Mathematics & Statistics Seminars
Northern Arizona University

Spring 2024 Department Colloquium

The talks will typically take place on Tuesdays at 4:00-5:00pm in Adel Room 164. Please contact Nandor Sieben if you would like to give a talk or have a question about the colloquium.

Tuesday 1/23 at 4:00-4:50

Speaker: Jaechoul Lee Title: An efficient least squares algorithm for periodic time series regression

Abstract: Periodic and autoregressive data like daily temperatures or sales of seasonal products can be seen in periods fluctuating between highs and lows throughout the year. Generalized least squares (GLS) estimators are frequently computed for such periodic data, because these estimators are minimum variance unbiased estimators. However, the GLS solution can require extremely demanding computations when the data is large. We develop an efficient GLS algorithm in several periodic regression settings. The algorithm can substantially simplify GLS computations by compressing large sets of data into smaller sets. This is accomplished by constructing a structured matching matrix for dimension reduction. Simulations show that the new computation methods using our algorithm can drastically reduce the GLS computing time. Our algorithm can be easily adapted to many big data that shows periodic characteristics often pertinent to economics, environmental studies, and engineering practices. This talk should be accessible to any audience with knowledge on simple matrix operations.

Tuesday 1/30

Most likely no colloquium because of faculty candidate visits. Please enjoy the candidate presentations.

Tuesday 2/6 at 4:00-4:50

Speaker: Shafiu Jibrin Title: On Second-Order Cone Functions

Abstract: We consider the second-order cone function (SOCF) $f: \mathbb{R}^n \to \mathbb{R}$ defined by $f(x)= c^T x + d -|A x + b |$, with parameters $c \in \mathbb{R}^n$, $d \in \mathbb{R}$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A$ and $b$ are not uniquely determined. We show that every SOCF can be written in the form $f(x) = c^T x + d -\sqrt{\delta^2 + (x-x_\star)^TM(x-x_\star)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $\delta$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above.

Our results have computational implications for convex optimization problems involving second-order constraints such as the problem of minimizing weighted barrier functions. They would allow one to recognize these optimization problems that can be solved efficiently, or to assist in reformulating those that are hard to solve.

This research is a joint work with Dr. James W. Swift.

Tuesday 2/13 at 4:00-4:50

Speaker: John Neuberger Title: A Bifurcation Lemma for Invariant Subspaces

Abstract: The Bifurcation from a Simple Eigenvalue (BSE) Theorem is the foundation of steady-state bifurcation theory for one-parameter families of functions. When eigenvalues of multiplicity greater than one are caused by symmetry, the Equivariant Branching Lemma (EBL) can often be applied to predict the branching of solutions. The EBL can be interpreted as the application of the BSE Theorem to a fixed point subspace. There are functions which have invariant linear subspaces that are not caused by symmetry. For example, networks of identical coupled cells often have such invariant subspaces. We present a generalization of the EBL, where the BSE Theorem is applied to nested invariant subspaces. We call this the Bifurcation Lemma for Invariant Subspaces (BLIS). We give several examples of bifurcations and determine if BSE, EBL, or BLIS apply. We extend our previous automated bifurcation analysis algorithms to use the BLIS to simplify and improve the detection of branches created at bifurcations.

Tuesday 2/20 at 4:00-4:50

Speaker: Jin Wang Title: Depth-Based Minimum Volume Ellipsoid Estimators for Multivariate Location, Spread, Skewness, and Kurtosis

Abstract: Multivariate descriptive measures for location, spread, skewness, and kurtosis are the foundation of multivariate statistics. Almost all multivariate statistical methods are based on them. The classical measures of these concepts are moment-based and are estimated by their sample versions. However, these estimators are not robust and are extremely sensitive to outliers. To eliminate or reduce the effects of outliers, we propose some depth-based minimum volume ellipsoid (DMVE) estimators for multivariate location, spread, skewness, and kurtosis. The important properties of the new estimators are studied.

