Mathematics Commons^™

An Introduction To Reverse Mathematics Through The Weakened Base System Rca_0^*., Kaden Dvorak

Boise State University Theses and Dissertations

The purpose of reverse mathematics, a field of mathematical logic, is to determine which axioms are required to prove particular mathematical theorems. Gödel's first incompleteness theorem states that within any standard consistent formal system of mathematics, there are statements for which neither themselves nor their negations can be proven. Thus, the goal of reverse mathematics cannot be the discovery of some formal system which underlies all mathematics, which was that of Hilbert's program. Instead, the questions lie in how much mathematical reasoning can be represented and how strong the formal systems are required to be to conduct said reasoning. In …

Multiple Monochromatic Subgraphs In Edge-Colored Graphs, Emma Felicity Jent

Dissertations

Ramsey theory, though a relatively young branch of mathematics, has captivated the attention of graph theorists, combinatorialists, and theoretical computer scientists alike through its raw beauty, versatility, and powerful applications. Before it emerged as a branch of mathematics, the central idea of Ramsey theory appeared in the form of three lemmas in three separate papers by three different mathematicians working on three distinct areas of research. The first such lemma was published by David Hilbert in 1892, followed by the second lemma published by Issai Schur in 1916. However, Frank Ramsey’s renowned lemma, published in 1930, compelled mathematicians to establish …

An Exploration Of The Structure Of The Linear Algebraic Model Of Color, Isaac Martin

University Honors Theses

This thesis surveys the mathematical grounding of linear algebraic models of color. It aims to build from the ground up the framework by which additive color is broadly understood in the digital age. Primarily building on the work of Jozef Cohen, Eric Dubois, David H. Krantz, and Günter Wyszecki, it aims to chart the construction of a model of color that underpins most modern understandings of color. While the construction is certainly established in colorimetric circles, the construction is, in the thesis author's opinion, either obtuse or non-rigorous. Ideally, this thesis serves to make the construction accessible to an audience …

Studies In Number Theory: Reciprocity Laws And Fundamental Domains, C. Xavier Parent

All Graduate Theses and Dissertations, Fall 2023 to Present

This thesis consists of two sections. The first section is an introductory survey of number theory discussing the reciprocity laws with a focus on accessibility. Number Theory has always been a fundamental area of mathematical study, with Gauss calling it “the queen of mathematics”. The reciprocity laws are a classical set of results from number theory which have driven number theory for quite a long time. Unfortunately, these results, while important, have always been very inaccessible to undergraduate students, making it hard to start studying the field. This survey attempts to help bridge that gap, giving a resource for novices …

(A,B,C) Tilings With Prescribed Symmetry Groups From Regular Triangle Or Hexagon Tiling, Mark D. Tomenes, Ma. Louise Antonette N. De Las Penas

Mathematics Faculty Publications

A tiling T of the Euclidean plane (E2) is a countable collection of closed topological disks called tiles T = {Ti : i ∈ N} that is a covering (Ui Ti = E2) as well as a packing (Int(Ti) ∩ Int(Tj) = ∅ if i ̸= j, Int(T) denotes the interior of tile T). One of the problems of interest in discrete geometry is the classification of tilings based on transitivity properties of their vertices, edges and tiles. This talk presents a family of tilings whose vertices, edges and tiles have exactly a, b and c orbits, respectively, under the …

Congruences For Quotients Of Klein Forms, Jeffery Opoku

Theses and Dissertations

This dissertation investigates the arithmetic properties of some modular forms and eta quotients, focusing on the divisibility properties and congruence relations satisfied by these. The first part examines quotients of the Rogers-Ramanujan and Rogers-Selberg functions, defined by \[ f(\tau) = q^{r} (q^5; q^5)^{a_0} (q, q^4; q^5)^{a_1} (q^2, q^3; q^5)^{a_2}= \sum_{n=r}^{\infty} P_{a_0,a_1,a_2}(n-r) q^n, \] and \[ g(\tau) = q^{s} (q^7; q^7)^{a_0} (q, q^6; q^7)^{a_1} (q^2, q^5; q^7)^{a_2} (q^3, q^4; q^7)^{a_3}= \sum_{n=s}^{\infty} P_{a_0,a_1,a_2,a_3}(n-s) q^n, \] respectively. We establish conditions on the exponents \(a_0, a_1, a_2, a_3\) and residue classes \(r\) and $s$ modulo $p$ such that \(P_{a_0, a_1, a_2}(pn - r) \equiv …

