Mathematics Commons^™

Computational Methods For Nonlinear Systems Analysis With Applications In Mathematics And Engineering, Geoffrey Kenneth Rose

Mechanical & Aerospace Engineering Theses & Dissertations

An investigation into current methods and new approaches for solving systems of nonlinear equations was performed. Nontraditional methods for implementing arc-length type solvers were developed in search of a more robust capability for solving general systems of nonlinear algebraic equations. Processes for construction of parameterized curves representing the many possible solutions to systems of equations versus finding single or point solutions were established. A procedure based on these methods was then developed to identify static equilibrium states for solutions to multi-body-dynamic systems. This methodology provided for a pictorial of the overall solution to a given system, which demonstrated the possibility …

Random Walks On Digraphs, J.J.P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Let V = {1, · · · n} be a vertex set and S a non-negative row-stochastic matrix (i.e. rows sum to 1). V and S define a digraph G = G(V, S) and a directed graph Laplacian L as follows. If (S)ij > 0 (in what follows we will leave out the parentheses) there is a directed edge j → i. Thus the ith row of S identifies the edges coming into vertex i and their weights. This set of vertices are collectively the neighbors of i, and is denoted by Ni . The diagonal elements Sii are chosen such …

Cybersecurity: Time Series Predictive Modeling Of Vulnerabilities Of Desktop Operating System Using Linear And Non-Linear Approach, Nawa Raj Pokhrel, Hansapani Rodrigo, Chris P. Tsokos

School of Mathematical & Statistical Sciences Faculty Publications

Vulnerability forecasting models help us to predict the number of vulnerabilities that may occur in the future for a given Operating System (OS). There exist few models that focus on quantifying future vulnerabilities without consideration of trend, level, seasonality and non linear components of vulnerabilities. Unlike traditional ones, we propose a vulnerability analytic prediction model based on linear and non-linear approaches via time series analysis. We have developed the models based on Auto Regressive Moving Average (ARIMA), Artificial Neural Network (ANN), and Support Vector Machine (SVM) settings. The best model which provides the minimum error rate is selected for prediction …

Almost-Bps Solutions In Multi-Center Taub-Nut, C. Rugina, A. Ludu

Publications

Microstates of multiple collinear black holes embedded in a non-collinear two-center Taub-NUT spacetime are sought in 4 dimensions. A set of coupled partial differential equations are obtained and solved for almost-BPS states, where some supersymmetry is preserved in the context of N = 2 supergravity in 4 dimensions. The regularity of solutions is carefully considered, and we ensure that no CTC (closed time-like curves) are present. The larger framework is that of 11-dimensional N = 2 supergravity, and the current theory is obtained by compactifying it down to 4 dimensions. This work is a generalization (to three non-collinear centers) of …

Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan

Dissertations, Theses, and Capstone Projects

The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff's celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of maps …

Interstructure Lattices And Types Of Peano Arithmetic, Athar Abdul-Quader

Dissertations, Theses, and Capstone Projects

The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA.

In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA^∗ that is pointwise definable. Enayat later showed that there are 2^ℵ₀ models with …

Some Results In Combinatorial Number Theory, Karl Levy

Dissertations, Theses, and Capstone Projects

The first chapter establishes results concerning equidistributed sequences of numbers. For a given $d\in\mathbb{N}$, $s(d)$ is the largest $N\in\mathbb{N}$ for which there is an $N$-regular sequence with $d$ irregularities. We compute lower bounds for $s(d)$ for $d\leq 10000$ and then demonstrate lower and upper bounds $\left\lfloor\sqrt{4d+895}+1\right\rfloor\leq s(d)< 24801d^{3} + 942d^{2} + 3$ for all $d\geq 1$. In the second chapter we ask if $Q(x)\in\mathbb{R}[x]$ is a degree $d$ polynomial such that for $x\in[x_k]=\{x_1,\cdots,x_k\}$ we have $|Q(x)|\leq 1$, then how big can its lead coefficient be? We prove that there is a unique polynomial, which we call $L_{d,[x_k]}(x)$, with maximum lead coefficient under these constraints and construct an algorithm that generates $L_{d,[x_k]}(x)$.

Lessons From Between The White Lines For Isolated Data Scientists, Benjamin Baumer

Mathematics Sciences: Faculty Publications

Many current and future data scientists will be “isolated”—working alone or in small teams within a larger organization. This isolation brings certain challenges as well as freedoms. Drawing on my considerable experience both working in the professional sports industry and teaching in academia, I discuss troubled waters likely to be encountered by newly minted data scientists and offer advice about how to navigate them. Neither the issues raised nor the advice given are particular to sports and should be applicable to a wide range of knowledge domains.

