Mathematics Commons^™

Calculation And Statistical Analysis Of Wins Above Replacement, Joshua Taylor

Departmental Honors & Graduate Capstone Projects

The Wins Above Replacement (WAR) statistic in Major League Baseball is a prominent metric used to estimate player value by quantifying all aspects of play in terms of wins added to a baseball team. We will use R to calculate WAR for all players from 1871 to 2012 and use data from those years to construct multivariate predictive models to attempt to estimate WAR for players from 2013 to 2024. We find strong correlations between predicted and actual WAR values for most models, with the exception of the polynomial predictive model for non-qualified pitchers.

Analysis Of Positive Solutions For Classes Of Nonlinear Boundary Value Problems, Bandar Alreshidi

Theses and Dissertations

In this dissertation, we study the existence and uniqueness of positive solutions for classes of nonlinear boundary value problems. In the first study, we the ��-superlinear case, we prove the existence of a large positive solution when a parameter is small and if, in addition, the reaction term satisfies a concavity-like condition at the origin, the existence of two positive solutions for a certain range of the parameter. In the ��-sublinear case, we establish the existence of a large positive solution when a parameter is large. We also investigate the number of positive solutions for the general ��-Laplacian with nonlinear …

A Supervised Machine Learning Statistical Design Of Experiment Approach To Modeling The Barriers To Effective Snakebite Treatment In Ghana, Eric Nyarko, Edmund F. Agyemang, Ebenezer Kwesi Ameho, Louis Agyekum, José María Gutiérrez, Eduardo Alberto Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

Background

Snakebite envenoming is a serious condition that affects 2.5 million people and causes 81,000–138,000 deaths every year, particularly in tropical and subtropical regions. The World Health Organization has set a goal to halve the deaths and disabilities related to snakebite envenoming by 2030. However, significant challenges in achieving this goal include a lack of robust research evidence related to snakebite incidence and treatment, particularly in sub-Saharan Africa. This study aimed to combine established methodologies with the latest tools in Artificial Intelligence to assess the barriers to effective snakebite treatment in Ghana.

Method

We used a MaxDiff statistical experiment design …

Analysis Of The Snake Cube Puzzle And Adjacency Criteria, Trey Matus

Honors Theses

The snake cube is a puzzle that consists of straight and turn pieces attached by a string that folds into a n x n x n cube. Finding solutions for large cubes is difficult, so finding necessary conditions for solutions is crucial. Using computational algorithms and mathematical proofs, I find improved bounds for the maximum number of straight pieces in a given puzzle size. In particular, I introduce and apply the adjacency criterion to identify groups of puzzles that are unsolvable. Additionally, I find a connection between the number of puzzles that adhere to the adjacency criterion and generalized Fibonacci …

The Uniform Triadic Transformation And Claude Debussy’S Music, Davielle Smith

Honors Theses

Beginning with Claude Debussy’s Berceuse Héroïque, I analyzed each triad using chord symbols along with their Roman numeral representation (where applicable).1 Next, I used integer notation and modular arithmetic, where each pitch name is given a number such that C is 0 and B is 11, creating a set, the integers (mod 12).

This leads to the development of pitch-class sets where the “mode” of a triad or arbitrary number of pitches is expressed in terms of interval spacings. In many cases, such as that of 12-tone or other serialized music, pitch class sets explain many of the composer’s artistic …

Conditional Cooperation With Longer Memory, Nikoleta E. Glynatsi, Ethan Akin, Martin A. Nowak, Christian Hilbe

Mathematics and Statistics Faculty Research & Creative Works

Direct reciprocity is a wide-spread mechanism for the evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well-known approach is given by reactive strategies, which respond to the coplayer's previous move. Here, we extend reactive strategies to longer memories. A reactive-n strategy takes into account the sequence of the last n moves of the coplayer. A reactive-n counting strategy responds to how often the coplayer cooperated during the last n rounds. We derive an algorithm to identify the partner strategies within these strategy sets. Partner strategies are those that ensure mutual cooperation without exploitation. …

Bifurcation And Multiplicity Results For Elliptic Problems With Subcritical Nonlinearity On The Boundary, S. Bandyopadhyay, M. Chhetri, B. B. Delgado, Nsoki Mavinga, R. Pardo

Mathematics & Statistics Faculty Works

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory.

