Mathematics Commons^™

Math 75: Introduction To Linear Algebra, Sarah K. Merz

Pacific Open Texts

This text is intended to use in a first course of Linear Algebra with a prerequisite of Calculus 1. Topics covered include systems of linear equations, matrix operations and inverses, linear transformations, Markov chains, determinants, eigenvalues and eigenvectors, diagonalization, vector geometry, projections and planes, homogeneous coordinates, subspaces, spanning sets, linear independence, orthogonality, fundamental subspaces, and least squares.

Gauss Newton Method For Solving Variational Problems Of Pdes With Neural Network Discretizaitons, Wenrui Hao, Qingguo Hong, Xianlin Jin

Mathematics and Statistics Faculty Research & Creative Works

The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by parameterizing a set of functions. Various approaches, such as the deep Ritz method and physics-informed neural networks, have been developed for numerical solutions. Training algorithms, including gradient descent and greedy algorithms, have been proposed to solve the resulting optimization problems. In this paper, we focus on the variational formulation of the problem and propose a Gauss–Newton method for computing the numerical solution. We provide a comprehensive analysis of the superlinear convergence properties …

If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many modern mathematical proofs are very complex, checking them is difficult; as a result, errors sneak into published proofs, even into proofs published in highly reputable journals. Sometimes, the errors are repairable, but sometimes, it turns out that the supposedly proven result is actually wrong. When the error is not noticed for some time, the published result is used to prove many other results -- and when the error is eventually found, all these new results are invalidated. This happened several times. Since it is not realistic to more thoroughly check all the proofs, and we want to minimize the …

Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

Recent observations have shown that the angles between the Galaxy Center filaments and the Galactic plane follow a bimodal distribution: a large number of filaments are approximately orthogonal to the Galactic plane, a large number of filaments are approximately parallel to the Galactic plane, and much fewer filaments have other orientations. In this paper, we show this bimodal distribution can be explained by natural geometric symmetries.

Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

For each geographic location, its seismicity level is usually determined by how close this location is to the boundaries of tectonic plates. However, there is one notable exception: while Ireland and Britain are at approximately the same distance from such boundaries, the seismicity level in Ireland is much lower than in Britain. A recent paper provided a partial explanation for this phenomenon: namely, it turns out that the lithosphere under Ireland is unusually thick, and this can potentially lead to lower seismicity. However, the current explanation of the relation between the lithosphere thickness and seismicity level strongly depends on the …

Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich

Departmental Technical Reports (CS)

In the State of Alaska there is no state income tax. Instead, there is a negative tex: every year every resident gets some money from the state. At present, every resident -- from the poorest to the richest -- gets the exact same amount of money: in 2024, it is expected to be around $1500. A natural question is: Is this fair? Maybe poor people should get more since their needs are greater? Maybe the rich people should get proportionally more, since fairness means equal added happiness for all, and for rich people, extra $1500 is barely noticeable? There have …

For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, for each of two classifications, we know the probabilities that a randomly selected object belong to different categories. For example, we know what proportion of people are below 20 years old, what proportion is between 20 and 30, etc., and we also know what proportion of people earns less than 10K, between 10K and 20K, etc. In such situations, we are often interested in proportion of people who are classified by two classifications into two given categories. For example, we are interested in the proportion of people whose age is between 20 and 30 and whose …

Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional decision theory recommendation about making a decision assume that we know both the probabilities of different outcomes of each possible decision, and we know the utility function -- that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we only have partial information about utility. Such cases are known as cases of stochastic dominance. In other cases, in addition to not knowing the utility function, we also only have partial information about the probabilities of different outcomes. For example, we may only known bounds on the outcomes (case of interval uncertainty) or bounds on the …

Mega-Influencers And Brand Dynamics: Shaping Attitudes Toward Leading And Challenger Brandsthrough Electronic Word Of Mouth, Riccardo Rialto, Lamberto Zollo, Kacy Kim, Sukki Yoon

