Physical Sciences and Mathematics Commons^™

Combinatorial Problems Related To Optimal Transport And Parking Functions, Jan Kretschmann

Theses and Dissertations

In the first part of this work, we provide contributions to optimal transport through work on the discrete Earth Mover's Distance (EMD).We provide a new formula for the mean EMD by computing three different formulas for the sum of width-one matrices: the first two formulas apply the theory of abstract simplicial complexes and result from a shelling of the order complex, whereas the last formula uses Young tableaux. Subsequently, we employ this result to compute the EMD under different cost matrices satisfying the Monge property. Additionally, we use linear programming to compute the EMD under non-Monge cost matrices, giving an …

Enumerative Problems Of Doubly Stochastic Matrices And The Relation To Spectra, Julia A. Vandenlangenberg

Theses and Dissertations

This work concerns the spectra of doubly stochastic matrices whose entries are rational numbers with a bounded denominator. When the bound is fixed, we consider the enumeration of these matrices and also the enumeration of the orbits under the action of the symmetric group.

In the case where the bound is two, we investigate the symmetric case. Such matrices are in fact doubly stochastic, and have a nice characterization when we are in the special case where the diagonal is zero. As a central tool to this investigation, we utilize Birkhoff's theorem that asserts that the doubly stochastic matrices are …

The Vanishing Discount Method For Stochastic Control: A Linear Programming Approach, Brian Hospital

Theses and Dissertations

Under consideration are convergence results between optimality criteria for two infinite-horizon stochastic control problems: the long-term average problem and the $\alpha$-discounted problem, where $\alpha \in (0,1]$ is a given discount rate. The objects under control are those stochastic processes that arise as (relaxed) solutions to a controlled martingale problem; and such controlled processes, subject to a given budget constraint, comprise the feasible sets for the two stochastic control problems.

In this dissertation, we define and characterize the expected occupation measures associated with each of these stochastic control problems, and then reformulate each problem as an equivalent linear program over a …

Collapsibility And Z-Compactifications Of Cat(0) Cube Complexes, Daniel L. Gulbrandsen

Theses and Dissertations

We extend the notion of collapsibility to non-compact complexes and prove collapsibility of locally-finite CAT(0) cube complexes. Namely, we construct such a cube complex $X$ out of nested convex compact subcomplexes $\{C_i\}_{i=0}^\infty$ with the properties that $X=\cup_{i=0}^\infty C_i$ and $C_i$ collapses to $C_{i-1}$ for all $i\ge 1$.

We then define bonding maps $r_i$ between the compacta $C_i$ and construct an inverse sequence yielding the inverse limit space $\varprojlim\{C_i,r_i\}$. This will provide a new way of Z-compactifying $X$. In particular, the process will yield a new Z-boundary, called the cubical boundary.

Non-Hyperbolic Right-Angled Coxeter Groups With Menger Curve Boundary, Cong He

Theses and Dissertations

We find a class of simplicial complexes as nerves of non-hyperbolic right-angled Coxetergroups, with boundary homeomorphic to the Menger curve. The nerves are triangulations of compact orientable surfaces with boundary. In particular, the nerves are non-graphs.

Random Quotients Of Hyperbolic Groups And Property (T), Prayagdeep Parija

Theses and Dissertations

What does a typical quotient of a group look like? Gromov looked at the density model of quotients of free groups. The density parameter $d$ measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for $d<1/2$, the typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov's result to show that for $d<1/2$, the typical quotient of many hyperbolic groups is also non-elementary hyperbolic.

Żuk and Kotowski--Kotowski proved that for $d>1/3$, a typical quotient of a free group has Property (T). We show that (in a closely related density model) for $1/3

An Optimal Decay Estimation Of The Solution To The Airy Equation, Ashley Scherf

Theses and Dissertations

In this thesis, we investigate the initial value problem to the Airy equation \begin{align} \partial_t u + \partial_{x}^3 u &= 0\\ u(0,x) &= f(x). \end{align}

Explorations In Baseball Analytics: Simulations, Predictions, And Evaluations For Games And Players, Katelyn Mongerson

Theses and Dissertations

From statistics being reported in newspapers in the 1840s, to present day, baseballhas always been one of the most data-driven sports. We make use of the endless publicly available baseball data to build models in R and Python that answer various baseball- related questions regarding predicting and optimizing run production, evaluating player effectiveness, and forecasting the postseason. To predict and optimize run production, we present three models. The first builds a common tool in baseball analysis called a Run Expectancy Matrix which is used to give a value (in terms of runs) to various in-game decisions. The second uses the …

Modeling Wlan Received Signal Strengths Using Gaussian Process Regression On The Sodindoorloc Dataset, Fabian Hermann Josef Fuchs

