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In 1941, F,L. Hitchcock presented a study entitled "The Distribution of a Product from Several Sources to Nu merous Localities." This remarkable presentation of a busi ness application is considered to be the first important contribution to the solution of transportation problems. Later in 1947 T.C. Koopmans presented a study under the title of "Optimum Utilization of the Transportation System." Transportation problems can be considered a subclass of those models that are applied to solve problems in which there are several activities to be enacted with the probability of making a selection among them when the resources are limited. In other words, the problem is to find the way to obtain maximum profit or minimum cost in combining acti vities and resources. Limited facilities represent tangible, real and mea surable situations, such as fixed production capacity; res tricted quantity of material, time and equipment; or any other sort of fixed means for manufacturing output, conversely, the amount of work that has to be done, using those limited resources can be solved.

<p> A spectrally based normalization methodology (Wpt normalization) for linearly transforming cone excitations or sensor values (sensor excitations) to a representation that preserves the perceptive concepts of lightness, chroma and hue is proposed resulting in a color space with the axes labeled <i> W, p, t.</i> Wpt (pronounced &ldquo;Waypoint") has been demonstrated to be an effective material color equivalency space that provides the basis for defining Material Adjustment Transforms that predict the changes in sensor excitations of material spectral reflectance colors due to variations in observer or illuminant. This is contrasted with Chromatic Adaptation Transforms that predict color appearance as defined by corresponding color experiments. Material color equivalency as provided by Wpt and Wpt normalization forms the underlying foundation of this doctoral research. A perceptually uniform material color equivalency space (&ldquo;Waypoint Lab" or WLab) was developed that represents a non-linear transformation of Wpt coordinates, and Euclidean WLab distances were found to not be statistically different from &Delta;<i>E</i>*₉₄ and &Delta;<i>E</i>₀₀ color differences. Sets of Wpt coordinates for variations in reflectance, illumination, or observers were used to form the basis of defining Wpt shift manifolds. WLab distances of corresponding points within or between these manifolds were utilized to define metrics for color inconstancy, metamerism, observer rendering, illuminant rendering, and differences in observing conditions. Spectral estimation and manipulation strategies are presented that preserve various aspects of &ldquo;Wpt shift potential" as represented by changes in Wpt shift manifolds. Two methods were explored for estimating Wpt normalization matrices based upon direct utilization of sensor excitations, and the use of a Wpt based Material Adjustment Transform to convert Cone Fundamentals to &rdquo;XYZ-like" Color Matching Functions was investigated and contrasted with other methods such as direct regression and prediction of a common color matching primaries. Finally, linear relationships between Wpt and spectral reflectances were utilized to develop approaches for spectral estimation and spectral manipulation within a general spectral reflectance manipulation framework &ndash; thus providing the ability to define and achieve &ldquo;spectrally preferred" color rendering objectives. The presented methods of spectral estimation, spectral manipulation, and material adjustment where utilized to: define spectral reflectances for Munsell colors that minimize Wpt shift potential; manipulate spectral reflectances of actual printed characterization data sets to achieve colorimetry of reference printing conditions; and lastly to demonstrate the spectral estimation and manipulation of spectral reflectances using images and spectrally based profiles within an iccMAX color management workflow.</p>

We study the DNLS and its dispersionless limit based on a family of matrices, named after Cantero, Moral, and Velazquez (CMV). The work is an analog to that of the Toda lattice and dispersionless Toda. We rigorously introduce the constants of motion and matrix symbols of the dispersionless limit of the DNLS. The thesis is an algebraic preparation for some potential geometry setup in the continuum sense as the next step.

