Mathematics Commons^™

Numerals And Other Means Of Expressing Plurality In Language And Thought: A Developmental Perspective Across Cultures., Giovanni Giuseppe G.G. Nicosia Prof.

Numeracy

Several ancient languages use linguistic structures to differentiate between singular, dual, and plural forms that are used when one, two, or many objects are referenced. (Some languages go further, adding a trial form to differentiate two from three.) In many modern languages, numerals are used to convey precise number. After reviewing a range of ways languages express number (including the interesting case of Pirahã which does not express precise number), the paper concludes with a proposed developmental path for the expression of quantity in languages.

Teaching Numeracy For Social Justice: Educational Equity, Esther Isabelle Wilder, Crystal C. Rodriguez, Caterina Shost, Eduardo Vianna, Frank Wang

Numeracy

The special collection, “Teaching Numeracy for Social Justice: Educational Equity,” focuses on how quantitative reasoning (QR) can function as a vehicle for equity, empowerment, and democratic participation. Building on a longstanding tradition that treats numeracy as inseparable from social justice, these contributions highlight how progressive pedagogies (e.g., active learning, authentic research, and equity-oriented assessment) have the potential to broaden access for students historically excluded from quantitative fields. The studies span a variety of conceptual frameworks, introductory and advanced quantitative instruction, and interdisciplinary applications, showcasing how inclusive QR practices can build confidence, agency, and real-world understanding. These articles demonstrate that when …

Bayes In The Brain: A Review Of Everything Is Predictable: How Bayesian Statistics Explain Our World, (2024) By Tom Chivers., Michael T. Catalano

Numeracy

Tom Chivers’ Everything is Predictable: How Bayesian Statistics Explain Our World, is an interesting and wide-ranging narrative on Bayesian thinking, its history, and its applicability to both our everyday lives and the pursuit of scientific truth. Although appropriate for the non-expert, afficionados and teachers of quantitative literacy should find the plethora of examples, links to psychology as it applies to how people reason about probabilities, and even Chivers’ philosophical musings informative and thought-provoking.

Exploring System Identification Of Non-Linear Dynamics Using The Weighted Composition Operator And The Liouville Operator, Chukwuebuka Amagwula

USF Tampa Graduate Theses and Dissertations

System identification is the process of determining mathematical models that describe the dynamics of a system from data. Dynamic Mode Decomposition (DMD) and Sparse Identification of Nonlinear Dynamical Systems (SINDy) are two distinct approaches used for this purpose.

DMD identifies dominant spatiotemporal modes and eigenvalues that describe the evo lution of a system. It assumes a near-linear representation of dynamics and is closely linked to the Koopman operator, making it ideal for analyzing fluid flows, oscillatory systems, and modal structures. The DMD method uses time series data where each data point is referred to as a snapshot and represents the …

Applications Of Contracting Self-Similar Groups To Cryptography And Scale Groups, Arsalan Akram Malik

USF Tampa Graduate Theses and Dissertations

Given their peculiar properties, self-similar groups are of great interest both from applications and theoretical standpoints. In this work we study the scope of their applications in post-quantum cryptography and in constructing scale groups via lifting maps.

We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). This class of groups admits fast polynomial-time algorithms for the word problem and element multiplication that can be used for effective encryption and decryption of messages. It contains extraordinary examples like the Grigorchuk group, which is known to be non-linear, thus making some of existing …

On Some Hypergraph Coloring Algorithms, Boyoon Lee

USF Tampa Graduate Theses and Dissertations

Of a given bipartite graph G = (V, E), it is elementary to construct a bipartition in timeO(|V |+|E|). For a given k-graph H = H(k) with k ≥ 3 fixed, Lovász proved that deciding whether H is bipartite is NP-complete. In this thesis, we consider the average running time of this problem. For that, let B_n = (B^(k))_n be the collection (family) of all bipartite k-graphs H on the fixed vertex set [n] = {1, ...., n}. We construct, of a given H ∈ B_n, a bipartition in time averaging O(n^k) over the class B_n. We provide two proofs …

The Only Constant Is Change: The Role Of Genetic Diversification In Cancer And Beyond, Malgorzata Tyczynska Weh

USF Tampa Graduate Theses and Dissertations

Genetic diversification, the process by which genetic variation arises in a population, is fundamental to evolution by natural selection. Mutation, a central diversification process, fuels adaptation across species, including in cancer. Yet most new mutations are neutral or deleterious on cell fitness, raising the question of how mutation-driven adaptation persists. To explore this paradox, I developed a spatial agent-based model (ABM) in which population fitness emerges from individual cells acquiring mutations at varied rates and with diverse fitness effects. I evaluated model behavior across adaptive states, mutation rates, and distributions of fitness effects. The results show that high mutation rates …

Topics In Knot, Quandle, And Quiver Theory, Brooke Jones

USF Tampa Graduate Theses and Dissertations

This dissertation explores the algebraic and combinatorial structures arising from the study of knot theory through the lens of quandles, their colorings, and associated quiver and ring constructions. It is organized around three central themes: the analysis of quandle coloring quivers for torus links, the development of polynomial invariants and quiver invariants for stuck knots and links, and the algebraic and graphical study of quandle rings. The first chapter begins with an introduction to relevant concepts throughout the dissertation.

