Digital Commons Network^™

Lorentzian Polynomials, Higher Hessians, And The Hodge-Riemann Property For Codimension Two Graded Artinian Gorenstein Algebras, Pedro Macias-Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

We study the Hodge-Riemann property (HRP) for graded Artinian Gorenstein (AG) algebras. We classify AG algebras in codimension two that have HRP in terms of higher Hessian matrices and positivity of Schur functions associated to certain rectangular partitions.

In this paper we introduce the Hodge Riemann property (HRP) on an arbitrary graded oriented Artinian Gorenstein (AG) algebra defined over R, and we give a criterion on the higher Hessian matrix of its Macaulay dual generator (Theorem 3.1). AG algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of complex manifolds, and the HRP is analogous to …

Mathematics For Biomedical Physics, Jogindra M. Wadehra

Open Textbooks

Mathematics for Biomedical Physics is an open access peer-reviewed textbook geared to introduce several mathematical topics at the rudimentary level so that students can appreciate the applications of mathematics to the interdisciplinary field of biomedical physics. Most of the topics are presented in their simplest but rigorous form so that students can easily understand the advanced form of these topics when the need arises. Several end-of-chapter problems and chapter examples relate the applications of mathematics to biomedical physics. After mastering the topics of this book, students would be ready to embark on quantitative thinking in various topics of biology and …

Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang

Department of Mathematics: Faculty Publications

We study sequences of Betti numbers (β^R_i (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I^☐ generated by the quadratic relations of the associated graded ring of R satisfies height I^☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, …

Low-Gain Integral Control For A Class Of Discrete-Time Lur’E Systems With Applications To Sampled-Data Control, Chris Guiver, Richard Rebarber, Stuart Townley

Department of Mathematics: Faculty Publications

We study low-gain (P)roportional (I)ntegral control of multivariate discrete-time, forced Lur’e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur’e system in feedback with sampled-data low-gain integral control. The theory …

A Comparison Of Statistical Methods For Modeling Count Data With An Application To Hospital Length Of Stay, Gustavo Fernandez, Kristina Vatcheva

School of Mathematical & Statistical Sciences Faculty Publications

Background

Hospital length of stay (LOS) is a key indicator of hospital care management efficiency, cost of care, and hospital planning. Hospital LOS is often used as a measure of a post-medical procedure outcome, as a guide to the benefit of a treatment of interest, or as an important risk factor for adverse events. Therefore, understanding hospital LOS variability is always an important healthcare focus. Hospital LOS data can be treated as count data, with discrete and non-negative values, typically right skewed, and often exhibiting excessive zeros. In this study, we compared the performance of the Poisson, negative binomial (NB), …

A Positivity Preserving, Energy Stable Finite Difference Scheme For The Flory-Huggins-Cahn-Hilliard-Navier-Stokes System, Wenbin Chen, Jianyu Jing, Cheng Wang, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we propose and analyze a finite difference numerical scheme for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. in the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term, explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection …

How Order And Disorder Affect People's Behavior: An Explanation, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

Experimental data shows that people placed in orderly rooms donate more to charity and make healthier food choices that people placed in disorderly rooms. On the other hand, people placed in disorderly rooms show more creativity. In this paper, we provide a possible explanation for these empirical phenomena.

Why Decision Paralysis, Sean Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

If a person has a small number of good alternatives, this person can usually make a good decision, i.e., select one of the given alternatives. However, when we have a large number of good alternatives, people take much longer to make a decision -- sometimes so long that, as a result, no decision is made. How can we explain this seemingly no-optimal behavior? In this paper, we show that this "decision paralysis" can be naturally explained by using the usual decision making ideas.

Why Five Stages Of Solar Activity, Why Five Stages Of Grief, Why Seven Plus Minus Two: A General Geometric Explanation, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent paper showed that the solar activity cycle has five clear stages, and that taking theses stages into account helps to make accurate predictions of future solar activity. Similar 5-stage models have been effective in many other application area, e.g., in psychology, where a 5-stage model provides an effective description of grief. In this paper, we provide a general geometric explanations of why 5-stage models are often effective. This result also explains other empirical facts, e.g., the seven plus minus two law in psychology and the fact that only five space-time dimensions have found direct physical meaning.

Why Micro-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanations With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, the larger the object, the larger should be its effect on the surroundings -- in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: micro-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we …

How To Make Inflation Optimal And Fair, Sean Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

A reasonably small inflation helps economy as a whole -- by encouraging spending, but it also hurts people by decreasing the value of their savings. It is therefore reasonably to come up with an optimal (and fair) level of inflation, that would stimulate economy without hurting people too much. In this paper, we describe how this can be potentially done.