Theses and Dissertations

While any wireless technology can be used for indoor localization purposes, WLANhas the advantage of having a huge existing infrastructure. A radio map that matches specific locations to received signal strength is needed, to enable most of these indoor localization methods. To create these radio maps, with enough detail to achieve sufficient localization accuracy, is expensive and time consuming. Therefore, methods to interpolate and extrapolate more detailed maps from sparse radio maps are being developed. One recent approach is to use Gaussian process regression. Even though some papers already studied Gaussian process regression, most studied only the basic model with …

Applying The Efficiency Gap To Wisconsin Politics, Joseph Robert Szydlik

Theses and Dissertations

Gerrymandering is a plague on modern democracy, blatantly violating the democratic principle of “one person, one vote.” Here we will methodically examine the 2018 Wisconsin state assembly election, and using a metric known as the efficiency gap demonstrate the extent to which gerrymandering played a role. Through this metric, and a probabilistic simulation of our own, we will show that in this election the Republican party benefited from systematic partisan gerrymandering. Additionally, we will use these findings to suggest methods for correcting this undemocratic practice that both parties utilize in order to disenfranchise opposition voters.

Modeling Brain Tumor Dynamics With The Help Of Cellular Automata, Paula Kathrin Viktoria Jaki

Theses and Dissertations

Primary brain tumors pose a serious threat to a person’s health. Gaining a deeper understanding of the dynamics of tumor growth is crucial for developing a proper treatment plan. Many computational models have been suggested to investigate the interaction between tumor cells and their surroundings. Using cellular automata is particularly promising since it integrates the features of self-organizing complex systems which allows them to properly depict local interactions between cells. The foundation of the thesis lies in the model proposed by Kansal et al. It uses four microscopic parameters, the maximum tumor extent, the base proliferative and necrotic thickness as …

Comparative Study Of Variable Selection Methods For Genetic Data, Anna-Lena Kubillus

Theses and Dissertations

Association studies for genetic data are essential to understand the genetic basis of complex traits. However, analyzing such high-dimensional data needs suitable feature selection methods. For this reason, we compare three methods, Lasso Regression, Bayesian Lasso Regression, and Ridge Regression combined with significance tests, to identify the most effective method for modeling quantitative trait expression in genetic data. All methods are applied to both simulated and real genetic data and evaluated in terms of various measures of model performance, such as the mean absolute error, the mean squared error, the Akaike information criterion, and the Bayesian information criterion. The results …

The Earth Mover's Distance Through The Lens Of Algebraic Combinatorics, William Quentin Erickson

Theses and Dissertations

The earth mover's distance (EMD) is a metric for comparing two histograms, with burgeoning applications in image retrieval, computer vision, optimal transport, physics, cosmology, political science, epidemiology, and many other fields. In this thesis, however, we approach the EMD from three distinct viewpoints in algebraic combinatorics. First, by regarding the EMD as the symmetric difference of two Young diagrams, we use combinatorial arguments to answer statistical questions about histogram pairs. Second, we adopt as a natural model for the EMD a certain infinite-dimensional module, known as the first Wallach representation of the Lie algebra su(p,q), which arises in the Howe …

Spline Modeling And Localized Mutual Information Monitoring Of Pairwise Associations In Animal Movement, Andrew Benjamin Whetten

Theses and Dissertations

to a new era of remote sensing and geospatial analysis. In environmental science and conservation ecology, biotelemetric data recorded is often high-dimensional, spatially and/or temporally, and functional in nature, meaning that there is an underlying continuity to the biological process of interest. GPS-tracking of animal movement is commonly characterized by irregular time-recording of animal position, and the movement relationships between animals are prone to sudden change. In this dissertation, I propose a spline modeling approach for exploring interactions and time-dependent correlation between the movement of apex predators exhibiting territorial and territory-sharing behavior. A measure of localized mutual information (LMI) is …

A Study Of Machine Learning Techniques For Dynamical System Prediction, Rishi Pawar

Theses and Dissertations

Dynamical Systems are ubiquitous in mathematics and science and have been used to model many important application problems such as population dynamics, fluid flow, and control systems. However, some of them are challenging to construct from the traditional mathematical techniques. To combat such problems, various machine learning techniques exist that attempt to use collected data to form predictions that can approximate the dynamical system of interest. This thesis will study some basic machine learning techniques for predicting system dynamics from the data generated by test systems. In particular, the methods of Dynamic Mode Decomposition (DMD), Sparse Identification of Nonlinear Dynamics …

Design Optimal Health Insurance Policies From Multiple Perspectives, Lianlian Zhou

