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One way that �鶹Ӱ�� provides opportunities for students to excel is through research opportunities. Below is a list of recent summer research projects conducted by students in the Math Department.

2019

Optimization of Inhomogeneous Buckled Rods and Plates

Alexa Bayangos ’21; Advisor: Ghassam Y. Sarkis​

Optimizing eigenvalues of biharmonic equations has many applications including frequency control of rods and plates based on density distribution and maximizing the critical buckling load. This research aims to find the optimal cross-sectional area of an elastic column with given length and volume to maximize the critical buckling load.  By using finite difference methods, we solved the forward problem on domains in one and two dimensions. We then evaluated the accuracy and robustness of our numerical approach by comparing our results to previous analytical studies. Lastly, we developed a numerical algorithm to identify the optimal cross-sectional area function which maximizes the k-th eigenvalue on any given general domain.

Existence of Solutions of Piecewise Linear, Two-Point Boundary Value Problems

Corina Oroz ’20; Advisor: Adolfo Rumbos​

A two-point boundary value problem (BVP) consists of an ordinary differential equation together with conditions at the end points of an interval. The goal of this project is to determine conditions that guarantee the existence of solutions of the piecewise linear, two-point BVP. We first study the linear BVP. In the homogeneous case, the linear BVP has nontrivial solution if and only if the lambda we are interested in is an eigenvalue. The set of eigenvalues is called the spectrum of the linear problem. In the nonhomogeneous case, the BVP has a unique solution if and only if lambda is not in the spectrum. If lambda is in the spectrum, the linear problem is solvable if and only if f is orthogonal to the eigenspace associated with lambda. For the piecewise linear problem, in the case in which f is equal to zero, we introduce the concept of the Dancer-Fucik Spectrum. This is the set of curves in which the piecewise linear problem has nontrivial solutions. We compute these curves. Understanding the Dancer-Fucik Spectrum will allow us to study the nonhomogeneous, piecewise linear BVP. Our goal is to determine the conditions that will guarantee existence of solutions in this case.

How to Write a Textbook: Representation Theory For Undergraduates

Bella Senturia ’20; Advisor: Gizem Karaali​

Representation Theory, often considered a graduate school course with multiple prerequisites, is generally not taught at an undergraduate level. When it is, the few textbook options can be inaccessible to students who haven’t taken numerous graduate-level preparation classes. Professor Karaali is writing an undergraduate Representation Theory textbook with only linear and abstract algebra prerequisites. The goal is that this book be accessible to students in their first years of mathematical training. I served as a student consultant in the textbook development process, combing through the current draft and giving Professor Karaali opinions and feedback on things as specific as her word choice and as general as thoughts about the structure of chapters or her style of mathematical writing. My previous representation theory course enabled me to give opinions as a student with previous material exposure for the first, more completed half of her manuscript. The second half was mostly concepts I had not seen before and I provided the perspective of a student seeing the material for the first time, trying to learn it directly from the book without assistance of a class curriculum or instruction. I had never had the chance to learn about or be involved with the textbook-writing process. My previous math courses gave me extensive exposure to various math textbook types, which coupled with my efficacy in conveying complex ideas, helped Dr. Karaali work towards book completion.

Oscillations of a Human Ponytail

Rafa Matinez-Avial Palazuelos ’21; Advisor: Dwight Whitaker

When a jogger runs down the street, her head moves up and down, yet her ponytail swings from side to side. This phenomenon was the focus of my research for 10 weeks in Cambridge, UK. The study of oscillations of human ponytails began in 2010 with Joseph Keller’s article “Ponytail Motion.” My research consisted of studying the models derived in the paper, extending them and quantifying their accuracy via a setup involving an oscillating support and an attached ponytail.

Understanding the theory of these models involved investigating partial differential equations, Floquet Theory and Perturbation Theory. I followed Keller’s analysis and concluded that the motion of the ponytail behaved according to the Mathieu Equation. To determine if lateral motion would occur, I needed to know the frequency and amplitude of motion of the head attached to the ponytail, as well as some parameters reflecting the ponytail’s material properties.

My work in the lab was threefold. I first looked at Keller’s models and determined two parameters of the ponytail: its drag coefficient and its natural frequency of oscillation. Secondly, I conducted experiments to obtain the values of these parameters. Finally, I used these values and the Mathieu Equation to discover that there existed a qualitative, not quantitative, agreement with the models. In other words, the real values of the parameters which cause lateral instability seem to agree with the models only up to a multiplying factor.’

