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This thesis focuses on three-dimensional thermally-driven boundary layers with three distinct types of (steady) thermal forcing (i) a streamwise-aligned heat source that is spanwise localised, (ii) a spanwise and streamwise localised heat source and (iii) a non-localised globally driven natural convection problem. What ties these problems together is that the induced three-dimensionality is on a spanwise lengthscale commensurate with the boundary-layer thickness, and so their leading-order flow is governed by the 'boundary-region' equations, attributed to Kemp (1951). A lot of work has been undertaken within this boundary-region framework in an isothermal context, for example forced by injection or roughness, but presently there has been limited work in which temperature variation has been applied. In our first problem (i), we embed a streamwise-aligned spanwise-localised heated strip, which has an O(Re^(-1/2)) spanwise scale, in a horizontal semi-infinite flat plate. At large spanwise distances from the forcing region, the flow takes the form of the two-dimensional Falkner--Skan boundary layer (1930). The three-dimensional flow response is determined numerically and a robust streak-roll structure forms downstream. We undertake an asymptotic analysis at large streamwise coordinates, and obtain the downstream transverse growth of the nonlinear response. We also determine the flow over a streamwise-aligned injection slot, and make qualitative comparisons with the thermally forced problem. A localised heat source with an O(1) streamwise scale and an O(Re^(-1/2)) spanwise scale is then considered (ii), and the corresponding nonlinear problem is solved. For weak thermal forcing, the isolated heated region results in a downstream response which has an algebraically developing velocity field. By seeking a linear bi-global eigenvalue problem in terms of the downstream algebraic spatial growth, we show that there is a new class of linear spanwise-localised algebraically developing disturbances in the Falkner--Skan boundary layer when thermal effects are included. The velocity field of the dominant mode grows (algebraically) downstream, and is driven by a decaying (weak) temperature field. Above a critical pressure gradient the dominant mode decays, and we determine the critical Prandtl number and pressure gradient parameter for downstream algebraic growth. The linear algebraically growing response ultimately gives rise to a nonlinear spanwise-localised streak-roll response downstream. We assess the stability of the streak response to time-harmonic (viscous) disturbances and inviscid Rayleigh modes. Our third problem (iii) is the natural convection flow in a heated vertical corner. This is a globally driven, non-localised problem, however in a high Grashof number regime the flow in a region near to the corner is governed by the boundary-region equations. Two self-similar solutions are determined: (A) a solution first discussed by Riley and Poots (1972), and (B) a new solution which, at large spanwise coordinates, has a coupled linearly-developing crossflow component. We show that the streamwise velocity and temperature fields determined by Riley and Poots (1972) for solution A were surprisingly accurate, given that their computational resources were limited. However, they concluded that there is an in-plane flow away from the corner which is not present in the properly resolved solution. The non-parallel stability of the self-similar profiles is tackled with the parabolised stability equations (PSE).

In this thesis, I investigate how the presence of mesoscopic colloidal inclusions distorts the locally-averaged molecular orientation of a thermotropic nematic shell in a state of thermodynamic equilibrium at a constant temperature. In the following analysis, I focus on three problems, utilising methods from different mathematical disciplines. Firstly, I consider the Robin problem for the Laplace operator in a two-dimensional domain with a small hole of radius ε > 0. I approximate the solution to this problem, in the spirit of compound asymptotics, by the sum of the so-called 'interior' and 'exterior' functions, which describe the behaviour of the solution in the bulk of the domain and in the neighbourhood of the small hole, respectively. I prove that the approximation converges uniformly to the solution of the Robin problem, and derive the rate of convergence as a function of the radius of the small hole and the Robin parameter. Secondly, I define a notion of convergence for sequences of integral energy functionals that are such that the pointwise limit of the related stored energy densities is non-integrable. To derive the limit of a given functional sequence, I consider an asymptotic expansion of each functional into the sum of a divergent lower-order term and convergent higher order term, similar to what is known as Γ-asymptotics. Firstly, I derive the subset of functions that minimise the rate at which the lower-order term diverges. As a consequence, I develop a heuristic algorithm predicting the strengths of defects of a thin film of nematic liquid crystal, given realistic boundary data and certain geometric parameters. Secondly, I then seek the Γ-limit of the convergent higher-order term in this subset. The sum of the lower-order term and the corresponding higher-order Γ-limit is referred to as the Γ-asymptotic limit. Finally, I study the limiting behaviour of the solutions to Poisson's equation subject to inhomogeneous boundary conditions, as the number of non-periodically distributed holes tends to infinity. I generalise existing convergence results for the case of Robin boundary conditions, by deriving the weak limit of the solutions to the inhomogeneous problems and investigate how the boundary data affects the ``strange term'' in the limit description.

Many environmental fluid flows are strongly influenced by the presence of bed topography, which we investigate in the context of idealised shallow water models considering a range of topographical effects. First, a barrier that controls the dynamics as the fluid overtops, which appears in models of a partially collapsed dam, and of a confined gravity current. Next, an edge over which the fluid drains, which is investigated in axisymmetric geometry demonstrating the spontaneous development of supercritical outflow, and then for a spreading current that drains from its rear. Finally, we consider obstacles that may be mobilised by the presence of the fluid, including an obstacle that slides under friction, and one that is destroyed by the current. Several of the problems feature the spontaneous development of a shock in the interior of the domain due to the sharpening of a gradient discontinuity, an unusual effect. We use a variety of analytical tools to elucidate the dynamics, including the method of characteristics and the hodograph transformation, similarity analysis, and multi-scale asymptotics, along with numerical simulations. The simulations were performed using a bespoke code with several novel developments which are also presented, including a custom implementation of the boundary conditions, and the ability to resolve steady states at high accuracy.

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Inspired by biological membrane shaping in the cell through means of curvature-inducing proteins, we investigate the interplay between membrane curvature and the distribution and movement of shape-inducing objects which are free to move as a consequence of the underlying shape. We initially study the self-assembly of a filament, taken as a proxy for the cross-section of a biomembrane, which is primarily driven by the chemical kinetics of attaching proteins and find that, under certain mechanical stiffness regimes of the attaching proteins, pattern formation occurs. Regions of high and low protein concentration form before spatially uniform filament shapes are obtained by means of protein adhesion and movement governed by diffusion and local curvature-seeking. However, noting that the curvature-mediated protein movement on membranes has been biologically observed to be long-range, we next study the self-assembly of embedded inclusions on a membrane as a result of the underlying geometry. We first derive an interaction law for the shape-mediated interaction of inclusions which break symmetry and find that there is a finite equilibrium distance to which the inclusions will aggregate. We derive corresponding equations of motion which describe this curvature-mediated aggregation mechanism and, using this framework, we investigate some of the properties of these self-assembled configurations, including their energy, stability, and their collective elastic behavior. Lastly, we consider the interaction energies of embedded inclusions on a periodic domain and determine that this mechanism may explain computational results of how proteins form rings to promote tubulation on cylindrical membranes.

