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Dissertations / Theses on the topic 'Topological invariants'
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Metzler, David S. (David Scott). "Topological invariants of symplectic quotients." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/43933.
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Mayer, Christoph. "Topological link invariants of magnetic fields." [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=969161964.
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Larsen, Nicholas Guy. "A New Family of Topological Invariants." BYU ScholarsArchive, 2018. https://scholarsarchive.byu.edu/etd/6757.
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We define an extension of the nth homotopy group which can distinguish a larger class of spaces. (E.g., a converging sequence of disjoint circles and the disjoint union of countably many circles, which have isomorphic fundamental groups, regardless of choice of basepoint.) We do this by introducing a generalization of homotopies, called component-homotopies, and defining the nth extended homotopy group to be the set of component-homotopy classes of maps from compact subsets of (0,1)n into a space, with a concatenation operation. We also introduce a method of tree-adjoinment for "connecting" disconnected metric spaces and show how this method can be used to calculate the extended homotopy groups of an arbitrary metric space.
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Woolf, Jonathan. "Some topological invariants of singular symplectic quotients." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299436.
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Roberts, Justin Deritter. "Quantum invariants via skein theory." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319336.
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Zach, Matthias [Verfasser]. "Topological invariants of isolated determinantal singularities / Matthias Zach." Hannover : Technische Informationsbibliothek (TIB), 2017. http://d-nb.info/1150664274/34.
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Brunnbauer, Michael. "Topological properties of asymptotic invariants and universal volume bounds." Diss., lmu, 2008. http://nbn-resolving.de/urn:nbn:de:bvb:19-87504.
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Mukherjee, Devarshi [Verfasser]. "Topological Invariants for Non-Archimedean Bornological Algebras / Devarshi Mukherjee." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2020. http://d-nb.info/121973179X/34.
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Guerville, Benoît. "Invariants Topologiques d'Arrangements de droites." Thesis, Pau, 2013. http://www.theses.fr/2013PAUU3033/document.
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Cette thèse est le point d’intersection entre deux facettes de l’étude des arrangements de droites : la combinatoire et la topologie. Dans une première partie nous avons étudié l’inclusion de la variété bord dans le complémentaire d’un arrangement. Nous avons ainsi généralisé le résultat d’E. Hironaka au cas de tous les arrangements complexes. Pour contourner les problèmes provenant des arrangements non réels, nous avons étudié le diagramme de câblage, dit wiring diagram, qui code la monodromie de tresses sous forme de tresse singulière. Pour pouvoir l'utiliser, nous avons implémenté un programme sur Sage permettant de calculer ce diagramme en fonction des équations de l’arrangement. Cela nous a permis de d’obtenir deux descriptions explicites de l’application induite par l’inclusion de la variété bord dans le complémentaire sur les groupes fondamentaux. Nous obtenons ainsi deux nouvelles présentations du groupe fondamental du complémentaire d’un arrangement. L’une d’entre elle généralise le théorème de R. Randell au cas des arrangements complexes. Pour continuer ces travaux, nous avons étudié l’application induite par l’inclusion sur le premier groupe d’homologie. Nous obtenons deux descriptions simples de cette application. En s’inspirant des travaux de J.I. Cogolludo, nous décrivons une décomposition canonique du premier groupe d’homologie de la variété bord comme produit de la 1-homologie et de la 2-cohomologie du complémentaire, ainsi qu'un isomorphisme entre la 2-cohomologie du complémentaire et la 1-homologie du graphe d’incidence. Dans la seconde partie de notre travail nous nous sommes intéressés à l’étude des caractères du groupe fondamental du complémentaire. Nous partons des résultats obtenus par E. Artal sur le calcul de la profondeur d’un caractère. Cette profondeur peut être décomposée en un terme projectif et un terme quasi-projectif. Un algorithme pour calculer la partie projective a été donné par A. Libgober. Les travaux de E. Artal concernent la partie quasi-projective. Il a obtenu une méthode pour la calculer en fonction de l’image de certains cycles particuliers du complémentaire par le caractère. En utilisant les résultats obtenus dans la première partie, nous avons obtenu un algorithme complet permettant le calcul de la profondeur quasi-projective d’un caractère. A travers l’étude de cet algorithme, nous avons obtenu une condition combinatoire pour admettre une profondeur quasi-projective potentiellement non combinatoire. Nous avons ainsi défini la notion de caractère inner-cyclic . Cette notion nous a permis de formuler des conditions fortes sur la combinatoire pour qu’un arrangement n’ait que des caractères de profondeur quasi-projective nulle. Enfin pour diminuer le nombre d’exemples à considérer nous avons introduit la notion de combinatoire première. Si une combinatoire ne l’est pas, alors les variétés caractéristiques de ses réalisations sont définies par celles d’un arrangement avec moins de droites. En parallèle à cette étude, nous avons observé que la composition de l’application induite par l’inclusion sur le premier groupe d’homologie avec un caractère nous fournit un invariant topologique de l'arrangement obtenu en désingularisant les points multiples (blow-up). De plus, nous montrons que cet invariant n’est pas de nature combinatoire. Il nous a ainsi permis de découvrir deux nouvelles nc-paires de ZariskiThis thesis is the intersection point between the two facets of the study of line arrangements: combinatorics and topology. In the first part, we study the inclusion of the boundary manifold in the complement of an arrangement. We generalize the results of E. Hironaka to the case of any complex line arrangement. To get around the problems due to the case of non complexified real arrangement, we study the braided wiring diagram. We develop a Sage program to compute it from the equation of the complex line arrangement. This diagram allows to give two explicit descriptions of the map induced by the inclusion on the fundamental groups. From theses descriptions, we obtain two new presentations of the fundamental group of the complement. One of them is a generalization of the R. Randell Theorem to any complex line arrangement. In the next step of this work, we study the map induced by the inclusion on the first homology group. Then we obtain two simple descriptions of this map. Inspired by ideas of J.I. Cogolludo, we give a canonical description of the homology of the boundary manifold as the product of the 1-homology with the 2-cohomology of the complement. Finally, we obtain an isomorphism between the 2-cohomology of the complement with the 1-homology of the incidence graph of the arrangement. In the second part, we are interested by the study of character on the group of the complement. We start from the results of E. Artal on the computation of the depth of a character. This depth can be decomposed into a projective term and a quasi-projective term, vanishing for characters that ramify along all the lines. An algorithm to compute the projective part is given by A. Libgober. E. Artal focuses on the quasi-projective part and gives a method to compute it from the image by the character of certain cycles of the complement. We use our results on the inclusion map of the boundary manifold to determine these cycles explicitly. Combined with the work of E. Artal we obtain an algorithm to compute the quasi-projective depth of any character. From the study of this algorithm, we obtain a strong combinatorial condition on characters to admit a quasi-projective depth potentially not determined by the combinatorics. With this property, we define the inner-cyclic characters. From their study, we observe a strong condition on the combinatorics of an arrangement to have only characters with null quasi-projective depth. Related to this, in order to reduce the number of computations, we introduce the notion of prime combinatorics. If a combinatorics is not prime, then the characteristics varieties of its realizations are completely determined by realization of a prime combinatorics with less line. In parallel, we observe that the composition of the map induced by the inclusion with specific characters provide topological invariants of the blow-up of arrangements. We show that the invariant captures more than combinatorial information. Thereby, we detect two new examples of nc-Zariski pairs
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Sweeney, Andrew. "A Study of Topological Invariants in the Braid Group B2." Digital Commons @ East Tennessee State University, 2018. https://dc.etsu.edu/etd/3407.
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The Jones polynomial is a special topological invariant in the field of Knot Theory. Created by Vaughn Jones, in the year 1984, it is used to study when links in space are topologically different and when they are topologically equivalent. This thesis discusses the Jones polynomial in depth as well as determines a general form for the closure of any braid in the braid group B2 where the closure is a knot. This derivation is facilitated by the help of the Temperley-Lieb algebra as well as with tools from the field of Abstract Algebra. In general, the Artin braid group Bn is the set of braids on n strands along with the binary operation of concatenation. This thesis also shows results of the relationship between the closure of a product of braids in B2 and the connected sum of the closure of braids in B2. Results on the topological invariant of tricolorability of closed braids in B2 and (2,n) torus links along with their obverses are presented as well.
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Newman-Gomez, Sharon Angela. "State sum invariants of three manifolds." CSUSB ScholarWorks, 1998. https://scholarworks.lib.csusb.edu/etd-project/1510.
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Leung, Wai-Man Raymond. "On spin c-invariants of four-manifolds." Thesis, University of Oxford, 1995. http://ora.ox.ac.uk/objects/uuid:a9790f36-748f-4574-a97c-4f416ca67207.
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The spinc-invariants for a compact smooth simply-connected oriented four-manifold, as defined by Pidstrigach and Tyurin, are studied in this thesis. Unlike the Donaldson polynomial invariants, they are defined by cutting down the moduli space M' of '1-instantons', which is the subspace of the moduli space M of anti-self-dual connections parametrizing coupled (spinc) Dirac operators with non-trivial kernel. Our main goal is to study the relationship between these spinc-invariants and the Donaldson polynomial invariants. The 'jumping subset' M' defined a cohomology class P of M which is given by the generalised Porteous formula. When the index l of the coupled Dirac operator is 1, the two smooth invariants are the same by definition. When l = 0 (or when M is compact), the spinc-invariants are expressable as a Donaldson polynomial evaluating the 'Porteous class' P. Our main results concern the first two non-trivial cases l = -1 and -2, when the generalised Porteous formula can not be applied directly. Using cut-and-paste arguments to the moduli space M, we show that for the former case the spinc-invariants and the contracted Donaldson invariants differ by a correction term. It is the number of points in the immediate lower stratum of the Uhlenbeck compactification times a universal 'linking invariant' on S4, which is obtained by computing an example (the K3 surface). The case when l = -2 and dimM = 8 is a parametrized version of the l = -1 situation and the correction term, which involves the same 'linking invariant', is obtained from a suitable obstruction theory.
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Atala, Marcos. "Measuring topological invariants and chiral Meissner currents with ultracold bosonic atoms." Diss., Ludwig-Maximilians-Universität München, 2014. http://nbn-resolving.de/urn:nbn:de:bvb:19-177350.
