Thesis (M.S.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains v, 17 p. : ill. Includes abstract. Includes bibliographical references (p. 17).

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Systems of quasilinear partial differential equations of the first order, known as hydrodynamic type systems, are one of the most important classes of nonlinear partial differential equations in the modern theory of integrable systems. They naturally arise in continuum mechanics and in a wide range of applications, both in pure and applied mathematics. Deep connections between the mathematical theory of hydrodynamic type systems with differential geometry, firstly revealed by Riemann in the nineteenth century, have been thoroughly investigated in the eighties by Dubrovin and Novikov. They introduced and studied a class of Poisson structures generated by a flat pseudo-Riemannian metric, called first-order Poisson brackets of hydrodynamic type. Subsequently, these structures have been generalised in a whole variety of different ways: degenerate, non-homogeneous, higher order, multi-dimensional, and non-local. The first part of this thesis is devoted to the classification of such structures in two dimensions, both non-degenerate and degenerate. Complete lists of such structures are provided for a small number of components, as well as partial results in the multi-component non-degenerate case. In the second part of the thesis we deal with deformations of Poisson structures of hydrodynamic type. The deformation theory of Poisson structures is of great interest in the theory of integrable systems, and also plays a key role in the theory of Frobenius manifolds. In particular, we investigate deformations of two classes of structures of hydrodynamic type: degenerate one-dimensional Poisson brackets and non-semisimple bi-Hamiltonian structures associated with Balinskii-Novikov algebras. Complete classification of second-order deformations are presented for two-component structures.

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A graph G is maximal nontraceable (MNT) (maximal nonhamiltonian (MNH)) if G is not traceable (hamiltonian), i.e. does not contain a hamiltonian path (cycle), but G+xy is traceable (hamiltonian) for all nonadjacent vertices x and y in G. A graph G is hypohamiltonian if G is not hamiltonian, but every vertex deleted subgraph G -u of G is hamiltonian. A graph which is maximal nonhamiltonian and hypohamiltonian is called maximal hypohamiltonian (MHH). Until recently, not much has appeared in the literature about MNT graphs, although there is an extensive literature on MNH graphs. In 1998 Zelinka constructed two classes of MNT graphs and made the conjecture, which he later retracted, that every MNT graph belongs to one of these classes. We show that there are many different types of MNT graphs that cannot be constructed by Zelinka's methods. Although we have not been able to characterize MNT graphs in general, our attempt at characterizing MNT graphs with a specified number of blocks and cut-vertices enabled us to construct infinite families of non-Zelinka MNT graphs which have either two or three blocks. We consider MNT graphs with toughness less than one, obtaining results leading to interesting constructions of MNT graphs, some based on MHH graphs. One result led us to discover a non-Zelinka MNT graph of smallest order, namely of order 8. We also present examples of MNTgraphs with toughness at least one, including an infinite family of 2-connected, claw-free graphs. We find a lower bound for the size of 2-connected MNT graphs of order n. We construct an infinite family of 2-connected cubic MNT graphs of order n, using MHH graphs as building blocks. We thus find the minimum size of 2-connected MNT graphs for infinitely many values of n. We also present a construction, based on MHH graphs, of an infinite family of MNT graphs that are almost cubic. We establish the minimum size of MNT graphs of order n, for all except 26 values of n, and we present a table of MNT graphs of possible smallest size for the excluded 26 values of n. / Mathematical Sciences / PHD (MATHEMATICS)

The goal of this dissertation has been to develop a method that enables one to calculate accurate, rigorous lower bounds to the eigenvalues of the standard nonrelativistic spin-free Hamiltonian for an atom with N electrons. Lower bounds are necessary in order to complement upper bounds obtained from the Hartree-Fock and Rayleigh-Ritz techniques. Without accurate lower bounds, it is impossible to estimate the error of the approximate values of the energies. By combining two heretofore distinct methods and using the symmetry properties of the Hamiltonian, this goal has been achieved. The first of the two methods is the method of intermediate problems. By beginning with an appropriately chosen "base operator" H⁰, one forms a sequence of intermediate Hamiltonians Hᵏ, k = 1,2,..., whose corresponding eigenvalues form a sequence of lower bounds to the eigenvalues of the original Hamiltonian H. Complications which occurred in this method due to the stability of essential spectra under compact perturbations were later surmounted with the use of abstract separation of variables by D. W. Fox. The second technique, the effective field method, provides a lower bound operator to the interelectron repulsion term in H that is of the form of a sum of separable potentials. This latter technique reduces the eigenvalue problem for H⁰ to a sum of single particle operators. With the use of a special potential, the Hulthen potential, one may construct an explicitly solvable base problem from the effective field method, if one uses the method of intermediate problems to calculate lower bounds to non-S states. This base problem is then suitable as a starting point for the method of intermediate problems with the Fox modifications. The eigenvalues of the new base problem are already comparable to the celebrated Thomas-Fermi energies. The final part of the dissertation provides a practical procedure for determining the physically realizable spectra of the intermediate operators. This is accomplished by restricting the Hamiltonian to subspaces of proper physical symmetry so that the resulting lower bounds will be to eigenvalues of physical significance.

