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The purpose of this paper is to explore some of the properties of the Cantor set and to extend the idea of this set to metric spaces, in general, and to other sets of real numbers and sets in N-dimensional Euclidean space, in particular.

在這篇論文裡，我們將要討論一類古典的問題，這類問題已經經由許多方法解決，例如：遞迴關係式、差分方程式、尤拉公式等等。接著我們歸納低維度的特性，並藉由定義出一組方程式-標準n維空間分割系統-來推廣這些特性到一般的$n$維度空間中。然後我們利用演算法來提供一個更直接的組合論證法。最後，我們會把問題再細分成有界區域與無界區域的個數。 / In this article, we will discuss a class of classical questions had been solved by Recurrence Relation, Difference Equation, and Euler's Formula, etc.. And then, we construct a system of equations -Standard Partition System of n-Dimensional Space- to generalize the properties of maximizing the number of regions made up by k partitioner in an n-dimensional space and look into the construction of each dimension. Also, we provide a more directly Combinatorial Argument by Algorithm for this kind of question. At last, we focus on the number of bounded regions and unbounded regions in sense of maximizing the number of regions.

Dimensional Stacking is a technique for displaying multivariate data in two dimensional screen space. This technique involves the discretization and recursive embedding of dimensions, each resulting N-dimensional bin occupying a unique position on the screen. This thesis describes the extension of this technique to a three dimensional projection. In addition to the visual enhancements, hashing was used to improve the scalability of records and dimensions. The resulting visualization was evaluated by a usability study.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-23T19:17:32Z No. of bitstreams: 1 DissPYSR.pdf: 744421 bytes, checksum: 298c0af1a2f63d6d09d684f123d5c614 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:44:32Z (GMT) No. of bitstreams: 1 DissPYSR.pdf: 744421 bytes, checksum: 298c0af1a2f63d6d09d684f123d5c614 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:44:39Z (GMT) No. of bitstreams: 1 DissPYSR.pdf: 744421 bytes, checksum: 298c0af1a2f63d6d09d684f123d5c614 (MD5) / Made available in DSpace on 2016-09-26T20:44:45Z (GMT). No. of bitstreams: 1 DissPYSR.pdf: 744421 bytes, checksum: 298c0af1a2f63d6d09d684f123d5c614 (MD5) Previous issue date: 2015-04-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The main aim of this work is to study a class of real vector elds de ned on a torus, where the concepts of global hypoellipticity, global C1 solvability and reduction to its normal form are equivalents. We also study a family of commuting vector fields that can be converted into a family of constant vector fields provided that there is one of them which its transpose operator is given by a hypoelliptic operator. We also provide some applications on the global hypoellipticity property for a class of sublaplaceans. This work was based on articles [11] and [12] from Gerson Petronilho. / O objetivo principal deste trabalho é estudar uma classe de campos vetoriais reais definidos sobre um toro, onde os conceitos de hipoelipticidade global, resolubilidade global C1 e redução à sua forma normal são equivalentes. Também estudamos que uma família de campos vetoriais reais que comutam entre si, pode ser transformada em uma família de campos vetoriais constantes com a condição de que um deles tenha o seu operador transposto dado por um operador hipoelíptico. Fornecemos ainda algumas aplicações sobre a propriedade de hipoelipticidade global para uma classe de sublaplaceanos. Este trabalho foi baseado nos artigos [11] and [12] de Gerson Petronilho.

Context. Big data visualization is a vital part of today's technological advancement. It is about visualizing different variables on a graph, map, or other means often in real-time. Objectives. This study aims to determine what challenges there are for big data visualization, whether significant amounts of data impact the visualization, and finding existing solutions for the problems. Methods. Databases used in this systematic literature review include Inspec, IEEE Xplore, and BTH Summon. Papers are included in the review if certain criteria are upheld. Results. 6 solutions are found to reduce large data sets and reduce latency when viewing 2D and 3D graphs. Conclusions. In conclusion, many solutions exist in various forms to improve visualizing graphs of different dimensions. Future grows of data might change this though and might require new solutions of the growing data.

This dissertation aims to expand the current one-dimensional game theory based model to a multidimensional model for multi-actor predictive analytics and generalize the concept of position to address problems where actors’ positions are distributed over a position spectrum. The one-dimensional models are used for the problems where actors are interacting in a single issue space only. This is less than an ideal assumption since, in most cases, players’ strategies may depend on the dynamics of multiple issues when dealing with other players. In this research, the one-dimensional model is expanded to N-Dimensional model by considering different positions, and separate salience values, across different axes for the players. The model predicts the outcome for a given problem by taking into account stakeholder’s positions in different dimensions and their conflicting perspectives. Furthermore, we generalize the concept of position in the model to include continuous positions for the actors throughout the position spectrum, enabling them to have more flexibility in defining their targets. We explore different possible functions to study the role of the position function and discuss appropriate distance measures for computing the distance between positions of actors. The proposed models are able to attain the same results as the previous one-dimensional models. In addition, to illustrate the capability of the proposed models, multiple case studies are designed and examined to assess the models’ capability and explainability.

A new N-dimensional digital modulation technique is proposed as a bandwidth efficient method for the transmission of digital data. The technique uses an antipodal scheme in which parallel binary data signs baseband orthogonal waveforms derived from Hermite polynomials. Orthogonality guarantees recoverability of the data from N simultaneously transmitted Hermite waveforms. The signed Hermite waveform is transmitted over a radio link using either amplitude or frequency modulation. The bandwidth efficiency of the amplitude Hermite method is found to be as good as filtered BPSK in practice, while the bit error rates for both modulations are identical. Hermite Keying (HK), the FM modulation version of the N-dimensional Hermite transmission, outperforms constant envelope FSK in terms of spectrum efficiency. With a simple FM detector, the bit error rate of HK is as good as that of non-coherent FSK. In a frequency selective fading channel, the simulation results suggest that specific data bits of HK are relatively secure from errors, which is beneficial in some applications. Symbol synchronization is critical to the system. An optimal synchronization method for the N-dimensional antipodal scheme in additive white Gaussian noise channel is derived. Simulation results confirm that the synchronizer can operate successfully at E/No of 3 dB. / Ph. D.