This thesis explores opportunities in the mathematical modelling of metal rolling processes, specifically asymmetrical sheet rolling. With the application of control systems in mind, desired mathematical models must make adequate predictions with short computational times. This renders generic numerical approaches inappropriate. Previous analytical models of symmetrical sheet rolling have relied on ad hoc assumptions about the form of the solution. The work within this thesis begins by generalising symmetric asymptotic rolling models: models that make systematic assumptions about the rolling configuration. Using assumptions that apply to cold rolling, these models are generalised to include asymmetries in roll size, roll speed and roll-workpiece friction conditions. The systematic procedure of asymptotic analysis makes this approach flexible to incorporating alternative friction and material models. A further generalisation of a clad-sheet workpiece is presented to illustrate this. Whilst this model was formulated and solved successfully, deterioration of the results for any workpiece inhomogeneity demonstrates the limitations of some of the assumptions used in these two models. Attention is then turned to curvature prediction. A review of workpiece curvature studies shows that contradictions exist in the literature; and complex non-linear relationships are seen to exist between asymmetries, roll geometry and induced curvature. The collated data from the studies reviewed were insufficient to determine these relationships empirically; and neither analytical models, including those developed thus far, nor linear regressions are able to predict these data. Another asymmetric rolling model is developed with alternative asymptotic assumptions, which shows non-linear behaviour over ranges of asymmetries and geometric parameters. While quantitative curvature predictions are not achieved, metrics of mechanisms hypothesised to drive curvature indicate these non-linear curvature trends may be captured with further refinement. Finally, coupling a curved beam model with a curvature predicting rolling model is proposed to model the ring rolling process. Both of these parts are implemented but convergence between them is not yet achieved. By analogy this could be extended with shell theory and a three-dimensional rolling model to model the wheeling process.

This thesis is concerned with the mathematical modelling of elastic-plastic deformation in regimes of stress far exceeding the yield stress. Such scenarios are typically encountered in violent impact testing, where millimetre-thick samples of metal are subjected to pressures on the order of the bulk modulus of the material. We begin with an overview of violent impact testing, with particular attention paid to a specific class of experiments known as isentropic compression experiments (ICEs), which will provide motivation for the mathematical modelling and analysis in subsequent chapters. In chapter 2, by appealing to sound notions from rational mechanics and thermodynamics, we construct a mathematical model which aims to encapsulate the essential phenomena involved in violent elastic-plastic deformation. This is followed in chapter 3 with a numerical analysis of the mathematical model in uniaxial strain, which is the geometry relevant ICEs. In chapters 4 and 5, we corroborate the observations made in chapter 3 via a systematic mathematical analysis. In particular, our focus will be on the elastic and plastic waves that can propagate through finite metal samples during isentropic compression. Finally, in chapter 6, we explore the applicability of our model to other geometries, specifically the radially axisymmetric expansion of a circular cavity embedded in an infinite elastic-plastic medium. We conclude with a summary of our findings and suggest some avenues for future investigation.

The retina is the ocular tissue responsible for the detection of light. Its extensive demand for oxygen, coupled with a concomitant elevated supply, renders this tissue prone to both hypoxia and hyperoxia. In this thesis, we construct mathematical models of the retina, formulated as systems of reaction-diffusion equations, investigating its oxygen-related dynamics in healthy and diseased states. In the healthy state, we model the oxygen distribution across the human retina, examining the efficacy of the protein neuroglobin in the prevention of hypoxia. It has been suggested that neuroglobin could prevent hypoxia, either by transporting oxygen from regions where it is rich to those where it is poor, or by storing oxygen during periods of diminished supply or increased uptake. Numerical solutions demonstrate that neuroglobin may be effective in preventing or alleviating hypoxia via oxygen transport, but that its capacity for oxygen storage is essentially negligible, whilst asymptotic analysis reveals that, contrary to the prevailing assumption, neuroglobin's oxygen affinity is near optimal for oxygen transport. A further asymptotic analysis justifies the common approximation of a piecewise constant oxygen uptake across the retina, placing existing models upon a stronger theoretical foundation. In the diseased state, we explore the effect of hyperoxia upon the progression of the inherited retinal diseases, known collectively as retinitis pigmentosa. Both numerical solutions and asymptotic analyses show that this mechanism may replicate many of the patterns of retinal degeneration seen in vivo, but that others are inaccessible to it, demonstrating both the strengths and weaknesses of the oxygen toxicity hypothesis. It is shown that the wave speed of hyperoxic degeneration is negatively correlated with the local photoreceptor density, high density regions acting as a barrier to the spread of photoreceptor loss. The effects of capillary degeneration and treatment with antioxidants or trophic factors are also investigated, demonstrating that each has the potential to delay, halt or partially reverse photoreceptor loss. In addition to answering questions that are not accessible to experimental investigation, these models generate a number of experimentally testable predictions, forming the first loop in what has the potential to be a fruitful experimental/modelling cycle.

Engineering assemblies are very frequently subject to fretting fatigue, which is a damage process that results when very small slip displacements arise at nominally stationary frictional interfaces. Fretting accelerates the initiation and early propagation of fatigue cracks, thereby causing significant reductions in the fatigue performance of many critical engineering components. A majority of the previous research on fretting fatigue has focused on incomplete (i.e. smooth-edged) contacts, while complete (i.e. sharp-edged) contacts have received less attention. The aim of this thesis is to contribute to the theoretical understanding of complete contacts, especially when they are subject to fatigue conditions. This problem is addressed in two separate ways. First, because fretting failures almost invariably initiate from the edge of contact, a detailed understanding of the conditions in this region should enable more accurate assessments of fatigue performance to be made. Thus, an asymptotic analysis is presented, which provides an accurate description of the contact edge under many conditions. This is done by using the elasticity solution for a semi-infinite notch to represent the state of stress near the contact edge in an asymptotic sense. Attention is then placed on the fact that cyclically loaded frictional contacts tend toward a steady-state response in which less frictional slip (and energy dissipation) occurs than in the first few load cycles. To investigate this effect, a numerical sub-structuring procedure is described, which significantly reduces the number of degrees of freedom in finite element models of frictional contact. This reduced model is then used to calculate the shakedown limit, i.e. the amplitude of cyclic load above which frictional slip is guaranteed to persist in the steady state. The sensitivity of the steady-state solution to the initial residual displacement state is then investigated, and it is shown that initial conditions can have a large influence on the steady-state behaviour of complete contacts.

In screen-printed silicon-crystalline solar cells, the contact resistance of a thin interfacial glass layer between the silicon and the silver electrode plays a limiting role for electron transport. The motivation of this project is to gain increased understanding of the transport mechanisms of the electrons across this layer, which can be exploited to provide higher performance crystalline silicon solar cells. Our methodology throughout is to formulate and analyse mathematical models for the electron transport, based on the drift diffusion equations. In the first chapter we outline the problem and provide a summary of relevant theory. In Chapter 2 we formulate a one-dimensional model for electron transport across the glass layer, that we solve both numerically and by employing asymptotic techniques. Chapter 3 extends the model presented in Chapter 2 to two dimensions. To solve the two-dimensional model numerically we devise and validate a new spectral method. The short circuiting of current through thinner regions of the glass layer enables us to find limiting asymptotic expressions for the average current density for two different canonical glass layer profiles. In Chapter 4 we include quantum mechanical effects into the one-dimensional model outlined in Chapter 2 and find that they have a negligible effect on the contact resistance of the glass layer. We model the boundary effects present at the silicon emitter-glass interface in Chapter 5. Finally, in Chapter 6 we summarise our key results, suggest possible future work, and outline the implications of our work to crystalline silicon solar cell manufacturers.

