Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 20 of 523

Multi-scale Modelling of Neurosteroid-mediated Seizure Trajectories in Childhood Absence Epilepsy
(University of Waterloo, 2025-06-20) Ahmed, Maliha
Childhood absence epilepsy (CAE) is a pediatric generalized epilepsy disorder characterized by brief episodes of impaired consciousness and distinctive 2.5--5 Hz spike-wave discharges (SWDs) on electroencephalography. With a well-established genetic aetiology, this condition tends to resolve spontaneously during adolescence in most cases. While several mechanisms have been proposed for remission, understanding remains insufficient to guide early intervention practices. In this thesis, we first utilize a conductance-based thalamocortical network model that exhibits characteristic SWDs, to demonstrate that allopregnanolone---a progesterone metabolite known to enhance GABAa receptor-mediated inhibition---has an ameliorating effect on SWDs. To investigate the divergence between this finding and clinical observations, we developed an enhanced thalamocortical model that incorporates a layered cortical structure to explore regional cortical heterogeneity and frontocortical connectivity as potential resistance factors to ALLO-mediated recovery. Our results suggest that non-resolving CAE may be due not only to increased frontocortical connectivity but also to the composition of cell types within the network. Specifically, a higher proportion of bursting-type cells may prevent the therapeutic effects of allopregnanolone. We extended our investigation to examine whether these findings apply to CAE caused by different genetic mechanisms, particularly mutations in sodium channel genes by modelling their effects at the individual neuron level. Furthermore, we examined the degree to which these alterations lead to network-level pathological activity, as well as the influence of ALLO on these genetically distinct networks. Our results demonstrate that ALLO facilitates recovery from SWDs regardless of the underlying mutation type. However, enhanced frontocortical connectivity prevents recovery in some mutation types, particularly when mutation effects are severe. Altogether, the multi-scale computational framework developed in this thesis demonstrates that CAE remission is determined by complex interactions between hormonal influences, genetic factors, and network connectivity patterns. The results suggest that certain genetic mutations may predispose individuals toward either remission or non-remission, which can be further modulated by connectivity profiles. In particular, enhanced frontocortical connectivity appears to be a significant factor in resistance to hormone-mediated remission. Additionally, this thesis develops techniques for analyzing transitions between distinct dynamical states in neural systems, incorporates genetic and hormonal factors into conductance-based models, and provides a computational framework to identify key parameters governing epileptic activity. These approaches not only advance our understanding of CAE specifically, but offer generalizable insights into the mathematical modelling of neurological conditions characterized by spontaneous shifts in brain dynamics.
Reduced-Order Modeling and Data Assimilation of the El Niño–Southern Oscillation
(University of Waterloo, 2025-05-16) Aydogdu, Yusuf
Simulations of complex fluid dynamics problems or climate models take weeks to complete even when run parallel in state-of-the-art supercomputers. Given computational resource constraints and the need for adaptable simulation settings, cost-efficient and accurate algorithms are essential. In this thesis, we explore stable, efficient, and accurate methodologies when applied to the El Niño–Southern Oscillation (ENSO), which integrates coupled atmosphere, ocean, and sea surface temperature (SST) mechanisms in the equatorial Pacific. ENSO is one of the most influential and complex climate phenomena, affecting weather patterns across the globe. ENSO consists of irregular oscillations between warm (El Niño) and cold (La Niña) phases in the Pacific Ocean, significantly impacting global weather patterns. Due to ENSO's inherent complexity and uncertainties, it is particularly suited for stochastic modeling. By modeling these uncertainties, stochastic simulations offer a more accurate representation of ENSO's variability, including its irregular periods and amplitudes. We first study the effects of stochastic perturbations on ENSO dynamics and introduce novel modeling and numerical schemes based on the Wiener Chaos Expansion (WCE). The key idea behind WCE is the explicit discretization of white noise through Fourier expansion. We also compare these methods with Monte Carlo (MC) simulations. Our findings demonstrate that the simulation of the linear stochastic ENSO model driven by the Ornstein-Uhlenbeck process using WCE requires far less computational resources and gives more accurate results compared to MC ensembles. This part of the thesis provides an alternative efficient approach for simulations of stochastic climate models and quantification of statistical moments,i.e, mean and variance. In the next stage of this research, we explore a reduced-order modeling (ROM) framework based on the POD-Galerkin method when applied to a nonlinear ENSO model. POD-Galerkin reduced order modeling aims to reduce the computational complexity and present high-dimensional problems~(usually PDEs) with reduced-order equations (ODEs). POD modes are optimal in capturing the system’s dominant features, making it particularly effective for reducing the dimensionality of systems governed by PDEs. By capturing the full-order ENSO (PDE) model with only four modes and four reduced-order equations, we achieve a substantial reduction in computational complexity without significant loss of accuracy. Due to the special properties of the model, we introduce a novel approach using different POD bases, but the same time coefficients for all model components. Moreover, we employ machine learning methods to explore different ROM and model discovery techniques in this part. The final part of this thesis focuses on the data assimilation of the nonlinear stochastic ENSO model, which forms the core results of this research. We first demonstrate the validity of POD-Galerkin reduced order modeling for the stochastic ENSO driven by the Ornstein-Uhlenbeck process. We project SPDEs onto POD modes derived from the deterministic model and introduce the reduced-order stochastic equations (SDEs). After setting up the filtering framework, we combine these equations and artificial observations in the Pacific ocean, based on realistic experiments, to estimate the ENSO-related SST anomalies. We employ particle filters and test the efficiency using different number of particles and ensembles. From novel ENSO modeling to uncertainty quantification, from reduced order modeling to nonlinear filtering, this thesis provides a promising approach for accurate and efficient predictions of ENSO-related climate variables.