In der gegebenen Arbeit werden hoeherdimensionale lokale Schiefkoerper, die natuerliche Verallgemeinerung von n-dimensionalen lokalen Koerpern, untersucht. Wir untersuchen nur Schiefkoerper mit kommutativem Restschiefkoerper. Wir geben eine hinreichende Bedingung fuer die Spaltbarkeit von Schiefkoerpern. Naemlich, ein lokaler Schiefkoerper ist spaltbar, falls er einen kanonischen Automorphismus unendlicher Ordnung hat. Wir klassifizieren alle Schiefkoerper, die diese Bedingung bis auf Isomorphie erfuellen. Die Ergebnisse sind unabhaengig von der Charakteristik des Schiefkoerpers. Wir klassifizieren auch alle lokalen spaltbaren Schiefkoerper von Charakteristik 0 mit kommutativem Restschiefkoerper und mit kanonischem Automorphismus von endlicher Ordnung. Unter anderem geben wir ein Kriterium, wann zwei Elemente aus einem solchen Schiefkoerper konjugiert sind. Als Folgerung beweisen wir, dass fast alle solche Schiefkoerper unendlichdimensional ueber ihrem Zentrum sind. Ausserdem beweisen wir, dass das Skolem-Noether Theorem nur in dem Fall des klassischen Ringes der Pseudodifferentialoperatoren richtig ist. Dann erhalten wir Anwendungen dieser Theorie auf die Krichever Korrespondenz. Naemlich, wir bekommen Verallgemeinerungen von klassischen KP-Gleichungen (Hierarchie). Die Untersuchung von lokalen Schiefkoerpern fuehrte zu einigen neuen unerwarteten Ergebnissen in der Bewertungstheorie auf endlichdimensionalen Algebren. Wir bekommen den Zerlegungssatz fuer wilde Divisionalgebren ueber Laurentreihen-Koerpern mit beliebigem Restkoerper der Charakteristik groesser als zwei. Dieses Theorem ist die Verallgemeinerung des Zerlegungssatzes fuer zahme Divisionalgebren von Jacob und Wadsworth. Als Folgerung bekommen wir die positive Antwort auf die folgende Vermutung: Fuer jede Divisionalgebra A ueber den Koerper F((t)), wo F ein quasialgebraisch abgeschlossener Koerper ist, muss der Exponent von A gleich dem Index von A sein. Dann erhalten wir Anwendungen dieser Theorie auf die Krichever Korrespondenz. Naemlich, wir bekommen Verallgemeinerungen von klassischen KP-Gleichungen (Hierarchie). Anderseits, fuehrt das Problem der Klassifizierung lokaler Schiefkoerper zu dem Problem der Klassifizierung der Konjugationsklassen in der Automorphismengruppe von n-dimensionalen lokalen (kommutativen) Koerpern. Wir loesen diese Aufgabe fuer die Gruppe der stetigen Automorphismen von 1- und 2-dimensionalen lokalen Koerpern. / In this work we study local skew fields, which are natural generalization of n-dimensional local fields, and their applications to the theory of central division algebras over henselian fields. We study mostly two-dimensional local skew fields with commutative residue skew field. The sufficient condition for a skew field to be split is given. Namely, a local skew field splits if the canonical automorphism has infinite order. We classify all the skew fields which posess this condition up to isomorphism. These results don't depend on the characteristic of a skew field. We classify all local splittable skew fields of characteristic 0 with commutative residue skew field and with the canonical automorphism of finite order as well. Some other properties of local skew fields are studied. In particular, we give a criterium when two elements from such a skew field conjugate. As a corollary we prove that almost all such skew fields are infinite dimensional over their center. Also we prove that the Scolem-Noether theorem holds only in the case of the classical ring of pseudo-differential operators. Studying of local skew fields leads to some new unexpected results in the valuation theory on finite dimensional division algebras. We get a decomposition theorem for some class of wild division algebras over a Laurent series field with arbitrary residue field of characteristic greater than two. This theorem is a generalization of the decomposition theorem for tame division algebras given by B.Jacob and A.Wadsworth. As a corollary we get the positive answer on the following conjecture: the exponent of a division algebra is equal to its index if the centre of this algebra is a Laurent series field with arbitrary quasialgebraically closed residue field. Using some ideas of A.N. Parshin, who raised a problem of classifying local skew fields, we get some applications of developed theory to the Krichever correspondence. Namely, we get some generalizations of the classical KP-equations (hierarchy). The problem of classification of local skew fields leads to the problem of classification of conjugacy classes in the automorphism group of an n-dimensional local (commutative) field. We solve this problem for the group of continuous automorphisms of one- and two- dimensional local fields.

Neural networks represent a powerful means capable of processing various multi-media data. Two applications of artificial neural networks to 3D surface models are examined in this thesis - detection of significant features in 3D data and model classification. The theoretical review of existing self-organizing neural networks is presented and followed by description of feed-forward neural networks and convolutional neural networks (CNN). A novel modification of existing model - N-dimensional convolutional neural networks (ND- CNN) - is introduced. The proposed ND-CNN model is enhanced by an existing technique for enforced knowledge representation. The developed theoretical methods are assessed on supporting experiments with scanned 3D face models. The first experiment focuses on automatic detection of significant facial features while the second experiment performs classification of the models by their gender using the CNN and ND-CNN.