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Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in Rn, but very little is known about the l2(Z) case or the l2(Zd) case. We establish some basic Gabor frame theory for l2(Z) and then generalize to the l2(Zd) case.

yes / The orthocomplemented modular lattice of subspaces L[H(d)] , of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)] ). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2) , which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2) , to the subspaces H 1, H 2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

no / The finite set of subsystems of a finite quantum system with variables in Z(n), is studied as a Heyting algebra. The physical meaning of the logical connectives is discussed. It is shown that disjunction of subsystems is more general concept than superposition. Consequently, the quantum probabilities related to commuting projectors in the subsystems, are incompatible with associativity of the join in the Heyting algebra, unless if the variables belong to the same chain. This leads to contextuality, which in the present formalism has as contexts, the chains in the Heyting algebra. Logical Bell inequalities, which contain "Heyting factors," are discussed. The formalism is also applied to the infinite set of all finite quantum systems, which is appropriately enlarged in order to become a complete Heyting algebra.

This thesis presents the integration of reproducing kernel Hilbert spaces (RKHSs) into the model reference adaptive control (MRAC) framework to enhance the control systems of quadcopters. Traditional MRAC systems, while robust under predictable conditions, can struggle with the dynamic uncertainties typical in unmanned aerial vehicle (UAV) operations such as wind gusts and payload variations. By incorporating RKHS, we introduce a non-parametric, data-driven approach that significantly enhances system adaptability to in-flight dynamics changes. The research focuses on the design, simulation, and analysis of an RKHS-enhanced MRAC system applied to quadcopters. Through theoretical developments and simulation results, the thesis demonstrates how RKHS can be used to improve the precision, adaptability, and error handling of MRAC systems, especially in managing the complexities of UAV flight dynamics under various disturbances. The simulations validate the improved performance of the RKHS-MRAC system compared to traditional MRAC, showing finer control over trajectory tracking and adaptive gains. Further contributions of this work include the exploration of the computational impact and the relationship between the configuration of basis centers and system performance. Detailed analysis reveals that the number and distribution of basis centers critically influence the system's computational efficiency and adaptive capability, demonstrating a significant trade-off between efficiency and performance. The thesis concludes with potential future research directions, emphasizing the need for further tests and implementations in real-world scenarios to explore the full potential of RKHS in adaptive UAV control, especially in critical applications requiring high precision and reliability. This work lays the groundwork for future explorations into scalable RKHS applications in MRAC systems, aiming to optimize computational resources while maximizing control system performance. / Master of Science / This thesis develops and tests an advanced flight control system for quadcopters, using a technique referred to as reproducing kernel Hilbert space (RKHS) embedded model reference adaptive control (MRAC). Traditional control systems perform well in stable conditions but often falter with environmental challenges such as wind gusts or changes in weight. By integrating RKHS into MRAC, this new controller adapts in real-time, instantly adjusting the drone's operations based on its performance and environmental interactions. The focus of this research is on the creation, testing, and analysis of this enhanced control system. Results from simulations show that incorporating RKHS into standard MRAC significantly boosts precision, adaptability, and error management, particularly under the complex flight dynamics faced by unmanned aerial vehicles (UAVs) in varied environments. These tests confirm that the RKHS-MRAC system performs better than traditional approaches, especially in maintaining accurate flight paths. Additionally, this work examines the computational costs and the impact of various RKHS configurations on system performance. The thesis concludes by outlining future research opportunities, stressing the importance of real-world tests to verify the ability of RKHS-embedded MRAC in critical real-world applications where high precision and reliability are essential.

