Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Albert Schueller

• Author: Elliot Granath

Abstract: In an elementary calculus course, we talk mostly, or exclusively, about in- tegrating continuous, real-valued functions. Since continuous functions on closed intervals are integrable, the Fundamental Theorem of Calculus gives us a method to calculate these integrals (given that we can find an antiderivative). Furthermore, the Fundamental Theorem of Calculus states that the integral can be used to define an antiderivative of a continuous function. In this pa- per, we will present a method for establishing the existence of antiderivatives of continuous functions without using any integration theory. In addition, we will explore the potentially counter-intuitive topic of derivatives which are not Riemann integrable. It is easy to find a function whose derivative is un- bounded, and thus not Riemann integrable; what is more surprising is that even bounded derivatives are not necessarily Riemann integrable. We will present two classes of functions, one conceived by Volterra and one by Pom- peiu, which are differentiable on closed intervals, and whose derivatives are not Riemann integrable. Finally, we will develop the Henstock integral as a tool which integrates all derivatives.

• Author: Tom Howe

Abstract: The permutations of the 15 puzzle have been a point of focus since the 1880's when Sam Lloyd designed a spin-off of the puzzle that was impossible to solve. In this paper, we explore which permutations of the 15 puzzle are obtainable by utilizing properties of permutations and results from graph theory. We begin our investigation by using brute force to obtain elementary permutations of the puzzle followed by simple proofs to show that exactly half of the permutations of the 15 puzzle may be obtained. While this approach is sufficient to demonstrate the properties of the 15 puzzle, a more elegant proof using properties of simple graphs yields us results that may be extrapolated onto other similar-style puzzles that may be represented as simple graphs. The portion of this paper that centers around graph theory is an exposition of Richard Wilson's 1973 paper "Graph Puzzles, Homotopy, and the Alternating Group".

• Author: Colin McCarthy

Abstract: Agent based Modeling in biological systems is a relatively new approach that has been used increasingly in examining environments that are difficult to describe through traditional statistical models. Studies in ecology have been particularly eager to employ this technique as the emphasis on the agents allow for studying conservation efforts with measurable effects on the population as a whole. In this paper, we showcase an agent based model of the process of salmon returning to their freshwater spawning ground as they pass through the Lower Snake River. Based on a model originally developed by Amy Steimke, we have updated the model to encompass more recent fish run data and provide a statistical analysis of the model that was not in Steimke’s original project. In the process of working with the model, we produced a distribution that suggested that the dam objects were causing the salmon population to spread out, a dimension of difficulties to salmon reproduction that is not captured purely in the threat to life that dams pose. After quantifying this observed spread, we developed an addition to the model for opportunity-based reproduction to capture the effects of such a spread on the reproductive success of the salmon population. We find that the submodel results suggest a clear impact of dams upon the reproductive success. This significant finding leads us to conclude with potential implications and expansions to the work presented here.

Abstract: Neural networks are powerful mathematical tools used for many purposes including data classification, self-driving cars, and stock market predictions. In this paper, we explore the theory and background of neural networks before progressing to different applications of feed-forward and auto-encoder neural networks. Starting with the biological neuron, we build up our understanding of how a single neuron applies to a neural network and the relationship between layers. After introducing feed-forward neural networks, we generate the error function and learn how we minimize it through gradient decent and backpropagation. Applying this, we provide examples of feed-forward neural networks in generating trend lines from data and simple classification problems. Moving to regular and sparse auto-encoders, we show how auto-encoders relate to the Singular Value Decomposition (SVD), as well as some knot theory. Finally, we will combine these examples of neural networks to discuss deep learning, as well as look at some examples of training networks and classifying data with these stacked layers. Deep learning is at the forefront of machine learning with applications in AI, voice recognition, and other advanced fields.

• Author: Lexi Perez

Abstract: In multiple linear regression models, covariates are sometimes correlated with one another. Multicollinearity can cause parameter estimates to be inaccurate, among many other statistical analysis problems. When these problems arise, there are various remedial measures we can take. Principal component analysis is one of these measures, and uses the manipulation and analyzation of data matrices to reduce covariate dimensions, while maximizing the amount of variation.

• Author: Andzu Schaefer

Abstract: Our approach to the topic of P vs NP is intuitive with regards to describing the classes P and NP, as well as to describing the inherent computational nature of these classes and the nature of NP-completeness. We also discuss and categorize sub-optimal solutions, as well as `progress in the field' such as proofs concerning proofs and the impact of other fields of study. We conclude by summarizing the far reaching consequences of both P$=$NP and P$\neq$NP. We do all this in a manner that is detailed enough for readers to move on to more formal sources.

• Author: Zongli Shi

Abstract: This paper is an introduction to graphics programming. This is a computer science field trying to answer questions such as how we can model 2D and 3D objects and have them displayed on screen. Researchers in this field are constantly trying to find more efficient algorithms for these tasks. We are going to look at basic algorithms for modeling and drawing line segments, 2D and 3D polygons. We are also going to explore ways to transform and clip 2D and 3D polygons. For illustration purpose, the algorithms presented in this paper are implemented in C++ and hosted at GitHub. Link to the GitHub repository can be found in the introduction paragraph.

• Author: Emma Twersky

Abstract: This paper is an exploration into centroidal Voronoi tessellations, or CVTs. A centroidal Voronoi tessellation is defined, and we specifically focus on 2-point CVTs. We also explore creating visualizations of the algorithms which generate CVTs, including the implementations of MacQueen’s and Lloyd’s methods. These visualizations are written using p5.js, hosted online at carrot.whitman.edu/CVT. We explore the algorithms and prove a method to test point inclusion in a convex polygon using the intersection of half planes. Finally, we explore characteristics and properties of 2-point CVTs in order to develop conjectures about their structure, including examining the stability of these methods and variations of these conjectures using alternative distance metrics.

• Author: Shengjun Wang

Abstract: Centuries ago, a French mathematician Henri Lebesgue noticed that the Riemann Integral does not work well on unbounded functions. It leads him to think of another approach to do the integration, which is called Lebesgue Integral. This paper will briefly talk about the inadequacy of the Riemann integral, and introduce a more comprehensive definition of integration, the Lebesgue integral. There are also some discussion on Lebesgue measure, which establish the Lebesgue integral. Some examples, like $F_{\sigma}$ set, $G_{\delta}$ set and Cantor function, will also be mentioned.