The classical method of solving random walk problems involves using Markov chain theory. When the particular random walk of interest is written in matrix form using Markov chain theory, the problem must then be solved using a digital computer. To solve all but the most trivial random walk problems by hand would be extremely difficult and time consuming. Very large random walk problems may even prove difficult to solve on the smaller digital computers. This paper intends to demonstrate a method that may be used to solve large random walk problems in a quick and economical manner. This alternate method uses resistive analogues and has the added feature of extracting particular solutions without having to completely solve the problem as would be necessary using a digital computer. Many analogues of random walks may also be quickly amended to include other random walks with relative ease using this alternate method of solution. Because this method uses nothing more than a power supply, a DC voltmeter and a set of resistors, the analogue of a particular random walk problem may be left set-up without incurring any loss of time or money on a digital computer. Once the resistors are mounted in a permanent fashion, the random walk analogues may also be used as an effective demonstration of random walk probabilities in the classroom.

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A model of non-Markovian random block copolymers is developed and used to study the effects of blockiness and compositional polydispersity on the phase behaviour of random block copolymers. The model approximates a random copolymer as a series of segments with equal lengths, while each segment is composed of sequences of different monomers drawn randomly from a distribution. The phase behaviour of the model random copolymers is first examined using the random phase approximation (RPA) to study the effects of blockiness and polydispersity on the order-disorder transition. It is observed that the critical point is inversely proportional to the blockiness. Compositional polydispersity is found to facilitate phase separation, and could induce macrophase separation. Next, the model is implemented into self-consistent field theory (SCFT) in order to elucidate the full phase behaviour of symmetric (A/B)-A random copolymers. Finally, the model is applied to the particular case of poly(styrenesulfonate-b-methylbutylene) (PSS-PMB) to study the effects of blockiness on the phase behaviour. In particular, the stability and structure of the `swollen gyroid' morphology predicted by previous Monte Carlo simulations is examined. / Thesis / Master of Science (MSc)

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This dissertation has two parts. In the first part, we focus on random inner product kernel matrices. Under various assumptions, many authors have proved that the limiting empirical spectral distribution (ESD) of such matrices A converges to the Marchenko- Pastur distribution. Here, we establish the corresponding rate of convergence. The strategy is as follows. First, we show that for z = u + iv ∈ C, v > 0, the distance between the Stieltjes transform m_A (z) of ESD of matrix A and Machenko-Pastur distribution m(z) is of order O (log n \ nv). Next, we prove the Kolmogorov distance between ESD of matrix A and Marchenko-Pastur distribution is of order O(3\log n\n). It is the less sharp rate for much more general class of matrices. This uses a Berry-Esseen type bound that has been employed for similar purposes for other families of random matrices. In the second part, random geometric graphs on the unit sphere are considered. Observing that adjacency matrices of these graphs can be thought of as random inner product matrices, we are able to use an idea of Cheng-Singer to establish the limiting for the ESD of these adjacency matrices. / Mathematics

My objective was to predict outcomes following police interactions with PMIs, and compare the predictive accuracy of logistic regression models and Random Forests learning algorithms. Additionally I evaluated if predictive accuracy of Random Forests changed when applied to merged versus region-specific data. I conducted a retrospective cohort study of reports completed by police in 13 communities between 2015 and 2018. 13,058 reports were analyzed. Random Forests learning algorithms were compared against logistic regression models for predictive accuracy in a merged dataset (13 communities) and 3 regional datasets. Outcomes for prediction were high risk of harm to self, risk of harm to others, and risk of failure to care for self within 24 and 72 hours following police contact. Random Forests learning algorithms were trained on merged and regional datasets, and compared against merged and regional holdout datasets. Performance was compared by area under the curve. For Random Forests learning algorithms, confusion matrix statistics were calculated for each outcome and predictive utility was examined by calculating conditional probabilities. Prediction accuracy was modest across all methods. Random Forests achieved better predictive accuracy than logistic regression. Random Forests accuracy varied between merged and regional holdout data. Sensitivity of Random Forests learning algorithms were moderate (74% average, 6 outcomes, merged holdout set). Specificity was low (53% average, 6 outcomes, merged holdout set). Conditional probabilities were modestly improved by the use of the Random Forests learning algorithm. The rareness of the target outcomes created a situation where even predictions with moderate likelihood ratios had only modest predictive value. Though the Random Forests learning algorithms did outperform the logistic regression learning algorithms, the clinical significance of those benefits were limited when conditional probabilities were calculated. These findings are limited to the outcomes considered, and may not apply to more common outcomes. / Thesis / Master of Health Sciences (MSc) / The study goal was to predict outcomes following police interactions with persons with mental illness (PMIs). Additionally we compare the predictive validity of logistic regression and Random Forests learning algorithms. Classification approaches were applied to outcomes following police interactions with PMIs, including: high risk of harm to self, high risk of harm to others, and high risk of failure to care for self within 24 hours and 72 hours of initial police contact. The study also sought to determine if the predictive accuracy of Random Forests was sensitive to the police service community. Variation in predictive accuracy was assessed between a merged data set (13 communities) and 3 community-specific data. The study found that the predictive accuracy of the classification approaches on outcomes was modest. Random Forests exhibited greater predictive validity than logistic regression. The performance of the Random Forests suggested that performance was not sensitive to police service context.