Thesis advisor: Kevin S. Bedell / We present several results based upon non-trivial extensions of Landau-Fermi liquid theory. First proposed in the mid-20th century, the Fermi liquid approach assumes an adiabatic “switching-on” of the interaction, which allows one to describe the collective excitations of the many-body system in terms of weakly-interacting quasiparticles and quasiholes. At its core, Landau-Fermi liquid theory is often considered a perturbative approach to study the equilibrium thermodynamics and out-of-equilibrium response of weakly-correlated itinerant fermions, and therefore non-trivial extensions and consequences are usually overlooked in the contemporary literature. Instead, more emphasis is often placed on the breakdown of Fermi liquid theory, either due to strong correlations, quantum critical fluctuations, or dimensional constraints. After a brief introduction to the theory of a Fermi liquid, I will first apply the Landau quasiparticle paradigm to the theory of itinerant Majorana-like fermions. Defined as fermionic particles which are their own anti-particle, traditional Majorana zero modes found in topological materials lack a coherent number operator, and therefore do not support a Fermi liquid-like ground state. To remedy this, we will apply a combinatorical approach to build a statistical theory of self-conjugate particles, explicitly showing that, under this definition, a filled Fermi surface exists at zero temperature. Landau-Fermi liquid theory is then used to describe the interacting phase of these Majorana particles, from which we find unique signatures of zero sound in addition to exotic, non-analytic contributions to the specific heat. The latter is then exploited as a “smoking-gun” signature for Majorana-like excitations in the candidate Kitaev material Ag3LiIr2O6, where experimental measurements show good agreement with a sharply-defined, “Majorana-Fermi surface” predicted in the underlying combinatorial treatment. I will then depart from Fermi liquid theory proper to tackle the necessary conditions for the applicability of Luttinger’s theorem. In a nutshell, Luttinger’s theorem is a powerful theorem which states that the volume of phase space contained in the Fermi surface is invariant with respect to interaction strength. In this way, whereas Fermi liquid only describes fermionic excitations near the Fermi surface, Luttinger’s theorem describes the fermionic degrees of freedom throughout the entire Fermi sphere. We will show that Luttinger’s theorem remains valid only for certain frequency and momentum-dependencies of the self-energy, which correlate to the exis- tence of a generalized Fermi surface. In addition, we will show that the existence of a power-law Green’s function (a unique feature of “un-particle” systems and a proposed characteristic of the pseudo-gap phase of the cuprate superconductors) forces Luttinger’s theorem and Fermi liquid theory to be mutually exclusive for any non-trivial power of the Feynman propagator. Finally, we will return to Landau-Fermi liquid theory, and close with novel out-of-equilibrium behavior and stability in unconventional Fermi liquids. First, we will consider a perfectly two- dimensional Fermi liquid. Due to the reduction in dimension, the traditional mode expansion in terms of Legendre polynomials is modified to an expansion in terms of Chebyshev polynomials. The resulting orthogonality conditions greatly modifies the stability and collective modes in the 2D system. Second, we will look at a Fermi liquid in the presence of a non-trivial gauge field. The existence of a gauge field will effectively shift the Fermi surface in momentum space, resulting in, once again, a modified stability condition for the underlying Fermi liquid. Supplemented with a modernized version of Mermin’s condition for the propagation of zero sound, we outline the full effects a spin symmetric or anti-symmetric gauge would have on a Fermi liquid ground state. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The aim of this paper was to investigate whether infrastructure could be modelled in order to find critical points using network science and graph theory. Since a lot of information about ourinfrastructures is publicly available, an attacker might exploit this to find vulnerabilities in our systems. With that in mind, the methods of this paper were implemented from an attackers point of view who’d want to maximize harm with minimal effort. The Swedish railway network served as an example and starting point for the optimization of the methods. The data for the network was obtained from the Swedish Transport Agency’s website and was implemented in Networkx using Python. Different centrality metrics were used to identify and remove critical nodes in the network. The centrality metrics were also used to rank the nodes and to remove them after order of importance as the size of the largest component was recorded. This was done both with no recalculation and with a recalculation after each removal. The results were compared with the random removal of nodes, and it showed that without a recalculation the random removal performed better but with recalculation all the centrality metrics performed significantly better. The Swedish railway network is a sensitive network in terms of how it is connected. Targeting a few nodes could completely cut off parts of the graph, creating two sub components with no way of traveling between them. Many different methods can identify these nodes, but only a few are suitable in terms of generic results that could work on other networks as well. The most prominent one being the one dealing with degree and connectivity. The nodes that had a higher degree than connectivity between themselves and the source, identified nodes that would completely disrupt the network, leaving two subcomponents. Infrastructures can be seen as highly intertwined systems, where the physical-, cyber- and human parts all affect and interact with each other.

