Mathematics Commons^™

The Construction Of Khovanov Homology, Shiaohan Liu

Master's Theses

Knot theory is a rich topic in topology that studies the how circles can be embedded in Euclidean 3-space. One of the main questions in knot theory is how to distinguish between different types of knots efficiently. One way to approach this problem is to study knot invariants, which are properties of knots that do not change under a standard set of deformations. We give a brief overview of basic knot theory, and examine a specific knot invariant known as Khovanov homology. Khovanov homology is a homological invariant that refines the Jones polynomial, another knot invariant that assigns a Laurent …

Complex Dimensions Of 100 Different Sierpinski Carpet Modifications, Gregory Parker Leathrum

Master's Theses

We used Dr. M. L. Lapidus's Fractal Zeta Functions to analyze the complex fractal dimensions of 100 different modifications of the Sierpinski Carpet fractal construction. We will showcase the theorems that made calculations easier, as well as Desmos tools that helped in classifying the different fractals and computing their complex dimensions. We will also showcase all 100 of the Sierpinski Carpet modifications and their complex dimensions.

A Survey On Varieties Generated By Small Semigroups And A Companion Website, João Araújo, João Pedro Araújo, Peter J. Cameron, Edmond W. H. Lee, Jorge Raminhos

Mathematics Faculty Articles

This paper presents new findings on varieties generated by small semigroups and groups, and offers a survey of existing results. A companion website is provided which hosts a computational system integrating automated reasoning tools, finite model builders, SAT solvers, and GAP. This platform is a living guide to the literature. In addition, the first complete and justified list of identity bases for all varieties generated by a semigroup of order up to 4 is provided as supplementary material. The paper concludes with an extensive list of open problems.

Every Relu-Based Neural Network Can Be Described By A System Of Takagi-Sugeno Fuzzy Rules: A Theorem, Barnabas Bede, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While modern deep-learning neural networks are very successful, sometimes they make mistakes, and since their results are "black boxes" -- no explanation is provided -- it is difficult to determine which recommendations are erroneous. It is therefore desirable to make the resulting computations explainable, i.e., to describe their results by using commonsense rules. In this paper, we use "fuzzy" techniques -- techniques developed by Lotfi Zadeh to deal with commonsense rules formulated by using imprecise ("fuzzy") words from natural language -- to show that such a rule-based representation is always possible. Our result does not yet provide the desired explainability, …

Smooth Non-Additive Integrals And Measures And Their Potential Applications, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we explain why non-additive integrals and measures are needed, how non-additive integrals and measures are related, how to use them in decision making, and how they can help in fundamental physics. These four topics are covered, correspondingly, in Sections 2-5 of this paper.

When Is A Single "And"-Condition Enough?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, there are several possible decisions. Any general recommendation means specifying, for each possible decision, conditions under which this decision is recommended. In some cases, a single "and"-condition is sufficient: e.g., a condition under which a patient is recommended to take aspirin is that "the patient has a fever and the patient does not have stomach trouble". In other cases, conditions are more complicated. A natural question is: when is a single "and"-condition enough? In this paper, we provide an answer to this question.

If We Add Axiom Of Choice To Constructive Analysis, We Get Classical Arithmetic: An Exercise In Reverse Constructive Mathematics, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent paper in Bulletin of Symbolic Logic reminded that the Axiom of Choice is, in general, false in constructive analysis. This result is an immediate consequence of a theorem -- first proved by Tseytin -- that every computable function is continuous. In this paper, we strengthen the result about the Axiom of Choice by proving that this axiom is as non-constructive as possible: namely, that if we add this axiom to constructive analysis, then we get full classical arithmetic.

Why Sigmoid Transformation Helps Incorporate Logic Into Deep Learning: A Theoretical Explanation, Chitta Baral, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional neural networks start from the data, they cannot easily handle prior knowledge -- this is one of the reasons why they often take very long to train. It is desirable to incorporate prior knowledge into deep learning. For the case when this knowledge consists of propositional statements, a successful way to incorporate this knowledge was proposed in a recent paper by van Krieken et al. That paper uses the fact that a neural network does not directly return a truth value, it returns a real value -- in effect, the degree of confidence in the corresponding statement -- from …

A History Of Complex Simple Lie Algebras, Avrila Frazier

Electronic Theses and Dissertations

In 1869, prompted by his work in differential equations, Sophus Lie wondered about categorizing what he called “closed systems of commutative transformations,” while around the same time, Wilhelm Killing’s work on non-Euclidean geometry encountered related topics. As mathematicians recognized this as a division of abstract algebra, the area became known as “continuous transformation groups," but we now refer to them as Lie groups.

