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Many studies measuring the occurrence of pharmaceuticals, understanding their environmental fate and the risk they pose to surface water resources have been published. However, very little is known about the relevant transformation products which result from the wide range of biotic and abiotic degradation processes that these compounds undergo in sewers, storage tanks, during engineered treatment and in the environment. Thus, the present study primarily investigated the degradation of the steroid progesterone (P4) in natural systems (rivers), with a focus on the identification and characterisation of transformation products. Initial work focussed on assessing the removal of selected compounds (Diclofenac, Fluoxetine, Propranolol and P4) from reed beds, with identification of transformation products in a field site being attempted. However, it was determined that concentrations of parent compounds and products would be too low to work with in the field, and a laboratory study was designed which focussed on P4. Focus on P4 was based on literature evidence of its rapid biodegradability relative to the other model compounds and its usage patterns globally. River water sampling for the laboratory-based degradation study was carried out at 1 km downstream of four south east England sewage works (Blackbirds, Chesham, High Wycombe and Maple Lodge) effluent discharge points. Suspected P4 transformation products were initially identified from predictions by the EAWAG Biocatalysis Biodegradation Database (EAWAG BBD) and from a literature review. At a later stage of the present work, a replacement model for EAWAG BBD (enviPath) which became available, was used to predict P4 degradation and results were compared. Samples were analysed using low resolution and accurate-mass time-of-flight mass spectrometers. Three degradation studies were conducted. Sampling for all studies was carried out at the same time in the year to minimize temporal variability in conditions and allow for effective comparison of results. Androgenic and progesterone yeast screens were carried out to assess the biological activity of transformation products.

The theory of Grobner bases has garnered the interests of a large number of researchers in computational algebra due to its applications not only in mathematics but also in areas like control systems, robotics, cryptography to name a few. It is well known that the computation of Grobner bases takes time doubly exponential in the number of indeterminates rendering it impractical in all but a few places.The current known algorithms for Grobner bases depend on the term order over which Grobner bases is computed. In this thesis, we study computational complexity of some problems in computational ideal theory. We also study the algebraic formulation of combinatorial optimization problems. Gritzmann and Sturmfels (1993) posed the following question: Given a set of generators, decide whether it is a Gr¨obner bases with respect to some term order. This problem, termed as the Grobner Basis Detection(GBD)problem, was introduced as an application of Minkowski addition of polytopes. It was shown by Sturmfels and Wiegelmann (1997) that GBD is NP-hard. We study the problem for the case of zero-dimensional ideals and show that the problem is hard even in this special case. We study the detection problem in the case of border bases which are an alternative to Grobner bases in the case of zero dimensional ideals. We propose the Border Basis Detection(BBD) problem which is defined as follows: Given a set of generators of an ideal, decide whether that set of generators is a border basis of the ideal with respect to some order ideal. It is shown that BBD is NP-complete. We also formulate the rainbow connectivity problem as a system of polynomial equations such that solving the polynomial system yields a solution to it. We give an alternate formulation of the rainbow connectivity problem as a membership problem in polynomial ideals.