Learning loss occurs during academic breaks, and this can be detrimental to student success especially in sequential classes like Arizona State University’s Engineering Calculus sequence in which retention of the topics taught in a prior class is expected. The Keeping in School Shape Program (KiSS) is designed as a cost effective, efficient, and accessible way of addressing this problem. The KiSS program uses push technology to give students a way to regularly review material over academic breaks while also fostering a growth mindset.Every day, during an academic break, students are sent a link via text message or email to access a multiple-choice daily review problem which represents material from a previous course that is requisite for success in an upcoming course. Before solving the daily problem, students use a 5-point scale to indicate how confident they are that they can solve the problem. Students then complete the daily review problem and have a variety of resources to support them as they do so, as well as options after they complete it. Students are able to view a hint and try a problem again, view a solution, and attempt a challenge problem. On Tuesdays (aka 2’s-Days) students are given the opportunity to complete either an additional daily review problem or an additional challenge problem, and on Sundays (aka Trivia Days) students can decide between completing only a mathematics trivia question or trivia along with the daily review problem. There is much to be learned from each individual student who participates in the KiSS program. Three surveys were conducted during the Winter Break 2020 KiSS program that gave insight into students’ experience in the KiSS program along with their personal background and mindset regarding mathematics. Ten students responded to all three of these surveys. This thesis will present a case study for each of these ten students based on their data from program participation and survey responses. Conclusions will be drawn regarding ways in which the KiSS program is helping students and ways in which it can be improved to help students be better prepared for their upcoming studies.

A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations.

The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.

Cryptocurrencies have become one of the most fascinating forms of currency and economics due to their fluctuating values and lack of centralization. This project attempts to use machine learning methods to effectively model in-sample data for Bitcoin and Ethereum using rule induction methods. The dataset is cleaned by removing entries with missing data. The new column is created to measure price difference to create a more accurate analysis on the change in price. Eight relevant variables are selected using cross validation: the total number of bitcoins, the total size of the blockchains, the hash rate, mining difficulty, revenue from mining, transaction fees, the cost of transactions and the estimated transaction volume. The in-sample data is modeled using a simple tree fit, first with one variable and then with eight. Using all eight variables, the in-sample model and data have a correlation of 0.6822657. The in-sample model is improved by first applying bootstrap aggregation (also known as bagging) to fit 400 decision trees to the in-sample data using one variable. Then the random forests technique is applied to the data using all eight variables. This results in a correlation between the model and data of 9.9443413. The random forests technique is then applied to an Ethereum dataset, resulting in a correlation of 9.6904798. Finally, an out-of-sample model is created for Bitcoin and Ethereum using random forests, with a benchmark correlation of 0.03 for financial data. The correlation between the training model and the testing data for Bitcoin was 0.06957639, while for Ethereum the correlation was -0.171125. In conclusion, it is confirmed that cryptocurrencies can have accurate in-sample models by applying the random forests method to a dataset. However, out-of-sample modeling is more difficult, but in some cases better than typical forms of financial data. It should also be noted that cryptocurrency data has similar properties to other related financial datasets, realizing future potential for system modeling for cryptocurrency within the financial world.

A thorough understanding of the key concepts of logic is critical for student success. Logic is often not explicitly taught as its own subject in modern curriculums, which results in misconceptions among students as to what comprises logical reasoning. In addition, current standardized testing schemes often promote teaching styles which emphasize students' abilities to memorize set problem-solving methods over their capacities to reason abstractly and creatively. These phenomena, in tandem with halting progress in United States education compared to other developed nations, suggest that implementing logic courses into public schools and universities can better prepare students for professional careers and beyond. In particular, logic is essential for mathematics students as they transition from calculation-based courses to theoretical, proof-based classes. Many students find this adjustment difficult, and existing university-level courses which emphasize the technical aspects of symbolic logic do not fully bridge the gap between these two different approaches to mathematics. As a step towards resolving this problem, this project proposes a logic course which integrates historical, technical, and interdisciplinary investigations to present logic as a robust and meaningful subject warranting independent study. This course is designed with mathematics students in mind, with particular stresses on different formulations of deductively valid proof schemes. Additionally, this class can either be taught before existing logic classes in an effort to gradually expose students to logic over an extended period of time, or it can replace current logic courses as a more holistic introduction to the subject. The first section of the course investigates historical developments in studies of argumentation and logic throughout different civilizations; specifically, the works of ancient China, ancient India, ancient Greece, medieval Europe, and modernity are investigated. Along the way, several important themes are highlighted within appropriate historical contexts; these are often presented in an ad hoc way in courses emphasizing technical features of symbolic logic. After the motivations for modern symbolic logic are established, the key technical features of symbolic logic are presented, including: logical connectives, truth tables, logical equivalence, derivations, predicates, and quantifiers. Potential obstacles in students' understandings of these ideas are anticipated, and resolution methods are proposed. Finally, examples of how ideas of symbolic logic are manifested in many modern disciplines are presented. In particular, key concepts in game theory, computer science, biology, grammar, and mathematics are reformulated in the context of symbolic logic. By combining the three perspectives of historical context, technical aspects, and practical applications of symbolic logic, this course will ideally make logic a more meaningful and accessible subject for students.

