Distance sampling is one of the most commonly used methods for estimating density and abundance. Conventional methods are based on the distances of detected animals from the center of point transects or the center line of line transects. These distances are used to model a detection function: the probability of detecting an animal, given its distance from the line or point. The probability of detecting an animal in the covered area is given by the mean value of the detection function with respect to the available distances to be detected. Given this probability, a Horvitz-Thompson- like estimator of abundance for the covered area follows, hence using a model-based framework. Inferences for the wider survey region are justified using the survey design. Conventional distance sampling methods are based on a set of assumptions. In this thesis I present results that extend distance sampling on two fronts. Firstly, estimators are derived for situations in which there is measurement error in the distances. These estimators use information about the measurement error in two ways: (1) a biased estimator based on the contaminated distances is multiplied by an appropriate correction factor, which is a function of the errors (PDF approach), and (2) cast into a likelihood framework that allows parameter estimation in the presence of measurement error (likelihood approach). Secondly, methods are developed that relax the conventional assumption that the distribution of animals is independent of distance from the lines or points (usually guaranteed by appropriate survey design). In particular, the new methods deal with the case where animal density gradients are caused by the use of non-random sampler allocation, for example transects placed along linear features such as roads or streams. This is dealt with separately for line and point transects, and at a later stage an approach for combining the two is presented. A considerable number of simulations and example analysis illustrate the performance of the proposed methods.

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function <strong>n</strong> ε W^1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state <strong>n</strong> to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV² (Ω,S<sup>2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.

Almost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation. The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calder&oacute;n. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L_&infin;(-1; 0; L^{3, &beta;}(B(1) &xcap; &Ropf;³ ₊)) with 3 &le; &beta; &lt; &infin;. What enables us to build upon the work of Escauriaza, Seregin and &Scaron;ver&aacute;k [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new &epsiv;-regularity criterion. Third, we show that if a weak Leray-Hopf solution in &Ropf;³ ₊&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim_t&uarr;T&verbar;&verbar;v(&middot;, t)&verbar;&verbar;_{L^3,&beta;(&Ropf;³ ₊)} = &infin; with 3 &lt; &beta; &lt; &infin;. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and &Scaron;ver&aacute;k [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in &Ropf;³, with solenoidal initial data in the critical Besov space ?^-1/4_4,&infin;(&Ropf;³), which has certain continuity properties with respect to weak&ast; convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in &Ropf;³&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim_t&uarr;T &verbar;&verbar;v(&middot;, t)&verbar;&verbar;_{L₃(&Ropf;³)} = &infin;. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

A comprehensive study of the High-pass/Low-pass topology has been performed, increasing the understanding of error sources arising from bit layout issues and fabrication tolerances. This included a detailed analysis of error sources in monolithic microwave phase shifters due to device size limitations, inductor parasitics, loading effects, and non-ideal switches. Each component utilized in the implementation of a monolithic high-low pass phase shifter was analyzed, with its influence on phase behavior shown in detail. An emphasis was placed on the net impact on absolute phase variation, which is critical to the system performance of a phased array radar system. The design of the individual phase shifter filter sections, and the influence of bit ordering on overall performance was also addressed. A variety of X-band four- and five-bit phase shifters were fabricated in a 200 GHz SiGe HBT BiCMOS technology platform, and further served to validate the analysis and design methodology. The SiGe phase shifter can be successfully incorporated into a single-chip T/R module forming a system-on-a-chip (SoC). Reduction in the physical size of transmission lines was shown to be a possibility with spinel magnetic nanoparticle films. The signal transmission properties of phase lines treated with nanoparticle thin films were examined, showing the potential for significant size reduction in both delay line and High-pass/Low-pass phase topologies. Wide-band, low-loss, and near-hermetic packaging techniques for RF MEMS devices were presented. A thermal compression bonding technique compatible with standard IC fabrication techniques was shown, that uses a low temperature thermal compression bonding method that avoids plastic deformations of the MEMS membrane. Ultimately, a system-on-a-package (SoP) approach was demonstrated that utilized packaged RF MEMS switches to maintain the performance of the SiGe phase shifter with much lower loss. The extremely competitive performance of the MEMS-based High-pass/Low-pass phase shifter, despite the lack of the extensive toolkits and commercial fabrication facilities employed with the active-based SiGe phase shifters, confirms both the effectiveness of the detailed phase error analysis presented in this work and the robust nature of the High-pass/Low-pass topology.