Theses and Dissertations

The majority of the literature about moral hazard focuses only on qualitative studies. If a health insurance plan imposes little copayment on the insured, the insured may be motivated to have more than necessary medical services, which would raise the insurer’s share of cost. This is referred to as moral hazard. Furthermore, the involvement of a third party–healthcare providers adds more complications on moral hazard. Healthcare providers and patients might choose to collaborate to benefit more from insurance reimbursement, which consequently result in unnecessary loss of the insurer. In this dissertation, we attempt to solve these issues and focus on …

Coarse Cohomology Of The Complement And Applications, Arka Banerjee

Theses and Dissertations

John Roe [15] introduced the notion of coarse cohomology of a metric space to studylarge scale geometry of the space. Coarse cohomology of a metric space roughly measures the way in which uniformly large bounded set in that space fit together. In the first part of this dissertation, we describe a joint work with Boris Okun that generalizes Roe’s theory to define coarse (co)homology of complement of any given subspace in a metric space. Inspired by the work of Kapovich and Kleiner [12], we introduce a notion of a manifold like object in the coarse category (called coarse PD(n) space) …

Robust Estimation Of Ornstein-Uhlenbeck Parameters, Timon Sebastian Kramer

Theses and Dissertations

The standard estimators of the parameter of the Ornstein-Uhlenbeck process are vulnerable to contamination in the data sets. In this thesis more robust estimators for the parameter of the Ornstein-Uhlenbeck process are proposed which use medians instead of means. The scaling for these estimators is more complex and numerical methods must be used. A possible numerical implementation is described. The performance of the standard estimators and the proposed robust estimators are compared on data sets with different levels of contamination and different kind of errors. This thesis shows that the proposed robust estimators can be considerably better than the standard …

Theoretical And Computational Modeling Of Contaminant Removal In Porous Water Filters, Aman Raizada

Theses and Dissertations

Contaminant transport in porous media is a well-researched problem across many scientific and engineering disciplines, including soil sciences, groundwater hydrology, chemical engineering, and environmental engineering. In this thesis, we attempt to tackle this multiscale transport problem using the upscaling approach, which leads to the development of macroscale models while considering a porous medium as an averaged continuum system.

First, we describe a volume averaging-based method for estimating flow permeability in porous media. This numerical method overcomes several challenges faced during the application of traditional permeability estimation techniques, and is able to accurately provide the complete permeability tensor of a porous …

Regime-Switching Jump Diffusion Processes With Countable Regimes: Feller, Strong Feller, Irreducibility And Exponential Ergodicity, Khwanchai Kunwai

Theses and Dissertations

This work is devoted to the study of regime-switching jump diffusion processes in which the switching component has countably infinite regimes. Such processes can be used to model complex hybrid systems in which both structural changes, small fluctuations as well as big spikes coexist and are intertwined. Weak sufficient conditions for Feller and strong Feller properties and irreducibility for such processes are derived; which further lead to Foster-Lyapunov drift conditions for exponential ergodicity. Our results can be applied to stochastic differential equations with non-Lipschitz coefficients. Finally, an application to feedback control problems is presented.

Two Counting Problems In Geometric Triangulations And Pseudoline Arrangements, Ritankar Mandal

Theses and Dissertations

The purpose of this dissertation is to study two problems in combinatorial geometry in regard to obtaining better bounds on the number of geometric objects of interest: (i) monotone paths in geometric triangulations and (ii) pseudoline arrangements.

\medskip(i) A directed path in a graph is monotone in direction of $\mathbf{u}$ if every edge in the path has a positive inner product with $\mathbf{u}$. A path is monotone if it is monotone in some direction. Monotone paths are studied in optimization problems, specially in classical simplex algorithm in linear programming. We prove that the (maximum) number of monotone paths in a …

The Gini Index In Algebraic Combinatorics And Representation Theory, Grant Joseph Kopitzke

Theses and Dissertations

The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the "Lorenz curve" of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we will define a Gini index on the set of integer partitions and prove some combinatorial results related to it; culminating in the proof of an identity for the expected value of the Gini index. …

Applications Of A U-Net Variant Neural Network: Image Classification For Vegetation Component Identification In Outdoors Images And Image To Image Translation Of Ultrasound Images, Adam Honts

Theses and Dissertations

Convolutional Neural Networks have been applied in many image applications, for both supervised and unsupervised learning. They have shown their ability to be used in an array of diverse use cases which include but are not limited to image classification, segmentation, and image enhancement tasks. We make use of Convolutional Neural Networks' ability to perform well in these situations and propose an architecture for a Convolutional Neural Network based on a network known as U-Net. We then apply our proposed network to two different tasks, a vegetation classification task for images of outdoors environment, and an image to image translation …

