In this paper we discuss the inverse problem of the identification of mechanical stresses by measuring the deformation of an electric potential field in a so called differential strain gauge (D-DMS). We derive a mathematical model, where the forward operator is given in terms of an elliptic boundary value problem. Derivatives of the forward operator are considered and the solution of the inverse problem via a least-squares minimization is introduced. Here, the discretized problem is solved with the Gauss-Newton method. Numerical studies of practical interest are presented.

The Laplace transform analytic element method (LT-AEM), applies the traditionally steady-state analytic element method (AEM) to the Laplace-transformed diffusion equation (Furman and Neuman, 2003). This strategy preserves the accuracy and elegance of the AEM while extending the method to transient phenomena. The approach taken here utilizes eigenfunction expansion to derive analytic solutions to the modified Helmholtz equation, then back-transforms the LT-AEM results with a numerical inverse Laplace transform algorithm. The two-dimensional elements derived here include the point, circle, line segment, ellipse, and infinite line, corresponding to polar, elliptical and Cartesian coordinates. Each element is derived for the simplest useful case, an impulse response due to a confined, transient, single-aquifer source. The extension of these elements to include effects due to leaky, unconfined, multi-aquifer, wellbore storage, and inertia is shown for a few simple elements (point and line), with ready extension to other elements. General temporal behavior is achieved using convolution between these impulse and general time functions; convolution allows the spatial and temporal components of an element to be handled independently.Comparisons are made between inverse Laplace transform algorithms; the accelerated Fourier series approach of de Hoog et al. (1982) is found to be the most appropriate for LT-AEM applications. An application and synthetic examples are shown for several illustrative forward and parameter estimation simulations to illustrate LT-AEM capabilities. Extension of LT-AEM to three-dimensional flow and non-linear infiltration are discussed.

The increasing concentration of CO_(2) has been linked to global warming and changes in climate. Geologic sequestration of CO_(2) in deep saline aquifers is a proposed greenhouse gas mitigation technology with potential to significantly reduce atmospheric emissions of CO_(2). Feasibility assessments of proposed sequestration sites require realistic and computationally efficient models to simulate the subsurface pressure response and monitor the injection process, and quantify the risks of leakage if there is any. This study investigates the possibility of obtaining closed form expressions for spatial distribution of CO_(2) injected in brine aquifers and gas reservoirs. Four new semi-analytical solutions for CO_(2) injection in brine aquifers and gas reservoirs are derived in this dissertation. Both infinite and closed domains are considered in the study. The first solution is an analysis of CO_(2) injection into an initially brine-filled infinite aquifer, exploiting self–similarity and matched asymptotic expansion. The second is an expanding to the first solution to account for CO_(2) injection into closed domains. The third and fourth solutions are analyzing the CO_(2) injection in infinite and closed gas reservoirs. The third and fourth solutions are derived using Laplace transform. The brine aquifer solutions accounted for both Darcyian and non-Darcyian flow, while, the gas reservoir solutions considered the gas compressibility variations with pressure changes. Existing analytical solutions assume injection under constant rate at the wellbore. This assumption is problematic because injection under constant rate is hard to maintain, especially for gases. The modeled injection processes in all aforementioned solutions are carried out under constant pressure injection at the wellbore (i.e. Dirichlet boundary condition). One major difficulty in developing an analytical or semi-analytical solution involving injection of CO_(2) under constant pressure is that the flux of CO_(2) at the wellbore is not known. The way to get around this obstacle is to solve for the pressure wave first as a function of flux, and then solve for the flux numerically, which is subsequently plugged back into the pressure formula to get a closed form solution of the pressure. While there is no simple equation for wellbore flux, our numerical solutions show that the evolution of flux is very close to a logarithmic decay with time. This is true for a large range of the reservoir and CO_(2) properties. The solution is not a formation specific, and thus is more general in nature than formation-specific empirical relationships. Additionally, the solution then can be used as the basis for designing and interpreting pressure tests to monitor the progress of CO_(2) injection process. Finally, the infinite domain solution is suitable to aquifers/reservoirs with large spatial extent and low permeability, while the closed domain solution is applicable to small aquifers/reservoirs with high permeability.

The main objective of this research is the study of the magnetization of ferromagnetic spheres in the presence of external magnetic fields. The exact analytical solutions derived in this thesis are benchmark solutions, valuable in testing the correctness and accuracy of various approximate models and numerical methods. The total scalar magnetic potential outside the spheres, related to the magnetic field intensity, is obtained by the superposition of the potentials due to all spheres and the potential corresponding to the external field. The translational addition theorems for scalar Laplacian functions are used to solve boundary value by imposing exact boundary conditions. The scalar magnetic potential inside each sphere, related to the magnetic flux density, also satisfies the Laplace equation, which is solved by imposing the boundary conditions known from the solution of the outside field. Finally, the expressions derived are used to generate numerical results of controllable accuracy for field quantities.

