Applied Mathematics Commons^™

(R1987) Hermite Wavelets Method For System Of Linear Differential Equations, Inderdeep Singh, Manbir Kaur

Applications and Applied Mathematics: An International Journal (AAM)

In this research paper, we present an accurate technique for solving the system of linear differential equations. Such equations often arise as a result of modeling in many systems and applications of engineering and science. The proposed scheme is based on Hermite wavelets basis functions and operational matrices of integration. The demonstrated scheme is simple as it converts the problem into algebraic matrix equation. To validate the applicability and efficacy of the developed scheme, some illustrative examples are also considered. The results so obtained with the help of the present proposed numerical technique by using Hermite wavelets are observed to …

(R1975) Map/Ph(1), Ph(2)/2 Queue With Multiple Vacation, Optional Service, Consultations And Interruptions, G. Ayyappan, S. Sankeetha

Applications and Applied Mathematics: An International Journal (AAM)

Two types of services are explored in this paper: regular server and main server, both of which provide both regular and optional services. Customers arrive using the Markovian Arrival Process (MAP), and service time is allocated based on phase type. The regular server uses the main server as a resource. Customers’ service at the primary server is disrupted as a result. When the queue size is empty, the main server can take several vacations. This system has been represented as a QBD Process that investigates steady state with the use of matrix analytic techniques, employing finite-dimensional block matrices. Our model’s …

On Colorings And Orientations Of Signed Graphs, Daniel Slilaty

Mathematics and Statistics Faculty Publications

A classical theorem independently due to Gallai and Roy states that a graph G has a proper k-coloring if and only if G has an orientation without coherent paths of length k. An analogue of this result for signed graphs is proved in this article.

Detecting Coherent Transport Structures In Ocean Surface Flows, Leah Hoogstra

Master's Theses

Ocean surface transport plays a critical role in marine ecosystems, influencing the complex spatiotemporal patterns of both marine species and pollutants. The theory of Lagrangian coherent structures (LCSs) aims to identify fundamental patterns within time-dependent, nonlinear fluid flows. LCSs are material surfaces that act as dividing lines which fluid does not cross for a relevant period of time. LCS theory is still under active development, and there are multiple proposed ways to mathematically determine an LCS. Each proposed mathematical definition aims to capture the same physical properties, and some capture those properties more successfully and consistently than others. Here we …

Analyzing Tortuosity In Patterns Formed By Colonies Of Embryonic Stem Cells Using Topological Data Analysis, Jackie Driscoll

Master's Theses

Pluripotent stem cells have been observed to segregate into Turing-like patterns during the early stages of Dox-inducible hiPSC differentiation. In this thesis, we de- velop a tool to quantify the tortuosity in the patterns formed by colonies of pluripo- tent stem cells using methods from topological data analysis. We use clustering techniques and the mapper algorithm to create simplicial complexes representing samples of cells and detail a method of evaluating the tortuosity of these complexes. We use the resulting persistence landscapes and their associated norms to evaluate experimental data and simulated data from an agent based model. This thesis finds …

Failure Distributions For Parallel Dependent Identical Weibull Components, Gina S. Sigler

Theses and Dissertations

For a parallel system, when one component fails, the failure distribution of the remaining components will have an increased failure rate. This dissertation takes a novel approach to finding the associated failure distribution of the full system using ordinal statistic distributions for correlated Weibull components, allowing for unknown correlations between the dependent components. A Taylor series approximation is presented for the two component; system failure time distributions are also derived for two failures in a two component system, two failures in an n component system, three failures in a three component system, and k failures in an n component system. …

Gradient-Based Trade-Off Design For Engineering Applications, Lena A. Royster, Gene Hou

Mechanical & Aerospace Engineering Faculty Publications

The goal of the trade-off design method presented in this study is to achieve newly targeted performance requirements by modifying the current values of the design variables. The trade-off design problem is formulated in the framework of Sequential Quadratic Programming. The method is computationally efficient as it is gradient-based, which, however, requires the performance functions to be differentiable. A new equation to calculate the scale factor to control the size of the design variables is introduced in this study, which can ensure the new design achieves the targeted performance objective. Three formal approaches are developed in this study for trade-off …

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

(R2032) Modeling The Effect Of Sanitation Effort On The Spread Of Carrier-Dependent Infectious Diseases Due To Environmental Degradation, Ram Naresh, Sandhya Rani Verma, J. B. Shukla, Manju Agarwal

Applications and Applied Mathematics: An International Journal (AAM)

In this present study, an SIS model is proposed and analyzed to study the effect of sanitation effort in controlling the spread of carrier-dependent infectious disease in a human habitat due to environmental degradation. The dynamics of the model consist of six dependent variables, the susceptible population density, infective population density, carrier population density, cumulative density of environmental degradation and the density of sanitation effort applied on carrier population and degraded environment. In the modeling process, the carrier population density and sanitation effort are modeled logistically and the degradation of the environment is assumed to be directly proportional to the …

(R2030) Generalized Quasilinearization Method For A Initial Value Problem On Time Scales, Şahap Çetin, Yalçın Yılmaz, Coşkun Yakar

Applications and Applied Mathematics: An International Journal (AAM)

We have investigated that the generalized quasilinearization method under some convenient conditions for nonlinear initial value problem (IVP) of dynamic equation on time scale constructed by monotone sequences of function by using comparison theorem that is the solution of linear IVP of dynamic equation on time scale which converge uniformly and monotonically to the unique solution of the original problem, and the convergence is quadratic.

(R2028) A Brief Note On Space Time Fractional Order Thermoelastic Response In A Layer, Navneet Lamba, Jyoti Verma, Kishor Deshmukh

Applications and Applied Mathematics: An International Journal (AAM)

In this study, a one-dimensional layer of a solid is used to investigate the exact analytical solution of the heat conduction equation with space-time fractional order derivatives and to analyze its associated thermoelastic response using a quasi-static approach. The assumed thermoelastic problem was subjected to certain initial and boundary conditions at the initial and final ends of the layer. The memory effects and long-range interaction were discussed with the help of the Caputo-type fractional-order derivative and finite Riesz fractional derivative. Laplace transform and Fourier transform techniques for spatial coordinates were used to investigate the solution of the temperature distribution and …

Distributed Control Of Servicing Satellite Fleet Using Horizon Simulation Framework, Scott Plantenga

Master's Theses

On-orbit satellite servicing is critical to maximizing space utilization and sustainability and is of growing interest for commercial, civil, and defense applications. Reliance on astronauts or anchored robotic arms for the servicing of next-generation large, complex space structures operating beyond Low Earth Orbit is impractical. Substantial literature has investigated the mission design and analysis of robotic servicing missions that utilize a single servicing satellite to approach and service a single target satellite. This motivates the present research to investigate a fleet of servicing satellites performing several operations for a large, central space structure.

This research leverages a distributed control approach, …

Hydrodynamic And Physicochemical Interactions Between An Active Janus Particle And An Inactive Particle, Jessica S. Rosenberg

Dissertations, Theses, and Capstone Projects

Active matter is an area of soft matter science in which units consume energy and turn it into autonomous motion. Groups of these units – whether flocks of birds, bacterial colonies, or even collections of synthetically-made active particles – may exhibit complex behavior on large scales. While the large-scale picture is of great importance, so is the microscopic scale. Studying the individual particles that make up active matter will allow us to understand how they move, and whether and under what circumstances their activity can be controlled.

Here we delve into the world of active matter by studying colloidal-sized (100 …

Analyzing Robustness Of An Agent Based Model On Action Potentials In Cardiac Tissue, Marion Jon Zollinger Lara

Master's Theses

An agent based model (ABM) is a computational model with ``agents'' that interact with each other in an ``environment.'' This paper analyzes a particular ABM simulating individual ions in cardiac tissue, with the goal of modelling the strength and consistency of the electrical signals needed for a healthy heartbeat. We build several frameworks based on work by M. A. Yereniuk and S. D. Olson to demonstrate robustness of the original model. We conclude a moderate level of robustness using those frameworks, through a combination of proofs and empirical evidence.

Learning The Game: Implementations Of Convolutional Networks In Automated Strategy Identification, Cameron Klig

Master's Theses

Games can be used to represent a wide variety of real world problems, giving rise to many applications of game theory. Various computational methods have been proposed for identifying game strategies, including optimized tree search algorithms, game-specific heuristics, and artificial intelligence. In the last decade, systems like AlphaGo and AlphaZero have significantly exceeded the performance of the best human players in Chess, Go, and other games. The most effective game engines to date employ convolutional neural networks (CNNs) to evaluate game boards, extract features, and predict the optimal next move. These engines are trained on billions of simulated games, wherein …

Continuum Modeling Of Active Nematics Via Data-Driven Equation Discovery, Connor Robertson

Dissertations

Data-driven modeling seeks to extract a parsimonious model for a physical system directly from measurement data. One of the most interpretable of these methods is Sparse Identification of Nonlinear Dynamics (SINDy), which selects a relatively sparse linear combination of model terms from a large set of (possibly nonlinear) candidates via optimization. This technique has shown promise for synthetic data generated by numerical simulations but the application of the techniques to real data is less developed. This dissertation applies SINDy to video data from a bio-inspired system of mictrotubule-motor protein assemblies, an example of nonequilibrium dynamics that has posed a significant …

Deep Hybrid Modeling Of Neuronal Dynamics Using Generative Adversarial Networks, Soheil Saghafi

Dissertations

Mechanistic modeling and machine learning methods are powerful techniques for approximating biological systems and making accurate predictions from data. However, when used in isolation these approaches suffer from distinct shortcomings: model and parameter uncertainty limit mechanistic modeling, whereas machine learning methods disregard the underlying biophysical mechanisms. This dissertation constructs Deep Hybrid Models that address these shortcomings by combining deep learning with mechanistic modeling. In particular, this dissertation uses Generative Adversarial Networks (GANs) to provide an inverse mapping of data to mechanistic models and identifies the distributions of mechanistic model parameters coherent to the data.

Chapter 1 provides background information on …

Domain Decomposition Methods For Linear And Non-Linear Elliptic Problems, Tadanaga Takahashi

Dissertations

The primary purpose of this dissertation is to expand upon the circle of domain decomposition methods (DDM) which are algorithms that reformulate a boundary value problem in terms of multiple localized problems on subdomains. The first project involves expanding upon DDMs in a relatively mature field: the Helmholtz equation for wave scattering applications. The proposed method is an adaptation of a continuous cross-point Finite Element Non-overlapping DDM algorithm. The usual unbounded computational domain is truncated and then the near-field wave pattern is solved with a parallelized finite element method. Several improvements over the standard transmission operator are discussed in this …

Bright Light Therapy And Depression: Assessing Suitability Using Entrainment Maps, Charles A. Mainwaring

Theses

Bright Light Therapy has been shown to be efficacious to mood disorders including Major Depression. Researchers use the Jewett-Forger-Kronauer model of the circadian rhythm with the Unified Model of melatonin including a mathematical term implementing feedback from the melatonin system into the circadian system to quantify the effects of bright light. Early investigations into intrinsic period, light sensitivity, and the circadian pacemaker's sensitivity to blood melatonin concentration may be indicators of subsets of patients with long intrinsic periods exhibiting symptoms of depression.

An Lp-Based Characterization Of Solvable Qap Instances With Chess-Board And Graded Structures, Lucas Waddell, Jerry Phillips, Tianzhu Liu, Swarup Dhar

Faculty Journal Articles

The quadratic assignment problem (QAP) is perhaps the most widely studied nonlinear combinatorial optimization problem. It has many applications in various fields, yet has proven to be extremely difficult to solve. This difficulty has motivated researchers to identify special objective function structures that permit an optimal solution to be found efficiently. Previous work has shown that certain such structures can be explained in terms of a mixed 0-1 linear reformulation of the QAP known as the level-1 reformulation-linearization-technique (RLT) form. Specifically, the objective function structures were shown to ensure that a binary optimal extreme point solution exists to the continuous …