Measurements of quantum fields with particle detectors
(University of Waterloo, 2025-05-14) Polo Gomez, Jose
This thesis aims to provide a working measurement theory for quantum fields built upon measurements using localized non-relativistic quantum systems, widely known as particle detectors. In Chapters 3 to 5, we focus on using detector-based schemes to carefully formulate a measurement theory that is compatible with relativity. To do so, we provide a rigorous analysis of the causal behaviour of the induced non-selective channels, as well as an update rule that is consistent with relativistic causality. In the process, we establish a body of results, including a characterization of localized causal channels in real scalar QFT, and the formulation of a formalism that allows a consistent treatment of non-relativistic multipartite systems in relativistic setups. In Chapters 6 to 8, we focus on verifying that the measurement theory formulated in the first part is a working measurement theory, meaning that it can actually be used in practice to measure specific targeted features of the quantum field. To ensure this, we introduce a measurement strategy where particle detectors always undergo the same measurement protocol—regardless of the quantity or feature of interest—and then are subjected to a tomography process. The resulting data is fed into a trained neural network that is capable of inferring the targeted quantity or feature from the detector's readings. This strategy has the potential to be applicable to measure any quantity of the field that is accessible through local measurements. More tangentially, we also introduce a method beyond perturbation theory that uses trains of delta couplings to efficiently approximate the final state of a detector undergoing a continuous interaction. Additionally, we apply the detector-based measurement theory to analyze the effect of measurements on the protocol of entanglement harvesting.
Cosmic Frontiers: From the Origins to Gravitational Wave Astronomy
(University of Waterloo, 2025-05-14) Dehghanizadeh, Amirhossein
In this thesis, we present different aspects of cosmology, focusing on the early universe, large scale structure formation, and gravitational wave astrophysics. We begin by introducing cosmological perturbation theory as a framework to describe the origin of the universe, emphasizing its observational implications. We discuss early universe models and, motivated by the avoidance of singularities, we propose bouncing cosmologies. In particular, we discuss the Cuscuton model; which modifies gravity to generate a bounce without introducing new degrees of freedom, provides a smooth transition between contraction and expansion, and generates scale-invariant entropy perturbations. We analyze scalar perturbations and demonstrate that the model remains stable and weakly coupled. Additionally, we compute three-point functions, showing negligible non Gaussianities on observable scales, with potential signals arising from the conversion of isocurvature to curvature perturbations. Building on this foundation, we investigate the formation of large scale structures and study bias parameters that describe the statistical distribution of dark matter halos and galaxies. We extend this analysis to gravitational wave (GW) sources, introducing the GW bias parameter. We propose a methodology to link binary black hole (BBH) mergers to their host galaxies using phenomenological GW host-galaxy probability functions. These functions are constructed based on observed astrophysical properties of galaxies, including stellar mass, star formation rate, and metallicity. We calculate the GW bias using the angular power spectrum for photometric surveys and the 3D power spectrum for spectroscopic surveys. We find that the GW bias depends on the parameters of the host-galaxy probability functions, which are linked to the formation channels of BBHs. Future measurements of the GW bias could provide insights into the astrophysical processes governing GW sources and their relation to large-scale structure. Lastly, we emphasize that cosmology may be entering a golden era, driven by a wealth of upcoming experiments, including the next generation of Cosmic Microwave Background (CMB) telescopes, galaxy surveys, and GW detectors. These experiments have the potential to provide hints for modifications to ΛCDM cosmology, the nature of dark matter and dark energy, black hole physics, and possible deviations from general relativity, paving the way for new theoretical and observational breakthroughs.
