This thesis examines certain aspects of diffraction and scattering of high frequency waves, utilising and extending upon the Geometrical Theory of Diffraction (GTD). The first problem considered is that of scattering of electromagnetic plane waves by a perfectly conducting thin body, of aspect ratio O(k^1/2), where k is the dimensionless wavenumber. The edges of such a body have a radius of curvature which is comparable to the wavelength of the incident field, which lies inbetween the sharp and blunt cases traditionally treated by the GTD. The local problem of scattering by such an edge is that of a parabolic cylinder with the appropriate radius of curvature at the edge. The far field of the integral solution to this problem is examined using the method of steepest descents, extending the recent work of Tew [44]; in particular the behaviour of the field in the vicinity of the shadow boundaries is determined. These are fatter than those in the sharp or blunt cases, with a novel transition function. The second problem considered is that of scattering by thin shells of dielectric material. Under the assumption that the refractive index of the dielectric is large, approximate transition conditions for a layer of half a wavelength in thickness are formulated which account for the effects of curvature of the layer. Using these transition conditions the directivity of the fields scattered by a tightly curved tip region is determined, provided certain conditions are met by the tip curvature. In addition, creeping ray and whispering gallery modes outside such a curved layer are examined in the context of the GTD, and their initiation at a point of tangential incidence upon the layer is studied. The final problem considered concerns the scattering matrix of a closed convex body. A straightforward and explicit discussion of scattering theory is presented. Then the approximations of the GTD are used to find the first two terms in the asymptotic behaviour of the scattering phase, and the connection between the external scattering problem and the internal eigenvalue problem is discussed.

The classical theory of channelized subglacial drainage,due orginally to Röthlisberger (1972) and Nye (1976), considers water flow in an ice channel overlying a rigid, impermeable bed. At steady flow, creep closure of the channel walls is counteracted by melt-back due to heat dissipation, and this leads to an equilibrium relation between channel water pressure and discharge. More generally, such a balance exhibits an instability that can be used to describe the mechanics of catastrophic flood events known as `jökulhlaups'. In this thesis, we substantiate these developments by exploring a detailed model where the channel is underlain by subglacial till and the flow supports a sediment load. Attention is given to the physics of bed processes and its effect on channel morphology. In particular, we propose a theory in which the channel need not be semi-circular, but has independently evolving depth and width determined by a local balance between melting and closure, and in which sediment erosion and deposition is taken into account. The corresponding equilibrium relation indicates a reverse dependence to that in the classical model, justifying the possibility of the subglacial canals envisaged by Walder and Fowler (1994). Theoretical predictions for sediment discharge are also derived. Regarding time-dependent flood drainage, we demonstrate how rapid channel widening caused by bank erosion can explain the abrupt recession observed in the flood hydrographs. This allows us to produce an improved simulation of the 1972 jökulhlaup from Grímsvötn, Iceland, and self-consistently, a plausible estimate for the total sediment yield. We also propose a mechanism for the observed flood initiation lake-level at Grímsvötn. These investigations expose the intimate interactions between drainage and sediment transport, which have profound implications on the hydrology, sedimentology and dynamics of ice masses, but which have received little attention.

This thesis is motivated by the need to calculate the electromagnetic fields produced by sources radiating in the presence of conductors. We begin by reviewing existing theory concerning sources in the presence of flat structures. Various extensions to the canonical Sommerfeld problem are considered. In particular we investigate the asymptotic solution for a finite source that focusses its energy at a point. In chapter 5 we review and extend the asymptotic results concerning illumination of a convex perfect conductor by an incident plane wave and outline the procedure for decoupling the electromagnetic surface field into two scalar modes. In chapter 6 we place a source on a perfect conductor and obtain a complete asymptotic solution for the fields. Special attention is paid to the asymptotic structure that smoothly matches between the leading order lit and shadow regions. We also investigate the degenerate case where one of the curvatures of the perfect conductor is zero. The case where the source is just off the surface is also investigated. In chapter 8 we use the Euler-Maclaurin summation formula to cheaply calculate the fields due to complicated arrays of point dipoles. The final chapter combines many earlier results to consider more general sources on the surface of a perfect conductor. In particular we must introduce new asymptotic regions for open sources. This then enables us to consider the focussing of the surface field due to a finite source. The nature of the surface and geometrical optics fields depends on the size of the source in comparison to the curvatures of the surface on which they lie. We discuss this in detail and conclude with the practical example of a spiral antenna.