Inversion of Laplace transform of a linear combination of point masses

Adam Guo ’22; Advisor: Adolfo Rumbos​

The objective of this research project is to investigate a new approach for the inverse problem of finding a finitely supported measure on the real line given samples of its Laplace transform. The problem appears, for example, in the analysis of magnetic resonance relaxometry data. This is a notoriously ill-posed problem. Existing algorithms in the literature require a prior knowledge of the number of points in the support of the measure. We propose an alternative method to determine this number as well as to determine the measure itself.

During our summer project, we used the samples of the Laplace transform at the zeros of the so-called Hermite polynomial of sufficiently large degree. Our project aimed at overcoming a number of numerical instability issues arising from the ill-posed nature of the problem.

We implemented code using the ongoing work of Mhaskar and Zhuang for a fast computation of Gauss quadrature formulas based on the zeros of Hermite polynomials, with time requirements analogous to those of the Fast Fourier Transform. With these, the data can be used efficiently to produce a power spectrum. Chui and Mhaskar have given an algorithm that can be used to determine in one stroke both the number of points in the support of the measure as well as the locations of points in this support. A comparison of the height of the peaks with the on-diagonal values of a reference spectrum yields the weights that the measure attaches to each of the points in its support.

Towards a Database of Belyi Maps

Myles Ashitey ’22 and Brian Bishop ’22; Advisor: Edray Goins​

A Belyĭ map β: P1(C) → P1(C) is a rational function with at most three critical values; we

may assume these are {0, 1, ∞}. A Dessin d’Enfant is a planar bipartite graph on the sphere

obtained by considering the preimage of a path between two of these critical values, usually

taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: β−1   ([0,1]) Í P1 (C) ~ S2(R). This project sought to either create or expand on a database of such Belyĭ pairs, their corresponding Dessins d’Enfant, and their monodromy groups. We did so for up to degree N = 5 in the hopes of generating an algorithm to generate Dessins from monodromy triples.

Analysis of Time Course RNA-Sequencing Data in T. brucei and E. coli

Ethan Ashby ’21; Advisor: Johanna Hardin

Time course RNA-Seq experiments permit study of dynamic transcriptomic responses to stimuli. However, temporal correlations inherent in time course experiments limit the application of conventional differential expression (DE) algorithms. Two time course RNA-Seq datasets were run through a thoughtfully-constructed differential expression pipeline combining two-sample and serial tools, and visualization and downstream analyses were conducted to address biological questions. The first dataset was a 150 minute time course RNA-Seq experiment of E. coli undergoing cell starvation. Analysis of the timing parameters of fitted sigmoid models to differentially-expressed gene (DEG) profiles demonstrated that graded sensitivity to the stress-responsive transcription factor, RpoS, did not describe the global transcriptomic trend in response to cell starvation. The second dataset was a 14 day time course RNA-Seq experiment of the insect-form of the protozoal parasite, Trypanosoma brucei, upon chemical knockout of an important class of chromatin-reader proteins. Gene ontology analysis of DEG clusters suggest these chromatin-readers play roles involved in parasite motility, glycosomal metabolism, and the maintenance of cytosolic stress granules in the insect-form of the parasite. These findings are suggestive of a possible developmental change in the parasite’s life cycle.

2017

Asymptotic Behavior of Mimic Numbers

Matthew Patterson ’19; Advisor: Ami Radunskaya; Collaborators: Jacob Fontana ’19, Oscar Galindo ’19

The mimic function maps multiples of a given divisor represented in a given base to lesser multiples of that same divisor. Iteration of this function allows one to recursively divide numbers, and the number of iterations required to reduce a given number to its divisor is its mimic number. In our project, we investigated the boundedness and asymptotic behavior of mimic numbers, as well as the conditions under which the mimic function is operable, by running computer simulations for sample data and following an analytic methodology in our proofs. We concluded that mimic numbers are unbounded and have an asymptotic growth rate of double logarithmic order. In a special case, we proved that mimic numbers corresponding to multiples of B-1 or B+1, where B is the fixed base, have an asymptotic growth rate of super-logarithmic order. We also slightly expanded the conditions under which the mimic function is operable. Finally, we began constructing a mimic counting function, that is, a function that inputs a mimic number and outputs the number of natural numbers with that mimic number, to study the density of mimic numbers over the naturals. There is potential for future research in these directions and in the cryptographic application of mimic numbers.
Funding Provided By: Rose Hills Foundation SURP Grant, The Class of 1971 Summer Undergraduate Research Fund