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Topologische Invarianten sind von zentraler Bedeutung für die Interpretation vieler Phänomene kondensierter Materie. In dieser Arbeit wird die erste Messung einer solchen Invarianten vorgestellt. Dazu wird ein neu entwickeltes Messprotokoll mit ultrakalten bosonischen Atomen in einem eindimensionalen optischen Gitter verwendet. Außerdem wird die Messung chiraler Meissner-Ströme in einer Leitergeometrie in einem künstlichen Magnetfeld sowie die Präparation sogenannter "Resonating Valence Bond"-Zustände (RVB) in vier Gitterplätze umfassenden Plaketten präsentiert. Das Hauptmerkmal des experimentellen Aufbaus ist ein Paar orthogonaler Übergitter-Potentiale, die es ermöglichen eine Vielzahl verschiedener Systeme zu simulieren. Die Modulation des Übergitters mit einem weiteren Paar interferierender Strahlen ermöglicht zu dem die Realisierung eines künstlichen Magnetfelds.
Die Zak-Phase ist eine Invariante, welche die topologischen Eigenschaften eines Energiebandes charakterisiert. Sie ist definiert als die Berry-Phase eines Teilchens bei adiabatischem Durchlaufen eines Pfades im Quasiimpulsraum durch die Brillouinzone. Ein einfaches Beispiel für ein System mit zwei verschiedenen topologischen Klassen ist eine eindimensionale Kette mit alternierender Tunnelkopplungsstärke. Im Experiment können diese Klassen durch Messung der Differenz zwischen ihren Zak-Phasen $\Deta\Phi_\text{Zak}\approx\pi$ unter Verwendung von Bloch-Oszillationen und Ramsey-Interferometrie in Übergittern unterschieden werden.
Der zweite Teil dieser Arbeit befasst sich mit der Messung chiraler Meissner-Ströme von Bosonen in einer Leitergeometrie mit magnetischem Fluss, welche eines der einfachsten Modelle zur Beobachtung von Orbitaleffekten ist. Obwohl die Atome ladungsneutral sind und daher keine Lorentzkraft auf sie wirkt, kann durch eine externe Modulation im Übergitter ein künstliches Magnetfeld erzeugt werden. Die dadurch hervorgerufenen Wahrscheinlichkeitsströme auf beiden Seiten der Leiter wurden separat mit einer Projektionsmethode gemessen. Beim Ändern der Tunnelkopplung entlang der Leitersprossen wurde, in Analogie zu einem Typ-II Supraleiter, ein Übergang zwischen einer Meissner-artigen Phase mit gesättigtem maximalen chiralen Strom und einer Vortex-Phase mit abnehmendem Strom beobachtet. Dieses System mit ultrakalten Atomen kann auch als Analogon zur Spin-Bahn-Kopplung betrachtet werden.
RVB-Zustände gelten als fundamental für das Verständnis von Hochtemperatursupraleitern. Der dritte Teil der Arbeit widmet sich mit der Realisierung eines Minimalbeispiels solcher Zustände auf einer Plakette bei halber Füllung. In diesem System wurden die zwei RVB-Zustände mit s- und d-Wellen-Symmetrie sowie Superpositionen der beiden Zustände präpariert.
Die in dieser Arbeit vorgestellten Experimente stellen einen neuen Ansatz dar, die topologischen Eigenschaften von Bloch-Bändern in optischen Gittern zu untersuchen; sie öffnen die Türen zur Erforschung von wechselwirkenden Teilchen in niedrigdimensionalen Systemen in einem homogenen Magnetfeld sowie der Eigenschaften des Grundzustandes des Heisenberg-Modells.The determination of topological invariants is of fundamental importance to interpret many condensed-matter phenomena. This thesis reports on the implementation of a newly developed protocol to measure these invariants for the first time, using ultracold bosonic atoms in one-dimensional optical lattices. In addition, it deals with the measurement of chiral Meissner currents in a ladder-like lattice geometry exposed to an artificial magnetic field, and presents results on the preparation of Resonating Valence Bond (RVB) states on plaquettes. The key feature of the experimental setup is a pair of orthogonal superlattice potentials that permit a rich variety of systems to be simulated, and that when combined with a pair of interfering beams which periodically modulate the lattice allow the realization of artificial magnetic fields. The Zak phase is an invariant that characterizes the topological properties of an energy band, and is defined as the Berry phase that a particle acquires as it adiabatically moves in the quasimomentum space across the Brillouin zone. A dimerized lattice -- a one-dimensional chain with alternating couplings -- is a simple example of a system that possesses two different topological classes. Using a combination of Bloch oscillations and Ramsey interferometry in superlattices we measured the difference of the Zak phase $\approx\pi$ for the two possible polyacetylene phases, which directly indicates that they belong to different topological classes. The second part of this thesis deals with the measurement of chiral Meissner currents in bosonic ladders with magnetic flux, one of the simplest models to observe orbital effects. Although charge neutrality prevents atoms from experiencing the Lorentz force when they are exposed to a magnetic field, employing lattice modulation techniques we implemented an artificial magnetic field on a ladder created with optical lattices. By using a projection technique, we were able to measure the probability currents on each side of the ladder. When changing the coupling strengths along the rungs of the ladder, we found, in analogy to type-II superconductors, a transition between a Meissner-like phase with saturated maximum chiral current and a vortex phase with decreasing currents. Additionally, the flux ladder realizes spin-orbit coupling with ultracold atoms. It is believed that RVB states are fundamental for the understanding of high-T$_c$ superconductivity. The third part of this work describes our measurements on the preparation of minimum instances of RVB states with bosonic atoms in isolated four-site plaquettes at half filling. These small systems possess two RVB states with $s$- and $d$-wave symmetry. Using atom manipulation techniques we prepared these two states, as well as a quantum resonance between them. The experiments in this thesis establish a new general approach for probing the topological structure of Bloch bands in optical lattices. Moreover, they open up the pathway to exploring interacting particles in low dimensions exposed to uniform magnetic fields and to studying ground state properties of the Heisenberg Hamiltonian.
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Araujo, Hamilton Regis Menezes de. "Sobre invariantes topológicos de folheações holomorfas com singularidade isolada." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/22884.
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ARAUJO, H. R. M. Sobre invariantes topológicos de folheações holomorfas com singularidade isolada. 2017. 62 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará,
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Considering the foliation induced by a complex holomorph vector field, we will look for topological invariants in the neighborhood of a singular point. At first, the Milnor Number of a vector field becomes important, in the sense that this number is topological invariant. In another discussion, we will emphasize vector fields in dimension two, in which case the leaves, whose foliation is induced by the field, will be integral curves of a 1-form. In this sense, we will deal with Desingularization, that is, after a finite number of processes, which we will call Blow-ups or explosions, we will turn the initial foliation into a foliation whose singularities are all simple. Finally, the Desingularization process of a field will give us tools that make it possible to relate the data obtained in this process to the objects treated throughout the work, with this we will present other topological invariants of foliations.
Considerando a folheação induzida por um campo vetorial complexo holomorfo, buscaremos exibir invariantes topológicos na vizinhança de um ponto singular. Num primeiro momento, ganha importância o Número de Milnor de um campo vetorial, no sentido desse número ser invariante topológico. Em outra discussão, daremos ênfase a campos vetoriais em dimensão dois, nesse caso, as folhas, cuja folheação é induzida pelo campo, serão curvas integrais de uma 1-forma. Nesse sentido, trataremos de Desingularização, ou seja, após um número finito de processos, que chamaremos de Blow-ups, ou explosões, transformaremos a folheação inicial em uma folheação cujas singularidades são todas simples. Por fim, o processo de Desingularização de um campo nos dará ferramentas que possibilitam relacionar os dados obtidos nesse processo com os objetos tratados ao longo de todo o trabalho, diante disto apresentaremos outros invariantes topológicos de folheações.
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Upreti, Lavi Kumar. "Periodically driven photonic topological gapless systems." Thesis, Lyon, 2020. http://www.theses.fr/2020LYSEN017.
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Propriétés topologiques de systèmes photoniques non gappés modulés périodiquement. La photonique est une plate-forme où les ondes électromagnétiques (ou photons) se propagent à l'intérieur d'un cristal (comme les ondes de Bloch) formé par les degrés de liberté discrets sous-jacents, par exemple des réseaux de guides d'ondes. Ces ondes ne peuvent pas se propager si la fréquence incidente se situe dans la bande interdite photonique, alors ces ondes sont connues sous le nom d'ondes évanescentes. Ainsi, le cristal se comporte comme un réflecteur de ces ondes. Cependant, s'il existe des modes pour lesquels il existe des ondes limites qui relient la bande interdite, alors ces ondes peuvent exister à la limite sans s'infiltrer dans la masse. Ceci est analogue au mouvement chiral des électrons aux bords du Hall quantique, avec un ingrédient supplémentaire de symétrie d'inversion du temps qui se brise dans les cristaux photoniques via certaines propriétés gyromagnétiques de l'échantillon, ou la dépendance inhérente au temps du système. Dans ce dernier cas, lorsque le système, en particulier, est commandé périodiquement, on peut également observer les phases de non équilibre plus exotiques dans ces réseaux.Dans ce travail, nous explorons les propriétés topologiques de ces réseaux photoniques à commande périodique. Par exemple, comment les symétries fondamentales, par exemple la symétrie particule-trou, peuvent être mises en oeuvre pour concevoir la topologie en 1D. Nous trouvons un lien entre les symétries cristallines et les symétries fondamentales, qui facilitent une telle mise en oeuvre. De plus, une dimension synthétique peut être introduite dans ces treillis qui simulent la physique des dimensions supérieures. La différence entre la dimension synthétique et la dimension spatiale devient apparente lorsqu'une symétrie cristalline spécifique, comme l'inversion, est rompue dans ces systèmes. Cette rupture transforme une bande interdite directe en une bande interdite indirecte qui se manifeste par l'enroulement de bandes dans le spectre de la bande quasi-énergétique. Si elle est rompue dans la dimension synthétique, il en résulte une interaction de deux propriétés topologiques : l'une est l'enroulement des bandes de quasi-énergie, et l'autre est la présence d'états de bord chiraux dans la géométrie finie. Cette ancienne propriété de l'enroulement se manifeste par des oscillations de Bloch des paquets d'ondes, où nous montrons que les points stationnaires de ces oscillations sont liés au nombre d'enroulements des bandes. Cette propriété topologique peut donc être sondée directement dans une expérience par la technologie de pointe. Cependant, si cette symétrie est rompue dans la dimension spatiale, l'enroulement des bandes se manifeste comme une dérive quantifiée de la position moyenne, qui est toujours caractérisée par un nombre d'enroulement des bandes.En outre, nous montrons qu'un régime sans lacune différent peut également être conçu tout en préservant la symétrie d'inversion. Dans ce régime, la topologie peut être saisie en enfermant les dégénérescences dans l'espace des paramètres et en calculant le flux de Berry qui traverse la surface enfermée. Dans ce cas, certaines des dégénérescences peuvent héberger des états chiraux de bord avec d'autres protégés à la même quasi-énergiePhotonics has emerged a platform where electromagnetic waves (or photons) propagate inside a crystal (likeBloch waves) formed by the underlying discrete degrees of freedom, e.g., waveguide arrays. These waves cannotpropagate if the incident frequency lies within the so-called photonic bandgap, then these waves are known asevanescent waves. Thus, the crystal behaves as a reflector to these waves. However, if there are modes for whichthere exist boundary waves that connect the bandgap, then these waves can exist at the boundary without leakinginto the bulk. This is analogous to the chiral motion of electrons at the quantum Hall edges, with an extraingredient of time-reversal symmetry breaking in photonic crystals via some gyromagnetic properties of thesample, or inherent time dependence of the system. In the latter case, when the system, specifically, drivenperiodically then the more exotic non-equilibrium phases can also be observed in these lattices.In this work, we explore the topological properties in these periodically driven photonic lattices. For instance,how fundamental symmetries, e.g., particle-hole symmetry, can be implemented to engineer topology in 1D. Wefind a connection between crystalline symmetries and the fundamental symmetries, which facilitate suchimplementation. Moreover, a synthetic dimension can be introduced in these lattices that simulate higherdimensional physics. The difference between synthetic and spatial dimension becomes apparent when a specificcrystalline symmetry, like inversion, is broken in these systems. This breaking changes a direct bandgap to anindirect one which manifests in the winding of bands in the quasienergy band spectrum. If it is broken in thesynthetic dimension, it results in an interplay of two topological properties: one is the winding of the quasienergybands, and the other one is the presence of chiral edge states in the finite geometry. This former property ofwinding manifests as Bloch oscillations of wavepackets, where we show that the stationary points in theseoscillations are related to the winding number of the bands. This topological property can thus be probed directlyin an experiment by the state-of-art technology. However, if this symmetry is broken in the spatial dimension, thewinding of bands manifest as a quantized drift of mean position, which is still characterized by a winding numberof the bands.Furthermore, we show that a different gapless regime can also be engineered while preserving the inversionsymmetry. In this regime, the topology can be captured by enclosing the degeneracies in parameter space andcalculating the Berry flux piercing through the enclosed surface. In this case, some of the degeneracies can hostchiral edge states along with other protected ones at the same quasienergy
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Blumen, Sacha Carl. "Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds." University of Sydney. School of Mathematics and Statistics, 2005. http://hdl.handle.net/2123/715.
