<p>We study how the preimages of a mapping f : X &rarr Y between manifolds vary under perturbations. First, we consider the preimage of a single point and track the history of its connected component as this point varies in Y. This information is compactly represented in a structure that is the generalization of the Reeb graph we call the Reeb space. We study its local and global properties and provide an algorithm for its construction. Using homology, we then consider higher dimensional connectivity of the preimage. We develop a theory quantifying the stability of each homology class under perturbations of the mapping f . This number called robustness is given to each homology class in the preimage. The robustness of a class is the magnitude of the perturbation necessary to remove it from the preimage. The generality of this theory allows for many applications. We apply this theory to quantify the stability of contours, fixed points, periodic orbits, and more.</p> / Dissertation

We construct Fuchsian groups [Gamma] of signature (0 : 2, ... ,2 ;1;0) so that the set of hyperbolic fixed points of [Gamma] will contain a given finite collection of elements in the boundary of the hyperbolic plane. We use this to establish that there are infinitely many non-commensurable non-cocompact Fuchsian groups [Delta] of finite covolume sitting in PSL₂(Q) so that the set of hyperbolic fixed points of [Delta] will contain a given finite collection of rational boundary points of the hyperbolic plane. We also give a parameterization of Fuchsian groups of signature (0:2,2,2;1;0) and investigate when particular hyperbolic elements have rational fixed points. Moreover, we include a detailed list of the group elements and their killer intervals for the known pseudomodular groups that Long and Reid found; in addition, the list contains a new list of killer intervals for a pseudomodular group not found by Long and Reid. / text

When many people think of fractals, they think of the beautiful images created by Mandelbrot’s set or the intricate dragons of Julia’s set. However, these are just the artistic stars of the fractal community. The theory behind the fractals is not necessarily pretty, but is very important to many areas outside the world of mathematics. This paper takes a closer look at various types of fractals, the fractal dimensionality of surfaces and chaotic dynamical systems. Some of the history and introduction of creating fractals is discussed. The tools used to prevent a modified Koch’s curve from overlapping itself, finding the limit of a curves length and solving for a surfaces dimensional measurement are explored. Lastly, an investigation of the theories of chaos and how they bring order into what initially appears to be random and unpredictable is presented. The practical purposes and uses of fractals throughout are also discussed. / text