Interaction of electromagnetic fields with two-dimensional materials at the interface between dielectric media
(University of Waterloo, 2025-05-08) Hansen, Asbjorn
This thesis investigates the interaction of electromagnetic plane waves with two-dimensional (2D) materials approximated by infinitesimal sheets positioned at interfaces between non-magnetic, isotropic, lossless dielectrics. On their own, 2D materials offer a host of useful properties, including high carrier mobility, dopant-free tuning through the application of external electromagnetic fields or stacking, as well as significant anisotropy. When combined with the excitation of plasmon modes to enhance light-matter interactions, the range of potential applications broadens greatly, including, for example, high-speed optical computation and enhanced light trapping in solar cells. In this thesis, a new derivation of the boundary condition for out-of-plane response is presented for a 2D material immersed in an infinite homogeneous dielectric. To account for the influence of distinct dielectric surrounding media, Fresnel coefficients are derived using a vacuum gap method. This approach yields novel dispersion relations for surface plasmons that incorporate the out-of-plane response of 2D materials. In the presence of distinct surrounding dielectrics, the out-of-plane mode hybridizes with the in-plane longitudinal plasmon mode. The hybridization is destroyed when the surrounding materials are made identical. Additionally, expressions for the conservation of energy and momentum are derived in terms of the Fresnel field amplitudes for the 2D materials described by singular current sheets. The vacuum gap method is also applied to solve the conservation equations in the presence of dielectric media, as this avoids controversial notions of electromagnetic momentum in media. The electromagnetic stress tensor derived through the vacuum gap method satisfies conservation of linear momentum, thus confirming that the forces on a 2D material can be self-consistently separated from those on surrounding dielectrics. This theoretical framework is applied to both graphene and phosphorene. For graphene, the reflectance, transmittance, and absorption characteristics are compared with and without considering the out-of-plane response. The analysis confirms that the influence of the out-of-plane component is minimal for plasmon modes at low frequencies as is often assumed, but becomes significant at high frequencies. Forces calculated around frequencies and wavenumbers defined by the plasmon dispersion relation showed significant amplification. However, the new non-monotonic dispersion profile of the hybridized plasmon mode introduces additional resonant and antiresonant behaviour in the forces. To emphasize the in-plane anisotropy of phosphorene, only in-plane response was considered, with a focus on forces in the plasmonic regime at low frequencies. Based on numerical examples, predictions are made for possible future experimental studies of both plasmon modes in the presence of the out-of-plane response and forces on 2D materials for possible optomechanical and sensing applications.
The Information Locally Stored in Quantum Fields: From Entanglement to Gravity
(University of Waterloo, 2025-05-01) Rick Perche, Tales
This thesis contains a local study of quantum field theory from fundamental, operational, and practical perspectives, with the primary goal of investigating the information that can be locally extracted from quantum fields. Central to this discussion is how the fundamental interactions of quantum fields give rise to the very objects that allow us to probe them. We approach this problem through the concept of localized quantum fields, which naturally reduce to local probes with finitely many degrees of freedom that can be accessed in realistic experiments. Building on this detailed description of localized probes, we apply these to explore two key aspects of the information locally stored in quantum fields: entanglement and gravity. In the study of entanglement, we explore the quantification of accessible vacuum entanglement between two finite regions of spacetime. Our discussion contains both a first-principles approach based on local field degrees of freedom and an operational framework, wherein we consider the entanglement that can be harvested by coupling local probes to independent degrees of freedom of the field. The study of entanglement in quantum field theory also leads us to classify the regimes where the quantum degrees of freedom of a field play an active role. Through the use of an effective quantum-controlled model, we show that the quantum degrees of freedom of mediating fields are only relevant in relativistic setups involving either high energies or interactions that are sufficiently localized in spacetime. In setups where these conditions are not met, a simplified effective model can accurately describe interactions while still incorporating some key relativistic elements. Finally, we will discuss the gravitational information locally stored in quantum fields. Specifically, we will show that the correlations of quantum fields contain full information about the geometry of spacetime, and how to physically access these degrees of freedom. While the fact that quantum fields store full gravitational information might suggest the possibility of a theory in which gravity emerges directly from quantum correlations, we speculate that gravity may instead be emergent from the entanglement in quantum field theory.
Theoretical Constraints on Phantom Dark Energy and Non-singular Bouncing Cosmology
(University of Waterloo, 2025-04-23) Moghtaderi, Elly
In general relativity, energy conditions can restrict the allowed geometries. One of the most prominent conditions is the classical null energy condition (NEC). However, quantum field theories can violate this condition which prompts the need for quantum null energy conditions. The smeared null energy condition (SNEC) is a proposed semilocal conjecture, that modifies NEC by averaging `energy density' along a null geodesic, with the averaging weight being a real, positive smearing function. The SNEC allows for `the null energy density' to be negative over short smearing scales. This can have insightful implications for phantom dark energy models and bouncing cosmologies by introducing bounds on dark energy equation-of-state parameters and an inequality between the duration of the bouncing phase and the growth rate of the Hubble parameter. In the research presented in this thesis, we studied the application of the SNEC on flat FLRW cosmology with pressureless matter and dark energy as sources and showed for which regions of parameter space the SNEC is compatible with an evolving phantom dark energy. What makes it interesting is that it provides us with a theoretical framework to see where it is in agreement with the recent Dark Energy Spectroscopic Instrument (DESI) observation findings, which suggest the violation of the classical NEC. Furthermore, we applied the SNEC to very early universe nonsingular bouncing cosmology and specifically found a bound where these models in general and more specifically the cuscuton model agree with the SNEC.