Tuesday 2/27 at 4:00-4:50

Speaker: Nandor Sieben Title: Problem solving for lazy mathematicians: Jim’s dress code puzzle and the magic of constraint solvers

Abstract: Constraint programming often provides simple formulations of a wide variety of problems. We can simply describe what we want in a solution without knowing how to get it. State-of-the-art solvers can do the rest. We learn how to use a few lines of Python code to easily handle Jim’s dress code puzzle, MAT 232 problems, Diophantine systems of inequalities, the send more money puzzle, and graph coloring. Einstein supposedly claimed that only 2% of the world could solve his zebra puzzle. This is no longer true. With constraint solvers, even lazy mathematicians can figure out who owns the zebra.

Tuesday 3/5 at 4:00-4:50

Speaker: Jeffrey Hovermill Title: STEM Education Partnerships

Abstract: School/University Partnerships allow for a bidirectional flow of information, cooperation, and support. This talk will describe several NAU STEM Education Partnerships including the goals, structure, and impact of these various initiatives.

Tuesday 3/12

Spring break

Tuesday 3/19 at 4:00-4:50

Speaker: Jeffrey Covington Title: Data Assimilation, the Arctic, and Bridging the Gap Between Theory and Practice

Abstract: Data assimilation, where observations and a mathematical model are combined to estimate the state of a dynamical system, is widely used in weather prediction, climate science, industrial processes, robotics, and many more fields. This talk will present an elementary introduction to data assimilation, give an overview of different approaches and techniques that have been developed, and explore the challenges of applying theoretical tools in practical applications. In particular, a real application of data assimilation to remote sensing of the Arctic will be discussed, as well as the concrete considerations of applying these techniques to a high-dimensional and highly nonlinear system.

Tuesday 3/26 at 4:00-4:50

Speaker: Rachel Neville Title: A Topological Approach to Snowpack Roughness

Abstract: Roughness of snowpack on mountainous terrain exhibits a high degree of spatiotemporal variance. Accurate estimation of the snow surface roughness is a parameter of fundamental importance for estimating turbulent fluxes and is an input into all existing numerical models of surface-atmosphere interactions. However, it proves difficult to estimate numerically from real data. We’ll describe geometric and topological techniques to estimate roughness from airborne LIDAR measurements and share some advantages of taking a topological perspective.

Tuesday 4/2 at 4:00-4:50

Speaker: Avery Bell Title: Exploring Foundations of Hierarchical Clustering with Applications

Abstract: Hierarchical clustering is a form of unsupervised machine learning that measures distances between observations to construct a hierarchy of clusters. The two variations of hierarchical clustering include agglomerative and divisive clustering and utilize a bottom-up or top-down approach to partition data. A one-dimensional agglomerative example is demonstrated, and the results of hierarchical clustering on thermal liquid biopsy is presented.

Speaker: Jayden Chrzanowski Title: Exploring XGBoost 2.0 with Optimal Feature Selection and Baseball Applications

Abstract: In this presentation, we explore the use of XGBoost 2.0 alongside modern feature selection methods like Boruta and SHAP, with a focus on baseball analytics. Through applying Boruta and BorutaSHAP for feature selection, we created and compared two models to predict the classification of pitches hit into play as either a ball or a strike. Our results highlight the potential over-simplification of models when using BorutaSHAP compared to Boruta. Furthermore, applications of our research integrate the idea of a theoretical strike zone for pitches hit into play.

Tuesday 4/9 at 4:00-4:50

Speaker: Keegan Line Title: Streamlined Thermogram Analysis: Automated Baseline Extraction and Processing

Abstract: Thermograms hold the potential to be crucial in assessing patient health status. However, a major obstacle lies in the time-intensive manual processing of vast amounts of raw thermogram data. To address this challenge, we present an algorithm designed to expedite data processing by automating endpoint detection and baseline subtraction, resulting in analysis-ready thermogram curves. Our approach enables laboratories to handle substantially larger quantities of thermograms compared to technician methods. With this algorithm, various statistical methods can be applied to the curves, such as signal detection, identification of significant regions, and potentially approximation and categorization using normal curves. This automated methodology represents a massive step forward in thermographic data processing and gives a solution for the baseline dilemma.