Image classification is a fundamental information extraction procedure in remote sensing that is used in land-cover and land-use mapping. Despite being considered as a replacement for manual mapping, it still requires some degree of analyst intervention. This makes the process of image classification time consuming, subjective, and error prone. For example, in unsupervised classification, pixels are automatically grouped into classes, but the user has to manually label the classes as one land-cover type or another. As a general rule, the larger the number of classes, the more difficult it is to assign meaningful class labels. A fully automated post-classification procedure for class labeling was developed in an attempt to alleviate this problem. It labels spectral classes by matching their spectral characteristics with reference spectra. A Landsat TM image of an agricultural area was used for performance assessment. The algorithm was used to label a 20- and 100-class image generated by the ISODATA classifier. The 20-class image was used to compare the technique with the traditional manual labeling of classes, and the 100-class image was used to compare it with the Spectral Angle Mapper and Maximum Likelihood classifiers. The proposed technique produced a map that had an overall accuracy of 51%, outperforming the manual labeling (40% to 45% accuracy, depending on the analyst performing the labeling) and the Spectral Angle Mapper classifier (39%), but underperformed compared to the Maximum Likelihood technique (53% to 63%). The newly developed class-labeling algorithm provided better results for alfalfa, beans, corn, grass and sugar beet, whereas canola, corn, fallow, flax, potato, and wheat were identified with similar or lower accuracy, depending on the classifier it was compared with. / vii, 93 leaves : ill., maps (some col.) ; 29 cm

Estimates of the value of manufacturing sector output enter into many economic indices, especially those measuring productivity. The South African Central Statistical Services has twice made substantial errors in the output series. Revisions to correct the first of these raised the growth rate in manufacturing over the period 1970-80 from 2,6 per cent per annum (compound) to 5 per cent. This episode is not common knowledge. After examining the conceptual difficulties involved in producing output stimates, a practical technique for detecting errors in the series , the Euler Consistency Test, is presented. Developed, refined, and then applied to the South African data, it predicted, retrospectively, the first set of errors (using only the information available at the time those errors were made), then detected another set of errors , not previously known to exist. The study records the process by which the CSS was made to concede this second error. Acknowledgement only came after protracted correspondence and an examination conducted by a special committee formed to investigate my complaints. With 1979 set equal to 100, the output level in 1988 was originally given as 113,8. After investigation, the CSS raised this to 126,1. The magnitude of this second error is equivalent to the omission of the total output of the two SASOL plants commissioned during the early 1980s. Estimates of productivity growth by the National Productivity Institute using these incorrect figures are shown to have created a misleading picture of the sector's performance, especially in the sensitive debate over the relationship between wage and productivity growth. An attempt is made to lay the groundwork of an analytical framework for comprehending (from a Marxist point of view) the activities of ideological state apparatusses like the NPI. A review of the literature on theory choice is conducted, and the necessarily political nature of this activity is explored. The relative impotence of I science' in the face of ideology in a conflict-ridden society is considered. The question of the significance of disagreements between economists is examined, and prospects for convergence and consensus on certain issues are weighed. / Thesis (Ph.D.)-Unversity of Natal, 1994.