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The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N > = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N > = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N > = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.
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Yildirim, Tuna. "Topologically massive Yang-Mills theory and link invariants." Diss., University of Iowa, 2014. https://ir.uiowa.edu/etd/1519.
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In this thesis, topologically massive Yang-Mills theory is studied in the framework of geometric quantization. This theory has a mass gap that is proportional to the topological mass m. Thus, Yang-Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern-Simons theory with level number k. The focus of this research is the near Chern-Simons limit of the theory, where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of two copies of Chern-Simons with level number k/2, very similar to the Chern-Simons splitting of topologically massive AdS gravity model. As m approaches to infinity, the split parts add up to give the original Chern-Simons term with level k. Also, gauge invariance of the split CS theories is discussed for odd values of k. Furthermore, a relation between the observables of topologically massive Yang-Mills theory and Chern-Simons theory is obtained. It is shown that one of the two split Chern-Simons pieces is associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang-Mills theory observables in the near Chern-Simons limit. Finally, motivated with the topologically massive AdS gravity model, Chern-Simons splitting concept is extended to pure Yang-Mills theory at large distances. It is shown that pure Yang-Mills theory acts like two Chern-Simons theories with level numbers k/2 and -k/2 at large scales. At very large scales, these two terms cancel to make the theory trivial, as required by the existence of a mass gap.
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Schwer, Brad. "Characterizing topological spaces using topological or algebraic invariants a thesis presented to the faculty of the Graduate School, Tennessee Technological University /." Click to access online, 2008. http://proquest.umi.com/pqdweb?index=35&did=1679674331&SrchMode=1&sid=1&Fmt=6&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1254145518&clientId=28564.
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Sacramento, Andrea de Jesus [UNESP]. "Sobre a equivalência de contato topológica." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/94226.
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O objetivo deste trabalho é estudar a equivalência de contato topológica dos germes de aplicações diferenciáveis tendo como plano de fundo o estudo da equivalência de contato clássica (ou C∞-K-equivalência). Neste sentido, apresentamos inicialmente uma análise detalhada sobre alguns invariantes e propriedades clássicas da equivalência de contato e, em seguida, introduzimos o estudo da versão topológica desta relação de equivalência. A equivalência de contato topológica (ou C0-K-equivalência) é um tema que recentemente ganhou o interesse de vários pesquisadores por se tratar de uma relação de equivalência cujos invariantes, propriedades e classi cações são pouco conhecidos ou inexistentes. Sob esta ótica, investigamos se alguns invariantes encontrados no caso clássico poderiam ser reproduzidos ou adaptados para o caso topológico. Como parte principal do trabalho, apresentaremos um invariante completo para a equivalência de contato topológica introduzido por T. Nishimura [22]. Este invariante é dado para germes de aplicações nitamente determinadas cujas dimensões da fonte e da meta coincidem
The goal of this work is to study the topological contact equivalence of smooth map germs having as background the study of the classical contact equivalence (or C∞-Kequivalence). In this sense, we rstly present a detailed analysis of some invariants and classical properties of the contact equivalence, and then we introduce the study of the topological version of this equivalence relation. Recently several researchers have been interested in this subject because it is an equivalence relation whose invariants, properties and classi cations are unknown or nonexistent. In this work we investigate if some invariants of contact equivalence could be reproduced or adapted for the topological case. In chapter 3 we present a complete invariant for the topological contact equivalence introduced by T. Nishimura [22]. This invariant is given to nitely determined map germs whose dimensions of the source and target are equal
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Atala, Marcos [Verfasser], and Immanuel [Akademischer Betreuer] Bloch. "Measuring topological invariants and chiral Meissner currents with ultracold bosonic atoms / Marcos Atala. Betreuer: Immanuel Bloch." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2014. http://d-nb.info/1065610068/34.
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Blanc, Anthony. "Invariants topologiques des espaces non-commutatifs." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2013. http://tel.archives-ouvertes.fr/tel-01012109.
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Dans cette thèse, on donne une définition de la K-théorie topologique des espaces non-commutatifs de Kontsevich (c'est-à-dire des dg-catégories) définis sur les nombres complexes. L'introduction de ce nouvel invariant initie la recherche des invariants de nature topologique des espaces non-commutatifs, comme "simplifications" des invariants algébriques (K-théorie algébrique, homologie cyclique, périodique comme étudiés dans les travaux de Tsygan, Keller). La motivation principale vient de la théorie de Hodge non-commutative au sens de Katzarkov--Kontsevich--Pantev. En géométrie algébrique, la partie rationnelle de la structure de Hodge est donnée par la cohomologie de Betti rationnelle, qui est la cohomologie rationnelle de l'espace des points complexes du schéma. La recherche d'un espace associé à une dg-catégorie trouve une première réponse avec le champ (défini par Toën--Vaquié) classifiant les dg-modules parfaits sur cette dg-catégorie. La définition de la K-théorie topologique a pour ingrédient essentiel le foncteur de réalisation topologique des préfaisceaux en spectres sur le site des schémas de type fini sur les complexes. La partie connective de la K-théorie semi-topologique peut être définie comme la réalisation topologique du champ en monoïdes commutatifs des dg-modules parfaits. Cependant pour atteindre la K-théorie négative, on réalise le préfaisceau donné par la K-théorie algébrique non-connective. Un de nos résultats principaux énonce l'existence d'une équivalence naturelle entre ces deux définitions dans le cas connectif. On montre que la réalisation topologique du préfaisceau de K-théorie algébrique connective pour la dg-catégorie unité donne le spectre de K-théorie topologique usuel. Puis que c'est aussi vrai pour la K-théorie algébrique non-connective, en utilisant la propriété de restriction aux lisses de la réalisation topologique. En outre, cette propriété de restriction aux schémas lisses nécessite de montrer une généralisation de la descente propre cohomologique de Deligne, dans le cadre homotopique non-abélien.La K-théorie topologique est alors définie en localisant par rapport à l'élément de Bott. Cette définition repose donc sur des résultats non-triviaux. On montre alors que le caractère de Chern de la K-théorie algébrique vers l'homologie périodique se factorise par la K-théorie topologique, donnant un candidat naturel pour la partie rationnelle d'une structure de Hodge non-commutative sur l'homologie périodique, ceci étant énoncé sous la forme de la conjecture du réseau. Notre premier résultat de comparaison concerne le cas d'un schéma lisse de type fini sur les complexes -- la conjecture du réseau est alors vraie pour de tels schémas. On montre ensuite que cette conjecture est vraie dans le cas des algèbres associatives de dimension finie.
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Riba, Garcia Ricard. "Trivial 2-cocycles for invariants of mod p homology spheres and Perron's conjecture." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/664243.
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L'objectiu principal d'aquesta tesis és l'estudi de la conjectura de Perron.
Aquesta conjectura afirma que certa funcio sobre el grup de Torelli mod p amb valors en Z/p és un invariant d'esferes d'homologia modul p.
Per tal d'abordar l'estudi d'aquesta conjectura, en aquesta tesis primer estudiem les esferes d'homologia modul p, les esferes d'homologia racional,
i donem un criteri per determinar quan una esfera d'homologia racional té un split de Heegaard amb aplicació d'enganxament un element del grup de Torelli mod p, el cual ve donat pel nucli de la representació Symplectica modul p del mapping class group.
A continuació estenem els resultats de l'article ''Trivial cocycles and invariants of homology 3-spheres'' obtenint una construcció d'invariants amb valors a un grup abelià sense restriccions, a partir d'una família adecuada de 2-cocycles sobre el grup de Torelli.
En particular, expliquem la influencia de l'invariant de Rohlin en la perdua de la unicitat en tal construcció.
Posteriorment, utilitzant les mateixes eines, obtenim una construcció d'invariants esferes d'homologia racional que tenen un split de Heegaard amb aplicació d'enganxament un element del grup de Torelli mod p, a partir d'una família adecuada de 2-cocycles sobre el grup de Torelli modul p.
A més, al llarg d'aquesta construcció obtenim un invariant d'esfers d'homologia modul p el qual no apareix en la literatura.