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Chee Shiu Chung = 阻尼振子的線空間結構與哈密頓理論 / 朱兆中. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 88). / Text in English; abstracts in English and Chinese. / Chee Shiu Chung = Zu ni zhen zi de xian kong jian jie gou yu ha mi dun li lun / Zhu Zhaozhong. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Conservative Systems --- p.4 / Chapter 2.1 --- General Formalism --- p.4 / Chapter 2.2 --- One Simple Harmonic Oscillator --- p.7 / Chapter 2.3 --- Two Coupled Harmonic Oscillators --- p.9 / Chapter 3 --- Dissipative Systems --- p.12 / Chapter 3.1 --- Elimination of Bath --- p.12 / Chapter 3.2 --- One Oscillator with Dissipation --- p.16 / Chapter 3.3 --- Two Oscillators with Dissipation --- p.19 / Chapter 4 --- Eigenvector Expansion and Bilinear Map --- p.21 / Chapter 4.1 --- Formalism --- p.21 / Chapter 4.2 --- Inner Product and Bilinear Map --- p.23 / Chapter 4.3 --- Normalization and Phase --- p.25 / Chapter 4.4 --- Matrix Representation --- p.25 / Chapter 4.5 --- Duality --- p.28 / Chapter 5 --- Applications and Examples of Eigenvector Expansion --- p.31 / Chapter 5.1 --- Single Oscillator --- p.31 / Chapter 5.2 --- Two Oscillators --- p.32 / Chapter 5.3 --- Uneven Damping --- p.33 / Chapter 6 --- Time Evolution --- p.36 / Chapter 6.1 --- Initial-Value Problem --- p.36 / Chapter 6.1.1 --- Green's Function --- p.37 / Chapter 6.2 --- Sum Rules --- p.39 / Chapter 7 --- Time-Independent Perturbation Theory --- p.41 / Chapter 7.1 --- Non-degenerate Perturbation --- p.41 / Chapter 7.2 --- Degenerate Perturbation Theory --- p.46 / Chapter 8 --- Jordan Block --- p.48 / Chapter 8.1 --- Jordan Normal Basis --- p.48 / Chapter 8.1.1 --- Construction of Basis Vectors --- p.48 / Chapter 8.1.2 --- Bilinear Map --- p.50 / Chapter 8.1.3 --- Example of Jordan Normal Basis --- p.55 / Chapter 8.2 --- Time Evolution --- p.56 / Chapter 8.2.1 --- Time Dependence of Basis Vectors --- p.56 / Chapter 8.2.2 --- Initial-Value Problem --- p.58 / Chapter 8.2.3 --- Green's Function --- p.59 / Chapter 8.2.4 --- Sum Rules --- p.60 / Chapter 8.3 --- Jordan Block Perturbation Theory --- p.61 / Chapter 8.3.1 --- Lowest Order Perturbation --- p.61 / Chapter 8.3.2 --- Higher-Order Perturbation --- p.65 / Chapter 8.3.3 --- Non-generic Perturbations --- p.66 / Chapter 8.4 --- Examples of High-Order Criticality --- p.66 / Chapter 8.4.1 --- Fourth-order JB --- p.67 / Chapter 8.4.2 --- Third-order JB --- p.74 / Chapter 8.4.3 --- Two Second-order JB --- p.79 / Chapter 9 --- Conclusion --- p.81 / Chapter A --- Appendix --- p.83 / Chapter A.l --- Fourier Transform and Contour Integration --- p.83 / Chapter B --- Degeneracy and Criticality --- p.86 / Bibliography --- p.88

Over the course of a college mathematics degree, students are inevitably exposed to elementary physics. The derivation of the equations of motion are the classic examples of applications of derivatives and integrals. These equations of motion are easy to understand, however they can be expressed in other ways that students aren't often exposed to. Using the Lagrangian and the Hamiltonian, we can capture the same governing dynamics of Newtonian mechanics with equations that emphasize physical quantities other than position, velocity, and acceleration like Newton's equations do. Building o of these alternate interpretations of mechanics and understanding gauge transformations, we begin to understand some of the mathematical physics relating to gauge theories. In general, gauge theories are eld theories that can have gauge transformations applied to them in such a way that the meaningful physical quantities remain invariant. This paper covers the buildup to gauge theories, some of their applications, and some computational approaches to understanding them.