Made available in DSpace on 2014-07-29T16:23:34Z (GMT). No. of bitstreams: 1 tese Alexandre Curcino.pdf: 1373269 bytes, checksum: 1a45fcde1db075780cddc670a30bd516 (MD5) Previous issue date: 2011-03-25 / In the fixed point methodology, the observer remains for a predetermined time, recording all birds registered by observation or hearing. Widely used in temperate regions, the methodology has been used in Brazil, where it has suffered adaptations through the years. Whereas the current studies show different sampling times of methodology (10, 15 and 20 min), the aim of this survey was to compare the birds richness estimated for the sampling times of 10 and 20 minutes, in cerrado, gallery forest and vereda, in the regions of Niquelândia and Barro Alto - Goias and verify the effectiveness of bird detections per hour of study for different sampling situations and travel times between fixed points. The survey in Niquelândia occurred in 2007 and 2008. Barro Alto, in the years 2008 and 2009. At each location were established 20 point counts, and sampled five points between 6h00min and 8h40min. The points were drawn for 10 min and 20 min sampling . The results suggest that the researcher who remains 10 min at each point obtain the same statistical results that the researcher who remains 20 min at each point. The species detection efficiency is related to the balance in the choice of sampling time and displacement in order to increase the chances of finding rare species, and at the same time, maximize the number of detections. / Na metodologia de pontos fixos, o observador permanece parado por um tempo pré-determinado, anotando todas as aves registradas por observação ou audição Amplamente utilizada em regiões de clima temperado, a metodologia tem sido utilizada no Brasil, onde sofreu adaptações ao longo dos anos. Considerando que os estudos atuais apresentam diferentes tempos de amostragem para a metodologia de pontos fixos (em geral 10, 15 e 20 min), nosso objetivo foi comparar a riqueza de aves estimada para os tempos de amostragem de 10 e 20 minutos de permanência em cada ponto, em seis fitofisionomias de Cerrado e estimar a eficiência de detecções de aves por hora de estudo, com base no tempo de deslocamento e no tempo de permanência nos pontos. Foram realizadas 12 campanhas nas regiões de Niquelândia GO e 12 campanhas em Barro Alto GO. Em Niquelândia, as coletas ocorreram em 2007 e 2008. Em Barro Alto, nos anos de 2008 e 2009. Em cada local foram estabelecidos 20 pontos de observação, sendo amostrados cinco pontos após sorteio prévio, entre 6h00min e 8h40min. Os pontos foram sorteados de modo a alternarem amostragens de 10 min e de 20 min. Os resultados sugerem que o pesquisador que opta por permanecer 10 min em cada ponto obterá, ao final da amostragem, um número estatisticamente equivalente de novas espécies que o obtido pelo pesquisador que opta por permanecer 20 min em cada ponto. A eficiência de detecções de espécies está relacionada ao equilíbrio na escolha do tempo de amostragem e deslocamento de modo a aumentar as chances de encontrar espécies raras e, ao mesmo tempo, maximizar o número de detecções como um todo.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22823号 / 理博第4633号 / 新制||理||1666(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Orientador: Prof. Dr. Alysson Fábio Ferrari / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017. / Na literatura encontramos argumentos tanto fenomenológicos quanto teóricos que favorecem o congelamento da constante de acoplamento da QCD a valores moderados no regime infravermelho. O acoplamento pode ser parametrizado em termos de uma massa efetiva para o gluon (mg) obtida dinamicamente através das equações de Schwinger- Dyson, cuja soluções são compatíveis com as simulações da QCD na rede. Primeiro nós consideramos o processo de aniquilação elétron-pósitron em hádrons Re+e- até O(3s) e adotamos o método de smearing sugerido por Poggio, Quinn e Weinberg para confrontar os dados experimentais com a teoria. Nós vamos usar como modelo teórico a QCD com uma constante de acoplamento finita no regime de baixas energias. Para encontrar o melhor fit entre os dados experimentais e teóricos, nós realizamos um test de 2, que dentro das incertezas do modelo , tem um valor mínimo quando mg=QCD está entre 1.2 - 1.4. Esses valores concordam com outras determinações fenomenológicas da razão mg=QCD e levam a uma carga efetiva s(0) 0.7. Nós comentamos como essas cargas efetivas poderiam afetar a escala de massa da dualidade global, a qual indica a fronteira entre a física perturbativa e não perturbativa. Calculamos tanto o potencial efetivo aprimorado no caso da QED escalar e da QCD com um escalar sem cor, como também a evolução do acoplamento escalar do Higgs () no Modelo Padrão. Em ambos os casos consideramos pontos fixos. No caso da QCD com o escalar sem carga de cor tanto a barreira associada ao polo de Landau quanto o mínimo do potencial mudam. Por outro lado, encontramos que a existência dos pontos fixos não perturbativos no infravermelho movem a evolução do acoplamento escalar na direcção da estabilidade. Para certos valores da constante de acoplamento da QCD no infravermelho, o potencial do Modelo Padrão pode ficar estável até a escala de Planck. / Several phenomenological and theoretical arguments favor a freezing of the Quantum Chromodynamics (QCD) coupling constant in the infrared region at one moderate value. This coupling can be parameterized in terms of an effective dynamical gluon mass (mg) which is determined through Schwinger-Dyson equations, whose solutions are compatible with QCD lattice simulations. First we consider the electron-positron annihilation process into hadrons Re+e- up to O(3s) and we adopt the smearing method suggest by Poggio, Quinn and Weinberg to confront the experimental data with theory. As a theoretical model we use the aforementioned QCD coupling constant frozen in the low energy regime. In order to find the best fit between experimental data and theory we perform a 2 study, that, within the uncertainties of the approach, has a minimum value when mg=QCD is in the range 1.2 - 1.4. These values are in agreement with other phenomenological determinations of this ratio and lead to an infrared effective charge s(0) 0.7. We comment how this effective charge may affect the global duality mass scale that indicates the frontier between perturbative and nonperturbative physics. We also compute the improved effective potential in the case of scalar QED and QCD with a colorless scalar and compute the Standard Model scalar boson Higgs coupling () evolution. In both cases we consider fixed points. In the case of QCD with a colorless scalar not only the barrier associated to the Landau pole is changed but the local minimum of the potential is also changed. On the other hand we find that the existence of such nonperturbative infrared fixed point moves the evolution towards stability. For the phenomenological preferred IR value of the QCD coupling constant the standard model Higgs potential may be stable up to the Planck scale.

