Regional Mathematics and Statistics Conference

Abstracts 2024

Name(s): Alexander Avarez
Affiliation(s): University of South Carolina Salkehatchie
Title: Generalization of A Sandwich Type Inequality
Abstract:
It is known that if $\frac{a}{b}\leq \frac{c}{d}$, then $\frac{a}{b}\leq \frac{a+c}{b+d}\leq \frac{c}{d}$. It can be proved using basic algebra. Applying this inequality as a known property, we can also prove that if $\frac{a}{b}\leq \frac{c}{d} \leq \frac{e}{f}$, then $\frac{a}{b}\leq\frac{a+c}{b+d}\leq\frac{a+c+e}{b+d+f}\leq\frac{c+e}{d+f}\leq\frac{e}{f}$. In this talk, we will present our proof of the above inequality, and generalize it to a sandwich type inequality for n fractions.

Name(s): Erik Blix
Affiliation(s): Elon University
Title: Using Data Visualization to Compare Ecoregion Classifications for Alaskan Deep-Sea Corals
Abstract:
Deep-sea corals provide critical habitats for marine life living deep in the ocean, yet their slow growth rate leaves them especially vulnerable to the impacts of climate change and trawling. This research focuses on reclassifying deep-sea coral data provided by the National Oceanic and Atmospheric Administration (NOAA) to match the marine ecoregions of North America, with the aim of gaining a clearer understanding of coral biodiversity in Alaskan waters. NOAA’s dataset assigns coral locations to specific ecoregions, and this study compares those to the marine ecoregions of North America. One key area of interest is Bowers Ridge, which NOAA considers part of the Aleutian Islands. However, the marine ecoregions of North America classify Bowers Ridge as part of the Bering Sea. Using data visualizations and multiple biodiversity metrics, this research compares how classification differences affect biodiversity assessments, which contributes to a larger effort investigating changes in coral biodiversity around Alaska over time. The findings will lead to a better understanding of where coral species are located and distributed in Alaska’s marine environments. Ultimately, this research will help inform future conservation efforts and provide valuable insights into the health and diversity of deep-sea corals in the region.

Name(s): Yan Liu
Affiliation(s): Wake Forest University
Title: Chromatic Quasisymmetric Function at q=-1
Abstract:
Symmetric functions are multivariable polynomials that remain invariant under any permutation of their variables, while quasisymmetric functions are invariant under certain ordered permutations of their variables, preserving relative order. So every symmetric function is quasisymmetric, but the converse is not true. In 1995, Stanley introduced the chromatic symmetric function of a graph, which is a generalization of the chromatic polynomial of a graph that counts the number of proper colorings of it. In 2012, Shareshian and Wachs defined a quasisymmetric refinement that keeps track of additional properties of proper colorings on labeled graphs with another variable q. In this talk, we will show that for all labelings of a unit interval graph, the chromatic quasisymmetric functions evaluated at q=-1 are symmetric, and moreover, the same across all labelings.

Name(s): Faustus D Maale
Affiliation(s): UNCC
Title: Second Generation P values and Sample Size Determination
Abstract:
Sample size calculations, estimating the number of patients needed in a clinical trial, are crucial in medical study planning. Typically, this sample size depends on the Type I error probability alpha, related to the p-value. The second-generation p-value (SGPV) extends this by using a composite null hypothesis to account for scientific relevance (Blume 2019).

SGPVs enhance transparency, rigor, and reproducibility by identifying practically meaningful hypotheses and providing reliable statistical summaries (Blume 2018). We are interested in how SGPV affects sample size calculations. To achieve desirable findings, a study with two groups of patients and normally distributed outcomes is needed to establish a sample size for SGPVs.

Results indicate that using SGPV for sample size determination may be more efficient and robust than traditional p-values, as a larger sample size can provide better statistical power to identify scientifically meaningful effects. This approach ensures that experiments or clinical trials are adequately powered to detect effects of practical significance.