Analysis Of Multi Grade Deep Learning, Ronglong Fang

Mathematics & Statistics Theses & Dissertations

Multi-Grade Deep Learning (MGDL) is a training framework that incrementally builds deep neural networks. It does this by dividing the training process into multiple “grades,” where each grade sequentially trains a shallow neural network to learn the residue from the previous one, using the outputs of prior grades as input. This approach progresses from shallow to deep architectures. This dissertation offers a comprehensive theoretical and numerical analysis of the MGDL methodology.

We first demonstrate that MGDL can effectively learn target functions within the sum-composition learning format. In this context, MGDL approximates high-frequency components by composing multiple low-frequency functions. This unique …

Certified Computation Of Julia Sets Via Numerical Methods, Hannah Kaufman

All Theses

The chaotic and fractal nature of Julia sets makes them difficult to graph. This research aims to provide graphical approximations of Julia sets with known and guaranteed levels of accuracy. We implement three methods to approximate Julia sets with c values chosen from the main cardioid of the Mandelbrot set. Each method utilizes different properties of these Julia sets. The exclusion method makes use of the fact that a Julia set of this type is topologically a circle. Attracting and repelling fixed points are used to find a region on the interior of the Julia set and a region on …

On The Garoufalidis-Kashaev State-Integral Invariant, Amelia Palmer Dusenbury

Boise State University Theses and Dissertations

This thesis explores a construction of the family of topological invariants for certain oriented 3-manifolds based on the state-integral approach developed by Andersen, Garoufalidis, and Kashaev in the Archimedean setting. Starting from an ideal triangulation of a 3-manifold equipped with angle data, variables are assigned to the faces and tetrahedra, taking values in a so-called 'Gaussian group'. The invariant is defined by integrating a distribution defined from the combinatorics of the triangulation and a special function over a product of the Gaussian group. The special function is a quantum dilogarithm, whose valuable feature, the pentagon relation, ensures the resulting integral …

Constructing Code-Based Zero-Knowledge Proofs Leveraging Generic Errors And Bounded Vectors, Freeman Slaughter

All Dissertations

Quantum computing is developing at an expeditious rate, and once fully scalable quantum computers become realized, classical cryptographic systems face obsolescence. This approaching peril has prompted a paradigm shift away from pre-quantum cryptography and towards post-quantum primitives, such as those that arise from the field of coding theory. Among these, zero-knowledge proofs have emerged as a dynamic tool instrumental in constructing quantum-resilient digital signature schemes.

We being by introducing HammR, a pre-quantum zero-knowledge proof protocol designed to verify Hamming weight and entry constraints of error vectors, and comprehensively establish its security. Subsequently, we extend HammR to the multi-party computation setting, …

Online Multiobjective Optimization, Kristen Joyce

All Dissertations

Online optimization (OO) is an iterative process of decision making under uncertainty. At every step, a decision is made before the outcome of this decision is known. For the online optimization model, the objective function is unknown at the time the decision is being made. It is very likely that the taken decision is not optimal, so the decision maker incurs a loss, called regret, in every iteration. The goal of the online optimization algorithm is to compute a decision at every step so that the overall regret cost is minimized. In particular, the average regret produced by an ideal …

Preservation Of The Bernstein Property For Sums Of Independent Random Variables, Iosif Pinelis

Michigan Tech Publications

It is shown that Bernstein-type conditions on independent random variables are preserved by their sum. Some optimality properties of such preservation are proved.

A Novel Diastolic Doppler Index Less Affected By Aortic Arch Anomalies Co-Existing With Coarctation, Arash Ghorbannia, Andrew D. Spearman, Shahd Sawalhi, Ronald K. Woods, Mehdi Maadooliat, John F. Ladisa Jr.

