Frieze And Tiling Groups In The Lorentz-Minkowski Plane,
2024
University of Central Florida
Frieze And Tiling Groups In The Lorentz-Minkowski Plane, Michael O. Lynch
Honors Undergraduate Theses
In this thesis, there is a presentation of the isometries from the Lorentz-Minkowski Plane and a solution to the Frieze Patterns. There is a suggestion for a solution for the Tiling Patterns. Since the construction of these mathematical structures is well understood in the Euclidean plane, one can follow a similar approach to the construction of such objects to find the unique number of groups that describe all possible frieze patterns while there is a suggestion of the number for the tiling case. There is a reflection of these results in a computational and cosmological context.
Enumeration Of Lattice Paths With Restrictions,
2024
Georgia Southern University
Enumeration Of Lattice Paths With Restrictions, Vince White
College of Graduate Studies: Theses & Dissertations
Lattice path enumeration, through the lens of Catalan numbers, plays a crucial role in combinatorics. This thesis delves into enumerations of some of the most common lattice paths – north-east paths, up-down paths, and Dyck paths – with restrictions applied. The first restriction is counting north-east lattice paths that only cross the diagonal line, y=x, once. The second form of lattice paths with restrictions is up-down paths that cross the x-axis exactly once and fall to a fixed depth of k. While working through this module, a novel proof for a known integer sequence was used, then applied to generate …
A Spiral Workbook For Discrete Mathematics 2nd Edition,
2024
SUNY Fredonia
A Spiral Workbook For Discrete Mathematics 2nd Edition, Harris Kwong
Milne Open Textbooks
This updated text covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions, relations, and elementary combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a different perspective or at a higher level of complexity. The goal is …
Torricelli's Law And The Turn-Up Number,
2024
Columbus State University
Torricelli's Law And The Turn-Up Number, Bianca K. Hampel
Theses and Dissertations
We investigate Torricelli's differential law for solids of revolution. Computing the ratio of times of drainage under the same circumstances (same existing requirements) but rotating the solids 180 degrees allows us to define a "Torricelli turn-up number" associated with the solid. For various solids, we calculate this turn-up number and construct solids that are not symmetric under the 180-degree rotation but with a turn-up number of 1. That allows us to construct nonsymmetric clepsydra.
Reducing Generalization Error In Multiclass Classification Through Factorized Cross Entropy Loss,
2024
Claremont McKenna College
Reducing Generalization Error In Multiclass Classification Through Factorized Cross Entropy Loss, Oleksandr Horban
CMC Senior Theses
This paper introduces Factorized Cross Entropy Loss, a novel approach to multiclass classification which modifies the standard cross entropy loss by decomposing its weight matrix W into two smaller matrices, U and V, where UV is a low rank approximation of W. Factorized Cross Entropy Loss reduces generalization error from the conventional O( sqrt(k / n) ) to O( sqrt(r / n) ), where k is the number of classes, n is the sample size, and r is the reduced inner dimension of U and V.
Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World,
2024
Claremont Colleges
Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World, Dylan Phares
CMC Senior Theses
Shor's Algorithm is an extremely powerful tool, in utilizing this tool it is important to understand how it works and why it works. As well as the vast implications it could have for cryptography
Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory,
2024
Claremont McKenna College
Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, Danzhe Chen
CMC Senior Theses
This thesis provides a comprehensive exploration of lattice theory, emphasizing its dual significance in both theoretical mathematics and practical applications, particularly within computational complexity and cryptography. The study begins with an in-depth examination of the fundamental properties of lattices and progresses to intricate lattice-based problems such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems are analyzed for their computational depth and linked to the Subset Sum Problem (SSP) to highlight their critical roles in understanding computational hardness. The narrative then transitions to the practical applications of these theories in cryptography, evaluating the shift from …
Classification In Supervised Statistical Learning With The New Weighted Newton-Raphson Method,
2024
Georgia Southern University
Classification In Supervised Statistical Learning With The New Weighted Newton-Raphson Method, Toma Debnath
College of Graduate Studies: Theses & Dissertations
In this thesis, the Weighted Newton-Raphson Method (WNRM), an innovative optimization technique, is introduced in statistical supervised learning for categorization and applied to a diabetes predictive model, to find maximum likelihood estimates. The iterative optimization method solves nonlinear systems of equations with singular Jacobian matrices and is a modification of the ordinary Newton-Raphson algorithm. The quadratic convergence of the WNRM, and high efficiency for optimizing nonlinear likelihood functions, whenever singularity in the Jacobians occur allow for an easy inclusion to classical categorization and generalized linear models such as the Logistic Regression model in supervised learning. The WNRM is thoroughly investigated …
A Recursive Framework For Evaluating Moments Using Zero-Suppressed Binary Decision Diagrams,
2024
Nara Institute of Science and Technology
A Recursive Framework For Evaluating Moments Using Zero-Suppressed Binary Decision Diagrams, Brian Godwin Lim, Renzo Roel P. Tan, Jun Kawahara, Shin Ichi Minato, Kazushi Ikeda
Quantitative Methods and Information Technology Faculty Publications
The zero-suppressed binary decision diagram (ZDD) is a compact data structure widely used for the efficient representation of families of sparse subsets. Its inherent recursive structure also facilitates easy diagram manipulation and family operations. Practical applications generally fall under discrete optimization, such as combinatorial problems and graph theory. Given its utility, summarizing the subsets represented in the diagram using key metrics is of great value as this provides valuable insights into the characteristics of the family. The paper proposes a recursive algorithm to extract information on moments from families represented as a ZDD. Given a value for every element in …
Utility In Time Description In Priority Best-Worst Discrete Choice Models: An Empirical Evaluation Using Flynn's Data,
2024
Old Dominion University
Utility In Time Description In Priority Best-Worst Discrete Choice Models: An Empirical Evaluation Using Flynn's Data, Sasanka Adikari, Norou Diawara
Mathematics & Statistics Faculty Publications
Discrete choice models (DCMs) are applied in many fields and in the statistical modelling of consumer behavior. This paper focuses on a form of choice experiment, best-worst scaling in discrete choice experiments (DCEs), and the transition probability of a choice of a consumer over time. The analysis was conducted by using simulated data (choice pairs) based on data from Flynn's (2007) 'Quality of Life Experiment'. Most of the traditional approaches assume the choice alternatives are mutually exclusive over time, which is a questionable assumption. We introduced a new copula-based model (CO-CUB) for the transition probability, which can handle the dependent …
Sparse Representer Theorems For Learning In Reproducing Kernel Banach Spaces,
2024
Jilin University
Sparse Representer Theorems For Learning In Reproducing Kernel Banach Spaces, Rui Wang, Yuesheng Xu, Mingsong Yan
Mathematics & Statistics Faculty Publications
Sparsity of a learning solution is a desirable feature in machine learning. Certain reproducing kernel Banach spaces (RKBSs) are appropriate hypothesis spaces for sparse learning methods. The goal of this paper is to understand what kind of RKBSs can promote sparsity for learning solutions. We consider two typical learning models in an RKBS: the minimum norm interpolation (MNI) problem and the regularization problem. We first establish an explicit representer theorem for solutions of these problems, which represents the extreme points of the solution set by a linear combination of the extreme points of the subdifferential set, of the norm function, …
Testing Informativeness Of Covariate-Induced Group Sizes In Clustered Data,
2024
Old Dominion University
Testing Informativeness Of Covariate-Induced Group Sizes In Clustered Data, Hasika K. Wickrama Senevirathne, Sandipan Duttta
Mathematics & Statistics Faculty Publications
Clustered data are a special type of correlated data where units within a cluster are correlated while units between different clusters are independent. The number of units in a cluster can be associated with that cluster’s outcome. This is called the informative cluster size (ICS), which is known to impact clustered data inference. However, when comparing the outcomes from multiple groups of units in clustered data, investigating ICS may not be enough. This is because the number of units belonging to a particular group in a cluster can be associated with the outcome from that group in that cluster, leading …
Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem,
2024
Old Dominion University
Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan
Mathematics & Statistics Faculty Publications
We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …
A Copula Discretization Of Time Series-Type Model For Examining Climate Data,
2024
Wake Forest University
A Copula Discretization Of Time Series-Type Model For Examining Climate Data, Dimuthu Fernando, Olivia Atutey, Norou Diawara
Mathematics & Statistics Faculty Publications
The study presents a comparative analysis of climate data under two scenarios: a Gaussian copula marginal regression model for count time series data and a copula-based bivariate count time series model. These models, built after comprehensive simulations, offer adaptable autocorrelation structures considering the daily average temperature and humidity data observed at a regional airport in Mobile, AL.
