The non-linear hyperbolic partial-differential equations governing long wave propagation in one and two plan dimensions are derived. By application of the Preissmann or 'box' finite difference scheme two numerical models of long wave behaviour are developed. The first, based on the one plan dimensional form of the partial differential equations, is intended for the solution of flood routing problems in natural river systems. The model has two constituent parts. A main channel algorithm reproducing flood wave behaviour in the main channel of the drainage system and a washland algorithm modelling the behaviour of lateral storage ponds on the river banks. The main channel algorithm possesses the ability to handle: natural channel cross-sections, variable distance increments, tributary inflows, calibration with both distance and stage, rating curve boundary conditions, the formation and drowning of controls and the analysis of controls. On completion of development trials the model was used to assess the effect a new road embankment would have on flood levels in the River Aire in Yorkshire. The second, based on the two plan dimensional partial differential equations, employs an alternating direction application of the Preissmann finite difference scheme to model tide and storm surge behaviour in estuaries and coastal seas. Special consideration was given to boundary conditions in the model and these include a moving shore line boundary condition permitting the flooding and drying of sand flat areas to be modelled and a "weir" flow boundary condition, enabling the overtopping of obstructions with a width considerably less than the grid size of the model to be represented. A practical assessment of the model's capabilities was accomplished by simulating tide and storm surge propagation in the Firth of Clyde and Humber Estuary.

No description available.

<p> A mathematical model is an abstraction that distills quantifiable behaviors and properties into a well-defined formalism in order to learn or predict something about a system. Such models may be as light as pencil-and-paper calculations on the back of an envelope or as heavy as to entail modern super computers. They may be as simple as predicting the trajectory of a baseball or as complex as forecasting the weather. By using macromolecular protein structures as substrates, the objective of this thesis is to improve upon and leverage mathematical models in order to address what is both a growing challenge and a burgeoning opportunity in the age of next-generation sequencing. The rapidly growing volume of data being produced by emerging deep sequencing technologies is enabling more in-depth analyses of protein conservation than previously possible. Increasingly, deep sequencing is bringing to light many disease-associated loci and localized signatures of strong conservation. These signatures in sequence space are the "shadows" of selective pressures that have been acting on proteins over the course of many years. However, despite the rapidly growing abundance of available data on such signatures, as well as the finer resolution with which they may be detected, an intuitive biophysical or functional rationale behind such genomic shadows is often missing (such intuition may otherwise be provided, for instance, by the need to engage in protein-protein interactions, undergo post-translational modification, or achieve a close-packed hydrophobic core). Allostery may frequently provide the missing conceptual link. Allosteric mechanisms act through changes in the dynamic behavior of protein architectures. Because selective evolutionary pressures often act through processes that are intrinsically dynamic in nature, static renderings can fail to provide any plausible rationale for constraint. In the work outlined here, models of protein conformational change are used to predict allosteric residues that either <i>a)</i> act as essential cavities on the protein surface which serve as sources or sinks in allosteric communication; or <i>b)</i> function as important information flow bottlenecks within the allosteric communication pathways of the protein interior. Though most existing approaches entail computationally expensive methods (such as MD) or rely on less direct measures (such as sequence features), the framework discussed herein is simultaneously both computationally tractable and fundamentally structural in nature &ndash; conformational change and topology are directly included in the search for allosteric residues &ndash; thereby enabling allosteric site prediction across the Protein Data Bank. Large-scale (i.e., general) properties of the predicted allosteric residues are then evaluated with respect to conservation. Multiple threads of evidence (using different sources of data and employing a variety of metrics) are used to demonstrate that the predicted allosteric residues tend to be significantly conserved across diverse evolutionary time scales. In addition, specific examples in which these residues can help to explain previously poorly understood disease-associated variants are discussed. Finally, a practical and computationally rapid software tool that enables users to perform this analysis on their own proteins of interest has been made available to the scientific public.</p>

<p> Modern general equilibria under uncertainty are modeled based on the recognition that all risks cannot be eliminated, perfect hedging is not possible, and some risk exposures must be tolerated. Therefore, we need to define the set of acceptable risks as a primitive of the financial economy. This set will be a cone, hence the word conic. Such a conic perspective challenges classical economics by introducing finance into the economic models and enables us to rewrite major chapters of classical micro- and macro-economics textbooks. </p>

<p>RNA viruses are an important cause of global morbidity and mortality. The rapid evolutionary rates of RNA virus pathogens, caused by high replication rates and error-prone polymerases, can make the pathogens difficult to control. RNA viruses can undergo immune escape within their hosts and develop resistance to the treatment and vaccines we design to fight them. Understanding the spread and evolution of RNA pathogens is essential for reducing human suffering. In this dissertation, I make use of the rapid evolutionary rate of viral pathogens to answer several questions about how RNA viruses spread and evolve. To address each of the questions, I link mathematical techniques for modeling viral population dynamics with phylogenetic and coalescent techniques for analyzing and modeling viral genetic sequences and evolution. The first project uses multi-scale mechanistic modeling to show that decreases in viral substitution rates over the course of an acute infection, combined with the timing of infectious hosts transmitting new infections to susceptible individuals, can account for discrepancies in viral substitution rates in different host populations. The second project combines coalescent models with within-host mathematical models to identify driving evolutionary forces in chronic hepatitis C virus infection. The third project compares the effects of intrinsic and extrinsic viral transmission rate variation on viral phylogenies.</p> / Dissertation

