Physical Sciences and Mathematics Commons^™

Spotting K-Tricaps In Spot It!, Jenarah Skyara Lara

Senior Projects Spring 2023

The card game “SPOT IT!” consists of 55 cards, with 8 symbols appearing on each card. Every pair of cards has exactly one symbol in common, and the goal of the game is to be the first person to find this symbol. An alternate way to play the game is to find sets of 3 cards that have the same symbol in common. We will use combinatorics, probability, and finite projective geometry to analyze the structure of the game. The game “SPOT IT!” can be viewed as the projective plane of order 7. However, we can construct a similar game …

Modeling Vascular Diffusion Of Oxygen In Breast Cancer, Tina Giorgadze

Senior Projects Spring 2023

Oxygen is a vital nutrient necessary for tumor cells to survive and proliferate. Oxygen is diffused from our blood vessels into the tissue, where it is consumed by our cells. This process can be modeled by partial differential equations with sinks and sources. This project focuses on adding an oxygen diffusion module to an existing 3D agent-based model of breast cancer developed in Dr. Norton’s lab. The mathematical diffusion module added to an existing agent-based model (ABM) includes deriving the 1-dimensional and multi-dimensional diffusion equations, implementing 2D and 3D oxygen diffusion models into the ABM, and numerically evaluating those equations …

Untouchable Money And Impossible Clones: Applications Of Quantum Picturalism And Zx-Calculus, Shea A. Roccaforte

Senior Projects Spring 2021

Quantum Picturalism allows a new technique for researchers and students alike in the areas of quantum computation and quantum information. This picturalistic method represents fundamental math concepts and quantum theory in a diagrammatic manner. This method is a high-level language that allows for the exploitation of quantum weirdness. Using these techniques, quantum processes and the composition of those processes are highlighted as a structure referred to as process theory. Viewing these processes in a purely diagrammatic language allows for an unambiguous universal language for qubits, and the manipulation of these diagrams is referred to as ZX-calculus. These concepts allow for …

Gibbs Phenomenon For Jacobi Approximations, Riti Bahl

Senior Projects Spring 2021

The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series "overshoots" (and "undershoots") the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials P_n^(a,b)(x). In particular, we show that for …

Electronic Properties Of Flat And Curved Graphene Sheets, Deng Yanpei

Senior Projects Spring 2021

This paper explored the electronic properties of the graphene sheet and also developed basis for understanding the electronic properties of the curved graphene sheet. This paper began with setting up basic knowledge about solid-state physics including introducing band structure, band gap, crystal structure, and reviews for quantum mechanical operators. Then this paper described two potential models that are suitable for considering periodic potential: the weak potential and the tight-binding model. We discovered the tight-binding model is better for our graphene case and by applying this model we find the energies of the graphene sheet. Next, we constructed the 1D and …

Chase-Escape On Sparse Networks, Emma Sylvie Bernstein

Senior Projects Spring 2020

Chase-escape is a competitive growth process in which prey spread through an environment while being chased and consumed by predators. The environment is typically modeled by a graph—such as a lattice, tree, or clique—and the species by particles competing to occupy sites. It is arguably more natural to study these dynamics in heterogeneous environments. To this end, we consider chase-escape on a canonical sparse random graph called the Erdo ̋s-R ́enyi graph. We show that if prey spreads too slowly then both species quickly die out. On the other hand, if prey spreads fast enough, then coexistence occurs. Concrete bounds …

Determining Tone Of A Body Of Text, Cole G. Hollant

Senior Projects Spring 2020

We will be looking into emotion detection and manipulation within a body of text based off of Robert Plutchik’s basic emotions. This project encompasses building probabilistic and lexical models, full-stack web development, and dataset creation and application. We will build our models off of Latent Dirichlet Allocation—a grouping model common in natural language processing (nlp) and lexicons compiled through crowdsourcing. User testing is undergone as a means of measuring the effectiveness of our models. We discuss the application of concepts and technologies including MongoDB, REST APIs, containerization, IaaS, and web frontends.

