The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

This master's thesis deals with the analysis of the optical power measurement using an optical fiber sensor. The thesis contains the theoretical background to understand the origin of polarization and its representation in space using Poincaré’s sphere and Stokes vectors. There is also a part describing optical performance measurement and the preview of analog measurements from various researches. Practical part focuses on verification of changes of fiber sensor parameters at temperature change in its surroundings and this part includes also additional suggestions for various types of measurements and verification of the function of the photodiode as an optical power transducer. Most measurements are based on the comparison of results obtained with the polarimeter. The results were plotted and discussed.

The first part of this Thesis deals with the analysis of piecewise-smooth mechanical systems and the definition of special stability criteria in presence of non-conservative follower forces. To illustrate the peculiar stability properties of this kind of dynamical system, a reference 2 d.o.f. structure has been considered, composed of a rigid bar, with one and constrained to slide, without friction, along a curved profile, whereas the other and is subject to a follower force. In particular, the curved constraint is assumed to be composed of two circular profiles, with different and opposite curvatures, defining two separated subsystems. Due to this jump in the curvature, located at the junction point between the curved profiles, the entire mechanical structure can be modelled by discontinuous equations of motion, the differential equations valid in each subsystem can be combined, leading to the definition of a piecewise-smooth dynamical system. When a follower force acts on the structure, an unexpected and counterintuitive behaviour may occur: although the two subsystems are stable when analysed separately, the composed structure is unstable and exhibits flutter-like exponentially-growing oscillations. This special form of instability, previously known only from a mathematical point of view, has been analysed in depth from an engineering perspective, thus finding a mechanical interpretation based on the concept of non-conservative follower load. Moreover, the goal of this work is also the definition of some stability criteria that may help the design of these mechanical piecewise-smooth systems, since classical theorems cannot be used for the investigation of equilibrium configurations located at the discontinuity. In the literature, this unusual behaviour has been explained, from a mathematical perspective, through the existence of a discontinuous invariant cone in the phase space. For this reason, starting from the mechanical system described above, the existence of invariant cones in 2 d.o.f. mechanical systems is investigated through Poincaré maps. A complete theoretical analysis on piecewise-smooth dynamical systems is presented and special mathematical properties have been discovered, valid for generic 2~d.o.f. piecewise-smooth mechanical systems, which are useful for the characterisation of the stability of the equilibrium configurations. Numerical tools are implemented for the analysis of a 2~d.o.f. piecewise-smooth mechanical system, valid for piecewise-linear cases and extendible to the nonlinear ones. A numerical code has been developed, with the aim of predicting the stability of a piecewise-linear dynamical system a priori, varying the mechanical parameters. Moreover, “design maps” are produced for a given subset of the parameters space, so that a system with a desired stable or unstable behaviour can easily be designed. The aforementioned results can find applications in soft actuation or energy harvesting. In particular, in systems devoted to exploiting the flutter-like instability, the range of design parameters can be extended by using piecewise-smooth instead of smooth structures, since unstable flutter-like behaviour is possible also when each subsystem is actually stable. The second part of this Thesis deals with the numerical analysis of an elastic cloak for transient flexural waves in Kirchhoff-Love plates and the design of special metamaterials for this goal. In the literature, relevant applications of transformation elastodynamics have revealed that flexural waves in thin elastic plates can be diverted and channelled, with the aim of shielding a given region of the ambient space. However, the theoretical transformations which define the elastic properties of this “invisibility cloak” lead to the presence of a strong compressive prestress, which may be unfeasible for real applications. Moreover, this theoretical cloak must present, at the same time, high bending stiffness and a null twisting rigidity. In this Thesis, an orthotropic meta-structural plate is proposed as an approximated elastic cloak and the presence of the prestress has been neglected in order to be closer to a realistic design. With the aim of estimating the performance of this approximated cloak, a Finite Element code is implemented, based on a sub-parametric technique. The tool allows the investigation of the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. Moreover, the Finite Element code is extended to investigate a meta-plate interacting with a Winkler foundation, to analyse how the substrate modulus transforms in the cloak region. This second topic of the Thesis may find applications in the realization of approximated invisibility cloaks, which can be employed to reduce the destructive effects of earthquakes on civil structures or to shield mechanical components from unwanted vibrations.