Tuesday 4/16 at 4:00-4:50

Speaker: Jeff Uyekawa Title: Robust Vector Equivalence of Synchronous and Bisynchronous Games

Abstract: Two player non-local games have gained attention in recent years because these games can be used as a helpful framework to study the potential advantages of using quantum entanglement as a resource. Synchronous and bisynchronous games are two types of non-local games with particularly nice structure, and hence have been well studied. When given a synchronous game with n inputs and k outputs, one can construct a bisynchronous game with nk inputs and nk outputs such that the players have a winning strategy in the synchronous game in one model of quantum mechanics if and only if they have a winning strategy in the bisynchronous game using the same model. We have found that this same result holds for a set of strategies called vector strategies. These are similar to strategies in quantum mechanics, but they are easier computationally. In addition, we present our finding that approximately winning strategies are also preserved between these two games so long as one considers synchronous strategies for the synchronous game and bisynchronous strategies for the bisynchronous game.

Tuesday 4/23 at 4:00-4:50

Speaker: Hannah Golab

Title: Pattern avoidance in Cayley permutations

Abstract: Any permutation of n may be written in one-line notation as a sequence of entries representing the result of applying the permutation to the sequence 1 2 ··· n. If p and q are two permutations, then p is said to contain q as a pattern if some subsequence of the entries of p has the same relative order as all of the entries of q. If p does not contain a pattern q, then p is said to avoid q. One of the first notable results in the field of permutation patterns was obtained by MacMahon in 1915 when he proved that the ubiquitous Catalan numbers count the 123-avoiding permutations. The study of permutation patterns began receiving focused attention following Knuth’s introduction of stack-sorting in 1968. Knuth proved that a permutation can be sorted by a stack if and only if it avoids the pattern 231 and that the Catalan numbers also enumerate the stack-sortable permutations. In the subsequent years, the notion of pattern avoidance has been extended to numerous combinatorial objects, including multiset permutations, set partitions, ordered set partitions, compositions, and modified ascent sequences. In this talk, we study pattern avoidance in the context of Cayley permutations, which were introduced by Mor and Fraenkel in 1983. A Cayley permutation is a finite sequence of positive integers that include at least one copy of each integer between one and its maximum value. When possible we will take a combinatorial species-first approach to enumerating Cayley permutations that avoid patterns of length two, pairs of patterns of length two, patterns of length three, and pairs of patterns of length three with the goal of providing species, exponential generating functions, and counting formulas. We also briefly study pattern avoidance in a special class of Cayley permutations known as primitive Cayley permutations. Throughout the talk, we include several conjectures and open problems.

Speaker: Ruth Schroeder Perry

Title: Braid Graphs in Coxeter Systems of Type $\Lambda$ are Median

Abstract: Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. Two reduced expressions are said to be braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In a recent paper, Awik et al. proved that when the Coxeter system is simply-laced and triangle free (i.e., the corresponding Coxeter graph has no three-cycles), the braid graph for a reduced expression is a partial cube (i.e., isometric to a subgraph of a hypercube). In her MS thesis, Barnes provided an alternate version of this fact and provided a description of the minimal dimension hypercube into which a braid graph can be isometrically embedded. In this talk, we prove that every braid graph in a simply-laced triangle-free Coxeter system is median, which is a strengthening of previous results. We conjecture that every braid graph of a link corresponds to the Hasse diagram for a distributive lattice.

Tuesday 4/30 at 4:00-4:50

Speaker: Peter Vadasz Title: Newtonian Gravitational Waves from a Continuum

Abstract: Gravitational waves are being shown to derive directly from Newtonian dynamics for a continuous mass distribution, e.g. compressible fluids or equivalent. It is shown that the equations governing a continuous mass distribution, i.e. the inviscid Navier-Stokes equations for a general variable gravitational field g(t,x), are equivalent to a form identical to Maxwell equations from electromagnetism, subject to a specified condition. The consequence of this equivalence is the creation of gravity waves that propagate at finite speed. The latter implies that Newtonian gravitation is not “spooky action at a distance” but rather is similar to electromagnetic waves propagating at finite speed, despite the apparent form appearing in the integrated field formula. In addition, this proves that in analogy to Maxwell equations the Newtonian gravitation equations are Lorentz invariant for waves propagating at the speed of light. Since gravitational waves were so far derived only from Einstein’s general relativity theory it becomes appealing to check if there is a connection between the Newtonian waves presented in this paper and the general relativity type of waves at least in a certain limit of overlapping validity. The latter is left for a follow-up research.