Applied Mathematics Commons^™

Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov

Publications and Research

We consider a complex representation of an arbitrary planar polygon P centered at the origin. Let P(1) be the normalized polygon obtained from P by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that P(1) is a normalized average of P. We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a complete and …

Creating Art Patterns With Math And Code, Boyan Kostadinov

Publications and Research

The goal of this talk is to showcase some visualization projects that we developed for a 3-day Code in R summer program, designed to inspire the creative side of our STEM students by engaging them with computational projects that we developed with the purpose of mixing calculus level math and code to create complex geometric patterns. One of the goals of this program was to attract more minority and female students into applied math and computer science majors.

The projects are designed to be implemented using the high-level, open-source and free computational environment R, a popular software in industry for …

Multi-Objective Optimization Under Uncertainty Using The Hyper-Volume Expected Improvement, Martin Figura, Piyush Pandita, Rohit K. Tripathy, Ilias Bilionis

The Summer Undergraduate Research Fellowship (SURF) Symposium

The design of real engineering systems requires the optimization of multiple quantities of interest. In the electric motor design, one wants to maximize the average torque and minimize the torque variation. A study has shown that these attributes vary for different geometries of the rotor teeth. However, simulations of a large number of designs cannot be performed due to their high cost. In many problems, design optimization of multi-objective functions is a very challenging task due to the difficulty to evaluate the expectation of the objectives. Current multi-objective optimization (MOO) techniques, e.g., evolutionary algorithms cannot solve such problems because they …

A Fast Model For The Simulation Of External Gear Pumps, Zechao Lu, Xinran Zhao, Andrea Vacca

The Summer Undergraduate Research Fellowship (SURF) Symposium

External gear pump is an important category of positive displacement fluid machines used to perform the mechanical–hydraulic energy conversions in many fluid power applications. An efficient numerical simulation program is needed to simulate the system in order to provide a direction for design purpose. The model consists of a lumped parameter fluid dynamic model and a model that simulates the radial micro-motions of the gear’s axes of rotation. The system consists of a set of ordinary differential equations related to the conservation on mass of the internal control volumes of the pump, which are given by the tooth space volumes …

Design Optimization Of A Stochastic Multi-Objective Problem: Gaussian Process Regressions For Objective Surrogates, Juan Sebastian Martinez, Piyush Pandita, Rohit K. Tripathy, Ilias Bilionis

The Summer Undergraduate Research Fellowship (SURF) Symposium

Multi-objective optimization (MOO) problems arise frequently in science and engineering situations. In an optimization problem, we want to find the set of input parameters that generate the set of optimal outputs, mathematically known as the Pareto frontier (PF). Solving the MOO problem is a challenge since expensive experiments can be performed only a constrained number of times and there is a limited set of data to work with, e.g. a roll-to-roll microwave plasma chemical vapor deposition (MPCVD) reactor for manufacturing high quality graphene. State-of-the-art techniques, e.g. evolutionary algorithms; particle swarm optimization, require a large amount of observations and do not …

Computing The Optimal Path In Stochastic Dynamical Systems, Martha Bauver, Eric Forgoston, Lora Billings

Department of Mathematics Faculty Scholarship and Creative Works

In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high- dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a …

Citizen Science Sensor Development - Smap | Soil Moisture Active Passive, Hagop Hovhannesian

STAR Program Research Presentations

“Detailed monitoring of soil moisture provides a view of how our whole Earth system works.”

The Soil Moisture Active Passive (SMAP) satellite mission was launched in January 2015; its main purpose is to acquire global measurements of soil moisture. SMAP partnered with the GLOBE program (Global Learning and Observations to Benefit the Environment), which is an international program where students collect environmental variables in a scientifically methodical way. SMAP readings and maps have various uses in various fields, which include monitoring drought, predicting floods, assisting in crop productivity, and linking water, energy and carbon cycles. The goal of this project …