Orientador: Maria Gorete Carreira Andrade / Banca: Denise de Mattos / Banca: Ermínia de Lourdes Campello Fanti / Resumo: Em Topologia, mais especificamente em Topologia Algébrica, temos alguns resultados clássicos que de alguma forma estão relacionados. No desenvolvimento deste trabalho, estudamos alguns desses resultados, a saber: Teorema de Lefschetz-Hopf, Teorema do Ponto Fixo de Lefschetz, Teorema do Ponto Fixo de Brouwer, Teorema da Curva de Jordan e o Teorema Clássico de Borsuk-Ulam. Além disso, tivemos como objetivo principal mostrar relações existentes entre esses teoremas a partir do Teorema de Lefschetz-Hopf. / Abstract: In Topology, more specifically in Algebraic Topology, we have some classical results that are in some way related. In developing this work, we studied some of these results, namely the Lefschetz-Hopf Theorem, the Lefschetz Fixed Point Theorem, the Brouwer Fixed Point Theorem, the Jordan Curve Theorem and the Classic Borsuk-Ulam Theorem. Moreover, our main objective was to show relationships among those theorems by using Lefschetz-Hopf Theorem. / Mestre

Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

Activities in the motor cortex are found to be dynamical in nature. Modeling these activities and comparing them with neural recordings helps in understanding the underlying mechanism for the generation of these activities. For this purpose, Recurrent Neural networks or RNNs, have emerged as an appropriate tool. A clear understanding of how the design choices associated with these networks affect the learned dynamics and internal representation still remains elusive. A previous work exploring the dynamical properties of discrete time RNN architectures (LSTM, UGRNN, GRU, and Vanilla) such as the fixed point topology and the linearised dynamics remains invariant when trained on 3 bit Flip- Flop task. In contrast, they show that these networks have unique representational geometry. The goal for this work is to understand if these observations also hold for networks that are more biologically realistic in terms of neural activity. Therefore, we chose to analyze rate networks that have continuous dynamics and biologically realistic connectivity constraints and on spiking neural networks, where the neurons communicate via discrete spikes as observed in the brain. We reproduce the aforementioned study for discrete architectures and then show that the fixed point topology and linearized dynamics remain invariant for the rate networks but the methods are insufficient for finding the fixed points of spiking networks. The representational geometry for the rate networks and spiking networks are found to be different from the discrete architectures but very similar to each other. Although, a small subset of discrete architectures (LSTM) are observed to be close in representation to the rate networks. We show that although these different network architectures with varying degrees of biological realism have individual internal representations, the underlying dynamics while performing the task are universal. We also observe that some discrete networks have close representational similarities with rate networks along with the dynamics. Hence, these discrete networks can be good candidates for reproducing and examining the dynamics of rate networks. / Aktiviteter i motorisk cortex visar sig vara dynamiska till sin natur. Att modellera dessa aktiviteter och jämföra dem med neurala inspelningar hjälper till att förstå den underliggande mekanismen för generering av dessa aktiviteter. För detta ändamål har återkommande neurala nätverk eller RNN uppstått som ett lämpligt verktyg. En tydlig förståelse för hur designvalen associerade med dessa nätverk påverkar den inlärda dynamiken och den interna representationen är fortfarande svårfångad. Ett tidigare arbete som utforskar de dynamiska egenskaperna hos diskreta RNN- arkitekturer (LSTM, UGRNN, GRU och Vanilla), såsom fastpunkts topologi och linjäriserad dynamik, förblir oförändrad när de tränas på 3-bitars Flip- Flop-uppgift. Däremot visar de att dessa nätverk har unik representationsgeometri. Målet för detta arbete är att förstå om dessa observationer också gäller för nätverk som är mer biologiskt realistiska när det gäller neural aktivitet. Därför valde vi att analysera hastighetsnätverk som har kontinuerlig dynamik och biologiskt realistiska anslutningsbegränsningar och på spikande neurala nätverk, där neuronerna kommunicerar via diskreta spikar som observerats i hjärnan. Vi reproducerar den ovannämnda studien för diskreta arkitekturer och visar sedan att fastpunkts topologi och linjäriserad dynamik förblir oförändrad för hastighetsnätverken men metoderna är otillräckliga för att hitta de fasta punkterna för spiknätverk. Representationsgeometrin för hastighetsnätverk och spiknätverk har visat sig skilja sig från de diskreta arkitekturerna men liknar varandra. Även om en liten delmängd av diskreta arkitekturer (LSTM) observeras vara nära i förhållande till hastighetsnäten. Vi visar att även om dessa olika nätverksarkitekturer med varierande grad av biologisk realism har individuella interna representationer, är den underliggande dynamiken under uppgiften universell. Vi observerar också att vissa diskreta nätverk har nära representationslikheter med hastighetsnätverk tillsammans med dynamiken. Följaktligen kan dessa diskreta nätverk vara bra kandidater för att reproducera och undersöka dynamiken i hastighetsnät.