References
Jeffrey D. Blume, Robert A. Greevy, Valerie F. Welty, Jeffrey R. Smith & William D. Dupont (2019) An Introduction to Second-Generation p-Values, The American Statistician, 73:sup1, 157-167, DOI: 10.1080/00031305.2018.1537893

Blume JD, D’Agostino McGowan L, Dupont WD, Greevy RA Jr (2018) Second-generation p-values: Improved rigor, reproducibility, & transparency in statistical analyses. PLoS ONE 13(3): e0188299. https://doi.org/10.1371/journal.pone.0188299

Name(s): Zachary Parker
Affiliation(s): UNCG
Title: Computations on Perfect Ternary Forms
Abstract:
A perfect form is uniquely determined by its arithmetic minimum and its set of minimal vectors. We computed perfect ternary forms over imaginary quadratic number fields with small discriminant. In this talk, I will present two examples: The first aims to develop intuition behind the underlying geometry and the algorithm, while the second serves to expound on the technical details involved. We conclude the presentation by viewing our data and asking number theoretic questions about the nature of some examples.

Name(s): Alex Paschal
Affiliation(s): University of North Carolina at Chapel Hill
Title: The Variational Principle for Locally Finite Countable State Shift Spaces with Specification
Abstract:
We define and discuss specification properties for countable state shift spaces, which are special cases of definitions from an upcoming paper by Climenhaga, Thompson, and Wang and generalize the well-studied compact specification property to non-compact shift spaces. We present an infinite class of examples of such shift spaces and prove the variational principle for these spaces. This gives the foundations for developing the theory of equilibrium states in this new setting.

Name(s): Vishva Patel
Affiliation(s): UNCG
Title: Analysis of trade-off between dispersal and intrinsic growth of a landscape ecological model
Abstract:
We analyze positive solutions to the steady state reaction diffusion equations:
\begin{equation*}
\left\lbrace \begin{matrix} – u” =\lambda r^*(\beta,m) f(u);~ x \in (0,1) \\
-u'(0)+\sqrt{\lambda} \gamma g(\beta,M,u(0)) u(0)=0\\
u'(1)+\sqrt{\lambda} \gamma g(\beta, M,u(1)) u(1)=0.
\end{matrix} \right.
\end{equation*}
where $\gamma >0$ is related to matrix hostility, $ \lambda >0$ is related to the patch size of the habitat, $f(s) = \frac{1}{a}s(1-s)(a+s)$ represents weak Allee effect growth ($a \in (0,1)$), $r^*(\beta,m) = r_1^*(\beta,m) = \frac{e^{-\beta}}{m + e^{-\beta}}$ or $r^*(\beta,m) = r_2^*(\beta,m) = \frac{e^{\beta}}{m + e^{\beta}}$ are intrinsic growth rates with $m \geq 0$ and $\beta \geq 0$, and $g(\beta,M,s) = 1 + \frac{1}{M} (e^{-\beta s}-1)$ with $M \geq 1$ controlling strength of the DDE (density dependent emigration). We will consider both the density independent dispersal case ($\beta = 0$) and the negative density dependent dispersal case ($\beta > 0$). We will discuss the effects of interaction of $r^*(\beta,m)$ and $g(\beta,M,s)$ when $\beta$ and $m$ vary on the existence and multiplicity of positive solutions.

Name(s): Thomas Payne
Affiliation(s): University of North Carolina at Greensboro
Title: The Automorphic Equivalency Problem for Graph Products of Groups
Abstract:
In his seminal 1936 paper, Whitehead provided an algorithm to decide automorphic equivalence between elements in a free group. The peak reduction theorem central to Whitehead’s original algorithm has since been generalized to free products of (certain) groups and right-angled Artin groups, extending the settings for which this classic decision problem is known to be solvable. We present work in progress developing an algorithm to solve the automorphic equivalency problem in graph products of groups, a natural generalization of right-angled Artin groups. A solution is presented for a subgroup generated by certain partial conjugations and an approach for the full partial conjugation subgroup is sketched.