Examining Social and Climate Feedbacks: Linking Climate Opinions and Climate System in Social Networks
(University of Waterloo, 2025-04-22) Satheesh Kumar, Athira
Climate change is one of the most critical challenges that humanity faces, particularly due to its impacts on ecosystems, economies, and societies around the globe. Usually, climate models focus on physical and economic factors, often overlooking social dynamics. Despite identifying human contribution to the climate crisis, behavioural factors, such as public opinion and decisions about mitigation, remain unintegrated in climate models. This research highlights the role of human actions in addressing climate challenges. Human actions are controlled by social factors such as climate rumours. The rising popularity of social media exposes people to unverified information about climate change and its impacts. In addition to rumours, other factors, such as the high costs to switch to climate-friendly alternatives make mitigation less appealing to people. Although several models consider integrating social behaviour into climate models, these models usually treat human behaviour as a binary choice between mitigation and non-mitigation. However, choices are not too simple when it comes to climate change. People can have more or less intense climate opinions, demonstrating the importance of having a continuous range of opinions when it comes to understanding the issue of climate change. We adopt various modelling approaches to identify the factors leading to reduced response towards the issue of climate change. Our analysis reveals the important role larger groups play in determining future climate scenarios. The behaviour (mitigating or non-mitigating) of these large groups determines the overall emission levels of the population. We identified the importance of mitigation strategies to achieve our current climate targets. Moreover, frequent rumors regarding climate change can also enhance mitigative behaviour and reduce emissions. Factors such as frequent and unexpected social or climate events, stubbornness in individuals, opinion polarization, and high mitigation costs can greatly influence future climate predictions. Ignorance towards climate issues due to delayed response in switching to climate-friendly alternatives or completely forgetting about these issues are some of the major reasons for falling behind in meeting the world's climate targets. The global nature of climate issues makes finding common ground to adopt mitigative strategies difficult. This thesis underscores incorporating social dynamics into climate models, a more comprehensive framework for predicting policy outcomes. Policymakers could leverage the model outcomes to identify factors that minimize emissions by maximizing public response to climate change. Ultimately, this research emphasizes the necessity of integrating behavioural insights into climate models to support informed, and effective climate policy development.
Learning-Based Safety-Critical Control Under Uncertainty with Applications to Mobile Robots
(University of Waterloo, 2025-02-13) Aali, Mohammad; Liu, Jun
Control theory is one of the key ingredients of the remarkable rise in robotics. Due to technological advancements, the use of automated robots, which was once primarily limited to industrial and manufacturing settings, has now expanded to impact many different parts of everyday life. Various control strategies have been developed to satisfy a wide range of performance criteria arising from recent applications. These strategies have different characteristics depending on the problem they solve. But, they all have to guarantee stability before satisfying any performance-driven criteria. However, as robotic technologies become increasingly integrated into everyday life, they introduce safety concerns. For autonomous systems to be trusted by the public, they must guarantee safety. In recent years, the concept of set invariance has been incorporated into modern control strategies to enable systematic safety guarantees. In this thesis, we aim to develop safety-critical control methods that can guarantee safety while satisfying performance-driven requirements. In the proposed strategies, we considered formal safety guarantees, robustness to uncertainty, and computational efficiency to be the highest design priorities. Each of them introduces new challenges which are addressed with theoretical contributions. We selected motion control in mobile robots as a use case for proposed controllers which is an active area of research integrating safety, stability, and performance in various scenarios. In particular, we focused on multi-body mobile robots, an area with limited research on safe operation. We provide a comprehensive survey of the recent methods that formalize safety for the dynamical systems via set invariance. A discussion on the strengths and limitations of each method demonstrates the capabilities of control barrier functions (CBFs) as a systematic tool for safety assurance in motion control. A safety filter module is also introduced as a tool to enforce safety. CBF constraints can be enforced as hard constraints in quadratic programming (QP) optimization, which rectifies the nominal control law based on the set of safe inputs. We propose a multiple CBF scheme that enforces several safety constraints with high relative degrees. Using the multi-input multi-output (MIMO) feedback linearization technique, we derive conditions that ensure all control inputs contribute effectively to safety. This control structure is essential for challenging robotic applications requiring multiple safety criteria to be met simultaneously. To demonstrate the capabilities of our approach, we address reactive obstacle avoidance for a class of multi-body mobile robots, specifically tractor-trailer systems. The lack of fast response due to poor maneuverability makes reactive obstacle avoidance difficult for these systems. We develop a control structure based on a multiple CBFs scheme for a multi-steering tractor-trailer system to ensure a collision-free maneuver for both the tractor and trailer in the presence of several obstacles. Model predictive control serves as the nominal tracking controller, and we validate the proposed strategy in several challenging scenarios. Although the CBF method has demonstrated a great potential for ensuring safety, it is a model-based method and its effectiveness is closely tied to an accurate system model. In practice, model uncertainty compromises safety guarantees and may lead to conservative safety constraints, or conversely, allow the system to operate in unsafe regions. To address this, we explore developing safety-critical controllers that account for model uncertainty. Achieving this requires combining the theoretical guarantees of model-based methods with the adaptability of data-driven techniques. For this study, we selected Gaussian processes (GPs) which bring together required capabilities. It provides bounds on the posterior distribution, enabling theoretical analysis, and producing reliable approximations even with a low amount of training data, which is common in data-driven control. The proposed strategy mitigates the adverse effects of uncertainty on high-order CBFs (HOCBFs). A particular structure of the covariance function is designed that enables us to convert the chance constraints of HOCBFs into a second-order cone constraint, which results in a convex constrained optimization as a safety filter. A discussion on the feasibility of the resulting optimization is presented which provides the necessary and sufficient conditions for feasibility. In addition, we consider an alternative approach that uses matrix variate GP (MVGP) to approximate unknown system dynamics. A comparative analysis is presented which highlights the differences and similarities of both methods. The proposed strategy is validated on adaptive cruise control and active suspension systems, common applications in mobile robots. This study next explores the safety of switching systems, focusing on cases where system stability is assured through control Lyapunov functions (CLFs) and CBFs are applied for safety. We show that the effect of uncertainty on the safety and stability constraint forms piecewise residuals for each switching surface. We introduce a batch multi-output Gaussian process (MOGP) framework to approximate these piecewise residuals, thereby mitigating the adverse effects of uncertainty. We show that by leveraging a specific covariance function, the chance constrained safety filter can be converted to a convex optimization, that is solvable in real-time. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through a simulation of a switching adaptive cruise control system.