In this dissertation, we study free boundary problems that describe equilibrium configurations of electromechanical systems consisting of a conducting elastic sheet deflected by an external charge distribution. Such systems are non-local in nature - the electrostatic pressure experienced by any individual point on the sheet depends on the entire deflection profile (as a result of the requirement that the deflected sheet must remain an equipotential). The magnitude of the electrostatic pressure varies quadratically with the magnitude of the local electric field. Similar non-local free boundary problems arise in two-layer fluid systems forced by withdrawal flows, but the normal viscous stress experienced by the fluid-fluid interface instead varies linearly with the local velocity gradients. The analysis presented focuses on two configurations in particular: the electromechanical system described above, forced by a point charge, and an artificially modified version of the same electromechanical system in which the induced electrostatic pressure varies linearly with the local electric field and the forcing is provided by an electric dipole. This second model is constructed as a crude approximation of the two-layer fluid flow forced by a point sink, and is primarily used to explore the influence of the forcing exponent on the bifurcation structure and solution types of the associated system. Our main contribution is the development of new techniques for the analysis and efficient numerical computation of large-deflection profiles for the true electromechanical system. The induced charge on such profiles accumulates near the interface tip, so that the geometry there is primarily determined by a balance between elastic and electrostatic forces. Away from the tip, the electrostatic pressure is low and the interface relaxes under the influences of gravity and elasticity only. Such interfaces exhibit features on widely disparate length scales. We exploit this separation of the interface into two regions dominated by different force balances to create a separate representation of each region (in appropriately rescaled coordinates), and then match the two representations together while ensuring that the relationship between local induced stress and global interface geometry is respected. This is achieved by combining tools and results from complex analysis and the method of matched asymptotic expansions.

In blood microvessels, the finite size of the red blood cell relative to the vessel diameter gives rise to non-continuum behavior. One such effect is the presence of a cell-free or cell-depleted layer of plasma near the vessel walls. This results from the tendency of red blood cells to migrate away from solid boundaries, towards the center of the blood vessel. In order to understand this and other flow behaviors of blood, it is fundamental to consider the motion and deformation of single red blood cells suspended in flows, including the effects of solid boundaries. In this dissertation, a two-dimensional model is used to simulate the motion and deformation of red blood cells in Stokes flow. First, the dynamics of a red blood cell in an unbounded shear flow is considered. Under such conditions, cells may execute a tanktreading motion, in which the cell maintains a stable orientation and the membrane continuously circulates around the cell, or a tumbling motion, in which the cell continuously flips. The motion depends on the viscosity of the suspending medium, with tank-treading in a high-viscosity medium and tumbling in a low-viscosity medium. Here, the effect of including an elastic energy potential dependent on the phase of the tank-treading motion is considered. It is found that the cell may then execute a swinging motion, in which the orientation oscillates. Furthermore, an intermittent regime is found in which the motion alternates between swinging and tumbling. These results are consistent with previous findings based on a model with a fixed cell shape. Next, the behavior of a freely suspended single red blood cell in a flow with solid boundaries is examined. Two cases are considered, channel flow and semi-infinite shear flow. A low-viscosity suspending medium is assumed, such that a cell in an unbounded shear flow would show continuous tumbling motion. In channel flow, the presence of solid boundaries and the curvature of the velocity profile tend to inhibit tumbling motion. Tumbling of an isolated cell is not seen in channels of width 8 and 10 μm, but can occur in larger channels. In semi-infinite shear flow, the cell typically executes a complicated non-periodic motion that involves both tumbling and lateral migration, and is sensitive to the assumed initial conditions. In normal blood, which contains 40 to 45% red blood cells, a central core region containing red blood cells is formed, and a cell at the outer edge of this core experiences frequent interactions with other cells in the core, which tend to drive it down the concentration gradient, towards the wall. This effect is known as shear-induced dispersion. Here, the effect of such interactions is modeled as a lateral force directed toward the wall, acting on a single red blood cell in a semi-infinite shear flow. For a large enough lateral force, a stable tank-treading motion is predicted. The expected lateral force is estimated based on the theory of shear-induced dispersion, and it is found that it may be large enough to stabilize the orientation of tank-treading red blood cells at the interface of the cell-depleted layer and the red blood cell core. In such cases, the interaction of the cell with the wall generates a lift force directed away from the wall. The mechanism of this lift generation is analyzed using lubrication theory and considering the typical profile of the gap between a tank-treading cell and the vessel wall. The studies described above assumed that the vessel wall is a solid impermeable boundary. Vessel walls in vivo are coated with a permeable macromolecular glycocalyx or endothelial surface layer that impedes fluid motion. Here, the motions of a red blood cell adjacent to such a layer lining the solid boundary are considered in confined channel flow and semi-infinite shear flow. The results indicate that the effect of the layer is similar to the effect of decreasing the width of the channel and increases the rate of migration away from the wall. In summary, the motion of a single red blood cell in shear or channel flow shows complex dynamical behaviors. Generally, interactions with the wall stabilize cell orientation and generate lift forces that lead to the formation and persistence of the cell-free layer in narrow blood vessels.

