Dans cette thèse j’étudie l’impact de la géométrie de micro et nano-domaines biologiques sur les propriétés d'électrodiffusion, ceci à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck. Je considère des domaines non-triviaux ayant une forme cuspide ou elliptique. Mon objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles. Dans la première partie j’étudie le voltage à l'équilibre pour un électrolyte dans un domaine borné, et ayant un fort excès de charges positives. Je montre que le premier temps de sortie dans une boule chargée dépend de la surface et non du volume. J’étudie ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Je construis ensuite une solution asymptotique pour le voltage dans les cas 2D et 3D et je montre qu’ils sont donnés au premier ordre par la même expression. Enfin, j’obtiens la même conclusion en considérant une géométrie formée d'une ellipse, dont je construis une solution asymptotique du voltage en 2D et 3D. La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. A partir de simulations numériques, je mets en évidence le lien entre la polarisation de concentration dans l'épine et sa géométrie. Je compare ensuite mon modèle à des données de microscopie. Je développe une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir des données de microscopie. Enfin j’estime la résistance du cou et montre que celle-ci ne suit pas la loi d'Ohm. / In this PhD I study how electro-diffusion within biological micro and nano-domains is affected by their shapes using the Poisson-Nernst-Planck (PNP) partial differential equations. I consider non-trivial shapes such as domains with cusp and ellipses. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales. In the first part I estimate the steady-state voltage inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. I show the mean first passage time in a charged ball depends on the surface and not on the volume. I further study a geometry composed of a ball with an attached cusp-shaped domain. I construct an asymptotic solution for the voltage in 2D and 3D and I show that to leading order expressions for the voltage in 2D and 3D are identical. Finally, I obtain similar conclusion considering an elliptical-shaped domain for which I construct an asymptotic solution for the voltage in 2D and 3D. In the second part, I model the electrical compartmentalization in dendritic spines. Based on numerical simulations, I show how spines non-cylindrical geometry leads to concentration polarization effects. I then compare my model to experimental data of microscopy imaging. I develop a deconvolution method to recover the fast voltage dynamic from the data. I estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius.

Yes / Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the 'probabilistic quadrant' , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables. / The full-text of this article will be released for public view at the end of the publisher embargo on 15 Oct 2020. / Research Development Fund Publication Prize Award winner, October 2019.

This thesis is devoted to life and work of writer named Arthur Charles Clarke. The first part introduces the author and his work, particulary the short story "The Wall of Darkness". The second part describes the creation of the autor´s book, including the ideological base. All of the parts are thematically connected to each other.

Die vorliegende Arbeit befasst sich mit einer Vermutung von Sarnak aus dem Jahre 2010 über die Orthogonalität von durch deterministische dynamische Systeme induzierte Folgen zur Möbiusschen μ-Funktion. Ihre Hauptresultate sind zum einen der Ergodensatz mit Möbiusgewichten, welcher eine maßtheoretische (schwächere) Version von Sarnaks Vermutung darstellt, und zum anderen die bereits gesicherte Gültigkeit der genannten Vermutung in Spezialfällen, wobei hier exemplarisch unter anderem der Thue–Morse Shift und Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis gewählt worden sind. Zum Zwecke der Motivation zeigen wir, dass eine gewisse Wachstumsabschätzung für die Mertensfunktion äquivalent ist zum Primzahlsatz und skizzieren ein Resultat, welches die Äquivalenz einer weiteren solchen Abschätzung zur Riemannschen Vermutung liefert, um auf diese Weise die Bedeutung der Möbiusfunktion für die Zahlentheorie herauszustellen. Da sie für das Verständnis von Sarnaks Vermutung unerlässlich ist, geben wir eine Einführung in die Theorie der Entropie dynamischer Systeme auf Grundlage der Definitionen von Adler–Konheim–McAndrew, Bowen–Dinaburg und Kolmogorov–Sinai. Ferner berechnen wir die topologische Entropie des Thue–Morse Shifts und von Schiefprodukterweiterungen von Rotatione auf dem Kreis. Wir studieren die ergodische Zerlegung T-invarianter Maße auf kompakten metrischen Räumen mit stetiger Transformation T, welche wir für den Beweis des Ergodensatzes mit Möbiusgewichten benötigen. Sodann beweisen wir den genannten gewichteten Ergodensatz. Wir geben eine hinreichende Bedingung an für das Erfülltsein von Sarnaks Vermutung in einem gegebenen dynamischen System, welche im anschließenden Kapitel Anwendung findet. So wird nachgewiesen, dass Sarnaks Vermutung im Falle des Thue–Morse Shifts und von Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis erfüllt ist. Abschließend wird gezeigt, dass Sarnaks Vermutung sich als Konsequenz aus einer Vermutung von Chowla ergibt. / The thesis in hand deals with a conjecture of Sarnak from 2010 about the orthogonality of sequences induced by deterministic dynamical systems to the Möbius μ-function. Its main results are the ergodic theorem with Möbius weights, which is a measure theoretic (weaker) version of Sarnak’s conjecture, and the already assured validity of Sarnak’s conjecture in special cases, where we have exemplarily chosen the Thue–Morse shift and skew product extensions of rational rotations on the significance of the Möbius function for number theory. Since it is essential for the understanding of Sarnak’s conjecture we give an introduction to the theory of entropy of dynamical systems based on the definitions of Adler–Konheim–McAndrew, Bowen–Dinaburg and Kolmogorov–Sinai. Furthermore, we calculate the topological entropy of the Thue–Morse shift and of skew product extensions of rotations on the circle. We study the ergodic decomposition for T-invariant measures on compact metric spaces with continuous transformations T, which we will need for the proof of the ergodic theorem with Möbius weights. Thereafter, we prove the namely weighted ergodic theorem. We give a sufficient condition for Sarnak’s conjecture to hold for a given dynamical system, which we make use of in the following chapter. Thereupon, it is varified that Sarnak’s conjecture holds for the Thue–Morse shift and for skew product extensions of rational rotations on the circle. Lastly, it is shown that Sarnak’s conjecture from one of Chowla.