In the second chapter, we classify quandle coloring quivers of torus links using dihedral quandles. We describe these quivers as weighted …

Fractional Calculus Approach For Learning Unknown Dynamic System, Haowei(Alice) Chen

USF Tampa Graduate Theses and Dissertations

The rapidly growing interest in data-driven modeling and fractional-order systems highlights a keychallenge in modern science and engineering: capturing long-memory effects and nonlinear behaviors in real-world dynamical processes. Conventional integer-order methods, while powerful, often overlook the history-dependent nature of many phenomena—ranging from viscoelastic materials to anomalous diffusion and hereditary feedback systems. This dissertation addresses that gap by blending fractional calculus with cutting-edge machine learning to create robust, memory-aware modeling and control frameworks.

Beginning with an extension of Dynamic Mode Decomposition (DMD) to fractional-order systems, we introduce Fractional Dynamic Mode Decomposition (F-DMD) as a data-driven tool that leverages Mittag- Leffler functions …

Telu Activation Function For Fast And Stable Deep Learning, Alfredo Fernandez

USF Tampa Graduate Theses and Dissertations

We propose the Hyperbolic Tangent Exponential Linear Unit (TeLU), a neural network hidden activation function defined as $TeLU(x)=x \cdot tanh(e^x)$. TeLU’s design is grounded in the core principles of key activation functions, achieving strong convergence by closely approximating the identity function in its active region while effectively mitigating the vanishing gradient problem in its saturating region. Its simple formulation enhances computational efficiency, leading to improvements in scalability and convergence speed. Unlike many modern activation functions, TeLU seamlessly combines the simplicity and effectiveness of ReLU with the smoothness and analytic properties essential for learning stability in deep neural networks. TeLU’s ability …

Quantitative Reasoning: What’S Math Got To Do With It?, Pamela Burdman

Numeracy

This keynote address explores the history and role of college math requirements with a focus on ensuring math courses serve to expand students’ horizons, rather than serve as gatekeepers. It discusses the advent of general education math courses, which brought more students into math departments, which ultimately contributed to broadening the scope of the courses to align with more students’ interests and majors, since their purpose was to advance quantitative reasoning, not mathematics skill per se. It also examines several practices to address calculus’ gatekeeping role: revising placement practices and prerequisites, redesigning courses, and updating instruction and assessment practices. Lastly, …

Improvements In Computational Techniques For Determining Ideal Class Groups And Class Numbers, Muhammed Rashad Erukulangara

USF Tampa Graduate Theses and Dissertations

The ideal class group is a fundamental concept in algebraic number theory, providing insights into the structure and factorization properties of the ring of integers of a number field. It measures the extent to which unique factorization fails in the ring of integers. Efficiently computing the ideal class group is crucial for exploring unproved heuristics in number theory and solving Diophantine equations. Additionally, the computation of the ideal class group has several cryptographic applications, such as schemes based on the Discrete Logarithm Problem (DLP), the computation of isogenies, and groups of unknown order. Despite its importance, much about the ideal …

Optimal Selection Of Good Polynomials And Constructions Of Locally Recoverable Codes Via Galois Theory, Austin Dukes

USF Tampa Graduate Theses and Dissertations

To keep up with the ever-growing demand for reliable and efficient availability of data,locally recoverable codes (LRCs) have been the focus of much study due to their applications in cloud and distributed storage systems. A fundamental construction of LRCs was given in [36] based on polynomials which relied on the existence of r-good polynomials. In the same paper some constructions of good polynomials were given, but these constructions did not cover every configuration of parameters. Naturally this led to research into constructing good polynomials for what was not addressed in [36], but new ponderings were also posed, such as the …

The Effect Of Fixed Time Delays On The Synchronization Phase Transition, Shaizat Bakhytzhan

USF Tampa Graduate Theses and Dissertations

Nature is full of synchronization phenomena, which are essential to many scientific fields like biology, chemistry, physics, and neuroscience. The Kuramoto model is a well-known theoretical model that helps explain the fundamental ideas behind synchronization dynamics [6]. Nevertheless, in practical situations, systems frequently display intrinsic latency, which can greatly impact their behavior during synchronization. This insight inspired our work, which looks at the results of adding temporal delays to the Kuramoto model. In particular, we investigate how the system’s synchronization dynamics are affected by delays. We shed light on the mechanisms underpinning synchronization in the face of temporal delays and …

Quandle Rings, Idempotents And Cocycle Invariants Of Knots, Dipali Swain

USF Tampa Graduate Theses and Dissertations

Quandles are sets with self-distributive binary operations that axiomatize the three Reidemeister movesin classical knot theory. In an attempt to bring ring theoretic techniques to the study of quandles, a theory of quandle rings analogous to the classical theory of group rings where several interconnections between quandles and their associated quandle rings have been explored. Functoriality of the construction implies that morphisms of quandle rings give a natural enhancement of the well-known quandle coloring and quandle 2 cocycle invariant of knots and links.

