The current study identifies the reasons why change in traditional perceptions of learning is needed by tracing the philosophies of traditional methods and their effect on the perception of learning, and proposes an alternative geodesic approach, the Mind-Mapping Approach (MMA). Inherent in the model is the implication that traditional methods do not facilitate effective holistic thinking and as a result, do not produce effective lifelong learners. Extrapolated from the MMA model and its assumptions, is a practical framework, the MMA, that, when implemented within learning environments, will foster geodesic thinking which is in natural compliance with the functioning of the brain and therefore to be preferred. The study tested the validity of the above assumption by providing training in an alternative geodesic approach, the MMA, to a group of 45 teachers and therapists that work with learning disabled pupils - this comprised the first experiment of this study. The teachers and therapists in turn used the MMA methods with their pupils (639) - this comprised the second experiment of the study. In this way both the MMA as a geodesic framework, and the actual effectiveness of the MMA training programme in conveying geodesic principles, were evaluated. The results indicated that although significant benefit was derived by the teachers and therapists from the MMA training, these were conservative. Furthermore, the overall longitudinal trends of the pupils' results also indicated that a significant positive change was experienced by the pupils with the introduction of the MMA methods, but not to the extent predicted. It is speculated that this conservative, although significantly positive improvement in the teachers, therapists and pupils is attributed to the fact that the MMA methods are geodesic facilitating improved thinking, problem-solving and research skills, and innovative learning. However traditional methods of teacher training, testing and evaluating do not facilitate these skills as their emphasis is on the accrual as opposed to creation of facts. It is possible therefore that the conservatively positive results of this study reflect the "carry-over" effect of geodesic training. It can be said that the partial application of the MMA methods by the teachers and therapists did improve the performance of the pupils and that this study was therefore successful, but that the results would have been more positive had the pupils been evaluated in a way that matched the geodesic training. In addition, the study provides valuable information regarding the effect of geodesic systems on traditional systems of learning. / Thesis (DPhil)--University of Pretoria, 2010. / Speech-Language Pathology and Audiology / unrestricted

A transformação de coordenadas entre Sistemas e/ou Redes Geodésicas é um procedimento amplamente utilizado por usuários de informações associadas a coordenadas. A posição de um móvel, feição ou corpo pode ser estabelecida em diferentes Redes, desde que se conheça os parâmetros que as interrelacionam. Do ponto de vista conceitual pode ser entendido como um problema trivial, mas na prática, sua solução é complexa. Isso decorre, basicamente, das diferentes maneiras que podem ser usadas para definir um Sistema, bem como das deformações existentes nas suas materializações, originadas pelas várias influências sistemáticas pertinentes à estimação das coordenadas. Uma metodologia para a transformação entre as duas materializações do Sistema Geodésico Brasileiro (SGB), associadas ao Sistema Geodésico SAD 69, é proposta nesta tese. A primeira, que vigorava desde o primeiro ajustamento, realizado na década de 70; e a segunda, referente ao primeiro reajustamento, terminado no ano de 1996. Doravante, para efeito somente desta pesquisa, as realizações serão denominadas SAD 69 e SAD 69/96, respectivamente. A metodologia proposta tem, como condicionante, a preservação da integridade daquelas duas realizações do SGB, de modo a garantir suas precisões e exatidões. Fundamenta-se, basicamente, em 3 etapas: 1) regionalização da Rede Geodésica Brasileira(RGB), produzida por uma Triangulação de Delaunay 3-D; 2) estimação, para cada região, dos parâmetros resultantes da aplicação de uma transformação geométrica 3-D entre os dois conjuntos de coordenadas; 3) aplicação de correções às coordenadas calculadas pela transformação geométrica Afim Geral 3-D. Adicionalmente, realizou-se um estudo sobre Sistemas e Redes de Coordenadas, enfatizando aquelas de natureza Geodésica, bem como sobre o SGB. Foi realizada ainda uma investigação sobre o reajustamento da RGB, quando foram abordados aspectos relativos à metodologia empregada e à sua exatidão. Finalmente, são apresentados os resultados dos testes executados para validação da metodologia, comprovando a viabilidade de sua aplicação. / The coordinate transformation between Reference Systems and/or Reference Frames is a procedure extensively used by users of information associate to coordinates. The position of any stationary or moving object can be established in different frames since their interrelation parameters are well known. Considering the conceptual point of view, it can be understood as a trivial problem, but in the practice, its solution is very complex. This reality results from the different possibilities in which one system can be defined, as well as of the existing deformations in the frames, originated from several influences related to the estimation of the coordinates. A methodology for transforming the coordinates between the two materializations of Brazilian Geodetic System (BGS), associated with the Geodetic System SAD 69 is proposed. The first materialization refers to the first adjustment, carried out in the 70's. The second one refers to the first readjustment, finished at the end of 1996. From now on, only for the purpose of this research, these materializations will be named SAD 69 and SAD 69/96, respectively. The proposed methodology preserves the integrity of the BGS realizations and assures their precisions and accuracies. Its foundation has three main stages: 1) partitioning of the Brazilian Geodetic Network, according to a Delaunay 3-D Triangulation; 2) estimating, for each region, parameters derived from the application of a geometric transformation between the two sets of coordinates; 3) application of corrections to the coordinates evaluated by the Affine Geometric Transformation. Additionally, a study on Geodetic Coordinate Systems and Frames and on the BGS has been carried out. An investigation on the readjustment of BGS has been achieved, where aspects refering to the used methodology and its accuracy have been taken into account.Finally, the results of the tests carried out for validation of the methodology are presented,emphasizing the reliability of its application.

