Analysis Commons^™

From Counting To Advanced Math: How Five Simple Axioms Shape The Mathematical Landscape, Joy Skaggs

ATU Scholars Symposium

In the late nineteenth and early twentieth centuries, mathematics faced a foundational crisis: Greog Cantor’s set theory led to an interesting self-referencing paradox in math and questions about the logical consistency of mathematics. From this crisis, two opposing viewpoints emerged: the Formalists and the Intuitionists. The Formalists praised Cantor’s work as a way to place math on a secure logical foundation, ensuring the discipline’s purity, however the Intuitionists despised Cantor’s work and heralded Cantor as a charlatan and corrupter of the youth. The leader of the Formalists, David Hilbert, proposed a formal system of rigorous proofs to build a complete …

Employing Effective Solution Methods For Caputo-Based Sequential Fractional Models, Eman A. A. Ziada, Mohamed F. Abouelenein, Hijaz Ahmad, Monica Botros

Mathematical Modelling and Numerical Simulation with Applications

This paper investigates a class of nonlinear sequential singular fractional differential equations (FDEs) involving Caputo derivatives. This type of equation has several key advantages that enhance its value, such as capturing memory and hereditary effects. Viscoelastic materials and anomalous diffusion, as well as biological systems, can take advantage of this feature. In addition, fractional derivatives possess a sequential structure that enables the implementation of multiscale processes and hierarchical memory responses. Moreover, it provides an effective and flexible framework for solving differential equations compared to classical differential equations. It can therefore be used to model complex systems in physics, biology, and …

On Links Between A Theorem Of Schoenberg, Rohlin Decompositions Of Measures, The Bochner-Minlos Theorem And The Fock Space, Daniel Alpay, Paula Cerejeiras, Palle Jorgensen, Uwe Kaehler

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main goal of this paper is to gain new results in stochastics by drawing on, and combining, different areas that are normally not considered to be related. Thus, in this paper we extend the previous class of Gaussian-like functions ML which will allow for future generalized stochastic processes in infinite dimensional analysis. We show that an approach similar to the one by the classical Bochner-Minlos theorem for the white-noise case can be achieved by using Gaussian-like functions belonging to a large family -the ML_r classes (0 < r ≤∞). We show how Schoenberg’s theorem for positive definite functions on a Hilbert space allows to go beyond the classical setting of Bochner-Milnos theorem. Furthermore, we show that the application of the Rohlin’s disintegration theorem allows for a decomposition of the associated probability measures , see Theorems 3.2 and 4.3. We end this paper with several important examples of functions in these classes ML_r and provide some interesting counterexamples, e.g. Theorem 7.4, to get a …

Empirical Validation Of Einstein’S Coefficient For The Deflection Of Light Due To Gravitational Forces, Alexandra R. Mcdowell

Honors College Theses

Einstein’s formula to calculate the deflection angle of light through space as it interacts with gravity was introduced in his 1916 publication on general relativity. This was not a new idea, but his equation was, and it was correct. Just 3 years after this publication, it was empirically validated by Sir Arthur Eddington and Sir Frank Dyson. Since that experiment in 1919, at least seven others have been performed that also gave definitive answers in support of Einstein’s deflection constant of 1.751 arcseconds. The two most recent ones made groundbreaking contributions to this effort. The 2017 eclipse showed reproducible results …

Norm-Variation Of Triple Ergodic Averages For Commuting Transformations, Polona Durcik, Lenka Slavíková, Christoph Thiele

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove an r-variation estimate, r>4, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all r≥2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r< ∞ for more than three commuting transformations.

Minimal Supersolution Of Bsdes Driven By Continuous Martingales In A General Filtration, Badr Elmansouri, Mohamed El Otmani

Journal of Stochastic Analysis

In this paper, we study minimal supersolutions of backward stochastic differential equations (BSDEs) driven by a continuous local martingale in a general filtration. We establish existence, uniqueness, and stability results under various mild conditions on the terminal value and the generator. Additionally, we explore the connection between the concept of non-linear expectation and minimal supersolutions, emphasizing the specific properties that are relevant to our framework. We also prove a general monotonic limit theorem and apply this result to determine the smallest constrained supersolution of a BSDE with a possibly non-convex constraint.

