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Wind power is one of the most promising renewable energy resources to replace conventional generation which carries high carbon footprints. Due to the abundance of wind and its relatively cheap installation costs, it is likely that wind power will become the most important energy resource in the near future. The successful development of wind power relies heavily on the ability to integrate wind power effciently into electricity grids. To optimize the value of wind power through careful power dispatches, techniques in forecasting the level of wind power and the associated variability are critical. Ideally, one would like to obtain reliable probability density forecasts for the wind power distributions. As wind is intermittent and wind turbines have non-linear power curves, this is a challenging task and many ongoing studies relate to the topic of wind power forecasting. For this reason, this thesis aims at contributing to the literature on wind power forecasting by constructing and analyzing various time series models and spatiotemporal models for wind power production. By exploring the key features of a portfolio of wind power data from Ireland and Denmark, we investigate different types of appropriate models. For instance, we develop anisotropic spatiotemporal correlation models to account for the propagation of weather fronts. We also develop twostage models to accommodate the probability masses that occur in wind power distributions due to chains of zeros. We apply the models to generate multi-step probability forecasts for both the individual and aggregated wind power using extensive data sets from Ireland and Denmark. From the evaluation of probability forecasts, valuable insights are obtained and deeper understanding of the strengths of various models could be applied to improve wind power forecasts in the future.

This thesis describes signal analysis methods for single-molecule fluorescence data. The primary factor motivating method development is the need to distinguish single-molecule FRET fluctuations due to conformational dynamics from fluctuations due to distance-independent FRET changes. Single-molecule Förster resonance energy transfer (FRET) promises a distinct advantage compared to alternative biochemical methods in its potential to relate biomolecular structure to function. Standard measurements assume that the mean transfer efficiency between two fluorescent probes, a donor and an acceptor, corresponds to the mean donor-acceptor distance, thus providing structural information. Accordingly, measurement analysis assumes that mean transfer efficiency fluctuations entail mean donor-acceptor distance fluctuations. Detecting such fluctuations is important in resolving molecular dynamics, as molecular function often necessitates structural changes. A problem arises, however, in that factors other than donor-acceptor distance changes may induce mean transfer efficiency fluctuations. We refer to these factors as distance-independent FRET changes. We present analysis methods to detect distance-independent photophysical dynamics and to determine their correlation with distance-dependent FRET dynamics. First, we review a theory of photon statistics and show how we can use the theory to detect FRET fluctuations. Second, we extend the theory to alternating laser excitation (ALEx) measurements and demonstrate how fluorophore stoichiometry, a measure of fluorophore brightness, reports on distance-independent photophysical dynamics. Next, we provide a measure to determine the extent to which stoichiometry fluctuations account for FRET dynamics. Finally, we use a framework similar to the preceding along with recent advances in the theory of total internal reflection fluorescence (TIRF) microscopy FRET measurements to detect TIRF FRET fluctuations which occur on a timescale faster than the measurement temporal resolution. We validate our methods with simulations and demonstrate their utility in delineating RNA polymerase open complex conformational dynamics.

n this thesis the analytic properties of power series over a class of non-Archimedean field extensions of the real numbers, a representative of which will be denoted by F, are investigated. In Chapter 1 we motivate the interest in said fields by recalling work done by K. Shamseddine and M. Berz . We first review some properties of well-ordered subsets of the rational numbers which are used in the construction of such a field F. Then, we define operations + and * which make F a field. Then we define an order under which F is non-Archimedean with infinitely small and infinitely large elements. We embed the real numbers as a subfield; and the embedding is compatible with the order. Then, in Chapter 2, we define an ultrametric on F which induces the same topology as the order on the field. This topology will allow us to define continuity and differentiability of functions on F which we shall show are insufficient conditions to ensure intermediate values, extreme values, et cetera. We shall study convergence of sequences and series and then study the analytical properties of power series, showing they have the same smoothness properties as real power series; in particular they satisfy the intermediate value theorem, the extreme value theorem and the mean value theorem on any closed interval within their domain of convergence. / October 2016