This thesis is devoted to creating a systematic way of determining all inequivalent orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero curvature. To achieve this, we represent the problem with Killing tensors and employ the recently developed invariant theory of Killing tensors. Killing tensors on the model spaces of spherical and hyperbolic space enjoy a remarkably simple form; even more striking is the fact that their parameter tensors admit the same symmetries as the Riemann curvature tensor, and thus can be considered algebraic curvature tensors. Using this property to obtain invariants and covariants of Killing tensors, together with the web symmetries of the associated orthogonal coordinate webs, we establish an equivalence criterion for each space. In the case of three-dimensional spherical space, we demonstrate the surprising result that these webs can be distinguished purely by the symmetries of the web. In the case of three-dimensional hyperbolic space, we use a combination of web symmetries, invariants and covariants to achieve an equivalence criterion. To completely solve the equivalence problem in each case, we develop a method for determining the moving frame map for an arbitrary Killing tensor of the space. This is achieved by defining an algebraic Ricci tensor. Solutions to equivalence problems of Killing tensors are particularly useful in the areas of multiseparability and superintegrability. This is evidenced by our analysis of symmetric potentials defined on three-dimensional spherical and hyperbolic space. Using the most general Killing tensor of a symmetry subspace, we derive the most general potential “compatible” with this Killing tensor. As a further example, we introduce the notion of a joint invariant in the vector space of Killing tensors and use them to characterize a well-known superintegrable potential in the plane. xiii

A computational fluid dynamics model has been developed to examine the separation of an oil film from a spherical oil-coated particle falling through quiescent water due to gravity. Using this model, the separation process was studied as a function of the viscosity ratio of oil to water, R, and the ratio of viscous forces to surface tension, represented by the Capillary number Ca. The governing equations of this flow-induced motion are derived in a non-inertial spherical coordinate system, and discretized using a finite volume approach. The Volume-of-Fluid method is used to capture the oil/water interface. The model predicts two mechanisms for oil separation: at R less than 1, the shear difference between the particle/oil interface and the oil/water interface is not significant and Ca determines whether separation occurs or not; at R larger than 1, the shear difference is considerable, and the Ca effect becomes less dominant.

Bakalauro darbą „Įstrižojo astigmatizmo kompiuterinis modeliavimas sferiniuose ir asferiniuose lęšiuose“ sudaro įvadas, 5 dėstymo skyriai, išvados ir priedai. Darbe pateikiami 50 paveikslai ir 2 lentelės, cituojami 16 literatūros šaltinių. Darbo apimtis 66 lapai. Darbe apibendrinta literatūra apie lęšių aberacijas, jų įtaką regėjimo kokybei, bei minimalizavimą asferiniais lęšiais. Taip pat pateikiami matematiniai sferinių ir asferinių lęšių modeliai, bei įstrižojo astigmatizmo analizė šiuose lęšiuose. Įvade iškeliami darbo tikslai ir uždaviniai. Pirmajame skyriuje pateikta teorinė medžiaga apie lęšių aberacijas, antrajame – apie sferinius ir asferinius lęšius. Išanalizuoti galimi asferinių lęšių paviršiai ir jų pranašumas prieš sferinius paviršius. Trečiajame skyriuje pateikiamas matematinis spindulių eigos modelis sferiniame paviršiuje. Ketvirtajame – matematinis spindulių eigos modelis asferiniame paviršiuje. Penktajame skyriuje lyginami kompiuterinių modelių pagalba gauti įstrižojo astigmatizmo apibūdinti rezultatai, tangentinėmis paklaidomis. / The bachelor’s thesis „The computer modeling of oblique astigmatism in spherical and aspherical lenses” consists of introduction, 5 chapters, conclusions and attachments. The thesis introduces 50 figures, 2 tables and refers to 16 literary sources. The volume of the work is 66 pages. The work summarizes the literature which describes the lens aberrations, its influence to quality of vision and reduction by using aspheric lenses. There is also included mathematical models of spherical and aspheric lenses, and analysis of oblique astigmatism in them. The introduction sets the aims and the goals of the study. The first chapter contains theoretic material about lens aberrations, the second – about spherical and aspheric lenses, the analysis about available surfaces of aspheric lenses and advantages of them. The third chapter contains mathematical model of light rays pathway in spherical lenses. The fourth chapter contains mathematical model of light rays pathway in aspheric lenses. The fifth chapter contains comparison of oblique astigmatism described as tangential errors in those two models.

