Mathematics Commons^™

On The Ratio-Type Family Of Copulas, Farid El Ktaibi, Rachid Bentoumi, Mhamed Mesfioui

All Works

Investigating dependence structures across various fields holds paramount importance. Consequently, the creation of new copula families plays a crucial role in developing more flexible stochastic models that address the limitations of traditional and sometimes impractical assumptions. The present article derives some reasonable conditions for validating a copula of the ratio-type form (Formula presented.). It includes numerous examples and discusses the admissible range of parameter (Formula presented.), showcasing the diversity of copulas generated through this framework, such as Archimedean, non-Archimedean, positive dependent, and negative dependent copulas. The exploration extends to the upper bound of a general family of copulas, (Formula presented.), …

New Examples Of Self-Dual Near-Extremal Ternary Codes Of Length 48 Derived From 2-(47,23,11) Designs, Sanja Rukavina, Vladimir Tonchev

Michigan Tech Publications

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

Mathematical Modeling Of An Epidemic In Scalefree Network With Imperfect Vaccination, Heba Hameed

Theses

In light of the recent COVID-19 pandemic, the mathematical epidemiological model has proven to be essential for understanding the disease dynamic and finding the best control tool to help contain the disease and minimize its impact. This thesis investigates the dynamics of an infectious disease spread with latent infection and vaccination. The aim is to study this model’s dynamic in a network that reflects the heterogeneity of the environment where the disease spreads. The vaccination is assumed to be not perfect, which means that vaccinated persons are likely to be infected and that the vaccinated person could lose their immunity, …

Cellular Automata Modeling Approach Of Addiction, Ruba Hameed

Theses

The purpose of this thesis includes the application of a mathematical model, as well as cell automata simulation, to shed light on the behavior of addiction spread in a population. The model describes the key determinants of drug addiction which includes peer influence, availability of substances, support networks, and awareness campaigns. The research aims to gain insights into the underlying mechanisms and patterns of addiction spread to develop effective prevention and intervention strategies.

The study begins by creating a discrete dynamical system that captures the transitions between 4 different states of substance use, including non-users (N), experimental users (E), recreational …

A Brief Introduction To General Topology, Richard P. Millspaugh

Open Educational Resources

The material in this text is intended to be accessible to undergraduates who have had an introduction to elementary set theory and proof techniques. It includes sufficient material from general topology to prove the two main topological results found in a standard first semester calculus course: the Intermediate Value Theorem and the Extreme Value Theorem. This material can be found in Chapters 2 through 6 and makes up the bulk of the text. Rather than approaching these topics through use of the standard euclidean metric, it defines the standard topology on R in terms of the usual order on R. …

Linear Ode Systems Having A Fundamental Matrix Of The Form F(Mt), Kevin L. Shirley, Vicky W. Klima

CODEE Journal

We interweave scaffolded problem statements with exposition and examples to support the reader as they explore specific linear systems of differential equations with variable coefficients, $\vec{x}'(t)=A(t)\vec{x}(t)$ and initial value $\vec{x}(0)$. We begin with a constant $n\times n$ matrix $M$ and a real or complex-valued function $f$, analytic at the eigenvalues of $M$ with $f(0) = 1$, and construct a linear system of differential equations with solutions $x(t)=f(Mt)\vec{x}(0)$, where $t$ is a parameter in some interval including zero. In general, the solutions to the resulting non-autonomous system are more difficult to analyze than solutions to the constant coefficient case. However, some …

Balanced N-Color Compositions, Jesus Omar Sistos Barron, Hua Wang

Mathematical Sciences: Faculty Publications

We introduce the notion of balanced n-color compositions, and consider the related enumeration problems. We also establish bijections with four other combinatorial objects, namely two types of Motzkin lattice paths, pairs of regular compositions with restricted parts, and regular compositions with a centered maximum.

A Novel Consumer-Centric Metric For Evaluating Hearing Device Audio Performance, Vinaya Manchaiah, Steve Taddei, Abram Bailey, De Wet Swanepoel, Hansapani Rodrigo

School of Mathematical & Statistical Sciences Faculty Publications

Background and Aim: The emergence of direct-to-consumer hearing devices has introduced confusion in making appropriate choices, highlighting the need for users to be well-informed for optimal device selection. Currently, no established metric offers insights into the sound performance of these devices. This study aimed to introduce and assess a novel consumer-centric metric (i.e., SoundScore) for hearing device audio performance.

