The Claremont Center for the Mathematical Sciences & Pomona College hosted a student poster session on Wednesday, September 13, 2017 in the Millikan Courtyard. The event included 14 Math Posters from students of The Claremont Colleges and abstracts for each poster.
2017 CCMS Poster Session Abstracts
CGU-1
Numerical Study of Stekolv Eigenvalue Problem
Weaam Alhejiali*
joint work with Chiu-Yen Kao (CMC)
We focus on the development of numerical approaches to the forward solver and the shape optimization solver for Steklov eigenvalue problems in two dimensions. This problem has wide range of applications in engineering and applied mathematics. We proposed numerical approaches via spectral methods or finite element methods. To apply spectral methods, we reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes k-th Steklov eigenvalue with a given k.
CGU-2
Brownian Mechanism for Cell Segregation
An Do*
joint work with Blerta Shtylla (Pomona)
The generation of the directed movement in bacterial chromosomal segregation frequently requires the Brownian motion. In Caulobacter crescentus and other bacteria, an active protein is employed to transport the chromosomal matter from one end to another of the cell during the segregation. How precisely this mechanism works still remains unclear. We have developed and analyze a mathematical model to examine the directed moment of the duplicated matter cargo then compared with the biological data for verification. We first employ Latin hypercube to uniformly sample from the parameters sample space to search for their sensitivities to the partition complexes’ average runlength and runspeed throughout the segregation period. Our study examine the effective range of diffusion coefficient and hydrolysis rate to yield the directed movement during the segregation period and also indicates the types and speed of the partition complex’s movement depends on their binding state with another protein. We also study the spatial formation pattern of the cell that closely relate the division site at the end of segregation period. In the future work, we would like to analyze the theoretical model as a system of reaction-advection-diffusion partial differential equations to understand this mechanism without simulation.
CGU-3
Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions
Casey Johnson*
joint work with Marina Chugunova (CGU)
Sturm-Liouville problems with eigenvalue-dependent boundary conditions originate from self-adjoint expansions of non-densely defined symmetric operators. These self-adjoint expansions can be defined in terms of a coupling of the boundary spaces of two symmetric operators. Properties of Nevanlinna-Pick functions are applied to study distribution of eigenvalues. Green’s functions are constructed for various examples of the operators coupling.
CGU-4
Minimization of Inhomogeneous Biharmonic Eigenvalue Problems
Di Kang*
joint work with Chiu-Yen Kao (CMC)
Biharmonic eigenvalue problems arise in the study of the mechanical vibration of plates. In this paper, we study the minimization of the first eigenvalue of a simplified model with clamped boundary conditions and Navier boundary conditions with respect to the coefficient functions which are of bang-bang type (the coefficient functions take only two different constant values). A rearrangement algorithm is proposed to find the optimal coefficient function based on the variational formula of the first eigenvalue. On various domains, such as square, circular and annular domains, the region where the optimal coefficient function takes the larger value may have different topologies. An asymptotic analysis is provided when two different constant values are close to each other. In addition, a symmetry breaking behavior is also observed numerically on annular domains.
CGU-5
Machine Learning Algorithms for Graph-Based Representations of Fracture Networks
Sean Matz*
joint work with Manuel Valera, Priscilla Kelly, Michael Guo, Adrian Cantu, Allon Percus, Jeffrey Hyman, Gowri Srinivasan, and Hari Viswanathan
Microstructural information such as fracture size, geometry, aperture, and material properties plays a key role in modeling the dominant physics for flow propagation through fractured rock. Discrete fracture network (DFN) computational suites, such as DFN Works have recently been developed to simulate flow and transport in such media. The methods allow for particle tracking, revealing a small backbone of fractures through which most transport occurs and therefore providing a significant reduction in the effective size of the fracture network. However, the simulations needed for particle tracking are computationally intensive, and may not be scalable to large systems.
