes regime. We investigate the efficacy of the motion by representing a pair of appendages as either a pair of rods or a pair of plates. What we discover is that both representations lead to a positive net displacement of particles in the desired direction and that increasing the maximum slope $a$ of the appendages lead to an increase in the displacement, specifically on the order of $a^2$ for small values of $a$.Item Topological Dynamics in the Study of C*-Algebras: Finite Approximations and Dynamic Dimension(University of Hawaii at Manoa, 2022) Pilgrim, Samantha Jane; Willett, Rufus; MathematicsWe investigate properties of dynamical systems motivated by the crossed product construction, specifically finite approximation properties and the dynamic asymptotic dimension. Chapter 1 provides some motivation and historical context with a more in depth overview of each chapter. In chapter 2, we show that equicontinuous actions on Cantor sets are profinite and that equicontinuous actions by finitely generated groups are residually finite. The latter requires some background on representation theory and Lie theory, which we also provide in this chapter. In chapter 3, we show that equicontinuous actions are quasidiagonal and use this to exhibit new examples of group actions whose crossed products have the MF property. Chapter 4 gives some background on the dynamic asymptotic dimen- sion and related concepts in geometric group theory. We show the dynamic asymptotic dimension of actions on profinite completions is closely related to the asymptotic dimension of the acting group’s box spaces. Chapter 5 contains proofs for Hurewicz-type theorems for the dynamic asymptotic dimension, which we use to describe the asymptotic dimension of box spaces of elementary amenable groups. In chapter 6, we give sharp bounds for the dimension of most isometric actions, and use these to completely describe the dimension of translation actions on compact lie groups in terms of the amenability and asymptotic dimension of the acting group.Item Topologies on Positive Type Functions on Groupoids, Weak Containment of Continuous Unitary Representations, and Property (T)(University of Hawaii at Manoa, 2022) Corea, Kenneth; Willett, Rufus; MathematicsWe introduce two new topologies on the space of normalized positive type functions on a groupoid with Haar system, the weak* topology and fiberwise compact convergence topology. We demonstrate these topologies are equivalent when the groupoid is second countable and locally compact, thereby extending Raikov's theorem. Using the fiberwise compact convergence topology we introduce a notion of weak contaiment of continuous unitary representations of a groupoid. We characterize weak containment of the trivial representation with almost invariant sections which extends the idea of almost invariant vectors in group representations. This weak containment naturally suggests a definition of property (T) for groupoids which extends Kazhdan's group property (T). We briefly explore this property and attempt to generalize some of the classic results from the theory of groups.Item Persistent Cohomology Of Cover Refinements(University of Hawaii at Manoa, 2022) Markovichenko, Oleksandr; Mileyko, Yuriy; MathematicsTopological data analysis (TDA) is a new approach to analyzing complex data which often helps reveal otherwise hidden patterns by highlighting various geometrical and topological features of the data. Persistent homology is a key in the TDA toolbox. It measures topological features of data that persist across multiple scales and thus are robust with respect to noise. Persistent homology has had many successful applications, but there is room for improvement. For large datasets, computation of persistent homology often takes a significant amount of time. Several approaches have been proposed to try to remedy this issue, such as witness complexes, but those approaches present their own difficulties. In this work, we show that one can leverage a well-known data structure in computer science called a cover tree. It allows us to create a new construction that avoids difficulties of witness complex and can potentially provide a significant computational speed up. Moreover, we prove that the persistence diagrams obtained using our novel construction are actually close in a certain rigorously defined way to persistence diagrams which we obtain using the standard approach based on Cech complexes. This quantifiable coarse computation of persistence diagrams has the potential to be used in many applications where features with a low persistence are known to be less important.