<p> The problem of the constrained partitioning of a set using quantitative relationships amongst the elements is considered. An approach based on constrained integer programming is proposed that permits a group objective function to be optimized subject to group quality constraints. A motivation for this problem is the partitioning of students, e.g., in middle school, into groups that target educational objectives. The method is compared to another grouping algorithm in the literature on a data set collected in the Poudre School District.</p>

A major component of quantitative risk assessment involves dose-response modeling. Therein, an appropriate statistical model that approximately quantifies the relationship between exposure level (dose) and response (adverse endpoint) is fit to experimental data. The objective of this dissertation is to estimate adverse risks encountered in settings when the statistical model is formally defined and developed. From this, statistical inferences on the risk are conducted.First introduced are eight parametric models. The advantage of parametric models is they can produce consistent result when the selected model fits the dose-response curve very well. The simplicity of knowing the expression of these models allows for the construction of a variety of lower confidence limits, based on the Wald approach.However, if the true dose-response curve deviates significantly from a posited parametric model, the result may perform poorly. Non-parametric methods are then needed. The percentile bootstrap method from linear splines with Pool Adjacent Violator is first introduced. The method appeals to an asymptotic approximation, hence there is interest in assessing the small-sample coverage properties of this method. These are addressed via Monte Carlo computer simulations. We find that this method with four doses operates reasonably well at large sample sizes except for the concave increasing dose-response curve. In practice, small sample sizes are more common, therefore we turn to increasing the number of doses. We do see that, in general, the coverage becomes better as the doses number increases.To study the most common four dose design, the biased-corrected and accelerated bootstrap method from linear spline with Pool Adjacent Violator and discrete delta approach are also introduced. Simulation results show that the coverage are similar from these methods and have no improvement over the concave increasing dose-response curve.A final quadratic spline is then considered. For four doses design, this is repeated at four different points, to find an averaged extra risk function. In order to understand the operating characteristics of the method, another Monte Carlo simulation study is undertaken. This study produces similar results to those found using the percentile bootstrap method from linear splines.

The methods for analysis of a three degree-of-freedom nonlinear optical support system, subject to periodic and random vibration, are presented. The analysis models were taken from those generated for the dynamic problems related to the NASA Space Infrared Telescope Facility (SIRTF). The models treat the one meter, 116 kilogram (258 pound) primary mirror of the SIRTF as a rigid mass, with elastic elements representing the mirror support structure. Both linear and nonlinear elastic supports are evaluated for the SIRTF. Advanced Continuous Simulation Language (ACSL), a commercially available software package for numerical solution of nonlinear, time-dependent differential equations, was used for all models. The methods presented for handling the nonlinear differential equations can be readily adapted for handling other similar dynamics problems.

<p> Much of the world's rapidly growing urban population relies upon water distribution systems to provide treated water through networks of pipes. Rather than continuously supplying water to users, many of these distribution systems operate intermittently, with parts of the network frequently losing pressure or emptying altogether. Such intermittent water supply deleteriously impacts water availability, infrastructure, and water quality for hundreds of millions of people around the world. In this work I introduce the problem of intermittent water supply through the lens of applied mathematics. I first introduce a simple descriptive mathematical model that captures some hydraulic features of intermittency not accounted for by existing water distribution system software packages. I then consider the potential uses of such a model in a variety of optimization examples motivated by real-world applications. In simple test networks, I show how to reduce pressure gradients while the network fills by changing either the inflow patterns or the elevation profile. I also show test examples of using measured data to potentially recover unknown information such as initial conditions or boundary outflows. I then use sensitivity analysis to better understand how various parameters control model output, with an eye to figuring out which parameters are worth measuring most carefully in field applications, and also which parameters may be useful in an optimization setting. I lastly demonstrate some progress in descriptively modeling a real network, both through the introduced mathematical model and through a fluid-mechanics-based method for identifying data where the model is most useful.</p>

<p> One of the reasons why predation is important is that no organism can live, grow, and reproduce without consuming resources. We have studied possible scenarios of pattern formation in three different predator-prey models; the Rozenweig-McArthur, and the Leslie-Gower model with alternative food for the predator with different functional responses; one uses a prey-dependent functional response while the other use ratio-dependent functional response. The Turing patterns observed for the Leslie-Gower model with prey dependence are of two types, hot spot patterns and cold spot patterns while the ratio dependent model exhibit only hot spot patterns. Also, the labyrinthine pattern is also observed for some choices of parameters values within the Turing-Hopf domain for both models.</p>

<p> In this dissertation we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.</p>

