Physical Sciences and Mathematics Commons^™

Staircase Packings Of Integer Partitions, Melody Arteaga

Mathematics, Statistics, and Computer Science Honors Projects

An integer partition is a weakly decreasing sequence of positive integers. We study the family of packings of integer partitions in the triangular array of size n, where successive partitions in the packings are separated by at least one zero. We prove that these are enumerated by the Bell-Like number sequence (OEIS A091768), and investigate its many recursive properties. We also explore their poset (partially ordered set) structure. Finally, we characterize various subfamilies of these staircase packings, including one restriction that connects back to the original patterns of the whole family.

A Brascamp-Lieb–Rary Of Examples, Anina Peersen

Mathematics, Statistics, and Computer Science Honors Projects

This paper focuses on the Brascamp-Lieb inequality and its applications in analysis, fractal geometry, computer science, and more. It provides a beginner-level introduction to the Brascamp-Lieb inequality alongside re- lated inequalities in analysis and explores specific cases of extremizable, simple, and equivalent Brascamp-Lieb data. Connections to computer sci- ence and geometric measure theory are introduced and explained. Finally, the Brascamp-Lieb constant is calculated for a chosen family of linear maps.

Framing How We Think About Curves, Megan Sattler

Mathematical Sciences Undergraduate Honors Theses

A frame defines a basis at a point in R^n, and we can frame a curve by placing one at every point along it. This thesis investigates adapted framed curves in R^2 and R^3 in which they are used to provide information about how a curve twists and turns. We derive differential information from the frames that describe how they change and consequently how the curve changes. We also deduce special properties of each framing and discuss how the differential information suffices to describe the shape of curves.

Modeling Wlan Received Signal Strengths Using Gaussian Process Regression On The Sodindoorloc Dataset, Fabian Hermann Josef Fuchs

Theses and Dissertations

While any wireless technology can be used for indoor localization purposes, WLANhas the advantage of having a huge existing infrastructure. A radio map that matches specific locations to received signal strength is needed, to enable most of these indoor localization methods. To create these radio maps, with enough detail to achieve sufficient localization accuracy, is expensive and time consuming. Therefore, methods to interpolate and extrapolate more detailed maps from sparse radio maps are being developed. One recent approach is to use Gaussian process regression. Even though some papers already studied Gaussian process regression, most studied only the basic model with …

Modeling Brain Tumor Dynamics With The Help Of Cellular Automata, Paula Kathrin Viktoria Jaki

Theses and Dissertations

Primary brain tumors pose a serious threat to a person’s health. Gaining a deeper understanding of the dynamics of tumor growth is crucial for developing a proper treatment plan. Many computational models have been suggested to investigate the interaction between tumor cells and their surroundings. Using cellular automata is particularly promising since it integrates the features of self-organizing complex systems which allows them to properly depict local interactions between cells. The foundation of the thesis lies in the model proposed by Kansal et al. It uses four microscopic parameters, the maximum tumor extent, the base proliferative and necrotic thickness as …

The Hilbert Sequence And Its Associated Jacobi Matrix, Caleb Beckler

Honors Theses

In this project, we investigate positive definite sequences and their associated Jacobi matrices in Hilbert space. We set out to determine the Jacobi matrix associated to the Hilbert sequence by methods described in Akhiezer’s book The Classical Moment Problem. Using methods in Teschl’s book Jacobi Operators and Completely Integrable Nonlinear Lattice, we determine the essential spectrum of the corresponding Jacobi matrix.

Enestr¨Om-Kakeya Type Results For Complex And Quaternionic Polynomials, Matthew Gladin

Electronic Theses and Dissertations

The well known Eneström-Kakeya Theorem states that: for P(z)=∑_i=0ⁿ a_i zⁱ, a polynomial of degree n with real coefficients satisfying 0 ≤ a₀ ≤ a₁ ≤ ⋯≤ a_n, all zeros of P(z) lie in |z|≤1 in the complex plane. In this thesis, we will find inner and outer bounds in which the zeros of complex and quaternionic polynomials lie. We will do this by imposing restrictions on the real and imaginary parts, and on the moduli, of the complex and quaternionic coefficients. We also apply similar restrictions on complex polynomials with …

Staircase Arrangements Of Pillars With Distinct Heights, Andrea L. Simmons

Mathematics, Statistics, and Computer Science Honors Projects

We study the family A_n of sequences (a₁, a₂, ..., a_n) where 0 ≤ a_k ≤ k and nonzero entries are distinct. We show that these sequences are in bijection with the set partitions of [n + 1]. These sequences have a natural poset structure, and we analyze the maximal chains within this poset. Finally, we explore various subfamilies of A_n, including sequences whose largest entry is k and sequences missing the value k.

Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance

Mathematics, Statistics, and Computer Science Honors Projects

A mixing measure is the expected length of a random walk in a graph given a set of starting and stopping conditions. We determine the tree structures of order n with diameter d that minimize and maximize for a few mixing measures. We show that the maximizing tree is usually a broom graph or a double broom graph and that the minimizing tree is usually a seesaw graph or a double seesaw graph.