Human cancers exhibit the common phenotype of elevated glycolytic metabolism, which causes acidification of the tissue microenvironment and may facilitate tumour invasion. In this thesis, we use mathematical models to address a series of open problems underlying the glycolytic tumour phenotype and its attendant acidity. We first explore tissue-scale consequences of metabolically-derived acid. Incorporating more biological detail into a canonical model of acidity at the tumour-host interface, we extend the range of tumour behaviours captured by the modelling framework. We then carry out an asymptotic travelling wave analysis to express invasive tumour properties in terms of fundamental parameters, and find that interstitial gaps between an advancing tumour and retreating healthy tissue, characteristic of aggressive invasion and comprising a controversial feature of the original model, are less significant under our generalised formulation. Subsequently, we evaluate a potential role of lactate---historically assumed to be a passive byproduct of glycolytic metabolism---in a perfusion-dependent metabolic symbiosis that was recently proposed as a beneficial tumour behaviour. Upon developing a minimal model of dual glucose-lactate consumption in vivo and employing a multidimensional sensitivity analysis, we find that symbiosis may not be straightforwardly beneficial for our model tumour. Moreover, new in vitro experiments, carried out by an experimental collaborator, place U87 glioblastoma tumours in a weakly symbiotic parameter regime despite their clinical malignancy. These results suggest that intratumoural metabolic cooperation is unlikely to be an important role for lactate. Finally, we examine the complex pH regulation system that governs expulsion of metabolically derived acid loads across tumour cell membranes. This system differs from the healthy system by expression of only a few key proteins, yet its dynamics are non-intuitive in the crowded and poorly perfused in vivo environment. We systematically develop a model of tumour pH regulation, beginning with a single-cell scenario and progressing to a spheroid, within a Bayesian framework that incorporates information from in vitro data contributed by a second experimental collaborator. We predict that a net effect of pH regulation is a straightforward transmembrane pH gradient, but also that existing treatments are unable to disrupt the system strongly enough to cause tumour cell death. Taken together, our models help to elucidate previously unresolved features of glycolytic tumour metabolism, and illustrate the utility of a combined mathematical, statistical, and experimental approach for testing biological hypotheses. Opportunities for further investigation are discussed.

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

In studies of the ocean it has become conventional to retain only the component of the Coriolis force associated with the radial component of the Earth’s rotation vector, the so-called “traditional approximation”. We investigate the role of the “non-traditional” component of the Coriolis force, corresponding to the non-radial component of the rotation vector, in transporting abyssal waters across the equator. We first derive a non-traditional generalisation of the multi-layer shallow water equations, which describe the flow of multiple superposed layers of inviscid, incompressible fluid with constant densities over prescribed topography in a rotating frame. We derive these equations both by averaging the three-dimensional governing equations over each layer, and via Hamilton’s principle. The latter derivation guarantees that conservation laws for mass, momentum, energy and potential vorticity are preserved. Within geophysically realistic parameters, including the complete Coriolis force modifies the domain of hyperbolicity of the multi-layer equations by no more than 5%. By contrast, long linear plane waves exhibit dramatic structural changes due to reconnection of the surface and internal wave modes in the long-wave limit. We use our non-traditional shallow water equations as an idealised model of an abyssal current flowing beneath a less dense upper ocean. We focus on the Antarctic Bottom Water, which crosses the equator in the western Atlantic ocean, where the bathymetry forms an almost-westward channel. Cross-equatorial flow is strongly constrained by potential vorticity conservation, which requires fluid to acquire a large relative vorticity in order to move between hemispheres. Including the complete Coriolis force accounts for the fact that fluid crossing the equator in an eastward/westward channel experiences a smaller change in angular momentum, and therefore acquires less relative vorticity. Our analytical and numerical solutions for shallow water flow over idealised channel topography show that the non-traditional component of the Coriolis force facilitates cross-equatorial flow through an almost-westward channel.

Fretting fatigue is a type of contact fatigue which causes premature failure in a number of engineering assemblies subjected to vibration or other forms of cyclic loading. It is concerned with the nucleation of cracks due to oscillatory micro slip between contacting bodies. Therefore, a detailed knowledge of the interface conditions and the means of quantifying crack nucleation are very important, and will be the ultimate goal of this thesis. The analysis of an incomplete contact (Herzian contact) is considered first followed by various complete contacts. Fretting fatigue tests employing a Hertzian contact are analysed accurately by introducing several modifications needed to the classical formulation. With the total state of stress in a strip established, the crack tip stress intensity factor for a crack growing inward from the trailing edge of the contact is determined by the distributed dislocation technique. The results are then correlated with local solutions for the contact stress field which enable an estimate of the crack nucleation life, and hence a characteristic material property quantifying initiation, to be found. The interfacial contact pressure distribution beneath a complete sliding contact between elastically similar components, in the presence of friction, has been studied in detail, with particular reference to contacts whose edge angles are 60 degree, 90 degree and 120 degree. The possible types of behaviour at the edge of contacts, namely power order singularity, power order bounded and square root bounded, are discussed. A full understanding of the behaviour requires a detailed study of a characteristic equation, and this shows the kinds of pressure distribution to be anticipated, which can vary very markedly. The transition from power order behaviour to local separation and bounded behaviour is examined, and an appropriate asymptotic form developed. The problem of trapezium shaped punches pressed into a frictional, elastically similar half-plane, and subject to sequential normal and shear loading, under partial slip, is studied. Detailed considerations have again been given to the specific cases of 60 degree, 90 degree and 120 degree punches, and maps have been developed showing the initial mix of stick, slip and separation regions, together with the steady state response when the shearing force is cycled. Conditions for full stick are established.

Sedimentary basins form when water-borne sediments in shallow seas are deposited over periods of millions of years. Sediments compact under their own weight, causing the expulsion of pore water. If this expulsion is sufficiently slow, overpressuring can result, a phenomenon which is of concern in oil drilling operations. The competition between pore water expulsion and burial is complicated by a variety of factors, which include diagenesis (clay dewatering), and different modes (elastic or viscous) of rheological deformation via compaction and pressure solution, which may also include hysteresis in the constitutive behaviours. This thesis is concerned with models which can describe the evolution of porosity and pore pressure in sedimentary basins. We begin by analysing the simplest case of poroelastic compaction which in a 1-D case results in a nonlinear diffusion equation, controlled principally by a dimensionless parameter lambda, which is the ratio of the hydraulic conductivity to the sedimentation rate. We provide analytic and numerical results for both large and small lambda in Chapter 3 and Chapter 4. We then put a more realistic rheological relation with hysteresis into the model and investigate its effects during loading and unloading in Chapter 5. A discontinuous porosity profile may occur if the unloaded system is reloaded. We pursue the model further by considering diagenesis as a dehydration model in Chapter 6, then we extend it to a more realistic dissolution-precipitation reaction-transport model in Chapter 7 by including most of the known physics and chemistry derived from experimental studies. We eventually derive a viscous compaction model for pressure solution in sedimentary basins in Chapter 8, and show how the model suggests radically different behaviours in the distinct limits of slow and fast compaction. When lambda << 1, compaction is limited to a basal boundary layer. When lambda >> 1, compaction occurs throughout the basin, and the basic equilibrium solution near the surface is a near parabolic profile of porosity. But it is only valid to a finite depth where the permeability has decreased sufficiently, and a transition occurs, marking a switch from a normally pressured environment to one with high pore pressures.

