We analyze the Schrödinger equation i𝜖 ¬²â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬^-4 and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬²). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems. / Ph. D.

The integration of deep learning models and classical techniques in robotics is constantly creating solutions to problems once thought out of reach. The issues arising in most models that work involve the gap between experimentation and reality, with a need for strategies that assess the risk involved with different models when applied in real-world and safety-critical situations. This work proposes the use of Bayesian approximations of uncertainty from deep learning in a robot planner, showing that this produces more cautious actions in safety-critical scenarios. The case study investigated is motivated by a setup where an aerial robot acts as a "scout'' for a ground robot when the below area is unknown or dangerous, with applications in space exploration, military, or search-and-rescue. Images taken from the aerial view are used to provide a less obstructed map to guide the navigation of the robot on the ground. Experiments are conducted using a deep learning semantic image segmentation, followed by a path planner based on the resulting cost map, to provide an empirical analysis of the proposed method. The method is analyzed to assess the impact of variations in the uncertainty extraction, as well as the absence of an uncertainty metric, on the overall system with the use of a defined factor which measures surprise to the planner. The analysis is performed on multiple datasets, showing a similar trend of lower surprise when uncertainty information is incorporated in the planning, given threshold values of the hyperparameters in the uncertainty extraction have been met. / Master of Science / Deep learning (DL) is the phrase used to refer to the use of large hierarchical structures, often called neural networks, to approximate semantic information from data input of various forms. DL has shown superior performance at many tasks, such as several forms of image understanding, often referred to as computer vision problems. Deep learning techniques are trained using large amounts of data to map input data to output interpretation. The method should then perform correct input-output mappings on new data, different from the data it was trained on. Robots often carry various sensors from which it is possible to make interpretations about the environment. Inputs from a sensor can be high dimensional, such as pixels given by a camera, and processing these inputs can be quite tedious and inefficient given a human interpreter. Deep learning has recently been adopted by roboticists as a means of automatically interpreting and representing sensor inputs, like images. The issue that arises with the traditional use of deep learning is twofold: it forces an interpretation of the inputs even when an interpretation is not applicable, and it does not provide a measure of certainty with its outputs. Many techniques have been developed to address this issue with deep learning. These techniques aim to produce a measure of uncertainty associated with DL outputs, such that even when an incorrect or inapplicable output is produced, it is accompanied with a high level of uncertainty. To explore the efficacy and applicability of these uncertainty extraction techniques, this thesis looks at their use as applied to part of a robot planning system. Specifically, the input to the robot planner is an overhead image taken by an unmanned aerial vehicle (UAV) and the output is a path from a set start and goal position to be taken by an unmanned ground vehicle (UGV) below. The image is passed through a deep learning portion of the system that performs what is called semantic segmentation, mapping each pixel to a meaningful class, on the image. Based on the segmentation, each pixel is given a cost proportionate to the perceived level of safety associated with that class. A cost map is thus formed on the entire image, from which traditional robotics techniques are used to plan a path from start to goal. A comparison is performed between the risk-neutral case which uses the conventional DL method and the risk-aware case which uses uncertainty information accompanying the modified DL technique. The overall effects on the robot system are envisioned by observing a metric called the surprise factor, where a high surprise factor signifies a poor prediction of the actual cost associated with a path. The risk-neutral case is shown to have a higher surprise factor than the proposed risk-aware setup, both on average and in safety-critical case studies.

<p> This thesis examines continued fraction expansions of the square root of nonsquare positive integers of periods one to six, and shows their relationships with Newton's method of approximation. It also contains known results concerning continued fractions.</p> / Thesis / Master of Science (MSc)

Chlodovsky showed that if x0 is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bn(f, x0) converge to the average of the one-sided limits on the right and on the left of the function f at the point x0. In 2009, Telyakovskii in (5) extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same result fails for the odd order case. Then in 2010, Tonkov in (6) found the right formulation and proved the result that was missing in the odd-order case. It turned out that the limit in the odd order case is related to the jump of the highest derivative. The proofs in these two cases look similar but have many subtle differences, so it is desirable to find out if there is a unifying principle for treating both cases. In this thesis, we obtain a unified formulation and proof for the asymptotic results of both Telyakovskii and Tonkov and discuss extension of these results in the case where the highest derivative of the function is only assumed to be bounded at the point under study.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.