Finalment, demostrem que la conjectura de Perrron és falsa donant una obstrucció que be donada pel fet que la primera classe caracteristica dels fibrats de superfícies reduida modul p no és nula.The main objective of this thesis is the study Perron's conjecture. This conjecture affirms that some function on the group of Torelli mod p, with values in Z/p, is an invariant of mod p homology spheres. In order to study this conjecture, in this thesis we first study the mod p homology spheres, the rational homology spheres and we give a criterion to determine whenever a rational homology sphere has a Heegaard splitting with gluing map an element of the Torelli group mod p, which is the group given by the kernel of the Symplectic representation modulo p of the mapping class group. Next, we extend the results of the article ''Trivial cocycles and invariants of homology 3-spheres'' obtaining a construction of invariants with values to an abelian group without restrictions, from a suitable family of 2-cocycles on the Torelli group. In particular, we explain the influence of the invariant of Rohlin in the lost of uniqueness in such construction. Later, using the same tools, we obtain a construction of invariants of rational homology spheres that have a Heegaard splitting with gluing map an element of the mod p Torelli group, from a suitable family of 2-coccycles on the mod p Torelli group. In addition, throughout this construction we obtain an invariant of mod p homology spheres which does not appear in the literature. Finally, we prove that Perrron's conjecture is false providing an obstruction that is given by the fact that the first characteristic class of surface bundles reduced modulo p does not vanish.
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Boros, Dan. "On ℓ2-homology of low dimensional buildings." Columbus, Ohio : Ohio State University, 2003. http://rave.ohiolink.edu/etdc/view?acc%5fnum=osu1062707630.
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Thesis (Ph. D.)--Ohio State University, 2003.Title from first page of PDF file. Document formatted into pages; contains vi, 77 p. Includes abstract and vita. Advisor: Michael Davis, Dept. of Mathematics. Includes bibliographical references (p. 75-77).
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Lamberti, Fabrice-Roland. "Opto-phononic confinement in GaAs/AlAs-based resonators." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC103/document.
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Ces travaux de thèse portent sur la conception et sur la caractérisation expérimentale de résonateurs opto-phononiques. Ces structures permettent le confinement simultané de modes optiques et de vibrations mécaniques de très haute fréquence (plusieurs dizaines jusqu’à plusieurs centaines de GHz). Cette étude a été effectuée sur des systèmes multicouches à l’échelle nanométrique, fabriqués à partir de matériaux semiconducteurs de type III-V. Ces derniers ont été caractérisés par des mesures de spectroscopie Raman de haute résolution. Grâce aux méthodes expérimentales et aux outils numériques développés, nous avons pu explorer de nouvelles stratégies de confinement pour des phonons acoustiques au sein de super-réseaux nanophononiques, à des fréquences de résonance de l’ordre de 350 GHz. En particulier, nous avons étudié les propriétés acoustiques de deux types de résonateurs planaires. Le premier est basé sur la modification adiabatique du diagramme de bande d’un cristal phononique unidimensionnel. Dans le deuxième système, nous utilisons les invariants topologiques caractérisant ces structures périodiques, afin de créer un état d’interface entre deux miroirs de Bragg phononiques. Nous nous sommes ensuite intéressés à l’étude de cavités opto-phononiques permettant le confinement tridimensionnel de la lumière et de vibrations mécaniques de haute fréquence. Nous avons mesuré par spectroscopie Raman les propriétés acoustiques de résonateurs phononiques planaires placés à l’intérieur de cavités optiques tridimensionnelles, de type micropiliers. Enfin, la dernière partie de cette thèse porte sur l’étude théorique des propriétés optomécaniques de micropiliers GaAs/AlAs. Nous avons effectué des simulations numériques par éléments finis, nous permettant d’expliquer les mécanismes de confinement tridimensionnel de modes acoustiques et optiques dans ces systèmes, et de calculer les principaux paramètres optomécaniques. Les résultats de cette étude démontrent que les micropilier GaAs/AlAs possèdent des caractéristiques prometteuses pour de futures expériences en optomécanique, telles que des fréquences de résonance acoustiques très élevées, de hauts facteurs de qualités mécaniques et optiques à température ambiante, ou encore de fortes valeurs pour les facteurs de couplage optomécaniques et pour le produit Q • fThe work carried out in this thesis addresses the conception and the experimental characterization of opto-phononic resonators. These structures enable the confinement of optical modes and mechanical vibrations at very high frequencies (from few tens up to few hundreds of GHz). This study has been carried out on multilayered nanometric systems, fabricated from III-V semiconductor materials. These nanophononic platforms have been characterized through high resolution Raman scattering measurements. The experimental methods and the numerical tools that we have developed in this thesis have allowed us to explore novel confinement strategies for acoustic phonons in acoustic superlattices, with resonance frequencies around 350 GHz. In particular, we have studied the acoustic properties of two nanophononic resonators. The first acoustic cavity proposed in this manuscript enables the confinement of mechanical vibrations by adiabatically changing the acoustic band-diagram of a one-dimensional phononic crystal. In the second system, we take advantage of the topological invariants characterizing one dimensional periodic structures, in order to create an interface state between two phononic distributed Bragg reflectors. We have then focused on the study of opto-phononic cavities allowing the simultaneous confinement of light and of high frequency mechanical vibrations. We have measured, by Raman scattering spectroscopy, the acoustic properties of planar nanophononic structures embedded in three-dimensional micropillar optical resonators. Finally, in the last sections of this manuscript, we investigate the optomechanical properties of GaAs/AlAs micropillar cavities. We have performed numerical simulations through the finite element method that allowed us to explain the three-dimensional confinement mechanisms of optical and mechanical modes in these systems, and to calculate the main optomechanical parameters. This work shows that GaAs/AlAs micropillars present very interesting properties for future optomechanical experiments, such as very high mechanical resonance frequencies, large optical and mechanical quality factors at room temperature, and high values for the vacuum optomechanical coupling factors and for the Q • f products
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De, Renzi Marco. "Construction of extended topological quantum field theories." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC114/document.
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La position centrale occupée par les Théories Quantiques des Champs Topologiques (TQFTs) dans l’étude de la topologie en basse dimension est due à leur structure extraordinairement riche, qui permet différentes interactions et applications à des questions de nature géométrique. Depuis leur première apparition, un grand effort a été mis dans l’extension des invariants quantiques de 3-variétés en TQFTs et en TQFT Étendues (ETQFTs). Cette thèse s’attaque à ce problème dans deux cadres généraux différents. Le premier est l’étude des invariants quantiques semi-simples de Witten, Reshetikhin et Turaev issus de catégories modulaires. Bien que les ETQFTs correspondantes étaient connues depuis un certain temps, une réalisation explicite basée sur la construction universelle de Blanchet, Habegger, Masbaum et Vogel apparaît ici pour la première fois. L’objectif est de tracer la route à suivre dans la deuxième partie de la thèse, où la même procédure est appliquée à une nouvelle famille d’invariants quantiques non semi-simples due à Costantino, Geer et Patureau. Ces invariants avaient déjà été étendus en TQFTs graduées par Blanchet, Costantino, Geer and Patureau, mais seulement pour une famille explicite d’exemples. Nous posons la première pierre en introduisant la définition de catégorie modulaire relative, un analogue non semi-simple aux catégories modulaires. Ensuite, nous affinons la construction universelle pour obtenir des ETQFTs graduées étendant à la fois les invariants quantiques de Costantino, Geer et Patureau et les TQFTs graduées de Blanchet, Costantino, Geer et Patureau dans ce cadre généralThe central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting
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Gäfvert, Oliver. "Algorithms for Multidimensional Persistence." Thesis, KTH, Matematik (Avd.), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-188849.
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The theory of multidimensional persistence was introduced in a paper by G. Carlsson and A. Zomorodian as an extension to persistent homology. The central object in multidimensional persistence is the persistence module, which represents the homology of a multi filtered space. In this thesis, a novel algorithm for computing the persistence module is described in the case where the homology is computed with coefficients in a field. An algorithm for computing the feature counting invariant, introduced by Chachólski et al., is investigated. It is shown that its computation is in general NP-hard, but some special cases for which it can be computed efficiently are presented. In addition, a generalization of the barcode for persistent homology is defined and conditions for when it can be constructed uniquely are studied. Finally, a new topology is investigated, defined for fields of characteristic zero which, via the feature counting invariant, leads to a unique denoising of a tame and compact functor.Teorin om multidimensionell persistens introduserades i en artikel av G. Carlsson och A. Zomorodian som en generalisering av persistent homologi. Det centrala objektet i multidimensionell persistens är persistensmodulen, som representerar homologin av ett multifilterat rum. I denna uppsats beskrivs en ny algoritm för beräkning av persistensmodulen i fallet där homologin beräknas med koefficienter i en kropp. En algoritm för beräkning av karaktäristik-räknings-invarianten, som introducerade av Chachólski et al., utforskas och det visar sig att dess beräkning i allmänhet är NP-svår. Några specialfall för vilka den kan beräknas effektivt presenteras. Vidare definieras en generalisering av stäckkoden för persistent homologi och kraven för när den kan konstrueras unikt studeras. Slutligen undersöks en ny topologi, definierad för kroppar av karaktäristik noll, som via karaktäristik-räknings-invarianten leder till en unik avbränning.
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Robert, Gilles. "Invariants topologiques et géométriques reliés aux longueurs des géodésiques et aux sections harmoniques de fibrés." Université Joseph Fourier (Grenoble), 1994. http://www.theses.fr/1994GRE10185.
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Cette these est composee de deux parties independantes. Le but poursuivi dans la premiere partie consiste a etudier differents invariants dynamiques lies au flot geodesique de maniere purement geometrique, afin de mieux apprehender leurs liens vis-a-vis de la metrique riemannienne. Un theoreme de fonctions implicites permet ainsi de calculer la derivee seconde par rapport a la metrique, au sens des espaces de banach, de la longueur de la geodesique minimisante dans une classe d'homotopie fixee. Une propriete de convexite pour l'entropie volumique par rapport a la metrique donne une preuve tres simple d'une inegalite due a a. Katok. Une generalisation de cette notion d'entropie volumique a un espace metrique quelconque permet de considerer en particulier le cas des groupes discrets, retrouvant certains resultats dus a m. Gromov. La seconde partie est le texte d'un article en commun avec m. Le couturier, dans lequel nous etablissons une inegalite de harnack pour les solutions d'une equation de schrodinger sur un fibre vectoriel riemannien dont la base est une variete compacte. Cette inegalite donne des conditions integrales sur le potentiel, sous lesquelles les solutions ne s'annulent pas. Des applications au cas des formes harmoniques, des champs de killing mais egalement dans le cadre des varietes kahleriennes, sont developpees
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Sarmiento, Ingrid Sofia Meza. "A topologia de folheações e sistemas integráveis Morse-Bott em superfícies." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11012016-112023/.
