Mathematics Commons^™

Level Sets For Lehmer Codes Of Pattern Avoiding Permutations, Avery Sinclair

Mathematics, Statistics, and Computer Science Honors Projects

We study the poset structures for two families of pattern avoiding permutations. An n-permutation is a list of the numbers [n]={1,2,...,n}. A permutation is 321-avoiding when it does not contain a decreasing subsequence of length 3. A poset (partially ordered set) is a set such that some elements can be compared with one another. Using Lehmer codes, we define a poset for 321-avoiding permutations. We then fully describe the six lowest levels of this poset. We then consider the analogous poset for 123-avoiding permutations (which don't contain an increasing subsequence of length 3) and fully describe the three lowest levels.

Ngram Data Social Media, William Zywiak

Data and Datasets

Data set for a NSF STEM grant.

Lag Correlation Variance, William Zywiak

Data and Datasets

Data set for a NSF STEM grant.

Modeling The Clonal Rosette Composition Of A Bromeliaceae Genet: A Combinatorial Approach, Layla K. Lammers, Erin N. Bodine

Spora: A Journal of Biomathematics

Bromeliaceae, a neo-tropical plant family encompassing over 3,000 species, exhibit two modes of reproduction: sexual reproduction via flowers and seeds, and asexual reproduction via genetically identical clonal rosettes. The vegetative bodies of bromeliads form rosettes with new leaves emerging from the center and clonal rosettes emerging above a single leaf in the rosette, resulting in a genetic individual consisting of a seed-grown rosette and multiple iterations of clonal rosettes. This research develops a combinatorial model of probability that a single genetic individual will include at least n clonal rosettes when a single rosette can produce at most 1 or 2 …

Learning Motion Primitive Selection And Environment Abstraction, Edison Alberto Martinez Samaniego, Natalie Alexander, Kaelyn Weddle

Discovery Day - Daytona Beach

Learning Motion Primitive Selection and Environment Abstraction Advanced Air Mobility (AAM) is emerging as a transformative solution for short and medium range transportation; however, it introduces an operational model that differs significantly from conventional aviation. AAM vehicles are expected to operate closer to populated areas, with increased autonomy, in dense urban and suburban environments. These settings present constrained maneuvering conditions which highlights the importance of maintaining safe operation under degraded flight conditions. Abnormal conditions may endanger onboard passengers, people on the ground, and surrounding infrastructure, making rapid detection and mitigation essential to prevent loss of control. Recent research has explored …

Advancements In Spacecraft Trajectory Generation Through Matrix Decomposition Techniques, David Stoev, Oshani Jayawardane, Kaitlyn Cavanaugh, Kristiyan Stefanov, Andrew Murphy

Discovery Day - Daytona Beach

The Circular Restricted Three-Body Problem (CR3BP) is renowned for its intricate and chaotic dynamics, leaving it without a closed-form solution. In this poster, we introduce an innovative approach to determine spacecraft trajectories within the CR3BP framework using matrix factorization techniques. We formulate a matrix equation where the right-hand side vector is constructed from the spacecraft's position and velocity data, while the coefficient matrix is derived from the spacecraft's temporal data. Subsequently, we apply several matrix decomposition techniques, including modified Gram-Schmidt, the Householder technique, and Givens Rotation, to analyze the coefficient matrix and derive the spacecraft trajectories. Finally, we evaluate the …

Asymptotic Enumeration Of K-Bounded Functions Using Toeplitz Matrices, Tiago Cavalcante Trindade, Pedro Martineli

Rose-Hulman Undergraduate Mathematics Journal

We introduce the concept of $(\alpha, \lambda)$-bounded functions, characterized by limited local variation, which is especially useful in discrete sets. Initially, we formally define these functions and investigate their fundamental properties, highlighting significant differences from continuous functions. The main result obtained is the asymptotic estimate of $a(n, k)$, representing the number of functions from $[n]$ to $[n]$ that are $k$-bounded with respect to the Manhattan distance. The proof of this result combines Toeplitz matrices with a well-known inequality from graph theory.

Set-Valued Conjugate Calculus And Duality In Optimization And Machine Learning Via Generalized Relative Interiors, Gary Lee Sandine

Dissertations and Theses

Convex analysis, optimization, and duality theory are broadly applicable to a multitude of real-world problems. Theoretical frameworks ensuring strong duality and optimality are fundamentally tied to convex separation under qualification conditions frequently involving topological or relative interiors. These can be empty in convex sets that appear naturally in infinite-dimensional models, yet guarantees of optimality and strong duality are often attainable. This dissertation presents frameworks for applying convex separation via generalized relative interiors to develop generalized calculus rules as well as conditions for strong duality and optimality for infinite-dimensional constrained convex optimization problems.

The first major contribution is the introduction of …

Determinants And Invertibility In Finite Modular Systems, Osasu Omobude

Discovery Day - Daytona Beach

This project investigates determinants and matrix invertibility in finite modular systems, focusing on matrices over Zn. Using the Hill cipher as context, it examines the algebraic conditions under which a matrix is invertible in modular arithmetic. In particular, the project studies how the determinant determines invertibility, showing that a matrix over Zn is invertible if and only if its determinant is coprime with n.   The project further compares invertibility over the real numbers with invertibility over modular systems, highlighting the distinction between prime moduli Zp and composite moduli. In the prime case, matrices behave similarly to those over fields, where …

