The avoidance of induced forests, or induced acyclic subgraphs, in $d$-dimensional grid graphs, or lattice graphs, has been studied in Alon _et al._ (2001) and later in Caragiannis _et al._ (2002), finding upper and lower bounds with respect to the number of vertices in a single dimension $n$ and the dimension $d$. In this work, we study the avoidance of induced $C_4$-free subgraphs, a superset of induced forests, of $2$-dimensional grid graphs $G$ and provide an upper bound on the size of maximal sets $S \subseteq V$ such that the induced subgraph $H_{S}$ of $G$ with vertex set $S$ is $C_4$-free. Additionally, we will show that the number of maximal $C_4$-free induced subgraphs with number of vertices slightly smaller than this upper bound is sufficiently small.
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Finding meaningful, realistic, but approachable data sets students can use to explore data science concepts is challenging. This talk will briefly review a few public data sets the presenter used in teaching an introductory data science course (aligned with Wickham, Cetinkaya-Rundel, and Grolemund's 2nd edition of [*R for Data Science*](https://r4ds.hadley.nz/) text) and the concepts the students explored with each.
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In Fall 2022, we switched our Statistics course from using graphing calculators to using Excel. This seemed to be going great until it became clear that the students had not spent enough time working with Excel when it came to the first test. That lead us to re-work our Statistics course into more of a flipped-style class. This gave students the opportunity to work on assignments and to build their skills and familiarity with Excel as they learned Statistics. This changed a bit of how content was presented and what the students did during class as how the tests were given. When we moved a section of the course on-line this semester, then this approach became the basis and skeleton of the course. We will talk about the changes we have made and what we have learned over the last few years.
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For more than a century, scientists and mathematicians have sought to understand and predict environmental change. Accurate forecasting is essential for improving preparedness for hurricanes and droughts, as well as anticipating shifts in food availability. In this lecture, I examine the mathematical tools—including systems of differential equations and numerical methods—that researchers use to study hurricanes, sea ice extent, temperature patterns, and precipitation changes, and to assess the resilience of these systems in a rapidly changing climate.
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We extend the study of inverse boundary value problems for quasilinear anisotropic conductivities from Euclidean domains to compact Riemannian manifolds with boundary. Given boundary voltage and current measurements, represented by the Dirichlet-to-Neumann (DN) map, we investigate whether the quasilinear anisotropic conductivity can be uniquely determined. Our main result establishes uniqueness for quasilinear anisotropic conductivities, where the conductivity tensor is given by a scalar function multiplied by a fixed Riemannian metric. Under natural geometric conditions, such as conformal flatness or boundary rigidity of the underlying manifold, we show that this scalar factor can be uniquely determined from the boundary measurements.
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An acyclic graph is defined to be a graph that contains no cycles. When extending the concept of acyclicity to hypergraphs, there are several nonequivalent definitions that, when reduced to graphs, are equivalent. We will look at some of these defintions.
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This workshop presents the successful design and launch of a Data Science Minor at Limestone University (2023-2025), developed by existing faculty without additional costs or new hires. The project demonstrates how mathematics faculty can lead data science initiatives that meet workforce demand, broaden student opportunities, and enhance institutional offerings within current academic structures and budgets. Participants will explore key steps in the program development process, including conducting market and peer-institution analyses, designing an interdisciplinary curriculum aligned with institutional mission, navigating approvals across campus offices, and selecting cost-effective digital learning systems. This session provides a replicable model for resource-conscious program innovation in higher education, particularly valuable for small colleges and liberal arts institutions.
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For a graph *G*, a *neighborhood balanced coloring* is a coloring of the vertices using red and blue such that each vertex has an equal number of red and blue vertices in its neighborhood. We introduce a variation called a *closed neighborhood balanced coloring* in which each closed neighborhood (which includes the vertex itself) has an equal number of red and blue vertices. A graph is *closed neighborhood balanced colorable* (CNBC) if such a coloring exists. In this talk, we will discuss various families of CNBC graphs, focusing on trees and classes relating to graph products.
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The Positive Mass Theorem is a fundamental result in differential geometry and general relativity. It states that if a space looks flat far away and has nonnegative scalar curvature (a measure of how the space bends), then its total mass must be nonnegative. Moreover, the only way the mass can be zero is if the space is completely flat. A natural question is what happens to this total mass when we deform the geometry in a conformal way i.e, when we rescale distances by a smooth function. In joint work with Alex Freire, we study this question for asymptotically flat manifolds that have a noncompact boundary (such as a half-space). Using harmonic functions, we prove that a certain weighted combination of the masses of two conformally related metrics remains nonnegative under corresponding curvature conditions.