Nightmare In The Library, Charles A. Coppin

Journal of Humanistic Mathematics

Students of real analysis and calculus find that the completeness property of the real numbers is difficult to understand, especially, its importance. The word numbers denote real numbers throughout this piece. After all, the real numbers are not any less imaginary than the so-called imaginary numbers. Although, sometimes, it does gain some mention in calculus courses, its presence as a topic is a mere will-o’-the-wisp of bygone days when teachers would often reach deep into the big ideas of calculus as a mainstay of their courses. For the sake of cultural literacy and the development of mathematical maturity, we believe …

Arbitrage-Free Pricing With Diffusion-Dependent Jumps, Hamza A. Virk, Yihren Wu, Majnu John

Journal of Stochastic Analysis

Standard jump-diffusion models assume independence between jumps and diffusion components. We develop a multi-type jump-diffusion model where jump occurrence and magnitude depend on contemporaneous diffusion movements. Unlike previous one-sided models that create arbitrage opportunities, our framework includes upward and downward jumps triggered by both large upward and large downward diffusion increments. We derive the explicit no-arbitrage condition linking the physical drift to model pa- rameters and market risk premia by constructing an Equivalent Martingale Measure using Girsanov’s theorem and a normalized Esscher transform. This condition provides a rigorous foundation for arbitrage-free pricing in models with diffusion-dependent jumps.

Evaluating Lunch Plan Data In The St. Charles School District (Scsd), Maddy Alexander, Guillermo Bilbao Olarreaga, Duncan Krige, Alyssa Schreiber, Nick Wintz, Wojciech Golik

The Confluence

The SCSD is a public school district in St. Charles, with, on average, 4500 students a year. The SCSD is subdivided into an early childhood center, six elementary schools, two intermediate (5-6,7-8) schools, and two high schools. Vocational schools are also within this district but were not included in this report. The SCSD is concerned with the impact of the Covid-19 pandemic on their district’s population and on the number of students that needed assistance with lunch. They have asked Lindenwood’s 2024-25 PIC Math group to analyze their data from the years 2020-25 and identify any trends. Identifying these trends …

Local Energy Decay For Non-Stationary Damped Wave Operators, Nicholas Dj Arsenault

Theses and Dissertations--Mathematics

This work establishes integrated local energy decay (ILED) estimates for the damped wave equation on certain non-stationary spacetimes. The main technical result is a high frequency estimate that holds in great generality, provided that null geodesics trapped in a compact region are sufficiently damped. This is combined with low- and medium-frequency estimates to establish full local energy decay. We conclude by providing a counterexample where the damping assumption fails and local energy decay does not hold.

Multivariate Quantile Autoregression-Mixed Data Sampling (Mvqar-Midas) Modeling Of Cost Of Living And Supply Chain Dynamics In Canada., Patrick Gbolonyo

Theses and Dissertations (Comprehensive)

In recent years, the rising cost of living as a result of persistent inflationary pressures, disruptions in the global supply chains, and changes in the macroeconomic landscape has become a critical topic of discussion. To address this, we move beyond a mean-based framework and employ a quantile regression approach. This allows the persistence of each series and the transmis- sion of shocks between the Consumer Price Index (CPI) (the total CPI which is a percentage change over the past 12 months), the Interest Rate (IR)(the target for the overnight rate), the New Housing Price Index (NHPI), and high-frequency supply chain …

Lie-Galois Theory, Giovanni Reed

Honors Undergraduate Theses

Differential equations are a much-studied topic in the field of mathematics, as well as other sciences, such as engineering, economics, and biology. While much is known concerning these, there is still a large gap in our knowledge about such equations. It is important therefore, both to mathematics and other sciences, that we gain a more complete knowledge of differential equations, in particular the nature of their solutions. In this research, we investigate the solution space of linear ordinary differential equations (ODEs) from the standpoint of differential algebra. Differential algebra allows an ODE to be treated similarly to a polynomial, allowing …

A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu

HMC Senior Theses

The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶^*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶^*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶^*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶^*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.

Spectral Decimation On The Schreier Graphs Of The Basilica Group: A Thesis In Fractal Analysis, Anne D. Bannon

Scripps Senior Theses

This thesis is intended to provide a comprehensive overview of the literature required to fully understand research conducted during the University of Connecticut's Fractals & Stochastics REU in the summer of 2025. The literature review includes a description of Robert Strichartz's seminal work pertaining to the Laplacian spectrum of the Sierpiński Gasket, which provides a framework for how we approach studying the spectrum of the basilica Julia set. Defining the basilica Julia set and the closely-related Basilica group involves graph theory, automata theory, iterated monodromy group theory, and amenable group theory. Further time is dedicated to defining the graph Laplacian …

Real Interpolation: An Approximate Introduction, Madeline Anderson

Scripps Senior Theses

This thesis provides an introduction to real interpolation. We establish

relevant notions in functional analysis first, and use these concepts to study

real interpolation using J. Peetre’s 𝐾-functional in some detail. We also

explore the basics of approximation theory, in particular the connection

between approximation and interpolation results.