Deep Learning Models of Cellular Decision-Making Using Single-Cell Genomic Data
(University of Waterloo, 2025-01-27) Sadria, Mehrshad; Layton, Anita
Cellular decision-making, essential to regenerative medicine, disease research, and developmental biology, relies on complex molecular mechanisms that guide cells in responding to stimuli and committing to specific fates. This thesis introduces several deep learning methods to analyze single-cell RNA sequencing data, uncover regulatory programs driving these processes, and predict the outcomes of gene perturbations. By applying representation learning and generative models, meaningful structures within high-dimensional data are identified, enabling tasks such as mapping cellular trajectories, reconstructing regulatory networks, and generating realistic synthetic data. Furthermore, integrating deep learning with dynamical systems theory enables the prediction of cellular decision timing and the identification of key regulatory genes involved in these processes. These methods enhance our understanding of gene activity dynamics, improve predictions of cellular behavior, and offer new avenues for progress in regenerative medicine, developmental biology, and disease research.
Mathematical Models of Kidney Function: Effects of Hypertension and Circadian Rhythm
(University of Waterloo, 2025-01-17) Zheng, Kaixin; Layton, Anita
Hypertension induced by chronic angiotensin II (Ang II) infusion serves as a valuable experimental model for studying blood pressure regulation and the kidney's role in electrolyte and fluid homeostasis. The kidney's function is modulated by the renin-angiotensin-aldosterone system (RAAS) and circadian rhythms, with notable differences observed between males and females in the former. Under normotensive conditions, female rat nephrons exhibit lower Na+/H+ exchanger 3 (NHE3) activity in the proximal tubule but higher Na+ transporter activities along distal segments compared to males. Chronic Ang II infusion reduces NHE3 activity, shifts Na+ transport downstream, and promotes vasoconstriction, anti-natriuresis, and hypertension. These effects are further influenced by diurnal oscillations in glomerular filtration, electrolyte transport, and renal transporter regulation by circadian clock genes. Using computational models of kidney function, this thesis explores two key areas: (i) the impact of Ang II infusion on segmental electrolyte transport and diuretic responses in male and female rat nephrons, and (ii) the influence of diurnal rhythms on the natriuretic and diuretic effects of loop, thiazide, and K+-sparing diuretics under normotensive and hypertensive conditions in male rats. Simulations suggest that NHE3 downregulation in the proximal tubule is a primary driver of natriuresis and diuresis, with stronger effects in males. In hypertension, the downstream shift in Na+ transport load amplifies the effects of diuretics, with hypertensive females exhibiting larger relative increases in Na+ excretion due to their higher distal transport load. Additionally, diuretic responses vary by time of day, with qualitatively similar diurnal oscillations observed in normotensive and hypertensive kidneys. These findings provide insights into sex-specific and time-dependent responses to hypertension and diuretic therapies, emphasizing the need to consider both physiological context and administration timing in treatment strategies.
Numerical Evolution of Correlation Functions With Applications to Dynamically Localized Quantum Fields
(University of Waterloo, 2025-01-14) Ragula, Boris; Martin-Martinez, Eduardo
This thesis presents two topics at the interface between computational physics and Quantum Field Theory (QFT). This first part of the thesis is a comprehensive study of a numerical evolution scheme for the correlation function of a scalar quantum field. In particular, it explores how one can numerically simulate a bi-scalar function that simultaneously satisfies a time dependent partial differential equation Partial Differential Equations (PDE) in two independent spacetime coordinates. We demonstrate an algorithm that is capable of performing time integration in two time coordinates and yielding convergent numerical results for not only the correlation function, but also for quantities of interest relating to the quantum field. Moreover, we demonstrate a number of methods that can be leveraged to optimize the speed along with the required memory of the algorithm. The second part of this thesis is concerned with the effects of dynamically localizing the vacuum state of scalar quantum field in (1 + 1)-dimensional Minkowski spacetime. Given recent develops in formulations to a measurement theory for quantum fields, localized field theories have emerged as a potential candidate in developing a relativistically consistent measurement theory. However, concerns have been raised regarding the use of these localized fields in realistic, experimental setups due to the fact that one must dynamically localize the field. The result of this localization would be a loss of purity in the experimentally accessible modes of the field, and thus would not be useful as a measurement device. Utilizing the methods presented in the first part of this thesis, we study the effect of localizing quantum field degrees of freedom by dynamically growing cavity walls through a time-dependent potential. We use our results to show that it is possible to do this without introducing non-negligible mixedness in localized modes of the field. We discuss how this addresses the concerns, raised in previous literature, that the high degree of entanglement of regular states in QFT may hinder relativistic quantum information protocols that make use of localized relativistic probes.