In fisheries, advice for the management of fish populations is based upon management quantities that are estimated by stock assessment models. Fisheries stock assessment is a process in which data collected from a fish population are used to generate a model which enables the effects of fishing on a stock to be quantified. This study determined the effects of various data sources, assumptions, error scenarios and sample sizes on the accuracy with which the age-structured production model and the Schaefer model (assessment models) were able to estimate key management quantities for a fish resource similar to the Cape hakes (Merluccius capensis and M. paradoxus). An age-structured production model was used as the operating model to simulate hypothetical fish resource population dynamics for which management quantities could be determined by the assessment models. Different stocks were simulated with various harvest rate histories. These harvest rates produced Downhill trip data, where harvest rates increase over time until the resource is close to collapse, and Good contrast data, where the harvest rate increases over time until the resource is at less than half of it’s exploitable biomass, and then it decreases allowing the resource to rebuild. The accuracy of the assessment models were determined when data were drawn from the operating model with various combinations of error. The age-structured production model was more accurate at estimating maximum sustainable yield, maximum sustainable yield level and the maximum sustainable yield ratio. The Schaefer model gave more accurate estimates of Depletion and Total Allowable Catch. While the assessment models were able to estimate management quantities using Downhill trip data, the estimates improved significantly when the models were tuned with Good contrast data. When autocorrelation in the spawner-recruit curve was not accounted for by the deterministic assessment model, inaccuracy in parameter estimates were high. The assessment model management quantities were not greatly affected by multinomial ageing error in the catch-at-age matrices at a sample size of 5000 otoliths. Assessment model estimates were closer to their true values when log-normal error were assumed in the catch-at-age matrix, even when the true underlying error were multinomial. However, the multinomial had smaller coefficients of variation at all sample sizes, between 1000 and 10000, of otoliths aged. It was recommended that the assessment model is chosen based on the management quantity of interest. When the underlying error is multinomial, the weighted log-normal likelihood function should be used in the catch-at-age matrix to obtain accurate parameter estimates. However, the multinomial likelihood should be used to minimise the coefficient of variation. Investigation into correcting for autocorrelation in the stock-recruitment relationship should be carried out, as it had a large effect on the accuracy of management quantities.

CAPES, Fundação Casimiro Montenegro Filho / No tratamento do movimento rotacional tridimensional de corpos rígidos é inevitável lidar-se com o fato de que rotações não são quantidades vetoriais. Elas podem, no entanto, ser tratadas como tais quando o ângulo de rotação é (muito) pequeno. Neste contexto, ou seja, o da análise infinitesimal, as derivadas temporais das variáveis de rotação mantêm um relacionamento simples (às vezes mesmo do tipo vetorial) com os componentes do vetor velocidade angular. Convencionalmente, esta distinta característica não pode ser associada a rotações grandes, nem mesmo medianas. Nesta tese é demonstrado que a relação diferencial entre o vetor rotação e o vetor velocidade angular pode, na realidade, ser expressa em termos de uma simples derivada temporal, desde que o ângulo de rotação seja mantido numa faixa moderada. O artifício permitindo tal simplicidade na equação cinemática (cinemática linear de atitude) com um ângulo de rotação moderado é a escolha criteriosa da base a partir da qual a derivada temporal é observada. Este resultado é utilizado vantajosamente em conjunto com uma versão generalizada das equações de movimento de Euler na construção de uma lei de controle simples. Essa lei realiza, concomitantemente, o rastreamento linear nominal de atitude e o rastreamento linear nominal de velocidade angular (rastreamento linear nominal de estado rotacional), dentro de uma faixa moderada de erro de rastreamento de atitude. O trabalho analítico apresentado é único no sentido em que este combina cinemática rotacional, dinâmica rotacional e controle de forma tal que linearidade nominal entre as variáveis de erro de estado é atingida mesmo para erros moderados de rastreamento de atitude. Pela primeira vez, uma lei de controle permite explicitamente que a dinâmica de erro de estado rotacional em malha fechada seja escolhida e motivada por conceitos físicos úteis da teoria linear de controle. O texto também inclui simulações numéricas que validam e ilustram os resultados teóricos obtidos. / In dealing with rigid body three-dimensional rotational motion, one is inevitably led to face the fact that rotations are not vector quantities. They may, however, be treated as such when the angle of rotation is (very) small. In this context, i.e. the infinitesimal case analysis, the time derivatives of the rotation variables hold simple (sometimes vector-like) relationships to the components of the angular velocity vector. Conventionally, this distinctive characteristic cannot be associated with general moderate-to-large rotations. In this thesis, it is demonstrated that the kinematical differential relationship between the rotation vector and the angular velocity vector may, in fact, be expressed in terms of a mere time derivative, provided that the angle of rotation is kept within moderate bounds. The key to achieve such simplicity in the kinematical equation (linear attitude kinematics) within moderate angles of rotation is a judicious choice of the basis from which the time derivative is observed. This result is used to advantage within a generalised version of Euler’s motion equations to construct a simple control law, which nominally realises both linear attitude tracking and linear angular velocity tracking (nominal linear attitude state tracking), within moderate attitude tracking errors. The analytical work presented here is unique in the sense that it combines attitude kinematics, dynamics and control in such a way that nominal linearity between the attitude state error variables is achieved within moderate attitude tracking errors. For the first time, an attitude control law explicitly enables the nominal closed-loop attitude state error dynamics to be chosen and motivated by useful physical concepts from linear control theory. The text also includes numerical simulations that validate and illustrate the theoretically achieved results.

