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Probabilistic bisimulation is a widely studied equivalence relation for stochastic systems. However, it requires the behavior of the states to match on actions with matching labels. This does not allow bisimulation to capture symmetries in the system. In this thesis we define lax probabilistic bisimulation, in which actions are only required to match within given action equivalence classes. We provide a logical characterization and an algorithm for computing this equivalence relation for finite systems. We also specify a metric on states which assigns distance 0 to lax-bisimilar states. We end by examining the use of lax bisimulation for analyzing Markov Decision Processes (MDPs) and show that it corresponds to the notion of a MDP homomorphism, introduced by Ravindran & Barto. Our metric provides an algorithm for generating an approximate MDP homomorphism and provides bounds on the quality of the best control policy that can be computed using this approximation.

This thesis is devoted to investigation of some properties of the permanent function over the set Omega_n of n-by-n doubly stochastic matrices. It contains some basic properties as well as some partial progress on Foregger's conjecture. CONJECTURE[Foregger] For every n\in N, there exists k=k(n)>1 such that, for every matrix A\in Omega_n, per(A^k)<=per(A). In this thesis the author proves the following result. THEOREM For every c>0, n\in N, for all sufficiently large k=k(n,c), for all A\in\Omega_n which minimum nonzero entry exceeds c, per(A^k)<=per(A). This theorem implies that for every A\in\Omega_n, there exists k=k(n,A)>1 such that per(A^k)<=per(A).

Cow-calf producers in western Canada are faced with many decisions throughout the production cycle. The choice of calving time impacts production rate, marketability of calves, income and expenses and net revenue. The purpose of this study was to determine whether June calving could increase net revenues and be a preferred choice across different risk aversion levels over March calving in western Canada. Data for this study were taken from a study carried out by Iwaasa et al. (2009), who collected information from three sites; Brandon, MB, Lanigan, SK and Swift Current SK. Stochastic budgets and a simulation model were used to study the economic impact of calving time. In Brandon and Lanigan, It was found that June calving increased net income and was the dominant alternative across all levels of risk aversion, and in Swift Current, June dominated at high-risk aversion levels.

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