Honey bees (Apis mellifera) are responsible for pollinating nearly 80\% of all pollinated plants, meaning humans depend on honey bees to pollinate many staple crops. The success or failure of a colony is vital to global food production. There are various complex factors that can contribute to a colony's failure, including pesticides. Neonicotoids are a popular pesticide that have been used in recent times. In this study we concern ourselves with pesticides and its impact on honey bee colonies. Previous investigations that we draw significant inspiration from include Khoury et Al's \emph{A Quantitative Model of Honey Bee Colony Population Dynamics}, Henry et Al's \emph{A Common Pesticide Decreases Foraging Success and Survival in Honey Bees}, and Brown's \emph{ Mathematical Models of Honey Bee Populations: Rapid Population Decline}. In this project we extend a mathematical model to investigate the impact of pesticides on a honey bee colony, with birth rates and death rates being dependent on pesticides, and we see how these death rates influence the growth of a colony. Our studies have found an equilibrium point that depends on pesticides. Trace amounts of pesticide are detrimental as they not only affect death rates, but birth rates as well.

In large part, the great success of eusocial insects is due to efficient division of labor (Duarte et al. 2011; Dornhaus 2008). Within ant colonies, the process of dividing labor is not clearly defined, but it may be key to understanding the productivity and success of these colonies. This study analyzed data from an experiment that was conducted with the goal of examining how finely division of labor is organized in ant colonies. The experiment considered the actions of all ants from three Temnothorax rugatulus colonies. The colonies were each carefully recorded during five distinct emigrations per colony. The experiment produced such a large quantity of data that it was challenging to analyze the results, a major obstacle for many studies of collective behavior. Therefore, I designed a computer program that successfully sorted all of the data and prepared it for an initial statistical analysis that was performed in R. The preliminary results suggest that while most of the ants perform little to no work, there is an overall pattern of elitism; it seems that division of labor in ants is not more finely divided than previously shown. Future studies should provide further analysis of the data and will be useful in forming a more complete understanding of the division of labor within the emigrations of Temnothorax rugatulus colonies.

Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.

Computer simulations are gaining recognition as educational tools, but in general there is still a line dividing a simulation from a game. Yet as many recent and successful video games heavily involve simulations (SimCity comes to mind), there is not only the growing question of whether games can be used for educational purposes, but also of how a game might qualify as educational. Endemic: The Agent is a project that tries to bridge the gap between educational simulations and educational games. This paper outlines the creation of the project and the characteristics that make it an educational tool, a simulation, and a game.

A model has been developed to modify Euler-Bernoulli beam theory for wooden beams, using visible properties of wood knot-defects. Treating knots in a beam as a system of two ellipses that change the local bending stiffness has been shown to improve the fit of a theoretical beam displacement function to edge-line deflection data extracted from digital imagery of experimentally loaded beams. In addition, an Ellipse Logistic Model (ELM) has been proposed, using L1-regularized logistic regression, to predict the impact of a knot on the displacement of a beam. By classifying a knot as severely positive or negative, vs. mildly positive or negative, ELM can classify knots that lead to large changes to beam deflection, while not over-emphasizing knots that may not be a problem. Using ELM with a regression-fit Young's Modulus on three-point bending of Douglass Fir, it is possible estimate the effects a knot will have on the shape of the resulting displacement curve.