There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

Prenant appui sur un ensemble d'entretiens auprès des protagonistes, la base de données des émissions de titres publics du Bureau National de Crédit Public et un large corpus d'articles de presse, cette recherche analyse la relation entre ressources économiques et pouvoir politique en étudiant l'impact de l'intégration semi-périphérique au système financier mondial sur la coalition sociopolitique au pouvoir en Argentine de 1989 à 2001. Cette coalition était composée des partis politiques au gouvernement, ce qui garantissait la légitimité démocratique, des technocrates des think tanks du libéralisme économique, des grandes entreprises locales et des acteurs privés du système financier mondial. Contrairement aux pays centraux qui sont au coeur du système financier mondial et les périphériques, qui en sont exclus, l'Argentine représente un cas d'intégration "semipériphérique", c'est-à-dire, avec un accès variable au crédit privé externe en fonction de la perception des principaux acteurs privés du système financier sur le risque de défaut des paiements de la dette. Cette intégration a joué un rôle important sur la formation, la consolidation et la dissolution de la coalition socio-politique au pouvoir. D'une part, le plan de Convertibilité implique une dépendance structurelle de l'afflux de devises pourassurer la stabilité économique. D'autre part, les caractéristiques de l'intégration du système financier mondial ont été modifiées par le plan Brady, qui implique une substitution de la dette publique de prêt bancaire par des titres, et le processus de mondialisation. Contrairement aux prêts bancaires, les titres sont des produits échangés sur un marché où les prêteurs prennent des décisions d'investissement de court terme en fonction de leur perception du risque de défaut de paiement de la dette. Cette perception, qui implique une surveillance permanente sur la politique économique, détermine la capacité de l'État à s'endetter et soutenir ainsi l'afflux de devises.

A constantly growing corpus of interdisciplinary studies support the idea that music is a complex multimodal medium that is experienced not only by means of sounds but also through body movement. From this perspective, musical instruments can be seen as technological objects coupled with a repertoire of performance gestures. This repertoire is part of an ecological knowledge shared by musicians and listeners alike. It is part of the engine that guides musical experience and has a considerable expressive potential. This thesis explores technical and conceptual issues related to the analysis and creative use of music-related body movements in instrumental music performance. The complexity of this subject required an interdisciplinary approach, which includes the review of multiple theoretical accounts, quantitative and qualitative analysis of data collected in motion capture laboratories, the development and implementation of technologies for the interpretation and interactive use of motion data, and the creation of short musical pieces that actively employ the movement of the performers as an expressive musical feature. The theoretical framework is informed by embodied and enactive accounts of music cognition as well as by systematic studies of music-related movement and expressive music performance. The assumption that the movements of a musician are part of a shared knowledge is empirically explored through an experiment aimed at analysing the motion capture data of a violinist performing a selection of short musical excerpts. A group of subjects with no prior experience playing the violin is then asked to mime a performance following the audio excerpts recorded by the violinist. Motion data is recorded, analysed, and compared with the expert’s data. This is done both quantitatively through data analysis xii as well as qualitatively by relating the motion data to other high-level features and structures of the musical excerpts. Solutions to issues regarding capturing and storing movement data and its use in real-time scenarios are proposed. For the interactive use of motion-sensing technologies in music performance, various wearable sensors have been employed, along with different approaches for mapping control data to sound synthesis and signal processing parameters. In particular, novel approaches for the extraction of meaningful features from raw sensor data and the use of machine learning techniques for mapping movement to live electronics are described. To complete the framework, an essential element of this research project is the com- position and performance of études that explore the creative use of body movement in instrumental music from a Practice-as-Research perspective. This works as a test bed for the proposed concepts and techniques. Mapping concepts and technologies are challenged in a scenario constrained by the use of musical instruments, and different mapping ap- proaches are implemented and compared. In addition, techniques for notating movement in the score, and the impact of interactive motion sensor systems in instrumental music practice from the performer’s perspective are discussed. Finally, the chapter concluding the part of the thesis dedicated to practical implementations describes a novel method for mapping movement data to sound synthesis. This technique is based on the analysis of multimodal motion data collected from multiple subjects and its design draws from the theoretical, analytical, and practical works described throughout the dissertation. Overall, the parts and the diverse approaches that constitute this thesis work in synergy, contributing to the ongoing discourses on the study of musical gestures and the design of interactive music systems from multiple angles.

