Mathematics Commons^™

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.

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 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.

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 …

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 …

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 …

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. …

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 …

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.