Uconn Baseball Batting Order Optimization, Gavin Rublewski, Gavin Rublewski

Honors Scholar Theses

Challenging conventional wisdom is at the very core of baseball analytics. Using data and statistical analysis, the sets of rules by which coaches make decisions can be justified, or possibly refuted. One of those sets of rules relates to the construction of a batting order. Through data collection, data adjustment, the construction of a baseball simulator, and the use of a Monte Carlo Simulation, I have assessed thousands of possible batting orders to determine the roster-specific strategies that lead to optimal run production for the 2023 UConn baseball team. This paper details a repeatable process in which basic player statistics …

Analyzing Real Life Scenarios Through Linear And Exponential Functions Using Open Pedagogy., Lili Grigorian

Open Educational Resources

This assignment is on linear and exponential growth, which is connected to real life scenarios from students’ everyday life as well as teaches them financial responsibility and awareness of economic issues. Project has five parts. In part 1, students would use digital communication ability by creating a video about the topic from the knowledge they had prior to this project. In part 2 (the Mathematical part) they will use problem-solving and inquiry learning to better understand linear and exponential growth, in part 3 students will use global learning through reading to then annotate article and watch video to then discuss …

Examining Factors Using Standard Subspaces And Antiunitary Representations, Paul Anderson

Undergraduate Honors Theses

In an effort to provide an axiomization of quantum mechanics, John von Neumann and Francis Joseph Murray developed many tools in the theory of operator algebras. One of the many objects developed during the course of their work was the von Neumann algebra, originally called a ring of operators. The purpose of this thesis is to give an overview of the classification of elementary objects, called factors, and explore connections with other mathematical objects, namely standard subspaces in Hilbert spaces and antiunitary representations. The main results presented here illustrate instances of these interconnections that are relevant in Algebraic Quantum Field …

An Integer Garch Model For A Poisson Process With Time-Varying Zero-Inflation, Isuru Panduka Ratnayake, V. A. Samaranayake

Mathematics and Statistics Faculty Research & Creative Works

A serially dependent Poisson process with time-varying zero-inflation is proposed. Such formulations have the potential to model count data time series arising from phenomena such as infectious diseases that ebb and flow over time. The model assumes that the intensity of the Poisson process evolves according to a generalized autoregressive conditional heteroscedastic (GARCH) formulation and allows the zero-inflation parameter to vary over time and be governed by a deterministic function or by an exogenous variable. Both the expectation maximization (EM) and the maximum likelihood estimation (MLE) approaches are presented as possible estimation methods. A simulation study shows that both parameter …

Volatility Modeling Of Time Series Using Fractal And Self-Similarity Models, William Kubin

Open Access Theses & Dissertations

The study uses various methods to compare financial and geophysical time series scaling parameters and long-term memory behavior. The Cantor Detrended Fluctuation Analysis (CDFA) method is proposed to provide more accurate estimates of Hurst exponents. The CDFA method is applied to real-time series and the results are verified. The study also analyzes the memory behavior of daily Covid-19 cases before and after the announcement of effective vaccines. Low and high-frequency dataâ??s influence on the Hurst Index estimation is investigated, and a new PCDFA method is proposed. The stability of the Dow Jones Industrial Average is analyzed using a multi-scale normalized …

An Investigation Into Optimal Descent Trajectories For Multipurpose Long Range Space Vehicles Under Advanced Conditions, John M. Levis

Theses and Dissertations

In this work, we investigate the problem of fuel-optimal control of space vehicle descent trajectories. The main tool we use to establish optimality is Pontryagin’s Maximum Principle. We present a variety of scenarios with increasing complexities, including drag, wind, and moving landing platforms in the context of differing atmospheric and gravitational conditions. Throughout the paper, we use a balance of analytical and numerical techniques. Finally, observations and conclusions drawn from the investigation form the basis for suggestions into additional areas of analysis.

An Analysis Of Antichimeral Ramanujan Type Congruences For Quotients Of The Rogers-Ramanujan Functions, Ryan A. Mowers

Theses and Dissertations

This paper proves the existence of antichimeral Ramanujan type congruences for certain modular forms These modular forms can be represented in terms of Klein forms and the Dedekind eta function. The main focus of this thesis is to introduce the necessary theory to characterize these specific Ramanujan type congruences and prove their antichimerality.

Data Science For Hospital Antibiotic Stewardship, Saikou Jawla

Theses and Dissertations

Antibiotics are widely used to treat bacterial infections, but their misuse leads to antibiotic resistance. Antibiotic resistance is one of the biggest threats to global health, food security, and development today. Antibiotic resistance leads to higher medical costs, prolonged hospital stays, and increased mortality. Antimicrobial stewardship is an approach to measure and improve the appropriate use of antibiotics in healthcare settings. Data science has the potential to support these programs by providing insights into antibiotic prescribing patterns, identifying areas for improvement, and predicting patient outcomes. We explored the role of data science in hospital antibiotic stewardship programs, including statistical methods …

Congruences For Quotients Of Rogers-Ramanujan Functions, Maria Del Rosario Valencia Arevalo

Theses and Dissertations

In 1919 the mathematician Srinivasa Ramanujan conjectured congruences for the partition function p(n) modulo powers of the primes 5,7,11. In this work, we study Ramanujan type congruences modulo powers of primes p = 7,11,13,17,19,23 satisfied by the Fourier coefficients of quotients the Rogers-Ramanujan Functions G(τ) and H(τ) and the Dedekind eta function η(5τ). In addition to deriving new congruences, we develop the foundational theory of modular forms to motivate and prove the results. The work includes proofs of congruences facilitated by Python/SageMath code.