<p> The solution for sequential Caputo linear fractional differential equations with variable coefficients of order q, 0 &lt; q &lt; 1 can be obtained from symbolic representation form. Since the iterative method developed in Chapter 2 is time-consuming even for the simple linear fractional differential equations with variable coefficients, the direct numerical approximation developed in Chapter 3 is very useful tool when computing the linear and non-linear fractional differential equations of a specific type. This direct numerical method is useful in developing the monotone method and the quasilinearization method for non-linear problems. As an application of this result, we have obtained the numerical solution for a special Ricatti, type of differential equation which blows up in finite time. The generalized monotone iterative method with coupled lower and upper solutions yields monotone natural sequence which converges uniformly and monotonically to coupled minimal and maximal solutions of Caputo fractional boundary value problem. We obtain the existence and uniqueness of sequential Caputo fractional boundary value problems with mixed boundary conditions with the Green's function representation.</p>

Music is an ever-changing cultural reflection. It is deeply integrated into our society, ubiquitous in movies, television shows, restaurants, sport venues, churches and a plethora of other places. This thesis proposes that we consider the lyrics in popular music, as determined by Billboards Hot 100 chart, as a natural medium to analyze the changes in culture over the past half-century. Using this collection of lyrics, we analyze the change in relative frequency of individual words over time, and compare to works of literature. Furthermore, we use the ranking in the Top 100 as a metric with which to explore the relationship between usage of particular words and the popularity of the respective songs. We find that our data coincides with a previous hypothesis that the relative happiness of lyrics has decreased over time, and find that this also applies to the relative happiness of popular music.

Various formulations of free Euclidean Markov spin one fields with m > 0 and m = 0 are studied. Attempts to construct Euclidean Markov spin two tensor fields with m > 0 and m = 0 are only partially successful.

<p> This thesis consists of two parts. The first part addresses the parameter estimation and calibration of the Generalized Hyperbolic (GH) distributions. In this part we review the classical expectation maximization (EM) algorithm and factor analysis for the GH distribution. We also propose a simple shrinkage estimator driven from the penalized maximum likelihood. In addition an on-line EM algorithm is implemented to the GH distribution; and its regret for general exponential family can be represented as a mixture of Kullback-Leibler divergence. We compute the Hellinger distance of the joint GH distribution to measure the performances of all the estimators numerically. Empirical studies for long-term and short-term predictions are also performed to evaluate the algorithms. </p><p> In the second part we applied the GH distribution to portfolio optimization and risk allocation. We show that the mean-risk portfolio optimization problem of a certain type of normal mixture distributions including the GH distribution can be reduced to a two dimensional problem by fixing the location parameter and the skewness parameter. In addition, we show that the efficient frontier of the mean-risk optimization problem can be extended to the three dimensional space. We also proposed a simple algorithm to deal with the transaction costs. The first and second derivatives of the CVaR are computed analytically when the underlying distribution is GH. With these results we are able to extend the effective number of bets (ENB) to general risk measures with the GH distribution. By diagonalizing the Hessian matrix of a risk measure we are able to extract locally independent marginal contributions to the risk. The minimal torsion approach can still be applied to get the local coordinators of the marginal contributions.</p>

<p> A system of nonlinear differential and algebraic equations governing diffusion and reaction is used to simulate a Film-Theory model of absorption of carbon dioxide and hydrogen sulfide in a tertiary-amine scrubber. Solutions to this system of equations are developed by applying various linearization methods.</p><p> First, the corresponding system of partial differential equations, with both spatial and temporal derivatives - i.e. which govern nonsteady-state behavior - is approximated with a finite-difference scheme, the results of which converge to a steady state (thereby providing what is believed to an exact &ldquo;numerical solution&rdquo;). However, its computation times make it unwieldy for optimization purposes.</p><p> Second, the method of van Krevelen and Hoftijzer (VKH) is applied to the ordinary differential equations (ODEs) and coupled algebraic equations governing steady-state behavior, and shown to provide reliable results over widely ranging conditions. However, the insertion of concentrations of nonvolatile reactants in the rate equation, whereby the VKH method linearizes the ODEs, is potentially problematic when applied to cases with higher pressures of hydrogen sulfide; in such cases, absorption rates based on the VKH lose accuracy when compared with those calculated using numerical methods. With that caveat, the VKH method is used to explore the role of the tertiary amine's pKa in optimizing selectivity for hydrogen sulfide.</p><p> Third, a regular perturbation method is applied to the steady-state problem. The resulting power series solution [with the Hatta number (a ratio of characteristic diffusion and reaction times) as the expansion parameter], truncated after the first order term, places an upper bound on the Hatta number range over which the zero-order solution may be considered valid. Inclusion of second order and third order terms does little to expand the range of accuracy.</p><p> Explicit solutions are also derived for the limiting cases of purely physical absorption (no reaction) and local reaction equilibrium. All models are applied over a range of conditions to calculate the enhancement factors by which chemical reactions multiply absorption rates, and in so doing, infer the approximate solutions' regimes of accuracy.</p>