Low-Complexity Polynomial Ring Learning For Quantum Space Assets, Lola Torres

Discovery Day - Daytona Beach

Secure communication for space-based systems requires cryptographic methods that remain both reliable and efficient under strict computational constraints. This work investigates a low-complexity polynomial ring learning algorithm designed for quantum space assets, including satellite–ground communication systems. The project focuses on post-quantum cryptographic principles, where encryption and decryption rely heavily on repeated polynomial operations; this can be computationally expensive with constrained platforms. This is addressed with reformulating polynomial multiplication as a structured linear transformation on coefficient vectors. By representing these operations as matrices with a cyclic structure, the structure allows the use of the discrete Fourier transform (DFT); this will simplify …

Complex Continued Fractions And Inadmissible Sequences, Carolina A. Rizzi, Holly Vanlooy

Rose-Hulman Undergraduate Mathematics Journal

Serret's Theorem says that real numbers are related by a type of Möbius transformation if and only if the tails of their regular continued fraction expansions are the same. Serret’s Theorem does not hold for Hurwitz Continued Fractions in the complex plane due to a counterexample of Lakein. By applying transformations that convert inadmissible sequences into their admissible forms, we explain Lakein's counterexample from an algorithmic perspective. We provide additional counterexamples and prove that there exists an uncountably infinite family of counterexamples.

The General Solution Analysis Of Homogeneous Linear Equations, Jacob Schwamb, Edward Whipple

Discovery Day - Daytona Beach

The general solution analysis of homogeneous linear equations are any systems of equations in which all constant terms are equal to zero is classified as a homogeneous linear equation. Some key characteristics of homogeneous linear equations are that there are “zero” solutions, where every system has at least a single solution where all variables are zero, also all solutions to any homogenous linear equation is linearly independent, along with having preserved homogeneity, where if any variable (x) may be added to the system, then any scalar multiple of the variable is also a solution. The General solution of any homogeneous …

A Compact Representation Of Oscillatory Limits Via Asymptotic Value Distributions, Ibrahim Arnous, Eric M. Rodarte, Chirag Kumar

Discovery Day - Daytona Beach

Classical limits describe asymptotic behavior through convergence to a single value, but many important oscillatory functions do not converge in this sense. Standard examples such as sin(𝑥) as 𝑥→∞ and sin(1/x) as x→0 instead display stable distributions of values over time. This project examines how such behavior can be described using a measure-theoretic framework, particularly through occupation measures and, in sequence-based settings, Young measures. The objective is to present this perspective in a clear and accessible way while introducing the Ansatz representation, a compact notation for recording the support and density of an asymptotic value distribution when the limiting measure …

Differentiating The Impossible: Feynman's Trick In Applications Of Modern Physics, Yaohua Zhao

Discovery Day - Daytona Beach

We discuss Feynman’s method of differentiating with respect to a parameter inside an integral and explore its significance on selective topics of modern physics. This powerful technique allows us to integrate functions that may seem impossible. Depending on the underlying parameters, the Feynman method becomes a unifying framework that connects mathematical concepts to many parameter-dependent equations in modern physics. In statistical mechanics, this appears directly in the partition function, where derivatives with respect to temperature-related parameters yield thermodynamic quantities such as internal energy and heat capacity; this demonstrates how parameter dependence gives rise to macroscopic behaviors observable at a larger …

Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes, Martyna Wojcik

Discovery Day - Daytona Beach

Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations       It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions …

Engineering Path Trajectories With Gravitational Fields, Eliane Dean, Aidan Hart, Isabel Noot, Nathan Browning, Valeria Villazon Fito

Discovery Day - Daytona Beach

This project explores how vector calculus concepts play a role in aerospace engineering though spacecraft trajectory design. In particular, the notion of vector fields is used to model the gravitational force, whose work done is expressed through line integrals. By taking the curl of the gravitational field and showing it is zero, the field is recognised as conservative, implying that the work done by gravity is path independent. This property is conceptually linked to gravitational potential energy and the principle of energy conservation. The results are then applied to spacecraft motion, where engineers use energy-base methods to determine efficient trajectories …

Visualizing The Invisible: Simulating Electromagnetic Field In 3d Space With Matlab, Aashman Gupta, Eliane Dean, Riley Esperanca, Hailey Grabinski, Jacob Summerhays, Ameya Sute

Discovery Day - Daytona Beach

Electromagnetic field behaviours in free space are defined by Maxwell’s Equations, which couple the temporal and spatial variations of electric and magnetic fields through partial derivatives. These derivatives quantify the rate of change of each field’s vector component with respect to position and time in a 3D lattice, forming the basis for numerical field analysis. This research will develop a mathematical and computational framework using multivariable calculus to model, simulate, and visualize electromagnetic wave propagation in free space using MATLAB. Gradient, divergence, and curl operations are implemented to compute local field variations and energy transfer. The resulting data are used …

Modelling Quadcopter Thrust And Torque Using Lu-Decomposition, Diya Patil

Discovery Day - Daytona Beach

3 & 4 motor systems have been in the world of aviation in many different forms as the field grows and evolves. To understand the complexities of an Unmanned Aerial Vehicle (UAV) and its stability, assessing the amount of thrust put into each motor can help generate the torque produced despite factors such as multidirectional movement. While a UAV does this multiple times a second, producing a simplified version of this calculation can aid in simpler models simulating UAV movement. Due to the popularity of the quadcopter drone, a simple algorithm depicting thrust through each spinning motor can aid in …

Underclosed Posets, Richard Ngo

McNair Summer Research Program

Underclosed complexes are a recent generalization of interval graphs to higher dimensions. Motivated by underclosed complexes, we define and study underclosed posets. Order ideals of these posets correspond to pure underclosed complexes. We classify which principal order ideals are rank-symmetric (and in fact are self-dual).

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …