Dynamical Systems Commons^™

Mathematical Modeling For Dental Decay Prevention In Children And Adolescents, Mahdiyeh Soltaninejad

Dissertations

The high prevalence of dental caries among children and adolescents, especially those from lower socio-economic backgrounds, is a significant nationwide health concern. Early prevention, such as dental sealants and fluoride varnish (FV), is essential, but access to this care remains limited and disparate. In this research, a national dataset is utilized to assess sealants' reach and effectiveness in preventing tooth decay, particularly focusing on 2nd molars that emerge during early adolescence, a current gap in the knowledge base. FV is recommended to be delivered during medical well-child visits to children who are not seeing a dentist. Challenges and facilitators in …

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

A Causal Inference Approach For Spike Train Interactions, Zach Saccomano

Dissertations, Theses, and Capstone Projects

Since the 1960s, neuroscientists have worked on the problem of estimating synaptic properties, such as connectivity and strength, from simultaneously recorded spike trains. Recent years have seen renewed interest in the problem coinciding with rapid advances in experimental technologies, including an approximate exponential increase in the number of neurons that can be recorded in parallel and perturbation techniques such as optogenetics that can be used to calibrate and validate causal hypotheses about functional connectivity. This thesis presents a mathematical examination of synaptic inference from two perspectives: (1) using in vivo data and biophysical models, we ask in what cases the …

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

Theses and Dissertations (Comprehensive)

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

Convolution And Autoencoders Applied To Nonlinear Differential Equations, Noah Borquaye

Electronic Theses and Dissertations

Autoencoders, a type of artificial neural network, have gained recognition by researchers in various fields, especially machine learning due to their vast applications in data representations from inputs. Recently researchers have explored the possibility to extend the application of autoencoders to solve nonlinear differential equations. Algorithms and methods employed in an autoencoder framework include sparse identification of nonlinear dynamics (SINDy), dynamic mode decomposition (DMD), Koopman operator theory and singular value decomposition (SVD). These approaches use matrix multiplication to represent linear transformation. However, machine learning algorithms often use convolution to represent linear transformations. In our work, we modify these approaches to …

Complex Dimensions Of 100 Different Sierpinski Carpet Modifications, Gregory Parker Leathrum

Master's Theses

We used Dr. M. L. Lapidus's Fractal Zeta Functions to analyze the complex fractal dimensions of 100 different modifications of the Sierpinski Carpet fractal construction. We will showcase the theorems that made calculations easier, as well as Desmos tools that helped in classifying the different fractals and computing their complex dimensions. We will also showcase all 100 of the Sierpinski Carpet modifications and their complex dimensions.

Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost

All Dissertations

In confocal single-molecule FRET experiments, the joint distribution of FRET efficiency and donor lifetime distribution can reveal underlying molecular conformational dynamics via deviation from their theoretical Forster relationship. This shift is referred to as a dynamic shift. In this study, we investigate the influence of the free energy landscape in protein conformational dynamics on the dynamic shift by simulation of the associated continuum reaction coordinate Langevin dynamics, yielding a deeper understanding of the dynamic and structural information in the joint FRET efficiency and donor lifetime distribution. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based …

Wavelet Compression As An Observational Operator In Data Assimilation Systems For Sea Surface Temperature, Bradley J. Sciacca

University of New Orleans Theses and Dissertations

The ocean remains severely under-observed, in part due to its sheer size. Containing nearly billion of water with most of the subsurface being invisible because water is extremely difficult to penetrate using electromagnetic radiation, as is typically used by satellite measuring instruments. For this reason, most observations of the ocean have very low spatial-temporal coverage to get a broad capture of the ocean’s features. However, recent “dense but patchy” data have increased the availability of high-resolution – low spatial coverage observations. These novel data sets have motivated research into multi-scale data assimilation methods. Here, we demonstrate a new assimilation approach …

Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu

Doctoral Dissertations

In this thesis, we study the mathematical model of one-dimensional microscopic heat conduction of gas particles, applying both both analytical and numerical approaches. The macroscopic law of heat conduction is the renowned Fourier’s law J = −k∇T, where J is the local heat flux density, T(x, t) is the temperature gradient, and k is the thermal conductivity coefficient that characterizes the material’s ability to conduct heat. Though Fouriers’s law has been discovered since 1822, the thorough understanding of its microscopic mechanisms remains challenging [3] (2000). We assume that the microscopic model of heat conduction is a hard ball system. The …

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

On The Spectrum Of Quaquaversal Operators, Josiah Sugarman

Dissertations, Theses, and Capstone Projects

In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …

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.

Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, Vincent Paul Flynn

Dartmouth College Ph.D Dissertations

Recently, there has been an explosion of interest in re-imagining many-body quantum phenomena beyond equilibrium. One such effort has extended the symmetry-protected topological (SPT) phase classification of non-interacting fermions to driven and dissipative settings, uncovering novel topological phenomena that are not known to exist in equilibrium which may have wide-ranging applications in quantum science. Similar physics in non-interacting bosonic systems has remained elusive. Even at equilibrium, an "effective non-Hermiticity" intrinsic to bosonic Hamiltonians poses theoretical challenges. While this non-Hermiticity has been acknowledged, its implications have not been explored in-depth. Beyond this dynamical peculiarity, major roadblocks have arisen in the search …

Photonic Sensors Based On Integrated Ring Resonators, Jaime Da Silva

Mechanical Engineering Research Theses and Dissertations

This thesis investigates the application of integrated ring resonators to different sensing applications. The sensors proposed here rely on the principle of optical whispering gallery mode (WGM) resonance shifts of the resonators. Three distinct sensing applications are investigated to demonstrate the concept: a photonic seismometer, an evanescent field sensor, and a zero-drift Doppler velocimeter. These concepts can be helpful in developing lightweight, compact, and highly sensitive sensors. Successful implementation of these sensors could potentially address sensing requirements for both space and Earth-bound applications. The feasibility of this class of sensors is assessed for seismic, proximity, and vibrational measurements.

Machine Learning-Based Data And Model Driven Bayesian Uncertanity Quantification Of Inverse Problems For Suspended Non-Structural System, Zhiyuan Qin

All Dissertations

Inverse problems involve extracting the internal structure of a physical system from noisy measurement data. In many fields, the Bayesian inference is used to address the ill-conditioned nature of the inverse problem by incorporating prior information through an initial distribution. In the nonparametric Bayesian framework, surrogate models such as Gaussian Processes or Deep Neural Networks are used as flexible and effective probabilistic modeling tools to overcome the high-dimensional curse and reduce computational costs. In practical systems and computer models, uncertainties can be addressed through parameter calibration, sensitivity analysis, and uncertainty quantification, leading to improved reliability and robustness of decision and …

A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman

HMC Senior Theses

We can think of a pixel as a particle in three dimensional space, where its x, y and z coordinates correspond to its level of red, green, and blue, respectively. Just as a particle’s motion is guided by physical rules like gravity, we can construct rules to guide a pixel’s motion through color space. We can develop striking visuals by applying these rules, called dynamical systems, onto images using animation engines. This project explores a number of these systems while exposing the underlying algebraic structure of color space. We also build and demonstrate a Visual DJ circuit board for …

Dynamical Aspects In (4+1)-Body Problems, Ryan Gauthier

Theses and Dissertations (Comprehensive)

The n-body problem models a system of n-point masses that attract each other via some binary interaction. The (n + 1)-body problem assumes that one of the masses is located at the origin of the coordinate system. For example, an (n+1)-body problem is an ideal model for Saturn, seen as the central mass, and one of its outer rings. A relative equilibrium (RE) is a special solution of the (n+1)-body problem where the non-central bodies rotate rigidly about the centre of mass. In rotating coordinates, these solutions become equilibria.