A transformação de coordenadas entre Sistemas e/ou Redes Geodésicas é um procedimento amplamente utilizado por usuários de informações associadas a coordenadas. A posição de um móvel, feição ou corpo pode ser estabelecida em diferentes Redes, desde que se conheça os parâmetros que as interrelacionam. Do ponto de vista conceitual pode ser entendido como um problema trivial, mas na prática, sua solução é complexa. Isso decorre, basicamente, das diferentes maneiras que podem ser usadas para definir um Sistema, bem como das deformações existentes nas suas materializações, originadas pelas várias influências sistemáticas pertinentes à estimação das coordenadas. Uma metodologia para a transformação entre as duas materializações do Sistema Geodésico Brasileiro (SGB), associadas ao Sistema Geodésico SAD 69, é proposta nesta tese. A primeira, que vigorava desde o primeiro ajustamento, realizado na década de 70; e a segunda, referente ao primeiro reajustamento, terminado no ano de 1996. Doravante, para efeito somente desta pesquisa, as realizações serão denominadas SAD 69 e SAD 69/96, respectivamente. A metodologia proposta tem, como condicionante, a preservação da integridade daquelas duas realizações do SGB, de modo a garantir suas precisões e exatidões. Fundamenta-se, basicamente, em 3 etapas: 1) regionalização da Rede Geodésica Brasileira(RGB), produzida por uma Triangulação de Delaunay 3-D; 2) estimação, para cada região, dos parâmetros resultantes da aplicação de uma transformação geométrica 3-D entre os dois conjuntos de coordenadas; 3) aplicação de correções às coordenadas calculadas pela transformação geométrica Afim Geral 3-D. Adicionalmente, realizou-se um estudo sobre Sistemas e Redes de Coordenadas, enfatizando aquelas de natureza Geodésica, bem como sobre o SGB. Foi realizada ainda uma investigação sobre o reajustamento da RGB, quando foram abordados aspectos relativos à metodologia empregada e à sua exatidão. Finalmente, são apresentados os resultados dos testes executados para validação da metodologia, comprovando a viabilidade de sua aplicação. / The coordinate transformation between Reference Systems and/or Reference Frames is a procedure extensively used by users of information associate to coordinates. The position of any stationary or moving object can be established in different frames since their interrelation parameters are well known. Considering the conceptual point of view, it can be understood as a trivial problem, but in the practice, its solution is very complex. This reality results from the different possibilities in which one system can be defined, as well as of the existing deformations in the frames, originated from several influences related to the estimation of the coordinates. A methodology for transforming the coordinates between the two materializations of Brazilian Geodetic System (BGS), associated with the Geodetic System SAD 69 is proposed. The first materialization refers to the first adjustment, carried out in the 70's. The second one refers to the first readjustment, finished at the end of 1996. From now on, only for the purpose of this research, these materializations will be named SAD 69 and SAD 69/96, respectively. The proposed methodology preserves the integrity of the BGS realizations and assures their precisions and accuracies. Its foundation has three main stages: 1) partitioning of the Brazilian Geodetic Network, according to a Delaunay 3-D Triangulation; 2) estimating, for each region, parameters derived from the application of a geometric transformation between the two sets of coordinates; 3) application of corrections to the coordinates evaluated by the Affine Geometric Transformation. Additionally, a study on Geodetic Coordinate Systems and Frames and on the BGS has been carried out. An investigation on the readjustment of BGS has been achieved, where aspects refering to the used methodology and its accuracy have been taken into account.Finally, the results of the tests carried out for validation of the methodology are presented,emphasizing the reliability of its application.

