Mathematics Commons^™

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. …

Continuous-Variable Quantum Computation Of The O(3) Model In 1+1 Dimensions, Raghav G. Jha, Felix Ringer, George Siopsis, Shane Thompson

Physics Faculty Publications

We formulate the O(3) nonlinear sigma model in 1+1 dimensions as a limit of a three-component scalar field theory restricted to the unit sphere in the large squeezing limit. This allows us to describe the model in terms of the continuous-variable (CV) approach to quantum computing. We construct the ground state and excited states using the coupled-cluster Ansatz and find excellent agreement with the exact diagonalization results for a small number of lattice sites. We then present the simulation protocol for the time evolution of the model using CV gates and obtain numerical results using a photonic quantum simulator. We …

Hyperparameter Estimation For Sparse Bayesian Learning Models, Feng Yu, Lixin Shen, Guohui Song

Mathematics & Statistics Faculty Publications

Sparse Bayesian learning (SBL) models are extensively used in signal processing and machine learning for promoting sparsity through hierarchical priors. The hyperparameters in SBL models are crucial for the model’s performance, but they are often difficult to estimate due to the nonconvexity and the high-dimensionality of the associated objective function. This paper presents a comprehensive framework for hyperparameter estimation in SBL models, encompassing well-known algorithms such as the expectation-maximization, MacKay, and convex bounding algorithms. These algorithms are cohesively interpreted within an alternating minimization and linearization (AML) paradigm, distinguished by their unique linearized surrogate functions. Additionally, a novel algorithm within the …

Pigeon Strike!: Solutions For Fermi Questions, October 2024, John Adam

Mathematics & Statistics Faculty Publications

This article, titled "Pigeon strike!: Solutions for Fermi Questions, October 2024," discusses the estimation of various physics-related questions regarding a pigeon striking a window. The author estimates the momentum and kinetic energy of the bird prior to hitting the window, as well as the force, pressure, and wingspan of the bird during impact. The author assumes an inelastic collision between the bird and the window, and provides calculations and explanations for their estimations. The article concludes with the author's observation that the estimated wingspan aligns with that of an adult pigeon.

The Weighty Words Of Lord Rayleigh: Solutions For Fermi Questions, January 2024, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Optimal Network Analysis Through Vertex Order Coloring Of Intuitionistic Fuzzy Graph Operations, A. Meenakshi, S. Dhanushiya, Hong Qin, Maniyandy Elangovan

Data Science Faculty Publications

Intuitionistic fuzzy graphs IFGs are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real-world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring IFVOC and analyzed the alpha-strong (alpha str), beta-strong (beta str), and gamma-strong (gamma str) vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of alpha str vertices, the chromatic number, and the weight …

Preface, Mohamed Mahdi Tekitek, Manfred Krafczyk, Li-Shi Luo

Mathematics & Statistics Faculty Publications

[Introduction] Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications, a macroscopic or continuum description has usually been sufficient due to the large disparity between the spatial and temporal scales relevant to these applications and the scales of the underlying molecular dynamics. In this case, the microscopic physics merely determines material properties such as the viscosity of a fluid or the elastic constants of a solid. These material properties cannot be derived within the macroscopic framework, but the qualitative nature of the macroscopic …

Cats, Dogs, And Roll Waves, John Adam

Mathematics & Statistics Faculty Publications

This article discusses the phenomenon of roll waves, which are shock-like patterns separated by smooth profiles that occur in flood waves. The author mentions that in a steady-state situation, the drag forces on the water in the channel are balanced by the down-slope gravitational force. The article also includes questions for readers to estimate the speed and length of the roll waves, as well as determine the stability of the flow. The author provides additional information about the Chezy formula, an empirical equation used in fluid mechanics to estimate the mean flow velocity of open channel flow.

Prospective Teachers' Conceptions Of Area, Betsy Mcneal, Sayonita Ghosh Hajra, Michael Battista, Ayse Ozturk

Teaching & Learning Faculty Publications

This study explored 10 prospective teachers’ (PTs’) understanding of the area of a rectangular region using square and non-square rectangular area-units. In an hour-long interview, each PT was first asked to explain how they would find the area of a given rectangular region in terms of a non-square notecard. For several PTs, this task prompted discussion of a square unit defined by the edge of the notecard. In a second task, PTs were presented with a rectangular array of squares and were asked to explain to a fictional child why multiplying length times width does not count the top left …

Snow Spheres And Ice Cubes: Solutions For Fermi Questions, February 2024, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Snow Spheres And Ice Cubes, John Adam

Mathematics & Statistics Faculty Publications

This article, titled "Snow spheres and ice cubes," discusses the melting of snowballs and ice cubes. It explains that in standard calculus textbooks, it is shown that a snowball melting at a rate proportional to its surface area will have its radius decrease at a constant rate, assuming it remains spherical as it melts. The article then poses several questions related to this concept, including the proportion of a smaller snow sphere that remains when half of a larger one has melted, the value of p for which the smaller sphere first melts, and the volume reduction factor for cubes …

Infusing Machine Learning And Computational Linguistics Into Clinical Notes, Funke V. Alabi, Onyeka Omose, Omotomilola Jegede

Mathematics & Statistics Faculty Publications

Entering free-form text notes into Electronic Health Records (EHR) systems takes a lot of time from clinicians. A large portion of this paper work is viewed as a burden, which cuts into the amount of time doctors spend with patients and increases the risk of burnout. We will see how machine learning and computational linguistics can be infused in the processing of taking clinical notes. We are presenting a new language modeling task that predicts the content of notes conditioned on historical data from a patient's medical record, such as patient demographics, lab results, medications, and previous notes, with the …

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, 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, 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, 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, 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.