The Al₂SiO₅ polymorphs have been examined to provide new insights into their chemical bonding, their crystal chemistry, their equations of state, and the incorporation of water in the form of hydroxyl in their structures. The Al₂SiO₅ polymorphs provide a unique structural assemblage for a crystal chemical examination due to the variation in Al coordination in the structures where Al is in 4-fold, 5-fold, and 6-fold in sillimanite, andalusite, and kyanite, respectively. Consequently, the Al₂SiO₅ polymorphs have been examined with a combination of experimental (high pressure X-ray diffraction and Polarized FTIR spectroscopy) and theoretical (VASP and Crystal 98) methods. An experimental high pressure X-ray diffraction study on andalusite and sillimanite has constrained their equation of state and the pressure derivatives of their bulk modulus with pressure. Additionally, the effect of pressure on the crystal structures has been examined, where the main structural response is compression of the AlO₆ octahedra. Comparatively, compression of the AlO₆ octahedra in andalusite is more anisotropic, while the major direction of axial compressibility in both structures is dependent on the orientation of the AlO6 octahedra. In order to better understand the crystal chemistry of the Al-O and Si-O bonds in the polymorphs, ELF isosurfaces were examined. ELF isosurfaces represent a graphical representation of the localized electron probability density. Six distinct types of ELF isosurfaces were observed in the Al₂SiO₅ polymorphs resulting from differences in the geometry, coordination, and coordinated cation atomic number surrounding the oxygens within the crystal structures. The ELF was also shown to be isostructurally related to electron density difference maps. In a combined experimental and theoretical investigation of the Al₂SiO₅ polymorphs, potential protonation sites within the crystal structures were determined at an atomic level with polarized FTIR spectroscopy and analysis of (3,-3) critical points of the negative Laplacian. The polarized FTIR spectra indicate the orientation of the OH dipole in the three polymorphs and the (3,-3) critical points indicate regions of locally concentrated electron density. Potential protonation sites were determined based on the value of the negative Laplacian, the underbonded nature of the oxygens, and the number of surrounding cations. / Ph. D.

