# Mathematics

## Intrinsic 2-component linking in complete graphs

**Spencer Johnson (2014); Mentor(s): Erica Flapan**

Abstract: We present an approach to finding the smallest n such that every embedding of the complete graph on n vertices in R^n contains a link of at least two disjoint cycles with linking number greater than one.

*Funding Provided by: Paul K. Richter and Evelyn E. Cook Richter Memorial Fund*

## Expanding DESeq to differential expression analysis of three or more conditions for high-throughput data

**Ciaran Evans (2016); Student Collaborator(s): Garrett Wong (2014 HMC); Additional Collaborator(s): Dan Stoebel (HMC); Mentor(s): Johanna Hardin**

Abstract: The software package DESeq is an important tool in differential gene expression analysis across two conditions. However, so far no one has extended the software to allow simultaneous analysis across three or more conditions. The goal of this project is to use statistical methods to analyze differential expression and overcome the issues that arise when dealing with more than two conditions.

*Funding Provided by: Howard Hughes Medical Institute*

## A robust extension of Sparse Canonical Correlation Analysis for the analysis of genomic data

**Joseph Replogle (2013); Student Collaborator(s): Jake Coleman (2013); Mentor(s): Johanna Hardin**

Abstract: Medical genomics seeks to explain complex phenotypes based on variations in genetic, epigenetic, and environmental elements. To this end, high-throughput genomic and molecular biology technologies generate vast biological datasets that provide for examination of many variables simultaneously. In order to illuminate the mechanisms and pathways underlying human traits and unveil novel therapeutic avenues, creative statistical techniques must help integrate these diverse genetic datasets. Canonical Correlation Analysis (CCA), a statistical method that maximizes the correlation between linear combinations of sets of variables, and particularly Sparse Canonical Correlation Analysis (SCCA), which performs CCA on a small subset of variables extracted using a penalty function, are fruitful techniques for analysis of the complex relationships found in genomic data. Here we extend SCCA using Spearman Rank Correlation to make the method more robust to outliers. We use a combination of simulated and real data to show that our method outperforms previously proposed SCCA methods in the presence of the noisy data commonly found in biology. Additionally, we propose a permutation test for assessing the significance of multiple canonical variates. We hope that our robust SCCA will allow biologists to better characterize the associations between genetic datasets in order to improve understanding of human disease.

*Funding Provided by: Howard Hughes Medical Institute*

## Averages in the Period 2 Region of the Logistic Map

**Maricela Cruz (2014); Mentor(s): Johanna Hardin; Ami Radunskaya**

Abstract: The logistic map is a nonlinear difference equation well studied in literature, used to model reproduction and starvation in certain populations. Here we study the distributional characteristics of the stochastic logistic map giving evidence that the map has a stable distribution over the period 2 region. We use simulations in R to support the claim that regardless of the initial distribution of x (for example, x = population size), the logistic map iterates to a unique stable distribution. That is, after 10,000 iterations of the logistic map we arrive at a unique stable distribution of x. We also examine the relationship between the mean of the stochastic logistic equation and the mean of the deterministic logistic equation for period 2. Our initial results show that the relationship between the two averages in period 2 is opposite of that in period one. In the period 2 case the mean of the stochastic logistic equation is greater than the mean of the deterministic logistic equation. We investigate this relationship as the parameter, λ, changes.

*Funding Provided by: Linares Family SURP*

## Quantum Algorithm for Markov Chain Monte Carlo Methods

**Gillian Grindstaff (2014); Student Collaborator(s): Kevin Wilson (2015 University of Oregon); Mentor(s): Yevgeniy Kovchegov (Oregon State University)**

Abstract: With quantum computers hopefully in development, the field of quantum algorithms is rapidly expanding. In my research I looked into a method of using quantum computation to drastically improve existing algorithms for sampling from a desired probability distribution. We exhibit a transformation that will produce a unitary matrix from a stochastic matrix, in particular matrices implemented in Markov chain Monte Carlo methods, as a means of defining a quantum dynamical system which parallels the Metropolis-Hastings algorithm. For the uniform cyclic walk on $n$ states, we give an explicit formula for the quantum operator which will, with use of a quantum Fourier transform to compute averages, converge on the desired distribution.

*Funding Provided by: National Science Foundation*

## Modeling the Effects of Angiogenesis and Macrophage Phenotype on Glioma Growth.

**Stephen Ragain (2014); Additional Collaborator(s): Lisette DePillis (HMC); Mentor(s): Ami Radunskaya**

Abstract: Glioma is cancer in the central nervous system arising from glial cells. We develop a novel compartment model featuring tumor cells, oxygen concentration, and two phenotypes of macrophages. The model is designed to focus on two major dynamics that affect tumor growth: the effects of angiogenesis as represented by oxygen influx in the model, and a conversion from phagocytic to reparative macrophage phenotype. Without angiogenesis, the tumor quickly depletes available nutrient and cells can no longer proliferate. Without changes to macrophage phenotype, the body's immune response effectively removes the tumor. Of those mechanisms that may potentially be affected by treatment, parameter sensitivity of the model shows that α, how strongly reparative macrophages promote oxygen influx, significantly impacts tumor growth.

*Funding Provided by: Howard Hughes Medical Institute*

## Solvability of non-linear two-point boundary value problems of second order

**Daria Drozdova (2014); Student Collaborator(s): Mary Kamitaki (2015); Mentor(s): Adolfo Rumbos**

Abstract: The goal of this project was to prove existence results for a general class of nonlinear two-point boundary value problems of second order. These problems are interesting from the point of view of the theory of differential equations. They also arise in many situations in the physical sciences and their study is fundamental to understanding the underlying physical problems. In addition to learning the theory of two-point boundary value problems, we went over the article “Boundary value problems for weakly nonlinear ordinary differential equations” by E. N. Dancer (Bulletin of the Australian Mathematical Society, Volume 15, 1976, pp. 321-328), which uses “shooting method” arguments to prove existence of solutions. Concentrating on the problem with Dirichlet boundary conditions, we considered a result in Dancer’s paper in which the nonlinearity is asymptotically linear at infinity and at resonance with respect to the Fucik spectrum. The key result of our project was proving that under certain conditions, the non-linear two-point boundary value problem has at least one solution. The observation that the gaps between zeros of a piecewise linear initial value problem and of the nonlinear initial value problem are small simplifies the task of finding the form of the solution to our problem; this implies that the shooting method can be implemented. The next step in the project is to look at the case in which the nonlinearity grows more than linearly.

*Funding Provided by: Pomona College SURP (DD); National Science Foundation # DMS-1016136 (MK)*