Applied Mathematics Commons^™

Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton

Mathematics Faculty Publications

This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton

Mathematics Faculty Publications

We study geodesics of the H¹ Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L² metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H¹ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C^∞ in the Banach topology C¹ ([0,1], R²), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, …

Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton

Mathematics Faculty Publications

For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.

Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi

Mathematics Faculty Publications

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Holloman’s power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and …

Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi

Mathematics Faculty Publications

In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …

On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei

Mathematics Faculty Publications

In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.

Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay

Mathematics Faculty Publications

In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.

Independent Dominating Sets In Triangle-Free Graphs, Wayne Goddard, Jeremy Lyle

Faculty Publications

The independent domination number of a graph is the smallest cardinality of an independent set that dominates the graph. In this paper we consider the independent domination number of triangle-free graphs. We improve several of the known bounds as a function of the order and minimum degree, thereby answering conjectures of Haviland.

R0 Analysis Of A Spatiotemporal Model For A Stream Population, H. W. Mckenzie, Y. Jin, J. Jacobsen, M. A. Lewis

Department of Mathematics: Faculty Publications

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem im- pacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R₀ for advection-diffusion-reaction equations and on related measures. We apply the measures to popula- tion persistence in rivers under various flow regimes. This …

Numerical Computation Of A Certain Dirichlet Series Attached To Siegel Modular Forms Of Degree Two, Nathan C. Ryan, Nils-Peter Skoruppa, Fredrik Stroemberg

Faculty Journal Articles

The Rankin convolution type Dirichlet series D-F,D-G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D-F,D-G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

Infeasible Full-Newton-Step Interior-Point Method For The Linear Complementarity Problems, Antré Marquel Drummer

College of Graduate Studies: Theses & Dissertations

In this tesis, we present a new Infeasible Interior-Point Method (IPM) for monotone Linear Complementarity Problem (LPC). The advantage of the method is that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. However, by suitable choice of parameters the iterates are forced to stay in the neighborhood of the central path, hence, still guaranteeing the global convergence of the method under strict feasibility assumption. The number of iterations necessary to find -approximate solution of the problem matches the best known iteration bounds for these types of methods. The preliminary implementation of the method …

Momcmc: An Efficient Monte Carlo Method For Multi-Objective Sampling Over Real Parameter Space, Yaohang Li

Computer Science Faculty Publications

In this paper, we present a new population-based Monte Carlo method, so-called MOMCMC (Multi-Objective Markov Chain Monte Carlo). for sampling in the presence of multiple objective functions in real parameter space. The MOMCMC method is designed to address the "multi-objective sampling" problem, which is not only of interest in exploring diversified solutions at the Pareto optimal front in the function space of multiple objective functions, but also those near the front. MOMCMC integrates Differential Evolution (DE) style crossover into Markov Chain Monte Carlo (MCMC) to adaptively propose new solutions from the current population. The significance of dominance is taken into …

Multi-Disciplinary Hands-On Desktop Learning Modules And Modern Pedagogies, Bernard J. Van Wie, David B. Thiessen, Marc Compere, Ximena Toro, Jennifer C. Adam, Et Al.

Publications

Our team’s research focuses on fundamental problems in undergraduate education in terms of how to expand use of well researched, yet still “new”, teaching pedagogies of ‘sensing’ or ‘hands-on’, ‘active’ and ‘problem-based learning’ within engineering courses. It is now widely accepted that traditional lectures ARE NOT best for students – yet that is what the community almost universally does.

To address this issue we are developing new Desktop Learning Modules (DLMs) that contain miniaturized processes with a uniquely expandable electronic system to contend with known sensor systems/removable cartridges, as well as, unknown expansions to the project. We have shown that …

Development Of New Mathematical Methods For Post-Pareto Optimality, Victor Manuel Carrillo

Open Access Theses & Dissertations

Many real-world applications of multi-objective optimization involve a large number of objectives. A multi-objective optimization task involving multiple conflicting objectives ideally demands finding a multi-dimensional Pareto-optimal front. Although the classical methods have dealt with finding one preferred solution with the help of a decision-maker, evolutionary multi-objective optimization (EMO) methods have been attempted to find a representative set of solutions in the Pareto-optimal front. Multiple objective evolutionary algorithms (MOEAs), which are biologically-inspired optimization methods, have become popular approaches to solve problems with multiple objective functions. With the use of MOEAs, multiple objective optimization becomes a two-part problem. First, the multiple objective …

Analytical And Numerical Solution To The Partial Differential Equation Arising In Financial Modeling, Pavel Bezdek

Open Access Theses & Dissertations

In this work we will present a self-contained introduction to the option pricing problem. We will introduce some basic ideas from the probability theory and stochastic differential equations. Later we will move to the partial differential equations since the option pricing problem arising in financial mathematics when asset is driven by a stochastic volatility process and assumed presence of transaction cost leads to solving non-linear partial dif- ferential equation. We will also present the complete process from deriving the desired partial differential equation to the proof of existence of a solution and also the numerical simulations. Using techniques form stochastic …

Granular Computing For Assessment Of Mild Traumatic Brain Injury, Melaku Ayenew Bogale

Open Access Theses & Dissertations

Mild traumatic brain injury (mTBI) is one of the most common neurological disorders. It is a serious public health problem in the United States. Although, penetrating (open) brain injuries that result in extended period of loss of consciousness (LOC) usually gets attention and well taken care of by the emergency departments, mild traumatic brain injury with no visible sign of damage, may be undetected or misdiagnosed. The clinical assessments and evaluations are mostly based on subjective cognitive and behavioral tests. Many people after suffering mTBI complain about decreased balance, coordination and stability even though the clinical evaluations show no sign …

Constrained Optimal Control For A Multi-Group Discrete Time Influenza Model, Paula Andrea Gonzalez Parra

Open Access Theses & Dissertations

During the last decades, mathematical epidemiological models have been used to understand the dynamics of infectious diseases and guide public health policy. In particular, several continuous models have been considered to study single in uenza outbreaks and the impact of dierent control policies. In this dissertation, a discrete time model is introduced in order to study optimal control strategies for in uenza transmission; since epidemiological data is collected on discrete units of time, a discrete formulation is more ecient. From a mathematical point of view, continuous time model are easier to analyze, however, the numerical solution of discrete-time models is …

High Tech High Touch: Lessons Learned From Project Haiti 2011, Yan Tang, Marc Compere, Yung Lun Wong, Jared Anthony Coleman, Matthew Charles Selkirk

Publications

In this paper, we will share our experiences and lessons learned from a design project for providing clean water to a Haitian orphanage (Project Haiti 2011). Supported by funds from a renewable energy company and the university president’s office, five engineering students and two faculty members from Embry-Riddle Aeronautical University successfully designed and installed a solar powered water purification system for an orphanage located in Chambellan, Haiti. This paper discusses the unique educational experiences gained from unusual design constraints, such as ambiguity of existing facilities due to limited communication, logistics of international construction at a remote village location, and cross-cultural …

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Stability Indices For Constrained Self-Adjoint Operators, Todd Kapitula, Keith Promislow

University Faculty Publications and Creative Works

A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called "energy spectrum", that is, the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified proof of the negative eigenvalue count for such constrained, self-adjoint operators, and extend the result to include an analysis of the location of the point spectra of the constrained operator relative to that of the unconstrained …