<p>For a field <i>F</i> and the polynomial ring <i>F</i> [<i>x</i>] in a single indeterminate, we define [special characters omitted] = {&alpha; &isin; End<i>_F</i>(<i>F</i> [x]) : &alpha;(<i>f</i>) &isin; <i>f F</i> [<i>x </i>] for all <i>f</i> &isin; <i>F</i> [<i> x</i>]}. Then [special characters omitted] is naturally isomorphic to <i>F</i> [<i>x</i>] if and only if <i>F</i> is infinite. If <i>F</i> is finite, then [special characters omitted] has cardinality continuum. We study the ring [special characters omitted] for finite fields <i>F.</i> For the case that <i>F </i> is finite, we discuss many properties and the structure of [special characters omitted].</p>

<p> The alcove model of Cristian Lenart and Alexander Postnikov describes highest weight crystals of semisimple Lie algebras in terms of so-called alcove walks. We present a generalization, called the quantum alcove model, which has been related to tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. </p><p> We also investigate Ram's version of Schwer's formula for Hall-Littlewood <i> P</i>-polynomials in type <i>A,</i> which is expressed in terms of the alcove model. We connect it to a formula similar in flavor to the Haglund-Haiman-Loehr formula, which is expressed in terms of fillings of Young diagrams.</p>

<p> Historically, electrostatic forces and capillary surfaces have been a main focus of scientific inquiry. Recently, with the move towards miniaturization in technology, systems that include the interplay of these two phenomena have become more relevant than ever. This is because at small scales, capillary and electrostatic forces come to dominate familiar macro scale forces and consequently, govern the behavior of many components used in modern technology. In particular, these <i>electro-capillary</i> systems have been applied to areas such self-assembly, &ldquo;lab-on-a-chip&rdquo; devices, microelectromechanical systems and mass spectrometry. </p><p> In this dissertation, we study two such systems. The first system involves subjecting a planar soap film to a vertically directed electric field. The second is an extension of the first that includes the small effect of gravity (or, similarly, a constant external pressure). Mathematical models for these systems are developed via variational techniques to describe the equilibrium deflection of the soap-film. In contrast to the standard theory, these models include the full effect of capillarity, yielding two prescribed mean curvature problems. These problems are then studied for general and specific domains, using a combination of analytic, asymptotic and numerical techniques. A detailed analysis of the solution set reveals several interesting bifurcation structures. Highlighted areas include a blow-up in the gradient, which occurs at the onset of strictly parametric solutions, and a prediction of the so-called pull-in instability with respect to the aspect ratio of the system, which provides an update to the standard theory. The work here illustrates the effect of including the mean curvature operator in such models and starts to build a general theory of electro-capillary surfaces.</p>

<p> A closed three manifold invariant Heegaard Floer homology was generalized to bordered Heegaard Floer homology, defined by Robert Lipshitz, Peter Ozsv&aacute;th and Dylan Thurston. Bordered Heegaard Floer homology is an invariant of three manifold with connected boundary, and its variant doubly bordered Floer homology is a bimodule defined on three manifold with two disconnected boundary components. In this thesis, we compute bordered Floer homology of (2,2<i>n</i>)-torus link complement.</p>

<p> We show the existence and uniqueness of an invariant measure for the kick-forced Navier-Stokes system on the 2-dimensional sphere, first without deterministic force and then with a time-independent deterministic force. The existence and uniqueness of an invariant measure for the white noise forced Navier-Stokes system on the 2- dimensional sphere without a deterministic forcing is also shown.</p><p> We examine the support of the invariant measure and give a description of the support of the measure in general, and in several special cases, for the kick-forced flow. The support of the invariant measure for the white noise forced equations is shown to be the entire space of admissible vector fields of the sphere.</p>

