Magnetic Resonance Spectroscopy is a powerful non-invasive tool for investigating liver metabolism in vivo. PRESS and STEAM localized 1H MRS can be used to provide liver lipid measurements calculated from the fat to water peak ratios. Surface coils can also be used to measure other metabolites using multinuclear MRS, 13C MRS directly measures glycogen levels and 31PMRS measures ATP and other phosphate metabolites. This thesis reports on studies undertaken to develop these techniques and describes a number of in vivo investigations that explored the effects of dietary interventions on hepatic metabolism.

The human brain can be divided into multiple areas, each responsible for different aspects of behaviour. For a century we have been developing techniques to non-invasively map these areas and their associated functions, a discipline now known as neuroimaging. In recent years the field has undergone a paradigm shift to investigate how the brain communicates with itself; it is widely regarded that healthy brain function relies upon efficient connectivity between different functional areas, and the neuroimaging field has been revolutionised by our ability to estimate this connectivity. Studies into communication between spatially separate locations in the brain have revealed a series of robust functional networks which govern mental processes. However these studies have been based on the temporal averaging of minutes or even hours of data to give us a generalised ’snapshot’ of connectivity. Increasing evidence shows us that these connections are dynamic in space, time and frequency and so the next generation of of neuroimaging methods, which capture this 5-dimensional connectivity will prove to be key tools in the investigation of brain networks and ultimately their breakdown in disease. In this thesis we introduce novel methods to capture non-stationarity using magnetoencephalography (MEG), an imaging modality which measures the changes in extracranial magnetic fields associated with neuronal current flow. MEG is a direct measurement of neural activity and has an excellent temporal resolution, which makes it attractive for non-invasively tracking dynamic functional connections. However there are many technical limitations which can confound assessment of functional connectivity which have to be addressed. In Chapters 2 and 3 we introduce the theory behind MEG; specifically how it is possible to measure the femtoTelsa changes in magnetic field generated by the brain and how to project these data to generate a 3-dimensional picture of current in the brain. Chapter 4 reviews some of popular methods of assessing functional connectivity and how to control for the influence of artefactual functional connections erroneously produced during source projection. Chapter 5 introduces a pipeline to assess functional connections across time, space and frequency and in Chapter 6 we apply this pipeline to show that resting state networks, measured using ’static’ metrics are in fact comprised of a series of rapidly forming and dissolving subnetwork connections. Finally, Chapter 7 introduces a pipeline to track dynamic network behaviour simultaneously across the entire brain volume and shows that networks can be characterised by their temporal signatures of connectivity.

Scanning Tunnelling Microscopy (STM) is the fastest possible method of imaging the molecular orbitals of the C[subscript]60 anions with resolution at the single atom level. For the particular anions of fullerene C[subscript]60, the splitting of the molecular orbitals due to the internal Jahn-Teller effects (JT) add further difficulties in understanding the published experimental images. In the current work, the effect of JT interaction on STM recorded images is studied. For higher charged states, the Coulomb interaction affects the distribution of electrons around the ion, and then as a consequence, the STM current. The external interaction between the molecule and the surface substrate is equally important. Symmetry analysis using group theory and Hückel molecular orbital (HMO) theory are applied in order to describe the influence of the surface interactions on JT minima associated with D[subscript]3d, D[subscript]5d, D[subscript]2h, and C[subscript]2h symmetries. It represents some fullerene anions, which are adsorbed to the surface with different orientations, such as pentagon, hexagon, and double-bond prone toward the surface. Several ions with higher charges are investigated, such as C2−60, C3−60, and C4−60. In case of high symmetry orientations, the JT minima of the ions on a surface are split into subgroups with equal energies, depending on the type of orientation. The interpretation of the experimental observations is always possible for any orientation from the JT minima distribution and the contribution to the images from different components of the degenerate molecular orbitals.

This is a mathematical study of certain aspects of the interacting electron system in very high perpendicular magnetic field. We analyse restrictions imposed upon the density correlation functions of this system and propose a set of sum rules which they must obey. We study the possibility of building a bosonisation scheme for the projected density operators in the lowest Landau level. We suggest a second order bosonisation, along with an approximation scheme, which may be useful for carrying out calculations in the lowest Landau level. We analyse the possible ground states of the system. We suggest a set of variational wavefunctions which can have lower energy than the Laughlin state for sufficiently soft interaction potentials. We study the collective excitations of the system, paying particular attention to its symmetries. We suggest a set of variational excited states and discuss their applicability to finite as well as infinite systems.