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In this talk, we investigate the enumeration of spanning forests in complete graphs with a fixed set of vertices in each tree using exponential generating functions. We begin with a brief overview of generating functions and demonstrate how the Tree function (Lambert W function) can be applied to count these forests. This approach leads to an elegant and unified solution that may have broader implications.
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A graph is a finite number of points(vertices) connected by edges. Graphs are used to model computer networks, business competitiveness, organic molecules, social media, logistics, computability, machine learning, etc. as well as being very useful in solving difficult theoretical problems in combinatorics and other mathematical fields. A spanning tree is a subgraph that contains all the vertices, but only some of the edges so that there are no cycles. This means that once you leave a vertex, you cannot return to that point unless you retrace your path. The Matrix-Tree Theorem states that if we represent the graph as an integer valued matrix, called a Laplacian, then its minor gives the number of spanning trees in the graph. This is useful not only in mathematics theories, but in questions about structure and reliability of the network. What is not very well known are questions such as: How many ways can you construct two disjoint subtrees (a 2-forest) where one vertex is in one tree and two others are in the other that together span all the vertices of the graph? There are many questions such as these that are unknown that can be answered by students as an undergraduate research project. We will explore what tools are needed to answer these questions using graph theory, linear algebra, abstract algebra, and mathematics software.
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Detecting changes in relationships over time is a central problem in longitudinal data analysis. While classical change-point methods focus on mean regression, structural shifts may occur differently across the distribution of the response. We propose a CUSUM-based testing framework for detecting structural changes in Bayesian quantile regression models, allowing quantile-specific shifts to be identified. Our approach leverages Bayesian quantile regression with a plug-in estimator for the regression coefficients and constructs test statistics based on cumulative sums of quantile score processes. We establish theoretical guarantees for validity under parameter stability and consistency under single-change alternatives. Simulation studies demonstrate that the method maintains good size control and strong power, effectively detecting structural changes across the response distribution. The framework opens new possibilities for uncovering nuanced dynamics in longitudinal studies.
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For fixed positive integers a and c, the prime sequences of the form ap_n + cb, where p_n is the nth prime and gcd(a,cb) = 1, can share terms as b varies. When the sequences share terms, we say that they overlap. Furthermore, when the sequences overlap with each other or another common prime sequence of a similar form, we say that these prime sequences are in the same family. We show that when a and cb have opposite parity, the number of families of prime sequences is Φ(a), where Φ(a) is Euler’s totient function. In other words, the number of families depends only on a. Several of these overlapping prime sequences, their families, and the pseudocode used to generate the sequences will be included in the presentation.
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This presentation highlights two key components of a joint undergraduate research project conducted by Duncan and Worley with undergraduate students. First, we will outline the collaborative research process and the roles students played in exploring the mathematical concept of laminations. Second, we will describe the method used to generate the corresponding Julia sets from specific laminations studied during the project. To provide context, we will introduce the foundational definitions of laminations and discuss the technology and tools employed to visualize and compute the associated Julia sets.
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Let $n$ be a positive integer. Suppose $(x_1, y_1), (x_2, y_2), \dots, (x_k, y_k)$ with $\sqrt{n} \le x_1 < x_2 < \dots < x_k$ are integer lattice points on the hyperbola $x y = n$ near the center $(\sqrt{n}, \sqrt{n})$. In this talk, we will discuss repulsion among these lattice points through a lower bound on $x_k - y_k$. It turns out that this lower bound is sharp when $k = 2$ and $k = 3$ but not $k \ge 4$. We apply elementary, Pell equation, and Diophantine approximation techniques. This is joint work with Jorge Jim\'{e}nez-Urroz.
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Why do we have to learn to integrate when Computer Algebra Systems (CAS) or Artificial Intelligence (AI) can do it for us? This talk looks at some tricky integration problems that even AI and CAS struggle with and how people can solve them. We also look at some tips for ensuring our answers are correct whatever method we use.
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For $x \ge 1\$, we define the expected value of the largest $k^{\text{th}}$-power divisor over integers $n \le x$ by $E_x(r^k) := \frac{1}{x} \sum_{n \le x} \max \{\{ r^k : r^k \mid n \}\}.$ Motivated by a question on MathOverflow concerning the square case ($k=2$) and a heuristic argument of Yuval Peres, we study the asymptotic behavior of $E_x(r^k)$ as $x \to \infty$. Peres’ heuristic predicts that $E_x(r^2)$ grows on the order of $\sqrt{x}$, but the associated error term is too large to determine the correct leading constant. We prove that $ E_x(r^2) = \frac{\zeta(3/2)}{3\zeta(3)} \sqrt{x} + O(\log x), $ where $\zeta(s) = \sum_{n=1}^{\infty} 1 / n^s$ is the Riemann zeta function. This result identifies the correct leading constant supported by numerical evidence. More generally, for any $k \ge 2$, we show that $ E_x(r^k) = \frac{\zeta\\left(\frac{k+1}{k}\right)}{(k+1)\zeta(k+1)}\, x^{1/k} + O(\log x). $ The constants arise naturally from zeta functions associated with $k$-free integers and quantify the average size of the largest $k^{\text{th}}$-power divisor.