Numerical Investigation And Statistical Analysis Of The Flow Patterns Behind Square Cylinders Arranged In A Staggered Configuration Utilizing The Lattice Boltzmann Method, M. Abid, N. Yasin, M. Saqlain, S. Ul-Islam, S. Ahmad

Mathematics & Statistics Faculty Publications

Flow past bluff bodies like square cylinders is important in engineering applications, but flow patterns behind staggered cylinder arrangements remain poorly understood. Existing studies have focused on tandem or side-by-side configurations, while offset orientations have received less attention. The aim of this paper is to numerically investigate flow dynamics and force characteristics behind two offset square cylinders using the single relaxation time lattice Boltzmann method. The effects of changing both the Reynolds number (Re = 1-150) and gap spacing ratio (g* = 0.5-5) between the cylinders are analyzed. Instantaneous vorticity contours, time histories of drag and lift coefficients, power spectra …

Shadow Hands: Solutions For Fermi Questions, September 2024, John Adam

Mathematics & Statistics Faculty Publications

The article titled "Shadow hands: Solutions for Fermi Questions, September 2024" discusses the enigmatic statement made by authors Dave Lynch and Bill Livingston in their book "Color and Light in Nature." The statement explains that the fuzzy outer edge of a shadow, known as the penumbra, is always a constant fraction of about 1/112 of its width divided by its distance from the object casting the shadow. The article provides a solution to Question 1, which asks for an explanation of this statement using elementary geometry. It also offers a solution to Question 2, which asks why the shadow of …

Shadow Hands, John Adam

Mathematics & Statistics Faculty Publications

The article titled "Shadow hands" discusses a statement made by authors Dave Lynch and Bill Livingston in their book "Color and Light in Nature." The authors state that when you cast a shadow with your hand onto a piece of paper, the sharpness of the shadow's edge diminishes as you raise your hand above the surface. This fuzzy outer edge of the shadow is called the penumbra, and its width divided by its distance from your hand is always a constant fraction of about 1/112. The article poses two questions related to this statement and asks readers to explain it …

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, 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, 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, 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, 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.

Cats, Dogs, And Roll Waves: Solutions For Fermi Questions, April 2024, John Adam

Mathematics & Statistics Faculty Publications

On a hot afternoon in western North Carolina, the heavens opened and (to mix metaphors) it started raining “cats and dogs.” The rain was torrential, and soon a river of water was flowing downhill into the parking lot. In rivers and dam spillways, these are known as flood waves. The picture shows a particular type of flood waves known s roll waves—shock-like patterns separated by smooth profiles. In a steady-state situation, the drag forces on the water in the channel are balanced by the down-slope gravitational force. The dimensionless friction coefficient is C, the stream depth is h, v is …

Earthquakes And Seismic Waves: Solutions For Fermi Questions, March 2024, John Adam

Mathematics & Statistics Faculty Publications

Solution to Question 1: To estimate h, we take the geometric mean of 1 km and 70 km, or ~8 km, and take ρ ≈ 2500 kg/m³.

Hence, δl ≈ 4 × 0.05 × (2.5 × 10³ kg/m³) × (10 m/s²) × (8 × 10³ m) × (2 × 10⁴ m) × 0.1 ÷ (3 × 10¹⁰ kg · m⁻¹ · s⁻²) ≈ 1.3 m. (Note that this is not meant to be an estimate of the width of the Icelandic rock fracture in the photograph, though it appears to be of the right order …

Earthquakes And Seismic Waves, John Adam

Mathematics & Statistics Faculty Publications

This article discusses earthquakes and seismic waves. It explains that earthquakes occur in the crust or upper mantle, with "shallow" earthquakes happening above a depth of 70 km in the crust. The article also introduces the slider-block model, which is a simple model used to understand earthquake behavior. It provides equations to calculate the displacement and maximum slip velocity of the fault based on various factors. The article then presents two questions related to earthquake calculations. Additionally, it mentions Fermi Questions, which are brief questions with estimation techniques.

Integrable Symplectic Maps With A Polygon Tessellation, T. Zolkin, Y. Kharkov, S. Nagaitsev

Physics Faculty Publications

Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a distinct form of area-preserving (symplectic) mappings derived from the stroboscopic Poincaré cross section of a kicked rotator—an oscillator subjected to an external force periodically switched on in short pulses. The significance of this class of problems extends to various applications in physics and mathematics, including particle accelerators, crystallography, and studies of chaos. Notably, Suris's theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this …