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Nesta tese estudamos os sistemas integráveis definidos em superfícies compactas possuindo uma integral primeira que é uma função Morse-Bott a valores em R. Estes sistemas são aqui chamados de sistemas integráveis Morse-Bott. Classificamos as curvas fechadas e oitos associados a pontos de selas imersos em superfícies compactas. Essa classificação é aplicada ao estudo das folheações Morse-Bott em superfícies e nos permite definir um invariante topológico completo para a classificação topológica global destas folheações. Como uma aplicação desse estudo obtemos a classificação dos sistemas Morse-Bott assim como a classificação topológica das funções Morse-Bott em superfícies compactas e orientáveis. Demonstramos ainda um teorema da realização baseado em duas transformações e numa folheação geradora. Para o caso das funções Morse-Bott também obtivemos um teorema de realização. Finalmente, investigamos a generalização de alguns dos resultados anteriores para sistemas definidos em superfícies não orientáveis.In this thesis we study integrable systems on compact surfaces with a first integral as a Morse-Bott function with target R. These systems are called here integrable Morse-Bott systems. Initially we present the classification of closed curves and eights associated to saddle points on compact surfaces. This classification is applied to the study of Morse- Bott foliations on surfaces allowing us to define a complete topological invariant for the global topological classification of these foliations. Then as an application of this study we obtain the classification of integrable Morse-Bott systems as well as the topological classification of Morse-Bott functions on compact and orientable surfaces. We also prove a realization theorem based on two transformation and a generating foliation (the foliation on the sphere with two centers). In the case of Morse-Bott functions we also obtain a realization theorem. Finally we investigate generalizations of previous results for systems defined on non-orientable surfaces.
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Baptista, Diogo Pedro Ferreira Nascimento. "Iteradas de aplicações do plano no plano." Doctoral thesis, Universidade de Évora, 2008. http://hdl.handle.net/10174/12257.
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Neste trabalho estudamos as iteradas de aplicações do plano no plano. Usando as técnicas da dinâmica simbólica em aplicações do plano no plano, tendo sempre por base a teoria de amassamento de Milnor e Thurston e o formalismo da dinâmica simbólica desenvolvido por Sousa Ramos, abordamos diferentes aspectos qualitativos da dinâmica das aplicações de Lozi.
Assim, através da dinâmica simbólica introduzida por Yutaka Ishii, começamos por refor-mular a fronteira do espaço dos parâmetros correspondente às aplicações de Lozi equivalentes à ferradura de Smale. No seguimento, apresentamos um método que permite a construção da bacia de atracção para o atractor de uma qualquer aplicação de Lozi.
Ainda usando a dinâmica simbólica para as aplicações de Lozi, apresentamos um método que fazendo uso de expansões em fracções contínuas, nos permite calcular o maior dos expoentes de Lyapunov de uma aplicação de Lozi.
Com a introdução do conceito de ponto crítico e subsequentemente de sequência de amassamento para as aplicações de Lozi, partimos para uma a construção de uma partição de Markov do seu espaço de fases. Desse modo, é possível a caracterização completa do espaço dos parâmetros através da introdução do conceito de curva de amassamento, que mostramos serem curvas isentrópicas. Consequentemente, obtemos a descrição em termos da entropia topológica da família das aplicações de Lozi.
### Abstract - In this work, we study the iterations of two dimensional maps. Using symbolic dynamics techniques for two dimensional maps, based on both the kneading theory of Milnor and Thurston and the formalism of symbolic dynamics developed by Sousa Ramos, we studied the qualitative aspects of the dynamics of Lozi maps.
Thus, through the symbolic dynamics introduced by Yutaka Ishii, through the correction of symbolic sequence that characterized the first tangency between stable and unstable manifolds, we reformulate the border for the Smale horseshoes. Following this work, we present a method that allows the construction of the basin of attraction for the Lozi attractor. Even using the symbolic dynamics, we introduce a new method, using continuous fractions expansions that allow us to compute the largest Lyapunov exponent.
Through the kneading sequence for Lozi map, we characterize the region in the parameter space were we have the kneading curves and we also give a method to the construction of a partition of Markov for the Lozi attractors. Consequently we characterize the topological entropy for the Lozi map, and costruct a new topological invariant, the second invariant.
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Dong, Zhaoyang. "Topological attractors of quasi-periodically forced one-dimensional maps." Doctoral thesis, Universitat Autònoma de Barcelona, 2019. http://hdl.handle.net/10803/666774.
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This thesis is concerned with topological attractors of some quasi-periodically
forced one-dimensional maps. The main aim of our study is to under-
stand the states of the attractors by analyzing the mechanisms which rule
the dynamics of the maps. Concretely we investigate two types of quasi-
periodically forced one-dimensional families. The first type consists of two
di erent quasi-periodically forced increasing systems. We present rigorous
proofs for the states of their attractors. The second type of systems that we
consider are those quasi-periodically forced S-unimodal systems. We pro-
pose the mechanism for their changes of periodicity according to the forced
terms, which is based on elaborate analysis of the S-unimodal maps and is
substantiated by numerical evidence.
The motivation of our research is the problem of Strange Nonchaotic
Attractors. we rst explore shortly the general topological properties of the
pinched closed invariant sets, which are of particular important for SNAs.
We prove that, the !-limit set of pinched points is the unique minimal set
in a pinched closed invariant set, and any continuous graph contained in a
pinched closed invariant set must be invariant. Our first main result shows
that, in a pinched system the !-limit set of pinched points is the only mini-
mal set which must be contained in every invariant sets. We prove rigorously
the states of attractors with respect to two parameters of two families which
are forced monotone increasing maps, with some concave or convex struc-
tures on bre maps.
The last chapter is devoted to the quasi-periodically forced S-unimodal
maps. In this chapter we propose the mechanism of the change of the pe-
riodicity of their attractors, which works with respect to a parameter who
controls the forcing term when the forced S-unimodal map is fixed. This
mechanism of forced systems is based on the topological structures of the
forced S-uniform maps. The similar merging and collision are also reported
for S-unimodal maps in physical context for decades, we prove this is true
later in this chapter. The facts that we present in the proof reveal the topo-
logical structures that rule the change of periodicity of the forced S-unimodal
systems.
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Moncao, de Carvalho Santana Hellen. "Euler obstruction and generalizations." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0641.
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Soit f, g : (X, 0)→ (C, 0) des germes des fonctions analytiques définies sur un espace analytique complexe X. Le nombre de Brasselet d’une fonction f décrit numériquement la topologie de la fibre de Milnor généralisée. Dans cette thèse, nous présentons des formules qui compare les nombres de Brasselet de f dans X et de f restreinte à X ∩ { g=0 } dans le cas où g a un ensemble critique stratifié de dimension un. Si, en plus, f a une singularité isolée à l’origine, nous déterminons le nombre de Brasselet de g dans X et nous le mettons en relation avec le nombre de Brasselet de f dans X. Par conséquence, nous obtenons des formules qui permet mesurer l’obstruction locale d’Euler de X e de X {g = 0} à l’origine, en comparant ces nombres avec des invariants locales associés à f et à g. Nous étudions aussi la topologie locale d’une déformation de g, {g} = g+f N, où N>>1. Nous donnons une relation des nombres de Brasselet de g et {g} dans X ∩ {f = 0}, dans le cas où f a une singularité isolée à l’origine. Nous présentons encore une nouvelle preuve pour la formule de Lê-Iomdine pour le nombre de BrasseletLet f, g : (X, 0) → (C, 0) be germs of analytic functions defined over a complex analyticspace X. The Brasselet number of a function f describes numerically the topology of its generalized Milnor fibre. In this thesis, we present formulas to compare the Brasselet numbers of f in X and of the restriction of f to X ∩ { g = 0 }, in the case where g has a one-dimensional stratified critical set and f has an arbitrary critical set. If, additionally, f has isolated singularity at the origin, we compute the Brasselet number of g in X and compare it with the Brasselet number of f in X. As a consequence, we obtain formulas to compute the local Euler obstruction of X and of X { g = 0 } at the origin, comparing these numbers with local invariants associated to f and g. We also study the local topology of a deformation of g, { g } = g+f N, for a positive integer number N>>1. We provide a relation between the Brasselet number of g and {g} in X ∩ { f=0 }, in the case where f has isolated singularity at the origin. We also provide a new proof for the Lê-Iomdine formula for the Brasselet number
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Gasparim, Elizabeth Terezinha 1963. "Tres cardinais invariantes topologicos." [s.n.], 1989. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306216.
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Orientador : Ofelia Teresa AlasDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica
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Resumo: Neste trabalho estudamos os conceitos de tightness, set-tightness e T-tightness. Investigamos o comportamento de tightnes sob as compactificações de Alexandroff e Stone-Cech em alguns exemplos específicos. Calculamos tightness, T-tightness e Set-tightness em alguns espaços prdouto e provamos o seguinte resultado: Se X e Y são espaços topológicos, então: ts(X x Y) min
Abstract: In this work we study the concepts of tightness, set-tightness and T-tightness. We investigate the behavior of tightness under Alexandroff and Stone-Cech compactifications, in some specific examples. We calculate tightness, T-tighness, T-tightness and set-tightness for some product spaces and prove spaces and prove the following result: If X and Y are topological spaces, then ts(X x Y) min
Mestrado
Mestre em Matemática
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Barbosa, Grazielle Feliciani. "Topologia de singularidades e o estudo de seus invariantes." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-08052008-135109/.