Some Results on the Convergence of Anderson Acceleration
(University of Waterloo, 2025-01-06) Smith, Adam; De Sterck, Hans
Anderson acceleration (AA), also known as Anderson mixing, is an extrapolation technique used to accelerate the convergence of fixed-point iterations $\bm{x}_{k+1}=\bm{q}(\bm{x}_k)$, $k=0,1,\dots$, with $\bm{q}:\R^n\to\R^n$, $\bm{x}_k\in\R^n$. AA was first introduced by D.G. Anderson in the context of solving integral equations but has since been adapted to fixed-point iteration problems in general. Despite relatively little being known about its convergence properties, AA has seen considerable usage in several areas such as electronic structure computations and machine learning. This thesis presents a broad overview of the current convergence literature for AA and introduces a variety of new results concerning properties of AA, such as its asymptotic convergence rate, the possibility of stagnation, and an analysis of its coefficients. Additionally, some variations of the AA iteration are proposed with an accompanying analysis and comparison to the classical AA algorithm.
Finite Automata Models: Algorithm, Application, and Semigroup Study
(University of Waterloo, 2024-12-17) Derets, Hanna; Nehaniv, Chrystopher
A finite automaton consists of states and state transitions labeled by letters of a finite alphabet. Every letter describes a transformation of the automaton's state space, generating a corresponding transformation semigroup. A stochastic version of automata has a probability distribution over the alphabet at every state, that allows assigning the likelihoods to generated sequences of letters. In this thesis, we explore two models: deterministic probabilistic finite automaton (DPFA) considered from the practical side, and the sandpile model considered from the theoretical side. For the first model, the presented study describes an algorithm for reconstructing DPFA from sequences of discrete observations using an n-gram merging method, including the practical implementation of the method, its experimental evaluation on case studies, and the application of this automata-based technique to the neurobiological data. For considered examples the performance is compared to the causal state splitting reconstruction (CSSR) technique: both methods achieve high quality in approximating the probability distribution over strings. Considering if transformation semigroups of reconstructed automata contain subgroups corresponding to those in examples, CSSR shows a good result for preserving cyclic permutation groups, but not the n-gram merging method, whose transformation semigroup is generally aperiodic. The application considered in this study uses electroencephalographic (EEG) microstate sequences and considers the question of distinguishing the participant groups (meditators and controls) and cognitive modes (mind-wandering, verbalization, visualization) by separating DPFA machines inferred from EEG data in a metric space. The separation between participant groups is achieved for many parameter settings with linear criterion (requiring non-overlapping clusters) and for a few instances with strict criterion (requiring dense distant clusters). Both criteria show great reliability when validated using permuted data. The separation of cognitive modes only demonstrated partial success with noticeably better performance within the group of controls and more instances of separation corresponding to the visualization condition. For the second, sandpile model, the presented study concentrates on the properties of their transformation semigroups for the standard Abelian sandpiles on circle graphs and the modified model, non-Abelian sandpiles on rooted trees. The exploration addresses the structure of recurrent configurations, the wreath product decomposition of semigroups, and the decomposition-based complexity measure. The identity and generator configurations of the recurrent group of Abelian sandpiles on circles are described, giving an explicit alternative way of understanding their well-known cyclic structure. The complexity of arbitrary finite Abelian semigroup is shown to be at most one. The embedding of the sandpile semigroup into the wreath product of flip-flop semigroups is constructed for non-Abelian sandpiles on rooted trees, implying its aperiodic complexity is the depth of the tree.
Designing and simulating a micro-robot for transporting filamentous cargos in Newtonian and viscoelastic fluids
(University of Waterloo, 2024-11-18) Ghadami, Sepehr; Shum, Henry
In recent years, there has been a notable surge in interest within the medical field towards minimally invasive procedures, with magnetic micro-robots emerging as a promising avenue. These micro-robots exhibit the capability to traverse various mediums, including viscoelastic and non-Newtonian fluids, facilitating targeted drug delivery and medical interventions. Many current designs, drawing inspiration from micro-swimmers found in biological systems such as bacteria and sperm, utilize a contact-based approach for transporting payloads. However, adhesion between the cargo and the carrier can complicate release at the intended site. In this thesis, our primary aim was to investigate the potential of a helical micro-robot for non-contact delivery of drugs or cargo. We conducted an extensive examination of the shape and geometrical parameters of the helical micro-robot, with a specific focus on its ability to transport passive filaments. Based on our analysis, we propose a novel design comprising three sections with alternating handedness, incorporating two pulling and one pushing microhelices, to improve the capture and transportation of passive filaments in Newtonian fluids using a non-contact approach. Subsequently, we simulated the process of capturing and transporting the passive filament and evaluated the functionality of the newly designed micro-robot. Initially concentrating on naturally straight filaments, we also demonstrated the micro-robot’s capability to capture filaments with intrinsic curvature and those with a spherical payload attached at one end. Our findings provide valuable insights into the mechanics of helical micro-robots and their potential applications in medical procedures and drug delivery. Furthermore, the proposed non-contact delivery method for filamentous cargo could pave the way for the development of more efficient and effective micro-robots for medical purposes. In the second phase of our project, recognizing that most biological fluids exhibit viscoelastic properties due to the presence of protein fibers, we proposed a viscoelastic model. Inspired by the viscoelastic structure of bovine vaginal fluid, we developed and initially characterized this model using creep and strain tests. Subsequently, in the proposed viscoelastic model, we investigated the functionality of the designed micro-robot in the viscoelastic fluid and compared its performance with that of the conventional single-handed helical micro-robot. It was observed that although the designed micro-robot is highly effective in transporting filamentous microcargo without contact in Newtonian fluid, its unique structure presents challenges when moving in viscoelastic fluid.
