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<p>Pages 8, 38, 70, 116 and 120 have no body of text in the hardcopy. All are end pages of sections with a title at the top.</p> / <p>Many symbols could not be replicated using the Special Characters list. Please download thesis to read abstract.</p> / Doctor of Philosophy (PhD)

<p> In premixed turbulent combustion, the normal speed of propagation of the flame front is enhanced by the turbulent velocity field. This project will focus on the method of computing the normal speed of propagation of the flame front in the Majda-Souganidis model of turbulent combustion. Solving this problem involves computing the eigenvalue of a nonlinear cell problem. Discussed in this thesis is a new, simple and direct numerical method for approximating the eigenvalue, also called the effective Hamiltonian.</p> / Thesis / Master of Science (MSc)

Energy-shaping techniques are used to expand the range of autonomous motion of unmanned aerial systems without prohibitively {color{black}increasing the computational cost of the resultant controller}. Passivity-based control presents a method to implement a static, nonlinear state feedback control law that stabilizes the motion of an aircraft with a large region of attraction. {color{black} The energy-based control scheme is applied to both multirotor and fixed-wing aircraft}. Multirotor aircraft dynamics are cast into a port-Hamiltonian System and the concept of trajectory tracking using canonical feedback transformation is implemented to construct a cross-track controller. Fixed-wing aircraft dynamics are cast in port-Hamiltonian form and a passivity-based nonlinear control law for steady, wings-level flight of a fixed-wing aircraft to a specified inertial velocity (speed, course, and climb angle) is constructed. Results in simulations and experiments suggest robustness, and a large region of attraction of the controller. The control law extended to support time-varying inertial velocity tracking that incorporates banking to turn. The results are extended by including a line-of-sight guidance law and varying the direction as a function of position relative to a desired path, rather than as a function of time. The control law and the associated proof of stability follow similarly to that of the time-varying directional stabilization problem. The results are supported with simulations as well as experimental flight tests. / Doctor of Philosophy / This dissertation presents an alternative but intuitive approach to regulate unmanned aerial vehicles' flight that would allow for more maneuverability {color{black} than conventional methods}. This scheme relies on modifying the energy of the system to achieve the desired motion and leverages the properties of the aircraft rather than eliminating them and imposing different properties. This approach is applied to both fixed-wing and aircraft and quadcopters. Simulations and experimental flights have show the efficacy of this approach compared to other more established methods.

The problem considered here is one of finding the minimal Hamiltonian chain of a graph. A single chain must traverse all 𝑛 vertices of a graph with the minimal distance. The proposed procedure reduces a large problem into several smaller problems and uses a branch and bound algorithm to find the minimal Hamiltonian chain of each partitioned subproblem. The graph is decomposed and partitioned into subproblems with the use of necessary conditions for the existence of a Hamiltonian chain. This process is only applicable to graphs with relatively few incident edges per vertex. The branch and bound algorithm makes use of concepts developed by Nicos Christofides. Hamiltonian chains are derived by using minimal spanning trees. / Master of Science

A new direct full variational approach is described. The approach exploits a tensor (Kronecker) product construction of the many-electron Hamiltonian and has a number of computational advantages. Explicit assembly and storage of the Hamiltonian matrix is avoided by using the Kronecker product structure to form matrix-vector products directly from the molecular integrals. Computation-intensive integral transformations and formula tapes are unnecessary. The wavefunction is expanded in terms of spin-free primitive kets rather than Slater determinants or configuration state functions and is equivalent to a full configuration interaction expansion. The approach suggests compact storage schemes and algorithms which are naturally suited to parallel and pipelined machines. Sample calculations for small two- and four-electron systems are presented. The preliminary ground state potential energy surface of the hydrogen molecule dimer is computed by the tensor product method using a small basis set. / Ph. D.

The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research. / Ph. D. / incomplete_metadata

Experimental methods based on the magnetic resonance phenomenon belong to the most widely used experimental techniques for investigations of molecular and electronic structure. The difficulty with such experiments, usually a proper interpretation of data obtained from high-resolution spectra, opens new challenges for pure theoretical methods. One of these methods is density functional theory (DFT), that now has an advanced position among a whole variety of computational techniques. This thesis constitutes an effort in this respect, as it presents theory and discusses calculations of electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) parameters of paramagnetic molecules. It is known that the experimental determination of the magnetic resonance parameters of such molecules, especially in the case of NMR, is quite complicated and requires special techniques of spectral detection. On the other hand, paramagnetics play an important role in many areas, such as molecular magnets, active centers in biological systems, and defects in inorganic conductive materials. Therefore, they have spurred great interest among experimentalists, motivating us to facilitate the interpretation of spectral data through theoretical calculations. This thesis describes new methodologies for the determination of magnetic properties of paramagnetic molecules in the framework of DFT, which have been developed in our laboratory, and their applications in calculations of a wide range of molecular systems. The first two papers of this thesis deal with the theoretical determination of NMRparameters, such as nuclear shielding tensors and chemical shifts, in paramagnetic nitroxides that form core units in molecular magnets. The developed methodology is aimed to realize a high calculational accuracy for these systems. The effects of hydrogen bonding are also described in that context. Our theory for the evaluation of nuclear shielding tensors in paramagnetic molecules is consistent up to second order in the fine structure constant and considers orbital, fully anisotropic dipolar, and isotropic contact contributions to the shielding tensor. The next projects concern electron paramagnetic resonance. The well-known EPR parameters, such as the g-tensors and the hyperfine coupling constants are explored. Calculations of electronic g-tensors were carried out in the framework of a spin-restricted open-shell Kohn-Sham method combined with the linear response theory recently developed in our laboratory and allowing us to avoid by definition the spin-contamination problem. The inclusion of solvent effects, described by the polarizable continuum model, extends the possibility to treat molecular systems often investigated in solution. For calculations of the hyperfine coupling constants a so-called restricted-unrestricted approach to account for the spin polarization effect has been developed in the context of DFT. To examine the validity of the approximations implicit in this scheme, the neglect ii of singlet operators, a generalized RU methodology was implemented, which includes a fully unrestricted treatment with both singlet and triplet operators. The small magnitude of the changes in hyperfine coupling constants confirms the validity of the original scheme. / QC 20100923

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.

abstract: In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2011