À l'automne 1914, la Première Guerre mondiale éclate. Pour la France et l'Allemagne, il s'agit de la deuxième guerre en moins d'un demi-siècle. Au terme du premier conflit en 1870, la Prusse victorieuse avait annexé l'Alsace- Lorraine. De ces événements est né en France le mythe des provinces perdues, et s'est développé en parallèle le culte de la Revanche. Raymond Poincaré, porté au pouvoir en 1913, sera donc le président de la République qui vena la Revanche se matérialiser en 1918 avec le retour de l'Alsace-Lorraine à la France. À quelle occasion, de quelle façon, dans quel contexte et dans quel but évoque-t-il le mythe dans ses discours de 1914 à 1919 ? Depuis la fin de la Grande Guerre, un débat fait rage en France à savoir si Poincaré a ou non souhaité une guerre de revanche. L'analyse des allusions au mythe alsacien-Lorraine dans ses discours, qui, à travers ce mémoire, nous est ici proposée, nous aidera à mieux comprendre comment Poincaré a participé à une telle guerre, notamment en utilisant la question de l'Alsace- Lorraine à des fins politiques dans le but de promouvoir son nationalisme haineux et revanchard.

Dizertační práce se zabývá elektronicky nastavitelnými oscilátory, studiem nelineárních vlastností spojených s použitými aktivními prvky a posouzením možnosti vzniku chaotického signálu v harmonických oscilátorech. Jednotlivé příklady vzniku podivných atraktorů jsou detailně diskutovány. V doktorské práci je dále prezentováno modelování reálných fyzikálních a biologických systémů vykazujících chaotické chování pomocí analogových elektronických obvodů a moderních aktivních prvků (OTA, MO-OTA, CCII ±, DVCC ±, atd.), včetně experimentálního ověření navržených struktur. Další část práce se zabývá možnostmi v oblasti analogově – digitální syntézy nelineárních dynamických systémů, studiem změny matematických modelů a odpovídajícím řešením. Na závěr je uvedena analýza vlivu a dopadu parazitních vlastností aktivních prvků z hlediska kvalitativních změn v globálním dynamickém chování jednotlivých systémů s možností zániku chaosu v důsledku parazitních vlastností použitých aktivních prvků.

Dans cette thèse on a étudié plusieurs problèmes<br />algébriques relatifs à une superalgèbre de Lie qui peuvent être<br />réduits à la résolution d'une équation fonctionnelle. Cette<br />technique a permis d'obtenir des résultats qui sont nouveaux<br />aussi pour une algèbre de Lie ordinaire et qui sont indépendants<br />de la classification des algèbres de Lie.

Les travaux de cette thèse portent sur des méthodes numériques pour la discrétisation d'équations aux dérivées partielles elliptiques et paraboliques de convection-réaction-diffusion non linéaires. Nous analysons ces méthodes et nous les appliquons à la simulation effective de l'écoulement et du transport de contaminants en milieux poreux et fracturés. Au chapitre 1, nous proposons un schéma permettant une discrétisation efficace, robuste, conservative et stable des équations de convection-réaction-diffusion non linéaires paraboliques dégénérées sur des maillages non structurés en dimensions deux ou trois d'espace. Nous discrétisons le terme de diffusion, qui contient en général un tenseur de diffusion inhomogène et anisotrope, par la méthode des éléments finis non conformes ou mixtes-hybrides et les autres termes par la méthode des volumes finis. La partie essentielle du chapitre est ensuite consacrée à montrer l'existence et l'unicité d'une solution discrète et sa convergence vers une solution faible du problème continu. La méthode de démonstration permet en particulier d'éviter des hypothèses restrictives sur le maillage souvent présentes dans la littérature. Nous proposons finalement une variante de ce schéma pour des maillages qui ne se raccordent pas, couplant cette fois la méthode des volumes finis avec celle des éléments finis conformes, et nous l'appliquons à la simulation du transport de contaminants en milieux poreux. Au chapitre 2, nous présentons une démonstration constructive des inégalités de Poincaré-Friedrichs discrètes pour une classe d'approximations non conformes de l'espace de Sobolev H1, indiquons les valeurs optimales des constantes dans ces inégalités et montrons l'inégalité de Friedrichs discrète pour des domaines bornés dans une direction uniquement. Ces résultats sont importants dans l'analyse de méthodes numériques non conformes, comme les méthodes d'éléments finis non conformes ou de Galerkin discontinu. Au chapitre 3, nous montrons que la méthode des éléments finis mixtes de Raviart-Thomas de plus bas degré pour des problèmes elliptiques en dimension deux ou trois d'espace est équivalente à un schéma de volumes finis à plusieurs points. Après avoir étudié ce schéma, nous l'appliquons à la discrétisation d'équations de convection-réaction-diffusion paraboliques non linéaires. Cette approche permet de réduire le temps de calcul de la méthode des éléments finis mixtes, tout en conservant sa très grande précision, ce qui est confirmé par les tests numériques. Enfin, au chapitre 4, nous proposons une version de la méthode des éléments finis mixtes de Raviart-Thomas de plus bas degré pour la résolution de problèmes elliptiques sur un système de polygones bidimensionnels placés dans l'espace tridimensionnel, démontrons qu'elle est bien posée et étudions sa relation avec la méthode des éléments finis non conformes. Ces résultats sont finalement appliqués à la simulation de l'écoulement de l'eau souterraine dans un système de polygones représentant un réseau de fractures perturbant un massif rocheux.