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Algumas relações entre A-invariantes de germes de aplicações de coposto 1 equidimensionais f : \'C POT. n\' , 0 \'SETA\' \"C POT.n\', 0 são descritas. O principal resultado estabelece que a soma alternada de números de Milnor dos fechos dos conjuntos Ai na fonte de f é igual a multiplicidade local de f menos n + 1. E existem fórmulas correspondentes para os s-tipos estáveis locais A(\'k IND.1\' ,...\'k IND.s\'). As relações nos garantem condiçõoes para a A-finitude de f e para a A-trivialidade topológica de deformações de f. Também classificamos os germes de aplicações A-simples f : \'C POT.2\', 0 \'SETA\' \'C POT.5\', 0, para multiplicidades 1, 2 e 3Some new relations between A-invariants of equidimensional corank-1 map germs f :\'C POT.n\', 0 \' \'ARROW\' \'C POT.n\', 0 are described. The main local result states that the alternating sum ofthe Milnor numbers of the closures of the Ai sets in the source of f is equal to the local multiplicity of f minus n + 1. And there are corresponding formulas for the s-local stable types A(\'k IND.1\' ,...,\'k IND.s\'). The realations provide simplified (or weaker) conditions for the A-finiteness of f and for the topological A-triviality of deformations of f. We also classify the A-simple germs f : \'C POT.2\', 0 \'ARROW\' \'C POT.5\', 0 for multiplicities 1, 2, and 3
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Kimme, Lukas. "Bound states and resistive edge transport in two-dimensional topological phases." Doctoral thesis, Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-212743.
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The subject of the present thesis are some aspects of impurities affecting mesoscopic systems with regard to their topological properties and related effects like Majorana fermions and quantized conductance. A focus is on two-dimensional systems including both topological insulators and superconductors.
First, the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal invariant superconductors from Altland-Zirnbauer (AZ) symmetry class DIII is addressed, and a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength is defined. These general results are applied to the time-reversal invariant p-wave phase of the doped Kitaev-Heisenberg model, where it is also demonstrated how a lattice of impurities can drive a topologically trivial system into the nontrivial phase.
Second, the result about the existence of zero-energy impurity states is generalized to all AZ symmetry classes. This is achieved by considering, for general Hamiltonians H from the respective symmetry classes, the “generalized roots of det H”, which subsequently are used to further explore the opportunities that lattices of nonmagnetic impurities provide for the realization of topologically nontrivial phases. The 1d Kitaev chain model, the 2d px + ipy superconductor, and the 2d Chern insulator are considered to show that impurity lattices generically enable topological phase transitions and, in the case of the 2d models, even provide access to a number of phases with large Chern numbers.
Third, elastic backscattering in helical edge modes caused by a magnetic impurity with spin S and random Rashba spin-orbit coupling is investigated. In a finite bias steady state, the impurity induced resistance is found to slightly increase with decreasing temperature for S > 1/2. Since the underlying backscattering mechanism is elastic, interference between different scatterers can explain reproducible conductance fluctuations. Thus, the model is in agreement with central experimental results on edge transport in 2d topological insulators.
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Schneider, Friedrich Martin, and Andreas Thom. "On Følner sets in topological groups." Cambridge University Press, 2018. https://tud.qucosa.de/id/qucosa%3A70711.
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We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.
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Duan, Zhihao. "Topological string theory and applications." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLEE011/document.
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Cette thèse porte sur diverses applications de la théorie des cordes topologiques basée sur différents types de variétés de Calabi-Yau (CY). Le premier type considéré est la variété torique CY, qui est intimement liée aux problèmes spectraux des différents opérateurs. L'exemple particulier considéré dans la thèse ressemble beaucoup au modèle de Harper-Hofstadter en physique de la matière condensée. Nous étudions d’abord les secteurs non perturbatifs dans ce modèle et proposons une nouvelle façon de les calculer en utilisant la théorie topologique des cordes. Dans la deuxième partie de la thèse, nous considérons les fonctions de partition sur des variétés de CY elliptiquement fibrées. Celles-ci présentent un comportement modulaire intéressant. Nous montrons que pour les géométries qui ne conduisent pas à des symétries de jauge non abéliennes, les fonctions de partition des cordes topologiques peuvent être reconstruites avec seulement les invariants de Gromov-Witten du genre zéro. Finalement, nous discutons des travaux en cours concernant la relation entre les fonctions de partitionnement des cordes topologiques sur les soi-disant arbres de Higgsing dans la théorie de FThis thesis focuses on various applications of topological string theory based on different types of Calabi-Yau (CY) manifolds. The first type considered is the toric CY manifold, which is intimately related to spectral problems of difference operators. The particular example considered in the thesis closely resembles the Harper-Hofstadter model in condensed matter physics. We first study the non-perturbative sectors in this model, and then propose a new way to compute them using topological string theory. In the second part of the thesis, we consider partition functions on elliptically fibered CY manifolds. These exhibit interesting modular behavior. We show that for geometries which don't lead to non-abelian gauge symmetries, the topological string partition functions can be reconstructed based solely on genus zero Gromov-Witten invariants. Finally, we discuss ongoing work regarding the relation of the topological string partition functions on the so-called Higgsing trees in F-theory
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Cecchi, Bernales Paulina Alejandra. "Invariant measures in symbolic dynamics : a topological, combinatorial and geometrical approach." Thesis, Sorbonne Paris Cité, 2019. https://theses.md.univ-paris-diderot.fr/CECCHI-BERNALES_Paulina_2_complete_20190626.pdf.
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Dans ce travail nous étudions quelques propriétés des systèmes symboliques, avec un accent particulier mis sur le rôle joué par les mesures invariantes de tels systèmes. Nous nous attachons à l'étude des mesures invariantes d'un point de vue topologique, combinatoire et géométrique. Du point de vue topologique, nous nous concentrons sur le problème de l'équivalence orbitale et l'équivalence orbitale forte entre des systèmes dynamiques donnés par des actions minimales de Z, par l'étude d'un invariant algébrique, à savoir, le groupe de dimension dynamique. Notre travail donne une description du groupe de dimension dynamique pour deux classes particulières de sous-shifts : les sous-shifts S-adiques et les sous-shifts dendriques. Du point de vue combinatoire, nous nous intéressons au problème de l'équilibre des sous-shifts minimaux et uniquement ergodiques donnés par des actions de Z. Nous étudions le comportement concernant l'équilibre pour des sous-shifts substitutifs, S-adiques et dendriques. Nous établissons des conditions nécessaires pour qu'un sous-shift substitutif minimal avec des fréquences rationnelles soit équilibré par rapport à ses facteurs, en obtenant comme corollaire le déséquilibre des facteurs de longueur supérieure à 2 dans le sous-shift engendré par la substitution de Thue-Morse. Enfin, du point de vue géométrique, nous étudions le problème de réalisation des simplexes de Choquet comme des ensembles de mesures de probabilité invariantes associés à des systèmes donnés par des actions minimales des groupes moyennables sur l'ensemble de Cantor. Nous introduisons la notion de groupe moyennable congruent-monopavable, nous montrons que tout groupe moyennable virtuellement nilpotent est congruent-monopavable, et que pour un group discret infini G avec cette propriété, tout simplexe de Choquet peut s'obtenir comme l'ensemble des mesures invariantes d'un G-sous-shift minimalIn this work we study some dynamical properties of symbolic dynamical systems, with particular emphasis on the role played by the invariant probability measures of such systems. We approach the study of the set of invariant measures from a topological, combinatorial and geometrical point of view. From a topological point of view, we focus on the problem of orbit equivalence and strong orbit equivalence between dynamical systems given by minimal actions of Z, through the study of an algebraic invariant, namely the dynamical dimension group. Our work presents a description of the dynamical dimension group for two particular classes of subshifts: S-adic subshifts and dendric subshifts. From a combinatorial point of view, we are interested in the problem of balance in minimal uniquely ergodic systems given by actions of Z. We investigate the behavior regarding balance for substitutive, S-adic and dendric subshifts. We give necessary conditions for a minimal substitutive system with rational frequencies to be balanced on its factors, obtaining as a corollary the unbalance in the factors of length at least 2 in the subshift generated by the Thue-Morse sequence. Finally, from the geometrical point of view, we investigate the problem of realization of Choquet simplices as sets of invariant probability measures associated to systems given by minimal actions of amenable groups on the Cantor set. We introduce the notion of congruent monotileable amenable group, we prove that every virtually nilpotent amenable group is congruent monotileable, and we show that for a discrete infinite group G with this property, every Choquet simplex can be obtained as the set of invariant measures of a minimal G-subshift
En este trabajo estudiamos algunas propiedades dinamicas de sistemas simbolicos, con especial enfasis en el rol que juegan las medidas de probabilidad invariantes de tales sistemas. Nuestra aproximacion al estudio de las medidas invariantes se realiza desde tres angulos: topologico, combinatorio y geometrico. Desde el punto de vista topologico, nos enfocamos en el problema de la equivalencia orbital y equivalencia orbital fuerte entre sistemas dinamicos dados por acciones minimales de Z, a traves del estudio de un invariante algebraico, a saber, el grupo de dimension dinamico. Nuestro trabajo presenta una descripcion del grupo de dimension dinamico para dos clases particulares de subshifts minimales: los subshifts S-adicos y los subshifts dendricos. Desde el punto de vista combinatorio, nos interesamos en el problema del equilibrio en subshifts minimales y unicamente ergodicos dados por acciones de Z. Investigamos el comportamiento en relacional equilibrio para subshifts substitutivos, S-adicos y dendricos. Establecemos condiciones necesarias para que un subshift substitutivo minimal con frecuencias racionales sea equilibrado en sus factores, obteniendo como corolario el desequilibrio en los factores de largo mayor o igual a 2 en el subshift generado por la substitucion de Thue–Morse. Finalmente, desde el punto de vista geometrico, investigamos la posibilidad de realizar sımplices de Choquet como conjuntos de medidas de probabilidad invariantes asociados a sistemas dados por acciones minimales de grupos promediables sobre el Cantor. Introducimos la nocion de grupo promediable congruente-monoembaldosable, probamos que todo grupo promediable virtualmente nilpotentees congruente-monoembaldosable, y mostramos que para un grupo discreto e infinito G con estapropiedad, todo sımplice de Choquet puede obtenerse como el conjunto de medidas invariantes de un G-subshift minimal
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Hagan, Scott. "Scale invariant and topological approaches to the cosmological constant problem." Thesis, McGill University, 1995. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=39926.
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The cosmological constant is historically reviewed from its introduction in classical and relativistic cosmology through its modern quantum guise where it appears as a vacuum energy density. Limits on the empirical value are in glaring contradiction to the expectations of field theoretical calculations.Motivated by the natural connection between dilatation invariance and the extinction of the vacuum energy density, a phenomenological realization of a global scale symmetry is constructed. A complete treatment of such a realization in the context of a supergravitational toy model is calculated to one loop using an effective potential formalism. Particular attention is paid to the quantization of both supersymmetric and general coordinate gauges and to the concomitant ghost structure since traditional treatments have introduced non-local operators in the ghost Lagrangian and generating functional. Contributions to the effective potentid from the gravity sector are thus determined that contradict the literature. A particular class of tree-level scalar potentials that includes the 'no-scale' case is studied in the that space limit. While it is found that scale invariance can be maintained at the one-loop level and the cosmological constant made to vanish for all potentials in the class this is directly attributable to supersymmetry. A richer form of the Kahler potential or an enlarged particle content may facilitate the breaking of supersymmetry.