Droplet combustion experiments were conducted in reduced gravity to investigate how various mass and energy transport mechanisms affect overall droplet combustion behaviors. In one study, individual heptane and methanol droplets in the $1.2-5.0$ \unit{mm} diameter range were burned in reduced gravity and in a controlled chamber, with air-inert mixtures at 0.1 and 0.07 MPa and about 298 K, where the monatomic gases helium and xenon were separately used as the added inert. Xenon is about 32 times heavier than helium, however, the thermal conductivity of helium is about 26 times higher compared to xenon at 300 K, 1 bar. Because the molecular weights and thermodynamic properties between xenon and helium are significantly different, the choice of these two gasses would be appropriate in order to investigate transport property effects, such as Lewis number variations on combustion phenomena. Xenon and helium flame suppressant effectiveness are also evaluated. The results indicate that ambient gas transport properties play an important role in determining limiting oxygen indices as well as burning rates and radiant heat output histories of flames. In the second study, the effects of liquid mixing on droplet combustion behaviors were investigated. To this end, reduced gravity combustion experiments were performed with individual propanol-glycerol and heptane-hexadecane mixture droplets. In the case of propanol-glycerol mixture droplets, the initial n-propanol mass fraction was $Y=0.95$ and droplets had initial diameters in the $2-5$ \unit{mm} range. Some droplets were fiber supported while others were free floating, and the environment was either an oxygen/nitrogen mixture at $1$ atm or an oxygen/helium mixture at pressures of $1$ and $3$ atm. In the case of heptane-hexadecane mixture droplets, the initial heptane mass fractions was either $Y=0.95$ or $Y=0.8$ with initial droplet diameters in the $1.7-4.8$ \unit{mm} range.The ambient in which heptane/hexadecane droplets were burned consisted of oxygen mixed with either helium or nitrogen at $1$ atm. The goals of these experiments are to evaluate the relationships between liquid mixing and droplet burning rates. Propanol is several orders of magnitude more volatile compared to glycerol. Likewise, heptane is also significantly more volatile compared to hexadecane. Differences in volatilities in these mixtures are related to liquid transport rates through theory developed using asymptotic analysis. Liquid diffusivities have been calculated from experimentally measured variables, and are used as a metric to quantify the degree of liquid mixing within droplets. Influences of the inert, initial droplet diameter, support fiber, and the delay time before droplet ignition on liquid diffusivities are also reported. Studies of bi-component droplets can potentially yield new insights on behaviors of droplets in sprays composed of practical fuels, which are generally mixtures.In the third study, the Nusselt number, $\Nu$, was evaluated for an isothermal fluid sphere immersed in a Stokes flow. The dimensionless convective-diffusive conservation equation is solved asymptotically for the case where the Peclet number based on the sphere radius, $\varepsilon$, is small relative to unity. The Nusselt number is evaluated by integrating the derivative of the asymptotic solution over the surface of the sphere. The conservation equation is also solved numerically. A comparison of the numerical and asymptotic solutions shows that the asymptotic expression for $\Nu$ is valid only for the Peclet number being small relative to unity. An alternative asymptotic expression for $\Nu$ is also developed, and it is shown that this expression is valid for Peclet numbers of order unity. The current work improves and corroborates prior correlations for $\Nu$, particularly those developed by Acrivos and Taylor (1962), as well as those by Brunn (1982), and may have application to evaporating droplets (e.g., as a numerical sub-model).

The jet-in-crossflow is a critical flowfield for a range of aerospace propulsion systems, both airbreathing and rocket engines. A jet-in-crossflow or transverse jet is typically composed of a jet issuing from a circular outlet perpendicularly to a uniform air crossflow. Previous experiments deploying hot-wire anememetry or laser-based optical diagnostics such as planar laser-induced fluorescence (PLIF) or particle image velocimetry (PIV) have found that there is a transition in the jet's upstream shear layer from convective to absolute instability as the jet-to-crossflow velocity ratio $R$ or momentum flux ratio $J$ decreases. Typically, convectively unstable jets-in-crossflow are correlated with more asymmetric cross-sections while more symmetric jet cross-sections are typically associated with absolutely unstable transverse jets at lower $R$ values, usually below $3$ (Gevorkyan et al., 2016). Besnard et al. (2022) demonstrated that even low-amplitude asymmetric acoustic excitation affected the symmetry of a convectively unstable jet-in-crossflow. Such a finding suggests that there may be a linear origin to symmetry-breaking in the face of a theoretically symmetric flow field. This dissertation describes a numerical investigation of the spatial linear stability characteristics of a jet-in-crossflow for the axisymmetric and helical azimuthal modes using a 2D-local viscous linear stability analysis. This study uses the viscous hyperbolic-tangent (tanh) and uniformly valid asymptotic solution (UVAS) base flows developed by Alves et al. (2008) and Alves & Kelly (2008) as viscous extensions of the cylindrical vortex sheet solution of Coelho & Hunt (1989), who performed asymptotic expansion-based linear stability analyses of the same base flows. This present work may be considered a `fully-coupled' extension of their work because we presently account for all base flow-eigenfunction azimuthal coupling.The eigenspectra from the present fully-coupled linear stability analysis indicate that the axisymmetric mode spatially stabilises as $R$ decreases - in contradiction with experimental measurements (Megerian et al., 2007; Davitian et al., 2010; Shoji et al., 2020a), direct numerical simulation results (Iyer & Mahesh, 2016), and with the prior asymptotic expansion-based linear stability analysis of Alves et al. (2008); Alves & Kelly (2008). The wavenumber and preferred Strouhal number trends, however, are in qualitative agreement with experimental measurements. The first, second, and third helical modes all stabilise as $R$ decreases, with the axisymmetric mode being more unstable than the helical modes. All helical modes exhibit degeneracy-breaking, as was found from multiple mode analysis of an inviscid base flow by Alves et al. (2007); these modes are symmetric or anti-symmetric with respect to the plane of symmetry found in the jet-in-crossflow base flows. A novel upwind-based treatment of the linearised convective term is developed to explore features of the eigenfunctions while eliminating non-physical oscillations in the high Reynolds number regime. This consists of a hybrid central-upwind finite difference discretisation scheme suitable for convection-dominated flows that allows for surgical suppression of non-physical numerical oscillations while still yielding similar eigenvalues to purely central difference schemes. Given that only the near-wake region needed this winding, the overall scheme consists of a $4^{\mathrm{th}}$-order central finite difference for all non-convective terms, a hybrid $2^{\mathrm{nd}}$-$1^{\mathrm{st}}$-order scheme for the near-wake region and a hybrid $4^{\mathrm{th}}$-$3^{\mathrm{rd}}$-order scheme elsewhere. A linear activation function is used to smooth the transition from one regime to another. This approach was effective in both Cartesian and polar coordinates. Given that the primary difference between the present fully-coupled and prior asymptotic expansion-based linear stability analyses lies in the number of supported base flow-eigenfunction azimuthal couplings, a weakly-coupled discrete Fourier-transformed linear stability analysis is developed to allow an exploration of the effect of base flow-eigenfunction azimuthal couplings on the eigenvalues. This weakly-coupled approach can highlight the essential velocity eigenfunctions or azimuthal interactions that affect the eigenvalues the most and may be considered as a sort of reduced-order model. By reproducing the base flow-eigenfunction couplings as closely as possible without repeating the same approach, the weakly-coupled linear stability analysis yields quantitatively similar axisymmetric eigenvalues as Alves & Kelly (2008). That is to say, the UVAS base flow spatial destabilises as $R$ decreases. As the number of Fourier modes $N_f$ increases, and therefore the number of base flow-eigenfunction couplings supported, the axisymmetric mode appears to become more similar to the fully-coupled results, with stabilisation as $R$ decreases. Hence, a mechanism is proposed wherein an inadequately-expanded asymptotic base flow leads to incomplete base flow-eigenfunction couplings that deleteriously affect the eigenvalues. Without those couplings, the surviving terms may contribute to eigenvalue stabilisation. To explore this hypothesis further, a spatial kinetic energy budget analysis is developed, showing the contribution of various physical mechanisms (`Advection', `Production', and `Pressure-velocity') to the spatial growth of the kinetic energy. The `Pressure-velocity' correlation term is unique to the spatial formulation. Typically, for example, for a one-dimensional parallel Blasius boundary layer, the `Pressure-velocity' term suppresses the energy when the disturbance is destabilising and supplies the energy when the disturbance is stabilising, per Hama et al. (1980). This implies that the `Pressure-velocity' term becomes an increasing percentage of the total energy budget as the flow destabilises. Indeed, in the present analysis for the jet-in-crossflow, the axisymmetric mode from the fully-coupled analysis exhibits a decreasing contribution from the `Pressure-velocity' term, consistent with our observed stabilisation when R is decreased. In contrast, the spatial kinetic energy budget for the weakly-coupled analysis found that the ‘Pressure-velocity’ contribution increases substantially for low R, suggesting destabilisation. All mode coupling terms are negative contributions to the energy budget - supporting the proposed mechanism. Based on the present studies, it is clear that future directions for research would include obtaining a more representative jet-in-crossflow base flow by extending the tanh or UVAS base flows to higher orders of $1/R$ or from using time-averaged experimental or numerical simulation data. Such a base flow could also be used to inform non-modal stability or resolvent analyses to better match impulsively or harmonically forced jets-in-crossflow. The wavemaker (in a 2D-local sense) could also be found, which may synergise with the passive tab disturbance of the jet-in-crossflow.

