Tesi sul tema "Topology"

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-10Bitstream added on 2014-06-13T18:30:56Z : No. of bitstreams: 1 dutra_acb_me_rcla.pdf: 707752 bytes, checksum: 003487414f094d392a97a22a4efb885b (MD5)<br>Neste trabalho tratamos do objeto matemático Grupo Topológico. Para este desenvolvimento, abordamos elementos básicos de Grupo e Espaço Topológico<br>In this work we consider the mathematical object Topological Group. For this development, we discuss the basic elements of the Group and Topological Space

Dans ce travail, on donne une description du feuilletage défini par une 1-forme logarithmique fermée, à poles sur un diviseur à croisements normaux. Le long de chaque strate du diviseur, il est localement trivial, ce qui nous permet de définir une notion de voisinage tubulaire adapté à cette forme. En exigeant de plus des conditions d'incidences entre ces différents voisinages tubulaires, nous obtenons une construction analogue à celle introduite par C. H. Clemens dans le cas des fonctions. Nous prouvons l'existence d'une telle "structure de Clemens adaptée" à la forme considérée, puis nous l'utilisons pour décrire la classification topologique des germes de formes logarithmiques génériques de c**(n). Nous nous ramenons à la situation considérée ci-dessus à l'aide d'une désingularisation des séparatrices de la forme.

Orientador: Elíris Cristina Rizziolli<br>Banca: Edivaldo Lopes da Silva<br>Banca: João Peres Vieira<br>Resumo: Neste trabalho tratamos do objeto matemático Grupo Topológico. Para este desenvolvimento, abordamos elementos básicos de Grupo e Espaço Topológico<br>Abstract: In this work we consider the mathematical object Topological Group. For this development, we discuss the basic elements of the Group and Topological Space<br>Mestre

<p>In the field of embedded computer and communication systems, the demands for the </p><p>interconnection networks are increasing rapidly. To satisfy these demands much advancement has </p><p>been made at the chip level as well as at the system level and still the research works are going </p><p>on, to make the interconnection networks more flexible to satisfy the demands of the real-time </p><p>applications. </p><p> </p><p>This thesis mainly focuses on the interconnection between the nodes in an embedded system via a </p><p>reconfigurable backplane. To satisfy the project goals, an algorithm is written for the </p><p>reconfigurable topology that changes according to the given traffic specification like throughput. </p><p>Initially the connections are established between pairs of nodes according to the given throughput </p><p>demands. By establishing all the connections, a topology is formed. Then a possible path is </p><p>chosen for traversing the data from source to destination nodes. Later the algorithm is </p><p>implemented by simulation and the results are shown in a tabular form. Through some application </p><p>examples, we both identify problems with the algorithm and propose an improvement to deal </p><p>with such problems.</p>

<p>In the last years, there has been done research in using topology as a new tool for studying data sets, typically high dimensional data. These studies have brought new methods for qualitative analysis, simplification, and visualization of high dimensional data sets. One good example, where these methods are useful, is in the study of microarray data (DNA data). To be able to use these methods, one needs to acquire knowledge of different topics in topology. In this paper we introduce simplicial homology, persistent homology, Mapper, and some simplicial complex constructions.</p>

<p>Today there is an immense production of data, and the need for better methods to analyze data is ever increasing. Topology has many features and good ideas which seem favourable in analyzing certain datasets where statistics is starting to have problems. For example, we see this in datasets originating from microarray experiments. However, topological methods cannot be directly applied on finite point sets coming from such data, or atleast it will not say anything interesting. So, we have to modify the data sets in some way such that we can work on them with the topological machinery. This way of applying topology may be viewed as a kind of discrete version of topology. In this thesis we present some ways to construct simplicial complexes from a finite point cloud, in an attempt to model the underlying space. Together with simplicial homology and persistent homology and barcodes, we obtain a tool to uncover topological features in finite point clouds. This theory is tested with a Java software package called JPlex, which is an implementation of these ideas. Lastly, a method called Mapper is covered. This is also a method for creating simplicial complexes from a finite point cloud. However, Mapper is mostly used to create low dimensional simplicial complexes that can be easily visualized, and structures are then detected this way. An implementation of the Mapper method is also tested on a self made data set.</p>

