Mathematics Commons^™

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 …

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 …

The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

Cybersecurity Undergraduate Research Showcase

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

Inference For Multiple Utility In Time-Dependent Choice Pairs Under Copula-Based Models, Sasanka Adikari

Mathematics & Statistics Theses & Dissertations

Models for discrete choice experiments (DCE) are frequently used to analyze consumer choices about products and services. A family of DCE, best-worst scaling experiments, offers more in-depth insights into consumer preferences by eliciting a best and worst choice from a set of options, rather than just a single preference. Traditional approaches often assume that choices are mutually exclusive over time, which may not always be the case. This dissertation proposes a novel model for DCE that takes into account the changing nature of consumer choices over time and the priority constraint of transition probabilities. The model introduces a copula combination …

Algebraic Tunnelling, Gaurab Sedhain

2023 REYES Proceedings

We study the quantum phenomenon of tunnelling in the framework of algebraic quantum theory, motivated by the tunnelling aspects of false vacuum decay. We see that resolvent C*-algebra, proposed relatively recently by Buchholz and Grundling rather than Weyl algebra provides an appropriate framework for treating the dynamics of non-free quantum mechanical system as an algebraic automorphism. At the end, we propose to investigate false vacuum decay in algebraic quantum field theoretic setting in terms of the two-point correlation function which gives us the tunneling probability, with the corresponding C*-algebraic construction.

An Adaptive Algorithm For `The Secretary Problem': Alternate Proof Of The Divergence Of A Maximizer Sequence, Andrew Benfante, Xiang Xu

OUR Journal: ODU Undergraduate Research Journal

This paper presents an alternate proof of the divergence of the unique maximizer sequence {𝑥∗ 𝑛} of a function sequence {𝐹𝑛(𝑥)} that is derived from an adaptive algorithm based on the now classic optimal stopping problem, known by many names but here ‘the secretary problem’. The alternate proof uses a result established by Nguyen, Xu, and Zhao (n.d.) regarding the uniqueness of maximizer points of a generalized function sequence {𝑆𝜇,𝜎 𝑛 } and relies on the strict monotonicity of 𝐹𝑛(𝑥) as 𝑛 increases in order to show divergence of {𝑥∗ 𝑛}. Towards this, limits of the exponentiated Gaussian CDF are …

More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder

Mathematics & Statistics Faculty Publications

We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, H^p (1 < p < ∞). For fixed f ∈ H^p and n ∈ N, the OPA of degree n associated to f is the polynomial which minimizes the quantity ∥qf −1∥_p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p ≠ 2, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions …

Oscillating Icebergs, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Another Angle On Perspective, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Exploding Haystacks, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Application Of Mixture Models For Doubly Inflated Count Data, Monika Arora, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

In health and social science and other fields where count data analysis is important, zero-inflated models have been employed when the frequency of zero count is high (inflated). Due to multiple reasons, there are scenarios in which an additional count value of k > 0 occurs with high frequency. The zero- and k-inflated Poisson distribution model (ZkIP) is more appropriate for such situations. The ZkIP model is a mixture distribution with three components: degenerate distributions at 0 and k count and a Poisson distribution. In this article, we propose an alternative and computationally fast expectation–maximization (EM) algorithm to obtain the parameter …

Generalized Sparse Bayesian Learning And Application To Image Reconstruction, Jan Glaubitz, Anne Gelb, Guohui Song

Mathematics & Statistics Faculty Publications

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are …

Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Dental Floss, Calculus, And Jail: Solutions For Fermi Questions, October 2023, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Ellipses, Leaves, And Solar Crescents, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Dental Floss, Calculus, And Jail, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Not Your Typical Tower Of Sauron: Solutions For Fermi Questions, September 2023, John Adam

Mathematics & Statistics Faculty Publications

The picture is of the tapering Chester Shot Tower, located in Chester, England. It was built in 1799 for the manufacture of lead shot for use in the Napoleonic Wars. Molten lead was poured through a sieve at the top of the tower, with the tiny droplets forming perfect spheres during the fall; these were then cooled in a vat of water at the base. This process was less labor-intensive than an earlier method using molds. It is the oldest of the three remaining shot towers in the UK. Using the parked van at the base, estimate (i) the height …

"Density" Of Light And Pi, John Adam

Mathematics & Statistics Faculty Publications

Question 1: What is the linear “density” of the spectral range on the ceiling? The crown molding is common for an older (circa 1925) house.

Question 2: The glass shown is a standard 16-oz. water glass engraved with many digits of pi. (The dimensions appear somewhat distorted because of the camera angle chosen to enhance the contrast of the numerals against the dark background.)

Shrub Sphericity, John Adam

Mathematics & Statistics Faculty Publications

Question 1: a. Show that α = 4.836, and hence find the sphericity index for (i) a cube and (ii) two identical "kissing" spheres, i.e., spheres in tangential contact. b. Estimate your sphericity index.

Question 2: Estimate χ for the yucca plant in the picture. It is about 1 m in diameter.

Exploding Haystacks: Solutions For Fermi Questions, March 2023, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Shrub Sphericity: Solutions For Fermi Questions, January 2023, John Adam

Mathematics & Statistics Faculty Publications

Answers the questions: Question 1: a. Show that α = 4.836, and hence find the sphericity index for (i) a cube and (ii) two identical "kissing" spheres, i.e., spheres in tangential contact. b. Estimate your sphericity index.