Ce travail consiste en une tentative de ré-examiner les propriétés électroniques du Fe, Cr et Au au sein de Ge, qui ont déjà été étudiées classiquement par DLTS (Deep Level Transient Spectroscopy). L'image générale qui en découle est que les métaux de transition dans Ge forment de manière prépondérante des centres accepteurs multiples, introduisant plusieurs niveaux profonds dans la bande de gap. A partir d'un modèle de liaison de valence simple, cette conclusion est en accord avec une survenue des impuretés sur des sites de substitution. Cependant, plusieurs questions demeurent ouvertes, comme le rôle de l'hydrogène en tant que contaminant dans l'élargissement des spectres DLTS. Notre contribution se base sur l'utilisation d'une approche plus performante nommée Laplace DLTS, en ce sens qu'elle autorise une meilleure résolution du signal. Nous présentons une analyse extensive par DLTS, MCTS et Laplace DLTS, afin d'étudier les propriétés électroniques des états accepteurs multiples, induits par les 4 métaux de transition sus-nommés. On distingue, parmis les paramètres étudiés, les barrières de capture des porteurs, les vraies sections efficaces de capture de sporteurs majoritaires (déterminées directement par la méthode de variation du pulse de remplissage), L'effet Pool-Frenkel (en lien avec la détermination de l'état de charge du niveau concerné). Ceci permet d'indiquer avec précision la position exacte des niveaux dans la bande interdite. Nous confirmons la plupart des résultats mis en évidence précédemment, tout en ajoutant quelques précisions sir le rôle de l'hydrogène dans la formation de nouveaux complexes. Une mise en parallèle avec le silicium. Dans le cas de Au, de nouveaux niveaux attribués aux complexes Au-Hn et Au-Sb sont observés. De manière générale, l'analyse des porteurs majoritaires et minoritaires par MCYS est toujours sujette à étude. En ce qui concerne le cas du Fe, la faible différence d'énergie entre ses deux niveaux soulève la possibilité d'un caractère de type U-négatif. L'ensemble de ses points devraient faire l'objet d'un travail approfondis dans un avenir proche.

The primate hand, a biomechanical structure with over twenty kinematic degrees of freedom, has an elaborate anatomical architecture. Although the hand requires complex, coordinated neural control, it endows its owner with an astonishing range of dexterous finger movements. Despite a century of research, however, the neural mechanisms that enable finger and grasping movements in primates are largely unknown. In this thesis, we investigate statistical models of finger movement that can provide insights into the mechanics of the hand, and that can have applications in neural-motor prostheses, enabling people with limb loss to regain natural function of the hands. There are many challenges associated with (1) the understanding and modeling of the kinematics of fingers, and (2) the mapping of intracortical neural recordings into motor commands that can be used to control a Brain-Machine Interface. These challenges include: potential nonlinearities; confounded sources of variation in experimental datasets; and dealing with high degrees of kinematic freedom. In this work we analyze kinematic and neural datasets from repeated-trial experiments of hand motion, with the following contributions: We identified static, nonlinear, low-dimensional representations of grasping finger motion, with accompanying evidence that these nonlinear representations are better than linear representations at predicting the type of object being grasped over the course of a reach-to-grasp movement. In addition, we show evidence of better encoding of these nonlinear (versus linear) representations in the firing of some neurons collected from the primary motor cortex of rhesus monkeys. A functional alignment of grasping trajectories, based on total kinetic energy, as a strategy to account for temporal variation and to exploit a repeated-trial experiment structure. An interpretable model for extracting dynamic synergies of finger motion, based on Gaussian Processes, that decomposes and reduces the dimensionality of variance in the dataset. We derive efficient algorithms for parameter estimation, show accurate reconstruction of grasping trajectories, and illustrate the interpretation of the model parameters. Sound evidence of single-neuron decoding of interpretable grasping events, plus insights about the amount of grasping information extractable from just a single neuron. The Laplace Gaussian Filter (LGF), a deterministic approximation to the posterior mean that is more accurate than Monte Carlo approximations for the same computational cost, and that in an off-line decoding task is more accurate than the standard Population Vector Algorithm.

This thesis studies the value of branding in the smartphone market. Measuring brand value with data available at product level potentially entails computational and econometric challenges due to data constraints. These issues motivate the three studies of the thesis. Chapter 2 studies the smartphone market to understand how operating system platform providers can grow one of the most important intangible assets, i.e., brand value, by leveraging the indirect network between two user groups in a two-sided platform. The main finding is that iPhone achieved the greatest brand value growth by opening its platform to the participation of third-party developers, thereby indirectly connecting the consumers and the developers via its app store effectively. Without the open app store, I find that iPhone would have lost its brand value by becoming a two-sided platform. Hence these findings provide an important lesson that open platform strategy is vital to the success of building platform brands. Chapter 3 solves a computational challenge in structural estimation of aggregate demand. I develop a computationally efficient MCMC algorithm for the GMM estimation framework developed by Berry, Levinsohn and Pakes (1995) and Gowrisankaran and Rysman (forthcoming). I combine the MCMC method with the classical approach by transforming the GMM into a Laplace type estimation framework, therefore avoiding the need to formulate a likelihood model. The proposed algorithm solves the two fixed point problems, i.e., the market share inversion and the dynamic programming, incrementally with MCMC iteration. Hence the proposed approach achieves computational efficiency without compromising the advantages of the conventional GMM approach. Chapter 4 reviews recently developed econometric methods to control for endogeneity bias when the random slope coefficient is correlated with treatment variables. I examine how standard instrumental variables and control function approaches can solve the slope endogeneity problem under two general frameworks commonly used in the literature.