Os números de ocorrências em acidentes de trânsito são assustadores. Avalia-se que o custo dos acidentes aproxime de 1% do PIB em países pobres e 2% nas nações desenvolvidas. Em termos globais, são gastos U$ 518 bilhões anuais. No Brasil, estudos apontam que houve uma queda de 20 mil vítimas fatais de acidentes de trânsito, de 1987 para 1998, apresentando, atualmente, em torno de 30 mil mortes/ano no trânsito. Essa redução pode ser em parte justificada pelo Código de Trânsito Brasileiro, que entrou em vigor em 22 de janeiro de 1998. Considerado como um desafio pelos órgãos gestores do planejamento urbano viário, a redução das ocorrências de acidentes de trânsito torna-se um tema discutido, incessantemente, na administração pública. Diante dessa leitura urbana, este estudo propõe conhecer as características dos acidentes de trânsito ocorridos na área urbana da cidade de Uberlândia entre os anos de 2000 e 2004, bem como espacializar as vias, os trechos e os cruzamentos com maior número de acidentes, com o intuito de identificá-los por meio de técnicas disponíveis em BRASIL (2002) e Gold (1998), que, adaptadas ao banco de dados disponíveis pela SETTRAN, determinaram quais são as vias, os trechos e os cruzamentos críticos. Por meio desses procedimentos, apresenta-se a identificação de vinte vias, vinte trechos e vinte cruzamentos que foram palco dos maiores números de acidentes ocorridos no trânsito e dos números mais altos de UPS dos acidentes de trânsito entre os anos de 2000 e 2004. Analisando a situação crítica dos segmentos analisados, as vias descritas a seguir foram consideradas importantes, por estarem enumeradas em todas as seleções hierarquizadas e agrupadas, bem como os trechos e os cruzamentos, sendo as Av. João Naves de Ávila e Av. Rondon Pacheco, seguidas da Av. Afonso Pena e Av. Floriano Peixoto. Para os trechos críticos, estas aparecem como sendo as vias principais e suas interseções mais significativas estão localizadas na área central e vias que dão acesso ao \"shopping center\". Quanto aos cruzamentos, as vias que fazem as interseções com as ora citadas possibilitam conflitos, que, pela alta freqüência dos acidentes de trânsito em um mesmo local, foram estabelecidos os pontos críticos pela significância que os números dos acidentes de trânsito apresentam e de suas especificações do local identificado. Para a redução das relações de conflitos no trânsito, bem como a prevenção de um grande número de ocorrências de acidentes de trânsito, devem ser consideradas medidas de educação que levem os cidadãos a atitudes e comportamentos de convivência ao utilizar o ambiente viário comum aos pedestres e veículos. Assim, o alcance deste objetivo é salutar, pois empenha-se na busca de estratégias que possam otimizar os recursos disponíveis e estabelecer parcerias que garantam a efetividade das estratégias definidas, afugentando esse mal que assola a cidade sem pedir licença / The numbers of traffic accidents occurrences are frightful. It is evaluated that the cost of the accidents is near to 1% of the GIP in poor countries and 2% in the developed nations. In global terms, U$ 518 billion/year are expensed. In Brazil, studies point that there was a fall in 20 thousand of fatal victims in traffic accidents between 1987 and 1998. Currently, we had around 30 thousand deaths/year in the traffic. This reduction can be partly justified for the Brazilian Transit Code, which works since 22 January 1998. Considered a challenge for the managing agencies of the road urban planning, the reduction of the traffic accidents occurrences become a subject incessantly discussed in the public administration. In front of this urban reading, this study proposes to know the characteristics of the traffic accidents occurred in the urban area of the Uberlândia-MG, considering the period 2004-2004, as well to space the streets, the stretches and the crossings with more accidents, intending to identify this streets through techniques available in BRAZIL (2002) and Gold (1998). This procedure was adapted from the available data base by SETTRAN and they had determined which are the critical streets, stretches and crossings in question. By these procedures, we present the identification of twenty streets, twenty stretches and twenty crossings where happens the biggest numbers of transit accidents and the highest numbers of UPS of the traffic accidents between 2000 and 2004. Considering the critical situation of the analyzed segments, the following described streets had been considered important because they are enumerated in all of the ranked and grouped selections, as well as the stretches and the crossings. The streets are: João Naves de Ávila Avenue, Rondon Pacheco Avenue, followed by Afonso Pena Avenue and Floriano Peixoto Avenue. For the critical stretches, these are the main streets and its more significant intersections are located in the central area and streets with access to the \"Shopping Center\". About the crossings, the streets intersected with those streets make possible the conflicts. On account of the high frequency of the traffic accidents in a same local, we established the critical points with basis in the significance that the numbers of the traffic accidents present and specifications presented by the identified place. For the reduction of the conflict relations in traffic, as well for to prevent a great number of traffic accidents occurrences, we considered education measures that take the citizens to attitudes and behaviors of social living when they use a street common to the pedestrians and vehicles. Thus, the reach of this objective is special, once it is pledged in the search of strategies that can optimize the available resources and establish partnerships for guarantee the effectiveness of the defined strategies, driving away this mad which devastates the city

Os números de ocorrências em acidentes de trânsito são assustadores. Avalia-se que o custo dos acidentes aproxime de 1% do PIB em países pobres e 2% nas nações desenvolvidas. Em termos globais, são gastos U$ 518 bilhões anuais. No Brasil, estudos apontam que houve uma queda de 20 mil vítimas fatais de acidentes de trânsito, de 1987 para 1998, apresentando, atualmente, em torno de 30 mil mortes/ano no trânsito. Essa redução pode ser em parte justificada pelo Código de Trânsito Brasileiro, que entrou em vigor em 22 de janeiro de 1998. Considerado como um desafio pelos órgãos gestores do planejamento urbano viário, a redução das ocorrências de acidentes de trânsito torna-se um tema discutido, incessantemente, na administração pública. Diante dessa leitura urbana, este estudo propõe conhecer as características dos acidentes de trânsito ocorridos na área urbana da cidade de Uberlândia entre os anos de 2000 e 2004, bem como espacializar as vias, os trechos e os cruzamentos com maior número de acidentes, com o intuito de identificá-los por meio de técnicas disponíveis em BRASIL (2002) e Gold (1998), que, adaptadas ao banco de dados disponíveis pela SETTRAN, determinaram quais são as vias, os trechos e os cruzamentos críticos. Por meio desses procedimentos, apresenta-se a identificação de vinte vias, vinte trechos e vinte cruzamentos que foram palco dos maiores números de acidentes ocorridos no trânsito e dos números mais altos de UPS dos acidentes de trânsito entre os anos de 2000 e 2004. Analisando a situação crítica dos segmentos analisados, as vias descritas a seguir foram consideradas importantes, por estarem enumeradas em todas as seleções hierarquizadas e agrupadas, bem como os trechos e os cruzamentos, sendo as Av. João Naves de Ávila e Av. Rondon Pacheco, seguidas da Av. Afonso Pena e Av. Floriano Peixoto. Para os trechos críticos, estas aparecem como sendo as vias principais e suas interseções mais significativas estão localizadas na área central e vias que dão acesso ao \"shopping center\". Quanto aos cruzamentos, as vias que fazem as interseções com as ora citadas possibilitam conflitos, que, pela alta freqüência dos acidentes de trânsito em um mesmo local, foram estabelecidos os pontos críticos pela significância que os números dos acidentes de trânsito apresentam e de suas especificações do local identificado. Para a redução das relações de conflitos no trânsito, bem como a prevenção de um grande número de ocorrências de acidentes de trânsito, devem ser consideradas medidas de educação que levem os cidadãos a atitudes e comportamentos de convivência ao utilizar o ambiente viário comum aos pedestres e veículos. Assim, o alcance deste objetivo é salutar, pois empenha-se na busca de estratégias que possam otimizar os recursos disponíveis e estabelecer parcerias que garantam a efetividade das estratégias definidas, afugentando esse mal que assola a cidade sem pedir licença / The numbers of traffic accidents occurrences are frightful. It is evaluated that the cost of the accidents is near to 1% of the GIP in poor countries and 2% in the developed nations. In global terms, U$ 518 billion/year are expensed. In Brazil, studies point that there was a fall in 20 thousand of fatal victims in traffic accidents between 1987 and 1998. Currently, we had around 30 thousand deaths/year in the traffic. This reduction can be partly justified for the Brazilian Transit Code, which works since 22 January 1998. Considered a challenge for the managing agencies of the road urban planning, the reduction of the traffic accidents occurrences become a subject incessantly discussed in the public administration. In front of this urban reading, this study proposes to know the characteristics of the traffic accidents occurred in the urban area of the Uberlândia-MG, considering the period 2004-2004, as well to space the streets, the stretches and the crossings with more accidents, intending to identify this streets through techniques available in BRAZIL (2002) and Gold (1998). This