Two new integral transforms and their applications

This thesis is in two parts. In Part I the independent variable θ in the trigonometric form of Legendre's equation is extended to the range ( -∞, ∞). The associated spectral representation is an infinite integral transform whose kernel is the analytic continuation of the associated Legendre function of the second kind into the complex θ-plane. This new transform is applied to the problems of waves on a spherical shell, heat flow on a spherical shell, and the gravitational potential of a sphere. In each case the resulting alternative representation of the solution is more suited to direct physical interpretation than the standard forms.

In Part II separation of variables is applied to the initial-value problem of the propagation of acoustic waves in an underwater sound channel. The Epstein symmetric profile is taken to describe the variation of sound with depth. The spectral representation associated with the separated depth equation is found to contain an integral and a series. A point source is assumed to be located in the channel. The nature of the disturbance at a point in the vicinity of the channel far removed from the source is investigated.

Bimetallic olefin polymerization catalysis : mechanisms and applications of proximal effects

This dissertation covers progress with bimetallic polymerization catalysts. The complexes we have designed were aimed at expanding the capabilities of homogeneous polymerization catalysts by taking advantage of multimetallic effects. Such effects were examined in group 4 and group 10 bimetallic complexes; proximity and steric repulsion were determined to be major factors in the effects observed.

Chapters 2 and 3 introduce the rigid p-terphenyl dinucleating framework utilized in most of this thesis. The permethylation of the central arene allows for the separation of syn and anti atropisomers of the terphenyl compounds. Kinetic studies were carried out to examine the isomerization of the dinucleating bis(salicylaldimine) ligand precursors. Metallation of the syn and anti bis(salicylaldimine)s using Ni(Me)2(tmeda) and excess pyridine afforded dinickel bisphenoxyiminato complexes with a methyl and a pyridyl ligand on each nickel. The syn and anti atropisomers of the dinickel complexes were structurally characterized and utilized in ethylene and ethylene/α-olefin polymerizations. Monometallic analogues were also synthesized and tested for polymerization activity. Ethylene polymerizations were performed in the presence of primary, secondary, and tertiary amines – additives that generally deactivate nickel polymerization catalysts. Inhibition of this deactivation was observed with the syn atropisomer of the bimetallic species, but not with the anti or monometallic analogues. A mechanism was proposed wherein steric repulsion of the substituents on proximal nickel centers disfavors simultaneous ligation of base to both of the metal centers. The bimetallic effect has been explored with respect to size and binding ability of the added base.

Chapter 4 presents the optimization of the bisphenoxyimine ligand synthesis and synthesis of syn and anti m-terphenyl analogues. Metallation with NiClMe(PMe₃)₂ yielded phosphine-ligated dinickel complexes, which have been structurally characterized. Ethylene/1-hexene copolymerizations in the presence of amines using Ni(COD)₂ as a phosphine scavenger showed significantly improved activity relative to the pyridine-ligated analogues. Incorporation of amino olefins in copolymerizations with ethylene was accomplished, and a mechanism was proposed based on proximal effects. Copolymerization trials with a variety of amino olefins and ethylene/1-hexene/amino olefin terpolymerizations were completed.

Early transition metal complexes based on the rigid p-terphenyl framework were designed with a variety of donor sets (Chapter 5 and Appendix B). Chapter 5 details the use of syn dizirconium di[amine bis(phenolate)] complexes for isoselective 1-hexene and propylene homopolymerizations. Ligand variation and monometallic complexes were studied to determine the origin of tacticity control. A mechanistic proposal was presented based on the symmetry at zirconium and the steric effects of the proximal metal center. Appendix B covers additional studies of bimetallic early transition metal complexes based on the p-terphenyl. Dititanium, dizirconium, and asymmetric complexes with bisphenoxyiminato ligands and derivatives thereof were targeted. Progress toward the synthesis of these complexes is described along with preliminary polymerization data. 1-hexene/diene copolymerizations and attempted polymerizations in the presence of ethers and esters with the syn dizirconium di[amine bis(phenolate)] complexes demonstrate the potential for further applications of this system in catalysis.

Appendix A includes work toward palladium catalysts for insertion polymerization of polar monomers. These complexes were based on dioxime and diimine frameworks with the intent of binding Lewis acidic metals at the oxime oxygens, at pendant phenolic donors, or at pendant aminediol moieties. The synthesis and structural characterization of a number of palladium and Lewis acid complexes is presented. Due to the instability of the desired species, efforts toward isolation of the desired complexes proved unsuccessful, though preliminary ethylene/methyl acrylate copolymerizations using in situ activation of the palladium species were attempted.

Multi-model inference ranking and applications to physics at the Large Hadron Collider

In the measurement of the Higgs Boson decaying into two photons the parametrization of an appropriate background model is essential for fitting the Higgs signal mass peak over a continuous background. This diphoton background modeling is crucial in the statistical process of calculating exclusion limits and the significance of observations in comparison to a background-only hypothesis. It is therefore ideal to obtain knowledge of the physical shape for the background mass distribution as the use of an improper function can lead to biases in the observed limits. Using an Information-Theoretic (I-T) approach for valid inference we apply Akaike Information Criterion (AIC) as a measure of the separation for a fitting model from the data. We then implement a multi-model inference ranking method to build a fit-model that closest represents the Standard Model background in 2013 diphoton data recorded by the Compact Muon Solenoid (CMS) experiment at the Large Hadron Collider (LHC). Potential applications and extensions of this model-selection technique are discussed with reference to CMS detector performance measurements as well as in potential physics analyses at future detectors.