Mathematics Commons^™

Smooth And Proper Maps With Respect To A Fibration, Mathieu Anel, Jonathan Weinberger

Engineering Faculty Articles and Research

This paper explain how the geometric notions of local contractibility and properness are related to the Σ-types and Π-types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Quantization Dimensions For The Bi-Lipschitz Recurrent Iterated Function Systems, Amit Priyadarshi, Mrinal Kanti Roychowdhury, Manuj Verma

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the quantization dimensions of the Borel probability measures supported on the limit sets of the bi-Lipschitz recurrent iterated function systems under the strong open set condition in terms of the spectral radius have been estimated.

Purely Pseudo-Anosov Subgroups Of The Genus Two Handlebody Group, Marissa E. Chesser, Christiopher J. Leininger

Faculty Work Comprehensive List

We prove that finitely generated, purely pseudo-Anosov subgroups of the genus 2 handlebody group are convex cocompact.

Non-Vanishing Modulo P Of Hecke L-Values Over Imaginary Quadratic Fields, Debanjana Kundu, Antonio Lei

School of Mathematical & Statistical Sciences Faculty Publications

Let p and q be two distinct odd primes. Let K be an imaginary quadratic field over which p and q are both split. Let Ψ be a Hecke character over K of infinity type (k, j) with 0 ≤ − j < k. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters κ over K which factor through the Z2q -extension of K, the p-adic valuation of the algebraic part of the L-value L(κΨ¯¯¯¯¯¯¯,k+j) is a constant independent of κ. In addition, when j = 0 and certain technical hypothesis holds, this constant is zero.

Math Is Good, Elisa Benthem

Student Work

"There are endless ways we can explore creation, each revealing something wonderfully distinct."

Posting about ­­­­­­­­the beauty of mathematics from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Math is Good

Book V Of The Mathematical Collection Of Pappus Of Alexandria, Translated By John B. Little, Pappus Of Alexandria, John B. Little

Holy Cross Bookshelf

John B. Little is the translator.

Book V of the Mathematical Collection is addressed to a certain Megethion, about whom we know nothing else. From the context he may have been a student or patron of Pappus in Alexandria. In a heading at the start, Pappus says that the general theme will be comparisons between different geometric figures. The overall structure brings interesting relations and connections to the fore. The book opens with a very well-known and charming discussion of how the importance of such comparisons can be seen by considering the structures built by non-human creatures such as bees. …

Magpy: A Python Package For Magmas, Skylar Korf

All NMU Master's Theses

There exist a multitude of computational tools available to mathematicians, such as interactive and automated theorem provers, finite counter-example generators, and computer algebra systems, most of which have a complicated installation process, unintuitive syntax, a lack of comprehensiveness for mathematical structures, or some combination of these. Computer algebra systems are indispensable tools due to their capacity for creating mathematical objects and performing computations on them. However, there is a lack of computational resources that focus on dealing with explicit examples, and extracting the properties thereof, especially for the most basic algebraic structures. Here, we will dive into several prominent computer …

Why Geological Angular Unconformity Is Usually Horizontal: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In several locations, geologists have observed the presence of two differently oriented rock masses, one horizonal (or almost horizontal) and the other somewhat inclined; this phenomenon is known as angular unconformity. Based on the detailed analysis of geophysical processes, geologists conclude that usually, horizontal rock masses are much newer. This is known as the law of original horizontality. From the fundamental viewpoint, it is desirable to take into account that geophysics is a developing science, its models get modified and adjusted as time progresses. It is therefore desirable to come up with an explanation of this phenomenon that would be …

Codes From Incidence Matrices Of Hypergraphs, Sudipta Mallik, Bahattin Yildiz

Mathematics Faculty Research

Binary codes are constructed from incidence matrices of hypergraphs. A combinatorial description is given for the minimum distances of such codes via a combinatorial tool called “eonv”. This combinatorial approach provides a faster alternative method of finding the minimum distance, which is known to be a hard problem. This is demonstrated on several classes of codes from hypergraphs. In particular the “eonv” method is used to prove that the minimum distance of codes from incidence matrices of complete 3-partite 3-uniform hypergraphs with partite set size of n is n2. A lower bound on the minimum distances of codes from …

A Complete Invariant For Shift Equivalence For Boolean Matrices And Finite Relations, Ethan Akin, Marian Mrozek, Mateusz Przybylski, Jim Wiseman

Mathematics and Statistics Faculty Research & Creative Works

We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those components.

The Near Normality Of The Commutant Of A Moufang Loop, Evan Phillips

All NMU Master's Theses

We investigate conditions under which the commutant of a Moufang loop is normal and related topics. We begin by giving relevant background on Moufang loops and normality. We then prove a theorem showing how close the commutant is to being normal: cR(x, y)R(x, y)R(x, y) = c. We then prove a couple of theorems emphasizing the importance of cubes in Moufang loops. Finally we prove a decomposition theorem for the multiplication group of a finite commutative Moufang loop.