Doutoramento em Matemática<br>A monitorização e avaliação do desempenho de uma rede são essenciais para detetar e resolver falhas no seu funcionamento. De modo a conseguir efetuar essa monitorização, e essencial conhecer a topologia da rede, que muitas vezes e desconhecida. Muitas das técnicas usadas para a descoberta da topologia requerem a cooperação de todos os dispositivos de rede, o que devido a questões e políticas de segurança e quase impossível de acontecer. Torna-se assim necessário utilizar técnicas que recolham, passivamente e sem a cooperação de dispositivos intermédios, informação que permita a inferência da topologia da rede. Isto pode ser feito recorrendo a técnicas de tomografia, que usam medições extremo-a-extremo, tais como o atraso sofrido pelos pacotes. Nesta tese usamos métodos de programação linear inteira para resolver o problema de inferir uma topologia de rede usando apenas medições extremo-a-extremo. Apresentamos duas formulações compactas de programação linear inteira mista (MILP) para resolver o problema. Resultados computacionais mostraram que a medida que o número de dispositivos terminais cresce, o tempo que as duas formulações MILP compactas necessitam para resolver o problema, também cresce rapidamente. Consequentemente, elaborámos duas heurísticas com base nos métodos Feasibility Pump e Local ranching. Uma vez que as medidas de atraso têm erros associados, desenvolvemos duas abordagens robustas, um para controlar o número máximo de desvios e outra para reduzir o risco de custo alto. Criámos ainda um sistema que mede os atrasos de pacotes entre computadores de uma rede e apresenta a topologia dessa rede.<br>Monitoring and evaluating the performance of a network is essential to detect and resolve network failures. In order to achieve this monitoring level, it is essential to know the topology of the network which is often unknown. Many of the techniques used to discover the topology require the cooperation of all network devices, which is almost impossible due to security and policy issues. It is therefore, necessary to use techniques that collect, passively and without the cooperation of intermediate devices, the necessary information to allow the inference of the network topology. This can be done using tomography techniques, which use end-to-end measurements, such as the packet delays. In this thesis, we used some integer linear programming theory and methods to solve the problem of inferring a network topology using only end-to-end measurements. We present two compact mixed integer linear programming (MILP) formulations to solve the problem. Computational results showed that as the number of end-devices grows, the time need by the two compact MILP formulations to solve the problem also grows rapidly. Therefore, we elaborate two heuristics based on the Feasibility Pump and Local Branching method. Since the packet delay measurements have some errors associated, we developed two robust approaches, one to control the maximum number of deviations and the other to reduce the risk of high cost. We also created a system that measures the packet delays between computers on a network and displays the topology of that network.

La integració de tots els components bàsics per a la generació, manipulació i detecció de llum en xips òptics està impulsant avenços científics i tecnològics, per exemple, en el desenvolupament de tecnologies de la informació o de dispositius de detecció per a les tecnologies quàntiques. Degut a la seva flexibilitat, escalabilitat i la possibilitat d’observar directament l’evolució de la funció d’ona utilitzant senzilles tècniques de tractament d’imatges, les estructures fotòniques integrades són una plataforma ideal per a la simulació quàntica, és a dir, per emular fenòmens quàntics que apareixen en altres branques de la física. A més, aquestes analogies òptiques-quàntiques també permeten dissenyar circuits fotònics integrats amb propietats excepcionals. En aquesta tesi aprofitem propietats no trivials de la física quàntica per dissenyar nous dispositius fotònics integrats amb funcionalitats avançades i rendiments millorats, així com nous simuladors fotònics. Específicament, explotem les similituds entre les equacions de Helmholtz i de Schrödinger, que permeten reproduir la dinàmica temporal d’una partícula atrapada en un potencial periòdic amb l’evolució espacial de la llum propagant-se en guies d’ona acoblades, per aplicar transformacions supersimètriques i processos adiabàtics així com explorar geometries topològiques no trivials en sistemes de guies d’ona òptiques acoblades. En aquesta línia, la primera part de la tesi està dedicada a introduir els conceptes físics i matemàtics que descriuen les guies d’ona òptiques acoblades, les analogies òptiques-quàntiques i la supersimetria en òptica. La segona part de la tesi engloba el disseny de nous dispositius fotònics integrats combinant l’aplicació de transformacions supersimètriques per manipular modes espacials amb tècniques de passatge adiabàtic per introduir la robustesa. Primer presentem un nou mètode per a la multiplexació de modes espacials basat en guies d’ona supersimetriques, que filtren els modes, en combinació amb la tècnica de passatge adiabàtic espacial que es fa servir per transmetre eficient i robustament els modes escollits entre guies. De manera similar, mantenint-nos en la idea d’aplicar protocols d’enginyeria quàntica per dissenyar nous dispositius fotònics amb rendiments millorats, proposem connectar de manera adiabàtica estructures supersimètriques al llarg de la distància de propagació. En particular, aquesta tècnica l’utilitzem per dissenyar guies d’ona còniques, filtres de modes, divisors de feixos i interferòmetres, eficients i robustos. Finalment, la tercera part de la tesi està dedicada a la simulació de diferents fenòmens quàntics utilitzant sistemes fotònics. Per començar aquesta part, explorem els efectes que les transformacions supersimètriques indueixen en sistemes amb propietats topologies no trivials, les quals estan intrínsecament lligades a les simetries internes del sistema. Amb aquest objectiu, considerem el sistema més simple amb propietats topològiques no trivials i demostrem en sistemes de guies d’ona acoblades com la protecció topològica d’un estat pot ser suspesa i restablerta utilitzant transformacions supersimètriques. A més, per accedir a aquestes fases topològiques no trivials, un element clau és la introducció de camps artificials gauge (AGF) que controlen la dinàmica de partícules no carregades que d’una altra manera eludeixen la influència dels camps electromagnètics estàndards. En aquesta línia, investiguem la possibilitat d’induir AGF utilitzant llum amb moment orbital angular en comptes de manipular la geometria del sistema. Específicament, mesurem l’efecte de gàbia d’Aharonov-Bohm que està lligat amb la presència d’un camp magnètic. Aquesta tècnica permet accedir a diferent règims topològics en una sola estructura, un pas important per a la simulació quàntica utilitzant sistemes fotònics.<br>La integración de todos los componentes básicos para la generación, manipulación y detección de luz en chips ópticos está impulsando avances científicos y tecnológicos, por ejemplo, en el desarrollo de tecnologías de la información o en los dispositivos de detección para las tecnologías cuánticas. Debido a su flexibilidad, escalabilidad y a la posibilidad de observar directamente la evolución de la función de onda utilizando senzillas técnicas de trata, las estructuras fotónicas son ideales para la simulación cuántica, es decir, para emular fenómenos cuánticos que aparecen en otras ramas de la física. Es más, estas analogías ópticas-cuánticas también permiten diseñar nuevos circuitos fotónicos integrados con propiedades excepcionales. En esta tesis, aprovechamos propiedades no triviales que emergen de la física cuántica para diseñar nuevos dispositivos fotónicos integrados con funcionalidades avanzadas y rendimientos mejorados, así como nuevos simuladores fotónicos. Específicamente, explotamos las similitudes entre las ecuaciones de Helmholtz y de Schrödinger, que permiten reproducir la dinámica temporal de una particula atrapada en un potencial periódico con la evolución espacial de la luz propagándose en guías de onda, para aplicar transformaciones supersimétricas y procesos adiabáticos así como explorar geometrías topológicas no triviales en sistemas de guías de onda ópticas acopladas. La primera parte de la tesis está dedicada a introducir los conceptos matemáticos y físicos que describen las guías de onda ópticas acopladas, las analogías ópticas-cuánticas y la supersimetria óptica. La segunda parte de la tesis engloba el diseño de nuevos dispositivos fotónicos integrados basados en combinar transformaciones supersimétricas para manipular los modos espaciales con las técnicas adiabáticas para introducir robustez. Primero presentamos un nuevo método para la multiplexación de modos espaciales basado en guías de onda supersimétricas, que filtran los modos, en combinación con la técnica de pasaje adiabático espacial que se usa para transmitir de manera eficiente y robusta los modos escogidos entre guías. De manera similar, manteniéndonos en la idea de aplicar protocolos de ingeniería cuántica para diseñar nuevos dispositivos fotónicos con rendimientos superiores, proponemos conectar de manera adiabática estructuras supersimétricas a lo largo de la propagación. En particular, ésta técnica la utilizamos para diseñar guías de onda cónicas, filtros modales, divisores de haz e interferómetros. Finalmente, la tercera parte de la tesis está dedicada a la simulación de diferentes fenómenos físicos utilizando sistemas fotónicos. Para empezar, exploramos los efectos que las transformaciones supersimétricas inducen en sistemas con propiedades topológicas no triviales, las cuales están intrínsecamente ligadas a las simetrías internas del sistema. Con este objetivo, consideramos el sistema más simple con propiedades topológicas no triviales y demostramos en un sistema de guías de onda acopladas cómo la protección topológica de un estado puede ser suspendida y restablecida utilizando transformaciones supersimétricas. Además, para acceder a las fases topológicas no triviales, un elemento clave es la introducción de campos artificiales de gauge (AGF) que controlan la dinámica de partículas no cargadas que de otra manera eluden la influencia de los campos electromagnéticos. Es esta línea, investigamos la posibilidad de inducir AGF utilizando luz con momento orbital angular en lugar de manipular la geometría del sistema. Específicamente, medimos el fenómeno de jaula de Aharonov-Bohm que está ligado a la presencia de un campo magnético sintético. Esta técnica permite acceder a diferentes regímenes topológicos en una sola estructura, un paso importante para la simulación cuántica utilizando sistemas fotónicos.<br>The integration of all the basic components for light generation, manipulation and detection in optical chips is boosting scientific and technological advances, for instance, in the development of information technology and data communications or of sensing devices for quantum technologies. Due to its flexibility, scalability and of the possibility of directly observing the wavefunction evolution using simple imaging techniques, integrated photonic structures are an ideal playground for quantum simulation i.e., for emulating quantum phenomena appearing in other branches of physics. Moreover, these quantum-optical analogies also allow to design novel integrated photonic circuits with exceptional properties. In this context, in this thesis we harness non-trivial properties stemming from quantum physics to design novel integrated photonic devices with advanced functionalities and enhanced performances as well as to engineer novel photonic simulators. Specifically, we exploit the similarities between the Helmholtz and the Schrödinger equations, which allow to mimic the temporal dynamics of a single particle trapped in a lattice potential with the spatial evolution of a light beam propagating in an array of optical waveguides, to apply supersymmetric (SUSY) transformations and adiabatic passage processes as well as to explore non-trivial topological geometries in systems of coupled optical waveguides. In this vein, the first part of the thesis is devoted to introduce the mathematical concepts and physical ideas behind coupled optical waveguides, quantum-optical analogies and optical SUSY. After that, the second part of the thesis encompasses the design of novel integrated photonic devices by combining the spatial modal content manipulation offered by SUSY transformations with the robustness supplied by adiabatic passage techniques. In this regard, we start by presenting a novel method for mode division (de)multiplexing rooted on SUSY waveguides, which provide the mode filtering capabilities, in combination with a Spatial Adiabatic Passage protocol, which is used to efficiently and robustly transfer the desired modes between waveguides. Similarly, keeping on the idea of applying quantum engineering protocols to design novel photonic devices with enhanced performances, we also propose to connect, in an adiabatic fashion, SUSY structures along the propagation direction. In particular, this technique is used to engineer efficient and robust tapered waveguides, mode filters, beam splitters and interferometers. Finally, the third part of the thesis is dedicated to the photonic simulation of different phenomena. We explore first the effect that SUSY transformations induce in systems with non-trivial topological properties, which are intrinsically connected with the system's internal symmetries. To this aim, we consider the simplest system with non-trivial topological properties and demonstrate in waveguide arrays how the topological protection of a targeted state can be suspended and reestablished by applying SUSY transformations. Moreover, to access these non-trivial topological phases, a key step is the introduction of Artificial Gauge Fields (AGF) controlling the dynamics of uncharged particles that otherwise elude the influence of standard electromagnetic fields. To this end, we investigate the possibility of inducing AGF by injecting light beams carrying Orbital Angular Momentum, rather than manipulating the geometry of the system. Specifically, we measure the Aharonov-Bohm caging effect, which is directly related with the presence of a synthetic magnetic flux, in an array of coupled optical waveguides. This technique paves the way towards accessing different topological regimes in one single structure, representing an important step forward for quantum simulation in photonic structures.

The aim of this thesis is to investigate how a biological structure changes its shape and boundary under different cases of load if flow of nutrients is included, since nutrient flow has not been taken into account in previous studies. In order to simulate such a scenario we construct a model by using topology optimization (the SIMP model) and a balance law which is suitable for biological structures. Moreover, the model is derived by using an analogy with the dissipation inequality and Coleman-Noll’s procedure. The model can be interpreted as bone or some other biological structure, where the growth and remodeling partly occurs due to nutrient flow. The theory is first investigated by selecting an MBB beam with a special boundary condition for the nutrient concentration and inflow of nutrients, and then with a bone-like model. For the analysis with different loads we have observed that the structure becomes thicker were the load is applied. Parameters like beta (β) (reflecting the relation between nutrients and material) and nutrient concentration (c) seem to play an important role in nutrient transport and building of the structure. The result for larger values of β and nutrient concentration (c) gives a thicker structure in the entire domain. We also made an assumption of Fick’s law of diffusion. Fick’s law of diffusion describes the flux from high concentration to low concentration. This phenomenon is observed in analysis with different nutrient concentrations (c): we can see that the structure tends to be built up where the concentration is high and continues to be built in the direction from high to low concentration. In analyses with mu-value (μ), which represents cost of material, the result gives a thinner structure for larger values of μ.

This master project examines whether there is an existing model fordescribing network topologies in abstract and generic manner. I alsolooked for networking protocols for exchanging network topologiesand handling of dynamically creation of circuit connections acrossdomains. I?ve also been working on a prototype for visualization ofnetwork topologies using Network Administration Visualized (NAV)as a data backend, and further to check the possibilities to use thefound topology model in my prototype.My findings shows that there is progress towards creating a stan-dard topology model to describe network topologies in an abstractand generic manner. There is also progress in creating a network ar-chitecture with networking protocols for exchanging network topolo-gies across domains and providing a connection reservation service tohandle creation of dynamically circuit connections. Prototype showsthere is lots of ideas for further works on what to implement in re-gards of the found network topology model and networking systemsthat was found.

In order to utilise network resources efficiently, we need a strong knowledge of how the resources are shared and provisioned. However,this information is often unavailable due to the complexity of modern networks, the restrictive access to information describing their configurations and accuracy/reliability issues regarding information provisioning methods. Here, we propose the concept of functional topologies to deduce how resources are shared between different traffic flows. A functional topology describes the dependencies between traffic flows as a graph of interactions; this is in contrast to typical network graphs that model the physical connections between network components (routers and hosts). Unlike other work relying on in-network data, this topology is constructed solely at end hosts by measuring interdependencies of traffic flows via cross-correlation analysis. In order to measure the complete sets of interdependencies of traffic flows, different time intervals are used for sampling time series data. It is shown that these time intervals are related to maximum delays of traffic flows in network. The results of cross-correlation analysis are validated using well-known inverse participation ratio (IPR). As a part of the validation process, the results are analysed and compared with dominant/important flows of the network obtained by a new technique that uses eigen decomposition and spanning tree algorithm. The methodology of measuring interdependencies of traffic flows is validated and evaluated using real world data from a sensor network,as well as detailed simulation modelling different network topologies e.g. local area network. All the dependency measurements of traffic flow results are fed into a novel algorithm to construct functional topology of the network. Result shows that the algorithm constructs accurate functional topology of the network. Functional topology simplifies network topology by considering only nodes that create dependencies among traffic flows. With the help of this topology, end hosts can gain insight into resource provisioning of a network without requiring ISP assistance.

This thesis concerns the study of string topology, a relatively new branch of algebraic topology. We begin with a survey of the background of string topology. In particular, this includes a summary of the papers [4] by Chas and Sulivan, [15] by Jones and [5] by Cohen and Jones that provide the background for the original work of this thesis. We then proceed to give a new e cient technique to do systematic computations of the full structure of the string topology for a large family of manifolds. For this, we rst use the results of Jones [15] and Cohen and Jones [5] to reduce the problem to calculating Hochschild homology and cohomology. Secondly, we use the concept of models to compute Hochschild homology and cohomology and obtain some further Hochschild structure. Thus, most of this work is devoted to developing this technique for calculating Hochschild homology and cohomology via models. This research contributes to the area by providing the rst general and systematic method of computing the full structure of string topology. In addition, we give multiple, transparent examples of our new theory.

Master of Science<br>Department of Mathematics<br>David Auckly<br>This report introduces the fiber bundles. It includes the definitions of fiber bundles such as vector bundles and principal bundles, with some interesting examples. Reduction of the structure groups, and covering homotopy theorem and some specific computation using obstruction classes, Cech cohomology, Stiefel-Whitney classes, and first Chern classes are included.