[pt] Lâmpadas de fita de tungstênio de alta estabilidade não são mais adequadas como artefatos de comparação para termometria de radiação de alta temperatura, por conta de serem frágeis, terem um tamanho de alvo pequeno, serem limitadas com relação a faixa de temperatura e não serem corpos negros. Este estudo desenvolveu protótipos de artefatos de comparação de alto desempenho, os quais podem superar os problemas existentes ao usar tais lâmpadas em comparações da Escala Internacional de Temperatura de 1990 (EIT-90) entre Institutos Nacionais de Metrologia (INMs). Este trabalho demonstrou o conceito de utilização de pontos fixos de alta temperatura (PFATs) que tenham temperaturas desconhecidas e assim sejam adequados como artefatos de comparações às cegas. Quatro destes novos PFATs foram projetados, construídos, preenchidos e medidos no trabalho aqui descrito. Inicialmente Co-C foi escolhido, mas devido a problemas de robustez, a liga eutética base selecionada foi Ni-C. As células Ni-C foram dopadas em duas concentrações diferentes com elementos selecionados em uma tentativa bem sucedida de modificar temperatura de transição do eutético puro em alguns décimos de graus Celsius. As temperaturas de realização das células eutéticas determinadas no Inmetro foram comparadas com aquelas previstas por simulação termoquímica, usando o programa Thermo-Calc e bancos de dados de propriedades termoquímicas adequados. Além disso, elas foram utilizadas para realizar uma comparação às cegas com o National Physical Laboratory (NPL - UK), o qual não sabia a priori quais eram essas temperaturas. Resultados muito bons foram alcançados (concordância das escalas e estabilidade das células), demonstrando que as células dopadas são artefatos de comparação de alta temperatura bem adequados para termometria de radiação. / [en] High stability tungsten strip lamps are no longer suitable comparison artefacts for high temperature radiation thermometry, because they are fragile, have a small target size, are restricted in temperature range and are not blackbodies. This study developed proof-of-concept high performance comparison artefacts, which overcome the problems encountered when using such lamps in comparisons of the International Temperature Scale of 1990 (ITS90) among National Metrology Institutes (NMIs). This work demonstrated the concept of using high temperature fixed points (HTFPs) that have unknown temperatures and hence suitable as blind comparison artefacts. Four of these novel HTFPs were designed, constructed, filled and measured in the work described here. Initially Co-C was chosen but due to robustness issues, Ni-C was the selected the base eutectic alloy. The Ni-C cells were doped in two different concentrations with selected elements in a successful attempt to change the pure eutectic transition temperature by some tenths of degrees Celsius. The realization temperatures of eutectic cells determined at Inmetro were compared to the ones predicted by thermochemical simulation, using Thermo-Calc software and thermochemical property databases. In addition they were used to perform a blind comparison with the National Physical Laboratory (UK), which did not know beforehand what their temperatures were. Very good results were achieved (scale agreement and cell stability), demonstrating that doped cells are very suitable high temperature comparison artefacts for radiation thermometry.