Automorphisms Of A Generalized Quadrangle Of Order 6, Ryan Pesak

Undergraduate Honors Theses

In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle Q can exist, we show that if it does, that Q cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle Q of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group G acts faithfully on …

Intrinsic Tame Filling Functions And Other Refinements Of Diameter Functions, Andrew Quaisley

Department of Mathematics: Dissertations, Theses, and Student Research

Tame filling functions are quasi-isometry invariants that are refinements of the diameter function of a group. Although tame filling functions were defined in part to provide a proper refinement of the diameter function, we show that every finite presentation of a group has an intrinsic tame filling function that is equivalent to its intrinsic diameter function. We then introduce some alternative filling functions—based on concepts similar to those used to define intrinsic tame filling functions—that are potential proper refinements of the intrinsic diameter function.

Adviser: Susan Hermiller and Mark Brittenham

Prefix-Rewriting: The Falsification By Fellow Traveler Property And Practical Computation, Ash Declerk

Department of Mathematics: Dissertations, Theses, and Student Research

The word problem is one of the fundamental areas of research in infinite group theory, and rewriting systems (including finite convergent rewriting systems, automatic structures, and autostackable structures) are key approaches to working on the word problem. In this dissertation, we discuss two approaches to creating bounded regular convergent prefix-rewriting systems.

Groups with the falsification by fellow traveler property are known to have solvable word problem, but they are not known to be automatic or to have finite convergent rewriting systems. We show that groups with this geometric property are geodesically autostackable. As a key part of proving this, we …

Invariants Of 3-Braid And 4-Braid Links, Mark Essa Sukaiti

Theses

In this study, we established a connection between the Chebyshev polynomial of the first kind and the Jones polynomial of generalized weaving knots of type W(3,n,m).
Through our analysis, we demonstrated that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices which allowed us to find an explicit formula for the Alexander polynomial of weaving knots of typeW(3,n).
In addition to confirming Fox’s trapezoidal conjecture, we also discussed the zeroes of the Alexander Polynomial of weaving knots of type W(3,n) as they relate to Hoste’s conjecture. In addition, …

Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov

Electronic Theses, Projects, and Dissertations

Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig's lemma …

A Case Study In Bivariate Spline Spaces, Ryann Lee Firestine

<strong> Theses and Dissertations </strong>

Loosely speaking, splines are piece-wise polynomial functions which are continuously differentiable of order r. The individual polynomial segments are degree less than or equal to d. Splines are typically associated with approximation theory, computer-aided graphic design, and partial differential equations. Due to the wide variety of applications, mathematicians became interested in better understanding the behavior of the vector spaces containing splines in various setting and values r and d. In particular, many focused on formulating the dimension of these vector spaces. In the 1980s, commutative algebra was first applied to the study of spline spaces. In our work, we use …

Let’S Make Patterns!: Symmetric Rubik’S Cube Permutations, Danny Anderson

Theses/Capstones/Creative Projects

Rubik’s cubes are well-known for having several different patterns, or permutations, that can be made from them. Meanwhile, cubes generally display a wide variety of symmetries. Naturally, these ideas can be combined to form a notion of "symmetric Rubik's cube patterns." The goal of this paper is to find an algorithm that can produce all of the permutations that display symmetries.

Comparative Study Of Variable Selection Methods For Genetic Data, Anna-Lena Kubillus

Theses and Dissertations

Association studies for genetic data are essential to understand the genetic basis of complex traits. However, analyzing such high-dimensional data needs suitable feature selection methods. For this reason, we compare three methods, Lasso Regression, Bayesian Lasso Regression, and Ridge Regression combined with significance tests, to identify the most effective method for modeling quantitative trait expression in genetic data. All methods are applied to both simulated and real genetic data and evaluated in terms of various measures of model performance, such as the mean absolute error, the mean squared error, the Akaike information criterion, and the Bayesian information criterion. The results …

Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun

Electronic Theses and Dissertations

The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we …

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

Doctoral Dissertations

Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …

Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson

Doctoral Dissertations

We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.

Long-Time Stability For An Imex Discretization Of The 1d Fujita Equation, Victoria Luongo

Honors College Theses

We study an efficient time-stepping scheme for the 1D Fujita equation that is implicit for the linear terms but explicit for the nonlinear terms. We analyze the long-time stability of the scheme for varying parameter values, which reveal parameter value regimes in which the method is stable. We provide numerical results that illustrate the theory and show the analytically derived stability conditions are sufficient to achieve long-time stability results.