In this thesis connections between categories of interior algebras and categories of topological spaces, and generalizations of topological concepts to interior algebras, are investigated. The following are some of the most significant results we obtain: The establishment of a duality between topological spaces and complete atomic interior algebras formalized in terms of a category-theoretic co-equivalence between the category of topological spaces and continuous maps and the category of complete atomic interior algebras and maps known as complete topomorphisms (Theorem 2.1.7). Under this co-equivalence, continuous open maps correspond to complete homomorphisms (Theorem 2.1.8). We also establish a duality between arbitrary interior algebras and structures known as Stone fields in terms of a co-equivalence between the category of interior algebras and topomorphisms (see Definition 1.1.8) and the category of Stone fields and their morphisms the field maps (Theorem 2.2.14). Under this co-equivalence weakly open field maps (see Definition 2.2.17) correspond to homomorphisms (Theorem 2.2.18). The well-known connection between pre-ordered sets and interior algebras is shown to be a special case of topological duality (see section 4 of chapter 2). The topological concepts of neighbourhoods, convergence and accumulation are generalized to interior algebras (Chapter 3), and are used to generalize the topological separation and compactness properties to interior algebras (Chapter 4 and Chapter 5). What is particularly interesting with regard to the separation properties is that most of them are first order properties of interior algebras (see Theorem 4.5.11). This should be contrasted with the situation for frames/locales [12] and topological model theory [10]. By generalizing the concept of α-separation to interior algebras we obtain an ω chain of strictly elementary classes of interior algebras all of which have hereditarily undecidable first order theories (Theorem 4.3.14). Characterizations of irreducibility properties for interior algebras are also found. These properties (subdirect irreducibility, finite subdirect irreducibility, direct indecomposability, simplicity and semi-simplicity) can be characterized in many different ways. Characterizations in terms of open elements (fixed points of the interior operator) are found (Theorem 1.3.18 and Theorem 1.3.21) and these are used to obtain further characterizations. In particular a characterization in terms of topological properties of Stone spaces of interior algebras is obtained (Theorem 2.3.9). We also find characterizations of the irreducibility properties in the power set interior algebras of topological spaces (Theorem 2.1.15) and in interior algebras obtained from pre-ordered sets (Theorem 2.4.16). What is particularly striking is that the irreducibility properties correspond to very natural topological properties. (Other results characterizing or related to the irreducibility properties are 2.4.11, 2.4.17, 5.1.13, and 5.1.15). Bibliography: pages 134-135.

Mestrado em Física da Matéria Condensada<br>The study of connectivity correlations between nodes has been somewhat neglected in the study of Complex Networks. We try to correct this by using the correlation function, combined with the concept of shell to calculate the connectivity distribution, P(d)(k) and the average connectivity for the neighbours, <knn (d)(k)> of a node as a function of distance d. With these results we create a better idea of how the Internet is organized and structured. We also determine that the time evolution of the Internet is coherent with the results obtained in the literature for the case of accelerated growth by the process of edge copying with a probability p=0.58.<br>O estudo das correlações de conectividade entre nodos tem sido algo negligenciado no estudo de Redes Complexas. Nós tentamos alterar esta situação usando funções de correlação em conjunto com o concenito de camada para calcular a distribuição de conectividades P(d)(k) e a conectividade média dos vizinhos <knn (d)(k)> de um nodo como função da distância d. Com estes resultados criamos uma melhor ideia acerca de como a Internet está organizada e estruturada. Concluimos também que a evolução da Internet é coerente com os resultados obtidos na literatura para o caso de crescimento acelerado devido a um processo de cópia de vértices com probabilidade p=0.58.

Doutoramento conjunto em Matemática - Matemática e Aplicações (PDMA)<br>Having as a starting point the characterization of probabilistic metric spaces as enriched categories over the quantale , conditions that allow the generalization of results relating Cauchy sequences, convergence of sequences, adjunctions of V-distributors and its representability are established. Equivalence between L-completeness and L-injectivity is also established. L-completeness is characterized via the Yoneda embedding, and injectivity is related with exponentiability. Another kind of completeness is considered and the formal ball model is analyzed.<br>Tendo como ponto de partida a caracterização de espaços métricos probabilísticos como categorias enriquecidas no quantal , estabelecemos condições que permitem a generalização de resultados que relacionam sucessões de Cauchy, convergência de sucessões, adjunções de Vdistribuidores e a sua representabilidade. Também estabelecemos a equivalência entre L-injectividade e L-completude. Caracteriza-se L-completude via a imersão de Yoneda, e injectividade é relacionada com exponenciabilidade. Considera-se outra forma de completude e analisa-se o modelo das bolas formais.