Method: The SoundScore metric was created based on five dimensions of hearing device audio performance (i.e., speech benefit in quiet and moderate, speech benefit in loud, own voice perception, feedback control, streamed music sound quality). Tests were conducted under lab conditions …

A Combinatorial Proof Of A Partition Perimeter Inequality, Hunter Waldron

Michigan Tech Publications

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher’s theorem: for all m ≥ 2 and n ≥ 1, there are at least as many partitions with perimeter n and parts repeating fewer than m times as there are partitions with perimeter n with parts not divisible by m. In this work, we provide a combinatorial proof of their theorem by relating the combinatorics of the partition perimeter to that of compositions. Using this technique, …

Composition-Theoretic Series And False Theta Functions, William Keith, Robert Schneider, Andrew V. Sills

Michigan Tech Publications

Many natural partition-theoretic series can be equally readily interpreted as compo-sition-theoretic series, but this viewpoint seems to have not been much employed in either theory. We consider some of the consequences of this viewpoint. As examples, we give results concerning the reciprocals of Ramanujan’s theta functions and of the false theta functions of L. J. Rogers, and raise an array of questions related to these. Part of this study may be considered a natural dual of the truncated pentagonal number theorem of Andrews and Merca.

On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, Brad Isaacson

Publications and Research

In a previous paper, we expressed three families of character sums by certain generalized Bernoulli functions which in turn were expressed by generalized Bernoulli numbers via a complicated and indirect process. In this paper, we generalize a theorem of Ibukiyama to directly express these generalized Bernoulli functions by generalized Bernoulli numbers. As a result, we can express the three families of character sums by generalized Bernoulli numbers in a more elegant fashion than was done before.

On The T-Degree Of Anderson Thakur Polynomials In The Case Of Fields Of Order P Where P Is A Prime, Joshua Bush

Master's Theses

The Anderson Thakur polynomials are of great interest in the investigation of zeta values. Here, we address the question of their degree in finite fields of the form Fp. We define a graph that is closely connected to Anderson Thakur polynomials and which represents a polynomial we denote as Anxn−1. Collections of terms of this polynomial are equating with the vertices of the graph and are vanishing or nonvanishing. The vanishing structure of the graph is analyzed and we conclude with a simple degree formula.

Some Hypergeometric Identities And Related Leonard Pairs, Taiyo Summers Terada

Dissertations and Theses

The notion of a Leonard pair was introduced by Terwilliger in 2001 to simplify Leonard's theorem, which classifies the orthogonal polynomials in the terminating branch of the Askey-Wilson scheme. In the same year, Kresch and Tamvakis made a conjecture about a certain ₄F₃ hypergeometric series while studying the arithmetic analogues of the standard conjectures for the Grassmanian G(2,n). The ₄F₃ series appearing in their conjecture is closely related to a family of orthogonal polynomials in the Askey-Wilson scheme. Consequently, the theory of Leonard pairs provides a useful framework for understanding their conjecture.

In …

A Computational Investigation Of Wood Selection For Acoustic Guitar, Jonah Osterhus

Senior Honors Theses

The acoustic guitar is a stringed instrument, often made of wood, that transduces vibrational energy of steel strings into coupled vibrations of the wood and acoustic pressure waves in the air. Variations in wood selection and instrument geometry have been shown to affect the timbre of the acoustic guitar. Computational methods were utilized to investigate the impact of material properties on acoustic performance. Sitka spruce was deemed the most suitable wood for guitar soundboards due to its acoustic characteristics, strength, and uniform aesthetic. Mahogany was deemed to be the best wood for the back and sides of the guitar body …

Engaging Students In Partial Differential Equations Through Modeling Fourier's Law Of Conduction, Justin G. Trulen, Kayla Keller, John Sinclair, Lauren Meagher

CODEE Journal

In an effort to make active learning exercises in a partial differential equations course, we present a student activity modeling Fourier's law of conduction under the framework of the heat equation. An overview of the heat equation, including several avenues of study, is provided. Then we give an intuitive way of constructing the heat equation from a couple of fundamental properties of physics including Fourier's law of conduction. We outline an experiment that can be run to collect their own data to model Fourier's law of conduction as well as provide data we collected. We conclude with a student activity …

Computational Tools For Exploring Eigenvector Localization, Robyn Ashley Markee Reid

Dissertations and Theses

We develop computational tools for exploring eigenvector localization for a class of selfadjoint, elliptic eigenvalue problems regardless of the cause for localization. The user inputs a desired region R (not necessarily connected), a tolerance for the amount localization in R, and the desired energy range [a,b]. The tool outputs eigenvectors concentrated within the tolerance inside R and within [a, b]. We develop ample theory that justifies our algorithm, which involves a complex, compact perturbation of the operator L, Ls = L+isχR, for some (small) s > 0. Our central idea can be summarized as follows: if (λ,ψ) is an eigenpair of …