Catalan And Hyper-Catalan Numbers: Combinatorial Applications To Polynomial Equations, Leilani Natale

Williams Honors College, Honors Research Projects

In this paper, we study Catalan numbers and their generalization, hyper-Catalan numbers, and explore how these sequences arise naturally in the context of solving polynomial equations using infinite power series. We begin by introducing the Catalan numbers through their combinatorial interpretation as triangulations of convex polygons. Using this geometric definition, we derive a relation whose recursive structure leads to a quadratic functional equation. Interpreting this relation as a formal power series equation allows us to express solutions to quadratic equations as infinite power series whose coefficients are given by the Catalan numbers. This framework is then extended by allowing polygon …

A Mathematical Frameworks For Singular, Nonlinear Phenomena: Applications To Nematocyst Firing And Inhomogeneous Nls With Coulomb Potential, Abdulrahman Alharbi

Theses and Dissertations

Nematocysts are specialized cellular organelles found in all cnidarians, including corals and jellyfish, as well as in some single-celled protists such as dinoflagellates. These organelles display remarkable diversity in morphology and function, playing roles in prey capture and defense. The firing of a nematocyst is one of the fastest accelerations in nature, yet the underlying physical mechanisms remain not fully understood. In this work, we address key questions: how sufficient force is generated to overcome the fluid boundary layer, whether fluid–structure interaction models can reproduce observed dynamics, and what mechanisms trigger discharge.

Our research investigates models based on osmotic pressure …

Spectral Properties And The Behavior Of The Current-Current Correlation Measure For The Luttinger-Sy Model, Jonathan Benoit

Theses and Dissertations--Mathematics

The Luttinger-Sy Model, sometimes referred to as the Pieces Model, is a Random Schrodinger Operator on L²(R) which is characterized in part by "pieces" whose endpoints are chosen by a Poisson Point Process. The Hamiltonian in this setting is then given as a direct sum of Laplacians with Dirchlet boundary conditions on each piece. In this work, we show several spectral properties of the Luttinger-Sy Model, including proving the deterministic spectrum is [0,infinity) and that a Wegner-type and Minami-type estimate both hold. Additionally, we show that the finite-volume Current-Current Correlation Measure is singular continuous with respect …

The Ellipsoidal Principal Semi-Axis Geometry Of The Solution To An Ivp For A Matrix Diffusion Pde, Brandon M. Fox

Electronic Theses & Dissertations (2024 - present)

We analyze the underlying geometry of the solution to an IVP for a matrix diffusion PDE. We first derive the fundamental solution to the PDE. We then determine the unique solution to the IVP. From there, we begin analyzing its underlying geometry. We first observe that the geometry exhibits an ellipsoidal nature. Furthermore, we observe that it is described by the principal semi-axis geometry of the ellipsoids associated with the solution. This conclusion follows from applying the Principal Axis Theorem to the associated ellipsoids to establish their principal semi-axis geometry, as governed by the eigenstructure of the matrix. This thesis …

Global Weak Solutions Of Optical Variational Wave System, Shahrazad Hamed Alnafie

Graduate Theses, Dissertations, and Problem Reports (ETD)

                                                       ABSTRACT

                   Global Weak Solutions of Optical Variational Wave System

                                        Shahrazad Hamed Mahal Alnafie

The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.

We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable …

Asymptotics Of Discrete Convolution Powers And Applications To Difference Schemes, Pedro Henrique Alves Silva Dos Santos

Honors Theses

In this thesis we provide Gaussian Estimates and Local Limit Theorems describing the asymptotic behavior of convolution powers of a class of complex-valued functions on $\mathbb{Z}^d$. Convolution powers arise naturally in the study of partial differential equations, as well as in random walks in probability theory. In particular, they are connected to the stability theory of difference schemes used to approximate solutions to partial differential equations. We take inspiration from the work of Vidar Thomée on stability theory to restrict our attention to convolution powers of functions whose Fourier Transforms satisfy certain local expansions. We then combine the Cauchy Integral …