Mathematical modeling and computer simulation of interactions of charged particles with 2D materials
(University of Waterloo, 2024-09-23) Moshayedi, Milad; Miskovic, Zoran
Recent interest in the nanophotonic and nanoplasmonic devices has led researechers to a detailed study of different aspects of the interactions of the moving charged particles with two dimensional (2D) materials. When a charged particle moves above a 2D material, electromagnetic forces due to the polarization of the charges in the target material cause energy dissipation. Analyzing the energy spectra and momentum change of the reflected particles provides valuable information about the internal structure of the target material. The analyses of this kind are extensively performed in the electron energy loss spectroscopy (EELS) and the high resolution EELS (HREELS) experiments. To model such interactions, we used the classical dielectric response theory in the non-retarded approximation. We obtained closed form expressions for the forces acting on the charged particles moving parallel to doped phosphorene using analytical models for its dielectric function, which expose the strongly anisotropic character of the electronic structure and dynamic response of this 2D material. The parameters of these models, which we call the optical and the semiclassical models, are supplied by the ab initio calculations. It was found that the force on the incident charge has three components, showing strong dependence on the direction of motion of the charged particle. Furthermore, we performed an analysis of the electric potential in the plane of phosphorene, revealing a rich variety of the plasmonic wake patterns induced by the moving charged particle. Our computations showed surprising analogies of these wakes with Kelvin’s ship wakes and atmospheric wakes regarding the asymmetry of the wake, slowing down of the plasmon dispersion, and the formation of Mach-like wake due to nonlocal effects in the semiclassical dielectric response function. In a related effort, we adopted the energy loss method (ELM) to compute the phosphorene carrier mobility tensor when the carrier scattering on charged impurities in the substrate is the main limiting factor of the mobility in the DC regime at low temperatures. The ELM provides a closed form expression for the phosphorene carrier’s mobility tensor in terms of a double integral, which is superior to the traditional approach for mobility calculation based on the Boltzmann transport equation that requires numerical solution of an integral equation. Using the ELM, we examined the mobilities for different statistical distributions of the charged particles, revealing strong effects of the inter-particle correlation distance on the mobilities. Finally, we evaluated the energy loss function of doped graphene when there exist a random distribution of charged impurities in the substrate. We modeled the resulting random potential landscape in graphene via local Fermi energy, with its in-plane spatial correlation governed by a Gaussian distribution, giving rise to an expression for the loss function that includes a memory function, which is the solution of a nonlinear integral equation. The overall effect of this random potential landscape was found to be a broadening and shifting of the plasmon peak in the energy loss function of doped graphene.
Continuous-Galerkin Summation-by-Parts Discretization of the Khokhlov-Zabolotskaya-Kuznetsov Equation with Application to High-Intensity-Focused Ultrasound
(University of Waterloo, 2024-09-18) Xie, Zhongyu; Del Rey Fernández, David; Sivaloganathan, Sivabal
Over the last two decades, High Intensity Focused Ultrasound (HIFU) has emerged as a promising non-invasive medical approach for locally and precisely ablating tissue, offering versatile applications in tumor treatment, drug delivery, and addressing brain disorders such as essential tremor. Its advantages include targeted energy delivery with no affect on skin integrity, low system maintenance costs, minimal impact on normal tissues, and swift recovery. Despite its’ merits, HIFU remains underutilized, primarily employed in specific breast cancer and prostate cancer treatments. To expand its range of applicability, a comprehensive understanding of the interaction between the ultrasound beam and local tissues at the focal point is essential. This thesis focuses on modeling critical nonlinear effects in the thermal modulation of local tissues by numerically solving the Khokhlov- Zabolotskaya-Kuznetsov (KZK) equation—which is an excellent model for the nonlinear acoustic field arising in HIFU. Constructing high-order stable discretizations of the KZK equations poses significant challenges due to the presence of polynomial nonlinear terms and a second derivative of an integral term within in this equation. Employing a continuous Galerkin approach, an operator is formulated to approximate the integral term, facilitating the construction of a modified second derivative operator. This establishes a clear correspondence between continuous and discrete stability proofs. Additionally, a skew-symmetric splitting technique is used to discretize the nonlinear advective term. The resulting semi-discrete scheme is proven to be stable. Numerical experiments using the method of manufactured solutions demonstrate the high-order accuracy and stability of the proposed numerical method. Finally, a HIFU verification test case demonstrates the applicability of the proposed scheme to investigate HIFU.