Mechanical and transport behavior of thin constrained layers has been an important area of research in solid mechanics. Emerging engineering applications, involving advanced materials, require one to model and analyze the response of thin layers in which mechanical and transport phenomena are coupled. Further, those thin layers are often integrated into larger complex systems, for which fundamental understanding is lacking. This gives rise to numerous challenging problems in applied mathematics, scientific computing, and modeling. We first analyze thin perfectly constrained linear elastic layers. A new asymptotic approach has been developed that, unlike all existing methods, does not invoke any assumptions other than the layer thinness and Saint-Venant's principle. Asymptotic solutions for the compressible and incompressible cases are relatively simple to construct, as they involve only one small parameter. However, these solutions are problematic for the nearly incompressible regime, which involves two small parameters. The problem is challenging both conceptually and in terms of matching accurate numerical solutions. Our approach closes the conceptual gaps, and is in excellent agreement with accurate numerical solutions across the entire physically admissible range of compressibilities. Next, we extend our asymptotic analysis to thin imperfectly constrained linear elastic layers. This problem is interesting due to the presence of interphases, whereby two additional small parameters are introduced. In particular, the interplay between up to four small parameters can significantly affect the mechanical response of thin constrained layers. Our new accurate asymptotic solution establishes conditions under which interphases associated with thin layers may be detected. Further, the solution for compressible layers leads to a new and remarkably simple experimental setup, which has yielded promising results. For thin linear chemo-elastic layers, we consider stress-assisted solid-state diffusion of solute through a deformable solid under isothermal conditions. We show that a thermodynamically-consistent characterization of stress-assisted diffusion should be in terms of the chemical potential, rather than the concentration, of the solute. Further, we formulate new thought experiments for measuring the chemo-elastic properties of solids. In particular, we propose a method for measuring the chemical potential of a solid, without imposing essential restrictions on its composition and external conditions. This enables the solid-state diffusion community to prescribe chemical potential boundary conditions, which as of yet is a non-intuitive task. Finally, we analyze a full-scale turning process involving a workpiece made of a difficult-to-machine advanced material. Lubrication was represented by varying the friction coefficient between the cutting tool and workpiece. Here we wanted to assess the existence of an anomalous trend observed experimentally by our project partners, whereby lubrication was less effective at certain cutting speeds. The extremely complex thermo-mechanical phenomena associated with this problem lead us to rely on a commercial finite element code for our simulations. Here our main challenge from the perspective of scientific computing was balancing available experimental data, computing resources, and accuracy of simulations.

Networks arise in a huge variety of real data scenarios: starting from social networks like Facebook or user product networks in recommendation systems to protein-protein interaction networks in biological systems, etc. In this thesis, we focus on developing fast and provable algorithms for some network inference problems. In the first part, we are interested in overlapping community detection problem under the popular Mixed Membership Stochastic Blockmodel (MMSB). We firstly establish sufficient conditions for the symmetric non-negative matrix factorization optimization to have a unique solution under MMSB, and propose a computationally efficient algorithm called GeoNMF that is provably optimal and hence consistent for a broad parameter regime. Then using the inherent geometry of MMSB, we link the inference of overlapping communities to the problem of finding corners in a noisy rotated and scaled simplex, for which consistent algorithms exist. We use this as a building block for our algorithm to infer the community memberships of each node, and provide uniform rates of convergence for the inferred community membership vector of each node in the network. As a byproduct of our analysis, we derive sharp row-wise eigenvector deviation bounds, and provide a cleaning step that improves the performance drastically for sparse networks. Our results hold over a broad parameter regime where the average degree only grows poly-logarithmically with the number of nodes. Using experiments with simulated and real datasets, we show that our method achieves better error with lower variability over competing methods, and processes real world networks of up to 100,000 nodes within tens of seconds. For the second part, we go beyond MMSB for overlapping community detection. Notice that many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with some small noise. Geometrically, each person is a point on a cone whose corners are these exemplars. This basic form encompasses the widely used MMSB of networks and its degree corrected variants, as well as topic models such as LDA. We show that a simple one-class SVM yields provably consistent parameter inference for all such models, and scales to large datasets. Experimental results on several simulated and real datasets show our algorithm (called SVM-cone) is both accurate and scalable. The final contribution of this thesis is novel nonparametric methods for network covariate estimation. Networks with node covariates are commonplace: for example, people in a social network have interests, or product preferences, etc. If we know the covariates for some nodes, can we infer them for the remaining nodes? We provide two provably consistent methods to solve this problem. For "low-rank" latent variable models, we develop SVD-RBF, which uses the top principal components of the network in a non-parametric regression. For general models, we present CN-VEC, which constructs a similarity measure between two nodes, based on the patterns of their 2-hop neighborhoods. CN-VEC then predicts node covariates by averaging the covariates of the top-k most similar nodes using this measure. SVD-RBF is consistent for low-rank models when the average degree grows with Õ(log n), while CN-VEC is consistent for a wide range of models when the degree grows with Õ(n¹ [superscript /] ³). To our knowledge, CN-VEC is the first provably consistent method for this problem under general models. Both methods are fast, and CN-VEC is also parameter-free. Experiments on 4 simulated network models and 3 real-world datasets show the effectiveness of our algorithms compared to the state of the art.