Algebraic Structures On Parallelizable Manifolds, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L , there exists a global trivialization of the tangent bundle, which defines a map ρ p : l ⟶ T p L for each point p ∈ L , where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l . Furthermore, flows of these vector fields give rise to a product between elements of l and L , which in turn induces …

Topology And The Quantum Hall Effects, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

The quantum Hall effects are an excellent example of physical systems where topology plays a major role in accounting for the physical observations. It is shown that the conductivity that appears in the quantum Hall effect is a topological invariant. It is illustrated how a fiber bundle over a torus can be constructed producing a geometry in which the system can be referred. The fractional effect can be studied by introducing homotopy and associated braid groups. Filling fractions can be obtained as a consequence of commensurability relations.

Exploring The Human Side Of Math, Mike Janssen

Faculty Work Comprehensive List

"Abstract mathematics is an important way to better understand and marvel at the beauty and intricacy of God’s creation."

Posting about ­­­­­­­­experiences in math research from In All Things, an online hub that offers insight into maintaining and faithful and orthodox Reformed Christian worldview while fearlessly engaging in every aspect of contemporary life – until all is made new.

Exploring the Human Side of Math

Exclusion Tests For Approximating All Solutions Of Nonlinear Equations, Yasmeen Omar Hamida

Theses

The exclusion test is a mathematical technique used to isolate and approximate the solutions of nonlinear equations within a specified interval. This thesis aims to construct a new test leveraging the principles of the exclusion test to achieve faster convergence to the zeros of any given interval within a compact domain. By selecting an alternative objective function and employing nonlinear equations, the proposed method significantly reduces the number of subintervals required for accurate approximation. The key finding of this research is the development of a more efficient algorithm that enhances the speed of convergence. The implications of these findings are …

On Fractional Dunkl-Type Laplacian, Saba Ibrahim Aldan

Theses

This thesis presents a comprehensive study of the fractional Dunkl-type Laplacian operator (−∥x∥ Δ��)^σ for 0<σ<1, through four equivalent key characterizations: the heat semigroup characterization, the pointwise characterization, the spherical mean characterization, and through an extension theorem. These characterizations offer different perspectives on the operator ∥x∥Δ_��, generalizing known results for the Euclidean Laplace operator by incorporating reflection symmetries, making it highly relevant in the harmonic analysis of root systems.

By exploring these characterizations, the thesis highlights the deep connections between the fractional operator (−∥x∥Δ_��)^σ and harmonic analysis, as well as its broader applicability in mathematical physics and partial differential equations.

The Implementation Of Stream In Mathematics Classrooms In Abu Dhabi Primary Schools: Prospects, Priorities, Processes, And Problems, Nadeia Rashed Alalawi

Dissertations

This study shed light on implementing science, technology, reading and writing, engineering, art, and mathematics (STREAM) in mathematics classrooms in Abu Dhabi primary schools. The study aimed to explore mathematics teachers’ views on the prospects, priorities, processes, and problems of the implementation of STREAM in their classrooms. The study employed qualitative methods to collect and analyze data to support the findings using interviews, classroom observations, and document analysis. Fifteen in-service mathematics teachers were interviewed to explore their views about the application of STREAM in their classrooms regarding prospects, priorities, processes, and problems (4 Ps). Then, three of them were observed …

Substitution Tilings With Transcendental Inflation Factor, Dirk Frettlöh, Alexey Garber, Neil Mañibo

School of Mathematical & Statistical Sciences Faculty Publications

For any λ>2, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor λ. In particular, we obtain the first class of examples of substitutive systems with transcendental inflation factors. We also show that both the associated subshift and tiling dynamical systems are strictly ergodic, which is related to the quasicompactness of the underlying substitution operator.

Cp Decomposition Initialization Schemes For Speeding-Up Of Convolutional Neural Networks, Mollee M. Swift

LSU Master's Theses

While machine learning and convolutional neural networks (CNNs) are making strides, a persistent effort remains to optimize classification techniques and target redundancies from a large number of parameters naturally present in CNNs. While CNNs have become more accessible across machine learning, the aim is to make their use optimal for central processing unit (CPU) schemes with varying degrees of computational power. Tensor decomposition methods, specifically canonical polyadic (CP) decomposition, look to reduce parameters by compressing specific layers in a CNN. However, they have certain inconsistencies, and their full potential remains untouched as decomposition research is endless, with many facets one …

Coloring Trivalent Graphs: A Defect Tft Approach, Amit Kumar

LSU Doctoral Dissertations

We show that the combinatorial matter of graph coloring is, in fact, quantum in the sense of satisfying the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. Defects can be considered as a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain bundle, distinguishing a coloring (global-sections) from a coloring …