Collaborative Research: Mathematics Of Doing, Understanding, Learning And Educating For Secondary Schools, Brynja R. Kohler

Funded Research Records

No abstract provided.

The Double Pivot Simplex Method, Fabio Vitor, Todd Easton

Mathematics Faculty Publications

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time …

On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez

Math Faculty Publications

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~k. Similar to the classical problem we show that the n = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to k[t] and give an algorithm to solve the Frobenius problem for polynomials over a field k of sufficiently large size.

“Wrong” Side Interpolation By Positive Real Rational Functions, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using polynomial interpolation, along with structural properties of the family of rational positive real functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m. Moreover we introduce an easy-to-find parametrization in R2m+3 of a large subset of these interpolating functions.

Categorical Models For Equivariant Classifying Spaces, Bertrand J. Guillou, J. Peter May, Mona Merling

Mathematics Faculty Publications

Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equi- variant infinite loop space theory and equivariant algebraic K–theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Evolution Of Superoscillations For Schrödinger Equation In A Uniform Magnetic Field, Fabrizio Colombo, Jonathan Gantner, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of …

Challenges In Modeling Complexity Of Neglected Tropical Diseases: A Review Of Dynamics Of Visceral Leishmaniasis In Resource Limited Settings, Swati Debroy, Olivia F. Prosper, Austin Mishoe, Anuj Mubayi

Mathematics Faculty Publications

Objectives: Neglected tropical diseases (NTD), account for a large proportion of the global disease burden, and their control faces several challenges including diminishing human and financial resources for those distressed from such diseases. Visceral leishmaniasis (VL), the second-largest parasitic killer (after malaria) and an NTD affects poor populations and causes considerable cost to the affected individuals. Mathematical models can serve as a critical and cost-effective tool for understanding VL dynamics, however, complex array of socio-economic factors affecting its dynamics need to be identified and appropriately incorporated within a dynamical modeling framework. This study reviews literature on vector-borne diseases and collects …

Ar(1) Sequence With Random Coefficients:Regenerative Properties And Its Application, Krishna B. Athreya, Koushik Saha, Radhendushka Srivastava

Communications on Stochastic Analysis

No abstract provided.

Applications Of The Lichnerowicz Laplacian To Stress Energy Tensors, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A generalization of the Laplacian for p-forms to arbitrary tensors due to Lichnerowicz will be applied to a 2-tensor which has physical applications. It is natural to associate a divergencefree symmetric 2-tensor to a critical point of a specific variational problem and it is this 2-tensor that is studied. Numerous results are obtained for the stress-energy tensor, such as its divergence and Laplacian. A remarkable integral formula involving a symmetric 2-tensor and a conformal vector field is obtained as well

Preventing Neurodegenerative Memory Loss In Hopfield Neuronal Networks Using Cerebral Organoids Or External Microelectronics, M. Morrison, P. D. Maia, J. N. Kutz

Mathematics Faculty Publications - Archive

Developing technologies have made significant progress towards linking the brain with brain-machine interfaces (BMIs) which have the potential to aid damaged brains to perform their original motor and cognitive functions. We consider the viability of such devices for mitigating the deleterious effects of memory loss that is induced by neurodegenerative diseases and/or traumatic brain injury (TBI). Our computational study considers the widely used Hopfield network, an autoassociative memory model in which neurons converge to a stable state pattern after receiving an input resembling the given memory. In this study, we connect an auxiliary network of neurons, which models the BMI …

Neuronal Correlation Parameter In The Idea Of Thermodynamic Entropy Of An N-Body Gravitationally Bounded System, Ioannis Haranas, Ioannis Gkigkitzis, Ilias S. Kotsireas, Carlos Austerlitz

Physics and Computer Science Faculty Publications

Understanding how the brain encodes information and performs computation requires statistical and functional analysis. Given the complexity of the human brain, simple methods that facilitate the interpretation of statistical correlations among different brain regions can be very useful. In this report we introduce a numerical correlation measure that may serve the interpretation of correlational neuronal data, and may assist in the evaluation of different brain states. The description of the dynamical brain system, through a global numerical measure may indicate the presence of an action principle which may facilitate a application of physics principles in the study of the human …

An Ancient Bankruptcy Solution Makes Economic Sense, Anh H. Ly, Michael Zakharevich, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While econometrics is a reasonable recent discipline, quantitative solutions to economic problem have been proposed since the ancient times. In particular, solutions have been proposed for the bankruptcy problem: how to divide the assets between the claimants? One of the challenges of analyzing ancient solutions to economics problems is that these solutions are often presented not as a general algorithm, but as a sequence of examples. When there are only a few such example, it is often difficult to convincingly extract a general algorithm from them. This was the case, for example, for the supposedly fairness-motivated Talmudic solution to the …