This dissertation focuses on two main areas: 1) data-driven stochastic modeling and applied probability, with applications to the sharing economy and networks; 2) machine learning, with a focus on sequential decision making for recommender systems and revenue management. Using tools from probability, statistics, and learning theory, this dissertation emphasizes fundamental contributions to both theory and methodology. Chapter 2 focuses on the design of stochastic models and mechanisms to increase efficiency of urban transportation and logistics systems. Trends in global urbanization require innovation, especially in the area of urban transportation networks. At the same time, the sharing economy provides opportunities to enhance efficient resource utilization. This Chapter addresses the new operational challenges that arise from emerging technologies within smart cities, focusing on last-mile delivery and smart mobility. Last-mile delivery may take up to 28% of the total transportation costs, and is arguably one of the biggest challenges in logistics management. We propose a new business model for optimizing the last-mile delivery of packages, using a strategy that combines the use of ride-sharing platforms (e.g. Uber or Lyft) with traditional in-house van delivery systems. To make the proposed solution tractable, we develop new theoretical results by approaching the problem from a probabilistic perspective. Our approach of determining the optimal reward to private drivers for delivering packages is computationally efficient. Using synthetic and real data, we show that our approach reduces cost by as much as 30% compared with the van-only strategy.In Chapter 3, motivated by the observation that overexposure to unwanted marketing activities can lead to customer dissatisfaction, we consider a setting where a platform offers a sequence of messages to its users and is penalized when users abandon the platform due to marketing fatigue. We propose a novel sequential choice model to capture multiple interactions taking place between the platform and its users: upon receiving a message, a user decides on whether to accept or reject the message. If she chooses to reject, she would then decide to either receive the next message in the sequence or abandon the platform. Based on user feedback, the platform dynamically learns users' abandonment distribution and the relevance of the recommended content. With a goal to maximize the cumulative payoff over a time horizon, the platform dynamically adjusts the sequence of messages and the order in which the messages are shown to a user. We refer to this online learning task as the sequential choice bandit (SC-Bandit) problem. For the offline combinatorial optimization problem, we show a polynomial-time algorithm. For the online problem, we consider two variants, depending on whether contexts are included, and propose algorithms that balance exploration and exploitation. Lastly, we evaluate the performance of our algorithms with both synthetic and real-world datasets. Complex networks appear in essentially all branches of science and engineering, and people from various fields have used random graphs to model, explain and predict some of the properties commonly observed in real-world networks. In Chapter 4, we study a family of directed random graphs whose arcs are sampled independently of each other, and are present in the graph with a probability that depends on the attributes of the vertices involved. In particular, this family of models includes as special cases the directed versions of the Erdos-Renyi model, graphs with given expected degrees, the generalized random graph, and the Poissonian random graph. We establish the phase transition for the existence of a giant strongly connected component and provide some other basic properties, including the limiting joint distribution of the degrees and the mean number of arcs. In particular, we show that by choosing the joint distribution of the vertex attributes according to a multivariate regularly varying distribution, one can obtain scale-free graphs with arbitrary in-degree/outdegree dependence.

Abstract: With increasing data rate demand, caching popular multimedia content at base stations or routers is emerging as a promising technology for 5G. The primary benefits of wireless caching in cellular networks are traffic offload from macro base stations and decreased delays at the user equipment. However, there are few points of concern that have yet to be addressed, such as optimal caching strategy, impacts of mobility and interference characterization. We address these concerns with three contributions. Firstly, we propose a caching strategy to enhance delay performance by utilizing the core routers. Secondly, we develop a mobility analysis for a small-cell network with caching abilities at small-cell base stations. Finally, the interference characterization of a cache network deviates from a traditional cellular network due to the opportunistic nature of association, and therefore, we develop an asymptotic analysis for the generalized fading channel that can be used for considered networks. Our analysis leads to following conclusions: firstly, static caching or simple content placement strategy can outperform state-of-the-art dynamic caching strategies; secondly, association with a cache source is ineffective for high velocity users; and finally, our developed asymptotic approximation for a Generalized-Gamma fading model provides better performance measurements over existing methods.

This dissertation studies the development of provably near-optimal real-time prescriptive analytics solutions that are easily implementable in a dynamic business environment. We consider several stochastic control problems that are motivated by different applications of the practice of pricing and revenue management. Due to high dimensionality and the need for real-time decision making, it is computationally prohibitive to characterize the optimal controls for these problems. Therefore, we develop heuristic controls with simple decision rules that can be deployed in real-time at large scale, and then show theirs good theoretical and empirical performances. In particular, the first chapter studies the joint dynamic pricing and order fulfillment problem in the context of online retail, where a retailer sells multiple products to customers from different locations and fulfills orders through multiple fulfillment centers. The objective is to maximize the total expected profits, defined as the revenue minus the shipping cost. We propose heuristics where the real-time computations of pricing and fulfillment decisions are partially decoupled, and show their good performances compared to reasonable benchmarks. The second chapter studies a dynamic pricing problem where a firm faces price-sensitive customers arriving stochastically over time. Each customer consumes one unit of resource for a deterministic amount of time, after which the resource can be immediately used to serve new customers. We develop two heuristic controls and show that both are asymptotically optimal in the regime with large demand and supply. We further generalize both of the heuristic controls to the settings with multiple service types requiring different service times and with advance reservation. Lastly, the third chapter considers a general class of single-product dynamic pricing problems with inventory constraints, where the price-dependent demand function is unknown to the firm. We develop nonparametric dynamic pricing algorithms that do not assume any functional form of the demand model and show that, for one of the algorithm, its revenue loss compared to a clairvoyant matches the theoretic lower bound in asymptotic regime. In particular, the proposed algorithms generalize the classic bisection search method to a constrained setting with noisy observations.

This dissertation establishes a mathematical framework for analyzing a viscoelastic model that displays thixotropic behavior as a model parameter gets very small. The model is the partially extending strand convection model, originally derived for polymeric melts that have long strands that get in the way of fully retracting. A Newtonian solvent is added. The uniaxial and equibiaxial extensional flows are studied using combined asymptotic analysis and numerical simulations. An initial value problem with a prescribed elongational stress is solved in the limit of large relaxation time. This gives rise to multiple time scales. If the initial stress is less than a critical value, the initial elastic elongation is followed by settling to an unyielded state at the slow time scale. If the initial stress is larger than the critical value, then yielding ensues. The extensional flows produce delayed yielding and hysteresis, both associated with thixotropy in complex fluids.