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Point cloud data have become increasingly vital due to their ability to capture detailed, multi-dimensional representations of physical objects and their surrounding environments. Point clouds are nowadays central to applications across industry and academia that range from robotics, navigation systems, and 3D printing to architecture, manufacturing, and agriculture. Despite its importance, the analysis of point cloud data faces several limitations, including high computational cost due to large data volume, contamination with localization noise and inaccuracies caused by sensor limitations, and missing points due to environmental factors or sensor positioning. Moreover, unavailable uncertainty quantification in reconstructed structures remains a critical gap in applications requiring reliable estimation. We present a fully Bayesian framework for point cloud data analysis and curve reconstruction. Our framework models a cloud’s points as noisy perturbations of latent positions constrained to lie on closed polylines, jointly inferring polyline vertices and connectivity, latent coordinates, and noise characteristics. Posterior inference is performed via a specialized Markov chain Monte Carlo sampler tailored to point cloud data processing. Experiments on synthetic data demonstrate accurate curve reconstruction while providing uncertainty quantification.
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Boolean networks (BNs) have become quite popular since their proposal as models of network regulation. Every BN can be decomposed into coordinate functions and thus defines a signed, directed graph, called a wiring diagram, that describes the variable dependencies. We will explore what it means for two BNs to be equivalent and how to define a structure-preserving map between them. In particular, maps that are topologically conjugate or semi-conjugate need not preserve locality of the functions or the wiring diagram. We will illustrate this with examples and non-examples, involving commutative diagrams with extra structure. Finally, we will discuss how to define a category of Boolean networks and explore some of its basic properties.
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Chemotherapy-induced neurological disorders (CIND) represent a serious side effect of cancer treatment. They can exist long after treatment ends and significantly impair functional independence and daily life. While this toxicity is well known in oncology, its epidemiological and cost burden features among various cohorts remain understudied. To better understand the current gap, we conducted a retrospective cohort study using de-identified administrative claims data from the IBM MarketScan research database. Our study included 258,410 adult patients undergoing chemotherapy and 4.7 million adult patients without chemotherapy. The chemotherapy group was stratified by cancer behaviors and treatment routes to examine CIND patterns and healthcare utilization. Survival analysis and Cox proportional hazards models showed that approximately one-quarter to over one-third of cancer patients receiving chemotherapy developed CIND. All malignant cohorts showed higher CIND probability than their benign counterparts across all treatment routes, with the highest rates observed in patients receiving both intravenous and oral chemotherapy. Cox modeling adjusted for demographic characteristics, comorbidities, and baseline medications showed that chemotherapy increased the risk of neuropathy by approximately 40% compared to cancer patients not exposed to chemotherapy, with female and older age as critical risk factors. A healthcare burden analysis showed that patients with CIND had significantly higher opioid use, CIND-specific medication use, and rehabilitation service needs compared to the general population, resulting in about 2 times higher healthcare costs per patient. These results indicate that CIND influences a considerable proportion of cancer patients experiencing chemotherapy, with chemotherapy administration route being a key determinant of risk. Our findings emphasize the clinical importance of comprehensive neurotoxicity monitoring and highlight the significant economic burden that CIND places on the healthcare system.
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Upper-level mathematics courses are widely regarded as challenging due to the complexity of their material. These courses often assess students’ understanding of course concepts through timed examinations. Because advanced mathematical concepts require critical thinking and problem-solving, students approach problems differently and may need varying amounts of time to reach solutions. Students with documented disabilities can request extended time; however, this accommodation often necessitates taking the test in a separate location, removing them from direct access to their professor. In this paper, we address the issue of test-time stress and explore strategies instructors can adopt to help students perform at their best. Our methodology involves gradually presenting test material over several days, eliminating standard time constraints for each component. We document student behavior and performance as compared to timed tests. Students are given a defined period over several days in which they can complete and submit individual components of the test as they progress. We observed student behavior, collected performance data, and analyzed the outcomes in comparison with classes from previous years that used standard timed testing methods. This paper discusses our findings and instructional strategies in detail, with the goal of providing fellow faculty members with practical approaches to reducing test-related anxiety and improving student outcomes.
