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# Student Research

Some of our students are involved in conducting original mathematics research. Research projects vary in nature and scope; they often take place during the summer, and often carry significant amounts of financial support. Students have participated in research that has led to presentations at conferences, and even publications in professional journals. Below are summaries of some of the most recent projects our students have worked on.

Research by Reed Acton, Blake Shirman, Daniel Toal, and Kyle Petersen (faculty)

Abstract: We study the problem of the Malicious Maitre d', as described in Peter Winkler's book Mathematical Puzzles: A Connoisseur's Collection. This problem, attributed to computer scientist Rob Pike, involves seating diners around a circular table with napkins placed between each pair of adjacent settings. As each diner sits, they take a napkin on the right or the left in a random fashion if both are available. If only one napkin is available, they take that napkin. If neither napkin is available because the napkins on both the left and right of their place already taken by their neighbors, the diner is napkinless. The goal of the maitre d' is to seat the diners in a way that maximizes the number of napkinless diners. Winkler proposes a solution to the problem that he claims is optimal. We analyze Winkler's solution using tools from enumerative combinatorics, then present a new strategy that performs better.

Preprint on arXiv: https://arxiv.org/abs/2209.11833

To appear in the American Mathematical Monthly

Research by Nick Kreissler (student), advised by Enrico Au-Yeung (faculty)

Abstract: The Ehrenfest Wind tree model was proposed as a means to explain gas dynamics in an environment with periodic obstacles. In the model, we consider an infinite grid of squares equally spaced and think about the way a billiard ball, or any particle, moves through the grid and collides with the squares. Under the setting of perfectly elastic collision, we observe the trajectory of the position can be chaotic or recurrent, depending on the initial slope of the position vector.

Research by Tim Komperda (student) and Enrico Au-Yeung (faculty)

Abstract: We implement the finite element method to solve a variational problem that is inspired by medical imaging. In our application, the domain of the image does not need to be a rectangle and can contain a cavity in the middle. The standard approach to solve a variational problem involves formulating the problem as a partial differential equation. Instead, we solve the variational problem directly, using only techniques available to anyone familiar with vector calculus. As part of the computation, we also explore how triangulation is a useful tool in the process.

Published as: Komperda, Tim and Au-Yeung, Enrico (2021) DePaul Discoveries, "Triangulation and finite element method for a variational problem inspired by medical imaging", Volume 10, Issue 1

Abstract: In this project we studied the mathematical concept of the Frobenius number and some curious patterns that come with it. One common example of the Frobenius number is the Coin Problem: If handed two denominations of coins, say 4¢ and 5¢, and asked to create all possible values, we will eventually find ourselves in a position where we can make any value. With 4¢ and 5¢ coins, we can create any value above 11¢, but not 11¢ itself. So, that makes 11 the Frobenius number of 4 and 5. What we explore in this paper is a pattern we call Frobenius symmetry: when all non-negative integers below the Frobenius number can be paired up such that one number is attainable, and the other is not. We looked at sets of two and three numbers and arrived at results about both.

Abstract: Clustering can be defined as the process of assembling objects into a number of groups whose elements are similar to each other in some manner. Given its many applications, such as face clustering, plant categorization, image segmentation, document classification etc., clustering is one of the most important unsupervised learning problems. Scientists have surveyed this problem for years and developed different techniques that can solve it, such as k-means clustering. In this project, we analyze one of these techniques: a powerful clustering algorithm called Sparse Subspace Clustering. We demonstrate several experiments using this method and then introduce a new approach that can reduce the computational time of the task.

Research by William J. Asztalos

Abstract: The efforts of this research project are best understood in the context of the subfield of dynamical combinatorics, in which one enumerates a set of combinatorial objects by defining some action to guide the search for underlying structures. While there are many examples with varying degrees of complexity, the necklace problem, which concerns the possible unique configurations of beads in a ring up to rotational symmetry, is a well-known example. Though this sort of approach to enumeration has been around for a century or more, activity in this area has intensified in the last couple of decades. Perhaps the most startling development was the discovery of the cyclic sieving phenomenon, in which polynomial generating functions produce information about the sizes of rotational symmetry classes of objects. This technique is an extension of the “q = -1” phenomenon which classifies objects on the basis of being a fixed point or an element of a “mirrored” pair. In this study, we are on the hunt for rotational symmetries in plane partitions, with the ultimate goal of recovering the “magic” polynomial that will allows us to count the symmetry classes of these objects. The unique characteristics of plane partitions under our devised operation portend that attaining such a goal is feasible.

Research by Greg Zanotti (student) and Enrico Au-Yeung (faculty)

Abstract: In dictionary learning, a matrix comprised of signals Y is factorized into the product of two matrices: a matrix of prototypical "atoms" D, and a sparse matrix containing coefficients for atoms in D, called X. Dictionary learning finds applications in signal processing, image recognition, and a number of other fields. Many algorithms for solving the dictionary learning problem follow the alternating minimization paradigm; that is, by alternating solving for D and X. In 2014, Agarwal et al. proposed a dictionary initialization procedure that is used before this alternating minimization process. We show that there is a modification to this initialization algorithm and a corresponding data generating process under which full recovery of D is possible without a subsequent alternating minimization procedure. Our findings indicate that the costly step of alternating minimization can be bypassed, and that other data models may enjoy the same features as the one we propose.

Research by Michael Dennis and Enrico Au-Yeung (faculty)

Abstract: In applications such as image processing, the data is given in a regular pattern with a known structure, such as a grid of pixels. However, it is becoming increasingly common for large data sets to have some irregular structure. In image recognition, one of the most successful methods is wavelet analysis, also commonly known as multi-resolution analysis. Our project is to develop and explore this powerful technique in the setting where the data is not stored in the form of a rectangular table with rows and columns of pixels. While the data sets will still have a lot of structure to be exploited, we want to extend the wavelet analysis to the setting when the data structure is more like a network than a rectangular table. Networks provide a flexible generalization of the rigid structure of rectangular tables.

Calculus and combinatorics overlap, in that power series can be used to study combinatorially defined sequences. In this project, we used exponential generating functions to study a curious refinement of the Euler numbers, which count the number of “up-down” permutations of length n.

Published in: Heneghan, Fiacha and Petersen, T. Kyle, "Power series for Up-Down Min-Max Permuations," College Mathematics Journal, Vol. 45, No. 2, March 2014, p.83-89.

Research by Charles Brittenham, Andrew Carroll (faculty), T. Kyle Petersen (faculty) , and Connor Thomas

We attempted to find combinatorial proofs of unimodality for various number sets, namely the q-analogue of n!, the q-binomial coefficients, and integer partitions with distinct parts of size at most n. We proceeded by attempting to find a sign-reversing involution on the gamma-vector expansions for each of these polynomials to show that the entries of these vectors were nonnegative, and hence that the polynomials modeled by those gamma-vectors are unimodal. While we were able to show this for the q-analogue of n!, further refining of the involutions for the remaining two number sets is needed to give a complete proof of unimodality.

Published as: *Unimodality via alternating gamma vectors*, Electron. J. Combin. **23** (2016), no. 2, Paper 2.40, 22 pp.