procedure was adapted from the available data base by SETTRAN and they had determined which are the critical streets, stretches and crossings in question. By these procedures, we present the identification of twenty streets, twenty stretches and twenty crossings where happens the biggest numbers of transit accidents and the highest numbers of UPS of the traffic accidents between 2000 and 2004. Considering the critical situation of the analyzed segments, the following described streets had been considered important because they are enumerated in all of the ranked and grouped selections, as well as the stretches and the crossings. The streets are: João Naves de Ávila Avenue, Rondon Pacheco Avenue, followed by Afonso Pena Avenue and Floriano Peixoto Avenue. For the critical stretches, these are the main streets and its more significant intersections are located in the central area and streets with access to the \"Shopping Center\". About the crossings, the streets intersected with those streets make possible the conflicts. On account of the high frequency of the traffic accidents in a same local, we established the critical points with basis in the significance that the numbers of the traffic accidents present and specifications presented by the identified place. For the reduction of the conflict relations in traffic, as well for to prevent a great number of traffic accidents occurrences, we considered education measures that take the citizens to attitudes and behaviors of social living when they use a street common to the pedestrians and vehicles. Thus, the reach of this objective is special, once it is pledged in the search of strategies that can optimize the available resources and establish partnerships for guarantee the effectiveness of the defined strategies, driving away this mad which devastates the city

With the purpose of verifying the quality of industrialization of yogurt without artifivcial preservatives with fruit pulp it was decided to use the Analysis of Risks and Critical Control Points, APPCC in SETREM agro industry. The work has started with an APPCC group first constitution were, hen in a second moment established the risks and ways of prevention. Subsequently, the points to be controlled and the points that demonstrate greater risk incidence were determined, the ones called priorities, the critical points of control that the critical limits determined. For the critical limits the control procedures had been established among them the statistics was used carrying out the statistical process control in temperature control during the fermentation of the yogurt where from the superior and inferior limits the control letters were built to follow the process. In order to verify the efficiency of the equipment and operators the application of the Measurement System Analysis (MSA) in ten yogurt samples during the determination of pH-gâmetro. Later wasandyzed necessary corrective actions were delimited to be applied in apply wag lines of the critical limits. Some procedures verification for proper system APPCC were needed, that is, procedures in addition to those used in the control to demonstrate if the system APPCC is being efficient. With the application of the tools, the guarantee of the quality of the final product was gotten decreasing the necessity of the finished product inspections / Com a finalidade de verificar a industrialização do iogurte sem conservante com polpa de fruta, com qualidade, é que se decidiu empregar a ferramenta de Análise de Perigos e Pontos Críticos de Controle, APPCC, na agroindústria/SETREM. O trabalho iniciou com a constituição da equipe APPCC, a qual inicialmente se estruturou para num segundo momento levantar os perigos e medidas preventivas. Subseqüentemente, determinaram-se os pontos a serem controlados e os pontos que demonstram maior incidência de risco, os assim chamados prioritários, os pontos críticos de controle, para os quais determinou-se os limites críticos. Para os limites críticos foram estabelecidos os procedimentos de monitoração entre os quais utilizou-se da estatística, realizando-se o controle estatístico de processo no controle da temperatura durante a fermentação do iogurte sem conservante com polpa de fruta onde, a partir dos limites superiores e inferiores, construíram-se os gráficos de controle. Para verificar a eficiência do equipamento/operador, utilizou-se a aplicação da análise do sistema de medição, MSA, em dez amostras de iogurte durante a determinação do pH, com a utilização do pH-gâmetro. Posteriormente, delimitaram-se as ações corretivas necessárias a serem aplicadas nos desvios dos limites críticos. Também foram necessários alguns procedimentos de verificação para o próprio sistema APPCC, ou seja, utilizaram-se procedimentos em adição àqueles aplicados na monitoração para evidenciar se o sistema APPCC está sendo eficaz. Com a aplicação da ferramenta, obteve-se a garantia da segurança do produto final, diminuindo-se a necessidade de testes dos produtos acabados

