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Full-Text Articles in Mathematics

Apex Calculus: Und Edition (June 2023), Gregory Hartman, Department Of Mathematics, University Of North Dakota Jun 2023

Apex Calculus: Und Edition (June 2023), Gregory Hartman, Department Of Mathematics, University Of North Dakota

Open Educational Resources

This text comprises a three–volume series on Calculus. The first part covers material taught in many “Calculus 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6. The second text covers material often taught in “Calculus 2”: integration and its applications, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 7 through 10. The third text covers topics common in “Calculus 3” or “Multivariable Calculus”: parametric equations, polar coordinates, vector–valued functions, and functions of more than one variable, found in Chapters 11 through 15. All three are available separately for free. …

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Jones Polynomial Obstructions For Positivity Of Knots, Lizzie Buchanan Jun 2023

Jones Polynomial Obstructions For Positivity Of Knots, Lizzie Buchanan

Dartmouth College Ph.D Dissertations

The fundamental problem in knot theory is distinguishing one knot from another. We accomplish this by looking at knot invariants. One such invariant is positivity. A knot is positive if it has a diagram in which all crossings are positive. A knot is almost-positive if it does not have a diagram where all crossings are positive, but it does have a diagram in which all but one crossings are positive. Given a knot with an almost-positive diagram, it is in general very hard to determine whether it might also have a positive diagram. This work provides positivity obstructions for three …

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Strongly Invertible Knots, Equivariant Slice Genera, And An Equivariant Algebraic Concordance Group, Allison N. Miller, M. Powell Jun 2023

Strongly Invertible Knots, Equivariant Slice Genera, And An Equivariant Algebraic Concordance Group, Allison N. Miller, M. Powell

Mathematics & Statistics Faculty Works

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let K be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum #n K is at least n/4. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

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Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe Jun 2023

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

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(R1965) Some More Properties On Generalized Double Fuzzy Z Alpha Open Sets, K. Jayapandian, A. Saivarajan, O. Uma Maheswari, J. Sathiyaraj Jun 2023

(R1965) Some More Properties On Generalized Double Fuzzy Z Alpha Open Sets, K. Jayapandian, A. Saivarajan, O. Uma Maheswari, J. Sathiyaraj

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a new class of sets termed as double fuzzy generalized Z alpha closed sets and double fuzzy generalized Z alpha open sets are introduced with the help of double fuzzy Z alpha open and double fuzzy Z alpha closed sets, respectively. Using these sets double fuzzy generalized Z alpha border, double fuzzy generalized Z alpha exterior and double fuzzy generalized Z alpha frontier of a fuzzy set in double fuzzy topological spaces are introduced. Also, the topological properties and characterizations of these sets and operators are studied. Furthermore, suitable examples have been provided to illustrate the theory.

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Deep Learning Recommendations For The Acl2 Interactive Theorem Prover, Robert K. Thompson, Robert K. Thompson Jun 2023

Deep Learning Recommendations For The Acl2 Interactive Theorem Prover, Robert K. Thompson, Robert K. Thompson

Master's Theses

Due to the difficulty of obtaining formal proofs, there is increasing interest in partially or completely automating proof search in interactive theorem provers. Despite being a theorem prover with an active community and plentiful corpus of 170,000+ theorems, no deep learning system currently exists to help automate theorem proving in ACL2. We have developed a machine learning system that generates recommendations to automatically complete proofs. We show that our system benefits from the copy mechanism introduced in the context of program repair. We make our system directly accessible from within ACL2 and use this interface to evaluate our system in …

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Groups Of Non Positive Curvature And The Word Problem, Zoe Nepsa Jun 2023

Groups Of Non Positive Curvature And The Word Problem, Zoe Nepsa

Master's Theses

Given a group $\Gamma$ with presentation $\relgroup{\scr{\scr{A}}}{\scr{R}}$, a natural question, known as the word problem, is how does one decide whether or not two words in the free group, $F(\scr{\scr{A}})$, represent the same element in $\Gamma$. In this thesis, we study certain aspects of geometric group theory, especially ideas published by Gromov in the late 1980's. We show there exists a quasi-isometry between the group equipped with the word metric, and the space it acts on. Then, we develop the notion of a CAT(0) space and study groups which act properly and cocompactly by isometries on these spaces, such groups …