Multiple-Input Multiple-Output (MIMO) systems are a pivotal solution for the significant enhancement of the band-limited wireless channels’ communication capacity. MIMO system is essentially a wireless system with multiple antennas at both the transmitter and receiver ends. Compared to the conventional wireless systems, the main advantages of the MIMO systems are the higher system capacity, more bit rates, more link reliability, and wider coverage area. All of these features are currently considered as crucial performance requirements in wireless communications. Additionally, the emerging new services in wireless applications have created a great motivation to utilize the MIMO systems to fulfil the demands these applications create. The MIMO systems can be combined with other intelligent techniques to achieve these benefits by employing a higher spectral efficiency. The MIMO system design is a multifaceted problem which needs both antenna considerations and baseband signal processing. The performance of the MIMO systems depends on the cross-correlation coefficients between the transmitted/received signals by different antenna elements. Therefore, the Electromagnetic (EM) characteristics of the antenna elements and wireless environment can significantly affect the MIMO system performance. Hence, it is important to include the EM properties of the antenna elements and the physical environment in the MIMO system design and optimizations. In this research, the MIMO system model and system performance are introduced, and the optimum MIMO antenna system is investigated and developed by considering the electromagnetic aspects within three inter-related topics: 1) Fast Numerical Analysis and Optimization of the MIMO Antenna Structures: An efficient and fast optimization method is proposed based on the reciprocity theorem along with the method of moment analysis to minimize the correlation among the received/transmitted signals in MIMO systems. In this method, the effects of the radio package (enclosure) on the MIMO system performance are also included. The proposed optimization method is used in a few practical examples to find the optimal positions and orientations of the antenna elements on the system enclosure in order to minimize the cross-correlation coefficients, leading to an efficient MIMO operation. 2) Analytical Electromagnetic-Theoretic Model for the MIMO Antenna Design: The first requirement for the MIMO antennas is to obtain orthogonal radiation modes in order to achieve uncorrelated signals. Since the Spherical Vector Waves (SVW) form a complete set of orthogonal Eigen-vector functions for the radiated electromagnetic fields, an analytical method based on the SVW approach is developed to excite the orthogonal SVWs to be used as the various orthogonal modes of the MIMO antenna systems. The analytic SVW approach is used to design spherical antennas and to investigate the orthogonality of the radiation modes in the planar antenna structures. 3) Systematic SVW Methodology for the MIMO Antenna Design: Based on the spherical vector waves, a generalized systematic method is proposed for the MIMO antenna design and analysis. The newly developed methodology not only leads to a systematic approach for designing MIMO antennas, but can also be used to determine the fundamental limits and degrees of freedom for designing the optimal antenna elements in terms of the given practical restrictions. The proposed method includes the EM aspects of the antenna elements and the physical environment in the MIMO antenna system, which will provide a general guideline for obtaining the optimal current sources to achieve the orthogonal MIMO modes. The proposed methodology can be employed for any arbitrary physical environment and multi-antenna structures. Without the loss of generality, the SVW approach is employed to design and analyze a few practical examples to show how effective it can be used for MIMO applications. In conclusion, this research addresses the electromagnetic aspects of the antenna analysis, design, and optimization for MIMO applications in a rigorous and systematic manner. Developing such a design and analysis tool significantly contributes to the advancement of high-data-rate wireless communication and to the realistic evaluation of the MIMO antenna system performance by a robust scientifically-based design methodology.