This dissertation investigates the tactical dynamic pricing decisions in industries where sellers sell multiple types of capacity-constrained products/services to their customers. Motivated by operational frictions posed by business considerations, I develop dynamic pricing heuristics that have both provably good revenue performance and nice features which can address these operational frictions. The first essay studies how to do effective dynamic pricing without too many price changes. In practice, many sellers have concerns about dynamic pricing due to the computational complexity of frequent re-optimizations, the negative perception of excessive price adjustments, and the lack of flexibility caused by existing business constraints. To address these concerns, I develop a pricing heuristic which is computationally easy to implement and only needs to adjust a small number of prices and do so infrequently to guarantee a strong revenue performance. In addition, when not all products are equally admissible to price adjustment, my heuristic can replace the price adjustment of some products by their similar products and maintain an equivalent revenue performance. These features allow the sellers to achieve most of the benefit of dynamic pricing with much fewer price changes and provide extra flexibility to manage prices. While the first essay assumes that the sellers know the underlying demand function, this information is sometimes unavailable to the sellers in practice. The second and the third essays study how to jointly learn the demand and dynamically price the products to minimize revenue loss compared to a standard revenue upper bound in the literature. The second essay addresses the parametric case where the seller knows the functional form of the demand but not the parameters; the third essay addresses the nonparametric case where the seller does not even know the functional form of the demand. There is a considerable gap between the revenue loss lower bound under any pricing policy and the performance bound of the best known heuristic in the literature. To close the gap, in my second essay, I propose a heuristic that exactly match the lower bound for the parametric case, and show that under a demand separation condition, a much sharper revenue loss bound can be obtained; in my third essay, I propose a heuristic whose performance is arbitrarily close to the lower bound for the nonparametric case. All the proposed heuristics are computationally very efficient and can be used as a baseline for developing more sophisticated heuristics for large-scale problems.

Asymptotic methods and experimental techniques have been used in combination with direct numerical simulations and stability analyses to investigate effects of buoyancy-driven motion in two different reactive problems, namely, flickering of nonpremixed diffusion flames in open atmospheres and slowly reacting combustion of confined mixtures. A brief description of these two separate topics is given below.Part I of this dissertation deals with the flickering of jet diffusion flames and the puffing of liquid-fuel pool fires. Both phenomena are a manifestation of an axisymmmetric global hydrodynamic instability driven by the interactions of the buoyancy force with thedensity differences induced by the chemical heat release, which occurs in the flame sheet separating the fuel and oxidizer domains. For these flows, the predictive capability of local quasi-parallel stability analyses is limited by the non-slender character of the resultingeigenmodes, so that a biglobal analysis is needed to accurately determine marginal instability conditions and resulting frequencies. The results for jet diffusion flames, including the Froude number/Reynolds number instability boundaries for different fuel-feed dilutions,are compared with direct numerical simulations, giving good agreement for the range of conditions explored in our study. For liquid-fuel pool fires, the stability analysis provides the critical value of the Rayleigh number at the onset of the puffing instability. The predictions for different liquid fuels are compared with the results obtained in small-scale laboratory experiments.Part II is concerned with the “slowly reacting” mode of combustion, and its thermal-explosion limits, of an initially cold gaseous mixture enclosed in a spherical vessel with a constant wall temperature, a relevant problem in connection with the safe storageand transportation of reactant gas mixtures. Following Frank-Kamenetskii’s seminal analysis of this problem, the strong temperature dependence of the effective overall reaction rate is taken into account by using a single-reaction model with an Arrhenius rate havinga large activation energy, resulting in a critical value Da_c of the controlling Damköhler number above which the slowly reacting mode of combustion no longer exists. A Rayleigh number Ra based on the relevant density difference is seen to measure the relative effect ofnatural convection. Our numerical computations indicate that the value of Da_c increases with Ra as a result of the enhanced heat-transfer rate. Specific consideration is given to the flow structure in the asymptotic limits Ra << 1 and Ra >> 1, which yield accuratepredictions of critical explosion conditions. For completeness, the application of Frank-Kamenetskii’s ideas to the problem of flow of a reactive mixture in a pipe is presented in an appendix for the case of buoyancy-free conditions.

Massive multiple-input multiple-output (MIMO) is the concept of deploying a very large number of antennas at the base stations (BS) of cellular networks. Frequency-division duplexing (FDD) massive MIMO systems in the downlink (DL) suffer significantly from the channel estimation overhead. In this thesis, we propose a minimum mean square error (MMSE)-based channel estimation framework that exploits the spatial correlation between the antennas at the BS to reduce the latter overhead. We investigate how the number of antennas at the BS affects the channel estimation error through analytical and asymptotic analysis. In addition, we derive a lower bound on the spectral efficiency of the communication system. Close form expressions of the asymptotic MSE and the spectral efficiency lower bound are obtained. Furthermore, perfect match between theoretical and simulation results is observed, and results show the feasibility of our proposed scheme.

This dissertation considers joint pricing and inventory control problems in which the customer's response to selling price and the demand distribution are not known a priori, and the only available information for decision-making is the past sales data. Data-driven algorithms are developed and proved to converge to the true clairvoyant optimal policy had decision maker known the demand processes a priori, and, for the first time in literature, this dissertation provides theoretical results on the convergence rate of these data-driven algorithms. Under this general framework, several problems are studied in different settings. Chapter 2 studies the classical joint pricing and inventory control problem with backlogged demand, and proposes a nonparametric data-driven algorithm that learns about the demand on the fly while making pricing and ordering decisions. The performance of the algorithm is measured by regret, which is the average profit loss compared with that of the clairvoyant optimal policy. It is proved that the regret vanishes at the fastest possible rate as the planning horizon increases. Chapter 3 studies the classical joint pricing and inventory control problem with lost-sales and censored demand. Major challenges in this study include the following: First, due to demand censoring, the firm cannot observe either the realized demand or realized profit in case of a stockout, therefore only biased data is accessible; second, the data-driven objective function is always multimodal, which is hard to solve and establish convergence for. Chapter 3 presents a data-driven algorithm that actively explores in the inventory space to collect more demand data, and designs a sparse discretization scheme to jointly learn and optimize the multimodal data-driven objective. The algorithm is shown to be very computationally efficient. Chapter 4 considers a constraint that only allows the firm to change prices no more than a certain number of times, and explores the impact of number of price changes on the quality of demand learning. In the data-driven algorithm, we extend the traditional maximum likelihood estimation method to work with censored demand data, and prove that the algorithm converges at the best possible rate for any data-driven algorithms.

Abstract: The role of interfacial slippage on the deformation and stress fields near an interfacial crack are investigated in detail. First, the limiting extents of interfacial bonded friction, defined as the `frictionless' and `no-slip' cases, were modeled as hyperelastic elastostatic boundary value problems in plane strain. Using ideas similar to a fracture mechanics cohesive zone model, the `finite friction' case is established as a shear stress threshold on the bonded surface which if exceeded, allows interfacial slippage. The solutions of all three of these boundary value problems are found in the near field of the crack front using asymptotic analysis. Comparison of these solutions confirm experimental results that the inhibition of interfacial slip in the no-slip case caused material to contract inward producing a non-vertical surface angle that is shown to be a ratio of the in-plane stress components. The finite friction and frictionless cases were found to be related in that they both showed interfacial slippage after loading and the surface angle was perpendicular to the bonded interface. Further, the frictionless case is achieved when the finite friction threshold is set to zero, and in the presence of interfacial friction on the bonded edge there was a reduction in movement and the crack opening profile became more blunted or sharpened depending on the direction of slip. Extensive numerical simulations were performed using commercial finite element analysis software. Simulations for all three friction cases were studied and the results showed strong agreement with the analytical near field findings. Using the numerical data, the effects of far-field loading conditions on the remaining constants in the near-field solutions were quantified and discussed. Through the course of these simulations, it was identified that the direction of loading also plays a significant role on the deformation and stress fields near the crack front. The blunting effect occurred when the loading direction was more perpendicular and the interfacial slip moved to the right. However, when the loading direction was more tangential to the surface away from the crack the deformation field was sharpened and material in the slip-zone moved to the left. This direction of transition was also affected by the total deflection of the far-field loading. Moreover, the amount of blunting was found to be directly related to the magnitude of the stress threshold, and depending on the shear stress direction on the bonded edge (linked to the direction of slippage), produces significantly different normal stress magnitudes and failure modes. From the results presented in this study, a method of quantifying interfacial friction through deflection geometry is introduced and design considerations for desired adhesive tearing mechanisms are provided.