<p> Traditional finite difference methods for solving the partial differential equations (PDEs) associated with wave and heat transport often perform poorly when used in domains that feature jump discontinuities in model parameter values (interfaces). We present a radial basis function-derived finite difference (RBF-FD) approach that solves these types of problems to a high order of accuracy, even when curved interfaces and variable model parameters are present.</p><p> The method generalizes easily to a variety of different problem types, and requires only the inversion of small, well-conditioned matrices to determine stencil weights that are applied directly to data that crosses an interface. These weights contain all necessary information about the interface (its curvature; the contrast in model parameters from one side to the other; variability of model parameter value on either side), and no further consideration of the interface is necessary during time integration of the numerical solution. </p>

The Darboux transformation is one of the main techniques for finding solutions of integrable equations. The Darboux transformation is not only powerful in the construction of muilti-soliton solutions, recently, it is found that the Darboux transformation, after some modification, is also effective in generating the rogue wave solutions. In this thesis, we derive the rogue wave solutions for the Davey-Stewartson-II (DS-II) equation in terms of Darboux transformation. By taking the spectral function as the product of plane wave and rational function, we get the fundamental rogue wave solution and multi-rogue wave solutions via the normal Darboux transformation. Last but not least, we construct a generalized Darboux transformation for DS-II equation by using the limit process. As applications, we use the generalized Darboux transformation to derive the second-order rogue waves. In addition, an alternative way is applied to derive the N-fold Darboux transformation for the nonlinear Schrödinger (NLS) equation. One advantage of this method is that the proof for N-fold Darboux transformation is very straightforward.

A Research Report submitted to the Faculty of Humanities, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements of the degree of Master of Arts, Applied Ethics for Professionals Johannesburg, 2016 / Balance, in the broadest sense, can be taken to be a desirable intermediary point or state between two or more opposing points or states. This thesis demonstrates and argues that the idea of balance is ubiquitous — it appears in nature, the sciences, religions and common sense beliefs. Furthermore, it goes beyond this by attempting to extract this idea from prominent ethical theories — both those in the West and East, namely, Kantian Ethics, Aristotelian Virtue Ethics, Utilitarianism, Islam, Taoism, Hinduism, Confucianism and Buddhism. In the sciences and natural world, molecules, objects or plants are simply forced towards the trajectory of balance. This descriptively gives us some forms of balance. Ethically, many theories, as I show in the essay, strive towards balance and thus normatively prescribe balance as the index of choice, conduct and action. I draw some conclusion from this, which is that a balanced conduct and action lead to ethical action, whilst an imbalanced conduct and action result in an unethical action. Objections to this notion are discussed and addressed in various stages of the essay. At the end of the essay, I apply this reasoning to issues in applied ethics, namely, terrorism, wealth inequality and the environment

A fluid layer of binary alloy is cooled from above with solidification occurring at the lower boundary. Some latent heat and light material is released at the freezing boundary. We assume, due to a small cooling rate and a large thermal diffusivity, that the net effect of thermal buoyancy is insignificant and convection is mainly driven by compositional buoyancy associated with the release of light material. The freezing interface advances upward at a slow speed as a result of solidified binary alloy. A stability problem is formulated for the eigenvalue R as a function of Q and S, where R is a ratio of the release rate of light material at the lower boundary to that diffused by pressure gradient, Q is associated with light material diffused by pressure gradient and S is a ratio of the specific volume change upon solidification to that due to compositional change. Before the onset of convective instability, material is diffused by the pressure and compositional gradients. Convective instability is possible provided R > 1. For infinite Schmidt number P(,L), instability sets in stationarily at the marginal state and the mode having the smallest minimum eigenvalue becomes dominant. Three different modes of instabilities, depending on Q and S, are shown: cellular convective modes of both long and short wavelength and morphological mode of short wavelength. Morphological instabilities, associated with the unstable growth of the freezing interface, occur when the conducting layer near the freezing interface is constitutionally supercooled. The results indicate that cellular convective modes require R 1 + S. Nonlinear analysis shows that disturbances just past the marginal state behave like (R-R(,c))(' 1/2), where R(,c) is the critical eigenvalue. Subcritical instabilities are possible for cellular convective modes of long wavelength other than rolls. Taking into / account the effect of the curved interface, the surface tension tends to suppress the unstable growth of the freezing interface. For fixed values of Q and S, morphological modes with surface tension have larger minimum eigenvalues than those without surface tension. / Source: Dissertation Abstracts International, Volume: 43-04, Section: B, page: 1170. / Thesis (Ph.D.)--The Florida State University, 1982.