The study investigated errors made by Grade 9 learners when simplifying algebraic expressions. Eighty-two (82) Grade 9 learners from a rural secondary school in Limpopo Province, South Africa participated in the study. The sequential explanatory design method which uses both quantitative and qualitative approaches was used to analyse errors in basic algebra. In the quantitative phase, a 20-item test was administered to the 82 participants. Learners’ common errors were identified and grouped according to error type. The qualitative phase involved interviews with selected participants. The interviews focused on each identified common error in order to establish the reasons why learners made the identified errors. The study identified six (6) common errors in relation to simplifying algebraic expressions. The causes of these errors were attributed to poor arithmetic background; interference from new learning; failure to deal with direction and operation signs; problems with algebraic notation and misapplication of rules. / Mathematics Education / M. Ed. (Mathematics Education)

This study investigates the possibility of applying Hatten’s theory of musical gesture to a formal system of musical analysis. Using historical antecedents and established musicological practice as a guide, a range of musical parameters in a motive length span of music are incorporated into a single gesture. This gesture forms the basic semantic unit upon which an analytical tableau structure is built, and a syntax is developed to allow derivations of new gestures; a large scale structure displaying fractal-like self-similarity is then proposed. The completed system is applied to the analysis of the ‘Adagio’ of Mozart’s Flute Quartet K.285 to test whether it can consistently be implemented and whether it produces falsifiable results while maintaining predictive power. It is found that these requirements are indeed met and that a set of inference rules can be derived suggesting that the proposed system has ample scope for further development. / Art History, Visual Arts and Musicology / M. Mus.

Text in Afrikaans / Summaries in Afrikaans and English / The supposition exists that the Transvaal Education Department's Senior Primary Class Music syllabus is not being implemented in all aspects and that this is connected to teacher training. An empirical investigation was launched using three questionnaires and a classroom field study. Two questionnaires, completed by Class Music teachers, were used to assess to what extent the school syllabus was being followed. Teachers also had to provide information about their training in Class Music teaching. The third questionnaire offered a structured interview with school principals. The actualisation of Class Music, extra curricular music activities, the role of the teacher and teacher training were examined. Lessons for standard two pupils at twenty schools, were presented for an in-depth investigation. The results were tabulated, summarised in bar graphs and interpretations rendered. According to the findings, recommendations with regard to curriculum, the implementation of the syllabus, and tertiary Class Music training were made. / Daar is vermoed dat die seniorprimere Klasmusieksillabus van die Transvaalse Onderwysdepartement gedeeltelik gelmplementeer word en dat df t verband hou met onderwysersopleiding. In die lig hiervan is 'n empiriese ondersoek geloods deur middel van drie vraelyste en deur veldwerk. In die twee vraelyste wat deur Klasmusiekonderwysers voltooi is, is nagevors in watter mate die skoolsillabus gelmplementeer word. Verder het die onderwysers hulle Klasmusiekopleiding in hierdie verband aangetoon. Die derde vraelys vorm die raamwerk vir 'n gestruktureerde onderhoud met skoolhoofde. Die realisering van Klasmusiek, buite-kurrikulere musiekaktiwiteit en die rol van die onderwyser en onderwysopleiding is nagegaan. In 'n diepte-ondersoek is lesse by twintig skole vir standerdtweeleerlinge aangebied. Die resultate word in tabelle uiteengesit en in staafdiagramme saamgevat, terwyl waarnemings weergegee word. Op grondhiervan word aanbevelings gemaak oor kurrikulering, sillabusimplementering en tersiere Klasmusiekopleiding. / Art History, Visual Arts and Musicology / M.Mus

The thesis is interested in the problem of musical sign. In which ways can we speak about musical sign and which elements of musical material can become signs? These are the basic questions which we are interested in. This thesis is divided into three parts. The first part attends to the common theory of sign. The historical survey with the main ideas of philosophers and thinkers will be helpful to find out basic terms and problems of semiotics. In the second part of thesis we look at the historical reflection of musical sign in order to find out in which way it was treated with music and musical sign in each historical period of European music. In the third part we focus on musical semiotics, but we will consider some problems of musical aesthetics and musical psychology, too. In the end we make recapitulation of answers to the aforesaid questions.