Cette thèse est rédigée par articles. Les articles sont rédigés en anglais et le reste de la thèse est rédigée en français. / Le travail établit une équivalence en termes de puissance entre les tests basés sur la alpha-distance de covariance et sur le critère d'indépendance de Hilbert-Schmidt (HSIC) avec fonction caractéristique de distribution de probabilité stable d'indice alpha avec paramètre d'échelle suffisamment petit. Des simulations en grandes dimensions montrent la supériorité des tests de distance de covariance et des tests HSIC par rapport à certains tests utilisant les copules. Des simulations montrent également que la distribution de Pearson de type III, très utile et moins connue, approche la distribution exacte de permutation des tests et donne des erreurs de type I précises. Une nouvelle méthode de sélection adaptative des paramètres d'échelle pour les tests HSIC est proposée. Trois simulations, dont deux sont empruntées de l'apprentissage automatique, montrent que la nouvelle méthode de sélection améliore la puissance des tests HSIC. Le problème de tests d'indépendance entre deux vecteurs est généralisé au problème de tests d'indépendance mutuelle entre plusieurs vecteurs. Le travail traite aussi d'un problème très proche à savoir, le test d'indépendance sérielle d'une suite multidimensionnelle stationnaire. La décomposition de Möbius des fonctions caractéristiques est utilisée pour caractériser l'indépendance. Des tests généralisés basés sur le critère d'indépendance de Hilbert-Schmidt et sur la distance de covariance en sont obtenus. Une équivalence est également établie entre le test basé sur la distance de covariance et le test HSIC de noyau caractéristique d'une distribution stable avec des paramètres d'échelle suffisamment petits. La convergence faible du test HSIC est obtenue. Un calcul rapide et précis des valeurs-p des tests développés utilise une distribution de Pearson de type III comme approximation de la distribution exacte des tests. Un résultat fascinant est l'obtention des trois premiers moments exacts de la distribution de permutation des statistiques de dépendance. Une méthodologie similaire a été développée pour le test d'indépendance sérielle d'une suite. Des applications à des données réelles environnementales et financières sont effectuées. / The main result establishes the equivalence in terms of power between the alpha-distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distribution of index alpha with sufficiently small scale parameters. Large-scale simulations reveal the superiority of these two tests over other tests based on the empirical independence copula process. They also establish the usefulness of the lesser known Pearson type III approximation to the exact permutation distribution. This approximation yields tests with more accurate type I error rates than the gamma approximation usually used for HSIC, especially when dimensions of the two vectors are large. A new method for scale parameter selection in HSIC tests is proposed which improves power performance in three simulations, two of which are from machine learning. The problem of testing mutual independence between many random vectors is addressed. The closely related problem of testing serial independence of a multivariate stationary sequence is also considered. The Möbius transformation of characteristic functions is used to characterize independence. A generalization to p vectors of the alpha -distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distributionof index alpha is obtained. It is shown that an HSIC test with sufficiently small scale parameters is equivalent to an alpha -distance covariance test. Weak convergence of the HSIC test is established. A very fast and accurate computation of p-values uses the Pearson type III approximation which successfully approaches the exact permutation distribution of the tests. This approximation relies on the exact first three moments of the permutation distribution of any test which can be expressed as the sum of all elements of a componentwise product of p doubly-centered matrices. The alpha -distance covariance test and the HSIC test are both of this form. A new selection method is proposed for the scale parameter of the characteristic kernel of the HSIC test. It is shown in a simulation that this adaptive HSIC test has higher power than the alpha-distance covariance test when data are generated from a Student copula. Applications are given to environmental and financial data.

<p>I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet.</p><p>Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar.</p> / <p>In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880.</p><p>The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.</p>

I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet. Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar. / In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880. The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.