Static compensators (STATCOM) based on high-power converters are widely used for utilitiesand industrial applications in order to enhance the power system reliability. Nowadays,the Chain-Link Modular Multilevel Converter is the best solution for such applications,providing high eciency and reliability, and good harmonic performance. However, thecurrent delta and wye congurations present diculties in controlling negative sequencein unbalanced networks, as well as high capacitance requirements, which results in bulkySTATCOMs.This thesis aims to analyse a new Chain-Link Modular Multilevel Converter for STATCOMapplications. The main feature of the proposed topology is the presence of a commondc-link that will allow the exchange of energy between phases, facilitating the compensationof negative sequence components. Consequently, the required zero sequence componentinjection for the current Chain-Link congurations are avoided, and thus, also theconsequent over-sizing of the converter. Moreover, it is expected to have lower capacitancerequirements, because of the elimination of the second order harmonic (100 Hz)component in most of the capacitors, as they are charged and discharged by a bi-phasecurrent instead of a single-phase one. Additionally, lower conduction losses are expectedby reducing the number of devices in the conduction path.The layout and operation of the new converter have been analysed in this thesis.A STATCOM Simulink model provided by ABB has been modied to satisfy the newtopology. The theoretical benets of the proposed converter are supported by dierentsimulations carried out in Matlab-Simulink. In particular, it is shown a 50 % of negativesequence capability without any zero sequence component injection. The total capacitanceof the converter can be reduced a 30 % in comparison with the delta Chain-LinkModular Multilevel Converter, which is the preferable topology in the market nowadays.Furthermore, the losses are kept quite low if SiC semiconductors are used.On the other hand, the proposed topology requires the usage of bi-directional switchesto do the commutation and allow the exchange of energy between phases. An in-depthanalysis of the commutation process is shown at the end of this thesis, concluding that BidirectionalControlled Thyristors would be a promising solution for this converter topology.<br>Reaktiv effektkompensering (STATCOM) baserade på högeffektomvandlare används vidaför samhällsservice och industriella användningar för att förbättra elsystemet pålitliga.Nuförtiden, Chain-Link Modular Multilevel Converter är den bästa lösningen för sådanaanvändningar, vilket ger hög verkningsgrad och mycket pålitliga, samt låg harmonisk distortion.Hursomhelst, den nuvarande delta och wye konfigurationer presentera svårigheteratt kontrollera negativ sekvens i obalanserade nätverk, vilket leder till överdimensioneringav omvandlaren. Vidare, dessa Chain-Link topologier har hög kapacitans kraven, vilketresulterar i skrymmande STATCOMs.Den här avhandlingen syftar till att analysera en ny Chain-Link Modular MultilevelConverter för STATCOM användningar. Det viktigaste kännetecken av den föreslagnatopologi är närvaron av en gemensam likspänningslänk som gör det möjligt utbytet avenergi mellan faserna, som underlättar ersättning av negativ sekvens komponenter. Pådetta sätt, den behövlig noll sekvens komponent injektion i de aktuella konfigurationernaundviks, och således, också den därav följande överdimensioneringen av omvandlaren.Dessutom, förväntas det att ha lägre kapacitans kraven, eftersom den första övertonens(100 Hz) komponent undviks i några av kondensatorerna, såsom de är laddas och urladdasav en bi-fas ström i stället för en enda-fas ett. Vidare, lägre ledningsförluster förväntasgenom att reducera antalet enheter i ledningsvågen.Layouten och drift av den nya omvandlaren beskrivs i denna avhandling. En Simulinkmodell från ABB har modifierats för att tillfredsställa den nya topologin. De teoretiskafördelarna med det föreslagna omvandlare topologi stöds av simuleringsresultat i Matlab-Simulink. I synnerhet, är det visas en 50 % av negativ sekvens kapacitet utan noll sekvenskomponent injektion. Den totala kapacitansen hos omvandlaren kan minskas på ett 30 %jämfört med den delta Chain-Link Modular Multilevel Converter, som är den föredragnatopologin på marknaden idag. Vidare är förlusterna hålls ganska låg om SiC halvledareanvänds.Å andra sidan, kräver den föreslagna topologi användningen av dubbelriktade omkopplareatt göra kommuteringen och möjliggöra utbytet av energi mellan faserna. Enfördjupad analys av kommuteringsförloppet visas i slutet av denna avhandling, slutsatsenatt Bi-directional Controlled Tyristors skulle vara en lovande lösning för topologin.

We consider a compact, oriented,smooth Riemannian manifold $M$ (with or without boundary) and wesuppose $G$ is a torus acting by isometries on $M$. Given $X$ in theLie algebra of $G$ and corresponding vector field $X_M$ on $M$, onedefines Witten's inhomogeneous coboundary operator $\d_{X_M} =\d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariantforms on $M$) and its adjoint $\delta_{X_M}$. First, Witten [35] showed that the resulting cohomology classeshave $X_M$-harmonic representatives (forms in the null space of$\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomologygroups are isomorphic to the ordinary de Rham cohomology groups ofthe set $N(X_M)$ of zeros of $X_M$. The first principal purpose isto extend Witten's results to manifolds with boundary. Inparticular, we define relative (to the boundary) and absoluteversions of the $X_M$-cohomology and show the classes haverepresentative $X_M$-harmonic fields with appropriate boundaryconditions. To do this we present the relevant version of theHodge-Morrey-Friedrichs decomposition theorem for invariant forms interms of the operators $\d_{X_M}$ and $\delta_{X_M}$; the proofinvolves showing that certain boundary value problems are elliptic.We also elucidate the connection between the $X_M$-cohomology groupsand the relative and absolute equivariant cohomology, followingwork of Atiyah and Bott. This connection is then exploited to showthat every harmonic field with appropriate boundary conditions on$N(X_M)$ has a unique corresponding an $X_M$-harmonic field on $M$to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of $X_M$-cohomology and then we definethe \emph{$X_M$-Poincar\' duality angles} between the interiorsubspaces of $X_M$-harmonic fields on $M$ with appropriate boundaryconditions.Second, In 2008, Belishev and Sharafutdinov [9] showed thatthe Dirichlet-to-Neumann (DN) operator $\Lambda$ inscribes into thelist of objects of algebraic topology by proving that the de Rhamcohomology groups are determined by $\Lambda$.In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?.Based on the results in the first part above, we define an operator$\Lambda_{X_M}$ on invariant forms on the boundary $\partial M$which we call the $X_M$-DN map and using this we recover the longexact $X_M$-cohomology sequence of the topological pair $(M,\partialM)$ from an isomorphism with the long exact sequence formed from thegeneralized boundary data. Consequently, This shows that for aZariski-open subset of the Lie algebra, $\Lambda_{X_M}$ determinesthe free part of the relative and absolute equivariant cohomologygroups of $M$. In addition, we partially determine the mixed cup product of$X_M$-cohomology groups from $\Lambda_{X_M}$. This shows that $\Lambda_{X_M}$ encodes more information about theequivariant algebraic topology of $M$ than does the operator$\Lambda$ on $\partial M$. Finally, we elucidate the connectionbetween Belishev-Sharafutdinov's boundary data on $\partial N(X_M)$and ours on $\partial M$.Third, based on the first part above, we present the(even/odd) $X_M$-harmonic cohomology which is the cohomology ofcertain subcomplex of the complex $(\Omega^{*}_G,\d_{X_M})$ and weprove that it is isomorphic to the total absolute and relative$X_M$-cohomology groups.