Flow separation is a source of aerodynamic in-efficiency; however by using vortex generators the issue of flow separation can be controlled. This is of particular benefit to flows around bluff bodies which are susceptible to large scale separated flows, such as bodies in ground effect. Previous studies concerning heat transfer applications focused on the ability of low aspect ratio spherical wells (dimples) to produce vortices for flow mixing. Dimpled surfacing on an inverted Tyrrell026 airfoil in ground effect (indicative of high performance automotive aerodynamic applications e.g. Formula One) has been investigated for similar vortex enhanced wake reductions. Experimental measurements using Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV) were taken inside a dimple to provide validation and verification of numerical analyses of dimple flows. The k-ω SST turbulence model showed good agreement to the experimental measurements. Additionally experiments were conducted using LDA and PIV with various configurations of dimple arrays placed from a fixed separation point of a 16˚ rearward facing ramp to determine how the array configuration influenced the large scale separation. The airfoil wake with numerous dimple configurations and placements were measured using LDA. Results showed that an array of dimples with close dimple to dimple spacing there was flow recovery in the airfoil wake from the velocity deficit with no dimples of u/Uo,min=-0.1, to u/Uo,min=0.4 with a dimple array, (at α=10˚, ground clearance h/c=0.313). At α=10˚ reductions in the wake size of 30%, 33%, 58% and 68% were found for the ground clearances of h/c=0.112, 0.134, 0.224 and 0.313 respectively. For numerous dimple array configurations, closely spaced dimple arrays were more effective in reducing the wake size, turbulence intensity and Reynolds stresses than those where dimple spacing was further apart. The chord wise location of the array on the wing affected the angle of incidence of the wing for which the wake was able to be reduced. Arrays placed towards the trailing edge improved wake losses at lower angles of incidence. Dimples placed further forward yielded the most improvement at higher angles of incidence, in part due to the increased venturi effect under the wing.

The acoustic scattering properties of the human head and torso match well with those of simple geometric shapes. Consequently, analytical scattering models can be utilised to account for the sound localisation cues introduced by these features. The traditional use of such models assumes that the head surface is completely rigid in nature. This thesis is concerned with modelling and understanding the effect of terminal scalp hair (i.e., a non-rigid head surface) on the auditory localisation cues. The head is modelled as a sphere, and the acoustical characteristics of hair are modelled using a locally-reactive equivalent impedance parameter. This allows the scattering boundary to be defined on the inner rigid surface of the head. The boundary assumptions are validated experimentally, through impedance measurement at oblique incidence and analysis of the near-field scattering pattern of a uniformly covered sphere. The impedance properties of human hair are also discussed, including trends with variations in sample thickness, bulk density, and fibre diameter. A general solution for the scattering of sound by a sphere with an arbitrarily distributed, locally reactive surface impedance is then presented. From this, an analytical solution is derived for a surface boundary that is evenly divided into two uniformly distributed hemispheres. For this boundary condition, cross-coupling is shown to exist between incoming and scattered wave modes of equi-order when the degrees are non-equal and opposite in parity. The overall effect of impedance on the resultant scattering characteristics is discussed in detail, both for uniform and for hemispherically divided surface boundaries. Finally, the analytical formulation and the impedance characteristics of hair are collectively utilised to investigate the effect of hair on human auditory localisation cues. The hair is shown to produce asymmetric perturbations to both the monaural and binaural cues. These asymmetries may help to resolve localisation confusions between sound stimuli positioned in the front and rear hemi-fields. The cue changes in the azimuth plane are characterised by two predominant features and remain consistent regardless of the decomposition baseline (i.e., the inclusion of a pinna offset, neck, etc). Experimental comparisons using a synthetic hair material show a good agreement with simulated results.