The dissertation is structured into two main parts. In the first part, we delve into quandle rings …

On The Subelliptic And Subparabolic Infinity Laplacian In Grushin-Type Spaces, Zachary Forrest

USF Tampa Graduate Theses and Dissertations

This thesis poses the ∞-Laplace equation in Grushin-type spaces. Grushin-type spaces G are defined by the vector fields which serve as a basis for their tangent spaces; by weighting the canonical (Euclidean) directional vectors {∂/∂x_i}ⁿ_i=1 by functions ρ_i that obey certain technical assumptions, we produce a class of metric spaces in which certain directions may not be accessible at all points in the space. We prove the existence and uniqueness of viscosity solutions to both Dirichlet problems and Cauchy-Dirichlet problems involving the∞-Laplacian over bounded Grushin-type domains. The main tool in proving uniqueness of …

The International Crisis In Numeracy Education, Nathan D. Grawe

Numeracy

The OECD recently released results from the 2022 administration of the Programme for International Student Assessment test. As other studies suggest, pandemic mitigation policies resulted in deep learning loss including in basic mathematics which forms the foundation of numeracy. Perhaps of greater concern, however, in many countries pandemic effects amplify declining performance that dates back a decade or more. Losses of two or more years' worth of mathematics education are not uncommon among developed countries. The editorial makes an urgent call for research that identifies practical steps to reverse these trends.

A Study On The Correlation Between A Star's Rayleigh-Taylor Characteristic Timescale And Stellar Wind Activity, Fiona Klett

Undergraduate Journal of Mathematical Modeling: One + Two

In this paper, we investigate the correlations between a star's internal dynamics due to the Rayleigh-Taylor instability and episodes of stellar wind activity, using both a theoretical model and observational data from the NOAA.% \cite{NOAA}. Besides its relevance as an astrophysics problem, this study is also informative for models of climate change which include secular perturbations in the Sun's internal dynamics, as a potential source of solar activity variability.

Quantifying Non-Primary Dna Formations Through Mechanical And Geometric Models, Sonia E. Teodorescu

Undergraduate Journal of Mathematical Modeling: One + Two

In this article, mechanical and geometric models for DNA strains (regarded as a helical structure in 3 dimensions, embedded into surfaces of various shapes (straight or curved cylinders, spheres, or projected into planes), are analyzed in order to obtain parameter estimates for DNA characteristics which can be used to detect the formation of secondary and tertiary formations in the presence of disorder. The models allow for the explicit representation of the DNA shape on constrained geometries and can therefore be implemented directly into {\it{ab initio}} or synthetic simulation studies.

An Introduction To The Algebra Revolution, Art Bardige

Numeracy

Bardige, Art. 2022. The Algebra Revolution: How Spreadsheets Eliminate Algebra 1 to Transform Education; (Bookbaby) 135 pp. UNSPSC 55111505.

The Algebra Revolution: How Spreadsheets Eliminate Algebra 1 to Transform Education argues that Algebra 1 can be eliminated by teaching mathematics through spreadsheets. Such a change would eliminate the greatest roadblock to student achievement.

Applied Analysis For Learning Architectures, Himanshu Singh

USF Tampa Graduate Theses and Dissertations

Modern data science problems revolves around the Koopman operator Cφ (or Composition operator) approach, which provides the best-fit linear approximator to the dynamical system by which the dynamics can be advanced under the discretization. The solution provided by Koopman in the data driven methods is in the sense of strong operator topology, which is nothing better then the point-wise convergence of data (snapshots) in the underlying Hilbert space. Chapter 2 provides the details about the aforementioned issues with essential counter-examples. Thereafter, provable convergence guarantee phenomena is demonstrated by the Liouville weighted composition operators Af,φ over the Fock space by providing …

Classification Of Finite Topological Quandles And Shelves Via Posets, Hitakshi Lahrani

USF Tampa Graduate Theses and Dissertations

The objective of this dissertation is to investigate finite topological quandles and topological shelves. Precisely, we give a classification of both finite topological quandles and topological shelves using the theory of posets. For quandles with more than one orbit, we prove the following Theorem.