CGU-6
Parallelization of Volume of Fluid Algorithms on Unstructured Meshes
Justin Sunu*
joint work with Donald Kruse, Alonso Navarro, Neil Carlson, and Zach Jibben
The development of advanced manufacturing techniques is a key contributor to US industrial competitiveness and energy efficiency. Conducting experiments on manufacturing processes can be both time consuming and costly, which calls for the use of computational simulation. Truchas is a software suite that is used to simulate a number of different manufacturing methods, such as metal casting and additive manufacturing, with the goal of obtaining crucial design information without performing physical experiments. A key and very time consuming component of Truchas deals with the advection of interfaces separating immiscible fluids. Our research is focused on improving the performance of this advection segment of the code using three optimization methods: implementation of high-level OpenMP parallelism, vectorization, and code restructuring. High-level OpenMP is implemented around computationally intensive routines operating on the entire mesh. Vectorization instructions are introduced in regions where multiple data undergo the same math operations. Lastly, some examples of code restructuring are reorganizing the code to remove idling time and repetitious tasks, and implementing design patterns that facilitate safe and efficient OpenMP and vectorization. Code performance was assessed using Intel Haswell and KNL architecture systems.
CGU-7
Reconstruction of Linear Multifractional Stable Motion
Ran Zhao*
joint work with Qidi Peng (CGU)
We provide a new almost surely uniformly convergent estimator of the Hurst function of the Linear Multifractional Stable Motion (LMSM). LMSM is the stochastic process with non-Gaussian increments. Simulation study shows that our estimator has smaller bias than the ones provided in existing literature. We apply the LMSM in modelling and forecasting financial times series and illustrate its outperformance over conventional models such as multifractional Brownian motion.
CMC-1
Asymptotic Behavior of Mimic Numbers
Ryan Chakmak*
joint work with Asuman Aksoy (CMC)
When discussing best approximation, we are concerned with its existence and uniqueness. For instance on a compact set, we know that the best approximation always exists. The purpose of this project is to ask these questions in the setting of metric trees, instead of an arbitrary space. The literature has already established the conditions on some functions required for the best approximation to be unique. We plan to dive further into the topic, investigating the connections between Chebyshev sets, metric trees, and best approximation.
Pomona-1
A Mean-Field Model for C. Elegans Embryo Localization
Erin Angelini*
joint work with Blerta Shtylla (Pomona)
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.
Pomona-2
Numbers Represented by a Finite Set of Binary Quadratic Forms
Chris Donnay*
joint work with Havi Ellers, Kate O’Connor, Katherine Thompson, and Erin Wood
Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.
Pomona-3
Modeling of Calcium Dynamics During Fertilization in C. Elegans
Alex Xiaotong Gui*
joint work with Kanyarat Jitmana (Pitzer College)
Fertilization induces calcium wave inside oocytes. This wave is important for embryonic development in most animals but mechanisms of how such wave is generated and propagated are not well understood. In this project, we pick Caenorhabditis elegans (C. elegans) as our experimental subject and use computational methods to understand possible mechanisms to propagate Ca2+ wave in C. elegans oocyte during fertilization based on well-studied Xenopus laevis oocyte model. Some information of Xenopus modeling has been analyzed to investigate and find best match to C. elegans Ca2+ wave dynamics.
Pomona-4
Asymptotic Behavior of Mimic Numbers
Matthew Patterson*
joint work with Jacob Fontana (Pomona) and Oscar Galindo (SMC)
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 testing computer simulations 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 cryptographic applications.
Pomona-5
A Statistical Approach to Analyzing Peer-Led Team Learning at CSU Channel Islands: Retention Rates
Lucero Ramirez
joint work with Angel Ramos, Mary DiCciccio, Emmanuel Rivera Guasco, and Dr. Delil Martinez (CSU Channel Islands)
Educational institutions across the country have turned to the Peer-Led Team Learning (PLTL) model to provide academic support to students. This project investigated the impact of the PLTL program at CSU Channel Islands with respect to retention or graduation through an in-depth analysis of masked individual records. Utilizing methods such as logistic regression and hypothesis testing, we found that from Fall 2012 to Spring 2016, college retention/graduation of PLTL participants was higher than that of non-PLTL students. We were also interested in exploring the demographic characteristics of participants in this program.