Patterns and structures emerged from their work, such as describing Lie groups in connection with their associated Lie algebras, which can be categorized in many important ways. In this paper, we focus on Lie algebras over the …

Combinatorial Problems Related To Optimal Transport And Parking Functions, Jan Kretschmann

Theses and Dissertations

In the first part of this work, we provide contributions to optimal transport through work on the discrete Earth Mover's Distance (EMD).We provide a new formula for the mean EMD by computing three different formulas for the sum of width-one matrices: the first two formulas apply the theory of abstract simplicial complexes and result from a shelling of the order complex, whereas the last formula uses Young tableaux. Subsequently, we employ this result to compute the EMD under different cost matrices satisfying the Monge property. Additionally, we use linear programming to compute the EMD under non-Monge cost matrices, giving an …

(R2054) Convergence Of Lagrange-Hermite Interpolation Using Non-Uniform Nodes On The Unit Circle, Swarnima Bahadur, Sameera Iqram, Varun .

Applications and Applied Mathematics: An International Journal (AAM)

In this research article, we brought into consideration the set of non-uniformly distributed nodes on the unit circle to investigate a Lagrange-Hermite interpolation problem. These nodes are obtained by projecting vertically the zeros of Jacobi polynomial onto the unit circle along with the boundary points of the unit circle on the real line. Explicitly representing the interpolatory polynomial as well as establishment of convergence theorem are the key highlights of this manuscript. The result proved are of interest to approximation theory.

Integrating Machine Learning Methods For Medical Diagnosis, Jazmin Quezada

Open Access Theses & Dissertations

Abstract:The rapid advancement of machine learning techniques has revolutionized the field of medical diagnosis by offering powerful tools to analyze complex data sets and make accurate predictions. In this proposed method, we present a novel approach that integrates machine learning and optimization models to enhance the accuracy of medical diagnoses. Our method focuses on fine-tuning and optimizing the parameters of machine learning algorithms commonly used in medical diagnosis, such as logistic regression, support vector machines, and neural networks. By employing optimization techniques, we systematically explore the parameter space of these algorithms to discover the most optimal configurations. Moreover, by representing …

Travel Time Theory For Traffic Conservation Laws With Applications, Sergio Contreras

UNLV Theses, Dissertations, Professional Papers, and Capstones

Travel time is an important concept in various intelligent transportation system (ITS) applications. The concept is used in a wide array of applications, such as system planning, system performance, and optimization. Reducing the time required to travel between different points on a network is an important goal. Benefits include reducing time wasted in traveling, and keeping travelers satisfied. Thus, studying and reducing travel time in ITS is beneficial in different applications.

The classic density-based Lighthill Whitman Richards (LWR) equation for modeling traffic flow is the starting point in this dissertation. A more recent travel time dynamics function built on top …

Generalized Vulnerability Measures Of Graphs, Julia Vanlandingham

All Theses

Several measures of vulnerability of a graph look at how easy it is to disrupt the network by removing/disabling vertices. As graph-theoretical parameters, they treat all vertices alike: each vertex is equally important. For example, the integrity parameter considers the number of vertices removed and the maximum number of vertices in a component that remains. We consider the generalization of these measures of vulnerability to weighted vertices in order to better model real-world applications. In particular, we investigate bounds on the weighted versions of connectivity and integrity, when polynomial algorithms for computation exist, and other characteristics of the generalized measures.

Measuring The Lengths Of Sperm Whales Of The Northern Gulf Of Mexico By Wavelet Analysis Of Their Usual Clicks, George Drouant

University of New Orleans Theses and Dissertations

Abstract

Acoustic recordings of underwater sounds produced by marine mammals present an attractive alternative to costly and logistically complex ship based visual surveys for collecting population data for various species.

The first reported use of underwater acoustic recordings in the long-term monitoring of sperm whale populations was by Ackleh et al. (Ackleh et al., 2012). The paper describes counting sperm whale clicks at different locations to track population changes over time.