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Abstract Algebra: An Inquiry-Based Approach, Second Edition, Supplemental Material, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom Jun 2023

Abstract Algebra: An Inquiry-Based Approach, Second Edition, Supplemental Material, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

Open Textbooks

This text is a supplement to Abstract Algebra: An Inquiry-Based Approach, Second Edition. It includes applications of algebra to RSA encryption, check digits, the games of NIM and the 15 Puzzle, and the determination of groups of small order. In addition, reference material that could be useful for some students appears in the appendices, such as background material on functions, methods of proof (including the equivalencies of the Well-Ordering Principle and different versions of mathematical induction), and complex roots of unity. The appendices also contain a complete proof that polynomial rings are rings, a proof of the Fundamental Theorem of …

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Discrete Wiener Algebra In The Bicomplex Setting, Spectral Factorization With Symmetry, And Superoscillations, Daniel Alpay, Izchak Lewkowicz, Mihaela Vajiac Jun 2023

Discrete Wiener Algebra In The Bicomplex Setting, Spectral Factorization With Symmetry, And Superoscillations, Daniel Alpay, Izchak Lewkowicz, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between bicomplex analysis and complex analysis with symmetry. We also write an application to superoscillations in this case.

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Flow Patterns For Newtonian And Non-Newtonian Fluids In A Cylindrical Pipe, Erick Sanchez, Dambaru Bhatta Jun 2023

Flow Patterns For Newtonian And Non-Newtonian Fluids In A Cylindrical Pipe, Erick Sanchez, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

A fully developed laminar steady flow of an incompressible, viscous fluid in a horizontal cylindrical pipe is considered here. Flow patterns for an incompressible, viscous fluid for both Newtonian and non-Newtonian fluids such as shear-thinning, shear-thickening and Bingham plastic fluids are analyzed in this study. Assuming that the flow is only due to the wall shear stress and the pressure drop, the velocity component in the axial direction for these cases is derived. Computational results of the velocity profiles for various cases are obtained using MATLAB and presented in graphical forms. It is observed that the velocity profile is parabolic …

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An Explicit Construction Of Sheaves In Context, Tyler A. Bryson Jun 2023

An Explicit Construction Of Sheaves In Context, Tyler A. Bryson

Dissertations, Theses, and Capstone Projects

This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology.

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Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo Jun 2023

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

Dissertations, Theses, and Capstone Projects

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set EX. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E2-terms coincide with certain …

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Quantifying Separability In Limit Groups, Keino Brown Jun 2023

Quantifying Separability In Limit Groups, Keino Brown

Dissertations, Theses, and Capstone Projects

We show that for any finitely generated non-abelian subgroup H of a limit group L, there exists a finite-index subgroup K which is fully residually H. This generalizes the result of Wilton that limit groups admit local retractions. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes results of Louder, …

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The Coupled Modified Yajima–Oikawa System: Model Derivation And Soliton Solutions, Junchao Chen, Bao-Feng Feng, Ken-Ichi Maruno Jun 2023

The Coupled Modified Yajima–Oikawa System: Model Derivation And Soliton Solutions, Junchao Chen, Bao-Feng Feng, Ken-Ichi Maruno

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we consider a coupled modified Yajima–Oikawa (YO) system which describes the nonlinear resonant interaction between one long wave (LW) and two short waves (SWs). It is shown that this coupled system can be derived from a three-component modified nonlinear Schrödinger equations through asymptotic reductions. Furthermore, the bright, dark multi-soliton and multi-breather solutions in terms of determinants are obtained respectively by virtue of the bilinear Kadomtsev–Petviashvili-hierarchy reduction technique. The detailed analysis of dynamical properties for one- and two-solitons and breathers is performed, which show the interesting collision properties for the bright and dark solitons. Particularly, differing from …

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Topics On Asymmetric Classification, Subhrasish Chakraborty May 2023