Time And Finance: Exploring Variance In The Black-Scholes Model, Edward Chase Skorupa

Senior Projects Spring 2019

In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published mathematical methods they had devised with the goal of accurately pricing European options. When using the model to predict future options prices, all input variables in the model can be empirically viewed, and calculated, at present time except for the future volatility of the underlying security. Retrospectively analyzing the volatility implied by the Black-Scholes model using price history shows that this implied volatility is an inaccurate estimate of actual future volatility. This project sought to explore the relationship between the implied future volatility …

Predicting How People Vote From How They Tweet, Rao B. Vinnakota

Senior Projects Spring 2019

In 2016 Donald Trump stunned the nation and not a single pollster predicted the outcome. For the last few decades, pollsters have relied on phone banking as their main source of information. There is reason to believe that this method does not present the complete picture it once did due to several factors--less landline usage, a younger and more active electorate, and the rise of social media. Social media specifically has grown in prominence and become a forum for political debate. This project quantitatively analyzes political twitter data and leverages machine learning techniques such as Naive-Bayes to model election results. …

Gerrymandering And The Impossibility Of Fair Districting Systems, Danielle Degutz

Senior Projects Spring 2019

Voting district boundaries are often manipulated, or gerrymandered, by politicians in order to give one group of voters an unfair advantage over another during elections. To make sure a system of voting districts is not gerrymandered, the population size, the shape, and the voting efficiency of each party in each district should be taken into consideration. Following recent work of Boris Alexeev and Dustin G. Mixon, we discuss mathematical criteria for each of these three aspects, and we prove how problems arise when attempting to apply all three at once to a districting system--first to a simplified districting system and …

The Mathematics Of Cancer: Fitting The Gompertz Equation To Tumor Growth, Dyjuan Tatro

Senior Projects Spring 2018

Mathematical models are finding increased use in biology, and partuculary in the field of cancer research. In relation to cancer, systems of differential equations have been proven to model tumor growth for many types of cancer while taking into account one or many features of tumor growth. One feature of tumor growth that models must take into account is that tumors do not grow exponentially. One model that embodies this feature is the Gomperts Model of Cell Growth. By fitting this model to long-term breast cancer study data, this project ascertains gompertzian parameters that can be used to predicts tumor …

Concerning The Construction Of Four-Bar Linkages And Their Topological Configuration-Spaces, Peter K. Servatius

Senior Projects Spring 2018

For a given linkage with one degree of freedom we can analyze the coupler curve created by any selected tracer point in relation to a driver link. The Watt Engine is a four-bar linkage constructed such that the tracer point draws an approximate straight line along a section of the coupler curve. We will explore the family of linkages that are created using Watt's parameters, along with linkages designed by other inventors; looking at methodologies of creating a linkage and the defining what we mean by approximate straight-line motion. Ultimately we will be creating our own linkage using what we …

Supersymmetry And The Math Of Adinkras, Ari Putt Spiesberger

Senior Projects Fall 2018

The purpose of this paper is to expand the dictionary of values related to parameterized supersymmetry values. These values, represented by Adinkras, are some of the most fascinating explanations of theoretical supersymmetry that exist. My goal was to approach and define an equivalence class on a specific value that had yet to be defined. I was able to do this, and in doing so, present information on a larger equivalence class in the field surrounding Adinkras.

Hyperplanes Equipartition With Cascading Makeev, Jialin Zhang

Senior Projects Spring 2018

Given a finite number of masses in the Euclidean space, one could ask is it possible to equipartition these masses into equal parts. Fixing the collection of masses, and the amount of hyperplanes, the equipartition-ability depends on the dimension, and there exists a dimension of such equipartition is possible. In this paper, topology and combinatorics method are used for estimating the lower bound and upper bound of the dimension. In particular, we are looking equipartition problem together with Cascading Makeev Constrain.

Quantifying The Effect Of The Shift In Major League Baseball, Christopher John Hawke Jr.

Senior Projects Spring 2017

Baseball is a very strategic and abstract game, but the baseball world is strangely obsessed with statistics. Modern mainstream statisticians often study offensive data, such as batting average or on-base percentage, in order to evaluate player performance. However, this project observes the game from the opposite perspective: the defensive side of the game. In hopes of analyzing the game from a more concrete perspective, countless mathemeticians - most famously, Bill James - have developed numerous statistical models based on real life data of Major League Baseball (MLB) players. Large numbers of metrics go into these models, but what this project …

The Expectation For The Center Of Mass Of Finite Integer Grids, Finnegan Maximilan Muller Hardy

Senior Projects Spring 2017

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

Modeling Purple Sea Urchin And California Sheephead Populations In Southern California Kelp Forests, Olivia Rachel Williams

Senior Projects Spring 2017

In this project I am modelling the predator-prey relationship between California sheephead and purple sea urchin populations, respectively, in kelp forests off the coast of southern California. The Lotka-Volterra equations explain predator-prey relationships in their most basic form. These equations incorporate a set of biological assumptions that can be unrepresentative of many ecological systems. I will consider alternate models that incorporate variations of the Lotka-Volterra model which may better represent the biology of the purple sea urchins and California sheephead. Using biological characteristics of both species in kelp forests, I will set possible and likely parameters and solve for unknown …