This dissertation examines models, methods, and applications of electric field pulse propagation in nonlinear optics. Standard nonlinear optical propagation models such as the NLS equation are derived using a procedure invoking a slowly-varying wave approximation which amounts to discarding second order derivatives in the propagation direction. This work follows a more intuitive procedure emphasizing unidirectionality, the core trait of laser light propagation, by projecting a nonlinear wave system onto a unidirectional subspace. The projection method is discussed as a general theory and then applied to a series of different electric field configurations. Two important full-field propagation models are examined. The unidirectional pulse propagation equations (UPPE's) are generated from Maxwell's equations with the sole approximation being that of unidirectionality. The second model studied is the MKP equation which is a canonical full-field propagation equation particularly amenable to mathematical analysis due to its status as a conserved system. Applications unique to full-field propagation including electric field shock and harmonic walk-off induced collapse arrest are studied through numerical simulations. An emphasis is placed on the mid-infrared to long-infrared wavelength regime where significant differences between envelope models and electric field models manifest as a result of extremely weak dispersion. Presented are the first embedded Runge-Kutta exponential time-differencing (RKETD) methods of fourth order with third order embedding and fifth order with third order embedding for non-Rosenbrock type nonlinear systems. A procedure for constructing RKETD methods that accounts for both order conditions and stability is outlined. In the stability analysis, the fast time scale is represented by a full linear operator in contrast to particular scalar cases considered before. An effective time-stepping strategy based on reducing both ETD function evaluations and rejected steps is described. Comparisons of performance with adaptive-stepping integrating factor (IF) are carried out on a set of canonical partial differential equations including the standard z-propagated UPPE.

<p> Ever since the introduction of the Anderson model in 1958, physicists and mathematicians alike have been interested in the effects of disorder on quantum mechanical systems. For example, it is known that transport is suppressed for an electron moving about in a random environment, which follows from localization results proven for the Anderson model. </p><p> Quantum spin systems provide a relatively simple starting point when one is interested in studying many-body systems. Here we investigate a random block operator arising from the anisotropic <i>xy</i>-spin chain model. Allowing for arbitrary nontrivial single-site distributions, we prove a zero-velocity Lieb-Robinson bound under the assumption of dynamical localization at all energies. </p><p> After a preliminary study of basic properties and location of the almost-sure spectrum of this random block operator, we apply a transfer matrix formalism and prove contractivity and irreducibility properties of the Furstenberg group and, in particular, positivity of Lyapunov exponents at all nonzero energies. Then in the general setting of random block Jacobi matrices, we establish a Thouless formula, and under contractivity and irreducibility assumptions, we conclude dynamical localization via multiscale analysis by proving a Wegner estimate and an initial length scale estimate. Finally we apply our general results to prove localization for the special case of the Ising model, and we discuss a critical energy that arises.</p>

<p> The goal of this thesis is to build hybrid deterministic/Monte Carlo algorithms for solving the neutron transport equation and associated <i> k</i>-eigenvalue problem. We begin by introducing and deriving the transport equation before discussing a series of deterministic methods for solving the transport equation. To begin we consider moment-based acceleration techniques for both the one and two-dimensional fixed source problems. Once this machinery has been developed, we will apply similar techniques for computing the dominant eigenvalue of the neutron transport equation. We'll motivate the development of hybrid methods by describing the deficiencies of deterministic methods before describing Monte Carlo methods and their advantages. We conclude the thesis with a chapter describing the detailed implementation of hybrid methods for both the fixed-source and <i>k</i>-eigenvalue problem in both one and two space dimensions. We'll use a series of test problems to demonstrate the effectiveness of these algorithms before hinting at some possible areas of future work.</p>

<p> The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. </p><p> In this dissertation, we consider computational issues arising for the MHD equations. We focus on developing fast computational algorithms for solving the algebraic systems that arise from finite element discretizations of the fully coupled MHD equations. Emphasis is on solvers for the linear systems arising from algorithms such as Newton's method or Picard iteration, with a main goal of developing preconditioners for use with iterative methods for the linearized systems. In particular, we first consider the linear systems arising from an exact penalty finite element formulation of the MHD equations. We then draw on this research to develop solvers for a formulation that includes a Lagrange multiplier within Maxwell's equations. We also consider a simplification of the MHD model: in the MHD kinematics model, the equations are reduced by assuming that the flow behavior of the system is known. In this simpler setting, we allow for epistemic uncertainty to be present. By mathematically modeling this uncertainty with random variables, we investigate its implications on the physical model.</p>