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The principal goal of this work was to synthesise and fully characterise a range of platinum group metal chelates of bis(pyridine-imine) ligands. These four-nitrogen donor Schiff base ligands are underdeveloped relative to their salen (ONNO donor) counterparts. The purified metal complexes were to be tested for their cytotoxicity against cancer cell lines and their mode of interaction (expected to be intercalation) studied. The syntheses, spectroscopic and structural properties of some novel and some already known bis(pyridine-imine) ligands are described. Furthermore, UV-Vis, IR and NMR spectroscopy, as well as electrospray ionisation mass spectrometry, have been used to characterize the ligands and comparisons were made with relevant literature. Fifteen Schiff base ligands were successfully condensed from a 2-formyl or 2-ketopyridine starting material and a diamine bridging group, while the attempted syntheses of a further three are described. Literature methods or variations thereof were employed in the syntheses of these derivatives, which generally resulted in good yields. X-ray quality crystals were obtained and X-ray structures were determined for four novel ligands and five unexpected cyclised hexahydropyrimidine- and imidazole-containing bidentate ligands, of which three were novel structures. The series of bicationic palladium and platinum complexes synthesised here were analysed by NMR, IR and UV-Vis techniques, as well as X-ray crystallography when possible. The complexes were prepared by reacting the free ligands with platinum group metal salts in refluxing acetonitrile. The complexes exhibit infrared bands for the imine C=N stretch between 1604–1670 cm-1; around 15–40 cm-1 lower than the free ligands. The 1H NMR spectra in the CH=N chemical shift region also display shifts (0.1 to 0.8 ppm for the palladium complexes or 0.4 to 0.9 ppm for the platinum complexes) which are consistent with metallation. X-ray crystal structures were obtained for eight novel metal complexes, which all crystallised in the monoclinic crystal system. The solid state analysis shows changes in the free ligands upon introduction of the metal ions, caused by the coordination process. Metallation of the free ligands led to twisting of the ligands due to size effects and the spatial restrictions of the coordination geometry of the central metal ions. The structures were generally solved by direct methods and refined to R1 = 0.0729 or less. Singlecrystal X-ray diffraction analysis confirmed that the mononuclear complexes exhibit a distorted square planar coordination sphere composed of the four donor nitrogen atoms (two imine and two pyridyl nitrogen atoms) from the Schiff base ligands. The complexes were generally very stable and differed by the type of metal ion (platinum(II) or palladium(II)) and the diamine bridging group (2-carbon or 3-carbon linking chain with various substituted groups). A range of unconventional F···H−C contacts is revealed to play an important role in the overall bonding and crystal packing of many complexes. To better understand our measured data and to separate the intrinsic properties of our molecules from intermolecular interactions, theoretical calculations at the DFT (B3LYP) level were carried out. These calculations predict the structural and spectroscopic properties for the free ligands and their metallated counterparts. DFT simulations were performed for the fifteen synthesised ligands (B3LYP functional, 6-31G** basis set), as well as three projected ligands that could not be synthesised, and for all proposed thirtysix metal complexes (B3LYP functional, SDD basis set). DFT simulations were used to obtain theoretical IR frequency data which was compared to the literature and used to prove the location of a local minimum energy structure in the geometry optimisation rather than a transition state. Our results collectively showed the B3LYP level of theory to be useful in the prediction of IR frequencies for these Schiff base ligands as well as the platinum(II) and palladium(II) complexes. Geometry optimisations performed using density functional theory for the free ligands and metal complexes were compared, where possible, with X-ray data. For the free ligands the main structural differences were observed for the position of the pyridyl-imine "arms" of the ligands with relatively good agreement between the two structures for the bridging groups. On the other hand, the metal complexes showed greater discrepancies for the bridging groups rather than for the pyridine rings. These differences were also affected by the length of the carbon bridge; the longer the carbon bridge, the less variation was observed. The trends in the variation of bond distances and angles with the metal ion identity and ligand structure delineated by DFT simulations were matched by similar trends in the X-ray data. Differences between the gas phase calculated geometries and those determined by Xray diffraction were attributed to packing effects in the solid state and intermolecular interactions not being accounted for during DFT computations. The impressive similarity observed between the structures obtained from X-ray crystallography and computed by DFT shows the applicability of these computations at the B3LYP level of theory. They were able to accurately predict structural and spectroscopic properties for the ligands and complexes presented in this work. The in vitro cytotoxicities of some of the metal complexes were evaluated against two mouse cancer cell lines. The compounds tested had lower than expected cytotoxicity towards the cell lines studied with IC50 values > 100 μM. The interaction of the complexes with calf thymus DNA has been explored by using absorption studies. From the observed changes in absorption possible modes of binding to DNA have been proposed for the metal complexes. Significant changes were observed in the absorption spectra upon interaction of the complexes with DNA, from which large binding constants were determined. The changes were, however, not as intense as expected nor were the spectral variations typical of intercalating drugs. There were also no bathochromic shifts nor any isobestic points observed for most of the metal complexes, except one. For the cationic PGM chelates studied, the lack of shift in the wavelength of the visible range MLCT band maximum and small changes in the band intensity were more indicative of weak electrostatic interactions. Our spectral data were thus interpreted as being consistent with non-intercalative binding; either simple electrostatic adduct formation or major/minor groove binding. The binding constant values (Kb) of the metal complexes estimated from the titration with calf thymus DNA monitored by absorption spectroscopy were all in the range of 105 M-1. This is within the range of those reported for other platinum(II) and palladium(II) metal complexes that have shown intercalation and/or groove binding. With the lack of inhibition of growth of the selected cancer cell lines it is possible that the compounds do not reach nuclear DNA in living cells or that such compounds are possibly substrates for efflux transporters. All the metal complexes in this work were determined to not be pure DNA intercalators as had been expected; however, in some cases partial intercalation cannot be ruled out from the spectral data. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

Les algèbres amassées sont des classes d'algèbres introduites dans les annés 2000 par Sergey Fomin et Andrei Zelevinsky, dans leurs recherches sur les bases canoniques duales et la positivité totale dans les groupes semi-simples (voir [FZ02] et [FZ03]). Il est possible de les étudier de façon combinatoire, en les illustrant par des carquois sans boucle, ni deux-cycles. Dans ce travail, nous introduirons d'abord les notions d'algèbres amassées et de carquois. Puis, nous nous servirons des carquois pour introduire et étudier les suites maximales de mutations vertes. Plus précisément, dans la dernière partie du travail, il sera question de l'existence des suites maximales vertes dans les carquois acycliques à trois points.

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Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

Thesis (M.S.)--University of Michigan, Ann Arbor, 1974. / Typescript (photocopy). Includes bibliographical references (leaves 34-36). Also issued in print.

Thèse (M.Sc.)--Université Laval, 2007. / Titre de l'écran-titre (visionné le 5 mai 2008). Bibliogr.

Mémoire (M.Eng.)--Univesité du Québec à Chicoutimi, 2000. / Document électronique également accessible en format PDF. CaQCU

Thèses (M.Sc.)--Université de Sherbrooke (Canada), 1999. / Titre de l'écran-titre (visionné le 20 juin 2006). Publié aussi en version papier.