This thesis applies a Partial Integral Differential Equation model, along with a Monte Carlo approach to quantitatively analyze the no arbitrage value of hedge fund performance fees. From a no-arbitrage point of view, the investor in a hedge fund is providing a free option to the manager of the hedge fund. The no-arbitrage value of this option can be locked in by the hedge fund manager using a simple hedging strategy. Interpolation methods, grid construction techniques and parallel computation techniques are discussed to improve the performance of the numerical methods for valuing this option.

It is known that orderings can be formed with settling time domination and strong settling time domination as relations on c.e. sets. However, it has been shown that no such ordering can be formed when considering computation time domination as a relation on $n$-c.e. sets where $n \geq 3$. This will be extended to the case of $2$-c.e. sets, showing that no ordering can be derived from computation time domination on $n$-c.e. sets when $n\geq 2$. Additionally, we will observe properties of the orderings given by settling time domination and strong settling time domination on c.e. sets, respectively denoted as $\mathcal{E}_{st}$ and $\mathcal{E}_{sst}$. More specifically, it is already known that any countable partial ordering can be embedded into $\mathcal{E}_{st}$ and any linear ordering with no infinite ascending chains can be embedded into $\mathcal{E}_{sst}$. Continuing along this line, we will show that any finite partial ordering can be embedded into $\mathcal{E}_{sst}$.

Partially Recursive Acceptance Rejection (PRAR) and bounding chains used in conjunction with coupling from the past (CFTP) are two perfect simulation protocols which can be used to sample from a variety of unnormalized target distributions. This paper first examines and then implements these two protocols to sample from the hardcore gas process. We empirically determine the subset of the hardcore process's parameters for which these two algorithms run in polynomial time. Comparing the efficiency of these two algorithms, we find that PRAR runs much faster for small values of the hardcore process's parameter whereas the bounding chain approach is vastly superior for large values of the process's parameter.

This thesis applies a Partial Integral Differential Equation model, along with a Monte Carlo approach to quantitatively analyze the no arbitrage value of hedge fund performance fees. From a no-arbitrage point of view, the investor in a hedge fund is providing a free option to the manager of the hedge fund. The no-arbitrage value of this option can be locked in by the hedge fund manager using a simple hedging strategy. Interpolation methods, grid construction techniques and parallel computation techniques are discussed to improve the performance of the numerical methods for valuing this option.

It is known that orderings can be formed with settling time domination and strong settling time domination as relations on c.e. sets. However, it has been shown that no such ordering can be formed when considering computation time domination as a relation on $n$-c.e. sets where $n \geq 3$. This will be extended to the case of $2$-c.e. sets, showing that no ordering can be derived from computation time domination on $n$-c.e. sets when $n\geq 2$. Additionally, we will observe properties of the orderings given by settling time domination and strong settling time domination on c.e. sets, respectively denoted as $\mathcal{E}_{st}$ and $\mathcal{E}_{sst}$. More specifically, it is already known that any countable partial ordering can be embedded into $\mathcal{E}_{st}$ and any linear ordering with no infinite ascending chains can be embedded into $\mathcal{E}_{sst}$. Continuing along this line, we will show that any finite partial ordering can be embedded into $\mathcal{E}_{sst}$.

This thesis presents a new algorithm for computing the Hermite form of a polynomial matrix. Given a nonsingular n by n matrix A filled with degree d polynomials with coefficients from a field, the algorithm computes the Hermite form of A in expected number of field operations similar to that of matrix multiplication. The algorithm is randomized of the Las Vegas type.

<p>This dissertation focuses on solving some common problems associated with ecological field studies. In the core of the statistical methodology, lies spatial modeling that provides greater flexibility and improved predictive performance over existing algorithms. The applications involve prevalence datasets for hundreds of plants over a large area in the Cape Floristic Region (CFR) of South Africa.</p><p>In Chapter 2, we begin with modeling the categorical abundance data with a multi level spatial model using background information such as environmental and soil-type factors. The empirical pattern is formulated as a degraded version of the potential pattern, with the degradation effect accomplished in two stages. First, we adjust for land use transformation and then we adjust for measurement error, hence misclassification error, to yield the observed abundance classifications. With data on a regular grid over CFR, the analysis is done with a conditionally autoregressive prior on spatial random effects. With around ~ 37000 cells to work with, a novel paralleilization algorithm is developed for updating the spatial parameters to efficiently estimate potential and transformed abundance surfaces over the entire region.</p><p>In Chapter 3, we focus on a different but increasingly common type of prevalence data in the so called <italic>presence-only</italic> setting. We detail the limitations associated with a usual presence-absence analysis for this data and advocate modeling the data as a point pattern realization. The underlying intensity surface is modeled with a point-level spatial Gaussian process prior, after taking into account sampling bias and change in land-use pattern. The large size of the region enforces using an computational approximation with a bias-corrected predictive process. We compare our methodology against the the most commonly used maximum entropy method, to highlight the improvement in predictive performance.</p><p>In Chapter 4, we develop a novel hierarchical model for analyzing noisy point pattern datasets, that arise commonly in ecological surveys due to multiple sources of bias, as discussed in previous chapters. The effect of the noise leads to displacements of locations as well as potential loss of points inside a bounded domain. Depending on the assumption on existence of locations outside the boundary, a couple of different models -- <italic>island</italic> and <italic>subregion</italic>, are specified. The methodology assumes informative knowledge of the scale of measurement error, either pre-specified or learned from a training sample. Its performance is tested against different scales of measurement error related to the data collection techniques in CFR.</p><p>In Chapter 5, we suggest an alternative model for prevalence data, different from the one in Chapter 3, to avoid numerical approximation and subsequent computational complexities for a large region. A mixture model, similar to the one in Chapter 4 is used, with potential dependence among the weights and locations of components. The covariates as well as a spatial process are used to model the dependence. A novel birth-death algorithm for the number of components in the mixture is under construction.</p><p>Lastly, in Chapter 6, we proceed to joint modeling of multiple-species datasets. The challenge is to infer about inter-species competition with a large number of populations, possibly running into several hundreds. Our contribution involves applying hierarchical Dirichlet process to cluster the presence localities and subsequently developing measures of range overlap from posterior draws. This kind of simultaneous inference can potentially have implications for questions related to biodiversity and conservation studies. .</p> / Dissertation

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This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.

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