Les visualisations, dans le sens général de représentations externes et physiques de données, sont plus anciennes que l'invention de l'écriture. De manière générale, les représentations externes encouragent la cognition et la pensée visuelle, et nous avons développé des savoir-faire pour les créer et les exploiter. La révolution informatique a augmenté la quantité de données qu'il est possible de collecter et de traiter, et a diversifié les façons de les représenter visuellement. Les systèmes de visualisation assistés par ordinateur, et étudiés dans le domaine de la visualisation d'information, sont aujourd'hui si puissants et complexes que nous avons besoin de techniques d'interaction très sophistiqués. Grâce au développement des possibilités technologiques au-delà des ordinateurs de bureau, un large éventail d'utilisations émerge. Non seulement des surfaces d'affichage de formes et de tailles variées permettent de montrer des visualisations plus riches, mais aussi des dispositifs d'entrée de nouvelle génération peuvent être utilisés qui exploitent les aptitudes humaines à manipuler les objets physiques. Cependant, ces technologies sont peu étudiées dans le contexte de la visualisation d'information. Tout d'abord, un premier problème découle d'une terminologie insuffisante. Dans cette thèse, je définis et étudie entre autres le concept de corporalisation (embodiment) pour la visualisation d'information. Concernant les visualisations, la corporalisation réfère à la congruence entre les éléments visuels d'une visualisation et leurs formes physiques. Ce concept intègre des concepts déjà connus tels que la tangibilité. Par exemple, l'interaction tangible s'attache à la représentation d'objets virtuels par des objets physiques. Mais en réalité, leur forme physique n'est pas nécessairement congruente avec l'objet virtuel. Un second problème découle du peu d'exemples convaincants d'interfaces tangibles appliquées à la visualisation d'information. Dans le domaine de la visualisation d'information, les écrans standard et les dispositifs d'entrée génériques tels que la souris, sont toujours les plus courants et considérés comme les plus efficaces. Cependant, aussi bien la partie affichage que la partie contrôle fournit des possibilités de corporalisation : les dispositifs d'entrée peuvent être spécialisés et adaptés de façon à ce que leur forme physique ressemble à leur fonction; les écrans peuvent être rendus déformables ou, dans l'avenir, être composés d'une matière programmable capable de prendre n'importe quelle forme imaginable. Mais la recherche sur les écrans et matières déformables est pour l'instant principalement dirigée par l'innovation technologique sans tenir compte des applications possibles à la visualisation d'information. Dans cette thèse, j'propose la corporalisation comme principe de conception pour la visualisation d'information. Je démontre l'efficacité et l'utilisabilité des dispositifs d'entrée corporalisés ainsi que des affichages corporalisés, en présentant trois expériences contrôlées. Par la suite, je présente un modèle d'interaction conceptuel et un système de notation visuelle pour décrire, comparer et critiquer différents types de systèmes de visualisation, et j'illustre l'utilisation de ce modèle à partir d'études de cas. Enfin, je présente un outil de conception pour aider à la création de visualisations physiques. Cet outil s'adresse à des utilisateurs novices en visualisation d'information et en fabrication numérique, et peut contribuer à sensibiliser ces utilisateurs à l'intérêt d'explorer des données qui les concernent dans leur vie quotidienne. En résumé, cette thèse contribue à la compréhension de la valeur ajoutée des interfaces physiques pour la visualisation d'information. / Visualizations in the most general sense of external, physical representations of information are older than the invention of writing. Generally, external representations promote external cognition and visual thinking, and humans developed a rich set of skills for crafting and exploring them. Computers immensely increased the amount of data we can collect and process as well as diversified the ways we can represent it visually. Computer-supported visualization systems, studied in the field of information visualization (infovis), have become powerful and complex, and sophisticated interaction techniques are now necessary to control them. With the widening of technological possibilities beyond classic desktop settings, new opportunities have emerged. Not only display surfaces of arbitrary shapes and sizes can be used to show richer visualizations, but also new input technologies can be used to manipulate them. For example, tangible user interfaces are an emerging input technology that capitalizes on humans' abilities to manipulate physical objects. However, these technologies have been barely studied in the field of information visualization. A first problem is a poorly defined terminology. In this dissertation, I define and explore the conceptual space of embodiment for information visualization. For visualizations, embodiment refers to the level of congruence between the visual elements of the visualization and their physical shape. This concept subsumes previously introduced concepts such as tangibility and physicality. For example, tangible computing aims to represent virtual objects through a physical form but the form is not necessarily congruent with the virtual object. A second problem is the scarcity of convincing applications of tangible user interfaces for infovis purposes. In information visualization, standard computer displays and input devices are still widespread and considered as most effective. Both of these provide however opportunities for embodiment: input devices can be specialized and adapted so that their physical shape reflects their functionality within the system; computer displays can be substituted by transformable shape changing displays or, eventually, by programmable matter which can take any physical shape imaginable. Research on such shape-changing interfaces has so far been technology-driven while the utility of such interfaces for information visualization remained unexploited. In this thesis, I suggest embodiment as a design principle for infovis purposes, I demonstrate and validate the efficiency and usability of both embodied visualization controls and embodied visualization displays through three controlled user experiments. I then present a conceptual interaction model and visual notation system that facilitates the description, comparison and criticism of various types of visualization systems and illustrate it through case studies of currently existing point solutions. Finally, to aid the creation of physical visualizations, I present a software tool that supports users in building their own visualizations. The tool is suitable for users new to both visualization and digital fabrication, and can help to increase users' awareness of and interest in data in their everyday live. In summary, this thesis contributes to the understanding of the value of emerging physical representations for information visualization.