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Dehn surgery is a topological process that creates new 3-dimensional manifolds from the 3-dimensional sphere by excising and regluing a thickened knot in interesting ways. We study when specific surgeries using fibered knots can contain a Klein bottle. We use Heegaard Floer topological invariants in their more tractable immersed curves form. This talk is aimed at undergraduates interested in accessible research topics in geometry and topology.
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We discuss the Navier-Stokes equations (NS equations) and the implementation of a well-known least squares finite element algorithm in an object-oriented framework. We review the conjugate gradient method (CG method) utilized in solving the least-squares formulation as well as the corresponding key components specific to the application of the CG method with respect to the NS equations and the natural application of the finite element method resulting from the components. We discuss the application of the implemented solver toward several standard benchmarks, we visualize benchmark results, and we discuss computational aspects of the implemented solver with respect to benchmark results.
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Bumpless pipe dreams (BPDs) are combinatorial objects used to study Schubert and Grothendieck polynomials. In 2025, Weigandt introduced a co-BPD object corresponding to each BPD and used them to prove change of bases formulas between these polynomials. She posed the open problem of characterizing which permutations have only reduced co-BPDs associated to their BPDs. In this talk, we present a pattern-avoidance characterization of these permutations. This is joint work with Josh Arroyo.
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In [Schwede 2010], uniformly $F$-compatible ideals were introduced as a generalization of centers of $F$-purity in prime characteristic, revealing deep connections to classical Matlis duality. When the underlying ring is Gorenstein, this notion coincides with the well-studied $F$-ideals of [Smith 1995] and [Kimura 2025]. In this talk, we will introduce the definition and core properties of uniformly $F$-compatible ideals. We will then present our recent result showing that uniformly $F$-compatible ideals cannot be generated by a regular sequence, refining known statements for $F$-ideals.
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After reading a few articles about the rational points that could be found in the well-known Cantor set, I wondered if I could similarly find rational points in generalized Cantor sets too. Many years ago, I worked with a student to see what we could find. In this talk, I will briefly describe the Cantor set and the motivation for this project. After that I will discuss two different types of generalizations of the Cantor set and how to think about finding what lies inside of these sets.
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A traditional configuration space of a metric graph *X* models *n* particles existing in a network of tracks with no collisions allowed. If instead of *n* particles, there are *n* "robots," then the resulting space's type depends on the robots' sizes and other distance restrictions given by a restraint parameter **r**. In this talk, we discuss the homotopy and homeomorphism types of these restricted configuration spaces $\{X^n_r\}$ over the domain of the parameter **r** and provide polynomial upper bounds (in the number of edges of the graph *X*) for the number of types.
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Hamilton proved that positive sectional curvature, sec > 0, is preserved under Ricci flow in dimensions 2 and 3. However, as shown by Bettiol and Krishnan, this is no longer true beginning in dimension 4. In fact, Cheung and Wallach and, later, González-Álvaro and Zarei showed that sec > 0 is not preserved under the added assumption that the starting metric is homogenous. We will show that, in contrast to these results, Ricci flow does in fact preserve the set of homogeneous sec > 0 metrics on a sphere of any dimension. This is joint work with David González-Álvaro and Masoumeh Zarei.
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We look at three results that use Euclid's algorithm as a backbone, producing side benefits apart from the computation of a GCD.
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The mathod of bracket is useful in evaluating some definite integrals. In this talk we present the method of brackets and provide some examples.
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Vibro-impact systems are central to many engineering applications, including energy harvesting; yet, unlike smooth dynamical systems, they still lack broadly applicable methods for global dynamical analysis. We develop a novel word-based first-return map framework to study the global dynamics of a vibro-impact pair, modeled as a ball moving up and down inside a harmonically forced capsule. Because impacts define the system’s evolution through discrete collision events, symbolic dynamics provides a natural representation. We encode impacts with the bottom and top of the capsule as B and T, respectively, and describe trajectories through finite words formed from these symbols. In our recent work (Bao et al., SIAM Journal on Applied Dynamical Systems, 24 , 1891, 2025), we focused on short word sequences associated with 1:1 responses (e.g., BTB). Here, we extend the approach to longer words (e.g., BTBB and BBTB) to capture regimes in which 1:1 and 2:1 solutions coexist. This extension demonstrates that the word-based first-return map remains effective in more complex settings and can be used to identify basins of attraction in bi-stable regimes. Moreover, by characterizing the global dynamics of energetically favorable states in the bi-stable regime, we identify parameter regions that maximize energy output in vibro-impact systems and clarify how noise may play a constructive role by triggering switches from low-output to high-output attractors.
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