Topics On Asymmetric Classification, Subhrasish Chakraborty

Dissertations

Asymmetric classification refers to a situation where the cost of misclassifying one class is significantly higher than the cost of misclassifying the other class. This problem is common in many real-world scenarios, such as medical diagnosis or fraud detection. In this dissertation two of the common types of asymmetric classification problems have been dealt with — imbalanced classification and ordinal classification. An example of imbalanced classification is to detect fraudulent credit card transactions where the distribution of the normal and fraud transactions are extremely skewed. On the other hand, ordinal classification, also known as ordinal regression, is widely used in …

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Bright Light Therapy And Depression: Assessing Suitability Using Entrainment Maps, Charles A. Mainwaring May 2023

Bright Light Therapy And Depression: Assessing Suitability Using Entrainment Maps, Charles A. Mainwaring

Theses

Bright Light Therapy has been shown to be efficacious to mood disorders including Major Depression. Researchers use the Jewett-Forger-Kronauer model of the circadian rhythm with the Unified Model of melatonin including a mathematical term implementing feedback from the melatonin system into the circadian system to quantify the effects of bright light. Early investigations into intrinsic period, light sensitivity, and the circadian pacemaker's sensitivity to blood melatonin concentration may be indicators of subsets of patients with long intrinsic periods exhibiting symptoms of depression.

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A Survey On Online Matching And Ad Allocation, Ryan Lee May 2023

A Survey On Online Matching And Ad Allocation, Ryan Lee

Theses

One of the classical problems in graph theory is matching. Given an undirected graph, find a matching which is a set of edges without common vertices. In 1990s, Richard Karp, Umesh Vazirani, and Vijay Vazirani would be the first computer scientists to use matchings for online algorithms [8]. In our domain, an online algorithm operates in the online setting where a bipartite graph is given. On one side of the graph there is a set of advertisers and on the other side we have a set of impressions. During the online phase, multiple impressions will arrive and the objective of …

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A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić May 2023

A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić

Mathematics, Physics, and Computer Science Faculty Articles and Research

For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x0, y0) such that A nontrivially intersects curves y − y0 = a(x −x0)β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …

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Decomposition Of Beatty And Complementary Sequences, Geremías Polanco May 2023

Decomposition Of Beatty And Complementary Sequences, Geremías Polanco

Mathematics Sciences: Faculty Publications

In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.

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Motion Planning Algorithm In A Y-Graph, David Baldi May 2023

Motion Planning Algorithm In A Y-Graph, David Baldi

Rose-Hulman Undergraduate Mathematics Journal

We present an explicit algorithm for two robots to move autonomously and without collisions on a track shaped like the letter Y. Configuration spaces are of practical relevance in designing safe control schemes for automated guided vehicles. The topological complexity of a configuration space is the minimal number of continuous instructions required to move robots between any initial configuration to any final one without collisions. Using techniques from topological robotics, we calculate the topological complexity of two robots moving on a Y-track and exhibit an optimal algorithm realizing this exact number of instructions given by the topological complexity.

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Δ-Small Intersection Graphs Of Modules, Ahmed H. Alwan May 2023

Δ-Small Intersection Graphs Of Modules, Ahmed H. Alwan

Al-Bahir

Let R be a commutative ring with unit and M be a unitary left R-module. The δ-small intersection graph of non-trivial submodules of , denoted by , is an undirected simple graph whose vertices are the non-trivial submodules of , and two vertices are adjacent if and only if their intersection is a -small submodule of . In this article, we study the interplay between the algebraic properties of , and the graph properties of such as connectivity, completeness and planarity. Moreover, we determine the exact values of the diameter and girth of , as well as give a formula …

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Using Deep Neural Networks To Classify Astronomical Images, Andrew D. Macpherson May 2023

Using Deep Neural Networks To Classify Astronomical Images, Andrew D. Macpherson

Honors Projects

As the quantity of astronomical data available continues to exceed the resources available for analysis, recent advances in artificial intelligence encourage the development of automated classification tools. This paper lays out a framework for constructing a deep neural network capable of classifying individual astronomical images by describing techniques to extract and label these objects from large images.