Mathematical Model Of Vaccine Noncompliance, Alex Capaldi, Jordan Bauer

Mathematics and Statistics Faculty Publications

Vaccine scares can prevent individuals from complying with a vaccination program. When compliance is high, the critical vaccination proportion is close to being met, and herd immunity occurs, bringing the disease incidence to extremely low levels. Thus, the risk to vaccinate may seem greater than the risk of contracting the disease, inciting vaccine noncompliance. A previous behavior-incidence ordinary differential equation model shows both social learning and feedback contributing to changes in vaccinating behavior, where social learning is the perceived risk of vaccinating and feedback repre- sents new cases of the disease. In our study, we compared several candidate models to …

How To Determine The Stiffness Of The Pavement's Upper Layer (Base) Based On The Overall Stiffness And The Stiffness Of The Lower Layer (Subgrade), Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In road construction, it is important to estimate difficult-measure stiffness of the pavement's upper layer based the easier-to-measure overall stiffness and the stiffness of the lower layer. In situations when the overall stiffness is not yet sufficient, it is also important to estimate how much more we need to add to the upper layer to reach the desired overall stiffness. In this paper, for the cases when a linear approximation is sufficient, we provide analytical formulas for the desired estimations.

A Two-Strain Tb Model With Multiple Latent Stages, Azizeh Jabbari, Carlos Castillo-Chavez, Fereshteh Nazari, Baojun Song, Hossein Kheiri

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving "arbitrary" distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence …

Local Polynomial Chaos Expansion Method For High Dimensional Stochastic Differential Equations, Yi Chen

Open Access Dissertations

Polynomial chaos expansion is a widely adopted method to determine evolution of uncertainty in dynamical system with probabilistic uncertainties in parameters. In particular, we focus on linear stochastic problems with high dimensional random inputs. Most of the existing methods enjoyed the efficiency brought by PC expansion compared to sampling-based Monte Carlo experiments, but still suffered from relatively high simulation cost when facing high dimensional random inputs. We propose a localized polynomial chaos expansion method that employs a domain decomposition technique to approximate the stochastic solution locally. In a relatively lower dimensional random space, we are able to solve subdomain problems …

Parametric Approaches To Fractional Programs: Analytical And Empirical Study, Chong Hyun Park

Open Access Dissertations

Fractional programming is used to model problems where the objective function is a ratio of functions. A parametric modeling approach provides effective technique for obtaining optimal solutions of these fractional programming problems. Although many heuristic algorithms have been proposed and assessed relative to each other, there are limited theoretical studies on the number of steps to obtain the solution. In this dissertation, I focus on the linear fractional combinatorial optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are binary that model certain combinatorial structures. Two parametric …

Implementing And Testing A Panel-Based Method For Modeling Acoustic Scattering From Cfd Input, S. Hales Swift

Open Access Dissertations

Exposure of sailors to high levels of noise in the aircraft carrier deck environment is a problem that has serious human and economic consequences. A variety of approaches to quieting exhausting jets from high-performance aircraft are undergoing development. However, testing of noise abatement solutions at full-scale may be prohibitively costly when many possible nozzle treatments are under consideration. A relatively efficient and accurate means of predicting the noise levels resulting from engine-quieting technologies at personnel locations is needed. This is complicated by the need to model both the direct and the scattered sound field in order to determine the resultant …

Conformally Invariant Operators In Higher Spin Spaces, Chao Ding

Graduate Theses and Dissertations

In this dissertation, we complete the work of constructing arbitrary order conformally invariant operators in higher spin spaces, where functions take values in irreducible representations of Spin groups. We provide explicit formulas for them.

We first construct the Dirac operator and Rarita-Schwinger operator as Stein Weiss type operators. This motivates us to consider representation theory in higher spin spaces. We provide corrections to the proof of conformal invariance of the Rarita-Schwinger operator in [15]. With the techniques used in the second order case [7, 18], we construct conformally invariant differential operators of arbitrary order with the target space being degree-1 …

Delay-Independent Stability Analysis Of Linear Time-Delay Systems Based On Frequency, Xianwei Li, Huijun Gao, Keqin Gu

SIUE Faculty Research, Scholarship, and Creative Activity

This paper studies strong delay-independent stability of linear time-invariant systems. It is known that delay-independent stability of time-delay systems is equivalent to some frequency-dependent linear matrix inequalities. To reduce or eliminate conservatism of stability criteria, the frequency domain is discretized into several sub-intervals, and piecewise constant Lyapunov matrices are employed to analyze the frequency-dependent stability condition. Applying the generalized Kalman–Yakubovich–Popov lemma, new necessary and sufficient criteria are then obtained for strong delay-independent stability of systems with a single delay. The effectiveness of the proposed method is illustrated by a numerical example.

