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This thesis discusses some of the invariants of rational elliptic surfaces, namely the Mordell-Weil Group, Mordell-Weil Lattice, and another lattice which will be called the Shioda Lattice. It will begin with a brief overview of rational elliptic surfaces, followed by a discussion of lattices, root systems and Dynkin diagrams. Known results of several authors will then be applied to determine the groups and lattices associated with a given rational elliptic surface, along with a discussion of the uses of these groups and lattices in classifying surfaces.

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A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed

This thesis is concerned with a solution theory for quasilinear singular stochastic partial differential equations. We approach the theory of regularity structures, a tool to tackle singular stochastic PDEs, from a new perspective which is well suited for, but not restricted to, quasilinear equations. In the first part of this thesis, we revisit the algebraic aspects of the theory of regularity structures. Although we approach regularity structures from a different perspective than originally done, we show that the same (Hopf-) algebraic structure is underlying. Trees do not play any role in our construction, hence the Hopf algebras underlying rough paths and regularity structures are not at our disposal. Instead, our alternative point of view gives a new (Lie-) geometric interpretation of the structure group, arising from simple actions on the nonlinearity of the equation and a parametrization of the solution manifold. In the second part of this thesis, we revisit the probabilistic aspects of the theory of regularity structures. We construct and stochastically estimate the centered model, which captures the local behaviour of the solution manifold. This is carried out under a spectral gap assumption on the driving noise, and based on a novel application of Malliavin calculus in regularity structures. In deriving the renormalized equation we are guided by symmetries, so that natural invariances of the model are built in. In the third part of this thesis, we make again use of the Malliavin derivative to obtain a robust characterization of the model, which persists for rough noise even as a mollification is removed. This allows for a simple derivation of invariances of the model that are not present at the level of approximations. Furthermore, we give a convergence result of models, which together with the characterization establishes a universality result in the class of noise ensembles satisfying uniformly a spectral gap assumption.

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematic areas. Despite they have been studied for decades, several aspects of their geometry are still mysterious, among which identifiability and singularity of their points. In this thesis we study the secant varieties of lines of Grassmannians and of Spinor varieties. As first result, we completely determine their posets of orbits under the action of the groups SL and Spin, respectively. Then we solve the problems of identifiability and tangential-identifiability of points in the secant varieties of lines: as a consequence, we also determine the second Terracini locus to a Grassmannian and to a Spinor variety. Our main result concerns the singular locus of the secant variety of lines: we completely determine it for Grassmannians, and we give lower and upper bounds for Spinor varieties. Finally, we partially describe the poset of orbits in the secant variety of lines of any cominuscule variety.

The current research investigates the use of solid-embedded thermocouples for determining accurate transient temperature measurements within a solid medium, with emphasis on measurements intended for use in inverse heat conduction problems. Metal casting experiments have been conducted to collect internal mold temperatures to be used, through inverse conduction methods, to estimate the heat exchange between a casting and mold. Inverse conduction methods require accurate temperature measurements for valid boundary estimates. Therefore, various sources of thermocouple measurement uncertainty are examined and some suggestions for uncertainty reduction are presented. Thermocouple installation induced bias uncertainties in experimental temperature data are dynamically corrected through the development and implementation of an embedded thermocouple correction (ETC) transfer function. Comparisons of experimental data to dynamically adjusted data, as well as the inverse conduction estimates for heat flux from each data set, are presented and discussed.

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