Phenomenological consequences of supergravity are investigated through a one-loop calculation of the electromagnetic form factor of the gravitino. Should such a form factor exist a signature of the gravitino might be found in processes with unlabeled products such as $e sp+e sp- to nothing.$ It is found that the form factor vanishes to this order, the Lorentz structures generated being too impoverished to withstand a constraining set of polarization conditions.
Finally the wormhole solution to the cosmological constant problem is examined in a semiclassical approximation. The notion that scalar field worm-holes must have associated conserved charges is questioned and a model of massive scalar field wormholes is delineated and proven to provide a counterexample. As the model allows baby universes nucleated with a certain eigenvalue of the scalar field momentum to classically evolve to a different value, competing semiclassical paths contribute to the same transition amplitude. Numerical simulations demonstrate that the novel semiclassical paths available to massive solutions cannot be overlooked in approximating the tunneling amplitude.
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Simmons, Skyler C. "Topological Properties of Invariant Sets for Anosov Maps with Holes." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/3101.
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We begin by studying various topological properties of invariant sets of hyperbolic toral automorphisms in the linear case. Results related to cardinality, local maximality, entropy, and dimension are presented. Where possible, we extend the results to the case of hyperbolic toral automorphisms in higher dimensions, and further to general Anosov maps.
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Borot, Gaëtan. "Quelques problèmes de géométrie énumérative, de matrices aléatoires, d'intégrabilité, étudiés via la géométrie des surfaces de Riemann." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112092/document.
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La géométrie complexe est un outil puissant pour étudier les systèmes intégrables classiques, la physique statistique sur réseau aléatoire, les problèmes de matrices aléatoires, la théorie topologique des cordes, …Tous ces problèmes ont en commun la présence de relations, appelées équations de boucle ou contraintes de Virasoro. Dans le cas le plus simple, leur solution complète a été trouvée récemment, et se formule naturellement en termes de géométrie différentielle sur une surface de Riemann : la "courbe spectrale", qui dépend du problème. Cette thèse est une contribution au développement de ces techniques et de leurs applications.Pour commencer, nous abordons les questions de développement asymptotique à tous les ordres lorsque N tend vers l’infini, des intégrales N-dimensionnelles venant de la théorie des matrices aléatoires de taille N par N, ou plus généralement des gaz de Coulomb. Nous expliquons comment établir, dans les modèles de matrice beta et dans un régime à une coupure, le développement asymptotique à tous les ordres en puissances de N. Nous appliquons ces résultats à l'étude des grandes déviations du maximum des valeurs propres dans les modèles beta, et en déduisons de façon heuristique des informations sur l'asymptotique à tous les ordres de la loi de Tracy-Widom beta, pour tout beta positif. Ensuite, nous examinons le lien entre intégrabilité et équations de boucle. En corolaire, nous pouvons démontrer l'heuristique précédente concernant l'asymptotique de la loi de Tracy-Widom pour les matrices hermitiennes.Nous terminons avec la résolution de problèmes combinatoires en toute topologie. En théorie topologique des cordes, une conjecture de Bouchard, Klemm, Mariño et Pasquetti affirme que des séries génératrices bien choisies d'invariants de Gromov-Witten dans les espaces de Calabi-Yau toriques, sont solution d'équations de boucle. Nous l'avons démontré dans le cas le plus simple, où ces invariants coïncident avec les nombres de Hurwitz simples. Nous expliquons les progrès récents vers la conjecture générale, en relation avec nos travaux. En physique statistique sur réseau aléatoire, nous avons résolu le modèle O(n) trivalent sur réseau aléatoire introduit par Kostov, et expliquons la démarche à suivre pour résoudre des modèles plus généraux.Tous ces travaux soulignent l'importance de certaines "intégrales de matrices généralisées" pour les applications futures. Nous indiquons quelques éléments appelant à une théorie générale, encore basée sur des "équations de boucles", pour les calculerComplex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory, … All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently : it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications.First, we consider all order large N asymptotics in some N-dimensional integrals coming from random matrix theory, or more generally from "log gases" problems. We shall explain how to use loop equations to establish those asymptotics in beta matrix models within a one cut regime. This can be applied in the study of large fluctuations of the maximum eigenvalue in beta matrix models, and lead us to heuristic predictions about the asymptotics of Tracy-Widom beta law to all order, and for all positive beta. Second, we study the interplay between integrability and loop equations. As a corollary, we are able to prove the previous prediction about the asymptotics to all order of Tracy-Widom law for hermitian matrices.We move on with the solution of some combinatorial problems in all topologies. In topological string theory, a conjecture from Bouchard, Klemm, Mariño and Pasquetti states that certain generating series of Gromov-Witten invariants in toric Calabi-Yau threefolds, are solutions of loop equations. We have proved this conjecture in the simplest case, where those invariants coincide with the "simple Hurwitz numbers". We also explain recent progress towards the general conjecture, in relation with our work. In statistical physics on the random lattice, we have solved the trivalent O(n) model introduced by Kostov, and we explain the method to solve more general statistical models.Throughout the thesis, the computation of some "generalized matrices integrals" appears to be increasingly important for future applications, and this appeals for a general theory of loop equations
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Gheisarieha, Mohsen. "Topological chaos and chaotic mixing of viscous flows." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/27768.
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Since it is difficult or impossible to generate turbulent flow in a highly viscous fluid or a microfluidic system, efficient mixing becomes a challenge. However, it is possible in a laminar flow to generate chaotic particle trajectories (well-known as chaotic advection), that can lead to effective mixing. This dissertation studies mixing in flows with the limiting case of zero Reynolds numbers that are called Stokes flows and illustrates the practical use of different theories, namely the topological chaos theory, the set-oriented analysis and lobe dynamics in the analysis, design and optimization of different laminar-flow mixing systems.
In a recent development, the topological chaos theory has been used to explain the chaos built in the flow only based on the topology of boundary motions. Without considering any details of the fluid dynamics, this novel method uses the Thurston-Nielsen (TN) classification theorem to predict and describe the stretching of material lines both qualitatively and quantitatively. The practical application of this theory toward design and optimization of a viscous-flow mixer and the important role of periodic orbits as "ghost rods" are studied.
The relationship between stretching of material lines (chaos) and the homogenization of a scalar (mixing) in chaotic Stokes flows is examined in this work. This study helps determining the extent to which the stretching can represent real mixing. Using a set-oriented approach to describe the stirring in the flow, invariance or leakiness of the Almost Invariant Sets (AIS) playing the role of ghost rods is found to be in a direct relationship with the rate of homogenization of a scalar. The mixing caused by these AIS and the variations of their structure are explained from the point of view of geometric mechanics using transport through lobes. These lobes are made of segments of invariant manifolds of the periodic points that are generators of the ghost rods.
A variety of the concentration-based measures, the important parameters of their calculation, and the implicit effect of diffusion are described. The studies, measures and methods of this dissertation help in the evaluation and understanding of chaotic mixing systems in nature and in industrial applications. They provide theoretical and numerical grounds for selection of the appropriate mixing protocol and design and optimization of mixing systems, examples of which can be seen throughout the dissertation.Ph. D.
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Mendonça, Hudson Kazuo Teramoto. "Teorias de 2-gauge e o invariante de Yetter na construção de modelos com ordem topológica em 3-dimensões." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-01082017-155641/.
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Ordem topológica descreve fases da matéria que não são caracterizadas apenas pelo esquema de quebra de simetria de Landau. Em 2-dimensões ordem topológica é caracterizada, entre outras propriedades, pela existência de uma degenerescência do estado fundamental que é robusta sobre perturbações locais arbitrarias. Com o proposito de entender o que caracteriza e classifica ordem topológica 3-dimensional o presente trabalho apresenta um modelo quântico exatamente solúvel em 3-dimensões que generaliza os modelos em 2-dimensões baseados em teorias de gauge. No modelo proposto o grupo de gauge é substituído por um 2-grupo. A Hamiltonia, que é dada por uma soma de operadores locais, é livre de frustrações. Provamos que a degenerescência do estado fundamental nesse modelo é dado pelo invariante de Yetter da variedade 4-dimensional Sigma × S¹, onde Sigma é a variedade 3-dimensional onde o modelo está definido.Topological order describes phases of matter that cannot be described only by the symmetry breaking theory of Landau. In 2-dimensions topological order is characterized, among other properties, by the presence of a ground state degeneracy that is robust to arbitrary local perturbations. With the purpose of understanding what characterizes and classify 3-dimensional topological order this works presents an exactly soluble quantum model in 3-dimensions that generalize 2-dimensional models constructed using gauge theories. In the model we propose the gauge group is replaced by a 2-group. The Hamiltonian, that is given by a sum of local commuting operators, is frustration free. We prove that the ground state degeneracy of this model is given by the Yetters invariant of the 4-dimensional manifold Sigma × S¹, where Sigma is the 3-dimensional manifold the model is defined.
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Gazon, Amanda Buosi [UNESP]. "Um estudo sobre certos invariantes homológicos relativos duais." Universidade Estadual Paulista (UNESP), 2012. http://hdl.handle.net/11449/92946.
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Baseado na teoria de cohomologia de grupos, Andrade e Fanti definiram um invariante algébrico, denotado por E(G;S;M), onde G é um grupo, S é uma família de subgrupos de G de índice finito e Mé um Z 2G-módulo. O objetivo deste trabalho é definir um invariante dual a E(G;S;M), que denotaremos por E (G;S;M), utilizando a homologia de grupos em vez da cohomologia. Com este invariante, obtemos diversos resultados e aplicações, principalmente nas teorias de grupos e pares de dualidade e de decomposição de grupos. Estes resultados fornecem uma maneira alternativa de obter aplicações e propriedades nestas teorias. E, para desenvolver este trabalho, estudamos as teorias de (co)homologia absoluta e relativa de grupos, bem como suas interpretações topológicas, e a teoria de grupos e pares de dualidade
Based on the cohomology theory of groups, Andrade and Fanti defined an algebraic invariant, denoted by E(G;S;M), where G is a group, S is a family of subgroups of G with nite index and M is a Z 2G-module. The purpose of this work is to define a dual invariant of E(G;S;M), which we denote by E (G;S;M), using the homology of groups instead of cohomology. With this invariant, we obtain many results and applications, especially in the duality and splitting theories of groups. These results provide an alternative way to get applications and properties in these theories. And to develop this work, we studied the absolute and relative (co)homology theories of groups, as well as their topological interpretations, and the theories of duality groups and pairs
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Lattanzi, Guemael Rinaldi. "Nós legendreanos e seus invariantes." Universidade Federal de Viçosa, 2013. http://locus.ufv.br/handle/123456789/4925.