In this thesis we study dynamical aspects of planar (4 + …

On The Spatial Modelling Of Biological Invasions, Tedi Ramaj

Electronic Thesis and Dissertation Repository

We investigate problems of biological spatial invasion through the use of spatial modelling. We begin by examining the spread of an invasive weed plant species through a forest by developing a system of partial differential equations (PDEs) involving an invasive weed and a competing native plant species. We find that extinction of the native plant species may be achieved by increasing the carrying capacity of the forest as well as the competition coefficient between the species. We also find that the boundary conditions exert long-term control on the biomass of the invasive weed and hence should be considered when implementing …

Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy

Doctoral Dissertations

Many ecological systems exhibit noisy period-2 oscillations and, when they are spatially extended, they undergo phase transition from synchrony to incoherence in the Ising universality class. Period-2 cycles have two possible phases of oscillations and can be represented as two states in the bistable systems. Understanding the dynamics of ecological systems by representing their oscillations as bistable states and developing dynamical models using the tools from statistical physics to predict their future states is the focus of this thesis. As the ecological oscillators with two-cycle behavior undergo phase transitions in the Ising universality class, many features of synchrony and equilibrium …

Machine Learning To Predict Warhead Fragmentation In-Flight Behavior From Static Data, Katharine Larsen

Doctoral Dissertations and Master's Theses

Accurate characterization of fragment fly-out properties from high-speed warhead detonations is essential for estimation of collateral damage and lethality for a given weapon. Real warhead dynamic detonation tests are rare, costly, and often unrealizable with current technology, leaving fragmentation experiments limited to static arena tests and numerical simulations. Stereoscopic imaging techniques can now provide static arena tests with time-dependent tracks of individual fragments, each with characteristics such as fragment IDs and their respective position vector. Simulation methods can account for the dynamic case but can exclude relevant dynamics experienced in real-life warhead detonations. This research leverages machine learning methodologies to …

Dynamical Systems And Matching Symmetry In Beta-Expansions, Karl Zieber

Master's Theses

Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.

In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when T_β,⍺ generates a subshift of finite type. We prove the set of ⍺ in …

The Butterfly Effect Of Fractals, Cody Watkins

Honors College Theses

This thesis applies concepts in fractal geometry to the relatively new field of mathematics known as chaos theory, with emphasis on the underlying foundation of the field: the butterfly effect. We begin by reviewing concepts useful for an introduction to chaos theory by defining terms such as fractals, transformations, affine transformations, and contraction mappings, as well as proving and demonstrating the contraction mapping theorem. We also show that each fractal produced by the contraction mapping theorem is unique in its fractal dimension, another term we define. We then show and demonstrate iterated function systems and take a closer look at …

Finite Subdivision Rules For Matings Of Quadratic Thurston Maps With Few Postcritical Points, Jeremiah Zonio

Undergraduate Theses

A finite subdivision rule is set of instructions for repeatedly subdividing a partitioning of a given space. This turns out to be incredibly useful when attempting to describe a process known as polynomial mating. Polynomial mating is a way of gluing together two spaces which two polynomials may act upon such that the glued borders of each space respects the dynamics described by each polynomial. For matings of Misiurewicz polynomials, the spaces we are gluing together are 1-dimensional and are thus all border. This poses a conceptual difficulty which this paper attempts to resolve by using finite subdivison rules to …

An Integrated Computational Pipeline To Construct Patient-Specific Cancer Models, Daniel Plaugher

Theses and Dissertations--Mathematics

Precision oncology largely involves tumor genomics to guide therapy protocols. Yet, it is well known that many commonly mutated genes cannot be easily targeted. Even when genes can be targeted, resistance to therapy is quite common. A rising field with promising results is that of mathematical biology, where in silico models are often used for the discovery of general principles and novel hypotheses that can guide the development of new treatments. A major goal is that eventually in silico models will accurately predict clinically relevant endpoints and find optimal control interventions to stop (or reverse) disease progression. Thus, it is …

Energy As A Limiting Factor In Neuronal Seizure Control: A Mathematical Model, Sophia E. Epstein