How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.

Le résultat principal de cette thèse est de donner une définition de géodésique sur les supervariétés riemanniennes $(\ca,g)$ paires (et aussi impaires) et de la justifier par un théorème reliant les courbes géodésiques avec le flot géodésique sur $\text{T}^*\ca$. Pour ce faire, nous construisons la 2-forme symplectique canonique sur $\text{T}^*\ca$ et l'analogue $H$ de la fonctionnelle énergie dans le contexte des supervariétés. Nous prenons ainsi le flot du champ de vecteurs hamiltonien associé à $H$ que nous nommons "flot géodésique''. Alors, nous relions les supergéodésiques, que nous définissons à l'aide de la dérivée covariante comme des courbes à vitesse auto-parallèle, avec le flot géodésique via des conditions initiales adaptées aux supervariétés. Une autre définition de géodésique a été proposée en 2006 par O. Goertsches mais ces courbes ne sont pas en bijection avec les courbes intégrales du flot géodésique que nous construisons. Notre définition de géodésique semble donc présenter plus d'avantages. Par ailleurs, nous pouvons, à l'aide du flot, construire l'application exponentielle. Nous en profitons pour démontrer le résultat, bien connu au cas de cadre des variétés classiques (non-graduées), de linéarisation des isométries en utilisant l'exponentielle. Dans la dernière partie, nous redémontrons un résultat de J. Monterde et O.M. Sánchez-Valenzuela concernant l'intégration des champs de vecteur pairs, impairs et aussi non homogènes dans le but d'éviter d'utiliser un modèle de Batchelor. Ceci permet par exemple, de généraliser leurs résultats aux supervariétés holomorphes. / We give a natural definition of geodesics on a Riemannian supermanifold $(\ca, g)$ and extend the usual geodesic flow on $T^*M$ associated to the underlying Riemannian manifold $(M,g)$ to a geodesic "superflow" on $T^*\ca$. Integral curves of this flow turn out to be in natural bijection with geodesics on $\ca$. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds. We give also a new proof of the Monderde et al. result about flows of non-homogeneous supervector fields. We give a treatment which allows extensions for instance to the holomorphic category. The original proof given by Monderde et al. is only applicable to split supermanifolds, since their proofs relied on Batchelor's Theorem. Finally, we reproves a characterization of vector fields whose flows are local $\sbb$-actions of an appropriate Lie supergroups structure

A review of geodesics and Busemann G-spaces is given. Aleksandrov curvature and the disjoint (0, n)-cells property are defined. We show how these properties are applied to and strengthened in Busemann G-spaces. We examine the relationship between manifolds and Busemann G-spaces and prove that all Riemannian manifolds are Busemann G-spaces, though not all metric manifolds are Busemann G-spaces. We show how Busemann G-spaces that also have bounded Aleksandrov curvature admit local closest-point projections to geodesic segments. Finally, we expound local properties of Busemann G-spaces and define a new property which we call the symmetric property. We show that Busemann G-spaces which have the disjoint (0,n)-cells property for every value of n cannot have the symmetric property.

We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between ``events''. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called ``geodesically complete'', or simply, ``complete''. It is easy to show that the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.