Mathematics and Economics Faculty Journal Articles

The aim of this paper is to explore how mega‐influencers' electronic word of mouth (eWOM) messages on social media influence consumers' brand attitudes in duopolistic markets. Through three experimental studies, we observe that when mega‐ influencers send positive (vs. negative) eWOM messages about a leading brand, followers form positive (vs. negative) brand attitudes, but these effects fail to occur when influencers back challenger brands. The findings are consistent across three duopolistic market rivals (Apple vs. Samsung; UPS vs. FedEx; Nike vs. Adidas), three social media platforms (Facebook, Instagram, and X), and four mega‐influencers (Marques Brownlee, Gary Vaynerchuk, Kanye West, and …

Mathematical Methods For Economics And Business Syllabus, Marian Melnyk

Open Educational Resources

This syllabus outlines an online asynchronous course introducing essential mathematical tools and techniques in economics. It spans 24 sessions covering Functions, Differentiation, Limits, Integration, and Multivariable Functions, grounded in college algebra and calculus. The course utilizes Open Educational Resources (OER) for all materials, available in text and video formats on Blackboard. Continuous assessment, including quizzes, homework, two 1-hour tests, and a final exam, ensures consistent progress. Weekly online office hours provide additional support. The primary objective is to equip students with the mathematical skills necessary for economic analysis and problem-solving.

Nonlocal Frameworks For Nonlinear Conservation Laws And Advection-Diffusion Processes, Anh Thuong Vo

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

Conservation laws are fundamental principles that play an important role in modeling various phenomena in physics, chemistry, and biology. However, their limitations, such as the development of shocks despite smooth initial conditions, are well known. The nonlocal model framework can be used to overcome these challenges. Nonlocal frameworks utilize integral operators that mimic differential operators but also incorporate long-range interactions within a finite horizon. This approach not only allows for non-smooth solutions, but also provides flexibility in modeling different phenomena. This study investigates the convergence of nonlocal divergence operators, defined with a general flux density function, to their classical counterparts. …

Differentiated Instruction In Geometry Using Low Floor, High Ceiling Mathematical Tasks, Franklin R. Falculan, Maria Alva Q. Aberin

Mathematics Faculty Publications

This study investigated the effects of using Low Floor High Ceiling (LFHC) mathematical tasks on students’ conceptual understanding and procedural fluency in seventh-grade Geometry by closely examining pre-test and posttest results. Two intact classes composed of thirty-two grade 7 students in each class participated in the study. The control group was taught and had practice using conventional, algorithmic tasks while the experimental group was taught and had practice using LFHC mathematical tasks. Data analysis revealed that, as compared to students exposed to algorithmic problems, students exposed to LFHC activities were much more mathematically proficient in Geometry, at the very least …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney

Theses and Dissertations

Bucklew and Wise (1982) established that the quantization dimension of an absolutely continuous probability measure on a Euclidean space equals the Euclidean dimension of the space, and that the quantization coefficient is finite and positive. This thesis explores the variability of quantization coefficients for uniform distributions on various geometric structures, including line segments, circles, and regular polygons. We derive the conditional optimal sets of n-points and calculate the n-th conditional quantization errors for uniform distributions under different conditions. On line segments, we identify optimal point distributions as either equally spaced or geometrically spaced towards fixed end-points. For circles, we analyze …

Torus Surgery, Fibrations, Multisections, And Spun 4-Manifolds, Nicholas Paul Meyer

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

A compact n-manifold X is fibered if it is a fiber bundle where the fiber F and base space B are manifolds. Fibered manifolds are particularly nice, as they are essentially classified by their monodromy maps. Two common examples of 4-dimensional fibered manifolds are surface bundles over surfaces and 3-manifold bundles over the circle.