Analysis of sperm whale clicks offers additional insight into sperm whale populations. The echo location clicks (usual clicks) of sperm whales can be used to give an estimate of …

Convolution And Autoencoders Applied To Nonlinear Differential Equations, Noah Borquaye

Electronic Theses and Dissertations

Autoencoders, a type of artificial neural network, have gained recognition by researchers in various fields, especially machine learning due to their vast applications in data representations from inputs. Recently researchers have explored the possibility to extend the application of autoencoders to solve nonlinear differential equations. Algorithms and methods employed in an autoencoder framework include sparse identification of nonlinear dynamics (SINDy), dynamic mode decomposition (DMD), Koopman operator theory and singular value decomposition (SVD). These approaches use matrix multiplication to represent linear transformation. However, machine learning algorithms often use convolution to represent linear transformations. In our work, we modify these approaches to …

Strategies Community College Mexican American Adult College Algebra Students Use When Graphing Function Transformations, Roxana Pamela Jimenez

Theses and Dissertations

This qualitative case study pursued to describe the different strategies Mexican American adult students in a local community college used to graph function transformations. Participants in the study were purposefully selected using a criterion sampling to ensure participants were atypical, above average students between the ages 18-22, and had a final course average of 89.5-100 in College Algebra. Three research questions were examined 1) In what ways do Mexican American adult college students graph a function transformation? 2) Which strategies do students implement when graphing a function transformation? Qualitative research methods using think aloud semi-structured interviews were used in this …

A Combinatorial Proof Of Supercranks For Partitions With A Fixed Number Of Parts, Jacob J. Gutierrez

Theses and Dissertations

In a previous paper by Eichhorn and Kronholm, a selection of supercranks for p(n,m) was established by generating functions. In this paper we will reprove this result with combinatorial witnesses for the selection of supercranks via integer lattice points.

An Automatic Solver For Optimal Control Problems, Marcel Efren Benitez

Theses and Dissertations

Optimal control theory is a study that is used to find a control for a dynamical system over a period of time such that a objection function is optimized. In this study we will be looking at optimal control problems for ordinary differential equations or ODEs and see that we can use an automatic solver using the forward-backward sweep using Matlab to solve for them from an 1 dimension to bounded cases and to nth dimension cases.

Attitudes Towards Mathematics Of Developmental Mathematics Students In A Community College, Benjamin Ortiz

Theses and Dissertations

Reformations to developmental mathematics aim to remove barriers for students entering higher education. Challenges like costly multi-course sequences and high failure rates prohibit students’ access to college-level math courses and prevent degree or certification completion. Understanding factors that foster student success is critical to increase student success. This study focuses on students’ attitudes towards mathematics, utilizing the novice-expert continuum through Code et al.’s Mathematical Attitude and Perceptions Survey (MAPS) instrument. Student expertise scores, including all MAPS dimensions and specific dimension scores, were assigned. Kruskal-Wallis Rank-Sum tests identified differences in student populations by course and attitude dimension. …

How An Instructor's Noticing For Equity Can Foster Students' Sense Of Belonging And Mathematical Confidence, Sthefania Espinosa

Theses and Dissertations

There are many aspects a teacher can notice inside the mathematics classroom, and the more a teacher notices, the more difficult it is to teach. In this study, I particularly focus on noticing for equity, which describes the role of the teacher in attending to students’ mathematical thinking through an equity lens that can allow the instructor to notice the aspects of classroom mathematical activity that can make students feel less or more empowered in their mathematical practices (van Es et al., 2017). There exists few research about how students perceive their instructor’s effort to promote equity and …

Quasipolynomials And The Unimodality Of Gaussian Polynomials, Paul Marsh

Theses and Dissertations

We illustrate a method to prove the unimodality of Gaussian polynomials ${N+m \brack m}$ for $m = 5$ and $6$, building upon Dr. Brandt Kronholm's work, which proved unimodality for $m = 2,3,$ and $4$. Our approach involves viewing coefficients $p(n,m,N)$ of Gaussian polynomials $N+m \brack m$ based on how far away $n$ is from the central coefficient $p(\lfloor\frac{mN}{2}\rfloor,m,N)$ and then creating generating functions for those coefficients. We then take the difference of neighboring generating functions and change those generating functions into quasipolynomials to verify that their coefficients are non-negative. While the generalization of these generating functions for the coefficients …

Case Studies Of Algebra 1 Teachers Selection And Implementation Of Mathematics Tasks Toward Situated Learning, Luis Román Sauceda