Vieta’S Formula, The Fabius Function And The Partition Function, Ahmed Sebbar

Mathematics, Physics, and Computer Science Faculty Articles and Research

We use an idea of Pólya and Szegö to give a common basis to Vieta’s formula, Fabius function and the partition function. Moreover our construction leads also to a function considered by Hallström, Bowen and Macintyre, which has, as a particular value, the Kepler-Bouwkamp constant, and to a function considered by Zondadari, that vanishes only at prime numbers.

Mathematics In Amusement Parks, Kacey Laumann

Honors Projects

This project focuses specifically on the Walt Disney World Park, Magic Kingdom. I started by collecting data through the MyDisneyExperience app. By recording the data, I was then able to create polynomial functions to the fifth the degree. Each attraction received a function which allowed me to predict the wait times for that attraction. Then, by graphing the functions and analyzing the graph using calculus the “best time” and “worst time” to go to the attraction were found. After the analysis the information is used to build a unique schedule for a guest. Then the guest receives this schedule after …

Applications Of Neural Networks In Parkinson’S Disease Diagnosis, Saladin Minhaaj

Theses

Parkinson's disease (PD) is a complex and debilitating neurodegenerative disorder that affects millions of people worldwide. Early and accurate diagnosis is crucial for effective treatment and management of PD. This thesis explores the application of neural networks in PD diagnosis, leveraging their ability to learn patterns from large datasets and make accurate predictions.

Thesis provides an overview of PD, including its symptoms, diagnosis, and current challenges in diagnosis. We then delve into the fundamentals of neural networks, including supervised learning, mathematical interpretations, and parametric models. This research focuses on the development of neural network models that can accurately diagnose PD …

Why Linear Faults Have Fewer Earthquakes: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Earthquakes usually occur in the vicinity of fault lines. Until recently, geophysical analysis implied that the fault shape should not strongly affect the frequency of its earthquakes. However, recent statistical analysis has shown that faults whose shape is close to linear experience much fewer earthquakes than faults of more complex shape. Based on this empirical fact, researchers have adjusted the corresponding geophysical models, so the updated models do explain this newly discovered phenomenon. The experience of geophysics shows that the updated model will probably need to be updated again when new data appears. It is therefore desirable to come up …

Fair Economic Division: How To Modify Shapley Value To Take Into Account That Different People Have Different Productivity, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Purpose: When several participants, working together, gained some amount of money, what is the fair way to distribute this amount between them? This is the problem that the future Nobelist Lloyd Shapley was working on when he proposed what is now called the Shapley value -- a division uniquely determined by natural fairness assumptions. However, this solutions is not universal: it assumes that all participants are equal -- in particular, that they have equal productivity. In practice, people have different productivity levels, and these productivity levels can differ a lot: e.g., some software engineers are several times more productive than …

How Shapley Value And Its Generalizations Can Help In The Analysis Of Complex Engineering Systems And What Next, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For a complex engineering system -- such as a city's street network -- it is important to predict how its functionality is decreased when some of these components break down, and, if repairs are needed and repairs budget is limited, which subset of the set of components should be repaired first to maximize the resulting functionality. For systems with a large number of components, the number of possible subsets is astronomical, we cannot try to simulate all these subsets. So, the natural idea is to approximate the actual dependence of functionality on the subset by a simple expression -- linear …

What Is Optimal Granularity When Estimating Reliability Of A Complex Engineering Systems, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For complex engineering systems, the usual way to estimate their reliability is to run simulations. If the resulting estimate does not satisfy the desired reliability level, we must replace some components with more reliable and again run simulations. This can take several iterations, so the required computation time often becomes unrealistically long. It is known that it is possible to speed up computations if components belong to a few types, and components of each type are identical. So, a natural idea to deal with the general case is to use the general granularity idea, i.e., to group components with similar …

Multiple And Nonexistence Of Positive Solutions For A Class Of Fractional Differential Equations With P-Laplacian Operator, Haoran Zhang, Zhaocai Hao, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

Research about multiple positive solutions for fractional differential equations is very important. Based on some outstanding results reported in this field, this paper continues the focus on this topic. By using the properties of the Green function and generalized Avery–Henderson fixed point theorem, we derive three positive solutions of a class of fractional differential equations with a p-Laplacian operator. We also study the nonexistence of positive solutions to the eigenvalue problem of the equation. Three examples are given to illustrate our main result.