<p> Game analytics is a fast growing field where game studios are allocating valuable resources to develop sophisticated statistical models to understand user behavior and monetization habits to optimize game play and performance. Game developers' ability to understand user retention allows for game features that will generate high engagement leading to stronger overall monetization and increased lifetimes of players. </p><p> One important industry adopted metric is the percentage of users who log back into the game one day after installation, otherwise known as a one-day retention. Although this is an important metric, game studios typically allocate little resources to determining what user transactions are typically conducted on the day of installation that drive a one-day retention. </p><p> In this project, we first conduct a cluster analysis in an attempt to uncover meaningful subgroups based on players' transaction history on their first day of installation. Secondly, we use various classification methods including decision trees, logistic regression, and k-Nearest Neighbor algorithm to determine which behaviors are important in identifying whether a new user will return the following day.</p>

<p> The motion of a viscous fluid flow is described by the well-known Navier-Stokes equations. The Navier-Stokes equations contain the conservation laws of mass and momentum, and describe the spatial-temporal change of the fluid velocity field. This thesis aims to investigate numerical solvers for the incompressible Navier-Stokes equations in two and three space dimensions. In particular, we focus on the second-order projection method introduced by Kim and Moin, which was extended from Chorin&rsquo;s first-order projection method. We apply Fourier-Spectral methods for the periodic boundary condition. Numerically, we discretize the system using central differences scheme on Marker and Cell (MAC) grid spatially and the Crank-Nicolson scheme temporally. We then apply the fast Fourier transform to solve the resulting Poisson equations as sub-steps in the projection method. We will verify numerical accuracy and provide the stability analysis using von Neumann. In addition, we will simulate the particles' motion in the 2D and 3D fluid flow.</p><p>

<p> In 2010 it was found that God&rsquo;s number is 20 in the face turn metric. That is, if the Rubik&rsquo;s cube hasn&rsquo;t been disassembled, it can always be solved in 20 twists or fewer, but sometimes requires 20 twists. However, the face turn metric only allows one face to be turned at a time for a total of 18 generators, or 18 possible twists at any time. This dissertation allows opposing, parallel faces to be twisted independent amounts at the same time and still get counted as 1 twist for a total of 45 generators. A new optimal-solving program was constructed, and the results so far show that God&rsquo;s number is at least 16 for the simultaneously-possible turn metric. </p><p> I note that in 3 dimensions the simultaneously-possible turn metric is the same as the axial turn metric (or robot turn metric), but not in 4 dimensions nor higher (e.g. 2&times;2&times;2&times;2, 3&times;3&times;3&times;3, 4&times;4&times;4&times;4, etc.&mdash;not to be confused with the 3-dimensional 4&times;4&times;4 cube). This difference is also described.</p><p>

<p> A new graph partitioning algorithm which makes use of a novel objective function and seeding strategy, Product Cut, frequently outperforms standard clustering methods. The solution strategy on solving this objective depends on developing a fast solution method for the systems of graph--based analogues of the screened Poisson equation, which is a well-studied problem in the special case of structured graphs arising from PDE discretization. </p><p> In this work, we attempt to improve the powerful Algebraic Multigrid (AMG) method and build upon the recently introduced Product Cut algorithm. Specifically, we study the consequences of incorporating a dynamic determination of the diffusion parameter by introducing a prior to the objective function. This culminates in an algorithm which seems to partially eliminate an advantage present in the original Product Cut algorithm's slower implementation. </p><p>

Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide powerful tools to investigate the spatial effects on the time evolutions of the agents' strategy choices. The deterministic equations allow us to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions. We introduce several methods of decomposition of two player normal form games in the third chapter. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal complements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from Rock-paper-scissors type games (in the case of symmetric games) or from Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decompositions by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the Hodge decomposition of vector fields for the replicator equations, (e) constructing Lyapunov functions for some replicator dynamics, (f) constructing Zeeman games—games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS. The hierarchical modeling of evolutionary games provides flexibility in addressing the complex nature of social interactions as well as systematic frameworks in which one can keep track of the interplay of within-group dynamics and between-group competitions. For example, it can model husbands and wives' interactions, playing an asymmetric game with each other, while engaging coordination problems with the likes in other families. In the fourth chapter, we provide hierarchical stochastic models of evolutionary games and approximations of these processes, and study their applications.