Exploring The Division Algorithm In Euclidean Domains With Exploding Dots, Nicholas Johnson

Theses and Dissertations

We will give an overview of the representation of place value and arithmetic known as Exploding Dots and use this idea to explore the division algorithm. It is well-known that the ring of integers, the ring of polynomials, and the ring of Gaussian integers are all examples of Euclidean domains and therefore possess a division algorithm. Exploding Dots beautifully illustrates how one can perform division in any base and how this naturally leads us to division of polynomials. We will show how this same idea of having a “base machine” can be used to perform division in the Gaussian integers. …

Examining Virtual Mathematics Instruction: A Comparative Case Study Of In-Service Elementary Teachers With Mathematics Anxiety And Mathematics Teaching Self-Efficacy, Telashay Swope-Farr

Theses and Dissertations

Mathematics Anxiety (MA) and Mathematics Teaching Self-Efficacy (MTSE) have been reported as factors related to teachers’ mathematics instruction. This study investigated MA and MTSE in in-service elementary teachers’ virtual mathematics instruction. A comparative case study design was used to understand the relationship between MA, MTSE, and their virtual mathematics instructional practices. Two in-service elementary teachers from an urban public charter school district in a large metropolitan city in the Midwest participated. I employed qualitative methods to examine the results from the Abbreviated Mathematics Anxiety Rating Scale (AMAS), an adapted version of a researcher-developed instrument called the Mathematics Teaching and Mathematics …

Development Of Novel Compound Controllers To Reduce Chattering Of Sliding Mode Control, Mehran Rahmani

Theses and Dissertations

The robotics and dynamic systems constantly encountered with disturbances such as micro electro mechanical systems (MEMS) gyroscope under disturbances result in mechanical coupling terms between two axes, friction forces in exoskeleton robot joints, and unmodelled dynamics of robot manipulator. Sliding mode control (SMC) is a robust controller. The main drawback of the sliding mode controller is that it produces high-frequency control signals, which leads to chattering. The research objective is to reduce chattering, improve robustness, and increase trajectory tracking of SMC. In this research, we developed controllers for three different dynamic systems: (i) MEMS, (ii) an Exoskeleton type robot, and …

Asymptotic Expansion Of The L^2 Norms Of The Solutions To The Heat And Dissipative Wave Equations On The Heisenberg Group, Preston Walker

Theses and Dissertations

Motivated by the recent work on asymptotic expansions of heat and dissipative wave equations on the Euclidean space, and the resurgent interests in Heisenberg groups, this dissertation is devoted to the asymptotic expansions of heat and dissipative wave equations on Heisenberg groups. The Heisenberg group, $\mathbb{H}^{n}$, is the $\mathbb{R}^{2n+1}$ manifold endowed with the law $$(x,y,s)\cdot (x',y',s') = (x+x', y+y', s+ s' + \frac{1}{2} (xy' - x'y)),$$ where $x,y\in \mathbb{R}^{n}$ and $t\in \mathbb{R}$. Let $v(t,z)$ and $u(t,z)$ be solutions of the heat equation, $v_{t} - \mathcal{L} v=0$, and dissipative wave equation, $u_{tt}+u_{t} - \mathcal{L}u =0$, over the Heisenberg group respectively, where …

The Second Law Of Thermodynamics And The Accumulation Theorem, Austin Maule

Theses and Dissertations

In Serrin's proof of the Accumulation Theorem, the presence of an ideal gas G is assumed.

In 1979 at the University of Naples, Serrin (allegedly) proved that the ideal system G can be replaced by a more general ideal system and still have the Accumulation Theorem hold.

In this paper, we attempt to reconstruct Serrin's proof and supply a proof for a more general theorem stated in a paper of Coleman, Owen and Serrin.

Asymptotic Probability Of Incidence Relations Over Finite Fields, Adam Buck

Theses and Dissertations

Given four generic lines in FP3, we ask, "How many lines meet the four?" The answer depends on the field. When F = C, the answer is two. When F = R, the answer is either zero or two.

If we work over a finite field there are only finitely many projective lines. We compute the probability four lines are met by two. The main result is that as q approaches infinity, this probability approaches 1/2. Asymptotically, the other half of the time zero lines will meet the four.

Local Connectedness Of Bowditch Boundary Of Relatively Hyperbolic Groups, Ashani Dasgupta

Theses and Dissertations

If the Bowditch boundary of a finitely generated relatively hyperbolic group is connected, then, we show that it is locally connected. Bowditch showed that this is true provided the peripheral subgroups obey certain tameness condition. In this paper, we show that these tameness conditions are not necessary.