Proposition 0.0.1. Let X be a finite quandle with n orbits X₁, ... , X_n. Then any right continuous poset on X is n-partite with vertex sets X₁, ... , X_n.

For connected quandles, we prove the following Theorem.

Theorem 0.0.2. There is no T …

Rational Functions Of Degree Five That Permute The Projective Line Over A Finite Field, Christopher Sze

USF Tampa Graduate Theses and Dissertations

Rational functions over a finite field Fq induce mappings from the projective line P1(Fq) to itself. Rational functions that permute the projective line are called permutation rational functions (PRs). The notion of permutation rational functions is a natural extension of the permutation polynomials which have been studied for over a century. Recently, PRs of degrees up to four have been determined. This dissertation is a project aimed at determining PRs of degree five.

Rational functions of degree five (excluding those that are equivalent to polynomials) are divided into five cases according to the factorization of their denominators. Our main results …

Data-Driven Learning Algorithm Via Densely-Defined Multiplication Operators And Occupation Kernels., John Kyei

USF Tampa Graduate Theses and Dissertations

Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this …

Matrix Models Of 2d Critical Phenomena, Nathan Hayford

USF Tampa Graduate Theses and Dissertations

The 2D Ising model has played an important role in the theory of phase transitions, as one of only ahandful of exactly solvable models in statistical mechanics. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been …

Recovering Generators Of Principal Ideals Using Subfield Structure And Applications To Cryptography, William Youmans

USF Tampa Graduate Theses and Dissertations

The principal ideal problem (PIP) is the problem of determining if a given ideal of a number field is principal, and if so, of finding a generator.Algorithms for resolving the PIP can be efficiently adapted to solve many hard problems in algebraic number theory, such as the computation of the class group, unit group, or $S$-unit group of a number field. The PIP is also connected to the search for approximate short vectors, known as the $\gamma$-Shortest Vector Problem ($\gamma$-SVP), in certain structured lattices called ideal lattices, which are prevalent in cryptography. We present an algorithm for resolving the PIP …

Exploring The Vulnerability Of A Neural Tangent Generalization Attack (Ntga) - Generated Unlearnable Cifar-10 Dataset, Gitte Ost

USF Tampa Graduate Theses and Dissertations

Nowadays, a massive amount of data is generated and stored on servers and cloudsfrom various applications daily. Preventing these data from unauthorized use often becomes necessary and critical in various real-world applications. Many researchers have studied this crucial problem and developed different methods for this purpose. Among them, Neural Tangent Generalization Attack (NTGA) is one of the most efficient methods to make a dataset unlearnable, which means that the dataset is not learnable by machine learning/deep learning methods. That is, the NTGA-generated dataset is protected against unauthorized use. In this thesis, we explore the vulnerability of an NTGA-generated unlearnable CIFAR-10 …

Accelerating Multiparametric Mri For Adaptive Radiotherapy, Shraddha Pandey

USF Tampa Graduate Theses and Dissertations

MR guided Radiotherapy (MRgRT) marks an important paradigm shift in the field of radiotherapy. Superior tissue contrast of MRI offers better visualization of the abnormal lesions, as a result precise radiation dose delivery is possible. In case of online treatment planning, MRgRT offers better control of intratumoral motion and quick adaptation to changes in the gross tumor volume. Nonetheless, the MRgRT process flow does suffer from some challenges that limit its clinical usability. The primary aspects of MRgRT workflow are MRI acquisition, tumor delineation, dose map prediction and administering treatment. It is estimated that the acquisition of MRI takes around …

“Math Talks Are Like An Alarm Clock Waking You Up”: Language’S Crucial Role In Mathematics, Gabriella M. Wasser

Journal of Practitioner Research

Whole group math talks, or number talks, are a common practice to get students talking about their own understanding of mathematical concepts. The purpose of this study was to implement math talks in small group settings to see what would happen, specifically to students’ conceptual understanding as well their general perceptions of math talks. This study took place in a fourth-grade math classroom, and math talks were implemented with the whole class for a week and then moved to small groups for the remaining three weeks of the study. During the study, a pre-and post-assessment was given, field notes were …

Boundary Behavior Of Analytic Functions And Approximation Theory, Spyros Pasias

USF Tampa Graduate Theses and Dissertations

In this Thesis we deal with problems regarding boundary behavior of analytic functions and approximation theory. We will begin by characterizing the set in which Blaschke products fail to have radial limits but have unrestricted limits on its complement. We will then proceed and solve several cases of an open problem posed in \cite{Da}. The goal of the problem is to unify two known theorems to create a stronger theorem; in particular we want to find necessary and sufficient conditions on sets $E_1\subset E_2$ of the unit circle such that there exists a bounded analytic function that fails to have …