Mathematics Commons^™

Transfunctions And Other Topics In Measure Theory, Jason Bentley

Electronic Theses and Dissertations, 2020-2023

Measures are versatile objects which can represent how populations or supplies are distributed within a given space by assigning sizes to subregions (or subsets) of that space. To model how populations or supplies are shifted from one configuration to another, it is natural to use functions between measures, called transfunctions. Any measurable function can be identified with its push-forward transfunction. Other transfunctions exist such as convolution operators. In this manner, transfunctions are treated as generalized functions. This dissertation serves to build the theory of transfunctions and their connections to other mathematical fields. Transfunctions that identify with continuous or measurable push-forward …

Providing Better Choices: An Exploration Of Solutions In Multi-Objective Optimization And Game Theory Using Variational Analysis, Glenn Matthew Harris

Graduate Research Theses & Dissertations

Multi-objective optimization problems and game theory problems have a wide array of

applications and because of this there are different types of solutions available. This dissertation

explores two areas of optimization and a solution type for each. First, substantial

efficiency (SE) as a type of solution to multi-objective optimization problems that extends

proper efficiency. Secondly, strong Nash equilibria (SNE) as a type of solution to game

theoretic problems that extends Nash equilibria. Substantial efficiency is demonstrated to

be a superior solution to the more rudimentary notion of proper efficiency in solving some

multi-objective financial market and economic problems. Using this …

The Sigma Chromatic Number Of The Sierpinski Gasket Graphs And The Hanoi Graphs, Agnes Garciano, Reginaldo M. Marcelo, Mari-Jo P. Ruiz, Mark Anthony C. Tolentino

Mathematics Faculty Publications

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a sigma coloring if σ(u) ≠ σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we determine the sigma chromatic numbers of the …

Twin Chromatic Indices Of Some Graphs With Maximum Degree 3, Jayson D. Tolentino, Reginaldo M. Marcelo, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by . In this paper, we determine the …

On Eigenvalue Bounds For The Finite-State Birth-Death Process Intensity Matrix, R.R.P Tan, K Ikeda, Len Patrick Dominic M. Garces

Mathematics Faculty Publications

The paper sets forth a novel eigenvalue interlacing property across the finite-state birth-death process intensity matrix and two clearly identified submatrices as an extension of Cauchy’s interlace theorem for Hermitian matrix eigenvalues. A supplemental proof involving an examination of probabilities acquired from specific movements across states and a derivation of a form for the eigenpolynomial of the matrix through convolution and Laplace transform is then presented towards uncovering a similar characteristic for the general Markov chain transition rate matrix. Consequently, the proposition generates bounds for each eigenvalue of the original matrix, easing numerical computation. To conclude, the applicability of 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.

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 …

Perfect 2-Colorings Of The Grassmann Graph Of Planes, Stefaan Dewinter, Klaus Metsch

Michigan Tech Publications, Part 1

We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG(n, q), n ≥ 5 odd, and show that the members of the family are the smallest possible examples if n ≥ 9 or q ≥ 25.

Cycle Double Covers And Integer Flows, Zhang Zhang

Graduate Theses, Dissertations, and Problem Reports (ETD)

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems.

Theory And Techniques Of Convergence Of Topological Transformation Group Actions, Murtadha Jaber Sarray

Graduate Theses, Dissertations, and Problem Reports (ETD)

n this dissertation, we present new set functions called strongly limit and strongly prolongation limit sets. We show the new sets, especially strongly prolongation limit sets, characterize proper action under an arbitrary setting. That is, we characterize proper action for wider class of proper ܩ-spaces. Also, we show the new version of the sets could be derived from strongly exceptional sets which have been used as a good technique for the characterization of a proper maps. Moreover, we review properties of well-known limit sets and prolongations and properties for the new version of limit sets under an arbitrary setting on …

Surviving Mathematics, Nathalie M. Luna

School of Mathematical & Statistical Sciences Faculty Publications

In this essay written in an informal voice, the author shares the ups and downs of her experience in academia. She shares her motivation to study mathematics, her undergraduate experience in Puerto Rico, and her graduate experience in South Texas.

Topological Pressure And Fractal Dimensions Of Cookie-Cutter-Like Sets, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero h of the pressure function. Moreover, it is shown that the h-dimensional Hausdorff measure and the h-dimensional packing measure are finite and positive.

