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. In the past, 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.
The Mystery of Frobenius Symmetry
Research by Maciej Piwowarczyk
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.
Sparse Subspace Clustering: Experiments and Randomization
Research by Mehmet Demirel (student) and Enrico Au-Yeung (faculty)
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.
A stochastic approach to Eulerian numbers
Research by Kiana Mittelstaedt
Abstract: We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of n particles perform random walks on the integers, beginning at the origin. Each particle walks until it reaches an unoccupied site, at which point it occupies that site and the next particle begins its walk. After all walks are complete, the set of occupied sites is an interval of length n containing the origin. We show the probability that k of the occupied sites are positive is given by an Eulerian probability distribution. Having made this connection, we use generating function techniques to compute the expected run time of the model, which is n^2(n+1)/12.
The Search for the Cyclic Sieving Phenomenon in Plane Partitions
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.
Published as: Asztalos, William J. (2018) "The Search for the Cyclic Sieving Phenomenon in Plane Partitions," DePaul Discoveries: Vol. 7 : Iss. 1 , Article 10.
Exact Recovery of Prototypical Atoms through Dictionary Initialization
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.
Published as: Zanotti, Greg and Au-Yeung, Enrico (2018) "Exact Recovery of Prototypical Atoms through Dictionary Initialization," DePaul Discoveries: Vol. 7 : Iss. 1 , Article 12.
Signal processing on graphs using Kron reduction and spline interpolation
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.
Published as: Dennis, Michael and Au-Yeung, Enrico (2017) "Signal Processing on Graphs Using Kron Reduction and Spline Interpolation," DePaul Discoveries: Vol. 6 : Iss. 1 , Article 7.
Power Series for Up-Down Min-Max Permutations
Research by Fiacha Heneghan and T. Kyle Petersen (faculty)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.
Real Time Bidding Optimization for Online Advertising
Research by Megan Davis
I participated in the Summer 2015 RIPS IPAM REU
at UCLA. This REU (Research Experience for Undergraduates) is founded on assigning industry-sponsored research projects for undergraduate students to complete by the end of the program. I worked with 3 other students from across the world and conducted research in the online advertising industry with respect to optimization algorithms. My team and I were able to come up with two different algorithms. At the end of the program, we were left with questions regarding clustering, algorithmic efficiency, and data mining techniques on different user aggregated data.
Unimodality via alternating gamma vectors
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.