Advances in classical and quantum wave dynamics on quasiperiodic lattices : a dissertation submitted for the degree of Doctor of Philosophy in Physics, Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University, Albany, New Zealand

Lattices and discrete networks are cornerstones of a number of scientific subjects. In condensed matter, optical lattices allowed the experimental realization of several theoretically predicted phenomena. Indeed, these structures constitute ideal benchmarks for light and wave propagation experiments involving interacting particles, such as clouds of ultra-cold atoms that Bose-Einstein condensate. Moreover, they allow experimental design of particular lattice topologies, as well as the implementation of several classes of spatial perturbations. For example, Anderson localization being observed for the first time in atomic Bose-Einstein condensate experiments and Aubry-André localization discovered with light propagating through networks of optical waveguide. This thesis considers different types of lattices in the presence of quasiperiodic modulations, mainly the celebrated Aubry-André potential. Particular attention will be given to spectral properties of models, localization features of eigenmodes and the transition from delocalized (metallic) eigenstates to localized (insulating) ones within the energy spectrum. We additionally discuss the relation between the model’s properties and the dynamics of particles hopping along the lattice. After introducing the linear discrete Schrödinger equation, we first discuss the spectral properties of the Aubry-André model. We then study the transition between metallic and insulating regimes of a class of quasiperiodic potentials constructed as an iterative superposition of periodic potentials with increasing spatial period. Next, we discuss the Aubry-André perturbation of flat-band topologies, their energy-dependent transition (mobility edge), which can be expressed in analytical forms in case of specific onsite energy correlations, highlighting existence of zeroes, singularities and divergences. We then discuss two cases of driven one-dimensional lattices, namely an Aubry-André chain with a weak time-space periodic driving and an Anderson chain with a quasiperiodic multi-frequency driving. We show anaytically and numerically how drivings can lift the respective localization and generate delocalization by design. Finally we discuss the problem of the possible generation of correlated metallic states of two interacting particles problem in one dimensional Aubry-André chains, under a coherent drive of the interaction

Detection of chromosomal regions showing differential gene expression in human skeletal muscle and in alveolar rhabdomyosarcoma

BACKGROUND: Rhabdomyosarcoma is a relatively common tumour of the soft tissue, probably due to regulatory disruption of growth and differentiation of skeletal muscle stem cells. Identification of genes differentially expressed in normal skeletal muscle and in rhabdomyosarcoma may help in understanding mechanisms of tumour development, in discovering diagnostic and prognostic markers and in identifying novel targets for drug therapy. RESULTS: A Perl-code web client was developed to automatically obtain genome map positions of large sets of genes. The software, based on automatic search on Human Genome Browser by sequence alignment, only requires availability of a single transcribed sequence for each gene. In this way, we obtained tissue-specific chromosomal maps of genes expressed in rhabdomyosarcoma or skeletal muscle. Subsequently, Perl software was developed to calculate gene density along chromosomes, by using a sliding window. Thirty-three chromosomal regions harbouring genes mostly expressed in rhabdomyosarcoma were identified. Similarly, 48 chromosomal regions were detected including genes possibly related to function of differentiated skeletal muscle, but silenced in rhabdomyosarcoma. CONCLUSION: In this study we developed a method and the associated software for the comparative analysis of genomic expression in tissues and we identified chromosomal segments showing differential gene expression in human skeletal muscle and in alveolar rhabdomyosarcoma, appearing as candidate regions for harbouring genes involved in origin of alveolar rhabdomyosarcoma representing possible targets for drug treatment and/or development of tumor markers

Intermittent many-body dynamics at equilibrium

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.We thank P. Jeszinszki and I. Vakulchyk for helpful discussions on computational aspects. The authors acknowledge financial support from IBS (Project Code No. IBS-R024-D1). (IBS-R024-D1 - IBS)Published versio