From the growing emphasis in the meat products security that gets to the consumer, it has been stimulated the identification of the ways to reduce or put down Salmonella spp before slaughter, since the reduction of the infection rate before slaughter results in an increase in security of the meat products. Salmonella spp highlights as the main pathogenic bacteria joined in the slaughter line by the swine itself and its ausence in the final product is extremely important to compete in the market, which presents a growing demand in relation to the quality of the products. The present paper was done in a slaughter plan of a slaughterhouse under Federal Inspection, situated in a town in Rio Grande do Sul, where 50% of the slaughter animals are produced by the integration system and 50% come from the other small farm available on the market. This paper aim was to evaluate the Salmonella spp incidence perceptual in the slaughter line in four predetermined steps by visual observation of the operation, through the microbiological analysis. A total of 128 superficial swine carcasses rubbing were collected from different steps in the slaughter line: skin after scalding and skinning; from abdominal cavity carcass after the evisceration, the carcass spine portion, including the skin of the swine loin; and the skin after the final shower. We also collected 32 cecal materials to microbiological analysis. Salmonella spp was found in 9,37% of the swine sampled. The presence of Salmonella spp was detected in 81,25% of the cecal analyzed samples in both groups. The integrated swine group presented greater perceptual of Salmonella spp in the different sampling points (5,47%) when linked with the studied groups, whereas the swine of the third showed 3,91% of the positive. In the skin collection point after-scalding and integrated swine skinning (ISS) it was found 18,75% of positive sample and in the third swine sample (TSS) 12,50%. In the abdominal cavity of the integrated swine after the evisceration step (ISE) it was detected no positive sample (0%), yet to the swine from the third (TSE) was found a positively (6, 25%). Sample from the spine portion after the carcass division step of the integrated swine (ISD) with saw showed 12,50% of the positive sample, while in the swine sample from the third (TSD) 6,25%. In the samples of the swine skin after the final shower with chlorine water of the integrated swine group (ISS), it was detected 12,50%, and in the swine from the third samples (TSS) 6,25%. Among the presented results there was no significant difference to the level of 5%. The achieved data confirm the scalding and skinning steps as the main Control Critical Points (CCP) to the analyzed slaughterhouse. In general, the integrated swine group revealed greater incidence of microorganisms in the different steps in the slaughter line, confirming that Salmonella spp epidemic in swine is complex, presenting a variety of determining factors in the microorganism transmission. / A partir da crescente ênfase na segurança de produtos cárneos que chegam ao consumidor, tem-se estimulado a identificação de meios para reduzir ou eliminar Salmonella spp antes do abate. Assim, a redução das taxas de infecção pré-abate resulta em aumento na segurança dos produtos cárneos. Salmonella spp destaca-se como a principal bactéria patogênica incorporada na linha de abate pelo próprio suíno e sua ausência no produto final é de extrema importância para competir no mercado, que apresenta uma crescente exigência em relação à qualidade dos produtos. O presente estudo foi realizado em uma planta de abate de um frigorífico sob Inspeção Federal, situado em um município do Rio Grande do Sul, onde 50% dos animais abatidos são produzidos pelo sistema de integração e 50% obtidos de outras granjas disponíveis no mercado (terceiros). Avaliou-se o percentual de incidência de Salmonella spp na linha de abate de suínos, em quatro etapas prédefinidas por observação visual das operações, através de análise microbiológica. Um total de 128 esfregaços superficiais, de carcaça suína, foram coletados de diferentes etapas da linha de abate: pele, após a escaldagem e depilação;da cavidade abdominal da carcaça, após a evisceração; da porção espinhal da carcaça, incluindo a pele da região lombar do suíno; e da pele após o chuveiro final. Também foram realizadas 32 coletas de material cecal (fezes) para análise microbiológica. Salmonella spp foi encontrada com uma freqüência de 9,37% dos suínos amostrados. Nas fezes de ambos os grupos a presença de Salmonella spp foi detectada em 81,25% das amostras analisadas. Relacionando com os grupos em estudo o grupo de suínos integrados apresentou percentual superior de presença para Salmonella spp nos diferentes pontos de amostragem (5,47%), enquanto os suínos de terceiros apresentaram 3,91% de positividade. No