A mean-field model for C. elegans embryo localization

Erin Angelini ’18; Advisor: Blerta Shtylla

Cell polarization is an essential process for differentiating cells. In early C. elegans embryos, this polarization is conducted by the precise localization of the pronuclear complex. The developing mitotic spindle drives the rotation and translation of the pronuclei through the forced-based interactions of its composite microtubules and the cell cortex. Building on previous models, both dynamic and steady state, we introduce a steady-state model that captures a reduced version of the pronuclear dynamics. By fitting this model to data from a dynamic computational model, we find that cell geometry determines both the final orientation of the spindle and how long it takes the system to reach this steady state. Moreover, the system tends to a steady state in the regions of the cell with highest curvature. Thus, our results indicate that the cell geometry necessary for proper pronuclear localization is highly specific and curvature-driven.
Funding Provided By: General SURP Fund

Modeling Transient Population and Crime Dynamics in LAPD Pacific Division

Amina Abdu ’18; Advisor: Ami Radunskaya

Although many studies have modeled general patterns of crime, we anticipate qualitative differences between various subpopulations, which have yet to be examined. Because of Los Angeles's large homeless population, the interaction between homelessness and crime is of particular interest in modeling the city's crime. Using data from the LAPD's Pacific Division and the Los Angeles Homeless Services Authority, this paper modifies methods used to study general crime in order to model transient crime dynamics and develops new models specific to the homeless population. Using the fixed and moving window methods, we test for event dependence for exact-repeat and near-repeat arrests. We develop a parametric model that describes the number of times transients are arrested and their recidivism rates. Motivated by spectral clustering and journey to crime models, we construct a probability distribution for the location of crime with kernel density estimation. We create a stochastic model, which predicts transient movement under ordinary circumstances and in the event of an encampment's dispersal. Our work is among the first to study the dynamics of transient crime as distinct from general crime. Much work remains on this subject, but we are able to characterize differences between crime dynamics within the transient and the general population.
Funding Provided By: Evelyn B. Craddock McVicar Memorial Fund

RpoS-Regulated Gene Expression Patterns in E. coli

Madison Hobbs ’19; Advisor: Johanna Hardin

There is biological interest in how RpoS-regulated gene expression in E. coli varies with RpoS level. Varying the level of RpoS, a key regulator of the bacteria’s general stress response, beyond knockout and wild type was largely unexplored prior to the work of Dr. Stoebel’s lab. Our research focuses on genes’ sensitivity to different RpoS levels, categorizing them by the slope of gene expression as RpoS increases. Dr. Stoebel's lab conducted an experiment with three RpoS levels (0%, 26%, 100%), and we classified genes using significance testing. This summer, with six RpoS levels (0%, 0.35%, 20.40%, 48.37%, 129.96%, 190.38%), we have compared our results to the prior classification and developed new ways to group genes by expression shape. We start by assigning genes using correlation to one of the six shapes identified in the study with three RpoS levels, but to get a better sense of the new observed expression shapes, we also implement Partitioning Around Medoids (Maechler et al. 2017) clustering analysis. We primarily identify sensitive positive, sensitive negative, linear positive, and non-monotonic shapes. A deeper level of RNA-sequencing and other concerns must be addressed however before definitively drawing conclusions from the six-level RpoS data.
Funding Provided By: HHMI

2015

3 Mathematicians, 400 Years, 1 Tradition

LeCuk Peral ’18; Mentor: Shahriar Shahriari; Collaborator: Malyq McElroy ‘18

While the word "algebra" may now evoke memories of early schooling, hundreds of years ago advances in the subject were vital to the development of global trade and mathematics as a whole. We studied the works of 3 mathematicians in this field spanning 6 centuries from the 700s to the late 1200s. The three mathematicians, the Persian Al-Khwarizmi, the Egyptian Abu-Kamil, and the Italian Leonardo of Pisano (better known as Fibonacci) all worked on similar problems and championed the use of the Hindu-Arabic numeric system. Delving into their works---as part of a larger project on dissemination of knowledge---we sought to figure out why many of their worked-out problems were nearly identical, considering the amount of time that separates them, and what this meant for the development of algebra in the medieval period.
Funding Provided By: Pomona Unrestricted (McElroy), Richter (Peral)