Frost heave describes the phenomenon whereby soil freezing causes upwards surface motion due to the action of capillary suction imbibing water from the unfrozen region below. The expansion of water on freezing is a small part of the overall surface heave and it is the flow of water towards the freezing front which is largely responsible for the uplift. In this thesis, we analyse a model of frost heave due to Miller (1972, 1978) which is referred to as `secondary frost heave'. Secondary frost heave is characterised by the existence of a `partially frozen zone', underlying the frozen soil, in which ice and water coexist in the pore space. In the first part of the thesis we follow earlier work of Fowler, Krantz and Noon where we show that the Miller model for incompressible soils can be dramatically simplified. The second part of the thesis then uses this simplification procedure to develop simplified models for saline and compressible soils. In the latter case, the development of the theory leads to the consideration of non-equilibrium soil consolidation theory and the formation of segregated massive ice within permafrost. The final part of the thesis extends the simplified Miller model to the analysis of differential frost heave and the formation of patterned ground (e.g. earth hummocks and stone circles). We show that an instability mechanism exists which provides a plausible theory for the formation of these types of patterned ground.

Title: Nonconvex stochastic programming problems - formulations, sample approximations and stability Author: RNDr. Martin Branda Author's e-mail address: branda@karlin.mff.cuni.cz Supervisor: Doc. RNDr. Petr Lachout, CSc. Supervisor's e-mail address: lachout@karlin.mff.cuni.cz Abstract: We deal with problems where integer variables may appear, hence no assumptions on convexity are made throughout this thesis. The goal of Chapter 2 is to introduce stochastic programming problems and to outline the most important tasks connected with solving the problems. In Chapter 3, we compare basic formulations of static stochastic programming problems with chance constraints, with integrated chance constraints and with penalties in the objective function. We show that the problems are asymptotically equivalent under mild conditions. We discuss solving the problems using sample approximation techniques and extend some results on rates of convergence. All the formulations and corresponding sample approximations are compared on an investment problem with real features with Value at Risk constraint, integer allocations and transaction costs. Then, stability of financial decision models where two-stage mixed-integer value function appears as a loss variable is studied. In Chapter 4, we study qualitative properties of the...

Les automates d'arbres et leurs applications à la vérification forment le tronc commun de cette thèse. Dans la première parie, nous définissons une plate forme de model-checking complète [...] La seconde partie se penche sur un aspect important des automates que nous utilisons: leur contraintes [...] Finalement, nous étudions également les automates d'arbres cheminants [...] Nous améliorons leur conversion en automates parallèles, et nous développons une procédure de semi décision de leur vacuité, à la fois efficace et précise / Tree automata, and their applications to verification from the common thread of this thesis In the first part, we definie a complete model-cheking framework.[...] The second part focus on an important aspect of the automata involved: constraints.[...] Finaly, we also study the very different variety of tree-walking automata which have tight connections with navigational languages on semi-structured documents.

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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 goals of the research presented in this dissertation were to create algorithms that produce images of complex phenomena, to study the efficacy of the algorithms, and to apply these algorithms to important mineralogical problems. The algorithms that were created include the Sphere Projection method, the Chicken Wire method, and methods for calculating the curvature at any point on a surface. The Sphere Projection method is best applied to roughly spherical surfaces. A theorem about the "fit" to a sphere determines the accuracy of the model in this special case and gives some insight into the limitations of this method. The Chicken Wire method was developed to model those surfaces for which the Sphere Projection method was ineffective. The effectiveness of the Chicken Wire method was also determined. The algorithms were used to produce images of equi-value surfaces of the Laplacian of the electron density function in selected molecules. The water molecule, H2O, was studied to demonstrate that these new methods are capable of reproducing known features. The disiloxane molecule, H6Si2O7, was studied because it serves as a model for bonding in quartz and other important silicates. Lastly, the molecule NaLi2Si2OF9 was examined as a molecular model for low albite. A new discovery suggests that these algorithms will be an important tool in mineralogy. / Ph. D.

Disertacijoje tiriamas operatorių pusgrupių Eulerio ir Josidos approximacijų konvergavimas. Gauti Eulerio aproksimacijų asimptotiniai skleidiniai ir optimalūs liekamųjų narių įverčiai. Taip pat pateiktos įvairios šių skleidinių koeficientų analizinės išraiškos. Josidos aproksimacijoms buvo rasti du optimalūs konvergavimo greičio įverčiai su optimaliomis konstantomis. Taip pat gauti Josidos aproksimacijų asimptotiniai skleidiniai ir liekamųjų narių įverčiai. / In this thesis we investigate the convergence of Euler's and Yosida approximations of operator semigroups. We obtain asymptotic expansions for Euler's approximations of semigroups with optimal bounds for the remainder terms. We provide various explicit formulas for the coefficients for these expansions. For Yosida approximations of semigroups we obtain two optimal error bounds with optimal constants. We also construct asymptotic expansions for Yosida approximations of semigroups and provide optimal bounds for the remainder terms of these expansions.