Nesse trabalho apresentaremos alguns exemplos clássicos que evidenciam as diferenças entre a geometria Riemanniana e a semi-Riemanniana (Lorentziana) quanto à completude de geodésicas. Para isso, revisitaremos conceitos básicos de Geometria, seguido de uma introdução aos espaços vetoriais de Lorentz e um estudo inicial sobre o grupo de Lorentz. Nos capítulos finais discutiremos sobre completude de geodésicas e como se distanciam do caso Riemanniano. / In this work we intend to present some classical examples that display the differences between Riemannian and semi-Riemannian (Lorentzian) geometry in relation to the completeness of geodesics. For this, we will revisit basic Geometry concepts followed by an introduction to the vector spaces of Lorentz and a simple study on the Lorentz group. In the final chapters we will discuss about the completeness of geodesics and how it distances itself from the Riemannian case.

The study of geodesics is of intrinsic significance in the study of the geometry of space-time. In this thesis null, space-like and time-like geodesics are studied in the case of the space-times of Schwarzschild, Reissner-Nordstrouml;m and Kerr black holes. These space-times have been investigated with varying degrees of thoroughness in many articles and some books. However, there are some significant gaps in these treatments and the central aim of this thesis is to fill these gaps where necessary. Moreover, the following topics are covered for the first time. 1. In Chapter 4 a thorough treatment of the space-like geodesics of the Schwarzschild solutions has been given. These geodesics are the trajectories of Tachyons (faster than light particles) and are treated in a complete manner. This has been done by obtaining exact solutions and solving them numerically. 2. In Part II all solutions for geodesics for a Reissner-Nordstrouml;m black hole have been given in complete detail, i.e. time-like, null and space-like geodesics and orbit of a charged particle. 3. In Chapter 14 all solutions for geodesics in the equatorial plane of a Kerr black hole have been given in complete detail, i.e. time-like, null and space-like geodesics. 4. The study of special types of non-equatorial geodesics for a Kerr black hole have been given in complete detail, i.e. time-like (Chapter 17), null (Chapter 15) and space-like (Chapter 16). This has been done in order to distinguish the qualitatively different types of solutions. Calculation of the explicit formulas, which describe these geodesics, as well as numerically computed diagrams representing the geodesics have been incorporated in these studies. The following subjects have been also treated: 5. Solutions for the geodesics in Reissner-Nordstrouml;m black holes with |Q_*| gt;= M, which are black holes with one (|Q_*| = M) or no horizon (|Q_*|gt; M) (Chapter 8). 6. Solutions of geodesics in extreme and fast Kerr black holes, i.e. black holes with a = M (extreme) and a gt; M (fast). As in the case of |Q_*| gt; M, fast black holes have naked singularities (Chapter 14). 7. Some general observations about orbit types of the Kerr black holes regarding relationships between parameters such as angular momentum, energy, Carter constant and mass and angular momentum of black holes (Chapter 13). 8. Some corrections to errors found in the literature. While it has not been possible to cover all different cases which occur for possible relations amongst the parameters specifying a general black hole, interesting geodesics have, however, been studied and a more thorough presentation of the properties of geodesics has now been given.

A roof over a manure reservoir eliminates the entry of precipitations, reduces odour and volatile gas emissions, lowers the storage and handling costs and alleviates environmental impacts. / Structural analysis and testing were carried out to establish the feasibility of geodesic domes as manure reservoir roofting. A stress analysis, conducted by computer was used to determine maximum compression loads on roof members. Compression tests were carried out in the laboratory to test two laminated designs and to evaluate the performance of a member joint. / Structural analysis showed that the Lamella type dome, with a diameter/height ratio of 4, offers the best geometry considering design criteria for manure roofs. Compression tests confirmed that laminated members, made of 38 x 89 mm and 38 x 140 mm pieces, can resist the design stress developed in a 22 meter diameter dome. / Loading tests revealed that non laminated wood members failed at 80% of the design load. However, the joints (formed by two 3,1 mm thick steel plates bolted to the members) were still able to resist an increase in loading. / Data obtained from the tests were used for a preliminary design of a geodesic roof taking into account specific conditions such as humidity and manure gas. However further study is required to adapt this concept to real conditions over a manure reservoir. / The cost of a 22 meter diameter dome, for covering manure reservoirs, is estimated at 79$/m$ sp2$ which is more expensive than the cost of the currently used, least expensive manure pit roof (wood truss design).