On Approximately Controlled Systems, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

No abstract provided.

Krylov Subspace Spectral Method With Multigrid For A Time-Dependent, Variable-Coefficient Partial Differential Equation, Haley Renee Dozier

Master's Theses

Krylov Subspace Spectral (KSS) methods are traditionally used to solve time-dependent, variable-coefficient PDEs. They are high-order accurate, component-wise methods that are efficient with variable input sizes.

This thesis will demonstrate how one can make KSS methods even more efficient by using a Multigrid-like approach for low-frequency components. The essential ingredients of Multigrid, such as restriction, residual correction, and prolongation, are adapted to the timedependent case. Then a comparison of KSS, KSS with Multigrid, KSS-EPI and standard Krylov projection methods will be demonstrated.

On Compactness And Closed-Rangeness Of Composition Operators, Arnab Dutta

Graduate Theses and Dissertations

Let $\phi$ be an analytic self-map of the unit disk $\mathbb{D}:=\{z:\lvert z\rvert

Mathematical Hybrid Models For Image Segmentation., Carlos M. Paniagua Mejia

Electronic Theses and Dissertations

Two hybrid image segmentation models that are able to process a wide variety of images are proposed. The models take advantage of global (region) and local (edge) data of the image to be segmented. The first one is a region-based PDE model that incorporates a combination of global and local statistics. The influence of each statistic is controlled using weights obtained via an asymptotically stable exponential function. Through incorporation of edge information, the second model extends the capabilities of a strictly region-based variational formulation, making it able to process more general images. Several examples are provided showing the improvements of …

Newsvendor Models With Monte Carlo Sampling, Ijeoma W. Ekwegh

Electronic Theses and Dissertations

Newsvendor Models with Monte Carlo Sampling by Ijeoma Winifred Ekwegh The newsvendor model is used in solving inventory problems in which demand is random. In this thesis, we will focus on a method of using Monte Carlo sampling to estimate the order quantity that will either maximizes revenue or minimizes cost given that demand is uncertain. Given data, the Monte Carlo approach will be used in sampling data over scenarios and also estimating the probability density function. A bootstrapping process yields an empirical distribution for the order quantity that will maximize the expected profit. Finally, this method will be used …

An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

Electronic Theses and Dissertations

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R³, but also to the general case of finding minimal functionals on hypersurfaces in Rⁿ associated with an arbitrary metric.

Numerical Computing With Functions On The Sphere And Disk, Heather Denise Wilber

Boise State University Theses and Dissertations

A new low rank approximation method for computing with functions in polar and spherical geometries is developed. By synthesizing a classic procedure known as the double Fourier sphere (DFS) method with a structure-preserving variant of Gaussian elimination, approximants to functions on the sphere and disk can be constructed that (1) preserve the bi-periodicity of the sphere, (2) are smooth over the poles of the sphere (and origin of the disk), (3) allow for the use of FFT-based algorithms, and (4) are near-optimal in their underlying discretizations. This method is used to develop a suite of fast, scalable algorithms that exploit …

Optimization Of Takeoffs On Unbalanced Fields Using Takeoff Performance Tool, Nihad E. Daidzic

International Journal of Aviation, Aeronautics, and Aerospace

Unbalanced field length exists when ASDA and TODA are not equal. Airport authority may add less expensive substitutes to runway full-strength pavement in the form of stopways and/or clearways to basic TORA to increase operational takeoff weights. Here developed Takeoff Performance Tool is a physics-based total-energy model used to simulate FAR/CS 25 regulated airplane takeoffs. Any aircraft, runway, and environmental conditions can be simulated, while complying with the applicable regulations and maximizing performance takeoff weights. The mathematical model was translated into Matlab, Fortran 95/2003/2008, Basic, and MS Excel computer codes. All existing FAR/CS 25 takeoff regulations are implemented. Average forces …

Theorems On Boundedness Of Solutions To Stochastic Delay Differential Equations, Youssef Raffoul, Dan Ren

Mathematics Faculty Publications

In this report, we provide general theorems about boundedness or bounded in probability of solutions to nonlinear delay stochastic differential systems. Our analysis is based on the successful construction of suitable Lyapunov functionals. We offer several examples as application of our theorems.