Pigeon Strike!,
2024
Old Dominion University
Pigeon Strike!, John Adam
Mathematics & Statistics Faculty Publications
This article, titled "Pigeon strike!" from the Physics Teacher journal, discusses an incident where a bird, likely a pigeon, struck a window leaving an imprint. The author went outside to check but found no trace of the bird, hoping it had flown away after receiving medical care. The article also includes questions for readers to estimate the momentum, kinetic energy, force, pressure, and wingspan of the bird. The answers to these questions can be found online in the current issue of the journal. The article concludes with information about Fermi Questions and how to submit ideas.
Refracting Ufos: Solutions For Fermi Questions, May 2024,
2024
Old Dominion University
Refracting Ufos: Solutions For Fermi Questions, May 2024, John Adam
Mathematics & Statistics Faculty Publications
This article, titled "Refracting UFOs: Solutions for Fermi Questions, May 2024," recounts the author's personal experience as a teenager observing a UFO sighting near their home in England. The author used their telescope to observe the object, which turned out to be lights from a distant airplane flying in a circular holding pattern. The article then presents two questions related to the observed phenomenon, including estimating the diameter of the aircraft's path and the banking angle of the aircraft at its speed. The article concludes with a solution to estimating the distance to the aircraft. The information is presented objectively …
Refracting Ufos,
2024
Old Dominion University
Refracting Ufos, John Adam
Mathematics & Statistics Faculty Publications
Question 2: Assuming θ ≈ 2° = π/90 rad, estimate the distance R to the aircraft. Assume that R ≫ d. Question 1: Suppose that the observed “period of oscillation” is 1 min, and the speed of the aircraft is 200 km/h. Estimate the diameter d of the aircraft path (assuming that it is circular). Estimate the banking angle of the aircraft at this speed. While still a teenager (and aspiring astronomer), I became interested in the subject of UFOs (more appropriately named UAPs—see Ref. 1). The region in which I lived (50 km west of London) was in the …
Sky Smiley Face,
2024
Old Dominion University
Sky Smiley Face, John Adam
Mathematics & Statistics Faculty Publications
The article discusses the formation of a circumzenithal arc (CZA) in the sky, caused by specific light ray paths through hexagonal ice crystals in cirrus clouds. The CZA is visible when the Sun's altitude is about 32° or less. The text includes questions for readers to explore the physics behind the formation of CZAs and circumhorizontal arcs (CHAs). The author acknowledges photographers for providing images and invites readers to find answers online.
Sky Smiley Face: Solutions For Fermi Questions, November 2024,
2024
Old Dominion University
Sky Smiley Face: Solutions For Fermi Questions, November 2024, John Adam
Mathematics & Statistics Faculty Publications
Question 2(a): By rotating Fig. 2 counterclockwise by 90°, explain why circumhorizontal arcs (CHAs) [Figs. 1(b) and (c)] form (under favorable conditions) only when the solar altitude exceeds 58° of arc. (In this case, the incoming ray enters a vertical face of an ice crystal and exits via the lower horizontal face.)Question 2(b): By considering the tilt of Earth’s imaginary N–S axis toward the plane of its orbit around the Sun (∼23.5°), explain why in the northern hemisphere CHAs cannot be seen above about latitude 56° N.
The History And Basics Of Fourier Series And Applications,
2024
Parkland College
The History And Basics Of Fourier Series And Applications, Syafino Yunalfian
A with Honors Projects
This research paper discusses the basics of the Fourier series, how to construct the series, and a brief explanation of Fourier Transform.