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Identifiability Of Phylogenetic Models, Thomas J. Yacovone May 2023

Identifiability Of Phylogenetic Models, Thomas J. Yacovone

Math and Computer Science Honors Theses

An algebraic statistical model is a parametrized family of probability distributions. Identifiability is a crucial property of a statistical model; when it holds, probability distributions in the model uniquely determine the parameters that produce them. In phylogenetics, models that are identifiable can be used to ascertain evolutionary relationships between species based on observed data, such as amino acid and DNA sequence data. However, global identifiability is a strong condition that may not always hold. A discrete parameter of a model is said to be generically identifiable if the set of probability distributions that do not uniquely determine the discrete parameter …

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Gap Junctions And Synchronization Clusters In The Thalamic Reticular Nuclei, Anca R. Radulescu, Michael Anderson May 2023

Gap Junctions And Synchronization Clusters In The Thalamic Reticular Nuclei, Anca R. Radulescu, Michael Anderson

Biology and Medicine Through Mathematics Conference

No abstract provided.

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Computing Brain Networks With Complex Dynamics, Anca R. Radulescu May 2023

Computing Brain Networks With Complex Dynamics, Anca R. Radulescu

Biology and Medicine Through Mathematics Conference

No abstract provided.

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Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo May 2023

Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we consider a family {Ht}t∈R of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations {(C2, πt)}t∈R of the hypercomplex system {Ht}t∈R, and study the realizations πt(h) of hypercomplex numbers h ∈ Ht, as (2 × 2)-matrices acting on C2, for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

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A Hörmander–Fock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa May 2023

A Hörmander–Fock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In a recent paper we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical ∂-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η= (ηn)n≥0 leading to a special entire function E(z) that is used to express the kernel function of the Hörmander-Fock space. We …

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Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn May 2023

Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn

Dartmouth College Ph.D Dissertations

Recently, there has been an explosion of interest in re-imagining many-body quantum phenomena beyond equilibrium. One such effort has extended the symmetry-protected topological (SPT) phase classification of non-interacting fermions to driven and dissipative settings, uncovering novel topological phenomena that are not known to exist in equilibrium which may have wide-ranging applications in quantum science. Similar physics in non-interacting bosonic systems has remained elusive. Even at equilibrium, an "effective non-Hermiticity" intrinsic to bosonic Hamiltonians poses theoretical challenges. While this non-Hermiticity has been acknowledged, its implications have not been explored in-depth. Beyond this dynamical peculiarity, major roadblocks have arisen in the search …

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Essays On Statistical Inference, Nonconvex Optimization And Machine Learning, Yanjie Zhong May 2023

Essays On Statistical Inference, Nonconvex Optimization And Machine Learning, Yanjie Zhong

Arts & Sciences Graduate Student Theses and Dissertations

Over the past decades, numerous optimization and machine learning (ML) algorithms have been proposed, many of which have demonstrated success in real-world applications and significantly impacted people's lives. Researchers have devoted considerable effort to understanding the theoretical underpinnings of these methods and to improving their performance, as well as designing algorithms compatible with demanding real-world constraints. This dissertation focuses on investigating the statistical properties of several mainstream optimization and ML algorithms, enabling us to make decisions with statistical guarantees.First, we examine the classical stochastic gradient descent algorithm (SGD) in a general nonconvex context. Utilizing the multiplier bootstrap technique, we design …

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Aging An Ancient Maya Population From Actuncan, Belize Using Dental X-Rays, Kaitlyn Nicole Cash May 2023

Aging An Ancient Maya Population From Actuncan, Belize Using Dental X-Rays, Kaitlyn Nicole Cash

Honors Theses

My goal is to determine an accurate age at death estimation of an ancient Maya population from the archaeological site of Actuncan, Belize. This was done by measuring the lengths of the coronal pulp cavities in the individuals’ teeth. I used X-Ray images to measure the coronal pulp cavities of the teeth and estimated age using multiple regression formulae for premolars, molars, and incisors to make age estimations. The formulae came from two studies, Ikeda et al. (1985) and Drusini (2008), that form the basis of my research. Ikeda et al. (1985) was the first to use this aging method, …

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