Optimal Control Analysis Of Ebola Disease With Control Strategies Of Quarantine And Vaccination, Muhammad Dure Ahmad, Muhammad Usman, Adnan Khan, Mudassar Imran

Mathematics Faculty Publications

The 2014 Ebola epidemic is the largest in history, affecting multiple countries in West Africa. Some isolated cases were also observed in other regions of the world.

Santa Clara Magazine, Volume 57 Number 3, Summer 2016, Santa Clara University

Santa Clara Magazine

26 - CAN’T THREAD A MOVING NEEDLE To tackle sexual assault on college campuses, a playwriting project comes to the screen. By Danae Stahlnecker ’15.

28 - MISSION CRITICAL When three students fell ill from meningitis-causing bacteria—which can be fatal—it meant the clock was ticking. And to get through this, it would take everybody’s help. By Harold Gutmann.

36 - “WHERE ARE THEY TAKING US?” A journal from the front lines of the Syrian refugee crisis in Greece. By Colleen Sinsky ’10.

40 - NO STRANGERS HERE Refugees, home, and work by Ameera Naguib ’16 from Jordan to Silicon Valley. …

Thermal Analysis In A Triple-Layered Skin Structure With Embedded Vasculature, Tumor, And Gold Nanoshells, Casey O. Orndorff

Doctoral Dissertations

In hyperthermia skin cancer treatment, the objective is to control laser heating of the tumor (target temperatures of 42-46 °C) so that the temperatures of the normal tissue surrounding the tumor remains low enough not to damage the normal tissue. However, obtaining accurate temperature distributions in living tissue related to hyperthermia skin cancer treatment without using an intruding sensor is a challenge. The objective of this dissertation research is to develop a mathematical model that can accurately predict the temperature distribution in the tumor region and surrounding normal tissue induced by laser irradiation. The model is based on a modified …

Computational Micro-Flow With Spectral Element Method And High Reynolds Number Flow With Discontinuous Galerkin Finite Element Method, Haibo Zhang

Doctoral Dissertations

In this dissertation, two numerical methods with high order accuracy, Spectral Element Method (SEM) and Discontinuous Galerkin Finite Element Method (DG-FEM), are chosen to solve problems in Computational Fluid Dynamics (CFD). The merits of these two methods will be discussed and utilized in different kinds of CFD problems. The simulations of the micro-flow systems with complex geometries and physical applications will be presented by SEM. Moreover, the numerical solutions for the Hyperbolic Flow will be obtained by DG-FEM. By solving problems with these two methods, the differences between them will be discussed as well.

Compressible Navier-Stokes equations with Electro-osmosis body …

An Interactive Likelihood For The Multi-Bernoulli Filter, Anthony B. Hoak

Master's Theses (2009 -)

In this thesis, a simple yet effective technique is presented for increasing the accuracy of multi-target tracking algorithms with a focus on sequential Monte-Carlo implementations of random finite set-based approaches. This technique, referred to throughout this work as an interactive likelihood, exploits the spatial information that exists in any given measurement, reducing the need for data association and allowing for more target interaction thereby increasing overall tracking accuracy. The interactive likelihood is constructed entirely within the random finite set framework and is integrated with a multi-Bernoulli filter. In addition, a state-of-the-art deep neural network for pedestrian detection is combined in …

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

Mathematics Faculty Publications

A basic kinetic model that incorporates a coupled dynamics of the carbon atoms and dimers ona copper surface is used to compute growth of a single-layer graphene island. The speed of theisland's edge advancement on Cu[111] and Cu[100] surfaces is computed as a function of the growthtemperature and pressure. Spatially resolved concentration pro les of the atoms and dimers aredetermined, and the contributions provided by these species to the growth speed are discussed.Island growth in the conditions of a thermal cycling is studied.