Mathematical and Statistical Science Faculty Research and Publications

Severity assessment for coarctation of the aorta (CoA) is challenging due to concomitant morphological anomalies (complex CoA) and inaccurate Doppler-based indices. Promising diagnostic performance has been reported for the continuous flow pressure gradient (CFPG), but it has not been studied in complex CoA. Our objective was to characterize the effect of complex CoA and associated hemodynamics on CFPG in a clinical cohort. Retrospective analysis identified discrete juxtaductal (n = 25) and complex CoA (n = 43; transverse arch and/or isthmus hypoplasia) patients with arm-leg systolic blood pressure gradients (BPG) within 24 h of echocardiography for comparison to BPG …

Optimal Quantization For Nonuniform Discrete Distributions, Russel Cabasag, Samir Huq, Eric Mendoza, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This paper explores the process of optimal quantization for several types of discrete probability distributions. Quantization is a technique used to approximate a complex distribution with a smaller set of representative points, which is important in fields such as data compression and signal processing. We begin by examining two specific nonuniform distributions over a finite set of values and identify the best representative points for different levels of approximation. We then extend our analysis to two infinite discrete distributions: one supported on the reciprocals of natural numbers and another on the natural numbers themselves. For these distributions, we compute the …

Interpolation In Weighted Projective Spaces, Shahriyar Roshan Zamir

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points.

After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this thesis, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space.

A main contribution of this work is the careful adaption of several classical algebro-geometric techniques to the …

On Kernels And Antiderivatives Of Nonlocal Derivatives, Alex John Heitzman

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Nonlocal operators are mathematical operators taking functions to other functions f → Df , where to evaluate the operator Df at a point x, one must know the value of f in some region around x, and that region cannot be arbitrarily small. Nonlocal derivatives are like derivatives in that they measure the deviation of a function f(z) from f(x) when z is close to x. In this thesis, we will study nonlocal operators of the form

D_kf(x) = [integral]_Ω [f(x) …

Twisted Equivariant Matrix Factorizations, Jan-Luca Spellmann

All Graduate Theses and Dissertations, Fall 2023 to Present

We introduce and study categories of twisted equivariant matrix factorizations MF^α_G(R, w), which are categories of matrix factorizations of a potential w over the local ring R = C[x₁, . . . , x_n] together with an action by a finite group G that is twisted via a 2-cocycle α. These categories provide rich examples of Z/2Z-differentially graded categories in the context of non-commutative geometry and come up naturally in the study of boundary conditions of 3d Rozansky–Witten theories. We prove that under certain assumptions on (R, w …

Mathematical Modeling Of Effects Of Tumor Location On Lung Function., Lamargaret Temukisa Johnson

Master of Engineering Theses

Lung cancer has the highest rates of incidence and mortality of all cancers. Most lung cancer tumors are Non-Small Cell Lung Cancer (NSCLC). NSCLC patients with lesions in the upper lobes are found to have better prognosis compared to those with lesions in the middle and lower lobes. Previous studies have suggested various causes for this discrepancy at both the organ-scale and tissue-scale. To model NSCLC growth in different locations within the lung, an organ scale lung model and tissue scale tumor model were coupled through the tissue pressure, and oxygen and carbon dioxide partial pressures. The coupling was used …

Properties Of A Class Of Analytic Functions Associated With Exponentially Convex Functions, K. R. Karthikeyan, Elangho Umadevi, G. Thirupathi, Dharmaraj Mohankumar

All Works

Studies in univalent function theory comprising the exponential of differential characterizations are rarely considered. The prominent study in this direction is the study of so-called α-exponentially convex functions. Here we study a class of analytic functions which satisfy an analytic characterization influenced by the definition of the multiplicative derivative and α-exponentially convex functions. Integral representation and coefficient inequalities of the defined function class are the main results of the paper.

Utilizing The Horseshoe Prior In Exploratory Factor Analysis And Gaussian Graphical Networks, James Thomas Roddy

Graduate Theses and Dissertations

High-dimensional data analysis frequently involves extracting meaningful structure from noisy, sparse signals. In recent years, Bayesian shrinkage priors—particularly global-local shrinkage priors—have emerged as powerful tools for inducing sparsity while preserving signal fidelity. Among these, the Horseshoe prior has gained notable attention for its capacity to simultaneously shrink irrelevant parameters and retain substantial signals. This dissertation explores the Horseshoe prior as a unified framework for sparse Bayesian inference across theory, simulation, and real-world application. The first component develops new theoretical results establishing the asymptotic Bayes optimality of the Horseshoe prior in Gaussian graphical models (GGMs). We consider sparse precision matrix estimation …