Thesis (doctoral)--Waseda University, 1988.<br>Accompanied by summary (6 p, ; 26 cm.) in Japanese. Includes bibliographical references (leaves 85-94). "List of papers by Takashi Nishimura": leaves 95.

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.

Thesis: Ph. D. in Organic Chemistry, Massachusetts Institute of Technology, Department of Chemistry, 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references.<br>Chapter 1: Introduction to Polymer Network Topology on a (Macro)Molecular Level Polymer network topology, comprising the ways in which strands and junctions are connected in polymer networks, plays a critical role in dictating many material properties. Here we discuss classical challenges in the field and review existing strategies to characterize and manipulate polymer network topology from a (macro)molecular level. Chapter 2: Semibatch Monomer Addition as a General Method to Tune and Enhance the Mechanics of Polymer Networks via Loop-defect Control In this chapter we introduce semibatch monomer addition as a general strategy to reduce/control an important topological feature at short length scale-primary loops, thus providing materials with tunable and significantly improved mechanical properties without changing their composition.<br>Chapter 3: Leaving Groups as Traceless Topological Modifiers for Controlling Topological Structure in Chemically Identical Polymer Networks Here we report "traceless topological modification" as a general approach to control an important long length-scale topological feature-junction distribution. Using self-assembled structures as templates that are not themselves incorporated into the network, our method enables us to synthesize truly topologically isomeric networks with drastically different macroscopic properties. Chapter 4: Photoswitching Topology in Polymer Networks with Metal-Organic Cages as Crosslinks Based on our works in Chapter 2 and Chapter 3, we further explored topology as the central design principle to create novel functional materials.<br>In this chapter we introduce topology switching via cooperative self-assembly as a design principle to reversibly alter multiple network properties simultaneously and enable the preparation of one material that can exist in multiple topological states. Chapter 5: Living Additive Manufacturing: Transformation of Parent Gels into Diversely Functionalized Daughter Gels Made Possible by Visible Light Photoredox Catalysis Our ability to control polymer network topology has been further enhanced by developing living additive manufacturing as an effective strategy to expand the original topology of parent networks in a photo-growth fashion. This approach enables us to transform the mechanical/physical properties of parent networks post-synthetically.<br>Chapter 6: polyMOF Nanoparticles: Dual Roles of a Multivalent polyMOF Ligand in Size Control and Surface Functionalization Here we present a novel approach to synthesizing well defined metal-organic framework nanoparticles (MOF NPs), where the size control and surface functionalization of MOF-5 NPs were simultaneously achieved using multivalent polyMOF ligands.<br>by Yuwei Gu.<br>Ph. D. in Organic Chemistry<br>Ph.D.inOrganicChemistry Massachusetts Institute of Technology, Department of Chemistry

Mémoire rédigé en vue de l'obtention de l'habilitation à diriger les recherches. Il donne un résumé de mon activité de recherche (anneaux de Chow, classes de Stiefel-Whitney, algèbres de Hopf, entrelacs, K-théorie de Milnor).

Orientador: Ariane Luzia dos Santos<br>Banca: Alice Kimie Miwa Libardi<br>Banca: Érica Regina Filletti Nascimento<br>Resumo: Este trabalho aborda em especial a análise dos pontos críticos de uma função diferenciável. Fazemos inicialmente uma abordagem sobre funções diferenciáveis com duas variáveis e outros temas necessários para a compreensão de algumas demonstrações e conceitos que serão abordados neste trabalho e em seguida apresentamos uma abordagem sobre curvas e superfícies. Depois, apresentamos um estudo sobre pontos críticos e as funções de Morse, que estão relacionadas ao estudo dos pontos críticos não degenerados de uma função diferenciável f: X → IR em uma superfície, e mostramos ainda que toda função diferenciável em torno de um ponto crítico não degenerado pode ser escrita como um polinômio quadrático. Para finalizar o trabalho, fazemos uma proposta de abordagem dos pontos críticos de uma função diferenciável destinada à 3ª série do ensino médio usando o conceito de derivada com uma variável<br>Abstract: This work deals in particular with the analysis of the critical points of a differentiable function. We make an initial approach on differentiable functions with two variables and other topics necessary for the understanding of the sampled concepts and concepts that will be approached in this work and next we present an approach on curves and surfaces.Then, we present a study on critical points and Morse functions, which are related to the study of the nondegenerate critical points of a differentiable f: X → IR function on a surface, and we show that any differentiable function around a nondegenerated critical point can be written as a quadratic polynomial.To finalize the work, we make a proposal to approach the critical points of a differentiable function destined for the 3rd grade of high school using the concept of derivative with one variable<br>Mestre

Neste trabalho estudamos alguns aspectos de geometria e topologia de variedades com o objetivo de aplicar tais resultados juntamente com dados observacionais para tentar determinar as variedades tridimensionais que possam servir de modelo para a parte espacial do universo.<br>In this work we study some aspects of geometry and topology of manifolds with the goal of applying such results with observational data to try to determine the tridimensional manifold that can serve as a model for the spatial part of the universe.