Les maladies neurodégénératives, telles que la maladie d'Alzheimer (AD) et Charcot Marie Tooth (CMT), sont des maladies complexes. Leurs mécanismes pathologiques ne sont toujours pas bien compris et les progrès dans la recherche et le développement de nouvelles thérapies potentielles modifiant la maladie sont lents. Les données catégorielles, comme les échelles de notation et les données sur les études d'association génomique (GWAS), sont largement utilisées dans les maladies neurodégénératives dans le diagnostic, la prédiction et le suivi de la progression. Il est important de comprendre et d'interpréter ces données correctement si nous voulons améliorer la recherche sur les maladies neurodégénératives. Le but de cette thèse est d'utiliser la théorie psychométrique moderne: théorie de la réponse d’item pour analyser ces données catégoriques afin de mieux comprendre les maladies neurodégénératives et de faciliter la recherche de médicaments correspondante. Tout d'abord, nous avons appliqué l'analyse de Rasch afin d'évaluer la validité du score de neuropathie Charcot-Marie-Tooth (CMTNS), un critère important d'évaluation principal pour les essais cliniques de la maladie de CMT. Nous avons ensuite adapté le modèle Rasch à l'analyse des associations génétiques pour identifier les gènes associés à la maladie d'Alzheimer. Cette méthode résume les génotypes catégoriques de plusieurs marqueurs génétiques tels que les polymorphisme nucléotidique (SNPs) en un seul score génétique. Enfin, nous avons calculé l'information mutuelle basée sur la théorie de réponse d’item pour sélectionner les items sensibles dans ADAS-cog, une mesure de fonctionnement cognitif la plus utilisées dans les études de la maladie d'Alzheimer, afin de mieux évaluer le progrès de la maladie. / Neurodegenerative diseases, such as Alzheimer’s disease (AD) and Charcot Marie Tooth (CMT), are complex diseases. Their pathological mechanisms are still not well understood, and the progress in the research and development of new potential disease-modifying therapies is slow. Categorical data like rating scales and Genome-Wide Association Studies (GWAS) data are widely utilized in the neurodegenerative diseases in the diagnosis, prediction and progression monitor. It is important to understand and interpret these data correctly if we want to improve the disease research. The purpose of this thesis is to use the modern psychometric Item Response Theory to analyze these categorical data for better understanding the neurodegenerative diseases and facilitating the corresponding drug research. First, we applied the Rasch analysis in order to assess the validity of the Charcot-Marie-Tooth Neuropathy Score (CMTNS), a main endpoint for the CMT disease clinical trials. We then adapted the Rasch model to the analysis of genetic associations and used to identify genes associated with Alzheimer’s disease by summarizing the categorical genotypes of several genetic markers such as Single Nucleotide Polymorphisms (SNPs) into one genetic score. Finally, to select sensitive items in the most used psychometrical tests for Alzheimer’s disease, we calculated the mutual information based on the item response model to evaluate the sensitivity of each item on the ADAS-cog scale.