Mathematics Commons^™

Pseudo-Contractions, Rigidity, Fixed Points And Related Questions, Daniel Alpay, Vladimir Bolotnikov, David Shoikhet

Mathematics, Physics, and Computer Science Faculty Articles and Research

The class of holomorphic self-mappings of the open unit disk (which are contractions with respect to the Poincaré metric) admits a natural extension to the class of holomorphic pseudo-contractions. In this paper, we study various inequalities involving the values of derivatives of holomorphic pseudo-contractions at fixed points (particularly, at the Denjoy–Wolff fixed point).

The Infinite Is The Chasm In Which Our Thoughts Are Lost: Reflections On Sophie Germain's Essays, Adam Glesser, Bogdan D. Suceavă, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

"Sophie Germain (1776–1831) is quite well-known to the mathematical community for her contributions to number theory [17] and elasticity theory (e.g., see [2, 5]). On the other hand, there have been few attempts to understand Sophie Germain as an intellectual of her time, as an independent thinker outside of academia, and as a female mathematician in France, facing the prejudice of the time of the First Empire and of the Bourbon Restoration, while pursuing her thoughts and interests and writing on them. Sophie Germain had to face a double challenge: the mathematical difficulty of the problems she approached and the …

From Biquandle Structures To Hom-Biquandles, Alissa Crans

Mathematics, Statistics and Data Science Faculty Works

We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun in [Hom quandles, J. Knot Theory Ramifications 23(2) (2014)], finding biquandle analogs of results therein. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.

Graphs Admitting Only Constant Splines, Alissa Crans, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

Shining A Light On A Hidden Figure: Dorothy Hoover, Lily S. Khadjavi

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

A Linear Optimal Feedback Control For Producing 1,3-Propanediol Via Microbial Fermentation, Yangping Ma

Mathematics, Statistics and Data Science Faculty Works

In this paper, we consider a multistage feedback control strategy for the production of 1,3-propanediol(1,3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1,3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The …

Elementary Hyperreal Analysis, Logan Cebula

Williams Honors College, Honors Research Projects

This text explores elementary analysis through the lens of non-standard analysis. The hyperreals will be proven to be implied by the existence of the reals via the axiom of choice. The notion of a hyperextension will be defined, and the so-called Transfer Principle will be proved. This principle establishes equivalence between results in real and hyperreal analysis. Sequences, subsequences, and limit suprema/infima will then be explored. Finally, integration will be considered.

Extrinsic Ricci Flow On Surfaces Of Revolution, Vincent E. Coll, Jeff Dodd, David L. Johnson

Research, Publications & Creative Work

An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in Rn+k, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R3. Our approach is to build the desired embedded surfaces of revolution S(t) in R3 into the flow at the outset by rewriting the Ricci …

The Ubiquity Of Phi In Human Culture & The Natural World, Jennifer Bressler

Masters Essays

No abstract provided.

Algebra I Topics Using Geogebra, Matthew Rancourt

Masters Essays

No abstract provided.

Testing Gene-Environment Interactions For Rare And/Or Common Variants In Sequencing Association Studies., Zihan Zhao, Jianjun Zhang, Qiuying Sha, Han Hao

Michigan Tech Publications, Part 1

The risk of many complex diseases is determined by a complex interplay of genetic and environmental factors. Advanced next generation sequencing technology makes identification of gene-environment (GE) interactions for both common and rare variants possible. However, most existing methods focus on testing the main effects of common and/or rare genetic variants. There are limited methods developed to test the effects of GE interactions for rare variants only or rare and common variants simultaneously. In this study, we develop novel approaches to test the effects of GE interactions of rare and/or common risk, and/or protective variants in sequencing association studies. We …

Uniformly Resolvable Decompositions Of Kv In 1-Factors And 4-Stars, Melissa S. Keranen, Donald L. Kreher, Salvatore Milici, Antoinette Tripodi

Michigan Tech Publications, Part 1

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. A uniformly resolvable {X, Y }-decomposition of the complete graph K_v is an edge decomposition of K_v into exactly r X-factors and s Y -factors. In this article we determine necessary and sufficient conditions for when the complete graph K_v has a uniformly resolvable decompositions into 1-factors and K_1,4-factors.

