Physical Sciences and Mathematics Commons^™

Oscillations In Neuronal Activity: A Neuron-Centered Spatiotemporal Model Of The Unfolded Protein Response In Prion Diseases, Elliot M. Miller, Tat Chung D. Chan, Carlos Montes-Matamoros, Omar Sharif, Laurent Pujo-Menjouet, Michael R. Lindstrom

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P …

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, Part 2

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.

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 and Statistical Sciences Faculty Publications and Presentations

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 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 …

Canonical Extensions Of Quantale Enriched Categories, Alexander Kurz

MPP Research Seminar

No abstract provided.

A Mceliece Cryptosystem, Using Permutation Error-Correcting Codes, Fiona Smith

CSB and SJU Distinguished Thesis

Using existing methods of cryptography, we can encrypt messages through the internet. However, these methods are vulnerable to attacks done by a quantum computer, which are a rising threat to security. In this thesis I discuss a possible method of encryption, secure against quantum attacks, using permutation groups and coding theory.

Representations Of Gender In Math-Related Films, Jacob Gathje

CSB and SJU Distinguished Thesis

This project analyzes how four popular math-related films - Hidden Figures, Mean Girls, Good Will Hunting, and A Beautiful Mind - either follow, resist, or reconfigure gender stereotypes in mathematics. It includes close readings of specific scenes in each of the films, along with broader analysis of the effects of how women and men are represented differently. It concludes forward-looking focus, providing suggestions for how future math-related movies can depict a more realistic and inclusive version of the field of mathematics. Ideally, this will help improve one part of the larger issue of gender disparities in math.

Weighted Ehrhart Theory: Extending Stanley's Nonnegativity Theorem, Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …

Local Existence Of Solutions To A Nonlinear Autonomous Pde Model For Population Dynamics With Nonlocal Transport And Competition, Michael R. Lindstrom

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Highlights

• Partial differential equation models are ubiquitous in applied sciences.

• A partial differential equation based in ecology is studied for solution existence.

• Energy methods and convergence analysis lead to local classical solutions.

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence …

The Future Of Brain Tumor Diagnosis: Cnn And Transfer Learning Innovations, Shengyuan Wang

Mathematics, Statistics, and Computer Science Honors Projects

For the purpose of improving patient survival rates and facilitating efficient treatment planning, brain tumors need to be identified early and accurately classified. This research investigates the application of transfer learning and Convolutional Neural Networks (CNN) to create an automated, high-precision brain tumor segmentation and classification framework. Utilizing large-scale datasets, which comprise MRI images from open-accessible archives, the model exhibits the effectiveness of the method in various kinds of tumors and imaging scenarios. Our approach utilizes transfer learning techniques along with CNN architectures strengths to tackle the intrinsic difficulties of brain tumor diagnosis, namely significant tumor appearance variability and difficult …

Bernstein Polynomials Method For Solving Multi-Order Fractional Neutral Pantograph Equations With Error And Stability Analysis, M. H. T. Alshbool

All Works

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series …

A Central Limit Theorem For The Number Of Excursion Set Components Of Gaussian Fields, Dmitry Beliaev, Michael Mcauley, Stephen Muirhead

Articles

For a smooth stationary Gaussian field f on R^d and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for ‘additive’ geometric functionals of the excursion/level sets (e.g. the volume or …

Asteroidal Sets And Dominating Targets In Graphs, Oleksiy Al-Saadi

Department of Computer Science and Engineering: Dissertations, Theses, and Student Research

The focus of this PhD thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the …

Analysis Of Nonsmooth Neural Mass Models, Cadi Howell

Honors College

Neural activity in the brain involves a series of action potentials that represent “all or nothing” impulses. This implies the action potential will only “fire” if the mem- brane potential is at or above a specific threshold. The Wilson-Cowan neural mass model [6, 28] is a popular mathematical model in neuroscience that groups excita- tory and inhibitory neural populations and models their communication. Within the model, the on/off behavior of the firing rate is typically modeled by a smooth sigmoid curve. However, a piecewise-linear (PWL) firing rate function has been considered in the Wilson-Cowan model in the literature (e.g., see …

An Investigation Into The Causes Of Home Field Advantage In Professional Soccer, Paige E. Tomer

Mathematics, Statistics, and Computer Science Honors Projects

Home-field advantage is the sporting phenomenon in which the home team outperforms the away team. Despite its widespread occurrence across sports, the underlying reasons for home-field advantage remain uncertain. In this paper, we employ a range of statistical methods to explore the causal relationships of potential determinants of home-field advantage. We measure home-field advantage using match outcomes and differential metrics (e.g., differences in yellow cards received). In an attempt to narrow the research disparity between men’s and women’s sports, we utilize data from the National Women’s Soccer League (NWSL) and the English Premier League (EPL) to investigate potential causes of …

Representation Theory And Burnside's Theorem, Nathan Fronk

Senior Seminars and Capstones

In this paper we give a brief introduction to the representation theory of finite groups, and by extension character theory. These tools are extensions of group theory into linear algebra, that can then be applied back to group theory to prove propositions that are based entirely in group theory. We discuss the importance of simple groups and the Jordan-Hölder theorem in order to prepare for the statement of Burnside’s pq theorem. Lastly, we provide a proof of Burnside’s theorem that utilizes the character theory we covered earlier in the paper.