Differential Equations For Modeling Pathways Leading To Diabetes Onset, Viktoria Savatorova, Aleksei Talonov

CODEE Journal

This paper presents a mathematical model that explains potential pathways leading to diabetes onset. By utilizing a system of nonlinear differential equations to describe the dynamics of the glucose regulatory system, the model can serve as a pedagogical tool for teaching and learning differential equations, dynamical systems, mathematical modeling, and introduction to biomathematics. Within this framework, students can analyze equilibrium solutions, investigate stability, assess parameter sensitivity, and explore the potential for bifurcations. Theoretical analysis is complemented by illustrative numerical examples. Instructors have the flexibility to adapt and incorporate suggested activities according to their teaching preferences and objectives.

Exchangeability And A Model Of Biological Evolution, Renee Haddad

Honors Scholar Theses

A sequence of random variables (RVs) is exchangeable if its distribution is invariant under permutations. For example, every sequence of independent and identically distributed (IID) RVs is exchangeable. The main result on exchangeable sequences of random variables is de Finetti's theorem, which identifies exchangeable sequences as conditionally IID. In this thesis, we explore exchangeability, provide an elementary proof of de Finetti's theorem, and present two applications: the classical Polya's urn model and a toy model for biological evolution.

Weakly Pseudo Primary 2-Absorbing Submodules, Omar Hisham Taha, Marrwa Abdulla Salih

Al-Bahir

Let be a commutative ring with identity. In this paper, we introduce the notion of a weakly pseudo primary 2-absorbing sub-module as a generalization of a 2-absorbing sub-module and a pseudo 2-absorbing sub-module. Moreover, we give many basic properties, examples, and characterizations of these notions.

Sustained Ifn Signaling Is Associated With Delayed Development Of Sars-Cov-2-Specific Immunity, Elsa Brunet-Ratnasingham, Sacha Sacha, Haley E. Randolph, Marjorie Labrecque, Justin Bélair, Raphaël Lima-Barbosa, Amélie Pagliuzza, Lorie Marchitto, Michael Hultström, Julia Niessl, Rose Cloutier, Alina M. Flores, Nathalie Brassard, Mehdi Benlarbi, Jérémie Prévost, Shilei Ding, Sai Priya Anand, Gérémy Sannier, Amanda Marks, Dick Wågsäter, Eric Bareke, Hugo Zeberg, Miklos Lipcsey, Robert Frithiof, Anders Larsson, Sirui Zhou, Tomoko Nakanishi, David Morrison, Dani Vezina, Catherine Bourassa, Gabrielle Gendron-Lepage, Halima Medjahed, Floriane Point, Jonathan Richard, Catherine Larochelle, Alexandre Prat, Janet L. Cunningham, Nathalie Arbour, Madeleine Durand, J. Brent Richards, Kevin Moon, Nicolas Chomont, Andrés Finzi, Martine Tétreault, Luis Barreiro, Guy Wolf, Daniel E. Kaufmann

Mathematics and Statistics Faculty Presentations

Plasma RNAemia, delayed antibody responses and inflammation predict COVID-19 outcomes, but the mechanisms underlying these immunovirological patterns are poorly understood. We profile 782 longitudinal plasma samples from 318 hospitalized patients with COVID-19. Integrated analysis using k-means reveals four patient clusters in a discovery cohort: mechanically ventilated critically-ill cases are subdivided into good prognosis and high-fatality clusters (reproduced in a validation cohort), while non-critical survivors segregate into high and low early antibody responders. Only the high-fatality cluster is enriched for transcriptomic signatures associated with COVID-19 severity, and each cluster has distinct RBD-specific antibody elicitation kinetics. Both critical and non-critical clusters with …

Struggling To Be Positive Problem Solvers – A Study On The Relationship Of Elementary Mathematical Self-Concept, Gender, Learner Identification, And Productive Struggle Trajectories On Constructed Response Questions When Being Taught With A Standardized Curriculum, Kali D. Rogers

LSU Doctoral Dissertations

Mathematical word problems are some of the most difficult problems for students to solve due to their complexity and relationship to real-world situations (Şahin et al., 2014; Verschaffel et al., 2020). With math being known for right-or-wrong answers, these open-ended questions cause much confusion and frustration, specifically for students of various subgroups. In the last several decades, educational teaching expectations have shifted from engaging, self-created learning to robotic and redundant curriculum expectations that induce a ‘teaching to the test’ environment (Shafiyeva, 2021). Mathematics is the powerhouse of where students can experience and succeed with productive struggle learning that teaches the …

