<div><div>The blockchain offers a decentralized way to provide security guarantees for financial transactions. However, this ability comes with the cost of storing a large (distributed) blockchain state and introducing additional computation and communication overhead to all participants. All these drawbacks raise a challenging scalability problem, especially for resource-constrained blockchain clients. On the other hand, some scaling solutions typically require resource-constrained clients to rely on other nodes with higher computational and storage capabilities. However, such scaling solutions often expose the data of the clients to risks of compromise of the more powerful nodes they rely on (e.g., accidental, malicious through a break-in, insider misbehavior, or malware infestation). This potential for leakage raises a privacy concern for these constrained clients, in addition to other scaling-related concerns. This dissertation proposes several cryptographic constructions and system designs enabling resource-constrained devices to participate in the blockchain network securely and efficiently. </div><div><br></div><div>Our first proposal concerns the storage facet for which we propose two add-on privacy designs to address the scaling issue of storing a large blockchain state. </div><div>The first solution is an oblivious database framework, called T³, that allows resource-constrained clients to obliviously fetch blockchain data from potential malicious full clients. The second solution focuses on the problem of using and storing additional private-by-design blockchains (e.g., Monero or ZCash) to achieve privacy. We propose an add-on tumbler design, called AMR, that offers privacy directly to clients of non-private blockchains such as Ethereum without the cost of storing and using different blockchain states.</div><div><br></div><div>Our second proposal addresses the communication facet with focus on payment channels as a solution to address the communication overhead between the constrained clients and the blockchain network. A payment channel enables transactions between arbitrary pairs of constrained clients with a minimal communication overhead with the blockchain network. However, in popular blockchains like Ethereum and Bitcoin, the payment data of such channels are exposed to the public, which is undesirable for financial applications. Thus, to hide transaction data, one can use blockchains that are private by design like Monero. However, existing cryptographic primitives in Monero prevent the system from supporting any form of payment channels. Therefore, we present <i>Dual Linkable Spontaneous Anonymous Group Signature for Ad Hoc Groups (DLSAG),</i> a linkable ring signature scheme that enables, for the first time, off-chain scalability solutions in Monero. </div><div><br></div><div>To address the computation facet, we address the computation overhead of the gossip protocol used in all popular blockchain protocols. For this purpose, we propose a signature primitive called <i>Flexible Signature</i>. In a flexible signature scheme, the verification algorithm quantifies the validity of a signature based on the computational effort performed by the verifier. Thus, the resource-constrained devices can partially verify the signatures in the blockchain transactions before relaying transactions to other peers. This primitive allows the resource-constrained devices to prevent spam transactions from flooding the blockchain network with overhead that is consistent with their resource constraints. </div></div>

The first part of this dissertation investigates the possibility of an output cut by a firm as a result of an increase in demand in industries with constrained capacities. We are specially interested in the crude oil industry, although the paper has implications beyond that market. Two simple closely related models are developed. In both models a firm cuts the output at some point solely because of an increase in demand. We use this fact to explain the sharp decline of the crude oil prices in 1986. There are price and quantity hysteresis in the second model. The price hysteresis has two implications. First, the price path when the demand increases might be different from the price path when the demand decreases. This in turn implies that a temporary shock in the demand for (or supply of) crude oil can cause permanent changes in the price. We claim that the temporary changes in the supply of crude oil in 1973 resulted in the price hysteresis phenomenon described in the second model in such a way that it kept the prices high even after the return of the producers to the market. The second part investigates the relationship between the taste for public expenditure and the size and distribution of social groups in a society. Societies with ethnic heterogeneity spend less on redistribution and welfare programs and impose lower tax rates relative to homogeneous societies. We construct a theoretical model to explain these facts. There are two social groups in the model: a minority group and a majority group. When members of one group feel empathy for each other but not for members of the other group, then taxes, and redistribution depend upon the size and distribution of those groups. At first, the equilibrium tax rate and redistribution decrease as the size of the minority group increases from zero, then eventually, the relationship between them becomes positive. / Ph. D.