Στην παρούσα εργασία σχεδιάστηκε και εφαρμόστηκε ένας νέος αλγόριθμος χειρισμού των ηλεκτροστατικών αλληλεπιδράσεων. Οι αλληλεπιδράσεις ηλεκτροστατικής φύσεως αποτελούν αντικείμενο εκτενούς μελέτης στον ερευνητικό κλάδο των μοριακών προσομοιώσεων. Ιδιαίτερα σε βιολογικά μοριακά συστήματα, τα οποία αποτελούνται από υδατικά διαλύματα πολυηλεκτρολυτών, οι ηλεκτροστατικές αλληλεπιδράσεις κυριαρχούν. Η ευρέως διαδεδομένη μέθοδος άθροισης κατά Ewald η οποία χαρακτηρίζεται για την ακρίβεια της, είναι ιδιαίτερα απαιτητική σε υπολογιστικό χρόνο δεδομένου ότι κλιμακώνεται ως Ν2, όπου Ν ο συνολικός αριθμός φορτίων του συστήματος. Στην παρούσα εργασία περιγράφεται η προσεγγιστική μέθοδος σφαιρικής αποκοπής για την περίπτωση πολυηλεκτρολυτών. Έως σήμερα, η μέθοδος της σφαιρικής αποκοπής έχει εφαρμοστεί σε μοριακά συστήματα ιοντικών κρυστάλλων. Σύμφωνα με τη μέθοδο αυτή, εισάγοντας μια σφαιρική αποκοπή στον υπολογισμό του ηλεκτροστατικού δυναμικού και παράλληλα επιτυγχάνοντας ηλεκτροουδετερότητα του υπό εξέταση σφαιρικού συστήματος, πραγματοποιείται μείωση του υπολογιστικού χρόνου σε δίχως σημαντικές απώλειες σε ακρίβεια. Στην παρούσα εργασία σχεδιάστηκε ο αλγόριθμος της σφαιρικής αποκοπής για δύο κατηγορίες μορίων: τους πολυηλεκτρολύτες χωρίς ενδομοριακές αλληλεπιδράσεις όπως είναι το μόριο του νερού και τους πολυηλεκτρολύτες με ενδομοριακές αλληλεπιδράσεις όπως είναι τα μόρια CiEj των αλκυλικών αιθέρων πολυ-(οξυ-αιθυλενίου). Επιπρόσθετα, επεκτείναμε τη μέθοδο άθροισης κατά Ewald για την εφαρμογή της σε πολυηλεκτρολύτες, χωρίς και με ενδομοριακές αλληλεπιδράσεις προκειμένου να προχωρήσουμε σε σύγκριση της ακριβής μεθόδου Ewald με τα αποτελέσματα του νέου αλγορίθμου της σφαιρικής αποκοπής. Συγκρίνοντας τα αριθμητικά αποτελέσματα των δύο μεθόδων για τα δύο διαφορετικά είδη πολυηλεκτρολυτών ( H2Ο και CiEj ) από ατομιστικές προσομοιώσεις μοριακής δυναμικής οδηγηθήκαμε σε πλήρη συμφωνία των δύο μεθόδων γεγονός που επιβεβαιώνει την εγκυρότητας της νέας μεθόδου. Επίσης, μέσω των αποτελεσμάτων των προσομοιώσεων μοριακής δυναμικής μελετήθηκαν οι θερμοδυναμικές, δομικές και δυναμικές ιδιότητες των δύο συστημάτων. Η παρούσα εργασία είναι δομημένη ως εξής: Στο πρώτο κεφάλαιο γίνεται μια εισαγωγή στο θεωρητικό υπόβαθρο προσομοιώσεων μοριακής δυναμικής. Στο δεύτερο κεφάλαιο περιγράφονται οι μέθοδοι χειρισμού των ηλεκτροστατικών αλληλεπιδράσεων που συναντούμε έως σήμερα στη βιβλιογραφία. Στη συνέχεια, στο κεφάλαιο 3 περιγράφεται ο αλγόριθμος της σφαιρικής αποκοπής. Στο τέταρτο κεφάλαιο παρουσιάζονται τα αποτελέσματα από τις προσομοιώσεις μοριακής δυναμικής με τις μεθόδους σφαιρικής αποκοπής και άθροισης κατά Ewald. / -

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No. of bitstreams: 1 EgmidioCF_DISSERT.pdf: 19081133 bytes, checksum: e987b9622c60ead5edc3617284f7df4e (MD5) Previous issue date: 2014-04-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents a model of a controlled actuating mechanism, with particular motion characteristics and control concepts. The model is based on the sums of concepts of mechanisms, electrical machinery, computer science and microcontrollers theory and it aims to make a computational control to be used in this new equipment. To this computational control, it is necessary to make an introductory mathematical modeling based on spherical coordinates, to propose a definition of points of motion for a mechanical system, characterized by spherical steps. This system is actuated by magnetic points distributed along an articulated spherical shell and it slides in the fixed sphere with only one magnetic spot. The mechanical system communicates two stems with angular relative positions. These positions are presented through the development of a matrix that represents the power status of all the points distributed in the spherical shell, determining whether they are on or off. This mathematical model identifies points in the spherical shell, where they are registered as a table row in a database, which has information of the neighborhood relation between points. So a software is used to update in database and an algorithm is used for will do control by using the JAVA programming language. The input data for the algorithm are the start and the end motion points and the steps speed. The result of this algorithm is a