The main focus of this dissertation is to investigate fibered 4-manifolds whose boundaries are the 3-torus and how these manifolds glue together to give new closed, fibered 4-manifolds. In particular, suppose W is diffeomorphic to S¹ × E_Y (K) where Y …

A Portable Numerical Library For The Calculation Of Multi-Dimensional Integrals, Ioannis Sakiotis

Computer Science Theses & Dissertations

Multi-dimensional numerical integration is a prevalent task in physics and other scientific fields, e.g., in the simulation of particle-beam dynamics and Bayesian parameter estimation. Scientific computing applications that simulate complex phenomena may require the solution to numerous multi-variate integrals. However, functions that have features such as sharp peaks or oscillations in high dimensional spaces, can result in an exorbitant number of computations. For many cases, convergence to accurate results in a reasonable amount of time is infeasible with existing numerical libraries. One approach towards making multi-dimensional integration viable is to parallelize existing algorithms. No commonly available algorithms or libraries exist …

Parallel-In-Time Implicit Schemes For Nonlinear Pdes, Subhash Paudel

Mathematics & Statistics Theses & Dissertations

An accurate prediction of unsteady physical phenomena arising in various applications (e.g., rotorcraft and turbomachinery flows, fluid-structure interaction, maneuvering flight conditions, etc.) requires a very large number of time steps, thus considerably increasing the total computational time, because conventional time integrators are inherently sequential. Parallel-in-time methods offer a promising direction for drastically reducing the computational time and achieving such scalability that is required for solving these unsteady problems on modern supercomputers with hundreds of thousands of computing cores. The parallel performance of existing parallel-in-time algorithms for nonlinear equations especially of the hyperbolic or mixed type is far from being satisfactory. …

On Laudenbach-Type Exact Sequences, Jorge Andres Robinson Arrieta

Graduate Theses and Dissertations

Laudenbach proved that the mapping class group of the connected sum of $n$ copies of $S^2 \times S^1$ is an extension of $\text{Out}(F_n)$ by a finite group. Brendle, Broaddus and Putman proved that this exact sequence splits. We provide an explicit section $s$ of this split exact sequence. Given a locally finite graph $\Gamma$, Udall proved that the mapping class group of the doubled handlebody associated to $\Gamma$ is an extension of $\map(\Gamma)$ by a possibly infinite direct product of $\Z_2$, and proved that this exact sequence splits. We provide an explicit formula for this section $s$ of $\Psi$ restricted …

Using Chatgpt As A Thought Partner In Writing Relevant Proportional Reasoning Word Problems, Andrea Berryhill, Lendon Chandler, Liza Bondurant, Bima Sapkota

School of Mathematical & Statistical Sciences Faculty Publications

In this article, we, two preservice mathematics teachers (PSTs) and two mathematics teacher educators (MTEs), explore the potential of using ChatGPT, a large language model (LLM), as a thought partner to create relevant proportional reasoning word problems. We detail our iterative process of refining our inputs and critically evaluating ChatGPT's outputs. Our takeaways from this journey offer valuable insights for educators seeking to leverage the capabilities of LLMs in their lesson-planning endeavors.

Application And Analysis Of Machine Learning And Deep Learning Algorithms In Detection Of Ddos Cyberattacks, Dipok Deb

Theses and Dissertations

A Distributed Denial-of-Service (DDoS) attack involves overwhelming a target system's data bandwidth or computational resources, often using multiple attack systems, aiming to slow down or disable the targeted system. Detecting and mitigating DDoS attacks effectively remains challenging due to their varying characteristics. One of the promising approaches involves developing an AI based Intrusion Detection System (IDS) against cyberattacks. In this study, we aim to develop an AI based Intrusion Detection System (IDS) for DDoS threat detection using Machine Learning, Deep Learning, or hybrid techniques. Different Machine Learning (ML) and Deep Learning (DL) algorithms like Random Forest (RF), Naïve Bayes (NB), …

On Angles In Higher Order Brillouin Tessellations And Related Tilings In The Plane, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian

School of Mathematical & Statistical Sciences Faculty Publications

For a locally finite set in R 2 , the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R 2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in …