Turbulent phenomena are found in both natural (e.g. the Earth's oceans, the Sun's corona) and artificial (e.g. flows through pipes, the plasma in a tokamak device) settings; evidence suggests that turbulence is usually the normal behaviour in most cases. Turbulence has been studied extensively for more than a century, but a complete and consistent theoretical description of it has not yet been proposed. It is in this context that the motion of particles under the influence of turbulent fields is studied in this work, with direct numerical simulations. The thesis is structured in three main parts. The first part describes the tools that are used. Methods of integrating particle trajectories are presented, together with a discussion of the properties that these methods should have. The simulation of magnetohydrodynamic (MHD) turbulence is discussed, while also introducing fundamental concepts of fluid turbulence. Particle trajectory integration requires information that is not readily available from simulations of turbulent flows, so the interpolation methods needed to adapt the fluid simulation results are constructed as well. The second part is dedicated to the study of two MHD problems. Simulations of Kolmogorov flow in incompressible MHD are presented and discussed, and also simulations of the dynamo effect in compressible MHD. These two scenarios are chosen because large scale structures are formed spontaneously by the turbulent flow, and there is an interest in studying particle transport in the presence of structures. Studies of particle transport are discussed in the third part. The properties of the overall approach are first analyzed in detail, for stationary predefined fields. Focus is placed on the qualitative properties of the different methods presented. Charged article transport in frozen turbulent fields is then studied. Results concerning transport of particles in fully developed, time-evolving, turbulent fields are presented in the final chapter.<p><p><p>\ / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

La combinatoire a permis de résoudre certains problèmes en Mathématiques, en Physique et en Informatique, en retour celles-ci inspirent des questions nouvelles à la combinatoire. Ce mémoire de thèse intitulé "Combinatoire et algorithme des factorisations tangentes à l'identité" regroupe plusieurs travaux sur la combinatoire des déformations du produit de Shuffle. L'objectif de cette thèse est d'écrire des factorisations dont le terme principal est l'identité à travers l'utilisation d'outils portant principalement sur la combinatoire des mots (ordres, graduation etc.). Dans le cas classique, soit F une algèbre libre. En raison du fait que F est une algèbre enveloppante, on a une factorisation exacte de l'identité de End(F) = F*⨶F comme un produit infini d'exponentielles (End(F) étant muni du produit de Shuffle sur la gauche et de la concaténation sur la droite, une représentation fidèle du produit de convolution). La procédure est la suivante : premièrement on commence avec une base de Poincaré-Birkhoff-Witt, deuxièmement on calcule la famille des formes coordonnées et alors les propriétés (combinatoires) non triviales de ces familles en dualité donne la factorisation. Si on part de l'autre côté, l'écriture pour le même produit ne donne exactement l'identité que sous des conditions très restrictives que nous précisons ici. Dans de nombreux autres cas (déformés), la construction explicite des paires de bases en dualité nécessite une étude combinatoire et algorithmique que nous fournissons dans ce mémoire. / Combinatorics has solved many problems in Mathematics, Physics and Computer Science, in return these domains inspire new questions to combinatorics. This memoir entitled "Combinatorics and algorithmics of factorization tangent to indentity includes several works on the combinatorial deformations of the shuffle product. The aim of this thesis is to write factorizations wich principal term is the identity through the use of tools relating mainly to combinatorics on the words (orderings, grading etc). In the classical case, let F be the free algebra. Due to the fact that F is an enveloping algebra, one has an exact factorization of the identity of End(F) = F⨶F as an infinite product of exponentials (End(F) being endowed with the shuffle product on the left and the concatenation on the right, a faithful representation of the convolution product) as follows : first on begins with a PBW basis, second one computes the family of coordinate forms and then non-trivial (combinatorial) properties of theses families in duality gives the factorization. Starting from the other side and writing the same product does give exactly identity only under very restrictive conditions that we clarify here. In many other (deformed) cases, the explicit construction of pairs of bases in duality requires combinatorial and algorithmic studies that we provide in this memoir.