Invariance And Invertibility In Deep Neural Networks, Han Zhang

Theses and Dissertations

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …

The Process And A Pitfall In Developing Biology And Chemistry Problems For Mathematics Courses, Mary Beisiegel, Lori Kayes, Devon Quick, Richard Nafshun, Michael Lopez, Steve Dobrioglo, Michael Dickens

Journal of Mathematics and Science: Collaborative Explorations

In this paper, we describe our process for developing applied problems from biology and chemistry for use in a differential calculus course. We describe our conversations and curricular analyses that led us to change from our initial focus on college algebra to calculus. We provide results that allowed us to see the overlaps between biology and mathematics and chemistry and mathematics and led to a specific focus on problems related to rates of change. Finally, we investigate the problems that were developed by the partner disciplines for use on recitation activities in calculus and how those problems were modified by …

A Mathematical Analysis Of The Game Of Santorini, Carson Clyde Geissler

Senior Independent Study Theses

Santorini is a two player combinatorial board game. Santorini bears resemblance to the graph theory game of Geography, a game of moving and deleting vertices on a graph. We explore Santorini with game theory, complexity theory, and artificial intelligence. We present David Lichtenstein’s proof that Geography is PSPACE-hard and adapt the proof for generalized forms of Santorini. Last, we discuss the development of an AI built for a software implementation of Santorini and present a number of improvements to that AI.

Domino Steganography, Robert Bosch, Aaron Kreiner

Faculty & Staff Scholarship

No abstract provided.

On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank

Faculty Journal Articles

We determine the ranks of string C-group representations of 4-dimensional projective symplectic groups over a finite field, and comment on those of higher-dimensional symplectic and orthogonal groups.

Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara

Faculty Journal Articles

No abstract provided.

Constructing And Writing Mathematical Proofs: A Guide For Mathematics Students, Ted Sundstrom

Open Textbooks

This little book is not intended to be a textbook for a course dealing with an introduction to constructing and writing mathematical proofs. It is intended to be a reference book for students who need to construct and write proofs in their upper division mathematics courses. So it is assumed that students who use this as a reference have already taken an “introduction to proofs” course.

With the exception of Chapter 1, each chapter in the book has a description of a proof technique along with some justification as to why it is a valid proof method. There are then …

Definable Modules Over Definable Rings Without Zero Divisors In O-Minimal Structures, Jaruwat Rodbanjong

Chulalongkorn University Theses and Dissertations (Chula ETD)

In an o-minimal structure, every definable group admits a definable group manifold. Moreover, one can also show an analogue of this statement for definable rings. In this work, we prove that every definable module over definable ring without zero divisors admits a definable module manifold. In addition, we also give the classification of definable modules over definable rings without zero divisors in o-minimal structures

Greybody Factors For Perfect Fluid Black Hole, Kunlapat Sansuk

Chulalongkorn University Theses and Dissertations (Chula ETD)

The Einstein equation is a field equation which describes gravity in terms of the curvature of spacetime. It states how matter curves spacetime. The solutions of the Einstein field equation are the metrics of spacetime from which the curvature of spacetime can be found. The field equations are non-linear, which are complicated to solve. Therefore, some assumptions are needed to reduce the complexity of the Einstein equation. One of these assumptions is a perfect fluid sphere. Perfect fluid sphere satisfies the following: no viscosity, no heat conduction, and isotropy. Perfect fluid black holes are black hole solutions of the Einstein …

A Two-Stage Model For Balancing Instructor Workload And Teaching Preference In University Course Timetabling, Nipitta Burana

Chulalongkorn University Theses and Dissertations (Chula ETD)

This thesis focuses on balancing instructor workload and maximizing preferences by using the data in the first semester of 2019 from Department of Mathematics and Computer Science,Faculty of Science, Chulalongkorn University as a case study. Since there are many instructors with over-workload in the department which directly affect their research qualities, balancing teaching workload is the main objective of this study. The proposed approach to balance workload is to split some basic courses into two parts: before midterm and after midterm and then assign each course to two instructors. Moreover, the preferences or the requests of teaching a course are …

Classification Of Main Components In Airports From Remote Sensing Images By Efficient Network, Pimpisa Charoenchittang