Theses and Dissertations

Mathematical tasks are vital in active learning, especially in situated learning. Adequate selection and appropriate implementation of tasks are steps toward success in engaging students for active learning. This study explored how a professional development (PD) workshop influences teacher participants’ capabilities in selecting, redesigning, implementing, and reflecting on mathematical tasks to promote situated and active learning. The teacher participants were Algebra 1 teachers from a South Texas secondary school. During the workshop, participants developed and implemented activities after being shown situated learning strategies to promote student-centered learning. They were required to design hypothetical dialogues to simulate their class practice before …

On Dyadic Parity Check Codes And Their Generalizations, Meraiah Martinez

Department of Mathematics: Dissertations, Theses, and Student Research

In order to communicate information over a noisy channel, error-correcting codes can be used to ensure that small errors don’t prevent the transmission of a message. One family of codes that has been found to have good properties is low-density parity check (LDPC) codes. These are represented by sparse bipartite graphs and have low complexity graph-based decoding algorithms. Various graphical properties, such as the girth and stopping sets, influence when these algorithms might fail. Additionally, codes based on algebraically structured parity check matrices are desirable in applications due to their compact representations, practical implementation advantages, and tractable decoder performance analysis. …

Unexpectedness Stratified By Codimension, Frank Zimmitti

Department of Mathematics: Dissertations, Theses, and Student Research

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore, and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose …

Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part Ii: Experiment Description, Victor Christianto, T. Daniel Chandra, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The present article was partly inspired by G. Pollack’s book, and also Dadoloff, Saxena & Jensen (2010). As a senior physicist colleague and our friend, Robert N. Boyd, wrote in a journal (JCFA, Vol. 1, No. 2, 2022), for example, things and Beings can travel between Universes, intentionally or unintentionally [4]. In this short remark, we revisit and offer short remark to Neil Boyd’s ideas and trying to connect them with geometry of musical chords as presented by D. Tymoczko and others, then to Escherian staircase and then to Jacob’s ladder which seems to pointto possibility to interpret Jacob’s vision …

Wavelet Compression As An Observational Operator In Data Assimilation Systems For Sea Surface Temperature, Bradley J. Sciacca

University of New Orleans Theses and Dissertations

The ocean remains severely under-observed, in part due to its sheer size. Containing nearly billion of water with most of the subsurface being invisible because water is extremely difficult to penetrate using electromagnetic radiation, as is typically used by satellite measuring instruments. For this reason, most observations of the ocean have very low spatial-temporal coverage to get a broad capture of the ocean’s features. However, recent “dense but patchy” data have increased the availability of high-resolution – low spatial coverage observations. These novel data sets have motivated research into multi-scale data assimilation methods. Here, we demonstrate a new assimilation approach …

Bioactivity And Structural Properties Of Hydroxyapatite On Ti6a|4v And Si(100) Surfaces By Pulsed Laser Deposition, Salizhan Kylchbekov

Masters Theses & Specialist Projects

Although biomedical implant technology is very advanced, there are still caveats in terms of biocompatibility properties because metals are inert to biological processes such as osseointegration, cell growth, and cell adhesion. This results in statistically significant complications and opens rooms for improvements. Among many techniques to improve this problem, coating the metal surface with biologically functional materials has resulted in the best performances. Hydroxyapatite (HAP), Ca_₁₀(PO_₄)_₆(OH)_₂, is the most stable form of calcium phosphates in human body environment and is functional in key processes such as ion exchange, osteoblast (bone cells) formation, …

Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part I: Theoretical Underpinning, Victor Christianto, T. Daniel Chandra, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

As a senior physicist colleague and our friend, Robert N. Boyd, wrote in a journal (JCFA, Vol. 1,. 2, 2022), Our universe is but one page in a large book [4]. For example, things and Beings can travel between Universes, intentionally or unintentionally. In this short remark, we revisit and offer short remark to Neil’s ideas and trying to connect them with geometrization of musical chords as presented by D. Tymoczko and others, then to Escher staircase and then to Jacob’s ladder which seems to point to possibility to interpret Jacob’s vision as described in the ancient book of Genesis …

Discrete Complementary Exponential And Sine Integral Functions, Samer Assaf, Tom Cuchta

Mathematics Faculty Research

Discrete analogues of the sine integral and complementary exponential integral functions are investigated. Hypergeometric representation, power series, and Laplace transforms are derived for each. The difficulties in extending these definitions to other common trigonometric integral functions are discussed.