From Random To Organized: The Architecture Of Neural Networks During Development, Avery Isabella Morris

Senior Projects Fall 2017

The brain is constantly changing during development as a result of various stimuli: memories, language, visual patterns and other sensory information. As a result, networks need to have specific learning rules to function being both plastic and stable. In this project, I’ve constructed a mathematical model based on a biological neural network during development. I’ve written differential equations to describe these specific learning rules as well as methods of visual input to the network. I’ve changed my model, using Euler’s method, to create a discrete-time version of this biological phenomenon to implement on the computer. I’ve successfully coded this, using …

Radical Recognition In Off-Line Handwritten Chinese Characters Using Non-Negative Matrix Factorization, Xiangying Shuai

Senior Projects Spring 2016

In the past decade, handwritten Chinese character recognition has received renewed interest with the emergence of touch screen devices. Other popular applications include on-line Chinese character dictionary look-up and visual translation in mobile phone applications. Due to the complex structure of Chinese characters, this classification task is not exactly an easy one, as it involves knowledge from mathematics, computer science, and linguistics.

Given a large image database of handwritten character data, the goal of my senior project is to use Non-Negative Matrix Factorization (NMF), a recent method for finding a suitable representation (parts-based representation) of image data, to detect specific …

The Facilitation Of Sound Waves Using Mathematical And Scientific Methods Of Digital Signal Processing, Benjamin Rocco Moss

Senior Projects Spring 2016

Mathematics and music have been lifelong partners since the beginning of time. Rhythm and time are two fundamental aspects of music which rely solely on counting, an under-appreciated skill in mathematics, yet recognized by all mathematically-minded people as the foundation of some of the most important mathematical findings; as John B. Fraleigh would say, “Never underestimate a theorem that counts something!” However, music recordings have evolved through the use of technology further than merely possessing the capabilities to quantify and archive the notes that were played in the recording. In the days before digital recordings, the only way to ensure …

Lose Big, Win Big, Sum Big: An Exploration Of Ranked Voting Systems, Erin Else Stuckenbruck

Senior Projects Spring 2016

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

Complex Semiclassics: Classical Models For Tunneling Using Complex Trajectories, Max Edward Meynig

Senior Projects Spring 2017

This project is inspired by the idea that black holes could explode due to a quantum process somewhat analogous to quantum mechanical tunneling. This idea was presented in recent research that also proposed that semiclassical physics could be used to investigate the so called black hole fireworks. Semiclassical physics connects quantum and classical physics and because of this it is a powerful tool for investigating gravity where the classical theory is known but there is no complete quantum theory. Unfortunately, the traditional tools in semiclassics that are needed fail to treat tunneling. However, if classical mechanics is extended to complex …

Photovoltaics: An Investigation Into The Origins Of Efficiency On All Scales, Jeremy Alexander Bannister

Senior Projects Spring 2016

This project is comprised of a set of parallel investigations, which share the common mo- tivation of increasing the efficiency of photovoltaics. First, the reader is introduced to core concepts of photovoltaic energy conversion via a semi-classical description of the phys- ical system. Second, a key player in photovoltaic efficiency calculations, the exciton, is discussed in greater quantum mechanical detail. The reader will be taken through a nu- merical derivation of the low-energy exciton states in various geometries, including a line segment, a circle and a sphere. These numerical calculations are done using Mathematica, a computer program which, due to …

Exploring A Generalized Partial Borda Count Voting System, Christiane Koffi

Senior Projects Spring 2015

The main purpose of an election is to generate a fair end result in which everyone's opinion is gathered into a collective decision. This project focuses on Voting Theory, the mathematical study of voting systems. Because different voting systems yield different end results, the challenge begins with finding a voting system that will result in a fair election. Although there are many different voting systems, in this project we focus on the Partial Borda Count Voting System, which uses partially ordered sets (posets), instead of the linearly ordered ballots used in traditional elections, to rank its candidates. We introduce the …

Welfare Versus Stability In "Stabilizing An Unstable Economy": A Minskyan Growth Model, Stergios Mentesidis

Senior Projects Spring 2012

The paper focuses on Minsky's financial fragility hypothesis incorporated in a growth model and investigates whether an inherently unstable economy can be stabilized by a big and proactive government. Using dynamical systems theory and expanding a supply-driven growth model developed by Lin, Day and Tse (1992), the paper explores how different government spending programs and financing paths can affect the growth, as well as the stability of a capitalist economy. The results and implications of the new frameworks are analyzed, using analytical and numerical methods of bifurcation, to examine the dependence of optimal government intervention on the economic environment. The …