CMC Senior Theses

The majority of seizures are self-limiting. Within a few minutes, the observed neuronal synchrony and deviant dynamics of a tonic-clonic or generalized seizure often terminate. However, a small epilesia partialis continua can occur for years. The mechanisms that regulate subcortical activity of neuronal firing and seizure control are poorly understood. Published studies, however, through PET scans, ketogenic treatments, and in vivo mouse experiments, observe hypermetabolism followed by metabolic suppression. These observations indicate that energy can play a key role in mediating seizure dynamics. In this research, I seek to explore this hypothesis and propose a mathematical framework to model how …

Role Of Inhibition And Spiking Variability In Ortho- And Retronasal Olfactory Processing, Michelle F. Craft

Theses and Dissertations

Odor perception is the impetus for important animal behaviors, most pertinently for feeding, but also for mating and communication. There are two predominate modes of odor processing: odors pass through the front of nose (ortho) while inhaling and sniffing, or through the rear (retro) during exhalation and while eating and drinking. Despite the importance of olfaction for an animal’s well-being and specifically that ortho and retro naturally occur, it is unknown whether the modality (ortho versus retro) is transmitted to cortical brain regions, which could significantly instruct how odors are processed. Prior imaging studies show different …

The Kepler Problem On Complex And Pseudo-Riemannian Manifolds, Michael R. Astwood

Theses and Dissertations (Comprehensive)

The motion of objects in the sky has captured the attention of scientists and mathematicians since classical times. The problem of determining their motion has been dubbed the Kepler problem, and has since been generalized into an abstract problem of dynamical systems. In particular, the question of whether a classical system produces closed and bounded orbits is of importance even to modern mathematical physics, since these systems can often be analysed by hand. The aforementioned question was originally studied by Bertrand in the context of celestial mechanics, and is therefore referred to as the Bertrand problem. We investigate the qualitative …

Structure-Dependent Characterizations Of Multistationarity In Mass-Action Reaction Networks, Galyna Voitiuk

Graduate Theses, Dissertations, and Problem Reports

This project explores a topic in Chemical Reaction Network Theory. We analyze networks with one dimensional stoichiometric subspace using mass-action kinetics. For these types of networks, we study how the capacity for multiple positive equilibria and multiple positive nondegenerate equilibria can be determined using Euclidian embedded graphs. Our work adds to the catalog of the class of reaction networks with one-dimensional stoichiometric subspace answering in the affirmative a conjecture posed by Joshi and Shiu: Conjecture 0.1 (Question 6.1 [26]). A reaction network with one-dimensional stoichiometric subspace and more than one source complex has the capacity for multistationarity if and only …

Dynamics Of Mutualism In A Two Prey, One Predator System With Variable Carrying Capacity, Randy Huy Lee

UNF Graduate Theses and Dissertations

We considered the livelihood of two prey species in the presence of a predator species. To understand this phenomenon, we developed and analyzed two mathematical models considering indirect and direct mutualism of two prey species and the influence of one predator species. Both types of mutualism are represented by an increase in the preys' carrying capacities based on direct and indirect interactions between the prey. Because of mutualism, as the death rate parameter of the predator species goes through some critical value, the model shows transcritical bifurcation. Additionally, in the direct mutualism model, as the death rate parameter decreases to …

A Connectivity Framework To Explore The Role Of Anthropogenic Activity And Climate On The Propagation Of Water And Sediment At The Catchment Scale, Christos Giannopoulos

Doctoral Dissertations

Anthropogenic disturbance in intensively managed landscapes (IMLs) has dramatically altered critical zone processes, resulting in fundamental changes in material fluxes. Mitigating the negative effects of anthropogenic disturbance and making informed decisions for optimal placement and assessment of best management practices (BMPs) requires fundamental understanding of how different practices affect the connectivity or lack thereof of governing transport processes and resulting material fluxes across different landscape compartments within the hillslope-channel continuum of IMLs. However, there are no models operating at the event timescale that can accurately predict material flux transport from the hillslope to the catchment scale capturing the spatial and …