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we classify the type of main components in airports from the remote sensing images. This datasets is considered as an interesting information since the same type of component may have different shape, size, and color. EfficientNet architecture is the deep learning architecture to use in this research due to the small number of parameters and computational time compared to the other architectures with similar accuracy. In our experiment, we apply the EfficientNet in versions B0, B1, B2, B3, and B4 to classify four types of components in the airport; the passenger terminal, the radio tower, the runway, …

Cops And Robbers On Hypergraphs, Pinkaew Siriwong

Chulalongkorn University Theses and Dissertations (Chula ETD)

Cops and robbers game is a game usually played on a finite connected graphwith two players, cop and robber. Recently, cops and robbers game played on hypergraphs was introduced. To give a better chance to a cop by allowing morethan one cop and at least one cop has to move, the cop-number, the least numberof cops to guarantee that they win the game, on graphs and hypergraphs is studied.This thesis provides (i) a characterization of a cop-win hypergraph (ii) some results on the products of hypergraphs and (iii) the cop-number of complete k-partite hypergraphs and n-prisms over a hypergraph. Moreover, …

Graph And Number Theoretic Properties Of Certain Maps Over Finite Field, Pratchayaporn Doemlim

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we study the graph over a finite field $\mathbb{F}_q$, where $q$ is a prime power, obtained by iterations of the map $g(x)=x^p$, where $p$ is a prime number. Some properties of the graphs such as a characterization of their vertices and the number of cycles with specific length are showed. Moreover, some statistical estimates about the tail and cycle lengths of the related graphs are established.

Graph Grabbing Games And Toucher-Isolator Games, Sopon Boriboon

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this research, we study the graph grabbing game and the Toucher-Isolator game. In the graph grabbing game, we partially confirm a conjecture of Seacrest and Seacrest which states that Alice wins the game on every weighted connected bipartite even graph. In the Toucher-Isolator game, we give a simple alternative proof of a result of Räty that determines the most suitable tree on n vertices for Toucher which answers a question of Dowden, Kang, Mikalački and Stojaković.

Stochastic Differential Equation Models On Sphere For Animal Migration, Viput Puttanugool

Chulalongkorn University Theses and Dissertations (Chula ETD)

Animal migration is a seasonal movement of a bundle of animals from place to place. A migration cycle is often annually and closely linked with the cyclic pattern of seasons. A short-distance migration can be modeled on a planar surface, but most animals have long-distance migrations, e.g., Arctic terns migrate from the North Pole to the South Pole; thus, their migration should be modeled on a spherical surface. In addition, the Brownian motion term is included in a model to represent noises and randomness of movements. This work aims to simulate their migration routes using stochastic differential equations (SDEs) on …

Knight's Tours On Ringboards, Deficient 4 X N Chessboards And Some L-Boards, Wasupol Srichote

Chulalongkorn University Theses and Dissertations (Chula ETD)

A (legal) knight's move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in a perpendicular direction. A closed knight's tour is a sequence of knight's moves that visits every square on a given chessboard exactly once and returns to its start square. A closed knight's tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n > 2r, an (m,n,r)-ringboard or RB(m,n,r) is defined to be an m x n chessboard, denoted by CB(m x n), with the middle part …

Tolerance-Localized And Control-Localized Solutions To System Of Interval Linear Equations, Kanokwan Burimas

Chulalongkorn University Theses and Dissertations (Chula ETD)

In this thesis, we are interested in a system of interval linear equations Ax=b whose coefficient A and right hand side b vary in some real intervals. We study two types of solutions called tolerance--localized and control--localized solutions of interval linear equations system. The characterizations of each solution are proposed in two main theorems. First, the proposed theorem is stated in terms of center and radius matrices which is directly proved by following their definitions. The other theorem is presented as magnitude sense with new notation. Based on the second theorem, the closed form of all solution sets is released. …

Discrete-Time Risk Model Based On Zero Inflated Poisson Time Series, Siwarak Sawongnam

Chulalongkorn University Theses and Dissertations (Chula ETD)

An important goal of actuary is to develop models for company portfolio and insurance products. Risk measurement is one of the essential measures that inform actuaries and risk managers about the degree to which the risk bearing entity. To have precise risk measure, we require an appropriate claim count process. The common claim count processes are usually constructed from the Poisson distribution. However, insurance data have generally excess zeros which causes the overdispersion. This violates the assumption of the Poisson distribution. Therefore, alternative distributions accommodating zero count are explored in literature. The zero inflated Poisson distribution is one of the …