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Turbulent flows occur in various fields and are a central, yet an extremely complex, topic in fluid dynamics. Understanding the behaviour of fluids is vital for multiple research areas including, but not limited to: biological models, weather prediction, and engineering design models for automobiles and aircraft. In this thesis, we study a number of fundamental problems that arise in 2D turbulent flows, using the 2D Navier-Stokes system. Introducing optimization techniques for systems described by partial differential equations (PDE), we frame these problems such that they can be solved using computational methods. We utilize adjoint calculus to build the computational framework to be implemented in an iterative gradient flow procedure, using the "optimize-then-discretize" approach. Pseudospectral methods are employed for solving PDEs in a numerically efficient manner. The use of optimization methods together with computational mathematics in this work provides an illuminating perspective on fluid mechanics. We first apply these techniques to better understand enstrophy dissipation in 2D Navier-Stokes flows, in the limit of vanishing viscosity. By defining an optimization problem to determine optimal initial conditions, multiple branches of local maximizers were obtained each corresponding to a different mechanism producing maximum enstrophy dissipation. Viewing this quantity as a function of viscosity revealed quantitative agreement with an analytic bound, demonstrating the sharpness of this bound. We also introduce an extension of this problem, where enstrophy dissipation is maximized in the context of kinetic theory using the Boltzmann equation. Secondly, these PDE-constrained optimization techniques were used to probe the fundamental limitations on the performance of the Leith eddy-viscosity closure model for 2D Large-Eddy Simulations of the Navier-Stokes system. Obtained by solving an optimization problem with a non-standard structure, the results demonstrate the optimal eddy viscosities do not converge to a well-defined limit as regularization and discretization parameters are refined, hence the problem of determining an optimal eddy viscosity is ill-posed. Further extending the problem of finding optimal eddy-viscosity closures, we consider imposing an additional nonlinear constraint on the control variable in the problem, in the form of requiring the time-averaged enstrophy be preserved. To address this problem in a novel way, we employ adjoint calculus to characterize a subspace tangent to the constraint manifold, which allows one to approximately enforce the constraint. Not only do we demonstrate that this produces better results when compared to the case without constraints, but this also provides a flexible computational framework for approximate enforcement of general nonlinear constraints. Lastly in this thesis, we introduce an optimization problem to study the Kolmogorov-Richardson energy cascade, where a pathway towards solutions is outlined. / Thesis / Doctor of Philosophy (PhD)