Improved Rational Approximation Of Near-To-Far Propagation Kernels For The Wave Equation, Sampson Owusu

Mathematics & Statistics ETDs

The 3-space, 1-time dimensional scalar wave equation, or 3+1 wave equation, describes the propagation of scalar or acoustic waves. The unforced homogeneous equation admits a class of outgoing solutions relative to a chosen fixed center, so called “multipole” solutions. This thesis examines near-to-far signal propagation in the context of these multipole solutions. Given a time-series (history of values) for a multipole solution recorded at a radius r1, near- to-far signal propagation recovers the corresponding time-series at larger radius r2 ≫ r1. This propagation takes into account both the appropriate time delay r2 − r1 and corrections to the wave shape. …

Nidus Idearum. Scilogs, Xiii: Structure / Neutrostructure / Antistructure, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this thirteenth book of scilogs – one may find topics on Neutrosophy, Plithogeny, Physics, Mathematics, Philosophy – email messages to research colleagues, or replies, notes, comments, remarks about authors, articles, or books, spontaneous ideas, and so on. It presents new types of soft sets and new types of topologies.

Exchanging ideas with Mohammad Abobala, Ishfaq Ahmad, Ibrahim M. Almanjahie, Fatimah Alshahrani, Nizar Altounji, Muhammad Aslam, Said Broumi, Victor Christianto, R. Diksh, Feng Liu, Frank Julian Gelli, Erick Gonzalez Caballero, Riad Hamido, Yaser Al-Hasan, Ahmed Hatip, Yasin Karmouta, Nivetha Martin, Preda Mihăilescu, V. Lakshmana Gomathi Nayagam, Ze Carlos Tiago de …

Landscape Influences On Thermal Sensitivity And Predicted Spatial Variability Among Brook Trout Streams In The Southeastern Usa, George P. Valentine, Xinyi Lu, C. Andrew Dolloff, Craig N. Roghair, Jacob M. Rash, Mevin B. Hooten, Yoichiro Kanno

Mathematics and Statistics Faculty Publications

Warming water temperatures as a result of climate change pose a major threat to coldwater organisms. However, the rate of warming is not spatially uniform due to surface-ground-water interactions and stream and watershed characteristics. Coldwater habitats that are most resistant to warming serve as thermal refugia and identifying their locations is critical to regional aquatic conservation planning. We quantified the thermal sensitivity of 203 streams providing current and potential habitat for brook trout (Salvelinus fontinalis) across nearly 1000 linear km of their native range in the southern and central Appalachian Mountains region, USA, and characterized their spatial variability …

Numerical Invariants Of Cohen-Macaulay Local And Graded Rings, Richard Francis Bartels

Dissertations - ALL

We characterize different classes of Cohen-Macaulay local rings (R,m, k) with positive Krull di?mension in terms of MCM approximations of finitely-generated R-modules. Assume R has a canonical module. For each finitely-generated R-module M, Auslander’s δ?invariant δR(M) equals the rank of a maximal free direct summand of the minimal MCM approxi?mation XM of M. We have δR(R/m) = 1 if and only if R is a regular local ring. Auslander defined the index of R, denoted index(R), as the infimum of positive integers n such that δR(R/mn ) = 1. When R is Gorenstein, we have index(R) ≤ gℓℓ(R) < ∞, where gℓℓ(R) denotes the generalized Loewy length of R, the smallest positive integer n such that mn ⊆ xR for some system of parameters x for R. We call such a system of parameters a witness to the generalized Loewy length of R. In Chapter 3, we generalize a theorem of Ding, who proved that if R is Gorenstein with infinite residue field k and Cohen-Macaulay associated graded ring grm(R), then gℓℓ(R) = index(R). We prove that if R is a one-dimensional Cohen-Macaulay local ring with finite index and nonzerodivi?sor x of order t with grm(R)-regular initial form x ∗ , then gℓℓ(R) ≤ index(R) +t −1. We use this estimate to derive a formula for the generalized Loewy length of a one-dimensional hypersurface R = kJx, yK/(f). If z is a witness to gℓℓ(R) such that z ∗ is grm(R)-regular, then gℓℓ(R) = ordR(z)+e(R)−1, where e(R) denotes the Hilbert-Samuel multiplicity of R. We compute the generalized Loewy lengths of several families {Rn} ∞ n=1 of one-dimensional hypersurfaces over finite and infinite fields such that gℓℓ(Rn) = index(Rn) for all n ≥ 1 or gℓℓ(Rn) = index(Rn) + 1 for all n ≥ 1. Lastly, we study a graded version of the generalized Loewy length of a Noetherian local ring for Noetherian k-algebras (R,m, k), where k is an arbitrary field and m is the irrelevant ideal of R. This invariant is called the generalized graded length of R and denoted ggl(R). After determining bounds for ggl(R) in terms of gℓℓ(R) and the degrees of generators for R, we compute the generalized graded length of numerical semigroup rings. We also characterize witnesses to the generalized graded length of numerical semigroup rings for semigroups with two generators. In Chapter 4, we study criteria for when an MCM module over a Gorenstein complete local ring R is stably isomorphic to an MCM approximation of a finitely-generated R-module of some fixed positive codimension r. If this condition holds for an MCM R-module M, we say with Kato that M satisfies the SCr-condition. If this condition holds for every MCM R-module, we say that R satisfies the SCr-condition. Only the SC1- and SC2- conditions have been characterized for Gorenstein complete local rings R. Kato proved that R satisfies the SC1-condition if and only if R is a domain, and R satisfies the SC2-condition if and only if R is a UFD. For rings of dimension d ≥ 3 and 3 ≤ r ≤ d, we prove an inductive criterion for when an MCM R-module satisfies the SCr-condition when its first syzygy module Ω1 R (M) satisfies the SCr−1-condition. We use this criterion to prove the equivalence of the SCd- and SCd−1-conditions for Gorenstein complete local rings of dimension d ≥ 3 that remain UFDs when factoring out certain regular sequences of length d −2.