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Previous issue date: 2013-07-31Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this work, we study the classical invariants of Legendrian Knots Theory and we show that these are not complet. To do this we introduce a notion of a Basic Knot Theory like their classical invariants, Thurston-Bennequin number and Maslov number. Then we discuss a new tool developed by Chekanon and denoted by DGA (Differential Graduated Algebra), wich will help us in the proof of the incompletness of classical invariants of legendrian knots.
Neste trabalho, estudaremos os invariantes clássicos da Teoria de Nós Legendreanos e mostraremos que estes não são completos. Para tal introduzimos uma noção básica da Teoria de Nós Legendreanos, assim como seus invariantes clássicos, o número de Thurston-Bennequin e o número de Maslov. Em seguida discutiremos uma nova ferramenta desenvolvida por Chekanov, a Álgebra Diferencial Graduada, denotada por DGA (Differential Graduated Algebra), que nos auxiliar na prova da incompletude dos invariantes clássicos de nós legendreanos.
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Gazon, Amanda Buosi. "Um estudo sobre certos invariantes homológicos relativos duais/." São José do Rio Preto : [s.n.], 2012. http://hdl.handle.net/11449/92946.
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Orientador: Maria Gorete Carreira AndradeBanca: Pedro Luiz Queiroz Pergher
Banca: Ermínia de Lourdes Campello Fanti
Resumo: Baseado na teoria de cohomologia de grupos, Andrade e Fanti definiram um invariante algébrico, denotado por E(G;S;M), onde G é um grupo, S é uma família de subgrupos de G de índice finito e Mé um Z 2G-módulo. O objetivo deste trabalho é definir um invariante dual a E(G;S;M), que denotaremos por E (G;S;M), utilizando a homologia de grupos em vez da cohomologia. Com este invariante, obtemos diversos resultados e aplicações, principalmente nas teorias de grupos e pares de dualidade e de decomposição de grupos. Estes resultados fornecem uma maneira alternativa de obter aplicações e propriedades nestas teorias. E, para desenvolver este trabalho, estudamos as teorias de (co)homologia absoluta e relativa de grupos, bem como suas interpretações topológicas, e a teoria de grupos e pares de dualidade
Abstract: Based on the cohomology theory of groups, Andrade and Fanti defined an algebraic invariant, denoted by E(G;S;M), where G is a group, S is a family of subgroups of G with nite index and M is a Z 2G-module. The purpose of this work is to define a dual invariant of E(G;S;M), which we denote by E (G;S;M), using the homology of groups instead of cohomology. With this invariant, we obtain many results and applications, especially in the duality and splitting theories of groups. These results provide an alternative way to get applications and properties in these theories. And to develop this work, we studied the absolute and relative (co)homology theories of groups, as well as their topological interpretations, and the theories of duality groups and pairs
Mestre
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Uggioni, Bruno Brogni. "Convergência da convolução de probabilidades invariantes pelo deslocamento." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/150847.
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Essa tese foi inspirada no artigo [10] de Lindenstrauss et al. e remete ao trabalho fundamental de Furstenberg [5]. Sejam (Z=pZ)N o produto cartesiano unlilateral de in nitas cópias de Z=pZ e a função shift em (Z=pZ)N.This thesis was inspired by the Lindenstrauss' article [10] and the fundamental work of Furstenberg [5]. Let (Z=pZ)N be the compact group which is the cartesian product of in nite copies of the nite group Z=pZ and be the shift function on (Z=pZ)N.
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Silva, Anderson Alves da. "Construção de uma teoria quântica dos campos topológica a partir do invariante de Kuperberg." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-26102015-133218/.
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Resumo Neste trabalho apresentamos, em detalhes, a construção de uma teoria quântica dos campos topológica (TQCT). Podemos definir uma TQCT como um funtor simétrico monoidal da categoria dos cobordismos para a categoria dos espaços vetoriais. Em duas dimensões podemos encontrar uma descrição completa da categoria dos cobordismos e classificar todas as TQCT\'s. Em três dimensões é possível estender alguns invariantes para 3-variedades e construir uma TQCT 3D. Nossa construção é baseada no invariante para 3-variedades de Kuperberg, o qual envolve diagramas de Heegaard e álgebras de Hopf. Começamos com a apresentação do invariante de Kuperberg definido para toda variedade 3D compacta, orientável e sem bordo. Para cada álgebra de Hopf de dimensão finita constrói-se um invariante. Por fim, apresentamos a TQCT associada com o invariante de Kuperberg. Isto é feito usando-se o fato de que o invariante de Kuperberg é definido como uma soma de pesos locais tal qual uma função de partição. A TQCT decorre dos operadores advindos de variedades com bordo.Abstract In this work we present in detail a construction of a topological quantum field theory (TQFT). We can define a TQFT as a symmetric monoidal functor from cobordism categories to category of vector spaces. In two dimension, we can give a complete description of cobordism categories and classify all TQFT\'s. In three dimension it is possible to extend some specific 3-manifold invariants and to construct a TQFT 3D. Our construction is based on the Kuperberg 3-manifold invariant which involves Heegaard diagrams and Hopf algebras. We start with the presentation of the Kuperberg invariant defined for every orientable compact 3-manifold without boundary. For each finite-dimensional Hopf algebra we can construct a invariant. Finally we presente the TQFT associated with the Kuperberg invariant. This is made using the fact that the Kuperberg invariant is defined like a sum of local weights in the same way as a partition function. The TQFT is constructed from the operators given by manifolds with boundary.
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Bianco, Giuseppe. "Studio di invarianti topologici attraverso applicazioni lisce e campi di vettori." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16412/.
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Uno studio di alcune proprietà delle varietà attraverso lo studio delle funzioni lisce fra varietà e dei campi di vettori. Su queste, in particolare si arriverà a provare, usando alcuni strumenti elementari ed altri più fini di topologia, il teorema del grado di Hopf ed il teorema di Poincarè-Hopf.
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Ngo, Hoai Diem Phuc. "Rigid transformations on 2D digital images : combinatorial and topological analysis." Thesis, Paris Est, 2013. http://www.theses.fr/2013PEST1091/document.
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Dans cette thèse, nous étudions les transformations rigides dans le contexte de l'imagerie numérique. En particulier, nous développons un cadre purement discret pour traiter ces transformations. Les transformations rigides, initialement définies dans le domaine continu, sont impliquées dans de nombreuses applications de traitement d'images numériques. Dans ce contexte, les transformations rigides digitales induites présentent des propriétés géométriques et topologiques différentes par rapport à leurs analogues continues. Afin de s'affranchir des problèmes inhérents à ces différences, nous proposons de formuler ces transformations rigides dans un cadre purement discret. Dans ce cadre, les transformations rigides sont regroupées en classes correspondant chacune à une transformation digitale donnée. De plus, les relations entre ces classes de transformations peuvent être modélisées par une structure de graphe. Nous prouvons que ce graphe présente une complexité spatiale polynômiale par rapport à la taille de l'image. Il présente également des propriétés structurelles intéressantes. En particulier, il permet de générer de manière progressive toute transformation rigide digitale, et ce sans approximation numérique. Cette structure constitue un outil théorique pour l'étude des relations entre la géométrie et la topologie dans le contexte de l'imagerie numérique. Elle présente aussi un intérêt méthodologique, comme l'illustre son utilisation pour l'évaluation du comportement topologique des images sous des transformations rigidesIn this thesis, we study rigid transformations in the context of computer imagery. In particular, we develop a fully discrete framework for handling such transformations. Rigid transformations, initially defined in the continuous domain, are involved in a wide range of digital image processing applications. In this context, the induced digital rigid transformations present different geometrical and topological properties with respect to their continuous analogues. In order to overcome the issues raised by these differences, we propose to formulate rigid transformations on digital images in a fully discrete framework. In this framework, Euclidean rigid transformations producing the same digital rigid transformation are put in the same equivalence class. Moreover, the relationship between these classes can be modeled as a graph structure. We prove that this graph has a polynomial space complexity with respect to the size of the considered image, and presents useful structural properties. In particular, it allows us to generate incrementally all digital rigid transformations without numerical approximation. This structure constitutes a theoretical tool to investigate the relationships between geometry and topology in the context of digital images. It is also interesting from the methodological point of view, as we illustrate by its use for assessing the topological behavior of images under rigid transformations
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Ekahana, Sandy Adhitia. "Investigation of topological nodal semimetals through angle-resolved photoemission spectroscopy." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:afed6156-7aa2-4ba9-afd1-af53d775494f.
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Nodal semimetals host either degenerate points (Dirac/Weyl points) or lines whose band topology in Brillouin zone can be classified either as trivial (normal nodal semimetals) or non trivial (topological nodal semimetals). This thesis investigates the electronic structure of two different categories of topological nodal semimetals probed by angleresolved photoemission spectroscopy (ARPES): The first material is Indium Bismuth (InBi). InBi is a semimetal with simple tetragonal structure with P4/nmm space group. This space group is predicted to host protected nodal lines along the perpendicular momentum direction at the high symmetry lines of the Brillouin zone boundary even under strong spin-orbit coupling (SOC) situation. As a semimetal with two heavy elements, InBi is a suitable candidate to test the prediction. The investigation by ARPES demonstrates not only that InBi hosts the nodal line in the presence of strong SOC, it also shows the signature of type-II Dirac crossing along the perpendicular momentum direction from the center of Brillouin zone. However, as the nodal line observed is trivial in nature, there is no exotic drumhead surface states observed in this material. This finding demonstrates that Dirac crossings can be protected in a non-symmorphic space group. The second material is NbIrTe4 which is a semimetal that breaks inversion symmetry predicted to host only four Weyl points. This simplest configuration is confirmed by the measurement from the top and bottom surface of NbIrTe4 showing only a pair of Fermi arcs each. Furthermore, it is found that the Fermi arc connectivity on the bottom surface experiences re-wiring as it evolves from Weyl points energy to the ARPES Fermi energy level. This change is attributed to the hybridisation between the surface and the bulk states as their projection lie within the vicinity of each other. The finding in this work demonstrates that although Fermi arcs are guaranteed in Weyl semimetals, their shape and connectivity are not protected and may be altered accordingly.