CAPES, Fundação Casimiro Montenegro Filho / No tratamento do movimento rotacional tridimensional de corpos rígidos é inevitável lidar-se com o fato de que rotações não são quantidades vetoriais. Elas podem, no entanto, ser tratadas como tais quando o ângulo de rotação é (muito) pequeno. Neste contexto, ou seja, o da análise infinitesimal, as derivadas temporais das variáveis de rotação mantêm um relacionamento simples (às vezes mesmo do tipo vetorial) com os componentes do vetor velocidade angular. Convencionalmente, esta distinta característica não pode ser associada a rotações grandes, nem mesmo medianas. Nesta tese é demonstrado que a relação diferencial entre o vetor rotação e o vetor velocidade angular pode, na realidade, ser expressa em termos de uma simples derivada temporal, desde que o ângulo de rotação seja mantido numa faixa moderada. O artifício permitindo tal simplicidade na equação cinemática (cinemática linear de atitude) com um ângulo de rotação moderado é a escolha criteriosa da base a partir da qual a derivada temporal é observada. Este resultado é utilizado vantajosamente em conjunto com uma versão generalizada das equações de movimento de Euler na construção de uma lei de controle simples. Essa lei realiza, concomitantemente, o rastreamento linear nominal de atitude e o rastreamento linear nominal de velocidade angular (rastreamento linear nominal de estado rotacional), dentro de uma faixa moderada de erro de rastreamento de atitude. O trabalho analítico apresentado é único no sentido em que este combina cinemática rotacional, dinâmica rotacional e controle de forma tal que linearidade nominal entre as variáveis de erro de estado é atingida mesmo para erros moderados de rastreamento de atitude. Pela primeira vez, uma lei de controle permite explicitamente que a dinâmica de erro de estado rotacional em malha fechada seja escolhida e motivada por conceitos físicos úteis da teoria linear de controle. O texto também inclui simulações numéricas que validam e ilustram os resultados teóricos obtidos. / In dealing with rigid body three-dimensional rotational motion, one is inevitably led to face the fact that rotations are not vector quantities. They may, however, be treated as such when the angle of rotation is (very) small. In this context, i.e. the infinitesimal case analysis, the time derivatives of the rotation variables hold simple (sometimes vector-like) relationships to the components of the angular velocity vector. Conventionally, this distinctive characteristic cannot be associated with general moderate-to-large rotations. In this thesis, it is demonstrated that the kinematical differential relationship between the rotation vector and the angular velocity vector may, in fact, be expressed in terms of a mere time derivative, provided that the angle of rotation is kept within moderate bounds. The key to achieve such simplicity in the kinematical equation (linear attitude kinematics) within moderate angles of rotation is a judicious choice of the basis from which the time derivative is observed. This result is used to advantage within a generalised version of Euler’s motion equations to construct a simple control law, which nominally realises both linear attitude tracking and linear angular velocity tracking (nominal linear attitude state tracking), within moderate attitude tracking errors. The analytical work presented here is unique in the sense that it combines attitude kinematics, dynamics and control in such a way that nominal linearity between the attitude state error variables is achieved within moderate attitude tracking errors. For the first time, an attitude control law explicitly enables the nominal closed-loop attitude state error dynamics to be chosen and motivated by useful physical concepts from linear control theory. The text also includes numerical simulations that validate and illustrate the theoretically achieved results.