The Eudaimonic turn: Well-being in Literary Studies (2013), a study by Pawelski et al, sheds light on a recent turn in literary studies. The eudaimonic approach entails that texts are examined with the help of a hermeneutic of affirmation rather than the sceptical, suspicious methods of the deconstruction and post-structuralism. Pawelski et al’s text is drawn upon in this study because it corresponds to the way in which Marlene van Niekerk utilises themes such as relationships, friendship and loss in Die sneeuslaper (2009). The eudaimonic turn focusses on the way in which complex interpersonal connections are able to add to an individual’s well-being through positive as well as negative processes. Die sneeuslaper is mainly a reflection on what it means to be an author. However, it also raises important questions about the nature of being. The four short stories provide different perspectives on friendship, on how friendship can sometimes be problematic and even a nuisance, but also how relations with others repeatedly prove to be beneficial to a person’s well-being. In my study, the relational theme of friendship in Die sneeuslaper is studied from a eudaimonic point of view. Kaja Silverman’s text, Flesh of my Flesh (2009), is referred to in order to shed light on the term relationality. The relational themes of finitude (or mortality) and interpersonal connection are clearly present in Van Niekerk’s text. Although the death of a beloved friend causes trauma in Die sneeuslaper, the trauma proves to have positive effects in the form of posttraumatic growth, comfort and acceptance as time goes by. Comfort is also construed through the creation and appreciation of a work of art like Die sneeuslaper. Cognitive narratology is referenced to show how Marlene van Niekerk overthrows and plays with fixed ideas regarding relationality and friendship, causing the reader to converse with the text. Views on friendship held by thinkers such as Aristotle, Montaigne, Lacan, Kierkegaard and Derrida are referenced to give Van Niekerk’s use of the theme in Die sneeuslaper a certain context. This context represents the fixed frames of thinking generally applicable with regard to friendship. When a reader is willing to critically interpret these as well as personal frames of reference, it provides him or her the opportunity to contemplate reality from new perspectives. In Die sneeuslaper the reader is continually challenged to question existing frames of reference by means of never ending methods (resembling a Möbius-strip) and strange notions. This study concludes with the notion that it is necessary for artists (like the writers in Die sneeuslaper) to reflect differently on reality, so that readers may be inspired to also view reality in a different light. This will result in a broader view of reality, which in turn will have a more defining influence on personal well-being. / MA (Afrikaans and Dutch), North-West University, Potchefstroom Campus, 2014

The Eudaimonic turn: Well-being in Literary Studies (2013), a study by Pawelski et al, sheds light on a recent turn in literary studies. The eudaimonic approach entails that texts are examined with the help of a hermeneutic of affirmation rather than the sceptical, suspicious methods of the deconstruction and post-structuralism. Pawelski et al’s text is drawn upon in this study because it corresponds to the way in which Marlene van Niekerk utilises themes such as relationships, friendship and loss in Die sneeuslaper (2009). The eudaimonic turn focusses on the way in which complex interpersonal connections are able to add to an individual’s well-being through positive as well as negative processes. Die sneeuslaper is mainly a reflection on what it means to be an author. However, it also raises important questions about the nature of being. The four short stories provide different perspectives on friendship, on how friendship can sometimes be problematic and even a nuisance, but also how relations with others repeatedly prove to be beneficial to a person’s well-being. In my study, the relational theme of friendship in Die sneeuslaper is studied from a eudaimonic point of view. Kaja Silverman’s text, Flesh of my Flesh (2009), is referred to in order to shed light on the term relationality. The relational themes of finitude (or mortality) and interpersonal connection are clearly present in Van Niekerk’s text. Although the death of a beloved friend causes trauma in Die sneeuslaper, the trauma proves to have positive effects in the form of posttraumatic growth, comfort and acceptance as time goes by. Comfort is also construed through the creation and appreciation of a work of art like Die sneeuslaper. Cognitive narratology is referenced to show how Marlene van Niekerk overthrows and plays with fixed ideas regarding relationality and friendship, causing the reader to converse with the text. Views on friendship held by thinkers such as Aristotle, Montaigne, Lacan, Kierkegaard and Derrida are referenced to give Van Niekerk’s use of the theme in Die sneeuslaper a certain context. This context represents the fixed frames of thinking generally applicable with regard to friendship. When a reader is willing to critically interpret these as well as personal frames of reference, it provides him or her the opportunity to contemplate reality from new perspectives. In Die sneeuslaper the reader is continually challenged to question existing frames of reference by means of never ending methods (resembling a Möbius-strip) and strange notions. This study concludes with the notion that it is necessary for artists (like the writers in Die sneeuslaper) to reflect differently on reality, so that readers may be inspired to also view reality in a different light. This will result in a broader view of reality, which in turn will have a more defining influence on personal well-being. / MA (Afrikaans and Dutch), North-West University, Potchefstroom Campus, 2014