Name(s): Reetika Sarkar
Affiliation(s): University of North Carolina at Greensboro
Title: Improved Apriori Method for Safety Signal Detection Using Post-Marketing Clinical Data
Abstract:
This paper investigates the implementation of the classical Apriori algorithm, a popular method in association rule mining, to identify frequently co-occurring drugs and AEs within safety data. We discuss previous applications of the Apriori algorithm for safety signal detection and conduct a detailed study of an improved Apriori method specifically tailored for this purpose. We present detailed comparative analyses between the performance of Confidence and disproportionality measures as the second parameter in the implementation of Apriori for safety signal detection. This enhanced approach refines the classical Apriori method to effectively reveal potential associations between drugs/vaccines and AEs from post-marketing safety monitoring datasets, especially when AEs are rare.

Name(s): Mauyon Wusu
Affiliation(s): Alabama Agricultural and Mechanical University
Title: Implementation of Numerical Methods in Electric Circuit Analysis using C++
Abstract:
Electric circuit analysis is the process of solving all the voltages and currents in a network of connected circuit components. Like every other analysis, potential problems arise when conducting Electric Circuit Analysis. One of these problems is finding the current or voltages in each circuit branch when the circuit includes more than one battery or DC Component and consists of only resistors- no capacitors or inductors included. However, it is essential to note that this problem can be solved. In this study, the Gaussian Elimination and Gauss-seidel methods were utilized using C++ to solve a linear system of equations for the current value in each circuit node. This solution was determined by the relationship of V=IR (Ohm’s Law) and some other basic laws like Kirchhoff’s Current and Voltage Law. In this research, circuits were designed to solve for the current and voltages, and particular test values were used to satisfy some conditions of the methods and establish the solution’s credibility. Based on my data, codes, and calculations, this implementation of numerical methods and application of linear systems in Circuit analysis using C++ proved to be efficient and authentic.

Name(s): Caroline Wyrick
Affiliation(s): University of North Carolina Greensboro
Title: An Optional Mixture Binary RRT Model that Accounts for Trust and Measurement Errors
Abstract:
In survey research, collecting accurate data on sensitive topics can be challenging, as respondents may be reluctant to provide truthful answers due to privacy concerns, fear of repercussions, or tendency to provide perceived socially desirable responses. The Randomized Response Technique (RRT) offers a way to mitigate these issues by protecting respondents’ anonymity, encouraging more honest reporting. Over time, various RRT models have been developed to balance the need for privacy protection with the efficient estimation of sensitive traits.

In this project, we extend the work of Meche et al. (2024), which estimates sensitive traits while accounting for both distrust in the model and measurement error (difference between the true value of the variable and its recorded value). It may be noted that while mistrust is intentional, measurement errors are not intentional. Our extension introduces optionality, allowing respondents to determine whether they perceive a question as sensitive. If a respondent does not find the question sensitive, they can answer it directly, bypassing the RRT mechanism, while maintaining the researcher’s blindness to their decision. Extensive simulations demonstrate that incorporating optionality not only produces efficient estimators but also reduces estimator variance.

Name(s): Flora Yi
Affiliation(s): Wake Forest University
Title: Searching for Points on Curves in Projective Space over Number Fields
Abstract:
In this project, we developed code that searches for K-rational points of a specified height (which measures the complexity of a point), on a given projective curve C defined over a number field K.

The algorithm takes the approach described in C.L. Turner’s PhD thesis (also M. Watkins’s notes), which involves lifting points modulo an ideal $p$, searching for vectors in integer lattices, and evaluating points on the curve. An improvement from the previous approach is optimizing the runtime by considering the ideal to work with. Our code computes a work estimate (heuristically, based on runtime data gathered) of lattice construction and vector search for each potential ideal. Selecting an ideal which minimizes the work estimate results in significant runtime reduction.

Further improvements can be made to the work estimation, including that of a more rigorous and involved computation. The code is implemented in Magma, and available on GitHub (PointSearchNumberFields).