The concept of simulation-based engineering has been embraced by virtually every research and industry sector (Sinha, Liang et al. 2001; Mocko and Fenves 2003). Engineering and science communities have become increasingly aware that computer simulation is an indispensable tool for resolving a multitude of scientific and technological problems. It is clearly desirable to gain a reliable perspective on the behaviour of a system early in the design stage, long before building costly prototypes (Chul and Ro 2002; Letherwood, Gunter et al. 2004; Makarand Datar 2007; Ersal, Fathy et al. 2008; Mueller, Ferris et al. 2009). Simulation tools have become a critical part of the automotive industry due to their ability to reduce the time and money spent in the development process. Terrain is the principle source of vertical excitation to the vehicle and must be accurately represented in order to correctly predict the vehicle response in simulation. In this dissertation, non-deformable terrain surfaces are defined as a sequence of vectors, where each vector comprises terrain heights at locations oriented perpendicular to the direction of travel. The evolution and implications of terrain surface measurement techniques and existing methods for correcting INS drift are reviewed as a framework for a new compensation method for INS drift in terrain surface measurements. Each measurement is considered a combination of the true surface and the error surface, defined on a Hilbert vector space, in which the error is decomposed into drift (global error) and noise (local error). It is also desirable to develop a compact, path-specific, terrain surface representation that exploits the inherent anisotropicity in terrain over which vehicles traverse. In order to obtain this, a set of analytic basis vectors is formed from Gegenbauer polynomials, parameterized to approximate the empirical basis vectors of the true terrain surface. It is also desirable to evaluate vehicle models and tire models over a wide range of terrain types, but it is computationally impractical to store long distances of every terrain surface variation. This dissertation examines the terrain surface, rather than the terrain profile, to maximize the information available to the tire model (i.e. wheel path data). A method to decompose the terrain surface as a combination of deterministic and stochastic components is also developed. / Ph. D.

Thesis advisor: Xiao Chen / The development of highly controllable quantum coherent simulators such as superconducting qubits and Rydberg atom arrays has stimulated the study of non-equilibrium quantum dynamics, opening the door to exciting topics including dynamical phase transitions, thermalization, transport, and quantum error correction. This thesis addresses various questions from non-equilbrium quantum dynamics, with a concentration on measurement-induced phase transitions (MIPT), adaptive dynamics with feedback mechanism, and Hilbert space fragmentation. In the first part, we study the hybrid quantum automaton (QA) circuits with different symmetries subject to local composite measurements. For $\mathbb{Z}_2$-symmetric hybrid QA circuits, there exists an entanglement phase transition from a volume-law phase to a critical phase by varying the measurement rate. The special feature of QA circuits enables us to interpret the entanglement dynamics in terms of a stochastic particle model. With the help of this stochastic model, we further investigate the entanglement fluctuations and quantum error correcting property of the volume-law phase in QA circuits with no symmetry, and study the entanglement dynamics in QA circuits with U(1) symmetry. Despite being a hallmark of non-unitary quantum dynamics, MIPT is absent in the density matrix averaged over measurement outcomes. In the second part, we introduce an adaptive quantum circuit subject to measurements with feedback. The feedback is applied according to the measurement outcome and steers the system towards a unique state above certain measurement threshold. We show that there exists an absorbing phase transition in both quantum trajectories and quantum channels. In the end, we turn to the phenomenon of Hilbert space fragmentation (HSF), whereby dynamical constraints fragment Hilbert space into many disconnected sectors, providing a simple mechanism by which thermalization can be arrested. However, little is known about how thermalization occurs in situations where the constraints are not exact. To study this, we consider a situation in which a fragmented 1d chain with pair-flip constraints is coupled to a thermal bath at its boundary. For product states quenched under Hamiltonian dynamics, we numerically observe an exponentially long thermalization time, manifested in both entanglement dynamics and the relaxation of local observables. To understand this, we study an analogous model of random unitary circuit dynamics, where we rigorously prove that the thermalization time scales exponentially with system size. Slow thermalization in this model is shown to be a consequence of strong bottlenecks in configuration space, demonstrating a new way of producing anomalously slow thermalization dynamics. / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The abstract kinetic equation Tψ’=-Aψ is studied with partial range boundary conditions in two geometries, in the half space x≥0 and on a finite interval [0, r]. T and A are abstract self-adjoint operators in a complex Hilbert space. In the case of the half space problem it is assumed that T is a (possibly) unbounded injection and A is a positive compact perturbation of the identity satisfying a regularity condition, while in the case of slab geometry T is a bounded injection and A is a bounded Fredholm operator with a finite dimensional negative part. Existence and uniqueness theory is developed for both models. Results are illustrated on relevant physical examples. / Ph. D.

No / We analyze the possibility and efficiency of nonholonomic control over quantum devices with exponentially large number of Hilbert space dimensions. We show that completely controllable devices of this type can be assembled from elementary units of arbitrary physical nature, and can be employed efficiently for universal quantum computations and simulation of quantum-field dynamics. As an example we describe a toy device that can perform Toffoli-gate transformations and discrete Fourier transform on 9 qubits.