Pomona-6
Bag of Little Random Forests (BLRF)
Zihao Xu*
joint work with Johanna Hardin (Pomona)
Random Forests are a successful ensemble method that utilizes a number of decision trees to make predictions robust in both regression and classification settings. However, the process of bootstrap aggregation, the mechanism underlying the Random Forest algorithm, requires each decision tree to physically store and perform computations on data sets of the same size as the input training set, a situation that is oftentimes impractical given the humongous sizes of data sets today. To address this problem, we introduce the Bag of Little Random Forests, a new algorithm that adapts the Bags of Little Bootstraps (Kleiner), aiming to achieve a better computational profile while producing predictions with comparable accuracy as those of Random Forest.
Scripps-1
RpoS-Regulated Gene Expression Patterns in E. Coli
Madison Hobbs*
joint work with Jo Hardin (Pomona) and Dan Stoebel (HMC)
There is biological interest in how the gene expression of all RpoS-regulated genes in E. coli change at different levels of RpoS, a protein which regulates the bacteria’s general stress response. Although many studies have investigated the genome-wide effects of turning RpoS on and off, varying the level of RpoS had been largely unexplored prior to the work done in Dr. Stoebel’s lab. Our prior research has focused on the sensitivity of genes to different RpoS levels, for instance whether gene expression is affected even at low levels of RpoS or whether higher RpoS levels are needed before gene expression changes. We design six gene expression profiles: sensitive positive regulation, sensitive negative regulation, insensitive positive regulation, insensitive negative regulation, linear positive regulation, and linear negative regulation. Dr. Stoebel’s lab conducted an experiment with three RpoS levels (0%, 26%, 100), and we classified genes into the six groups above. This summer, with six levels of RpoS (0%, 0.35%, 20.40%, 48.37%, 129.96%, 190.38%), we have been comparing our results to the earlier profile assignments and developing new ways to group genes according to expression shape. The vast majority of genes in the six-level RpoS dataset show a sensitive positive or sensitive negative regulation shape when using the profiles we designed, however the genes are generally poorly correlated and lack differentiability between the six profiles. To get a better sense of the primarily observed expression shapes, we implement Partitioning Around Medoids (Maechler et al. 2017) clustering analysis. The most prevalent shapes in the data identified by Partitioning Around Medoids clustering include shapes that appear sensitive positive, sensitive negative, linear positive, and non-monotonic. A deeper level of sequencing and other concerns regarding technical error need to be addressed however before we draw conclusions from the six-level RpoS data.
Scripps-2
Integrated Metabolic Regulatory Model (IMRM): Leveraging Statistical Uncertainty to Improve Outcome Predictions
Lauren Stanislaw*
joint work with Shihua Wang, Jacob Alfieri, Craig Disselkoen, and Nathan Tintle (Dordt College)
A number of papers have described methods to include transcriptional regulatory networks (TRNs) in the development of metabolic models. However, in general, these models do not allow for statistical uncertainty in TRN interactions. Furthermore, methods which do allow for uncertainty do so in a non-rigorous manner, which effectively downweights prior transcriptomics data to the point where it has little impact on the resulting model. To combat these limitations, we have developed a novel integrated metabolic regulatory model (iMRM). This novel approach explicitly models the statistical uncertainty in gene state activity inferences by utilizing gene activity state estimates. This allows for benefits such as differential thresholds for gene activity for different genes, a stronger correlation with experimental flux data, and the application of a non-Boolean confidence metric. Furthermore, this is the first such metabolic model to be grounded on a statistically rigorous foundation.
* The presenter of the poster.