(R2100) Optimality Conditions Of A Topsis Optimization Model And Its Application On Interval-Valued Data, Sudipta Roy, Sandip Chatterjee

Applications and Applied Mathematics: An International Journal (AAM)

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is widely used in the field of multi-criteria decision analysis. Despite its popularity and widespread application, little attention has been given to the mathematical foundation that underlies the TOPSIS algorithm. The existing literature on this subject is far from comprehensive, leaving many aspects of the algorithm unexplored. This paper aims to address this gap in the literature by delving into the optimization problem associated with TOPSIS. Unlike traditional interval analysis theory, which only covers a limited scope, our approach extends to a broader range of scenarios and offers …

(R2096) Pbib-Designs Associated With Products Of Graphs, Medha Itagi Huilgol, Vidya M.D

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we introduce a design, called “geodetic-design" arising from geodetic sets in a graph. A geodetic-design, over a regular graph is an ordered pair D = (V, B), where V = V (G) and B, the set of all geodetic sets, called blocks, containing vertices belonging to geodetic sets, such that every pair of non-adjacent vertices appears in exactly µ blocks. We first find governing results of a geodetic-design, if it exists, and then get such PBIB-designs for different products of graphs. It is common to have geodetic sets of graphs to be independent sets, hence we extend …

(R2109) Characterizations Of Tzitzeica Curves Using Conformable Frenet Frame, Aykut Has, Beyhan Yılmaz, Kebire Hilal Ayvacı

Applications and Applied Mathematics: An International Journal (AAM)

The aim of this study, the conditions for a conformable curve and its spherical indicator curves to be a Tzitzeica curve, will be examined. Thus, by understanding the behavior of the Tzitzeica curve, researchers can gain insight into complex systems and make more accurate predictions about their behavior.

Is Energy Local? Counterintuitive Non-Locality Of Energy In General Relativity Can Be Naturally Explained On The Newtonian Level, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describe how much work this field can perform. However, there is an exception -- gravitational field in General Relativity. The known formalism to compute its energy density leads to 0 -- and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the …

Logarithmic Number System Is Optimal For Ai Computations: Theoretical Explanation Of Empirical Success, Olga Kosheleva, Vladik Kreinovich, Christoph Lauter, Kristalys Ruiz-Rohena

Departmental Technical Reports (CS)

Everyone knows the success story of machine-learning AI. However, the current AI tools are not perfect. We know how to make them better: every time we increase the amount of computations by the order of magnitude, we get a drastic improvement in the performance of the resulting machine learning tools. Training modern AI system requires a tremendous amount of computations -- that already take a lot of time. So, to increase the number of computations, we need to make each computation step faster. One way to do that is to use low-precision arithmetic operations, e.g., with 1 byte per real …

Algotric: Symmetric And Asymmetric Encryption Algorithms For Cryptography – A Comparative Analysis In Ai Era, Naresh Kshetri, Mir Mehedi Rahman, Md Masud Rana, Omar Faruq Osama, James Hutson

Faculty Scholarship

The increasing integration of artificial intelligence (AI) within cybersecurity has necessitated stronger encryption methods to ensure data security. This paper presents a comparative analysis of symmetric (SE) and asymmetric encryption (AE) algorithms, focusing on their role in securing sensitive information in AI-driven environments. Through an in-depth study of various encryption algorithms such as AES, RSA, and others, this research evaluates the efficiency, complexity, and security of these algorithms within modern cybersecurity frameworks. Utilizing both qualitative and quantitative analysis, this research explores the historical evolution of encryption algorithms and their growing relevance in AI applications. The comparison of SE and AE …

Machine Learning Methods For Quantification Of Glacier Variations Through Satellite Imagery, Robert D. Breininger

Theses and Dissertations

Glaciers around the world have experienced a trend of recession within the past century. Quantification of glacier variations using satellite imagery is of great interest due to the importance of glaciers as freshwater resources and as indicators of climate change. The potential methods to quantify glacier variations with increasing complexity include detecting the terminus location, quantifying the glacier surface area, and measuring glacier volume. Although there are methods in literature designed purposefully for glacier area segmentation that have achieved acceptable results, they are often localized to the region where their training data were acquired and further rely on training sets …

Latent Characterization Of The Complete Batse Gamma-Ray Bursts Catalogue Using Gaussian Mixture Of Factor Analysers And Model-Estimated Overlap-Based Syncytial Clustering, Fan Dai, Ranjan Maitra

Michigan Tech Publications

Characterizing and distinguishing gamma-ray bursts (GRBs) has interested astronomers for many decades. While some authors have found two or three groups of GRBs by analysing only a few parameters, recent work identified five ellipsoidally shaped groups upon considering nine parameters T50, T90, F1, F2, F3, F4, P64, P256, P1024. Yet others suggest subclasses within the two or three groups found earlier. Using a mixture model of Gaussian factor analysers, we analysed 1150 GRBs, that had nine parameters observed, from the current Burst and Transient Source Experiment (BATSE) catalogue, and again established five ellipsoidal-shaped groups to describe the GRBs. These five …

An Examination Of Hilbert Spaces, Fnu Sumaiya

Theses and Dissertations

This independent study focuses on the exploration of Hilbert spaces, a cornerstone of modern analysis and a fundamental construct in mathematics. It starts by introducing vector space, metric and metric spaces and then Hilbert spaces and its properties.