script with sequence of movement, using best route criteria the algorithm selects the most efficient within several possibilities. The program’s output is a file that is sent to an electronic device and it receives the information, controls the power state for the coils of the actuating rod, to promote a sequence of desired movement. So this modeled mechanism, plans to make possible a device that offers new possibilities for solutions for controls as actuators positioners that can be applied to robotics and for the most varied of automation problem / Este trabalho apresenta um modelo de um mecanismo atuador controlado, com características particulares de movimento e de conceitos de controle. O modelo é baseado em somas de conceitos de mecanismos, máquinas elétricas, computação e microcontroladores, e tem o objetivo de fazer um controle computacional para ser usado de forma prática nesse novo equipamento. Para fazer o controle computacional, é necessário fazer uma modelagem matemática introdutória, baseada em coordenadas esféricas, para propor uma definição dos pontos de movimento de um sistema mecânico, caracterizados por passos esféricos. Esse sistema é atuado por zonas magnetizáveis ao longo de uma casca esférica rotulável e uma esfera fixa com apenas um ponto magnético. Essa rótula comunica duas hastes com posições angulares relativas entre si. Essas posições são definidas nesse trabalho por meio da elaboração de uma matriz que representa o estado de ligação de todas as zonas magnetizáveis por bobinas distribuídas na casca esférica, determinando se elas estão ligadas ou desligadas. Esse modelo matemático identifica os pontos na casca esférica, onde são cadastrados como registro de tabela em um banco de dados, que possuem a informação da relação de vizinhança entre pontos. Em seguida, um software gerencia a manutenção nesse banco de dados para fazer o controle usando algoritmo através de programação em linguagem JAVA. Os dados de entrada de informações para o algoritmo são os pontos de inicio e fim de movimento e a velocidade de acionamento das bobinas. O resultado desse algoritmo é um roteiro de sequencia de movimento, onde, usando critério de melhor rota, o algoritmo escolhe a mais eficiente dentro de várias possibilidades. O resultado do programa é um arquivo que é enviado para um dispositivo eletrônico e que ao receber as informações, comanda o estado das ligações das bobinas da rótula atuadora de forma sequenciada que promove o movimento desejado. Por fim, esse mecanismo assim modelado, pretende viabilizar um equipamento que ofereça novas possibilidades de soluções tanto para controle, como para soluções em atuadores posicionadores que podem ser aplicáveis a robótica e para os mais variados ramos da automação / 2017-07-12

Esta dissertação traz ideias para a inserção de novos conteúdos na matemática escolar. Ela trata da exploração de geometrias não-euclidianas, através de dois ambientes de geometria dinâmica, o "Spherical Easel” e o "Disco de Poincaré". O primeiro é um software livre e o segundo foi desenvolvido utilizando-se o recurso de macro-construção do GeoGebra. Na concepção das atividades tratamos as idéias que correspondem ao mundo não-euclidiano fazendo comparações com aquelas que fazem parte da geometria euclidiana e para cada atividade há um comentário que explica a sua intenção de aprendizagem. É a partir de considerações teóricas sobre a natureza da geometria e sua evolução histórica, bem como sobre o processo de aprendizagem da geometria, que é feita a apresentação da proposta. / This dissertation brings ideas to the inclusion of new contents in school mathematics. They are related to the exploitation of non-Euclidean geometries through two dynamic geometry environments, the "Spherical Easel" and the "Poincaré Disk". The first one is a free software and the second one was developed using the GeoGebra macro-construction. In the design of the activities the approach of ideas that correspond to non-euclidian worlds was made through comparison with the euclidian world and for each activity there is a comment that explain its learning objective. The proposal is supported by theoretical considerations about the nature of geometry and its historical evolution, as well as about the geometry learning process.