We present works on theory and applications for Mixed Integer Quadratically Constrained Quadratic Programs (MIQCQP). We introduce new mixed integer programming (MIP)-based relaxation and approximation schemes for general Quadratically Constrained Quadratic Programs (QCQP's), and also study practical applications of QCQP's and Mixed-integer QCQP's (MIQCQP). We first address a challenging tank blending and scheduling problem regarding operations for a chemical plant. We model the problem as a discrete-time nonconvex MIQCP, then approximate this model as a MILP using a discretization-based approach. We combine a rolling horizon approach with the discretization of individual chemical property specifications to deal with long scheduling horizons, time-varying quality specifications, and multiple suppliers with discrete arrival times. Next, we study optimization methods applied to minimizing forces for poses and movements of chained Stewart platforms (SPs). These SPs are parallel mechanisms that are stiffer, and more precise, on average, than their serial counterparts at the cost of a smaller range of motion. The robot will be used in concert with several other types robots to perform complex assembly missions in space. We develop algorithms and optimization models that can efficiently decide on favorable poses and movements that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger workspace and a greater overall payload capacity. In the third work, we present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions and formulate this approximation using mixed-integer programming (MIP). Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. We computationally demonstrate that our model outperforms existing MIP relaxations, and on hard instances can compete with state-of-the-art solvers. Finally, we study piecewise linear relaxations for solving quadratically constrained quadratic programs (QCQP's). We introduce new relaxation methods based on univariate reformulations of nonconvex variable products, leveraging the relaxation from the third work to model each univariate quadratic term. We also extend the NMDT approach (Castro, 2015) to leverage discretization for both variables in a bilinear term, squaring the resulting precision for the same number of binary variables. We then present various results related to the relative strength of the various formulations. / Doctor of Philosophy / First, we study a challenging long-horizon supply acquisition problem for a chemical plant. For this problem, constraints with products of variables are required to track raw material composition from supply carriers to storage tanks to the production feed. We apply a mixed-integer nonlinear program (MIP) approximation of the problem combined with a rolling planning scheme to obtain good solutions for industry problems within a reasonable time frame. Next, we study optimization methods applied to a robot designed as a stack of Stewart platforms (SPs), which will be used in concert with several other types robots to perform complex space missions. When chaining these SPs together, we obtain a robot that is generally stiffer more precise than a classic robot arm, enabling their potential use for a variety of purposes. Our methods can efficiently decide on favorable poses and movements for the robot that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger usable range of motion and a greater overall payload capacity. Our final two works focus on MIP-based techniques for nonconvex QCQP's. In the first work, we break down the objective into an easy-to-handle term minus some squared terms. We then introduce an elegant new MIP-based approximation to handle these squared terms. We prove that this approximation has strong theoretical guarantees, then demonstrate that it is effective compared to other approximations. In the second, we directly model each variable product term using a MIP relaxation. We introduce two new formulations to do this that build on previous formulations, increasing the accuracy with the same number of integer variables. We then prove a variety of useful properties about the presented formulations, then compare them computationally on two families of problems.

The densification and shear viscosity of borosilicate glass (BSG) + silica films on rigid substrates were studied. Optical measurement techniques were devised to determine the shrinkage profiles of the free and constrained films and in-plane stress in the constrained films. These films were prepared from slurries of the powder by tape casting. Sintering was carried out isothermally in a hot stage between 665°C and 775°C. The densification rates of both films were observed to be the same in the early stage of sintering but slowed down in the constrained film during the later stage resulting in a lower density. The activation energies for both free and constrained sintering were found to be 385 ± 10 kJ/mol. In-plane stress measurements in constrained films of the pure glass showed the stress to rapidly rise to a maximum level of 20 kPa during the initial stage of sintering and gradually decreased back to zero during the final stage. Densification rates can be given as a product of mobility and driving pressure. Activation energy determinations did not indicate a change in the densification mechanism such that a change in mobility can be ruled out as the reason for the reduced densification rate in constrained films. The stresses are substantially smaller than the estimated lower limit of the sintering pressure and had no observed effect on the densification of the constrained film during the early stages of sintering. However, the stresses could have prevented a few large pores from shrinking during the early stage of sintering leading to the lower density and larger pores observed in the constrained film after sintering. Shear viscosity determinations were also done using data obtained from the sintering of constrained films. The results showed that the density dependence of the shear viscosity is consistent with other work in sinter-forging experiments. However, the results also indicated that the shear viscosity is strongly dependent on sintering temperature. This can be attributed to the different microstructures that evolved when the films were sintered at different temperatures. / Ph. D.

The focus of Location-Aware Adaptive Vehicle Dynamics System (LAAVDS) research is to develop a system to avoid situations in which the vehicle exceeds its handling capabilities. The proposed method is predictive, estimating the ability of the vehicle to successfully navigate upcoming terrain, and it is assumed that the future vehicle states and local driving environment is known. An Intervention Strategy must be developed such that the vehicle is navigated successfully along a road via modest changes to the driver's commands and do so in a manner that is in harmony with the driver's intentions and not in a distracting or irritating manner. Clearly this research relies on the numerous new technologies being developed to capture and convey information about the local driving environment (e.g., bank angle, elevation changes, curvature, and friction coefficient) to the vehicle and driver. / Ph. D.