Academic Integration And Self-Regulation Strategies In Precalculus And Calculus I, Kyle Russell Turner

Mathematics Dissertations - Archive

This case study examines the self-regulation strategies and academic integration of first-semester undergraduate students enrolled in precalculus and first-semester calculus at a large urban university in the southwestern United States. I use a sequence of interviews to examine the relationship of mathematics self-efficacy and mathematics identity on students’ use of these strategies. Five interviews occurred during the Fall 2021 semester with three precalculus and six calculus students, and I distributed initial surveys to thirteen first-semester calculus sections and five precalculus sections. To analyze the data, I integrated frameworks from Zimmerman and Pons (1986) and Wolters (1998) on the self-regulation strategies …

Decomposition Of Modules And Tensor Products Over Principal Subalgebras Of Truncated Polynomial Rings, Kevin Steine Jr. Harris

Mathematics Dissertations - Archive

The topic of my dissertation is to investigate the behavior of modules and tensor products over a truncated polynomial ring with prime characteristic. This investigation utilizes principal subalgebras of the truncated polynomial ring as the main tool for studying these objects. Then, we investigate if these modules and their tensor products have a similar behavior when viewed over more general truncated polynomial rings. In particular, we aim to investigate the behavior of these objects when we replace principal subalgebras over a field with prime characteristic by hypersurfaces over a field with no characteristic restriction.

Abrading The Enigma Of The Wound Healing Process: Modeling The Inflammation, Proliferation, And Maturation Stage, Amanda Patrick

Mathematics Dissertations - Archive

Wound healing encompasses a group of processes categorized into overlapping stages known as the inflammation, proliferation, and maturation/remodeling stage. The dynamics of these processes are important in studying outcomes of wound care and determining factors that contribute to certain wound outcomes. A system of ordinary differential equations is constructed for the inflammation, proliferation, and remodeling stage. Parameter sets for this model are investigated based on output dynamics according to the literature and based on experimental data. A bifurcation analysis is conducted to determine sudden changes that can occur in the inflammation system. Fourier Amplitude Sensitivity Test (FAST) is implemented to …

Properties And Classifications Of Certain Lcd Codes., Dalton Seth Gannon

Electronic Theses and Dissertations

A linear code $C$ is called a linear complementary dual code (LCD code) if $C \cap C^\perp = {0}$ holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code $C$ over $\Z_4$ of odd length to be an LCD code is that $C=\big( f(x) \big)$ where $f$ is a self-reciprocal polynomial in $\Z_{4}[X]$ which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code $C$ of length …

Developing Confidence And Interest In Teaching Relevant Mathematical Modeling Lessons, Jacy Beck

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

What is mathematical modeling and how can inservice and pre-service teachers develop the skills and competencies necessary to increase confidence and interest in teaching relevant mathematical modeling lessons? Mathematical modeling is “the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions” (CSSM, 2010, p. 72). By providing students with an opportunity to engage in relevant mathematical modeling prompts, we provide them with transferable skills and knowledge. The aim of this paper will be to provide insight into the relevance of teaching mathematical modeling, provide resources for integrating modeling …

Dynamic System Discovery With Recursive Physics-Informed Neural Networks, Jarrod Mau

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This thesis presents a novel method, recursive Physics informed neural network, to learn the right hand side of differential equations. The neural network takes in data, then trains, and then acts as a proxy for the differential equation which can be used for modeling. We show the theoretical superiority of the recursive approach. We also use computer simulations to demonstrate the proved properties.

Joint Invariants Of Primitive Homogenous Spaces, Illia Hayes

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Joint invariants are motivated by the study of congruence problems in Euclidean geometry, where they provide necessary and sufficient conditions for congruence. More recently joint invariants have been used in computer image recognition problems. This thesis develops new methods to compute joint invariants by developing a reduction technique, and applies the reduction to a number of important examples.

Using The Reshetikhin-Turaev Link Invariant Approach With Non-Semisimple Categories, Adam Robertson

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Invariants of knots and links are useful because they give rise to invariants of 3-manifolds. In particular, combinatorial link invariants give rise to combinatorial invariants of 3-manifolds, which are hard to come by using traditional methods from classical topology. The Reshetikhin–Turaev approach, which is based in quantum topology, develops link invariants using semisimple ribbon categories. However, a large class of algebraically interesting ribbon categories are non-semisimple and so give trivial link invariants via the Reshetikhin–Turaev method. We modify the Reshetikhin–Turaev method to make it suitable for non-semisimple ribbon categories. We discuss explicitly the following three examples: semisimple modules for the …

Extensions To The Syrjala Test With Eye-Tracking Data Analysis Applications In R, Eric D. Mckinney

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Eye tracking is a process for measuring the movement of an individual’s eye(s) when that individual is looking at something. Many eye-tracking technologies exist to aid in calculating and recording data associated with what a person focuses their visual attention on. For example, eye-tracking technology can record points on an image that a person is looking at. Often the question arises as to whether two people, or groups of people, are looking at the same thing(s). This dissertation presents a new way (or test) to quantify those differences while taking into consideration the randomness associated with such data. Hence, the …

Contributions To Random Forest Variable Importance With Applications In R, Kelvyn K. Bladen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A major focus in statistics is building and improving computational algorithms that can use data to predict a response. Two fundamental camps of research arise from such a goal. The first camp is researching ways to get more accurate predictions. Many sophisticated methods, collectively known as machine learning methods, have been developed for this very purpose. One such method that is widely used across industry and many other areas of investigation is called Random Forests.