Blackbox Optimization for Free-space Quantum Key Distribution
(University of Waterloo, 2024-09-16) Maierean, Alexandra; Lütkenhaus, Norbert; Jennewein, Thomas
The Quantum Encryption and Science Satellite is an experimental proof-of-concept of free-space quantum key distribution. As part of the mission conception, it is imperative to have an optimal design of all elements of the communication protocol. That is, given all of the possible parameters, we ask which combination will achieve the most secure link? This thesis explores the answer to this question for some specific parameters. A detailed model of the mission is simulated in MATLAB to obtain data points describing the security of the link over the possible set of parameter values. Then, applying appropriate optimization algorithms, we seek to maximize the key rate and minimize the quantum bit error rate. Mathematically, this task reduces to an optimization problem where the objective is a 2-tuple of key rate and quantum bit error rate, which are indicators of protocol security; and the search space is the set of parameter values input to the simulation. The MATLAB simulation cannot be described analytically and thus represents an oracle or blackbox objective function. In this thesis, a comparison of optimization protocols is conducted between Mesh Adaptive Direct Search (MADS), and Model-based Trust Region (MBTR) methods. The investigation of optimization algorithm performance is applied to a subset of real Quantum EncrYption and Science Satellite (QEYSSat) parameters: excess voltage supplied to the detectors, and the mean photon number per pulse for the signal.
Degenerate Parabolic Diffusion Equations: Theory and Applications in Climatology
(University of Waterloo, 2024-08-27) Vlachos, Panagiotis; Guglielmi, Roberto
Diffusion models describe the spread of particles, energy, or other entities within a medium. Perturbations of mechanical systems, random walks(discrete case), and Brownian motion(continuous-time stochastic process) are some classical methods used to model diffusion. Among these, those generated by stochastic processes have been extensively studied by employing the Fokker-Planck equation—a one-dimensional parabolic partial differential equation—to examine these systems by analyzing the probability density function. Given the incomplete theory surrounding degenerate diffusion equations, our objective is to generalize and expand existing results for degenerate diffusion processes by examining cases where weak degeneracy occurs at the boundaries, utilizing a Fokker-Planck-like equation. More precisely, we first address the well-posedness results, which ensure the existence and uniqueness of a solution and are critical for investigating other qualitative properties such as controllability, observability, stabilization, and optimal control. Additionally, we explore the interval of the existence or absence of stationary states, which is fundamental in the analysis of mechanical or physical systems. To this end, we examine sufficient conditions for both the non-existence and existence of stationary points. Furthermore, to verify and illustrate our analytical results, we delve into the Budyko-Sellers model, a climate model, providing results on its well-posedness and addressing the inverse problem of determining the insolation function. Throughout this study, we primarily employ semigroup theory, op- erator and functional analysis, and weighted Sobolev spaces to manage the non-ellipticity of the diffusion and well-posedness of the parabolic equation, while using the theory of Lyapunov functions to ensure the existence of stationary states.
Systems biology models for cancer immunotherapy
(University of Waterloo, 2024-08-19) Cotra, Sonja; Kohandel, Mohammad; Przedborski, Michelle
Cancer is a complex disease that continues to affect millions of people around the world every year. With ever-improving science and technology, several forms of treatment have been introduced within the past century and continue to be developed so as to provide increasing chances of survival and comfort to patients. Particularly, the 21st century has seen the blossoming of immunotherapy methods, which exploit the natural immune system's ability to kill tumor cells. Several varieties of immunotherapy exist in order to use all sorts of immune cells, targeting specific antigens expressed on tumors or blocking checkpoints which inhibit necessary immune responses. Unfortunately, there is no perfect immunotherapy that can provide a safe and effective path to remission for every patient. Traditional clinical experimentation, while providing important insight, remains a costly option in increasing our understanding of immunotherapies against cancer. Systems biology methods provide a unique and effective channel for exploring the complex dynamics involved in tumor micro-environments between cancer cells, native immune cells and administered drugs. Resulting insight may be used to inform drug development leading to safe, effective, and personalized therapeutic routines. In this thesis, we start by providing a general overview of cancer biology starting from the cell, and systems biology. We then detail equations and parameters comprising a particular systems biology model for nivolumab, an anti-PD-1 immune checkpoint inhibitor, informed by ex vivo data extracted from patients suffering from head and neck squamous cell carcinoma. We then present results of an examination of sex differences in regards to patient response to nivolumab monotherapy as well as combination therapy with recombinant IL12. Here, the aforementioned model was used alongside basal immune differences between the sexes from the literature to generate virtual cohorts of male and female patients receiving these treatments. Finally, we conclude with a general summary as well as potential future directions involving a similar systems biology model describing cytokine release syndrome as a side-effect of CAR-T cell therapy.

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