Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, 2024 Portland State University

#### 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, 2024 The Graduate Center, City University of New York

#### 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, 2024 Wilfrid Laurier University

#### 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 …

Reducing Food Scarcity: The Benefits Of Urban Farming, 2023 Brigham Young University

#### Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia

*Journal of Nonprofit Innovation*

Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value, eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.

Imagine Doris, who is …

Convolution And Autoencoders Applied To Nonlinear Differential Equations, 2023 East Tennessee State University

#### 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, 2023 California Polytechnic State University, San Luis Obispo

#### 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, 2023 Clemson University

#### 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, 2023 University of New Orleans, New Orleans

#### 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 …

Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, 2023 University of Washington

#### Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu

*Rose-Hulman Undergraduate Mathematics Journal*

In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.

Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, 2023 University of Massachusetts Amherst

#### 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 …

Langevin Dynamic Models For Smfret Dynamic Shift, 2023 Clemson University

#### Langevin Dynamic Models For Smfret Dynamic Shift, David Frost, Keisha Cook Dr, Hugo Sanabria Dr

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Geometry Of Competition And Stability For One-Host, Two-Parasitoid Systems With Application To Biocontrol, 2023 University of Richmond

#### Geometry Of Competition And Stability For One-Host, Two-Parasitoid Systems With Application To Biocontrol, Michael Kerckhove

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Reducing Uncertainty In Sea-Level Rise Prediction: A Spatial-Variability-Aware Approach, 2023 University of Minnesota - Twin Cities

#### Reducing Uncertainty In Sea-Level Rise Prediction: A Spatial-Variability-Aware Approach, Subhankar Ghosh, Shuai An, Arun Sharma, Jayant Gupta, Shashi Shekhar, Aneesh Subramanian

*I-GUIDE Forum*

Given multi-model ensemble climate projections, the goal is to accurately and reliably predict future sea-level rise while lowering the uncertainty. This problem is important because sea-level rise affects millions of people in coastal communities and beyond due to climate change's impacts on polar ice sheets and the ocean. This problem is challenging due to spatial variability and unknowns such as possible tipping points (e.g., collapse of Greenland or West Antarctic ice-shelf), climate feedback loops (e.g., clouds, permafrost thawing), future policy decisions, and human actions. Most existing climate modeling approaches use the same set of weights globally, during either regression or …

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, 2023 Embry-Riddle Aeronautical University

#### 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, 2023 The Graduate Center, City University of New York

#### 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, 2023 New Jersey Institute of Technology

#### 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.

Computing Brain Networks With Complex Dynamics, 2023 State University of New York at New Paltz

#### Computing Brain Networks With Complex Dynamics, Anca R. Radulescu

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Gap Junctions And Synchronization Clusters In The Thalamic Reticular Nuclei, 2023 State University of New York at New Paltz

#### Gap Junctions And Synchronization Clusters In The Thalamic Reticular Nuclei, Anca R. Radulescu, Michael Anderson

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Effective Non-Hermiticity And Topology In Markovian Quadratic Bosonic Dynamics, 2023 Dartmouth College

#### 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, 2023 Southern Methodist University

#### 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.