Question 2: Estimate χ for the yucca plant in the picture. It is about 1 m in diameter.

Not Your Typical Tower Of Sauron, John Adam

Mathematics & Statistics Faculty Publications

The picture is of the tapering Chester Shot Tower, located in Chester, England. It was built in 1799 for the manufacture of lead shot for use in the Napoleonic Wars. Molten lead was poured through a sieve at the top of the tower, with the tiny droplets forming perfect spheres during the fall; these were then cooled in a vat of water at the base. This process was less labor intensive than an earlier method using molds. It is the oldest of the three remaining shot towers in the UK.

Question 1: Using the parked van at the base, estimate …

Machine-Assisted Discovery Of Integrable Symplectic Mappings, T. Zolkin, Y. Kharkov, S. Nagaitsev

Physics Faculty Publications

We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e., integrals of motion. The core idea of the algorithm is based on the knowledge that the evolution of an integrable system in the phase space is restricted to a lower-dimensional submanifold. Limiting ourselves to polygon invariants of motion, we analyze the shape of individual trajectories thus successfully distinguishing integrable motion from chaotic cases. For example, our method rediscovers some of the famous McMillan-Suris integrable mappings and ultradiscrete Painlevé equations. In total, …

Jet Noise Reduction: A Fresh Start, Christopher K. Tam, Fang Q. Hu

Mathematics & Statistics Faculty Publications

Attempts to reduce jet noise began some 70 years ago. In the literature, there have been many publications written on this topic. By now, it is common knowledge that jet noise consists of a number of components. They possess different spectral and radiation characteristics and are generated by different mechanisms. It appears then that one may aim at the suppression of the noise of a single component instead of trying to reduce jet noise overall. The objective of the present project is to reduce large turbulence structures noise. It is the most dominant noise component radiating in the downstream direction. …

Inexact Fixed-Point Proximity Algorithms For Nonsmooth Convex Optimization, Jin Ren

Mathematics & Statistics Theses & Dissertations

The aim of this dissertation is to develop efficient inexact fixed-point proximity algorithms with convergence guaranteed for nonsmooth convex optimization problems encountered in data science. Nonsmooth convex optimization is one of the core methodologies in data science to acquire knowledge from real-world data and has wide applications in various fields, including signal/image processing, machine learning and distributed computing. In particular, in the context of image reconstruction, compressed sensing and sparse machine learning, either the objective functions or the constraints of the modeling optimization problems are nondifferentiable. Hence, traditional methods such as the gradient descent method and the Newton method are …

Chen-Fliess Series For Linear Distributed Systems, Natalie T. Pham

Electrical & Computer Engineering Theses & Dissertations

Distributed systems like fluid flow and heat transfer are modeled by partial differential equations (PDEs). In control theory, distributed systems are generally reformulated in terms of a linear state space realization, where the state space is an infinite dimensional Banach space or Hilbert space. In the finite dimension case, the input-output map can always be written in terms of a Chen-Fliess functional series, that is, a weighted sum of iterated integrals of the components of the input function. The Chen-Fliess functional series has been used to describe interconnected nonlinear systems, to solve system inversion and tracking problems, and to design …

Machine Learning Classification Of Digitally Modulated Signals, James A. Latshaw

Electrical & Computer Engineering Theses & Dissertations

Automatic classification of digitally modulated signals is a challenging problem that has traditionally been approached using signal processing tools such as log-likelihood algorithms for signal classification or cyclostationary signal analysis. These approaches are computationally intensive and cumbersome in general, and in recent years alternative approaches that use machine learning have been presented in the literature for automatic classification of digitally modulated signals. This thesis studies deep learning approaches for classifying digitally modulated signals that use deep artificial neural networks in conjunction with the canonical representation of digitally modulated signals in terms of in-phase and quadrature components. Specifically, capsule networks are …

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

Exploring The Complex Relationship Between Engineering Students Math Experiences And Identity Formation, Jill Davishahl, Joseph Brobst, Elizabeth Litzler, Andrew Klein, Sura Alqudah

Division of Research and Economic Development Faculty & Staff Publications

This paper shares the experiences of a group of S-STEM scholarship students as they progressed through their first year of undergraduate mathematics education including a math placement exam, a math focused bridge program, and calculus course(s). In addition, connections between student experiences and their math identity development were investigated with the goal of better understanding math identity formation. The authors used a mixed-methods approach to explore the math-related experiences of 12 students over the first year of participation in an NSF-funded S-STEM program. Data includes answers to select survey questions and transcripts of student focus groups. First, each student case …

Review Of Copula For Bivariate Distributions Of Zero-Inflated Count Time Series Data, Dimuthu Fernando, Mohammed Alqawba, Manar Samad, Norou Diawara

Mathematics & Statistics Faculty Publications

The class of bivariate integer-valued time series models, described via copula theory, is gaining popularity in the literature because of applications in health sciences, engineering, financial management and more. Each time series follows a Markov chain with the serial dependence captured using copula-based distribution functions from the Poisson and the zero-inflated Poisson margins. The copula theory is again used to capture the dependence between the two series.

However, the efficiency and adaptability of the copula are being challenged because of the discrete nature of data and also in the case of zero-inflation of count time series. Likelihood-based inference is used …