Three-dimensional scans and other volumetric data sources often result in representations that are more complex topologically than the original model. The extraneous critical points, handles, and components are called topological noise. Many algorithms in computer graphics require simple topology in order to work optimally, including texture mapping, surface parameterization, flows on surfaces, and conformal mappings. The topological noise disrupts these procedures by requiring each small handle to be dealt with individually. Furthermore, topological descriptions of volumetric data are useful for visualization and data queries. One such description is the contour tree (or Reeb graph), which depicts when the isosurfaces split and merge as the isovalue changes. In the presence of topological noise, the contour tree can be too large to be useful. For these reasons, an important goal in computer graphics is simplification of the topology of volumetric data. The key to this thesis is that the global topology of volumetric data sets is determined by local changes at individual points. Therefore, we march through the data one grid cell at a time, and for each cell, we use a local check to determine if the topology of an isosurface is changing. If so, we change the value of the cell so that the topology change is prevented. In this thesis we describe variations on the local topology check for use in different settings. We use the topology simplification procedure to extract a single component with controlled topology from an isosurface in volume data sets and partially-defined volume data sets. We also use it to remove critical points from three-dimensional volumes, as well as time-varying volumes. We have applied the technique to two-dimensional (plus time) data sets and three dimensional (plus time) data sets.

The goal of this thesis is to understand the topology of representation varieties. To be more precise, let G be a complex reductive linear algebraic group and let K ⊂ G be a maximal compact subgroup. Given a finitely generated nilpotent group Γ, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) endowed with the compact-open topology. Our main result shows that there is a strong deformation retraction of Hom(Γ,G) onto Hom(Γ,K). We also obtain a strong deformation retraction of the geometric invariant theory quotient Hom(Γ,G)//G onto the ordinary quotient Hom(Γ,K)/K. Using these deformations, we then describe the topology of these spaces.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate

This thesis investigates topology optimization techniques for periodic continuum structures at the macroscopic level. Periodic structures are increasingly used in the design of structural systems and sub-systems of buildings, vehicles, aircrafts, etc. The duplication of identical or similar modules significantly reduces the manufacturing cost and greatly simplifies the assembly process. Optimization of periodic structures in the micro level has been extensively researched in the context of material design, while research on topology optimization for macrostructures is very limited and has great potential both economically and intellectually. In the present thesis, numerical algorithms based on the bi-directional evolutionary structural optimization method (BESO) are developed for topology optimization for various objectives and constraints. Soft-kill (replacing void elements with soft elements) formulations of topology optimization problems for solid-void solutions are developed through appropriate material interpolation schemes. Incorporating the optimality criteria and algorithms for mesh-independence and solution-convergence, the present BESO becomes a reliable gradient based technique for topology optimization. Additionally, a new combination of genetic algorithms (GAs) with BESO is developed in order to stochastically search for the global optima. These enhanced BESO algorithms are applied to various optimization problems with the periodicity requirement as an extra constraint aiming at producing periodicity in the layout. For structures under static loading, the present thesis addresses minimization of the mean compliance and explores the applications of conventional stiffness optimization for periodic structures. Furthermore, this thesis develops a volume minimization formulation where the maximum deflection is constrained. For the design of structures subject to dynamic loading, this thesis develops two different approaches (hard-kill and soft-kill) to resolving the problem of localized or artificial modes. In the hard-kill (completely removing void elements) approach, extra control measures are taken in order to eliminate the localized modes in an explicit manner. In the soft-kill approach, a modified power low material model is presented to prevent the occurrence of artificial and localized modes. Periodic stress and strain fields cannot be assumed in structures under arbitrary loadings and boundaries at the macroscopic level. Therefore being different from material design, no natural base cell can be directly extracted from macrostructures. In this thesis, the concept of an imaginary representative unit cell (RUC) is presented. For situations when the structure cannot be discretized into equally-sized elements, the concept of sensitivity density is developed in order for mesh-independent robust solutions to be produced. The RUC and sensitivity density based approach is incorporated into various topology optimization problems to obtain absolute or scaled periodicities in structure layouts. The influence of this extra constraint on the final optima is investigated based on a large number of numerical experiments. The findings shown in this thesis have established appropriate techniques for designing and optimizing periodic structures. The work has provided a solid foundation for creating a practical design tool in the form of a user-friendly computer program suitable for the conceptual design of a wide range of structures.

State-of-the-art methods for topology of α-helical membrane proteins are based on the use of time-consuming multiple sequence alignments obtained from PSI-BLAST or other sources. Here, we examine if it is possible to use the consensus of topology prediction methods that are based on single sequences to obtain a similar accuracy as the more accurate multiple sequence-based methods. Here, we show that TOPCONS-single performs better than any of the other topology prediction methods tested here, but ~6% worse than the best method that is utilizing multiple sequence alignments. AVAILABILITY AND IMPLEMENTATION: TOPCONS-single is available as a web server from http://single.topcons.net/ and is also included for local installation from the web site. In addition, consensus-based topology predictions for the entire international protein index (IPI) is available from the web server and will be updated at regular intervals.