<p>A possible mathematical formulation of the practical problem of computer - aided design for any system or any engineering design subject to tolerances on the n independent parameters is used to solve some special cases under certain restrictions. The sequential unconstrained minimization technique of Fiacco - McCormick is used to get the optimum solution. The scheme used is : starting from arbitrary initial acceptable or unacceptable designs and culminating in designs which under reasonable restrictions are acceptable.</p> / Master of Science (MSc)

<p>Goodness-of-fit testing of simple hypotheses has a long, well known history. When the Null Hypothesis specifies only the form of the Null Cumulative Distribution Function (CDF), with values for one or more parameters unspecified, the problem is not so clear cut. This project examines several methods of testing fit in the presence of unknown parameters. The methods, briefly described below, are all based on the Empirical Distribution Function (EDF).</p> <p>(1) The unknown parameters are estimated from the sample. Modified EDF statistics using these sample estimates, are computed, and compared to the significance points which have been obtained by computer simulation.</p> <p>(2) The unknown parameters are estimated from the sample. Transformations are applied to the observations to obtain transformed variates which, under the Null Hypothesis, are distributed as dependent uniform variates. These transforms are tested for uniformity by the EDF statistics.</p> <p>(3) A series of transformations is applied to the data to obtain transformed variates which under the Null Hypothesis follow a completely specified Distribution Function; the nuisance parameters have been eliminated. This new, simple hypothesis is then tested by the EDF statistics.</p> <p>(4) For various parameter values throughout the parameter space, the EDF statistics are computed. A region in the parameter space for acceptance of the corresponding simple hypotheses is determined.</p> / Master of Science (MS)

<p>The Hewlett-Packard Cross Assembler is a program available on CDC-6400 computer and is used to assemble HP-2000 series assembly language programs. The binary code for the assembled programs can be punched on cards for execution on HP 2100A computer.</p> <p>This report discribes of the design and working of the Cross-Assembler. A program listing (in PASCAL) and a few sample runs are also included.</p> / Master of Science (MS)

<p> Inverse problems have been studied in great detail and optimization methods using objective functionals such as output least-squares (OLS) and modified output least-squares (MOLS) are well understood. However, the existing literature has only dealt with identifying parameters that appear linearly in systems of partial differential equations. We investigate the changes that occur in the identification process if the parameter appears nonlinearly. We extend the OLS and MOLS functionals to this nonlinear case and give first and second derivative formulas. We further show that the typical convexity of the MOLS functional can not be guaranteed when identifying nonlinear parameters. To numerically verify our findings we employ a C++ based computational framework. Discretization is done via the finite element method, and details are given for the new results of the functionals and their derivatives. Since we consider nonlinear parameters, gradient methods such as adjoint stiffness are not applicable to the OLS functional and we instead show computation methods using the adjoint approach.</p>

For matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death. Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties. We show that this dynamic dichotomy is a general feature of synchrony models which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).

<p> For matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death. </p><p> Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties. </p><p> We show that this dynamic dichotomy is a general feature of <i> synchrony models</i> which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).</p>

<p> This dissertation presents a collection of mathematical models for cellular response to the most common forms of anti-cancer therapy (radiation and chemotherapy) and the role of the cell cycle in this response. These models have application to improving cancer therapy through: optimization of dose and scheduling of agents already in clinical use; better predictive methods for selecting the most promising among new drug candidates or candidate combinations; and combination therapy with newer agents that target the cell cycle or cellular metabolism. The cellular pharmacodynamic models are based on the key concept of ''cellular damage'' that results from exposure to therapy and has distinct kinetics from cell kill. Damage is lethal only if it exceeds a cell's tolerance threshold at one or more checkpoints in the cell cycle (for apoptosis) or at any time point (for necrosis). This is the ''peak damage'' model. Each mechanism of cell kill, each agent in combination therapy, and each cycle in fractionated therapy increases the damage function (the ''additive damage'' model). The overall framework is termed the ''peak additive damage model.'' This model framework is first tested on 128 independent <i>in vitro</i> dose-response data sets for radiation alone. It outperforms previously proposed models, including the widely-used linear-quadratic model. The peak additive damage model is then applied to radiochemotherapy. Its performance is superior to the previously proposed independent cell kill model when tested with 218 data sets for fixed-schedule exposure against. For varying-schedule exposures, cell heterogeneity and cycle asynchrony necessitate a cell-cycle-phase-structured model that divides the population into cohorts with different responses, still determined for each cohort by the peak additive damage model, that are then averaged. This model simultaneously predicts dose-response and cell cycle distribution data, and is tested with such data from the literature. Finally, a first step is taken towards allowing for microenvironmental input into cellular response, by developing a model for the cell cycle that is driven by metabolic inputs and external growth factors. Overall, the models presented here have great flexibility to be extended to complex schedules, any number of agents, and agents with metabolic or cell-cycle targets.</p>

<p> The main subject of this dissertation is smooth incompressible fluids. The emphasis is on the incompressible Euler equations in all of <b>R</b>² or <b>R</b>³, but many of the ideas and results can also be adapted to other hydrodynamic systems, such as the Navier-Stokes or surface quasi-geostrophic (SQG) equations. A second subject is the modeling of moving contact lines and dynamic contact angles in inviscid liquid-vapor-solid systems under surface tension. </p><p> The dissertation is divided into three independent parts: First, we introduce notation and prove useful identities for studying incompressible fluids in a pointwise Lagrangian sense. The main purpose is to provide a unified treatment of results scattered across the literature. Furthermore, we prove several analogs of Constantin&rsquo;s local pressure formula for other nonlocal operators, such as the Biot-Savart law and Leray projection. Also, we define and study properties of a Lagrangian locally compact Abelian group in terms of which some nonlocal formulas encountered in fluid dynamics may be interpreted as convolutions. </p><p> Second, we apply the algebraic theory of scalar polynomial orthogonal invariants to the incompressible Euler equations in two and three dimensions. Using this framework, we give simplified proofs of results of Chae and Vieillefosse. We also investigate other uses of orthogonal transformations, such as diagonalizing the deformation tensor along a particle trajectory, and comment on relative advantages and disadvantages. These techniques are likely to be useful in other orthogonally invariant PDE systems as well. </p><p> Third, we propose an idealized inviscid liquid-vapor-solid model for the macroscopic study of moving contact lines and dynamic contact angles. Previous work mostly addresses viscous systems and frequently ignores a singular stress present when the contact angle is not at its equilibrium value. We also examine and clarify the role that disjoining pressure plays and outline a program for further research.</p>

<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.