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 …

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 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 …

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 …