Pedagogical Moves Related To Analogy That Support A Unified Understanding Of Eigentheory Concepts In A Quantum Mechanics Class, Kaitlyn Stephens Serbin, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

It is beneficial for quantum mechanics students to have a unified understanding of eigentheory concepts, so they can recognize the shared structure of mathematized phenomena from the different quantum mechanical systems of spin, energy, or position and recognize those as instantiations of the same overarching concept. Quantum mechanics instructors should, therefore, provide opportunities for their class community to develop a shared unified understanding of eigentheory concepts. One such opportunity can arise by engaging students in analogizing eigentheory concepts in one context with those from another context. We investigate the pedagogical moves related to analogies that can be used by a …

Pricing Variance Swaps For The Discrete Bn-S Model, Semere Gebresilasie

Journal of Stochastic Analysis

No abstract provided.

Group Representations & Quantum Theory, Vehbi Paksoy

Mathematics Colloquium Series

What is the common thing between some integers and symmetries of an equilateral triangle? How studying rotations on a circle helps us understand quantum mechanics? Groups and their representations provide an amazing framework for many physics and chemistry theories. Understanding how certain groups act on physical systems, both classical and quantum, helps us to build models and make predictions. In this talk, we discover how abstract mathematical notions such as groups and representation theory and counterintuitive nature of quantum mechanics come together to guide us in our journey to explore the world we live in. This talk is designed to …

A Phenomenological Study Of Persistence For Black Women In Mathematics Doctoral Programs, Shushannah Marie Smith

Doctoral Dissertations and Projects

The purpose of this phenomenological study was to describe the doctoral persistence experiences of Black women mathematicians. Guided by McGee’s fragile and robust mathematical identity framework, the research design followed Moustakas’ qualitative transcendental phenomenological approach. The researcher explored the contributing factors to the phenomenon guided by the central research question, “What are the doctoral persistence experiences of Black women mathematicians?” The study involved 11 Black women participants who hold or were within one year of holding a doctoral degree in mathematics and were selected to ensure diversity and representativeness. Data collection encompassed three methods: a mathematical identity development timeline, one-on-one …

Uniqueness Of Maximum Points Of A Sequence Of Functions Arising From An Adapted Algorithm For ‘The Secretary Problem’, Boning Wang, Giang Vu Thanh Nguyen

OUR Journal: ODU Undergraduate Research Journal

This paper is aimed at a sequence of functions that is extended from an adaptive algorithm of the classical ‘secretary problem’. It was proved in Nguyen et al. (2024) the uniqueness of maximizers of a function sequence that represents the expected score of an element in a ‘candidate’ sequence. This function sequence is indeed considered as a special case of an extended function sequence that corresponds to the case α = 0. More specifically, we are motivated to prove the uniqueness of maximizers for this extended function sequence in the case α = 1. Nevertheless, the corresponding proof is rather …

Scaled Global Operators And Fueter Variables On Non-Zero Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers H_t, t ∈ R^∗ , of which the H₋₁ = H is the space of quaternions and H₁ is the space of split quaternions.We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on H_t. Counterparts of the Hardy space and of the …

Integrable Semi-Discretization For A Modified Camassa-Holm Equation With Cubic Nonlinearity, Bao-Feng Feng, Heng-Chun Hu, Han-Han Sheng, Wei Yin, Guo-Fu Yu

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by …

Counting The Classes Of Projectively-Equivalent Pentagons On Finite Projective Planes Of Prime Order, Maxwell Hosler

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we examine the number of equivalence classes of pentagons on finite projective planes of prime order under projective transformations. We are interested in those pentagons in general position, meaning that no three vertices are collinear. We consider those planes which can be constructed from finite fields of prime order, and use algebraic techniques to characterize them by their symmetries. We are able to construct a unique representative for each pentagon class with nontrivial symmetries. We can then leverage this fact to count classes of pentagons in general. We discover that there are (1/10)((p+3)(p-3)+4 …

Nano Topology And Decision Making In Medical Applications, Samir Mukhtar, Mohamed Shokry, Manar Omran

Journal of Engineering Research

Nano Topology is one of the essential topics that receive special attention from some athletes in the field of General Topology, Operations Research, and Computer Science, because it has a vital role in the generalizing most of the various mathematical concepts. Recently, many efforts have been made to study many types of Nano Topology, as the previous studies lacked real applications in Engineering, Medicine, Pharmacy, and Social Sciences. In this paper, we present some different applications of these studies. The paper is divided into two parts: Firstly, we study the theory of The Nano Topology and investigate its relation with …

New Operation Defined Over Dual-Hesitant Fuzzy Set And Its Application In Diagnostics In Medicine, Manar Mohamed Omran, Reham Abdel-Aziz Abo-Khadra

Journal of Engineering Research

In recent decades, several types of sets, such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, type n fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multi-sets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, and then propose an extension principle of DHFSs. Additionally, we give an example to illustrate …