In this thesis we study the central limit theorem (CLT) for nonuniformly hyperbolic dynamical systems. We examine cases in which polynomial decay of correlations leads to a CLT with a non-standard scaling factor of √ n ln n. We also formulate an explicit expression for the the diffusion constant σ in situations where a return time function on the system is a certain class of supermartingale. We then demonstrate applications by exhibiting the CLT for the return time function in four classes of dynamical billiards, including one previously unproven case, the skewed stadium, as well as for the linked twist map. Finally, we introduce a new class of billiards which we conjecture are ergodic, and we provide numerical evidence to support that claim.

In this episode, we add one more tool to our Boolean algebra toolbox: DeMorgan’s Theorem. We then use it, along with some of our other tools, to modify an expression down to its simplest form.

A topological group G is known as a B(𝑎) group if every continuous and almost open homomorphism from G onto a Hausdorff group is open. The permanence properties of the category of B(𝑎) groups are investigated and an internal characterization of such groups is established. Extensions of the closed graph and open mapping theorem are proved, employing this and related categories of groups. A similar concept is defined for topological semigroups, and further extensions of the open mapping and closed graph theorem are proved for them. / Thesis / Doctor of Philosophy (PhD)

<p> The electron density approach in conjunction with the Hellmann-Feynman theorem is used for a systematic analysis of binding characteristics of the two isoelectronic molecular series: N₂, CO, BF, and LiF, BeO. Electron density distributions, forces and field gradients corresponding to static properties of electron densities, have been calculated from Hartree-Fock wavefunctions (obtained from the work of other authors) for these molecules. Correlation of these static properties with binding characteristics are presented. Covalent and ionic characteristics are made evident by an analysis of the density distributions, density difference maps obtained by subtracting atomic from molecular distributions, and the forces exerted on nuclei by these distributions. A discussion of the field gradients, as related to quadrupole polarizations of the electron densities, is presented and the relevance of these polarizations to the interpretation of nuclear quadrupole coupling constants is indicated. </p> <p> Dynamic properties, as reflected by the magnitude of force constants, are analyzed in terms of functionals of the one-electron density. Force constant expressions are derived from the Hellmann-Feynman theorem. Any relation of force constants to field gradients is shown to be not unique as a result of cancellation of static and dynamic electron contributions to the total force constant. The total electronic contribution is shown to arise from a relaxation of density after a displacement of a certain nucleus. Relaxation of density with respect to one nucleus but which remains localized on some other nucleus in a molecule is shown to be equivalent to a field gradient. Thus, such density is separated from other density and its contribution to the force constant is treated as a field gradient. All contributions are computed from polynomial fits of the corresponding forces calculated at a number of internuclear distances. Relaxation density maps for the remaining atomic and overlap densities centered on a specific nucleus are presented. These maps are calculated as the difference between densities of the extended and equilibrium configurations of a molecule. The relaxation densities are correlated to the magnitude of the corresponding electronic force constant components. Thus, for the first time, there is demonstrated the concrete relation between covalent and ionic characteristics of electron densities in molecules and their dynamic properties which result in the magnitude of force constants. </p> / Thesis / Doctor of Philosophy (PhD)

<p> This thesis establishes the general Principle of Complementary Potential Energy for the finite deformations of an elastic continuum, in which the Lagrange stress tensor is employed as the stress variable. It is demonstrated that constitutive relations, formulated in terms of the Lagrange stress tensor and the deformation gradient, will admit inversion. Consequently, the present theorem and the theorem proposed by LEVINSON are established as valid principles. The complementary strain energy density of the present theorem, however, is shown to be Independent of rigid displacements, in contrast to that of the LEVINSON formulation. The general Principle is reduced to the form appropriate to finite elastic systems, and it is established that the present theorem reduces to, and therefore contains as a special case, the LIBOVE Theorem.</p> / Thesis / Doctor of Philosophy (PhD)

The relative polarization of the two photons emitted when a positron annihilates at rest has been re-investigated with high precision and a different method of data analysis. An experiment using a pair of ideal polarization analyzers to measure this relative polarization would be a special case of the general class of thought experiments discussed by Einstein, Podolsky, and Rosen (EPR). EPR argued from these thought experiments that a physical system can exist in a state with definite values for two non-commuting variables. Since quantum mechanics can not describe such a state, EPR called quantum mechanics "incomplete". But EPR believed a complete theory -sometimes called a hidden variable theory- is possible. (This argument of EPR is sometimes called the Einstein-Podolsky-Rosen "paradox".) Our experimental results, together with a theorem due to Bell, provide strong evidence that a local "hidden variable" theory is not possible. The results also rule out a hypothetical modification of quantum mechanics, suggested by Bohm and Aharonov, which was motivated by the EPR thought experiments. Compton scattering was used to analyze the linear polarization. But the theorem of Bell, mentioned above, applies to relatively "ideal" polarization measurements. Therefore, it was necessary to prove the existence, and find the explicit form of the function f relating Compton and ideal linear polarization measurements. The existence of f is shown here to follow from general principles of quantum mechanics, plus parity and angular momentum conservation; the explicit form of f is deduced from the Klein-Nishina equation. Experimental evidence is cited against the argument that f may be different in a local "hidden variable" theory.

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