The Functoriality Of Odd Khovanov Homology Up To Sign And Applications, Jacob Migdail-Smith

Dissertations - ALL

Odd Khovanov Homology is a homological invariant of knots and links that permits a Bar-Natan category presentation. In this dissertation, we extend the odd Khovanov bracket to link cobordisms and prove that our construction is functorial up to sign. We then build an odd Khovanov theory for dotted link cobordisms. Out of the dotted theory, a module structure on the odd Khovanov homology of a diagram over the exterior algebra of the diagram’s coloring group arises. We finish by using our functoriality result to prove that if n is even or if the knot has even framing, then the odd …

On The Limitations And Restrictions Of The Hardy-Littlewood Circle Method, Daniel W. Havens

Mathematics & Statistics ETDs

We discuss herein the history, layout, and philosophy of the Hardy-Littlewood Circle method, as well as the more modern renditions thereof. The limitations and scope of each method presented is discussed in detail, providing examples of cases where the failure of the circle method is of relevance. We include a summary of famous problems which have been resolved using each methodology, as well as what limitations each methodology showcases.

Exploring The Mandelbrot Set, James Shirley

Electronic Theses and Dissertations

The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-
tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases
its usefulness in fields across the many realms of math—ranging from physics to nu-
merical methods and even biology. While typically defined in terms of its bounded
sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-
eterization of connectivity itself, specifically that of complex-valued rational maps
of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges
the worlds of intuition and theory with a series of …

Robust Prediction Of Charpy Toughness Of Additively Manufactured Kovar Using Deep Convolutional Neural Networks, Nathan R. Bianco

Mathematics & Statistics ETDs

Understanding the reason for mechanical failures of manufactured parts in their operating environments is critical to prevention of future failures. However, in-situ post-mortem evaluation of physical properties, such as fracture toughness, is time consuming and alters the condition of the material, leading to potentially misleading findings. In this study, additively manufactured test coupons were produced over a wide range of process conditions to test the impact toughness of a material. The Charpy V-Notch toughness was measured on over 200 samples alongside corresponding optical images of both sides of the fracture surface. Convolutional neural network models were trained to correlate fracture …

A Novel Method For Multiple Phenotype Association Studies Based On Genotype And Phenotype Network, Xuewei Cao, Shuanglin Zhang, Qiuying Sha

Michigan Tech Publications

Joint analysis of multiple correlated phenotypes for genome-wide association studies (GWAS) can identify and interpret pleiotropic loci which are essential to understand pleiotropy in diseases and complex traits. Meanwhile, constructing a network based on associations between phenotypes and genotypes provides a new insight to analyze multiple phenotypes, which can explore whether phenotypes and genotypes might be related to each other at a higher level of cellular and organismal organization. In this paper, we first develop a bipartite signed network by linking phenotypes and genotypes into a Genotype and Phenotype Network (GPN). The GPN can be constructed by a mixture of …