Due to its relatively low computational cost, the neutron diffusion approximation remains one of the most commonly-used computational tools for reactor analysis. Although diffusion requires more approximations than higher-order "transport" methods such as SN and Monte Carlo, the computational cost of these methods prohibits their widespread general use. It has long been known that diffusion is anisotropic in heterogeneous reactors, i.e., neutrons diffuse more rapidly in some directions than others. While in many reactors the anisotropic diffusion effects are negligible, in lattices containing voided or optically-thin channels these effects are significant. It is therefore desirable that the homogenized diffusion coefficient be a tensor. In this thesis, a multigroup, homogenized, anisotropic diffusion equation is derived asymptotically from the exact Boltzmann transport equation for large, 3-D, multiplying systems with a periodic lattice structure. The primary mathematical assumption is that the system is a large, periodic lattice, and that the length scale of a lattice element is small relative to the total system size. The leading-order term of the asymptotic flux has a standard form, i.e., it is the product of a homogenized diffusion solution and a transport solution for an infinite, periodic system. The first-order correction term significantly improves the accuracy of the reconstructed fluxes. The goal of this research is to derive a complete homogenized, multigroup diffusion theory using rigorous mathematical analyses, rather than using ad hoc approximations, as have typically been done previously. Specifically, we use an asymptotic method to derive the homogenized diffusion equation, and we use a variational analysis to obtain "Marshak"- and ``variational''-like boundary conditions specifically for lattice geometries, and interface conditions that include flux and current discontinuity factors. We also provide extensive numerical simulations that demonstrate the high accuracy of the asymptotic method. Our results indicate that when the assumptions of the asymptotic analysis are met, the use of the asymptotic diffusion coefficient can lead to significantly improved estimates of the reactor eigenvalue and the neutron flux, otherwise it is comparable in accuracy. The asymptotic flux reconstruction is significantly more accurate than the standard flux reconstruction, especially near the outer regions of the reactor core.

This dissertation addresses some of the geometric properties of orbits of integral operators on the Banach spaces C[0, 1] and Lp[0, 1]. It will be shown that, under very general conditions on the starting element, an orbit of the Volterra operator cannot be a Schauder basis for its closed linear span. However, lacunary subsequences of the orbit will be seen to be Schauder bases for their closed linear span. Bounds on the norm of the iterates and a monotonicity result for a certain class of functions will be established. Moreover, exact asymptotic constants arising from the analysis will be exhibited.

The Benjamin-Ono equation is an integrable partial integro-differential equation which attracts much attention due to its application in modeling of the internal gravity waves in deep water and the ``morning glory cloud" in Northeastern Australia. In this dissertation, we first analyze the zero dispersion limit of the Cauchy problem of the Benjamin-Ono equation and give the first rigorous results. We demonstrate existence of the zero dispersion limit in the weak L2(R) sense and show this limit is equal to the signed sum of the branches of the multivalued solution of the inviscid Burgers equation with the same initial condition. Generalizations of these results are also given in this dissertation by using the formula of the $N$-soliton solutions of the higher order BO equations obtained by Matsuno. Moreover, we study three different numerical methods: the Fourier pseudospectral method, the Radial Basis Function method and the Christov method which are applied to solve the Benjamin-Ono equation. A comparison among the three methods is included. In the end, we also numerically illustrate and verify our theoretical results and study the traveling wave solutions of the cubic Benjamin-Ono equation.

The non-parametric MLE for an isotonic density was first developed by Grenander [8], and the estimator is a step function. A smoothed NPMLE hasn't been studied in the statistical literature. The problem considered in this thesis is to find a smoothed NPMLE for a continuous and non-increasing density in the class of piecewise linear densities. A piecewise linear NPMLE is found for grouped data. An EM-Algorithm for computing the estimator is developed, and the iterates in the algorithm are shown to converge to the NPMLE. Potential applications to distance sampling models are described and illustrated with a numerical example. Let h be the class width. It is shown that the NPMLE is not only consistent at x > 0 but also at 0 with a mild condition on h. The asymptotic distribution of the NPMLE is normal with convergence rate n-1/3 if nh3→infinity ; With h=Oe&parl0;n-1/3&parr0; , it can be described by the left and right derivatives of the Least Concave Majorant of a shifted Brownian motion; When h=o&parl0;n-1/3&parr0; , it is same as that of the Grenander's estimator. For ungrouped data, a linear spline, which is called SNPMLE, is found by maximizing the penalized likelihood. An algorithm is developed and shown to converge to the SNPMLE. The algorithm is illustrated with simulated random numbers and a real data set. A simulation study shows that the SNPMLE performs better than Grenander's estimator. It is shown that the SNPMLE is consistent at x > 0, and the Hellinger distance between SNPMLE and the underlying density f converges to 0 in probability with rate n-1/3 if f has a finite support.

Lattice-based quantization is an attractive method of vector quantization, because of its potentially low complexity and its ability to form nearly spherical cells. In this dissertation, three lattice-based quantization methods are optimized using high resolution analysis. In the first part of this dissertation, fixed-rate lattice quantization for a class of generalized Gaussian sources is considered. Asymptotic expressions for the optimal scaling factor and resulting minimum distortion are presented. These expressions are derived by minimizing upper and lower bounds to distortion. It is also shown that for scale-optimized lattice quantization, granular distortion asymptotically dominates overload distortion, and the ratio of optimal lattice quantizer distortion to optimal vector quantizer distortion is shown to increase without bound as rate increases. In the second part, multidimensional companding, which utilizes a nonlinear compressor function and a lattice quantizer, is optimized for memoryless sources. Under certain technical assumptions on the compressor function, it is shown that the best compressor function consists of the best scalar compressor functions for each component of the source vector. The gain of optimal companding compared to scalar quantization is shown to be the gain in normalized moment of inertia of the lattice cell compared to a cube. The third part considers uniform polar quantization, which utilizes a polar transformation and a two dimensional integer lattice quantixer. It is shown that the optimal rate allocation between magnitude and phase gives increasingly more rate to the magnitude as total rate increases, compared to nonuniform polar, where the difference between optimal magnitude and phase rates is a constant. In addition, a unified analysis of several nonuniform polar quantization schemes is developed by focusing on their point densities and inertial profiles and using Bennett's integral to express the mean-squared error. The subsequent analysis is straightforward and leads to new insights into the relationship between polar quantization and Cartesian quantization.