Effect Of Hydraulic Resistivity On A Weakly Nonlinear Thermal Flow In A Porous Layer, Dambaru Bhatta, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

Heat and mass transfer through porous media has been a topic of research interest because of its importance in various applications. The flow system in porous media is modelled by a set of partial differential equations. The momentum equation which is derived from Darcy’s law contains a resistivity parameter. We investigate the effect of hydraulic resistivity on a weakly nonlinear thermal flow in a horizontal porous layer. The present study is a realistic study of nonlinear convection flow with variable resistivity whose rate of variation is arbitrary in general. This is a first step for considering more general problems in …

On Cohen-Macaulay Hopf Monoids In Species, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen- Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative h-vector. We apply our results to the weak and strong chromatic polynomials of …

Screening Potential Citrus Rootstocks For Phytophthora Nicotianae Tolerance, Madhurababu Kunta, Sandy Chavez, Zenaida Viloria, Hilda S. Del Rio, Madhavi Devanaboina, George Yanev, Jong-Won Park, Eliezer S. Louzada

School of Mathematical & Statistical Sciences Faculty Publications

Seeds from four citrus rootstocks including sour orange, Bitters-C22 citrandarin, Sarawak pummelo 3 Rio Red grapefruit, and Sarawak pummelo 3Bower mandarin were exposed to high inoculum levels of Phytophthora nicotianae to screen for tolerance. Inoculation of pregerminated seeds (PGIS) and non-PGIS was carried out. The average P. nicotianae propagule counts from the soil samples where these seedlings were raised ranged from 424 to 1361 colony forming units/cm3. The proportion of live to dead plants was recorded at 11months postinoculation, which showed that Sarawak3Bower performed significantly better than other rootstocks. Evaluation of the rootstocks 18 months postinoculation resulted in only one …

Stability Of Anisotropic Parabolic Equations Without Boundary Conditions, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

"Lagrangian Disks" In M-Theory, Sebastián Franco, Sergei Gukov, Sangmin Lee, Rak-Kyeong Seong, James Sparks

Publications and Research

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G₂ holonomy spaces and to Spin(7) metrics on 8-manifolds with T² fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T[M₄] on …

Forced (2 + 1)-Dimensional Discrete Three-Wave Equation, Junyi Zhu, Sishou Zhou, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

We generalize the ∂-dressing method to investigate a (2 + 1)-dimensional lattice, which can be regarded as a forced (2 + 1)-dimensional discrete three-wave equation. The soliton solutions to the (2 + 1)-dimensional lattice are given through constructing different symmetry conditions. The asymptotic analysis of one-soliton solution is discussed. For the soliton solution, the forces are zero.

Professional Development For Mathematics Teacher Education Faculty: Need And Design, Billy Jackson, Shandy Hauk, Jenq Jong Tsay, Alma Ramirez

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of this report is to share a conceptual model useful in the design of professional learning about teaching for university mathematics faculty. The model is illustrated by examples from a particular design effort: the development of an online shortcourse for faculty new to teaching mathematics courses for prospective primary school teachers. How novice mathematics teacher educators grow as instructors is an emerging area of research and development in the United States. At the same time, it is well established that effective instructional design of any course, including a course for faculty, requires breadth first: understanding and anticipating the …

Flows Of Co-Closed G_{2}-Structures, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

We survey recent progress in the study of G_{2}-structure Laplacian coflows, that is, heat flows of co-closed G_{2}-structures. We introduce the properties of the original Laplacian coflow of G_{2}-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified coflow and well as recent Shi-type estimates that apply to a more general class of G_{2}-structure flows.

Bayesian Modelling Of Nonlinear Poisson Regression With Artificial Neural Networks, Hansapani Rodrigo, Chris P. Tsokos

School of Mathematical & Statistical Sciences Faculty Publications

Modelling and prediction of count and rate responses have substantial usage in many fields, including health, finance, social, etc. Conventionally, linear Poisson regression models have been widely used to model these responses. However, the linearity assumption of the systematic component of linear Poisson regression models restricts their capability of handling complex data patterns. In this regard, it is important to develop nonlinear Poisson regression models to capture the inherent variability within the count data. In this study, we introduce a probabilistically driven nonlinear Poisson regression model with Bayesian artificial neural networks (ANN) to model count and rate data. This new …

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 …

Analysis On Sharp And Smooth Interface, Elizabeth V. Hawkins

College of Graduate Studies: Theses & Dissertations

In biology, minimizing a free energy functional gives an equilibrium shape that is the most stable in nature. The formulation of these functionals can vary in many ways, in particular they can have either a smooth or sharp interface. Minimizing a functional can be done through variational calculus or can be proved to exist using various analysis techniques. The functionals investigated here have a smooth and sharp interface and are analyzed using analysis and variational calculus respectively. From the former we find the condition for extremum and its second variation. The second variation is commonly used to analyze stability of …

Explicit Pseudo-Kähler Metrics On Flag Manifolds, Thomas A. Mason Iii

College of Graduate Studies: Theses & Dissertations

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of certain line bundles (Borel-Weil theorem). Less well-known are the (indefinite) invariant pseudo-Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott), and whose analogues in a non-compact setting lead to new representations (Kostant-Langlands). The purpose of this thesis is to give an explicit description of these metrics in the case of the unitary group G=U_n.