Transdisciplinarity emerged from the urge to grasp the elusive knowledge in the most fertile zone in between and beyond disciplines that escapes even the most elaborate interdisciplinary operations. While interdisciplinary protocol enables experts to operate within foreign disciplines, in the extreme case as diverse as art and science (by inviting artists into scientific departments and vice versa), the production of knowledge remains confined to particular domains. To transcend these confinements and access the knowledge that evades institutionalisation Basarab Nicolescu’s Manifesto of Transdisciplinarity sets up conditions for an open structure to be grown outside the current compartmentalisation into a living knowledge. This thesis imagines a possible evolution of transdisciplinarity into knowledge to be lived internally rather than learnt externally in order to overcome the anxiety in transcending the established culture of disciplinary research. By entering the transdisciplinary zone, the identity of experts-specialists dissolves, even the crudest separation into artists and scientists becomes obsolete. From the illusion of losing control over knowledge arises the fear of a return to archaic, mystic or even shamanic ways of knowing. Far from proposing a return to shamanism in its ancient forms this thesis imagines the way of polyphibianism – an imaginary solution to navigate efficiently the protoplasmic state of knowledge that would be indigenous to culture of disciplinary researchers. With every significant discovery the disciplinary researchers already intuitively trespass into the very zone that the Manifesto of Transdisciplinarity invites them to enter intentionally. From examination of documented introspective inquiries into their act of discovery the thesis infers the necessary sensibilities and adaptabilities of the individuals to cross the borders of their disciplines. Their seemingly lost identity is temporarily restored with the term polyphibian (analogous to amphibian) designating their ability to survive and explore multiple environments. With each change of circumstances in research a polyphibian adapts by swiftly reinventing its instinctive instruments, mutating its organs of knowing, indifferently to conventional habits of thought. Through their introspective writings this thesis investigates the polyphibic aptitude of Henri Poincaré, Henri Bergson and Marcel Duchamp to scout at the periphery of physics, metaphysics and ‘pataphysics, to intuitively anticipate the role of chance, chaos and complexity in both arts and sciences. A threshold of complexity has to be surpassed in order to bring the current apparatus of knowledge to life. Bergson’s insight on laughter and dreams suggests how intellect could transcend itself. The thesis proposes to consider laughter as faculty that could induce self-awareness in the intellectual apparatus while dreams are considered to facilitate self-organisation of intellect on higher orders of awareness. In Deleuzian manner of mutating Bergson’s work into Bergsonism, polyphibianism is a mutation in transcribing the code of Creative Evolution where Bergson insisted on interdependency between the theory of knowledge and the theory of evolution. The scholarly dispute on Bergsonian and anti-Bergsonian tendencies present in Marcel Duchamp’s work is revisited in the thesis by interpreting the higher dimensional Bride as a polyphibic organism of living knowledge with access to higher orders of awareness, able to guide the Bachelor’s apparatus of mechanical production and preservation of knowledge out of its predicament. Informed by peculiar Duchampian experiments that challenged both the domain of art and science the research projects in this thesis consist of an intervention at CERN that tested the impenetrability of institutionalised art-science collaborations and installation of the Interval of Suspended Judgement with high mathematical precision at the threshold between physics and ‘pataphysics. With these projects the problems of categorising researchers into artists and scientists are revealed. As Deleuze suggested, to effectively formulate the problem, to realize it in multiplicity of contexts, a new concept must be invented, a new organism must be conceived. This thesis gave birth to an imaginary organism of living knowledge in order to relieve the unnecessary anxieties and to fully engage in transdisciplinary research.