A Walk Through the Woods: Growing Random Forests for Big Data

John Bryan ’16; Mentor: Johanna Hardin: Collaborator: Yenny Zhang ‘17

The Bag of Little Bootstraps (BLB) is a relatively new methodology that is designed to implement bootstrapping on large datasets. It relies on the bootstrap method, a way of estimating sampling distributions that are not well-characterized by statistical theory. BLB utilizes parallel computing architectures and a multinomial method to drastically reduce computational costs while preserving the appropriate level of error and statistical correctness. The multinomial method is used to resample to a given original sample size N while only using data vectors of size B << N. We combined the ideas of BLB with Random Forests (RF), which is a type of widely-used model for classification and prediction. Applying BLB to RF required that we study and modify the original randomForest R package. The original package lacked the capability to employ a multinomial method like BLB, so we implemented the multinomial method into the package. Through these modifications, we constructed a novel randomForest package that performs the new algorithm. Looking to the future, we plan to run simulations using this new algorithm to assess the reduction in computational cost as well as the algorithm’s reliability in producing confidence estimates of the predicted values. Additionally, we hope to investigate the infinitesimal jackknife estimator to measure the variability of the estimates. Ultimately, we hope to produce a BLBRF algorithm that is optimized for employing RF models in big data situations.
Funding Provided By: Richter (Bryan), �鶹Ӱ�� Mathematics Department (Zhang)

Alternating Means: The Impact of Noise in the Logistic Family

Austin Wei ’18; Mentors: Ami Radunskaya and Johanna Hardin

The logistic map is a one-parameter model for population growth with limited resources. We can model the effects of noise in the environment by replacing the fixed parameter value with a random value chosen according to a distribution centered at that value. We call this the stochastic logistic map. The question arises of whether the introduction of noise tends to reduce or increase the long term average of successive iterations of the logistic map. We show that the introduction of randomness decreases the mean value for parameter values where the deterministic map has an attracting fixed point. Surprisingly, however, the mean value actually increases when the deterministic map has an attracting period 2 cycle. We further conjecture that this alternation of optimal means continues through the period doubling bifurcations of the deterministic logistic map.
Funding Provided By: Fletcher Jones

False discovery rate control in a two-step dependent filtering procedure for differential gene expression analysis

Ciaran Evans ’16; Mentors: Johanna Hardin and Daniel Stoebel (HMC)

Analysis of genetic data to investigate differential gene expression often involves performing thousands of hypothesis tests. When multiple tests are performed, it is necessary to control the quantity of false positives that are produced, so that the test results are meaningful. The standard error rate used in differential expression analysis is the false discovery rate (FDR), which is the expected ratio of the number of false positives to the total number of tests declared significant. A common goal of research in this field is to increase the power of multiple-testing procedures to detect true differences while maintaining FDR control. A collection of proposed methods to increase power involve data-based approaches to filtering out true null hypotheses. The method we consider in this research uses a two-step testing procedure for differential gene expression, in which a global test is performed to test for any differences in gene expression between all experimental conditions, and then pairwise tests to determine the nature of differential expression are performed on those genes for which the global null hypothesis is rejected. To attain FDR control, an estimate of the distribution of p-values in the second step of the procedure is required. We examine a mixture-model approach to estimate this distribution, using a mixture of beta distributions which is estimated through the Expectation-Maximization (EM) algorithm, and investigate its ability to aid in FDR control.
Funding Provided By: Howard Hughes Medical Institute

Parity Biquandle Invariants of Virtual Knots

Leo Selker ’17; Mentor: Sam Nelson (CMC); Collaborator: Aaron Kaestner (North Park University)

Virtual knots are a generalization of classical knots. We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle $2$-cocycle $\phi^0$ and a map $\phi^1$ with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants.
Funding Provided By: �鶹Ӱ�� Mathematics Department

What’s in a Name? Defining Quantitative Literacy

Edwin Villafane Hernandez ’18; Mentor: Gizem Karaali; Collaborator: Jeremy Taylor ‘18

This project aims to bring together different threads in the eclectic literature that make up the scholarship around the theme of Quantitative Literacy. In investigating the meanings of terms like "quantitative literacy", "quantitative reasoning" and "numeracy", we seek common ground, common goals and aspirations of a community of practitioners, and common themes. A decade ago, these terms were relatively new; today accrediting agencies are using them and inserting them in general education conversations. Having good, representative, and perhaps even compact and easily digestible definitions of these terms might come in handy in public relations contexts as well as in other situations where practitioners need to communicate their goals and practice to others (journalists, policy makers, funding agencies) and even assess their own success (how do you measure something if you cannot even define it?).
Funding Provided By: Pomona Unrestricted (Villafane Hernandez)