The second camp of research is that of improving the interpretability of machine learning methods. This is worthy of attention when analysts desire to optimize …

Defining Areas Of Interest Using Voronoi And Modified Voronoi Tesselations To Analyze Eye-Tracking Data, Joanna D. Coltrin

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Eye tracking is a technology used to track where someone is looking. Eye-tracking technology is often used to study what people focus on when looking at a photo of another person. The eye-tracking technology records points on a photo that a person is looking at. When the photo being looked at shows a person, the points can be categorized by body part such as head, right hand, left hand, and torso. This thesis presents the use of partially circular areas to define the body parts of the person in the photo and therefore categorize the points collected by the eye-tracker. …

The On-Line Width Of Various Classes Of Posets., Israel R. Curbelo

Electronic Theses and Dissertations

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. We prove …

Minimal Differential Graded Algebra Resolutions Related To Certain Stanley-Reisner Rings, Todd Anthony Morra

All Dissertations

We investigate algebra structures on resolutions of a special class of Cohen-Macaulay simplicial complexes. Given a simplicial complex, we define a pure simplicial complex called the purification. These complexes arise as a generalization of certain independence complexes and the resultant Stanley-Reisner rings have numerous desirable properties, e.g., they are Cohen-Macaulay. By realizing the purification in the context of work of D'alì, et al., we obtain a multi-graded, minimal free resolution of the Alexander dual ideal of the Stanley-Reisner ideal. We augment this in a standard way to obtain a resolution of the quotient ring, which is likewise minimal and multi-graded. …

Symbolic Power Containments In Singular Rings In Positive Characteristic, Eloísa Grifo, Linquan Ma, Karl Schwede

Department of Mathematics: Faculty Publications

The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I(a)⊆Ib. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use …

Estimating The Parameters Of A Simple Linear Regression Model Without Using Differential Calculus, Mamunur Rashid, Jyotirmoy Sarkar

Mathematics Faculty Publications

To estimate the parameters of a simple linear regression model, students who already know calculus can minimize the total squared deviations by setting its first-order partial derivatives to zero and solving simultaneously. For students who do not know calculus, most teachers/textbooks simply state the formulas without justifying them. Students accept the formulas on faith; and for given data, they evaluate the estimates using a calculator or a statistical software. In this paper, we justify the formulas without invoking calculus. We hope the users of statistics will benefit from our proposed justifications.

On Some Problems In Sparse Hybrid Imaging, Non-Standard Finite Difference Methods, And Fokker-Planck Frameworks In Esophageal Cancer, Madhu Gupta

Mathematics Dissertations - Archive

In this thesis, we first discuss nonlinear optimization frameworks for the sparsity- based on nonlinear reconstruction of parameters in hybrid imaging modalities such as current density impedance imaging (CDII) and two-photon photoacoustic computed tomography (2P-PACT). The framework comprises minimizing an objective functional involving a least square fit and some regularization terms that promote sparsity patterns and enhance the edges to facilitate high contrast and resolution. Next, we show the construction and analysis of the second-order nonstandard finite difference methods (NSFD) scheme for theta methods and explicit Runge-Kutta method. Finally, we present an application of the NSFD scheme for Fokker-Planck (FP) …

Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of Θ-Functions In Genus One And Two: A Geometric Perspective, Shantel Spatig

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Anxiety and mathematics come hand in hand for many individuals. This is due, in
part, to the fact that the only experience they have with mathematics is what some
mathematics educators refer to as "schoolmath," which uses a somewhat different
language than real mathematics. The language of schoolmath can cause individu-
als to have confusion and develop misconceptions related to several mathematical
concepts. One such concept is a fraction. In chapter one of this report, one possible
reason for this is discussed and a possible solution is purposed.
In chapter three of this report, genus-two curves admitting an elliptic involution …

The Hfd Property In Orders Of A Number Field, Grant Moles

All Theses

We will examine orders R in a number field K. In particular, we will look at how the generalized class number of R relates to the class number of its integral closure R. We will then apply this to the case when K is a quadratic field to produce a more specific relation. After this, we will focus on orders R which are half-factorial domains (HFDs), in which the irreducible factorization of any element α∈R has fixed length. We will determine two cases in which R is an HFD if and only if its ring of …

Lyubeznik Ideals Minimally Generated By Four Or Fewer Elements, Nathan S. Fontes

All Theses

Free resolutions for an ideal are constructions that tell us useful information about the structure of the ideal. Every ideal has one minimal free resolution which tells us significantly more about the structure of the ideal. In this thesis, we consider a specific type of resolution, the Lyubeznik resolution, for a monomial ideal I, which is constructed using a total order on the minimal generating set G(I). An ideal is called Lyubeznik if some total order on G(I) produces a minimal Lyubeznik resolution for I. We investigate the problem of characterizing whether an ideal I is Lyubeznik …