This thesis studies the value of branding in the smartphone market. Measuring brand value with data available at product level potentially entails computational and econometric challenges due to data constraints. These issues motivate the three studies of the thesis. Chapter 2 studies the smartphone market to understand how operating system platform providers can grow one of the most important intangible assets, i.e., brand value, by leveraging the indirect network between two user groups in a two-sided platform. The main finding is that iPhone achieved the greatest brand value growth by opening its platform to the participation of third-party developers, thereby indirectly connecting the consumers and the developers via its app store effectively. Without the open app store, I find that iPhone would have lost its brand value by becoming a two-sided platform. Hence these findings provide an important lesson that open platform strategy is vital to the success of building platform brands. Chapter 3 solves a computational challenge in structural estimation of aggregate demand. I develop a computationally efficient MCMC algorithm for the GMM estimation framework developed by Berry, Levinsohn and Pakes (1995) and Gowrisankaran and Rysman (forthcoming). I combine the MCMC method with the classical approach by transforming the GMM into a Laplace type estimation framework, therefore avoiding the need to formulate a likelihood model. The proposed algorithm solves the two fixed point problems, i.e., the market share inversion and the dynamic programming, incrementally with MCMC iteration. Hence the proposed approach achieves computational efficiency without compromising the advantages of the conventional GMM approach. Chapter 4 reviews recently developed econometric methods to control for endogeneity bias when the random slope coefficient is correlated with treatment variables. I examine how standard instrumental variables and control function approaches can solve the slope endogeneity problem under two general frameworks commonly used in the literature.

A valuable analytical tool to understand the event of ruin is a Gerber-Shiu discounted penalty function. It acts as a unified means of identifying ruin-related quantities which may help insurers understand their vulnerability ruin. This thesis provides an introduction to the basic concepts and common techniques used for the Gerber-Shiu analysis. Chapter 1 introduces the insurer's surplus process in the ordinary Sparre Andersen model. Defective renewal equations, the Dickson-Hipp transform, and Lundberg's fundamental equation are reviewed. Chapter 2 introduces the classical Gerber-Shiu discounted penalty function. Two framework equations are derived by conditioning on the first drop in surplus below its initial value, and by conditioning on the time and amount of the first claim. A detailed discussion is provided for each of these conditioning arguments. The classical Poisson model (where interclaim times are exponentially distributed) is then considered. We also consider when claim sizes are exponentially distributed. Chapter 3 introduces the Gerber-Shiu function in the delayed renewal model which allows the time until the first claim to be distributed differently than subsequent interclaim times. We determine a functional relationship between the Gerber-Shiu function in the ordinary Sparre Andersen model and the Gerber-Shiu function in the delayed model for a class of first interclaim time densities which includes the equilibrium density for the stationary renewal model, and the exponential density. To conclude, Chapter 4 introduces a generalized Gerber-Shiu function where the penalty function includes two additional random variables: the minimum surplus level before ruin, and the surplus immediately after the claim before the claim causing ruin. This generalized Gerber-Shiu function allows for the study of random variables which otherwise could not be studied using the classical definition of the function. Additionally, it is assumed that the size of a claim is dependant on the interclaim time that precedes it. As is done in Chapter 2, a detailed discussion of each of the two conditioning arguments is provided. Using the uniqueness property of Laplace transforms, the form of joint defective discounted densities of interest are determined. The classical Poisson model and the exponential claim size assumption is also revisited.

In this thesis, we explore two important aspects of study of differential equations: analytical and computational aspects. We first consider a partial differential equation model for a static liquid surface (capillary surface). We prove through mathematical analyses that the solution of this mathematical model (the Laplace-Young equation) in a cusp domain can be bounded or unbounded depending on the boundary conditions. By utilizing the knowledge we have obtained about the singular behaviour of the solution through mathematical analysis, we then construct a numerical methodology to accurately approximate unbounded solutions of the Laplace-Young equation. Using this accurate numerical methodology, we explore some remaining open problems on singular solutions of the Laplace-Young equation. Lastly, we consider ordinary differential equation models used in the pharmaceutical industry and develop a numerical method for estimating model parameters from incomplete experimental data. With our numerical method, the parameter estimation can be done significantly faster and more robustly than with conventional methods.