This study includes good amount of mathematical proofs, theorems and examples. Through extensive research and analysis, this independent study aims to offer a comprehensive examination of Hilbert spaces, detailing their properties and their pivotal role in modern analysis.

An Analysis Of The Properties Of Polar Codes, Luke Szramowski

All Theses

Polar Codes have risen to the forefront of practical coding theory, due to their incredible efficiency and ease of construction. Originally introduced by Arikan in his 2009 paper, they are the first code defined with an explicit construction that achieved channel capacity. Moreover, polar codes possess some physically practical properties that make their implementation alluring. In the same paper as mentioned above, Arikan elaborated on his construction and noted that the construction given was one specific instance of a polar code and that there is a family of polar codes that can be produced by the same method. Since this …

Analysis Of Impulsive Differential Equation Models Of Cell Populations Undergoing Radiation Therapy, Abigail D'Ovidio Long

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Radiation therapy is a mode of treatment which is implemented for approximately 50% of cancer patients. Treatment needs to be able to kill cancer cells, but also do minimal damage to surrounding healthy tissue. We propose two main impulsive differential equation models of radiation therapy to capture the periodic nature of the treatment. These models build off of previous studies using clinical data to ensure biological relevance. The first model incorporates only cancer cell populations, and we provide parameter relationships which theoretically ensure treatment outcomes of cancer eradication, cancer approaching a carrying capacity, and cancer approaching a periodic solution. We …

Nestings Of Bibds With Block Size Four, Marco Buratti, Donald L. Kreher, Douglas R. Stinson

Michigan Tech Publications

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a (Formula presented.) -BIBD in such a way that we end up with a partial (Formula presented.) -BIBD. In the case where the partial (Formula presented.) -BIBD is in fact a (Formula presented.) -BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if (Formula presented.). Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., (Formula presented.) -BIBDs) when (Formula presented.), as well as …

Surgical Suturing Skill Assessment Using Estimated Hand Roll Angle From A Deep-Learning Computer Vision Algorithm, Jianxin Gao, Amir Mehdi Shayan, Simar P. Singh, Joe Bible, Ravikiran Singapogu, Richard E. Groff

Publications

This paper proposes a deep-learning computer vision algorithm to estimate hand roll angles for metric-based assessment of surgical suturing skills. The number of rolls metric, previously calculated directly from IMU data, counts the number of hand roll reversals during a single suture. To calculate this metric using computer vision, we apply a deep-learning algorithm that can reliably estimate hand roll angles after training on suturing videos collected on the SutureCoach simulator. Results show that the estimation accuracy of the deep-learning algorithm is robust to different video backgrounds. The number of rolls metrics were used to analyze suturing performance in the …

Computational Representation, Analysis And Verification Of Requirements In Engineering Design And Systems Engineering, Chandan Kumar Sahu

All Dissertations

Systems are developed to satisfy a set of requirements derived from stakeholders’ needs, defining the problem space for which the system is created as a feasible solution. The system design process begins with eliciting these requirements and concludes with validating whether the created system meets them. Requirements engineering (RE) encompasses elicitation, representation, analysis, documentation, verification, and validation. However, challenges in RE, such as imprecision in natural language (NL), proprietary restrictions, and a lack of standardized quality metrics, hinder the creation of well-formed and comprehensive requirements. These challenges complicate formalization and analysis of requirements.

This dissertation addresses these challenges by proposing …

Andre-Quillen Homology And Special Classes Of Ring Homomorphisms, Hossein Faridian

All Dissertations

This thesis is comprised of three chapters. The first chapter deals with a purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. The central focus of our approach is on the study of shuffle product of connective chain complexes that provides a bridge to translate the constructions in the simplicial algebra world to the chain complex world.

The second chapter delves into Quillen's fundamental spectral sequences that relate Andre-Quillen homology and cohomology to Tor and Ext functors. Our comprehensive treatment develops and streamlines the …