Caterpillar, Lobster, X Graphs, Gerald Melin, Landon Seward, Will Mahowald, Xavier Jones

Celebrating Scholarship and Creativity Day (2018-)

We studied a combinatorial game played between two players ("Alpha", who goes first, and "Beta", who goes second). The idea is that there are a lot of lightbulbs in a large warehouse, and they take turns turning a light bulb on. When a light bulb is turned on, it illuminates the area directly by it as well as the areas immediately surrounding it. The player who is the one to make all of the warehouse illuminated is the winner. This can be modeled on a graph. The two players take turns (1) selecting a vertex that has not yet been …

Statistics For Iwasawa Invariants Of Elliptic Curves, Ii, Debanjana Kundu, Anwesh Ray

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.

Generalized Q-Fock Spaces And Structural Identities, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using 𝑞-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their 𝑞-calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

On Axially Rational Regular Functions And Schur Analysis In The Clifford-Appell Setting, Daniel Alpay, Fabrizio Colombo, Antonino De Martino, Kamal Diki, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in . The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a …

A Note On Umbilic Points At Infinity, Brendan Guilfoyle

Publications

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric …

Study Of Hybrid Nanofluid Flow In A Stationary Cone-Disk System With Temperature-Dependent Fluid Properties, A. S. John, Mahanthesh Basavarajappa, G. Lorenzini

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation …

How Generative Ai Models Such As Chatgpt Can Be (Mis)Used In Spc Practice, Education, And Research? An Exploratory Study, Fadel M. Megahed, Ying-Ju (Tessa) Chen, Joshua A. Ferris, Sven Knoth, L. Allison Jones-Farmer

Mathematics Faculty Publications

Generative Artificial Intelligence (AI) models such as OpenAI's ChatGPT have the potential to revolutionize Statistical Process Control (SPC) practice, learning, and research. However, these tools are in the early stages of development and can be easily misused or misunderstood. In this paper, we give an overview of the development of Generative AI. Specifically, we explore ChatGPT's ability to provide code, explain basic concepts, and create knowledge related to SPC practice, learning, and research. By investigating responses to structured prompts, we highlight the benefits and limitations of the results. Our study indicates that the current version of ChatGPT performs well for …

Normalized Ground State Of A Mixed Dispersion Nonlinear Schrodinger Equation With Combined Power-Type Nonlinearities, Zhouji Ma, Xiaojun Chang, Zhaosheng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.

Variable-Order Fractional Laplacian And Its Accurate And Efficient Computations With Meshfree Methods, Yixuan Wu, Yanzhi Zhang

Mathematics and Statistics Faculty Research & Creative Works

The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian (-Δ) α (x) / 2 with 0 < α (x) ≤ 2, which will also be referred as the variable-order fractional Laplacian if α(x) is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian (-Δ) α (x) / 2. Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension d ≥ 1. Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen–Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives.

Mediating Effect Of Bmi On The Association Of Economic Status And Coexistence Of Hypertension And Diabetes In Bangladesh: A Counterfactual Framework-Based Weighting Approach, Foyez , Md. Jamal Hossain Ahmmed, Md. Jamal Hossain, Md Tareq Ferdous Khan, Muhammad Mahabub Rahaman Manik, Saimon Shahriar, Dulal Chandra Nandi, Md Parvej Hussain

Mathematics and Statistics Faculty Publications

Background and Aims

Non-communicable diseases such as hypertension and diabetes are matters of huge concern worldwide, with an increasing trend in prevalence over the previous decade. First of all, this study aimed to evaluate the association between economic status (ES) and body mass index (BMI), ES and comorbidity of hypertension and diabetes, and BMI and comorbidity independently. Second, it explored the mediating role of BMI in the association between ES and comorbidity of hypertension and diabetes. Finally, it investigated whether the mediating effect differs with the place of residence, gender, and education levels.

Methods

A total of 11,291 complete cases …

Visualizing The Standard Deviation Via Revolution Using R/Rstudio, Hieu Nguyen '25, Trung Pham '25, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

The standard deviation is a commonly used statistical measure to quantify the level of variation present in a set of numbers or in a random variable. Sarkar and Rashid (2016) introduced an interpretation of the population standard deviation as the radius of a cylinder with a volume equivalent to that of the solid of revolution when the 2-D graph of the empirical cumulative distribution function is revolved about the vertical line through the mean. This article demonstrates step-by-step how to use the RevSD package in R/RStudio to visualize the standard deviation of data using this innovative technique. The RevSD package …

How Difficult Is It To Comprehend A Program That Has Significant Repetitions: Fuzzy-Related Explanations Of Empirical Results, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In teaching computing and in gauging the programmers' productivity, it is important to property estimate how much time it will take to comprehend a program. There are techniques for estimating this time, but these techniques do not take into account that some program segments are similar, and this similarity decreases the time needed to comprehend the second segment. Recently, experiments were performed to describe this decrease. These experiments found an empirical formula for the corresponding decrease. In this paper, we use fuzzy-related ideas to provide commonsense-based theoretical explanation for this empirical formula.

Mcfadden's Discrete Choice And Softmax Under Interval (And Other) Uncertainty: Revisited, Bartlomiej Jacek Kubica, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Studies of how people actually make decisions have led to an empirical formula that predicts the probability of different decisions based on the utilities of different alternatives. This formula is known as McFadden's formula, after a Nobel prize winning economist who discovered it. A similar formula -- known as softmax -- describes the probability that the classification predicted by a deep neural network is correct, based on the neural network's degrees of confidence in the object belonging to each class. In practice, we usually do not know the exact values of the utilities -- or of the degrees of confidence. …