ponto de coleta de pele pós-escaldagem e depilação de suínos integrados (PSI) foi encontrado 18,75% de amostras positivas e nas amostras de suínos de terceiros (PST) 12,50%. Na cavidade abdominal de suínos integrados após a etapa de evisceração (ESI) não foi detectada nenhuma amostra positiva (0%), já para os suínos de terceiros (EST) foi encontrada uma positividade (6,25%). Amostras da porção espinhal após a etapa de divisão da carcaça de suínos integrados (SSI) com serra apresentaram 12,50% de amostras positivas, enquanto nas amostras de suínos de terceiros (SST) 6,25%. Nas amostras de pele dos suínos após a lavagem final da carcaça com água clorada do grupo de suínos integrados (CSI) detectou-se 12,50% e nas amostras de suínos de terceiros (CST) 6,25%. Entre os resultados apresentados não houve diferença significativa ao nível de 5%. Os dados obtidos confirmam as etapas de escaldagem e depilação, como o principal Ponto Crítico de Controle (PCC) para o frigorífico estudado. A previsão inicial e motivo deste trabalho de confirmar os suínos de terceiros como principais veículos de Salmonella spp para o interior do frigorífico não foram confirmados. De modo geral, o grupo de suínos integrados revelou maior incidência do microrganismo nas diferentes etapas da linha de abate, confirmando que a epidemiologia por Salmonella spp em suínos é complexa, apresentando múltiplos fatores determinantes na transmissão do microrganismo.

This thesis studies the relationship between the zeroes of complexpolynomials in one variable and the critical points of those polynomials. Our methods are both analytical and statistical in nature, usingtechniques from both complex analysis and probability theory. Wepresent an alternative proof for the famous Gauss-Lucas theorem aswell as proving that the distribution for the critical points of a randompolynomial with real zeroes will converge in probability to the distribution of the zeroes. A simulation of the case with complex zeroesis also presented, which gives statistical support that this holds forrandom polynomials with complex zeroes as well. Lastly, the previous results are then applied to Sendov’s conjecture where we take aprobabilistic approach to this problem.

In an era of rapidly growing data sets, information reduction techniques such as extracting and highlighting characteristic features, are becoming increasingly important for efficient data analysis. Particularly relevant features of scalar fields are their critical points since they mark locations in the domain where a field's level set undergoes fundamental topological changes. There are well-established methods for locating and relating such points in a deterministic setting. However, many real-world phenomena studied in the computational sciences today are the result of a chaotic system that cannot be fully described by a single scalar field. Instead, the variability of such systems is typically captured with ensemble simulations, which generate a variety of possible outcomes of the simulated process. The topological analysis of such ensemble data sets, and uncertain data in general, is less well studied. In particular, there is no established definition for critical points of uncertain scalar fields. This thesis therefore aims to generalize the concept of critical points to uncertain scalar fields. While a deterministic field has a single set of critical points, each outcome of an uncertain scalar field has its own set of critical points. A first step towards finding an appropriate analog for critical points in uncertain data is to look at the distribution of all these critical points. In this work, different methods for analyzing this distribution are presented, which identify and track the likely locations of critical points over time, estimate their local occurrence probabilities, and eventually characterize their spatial uncertainty. A driving factor of winter weather in western Europe is the North Atlantic Oscillation (NAO), which is manifested by fluctuations in the sea level pressure difference between the Icelandic Low and the Azores High. Several methods have been developed to describe the strength of this oscillation. Some of them are based on certain assumptions, such as fixed positions of these two pressure systems. It is possible, however, that climate change will affect the locations of the main pressure variations and thus the validity of these descriptive methods. An alternative approach is based on the leading empirical orthogonal function (EOF) computed from the sea level pressure fields over the North Atlantic. The critical points of these fields indicate the actual locations of maximum pressure variations and can thus be used to assess how climate change affects these locations and to evaluate the validity of methods that use fixed locations to characterize the strength of the NAO. Because the climate is described by a chaotic system, such an analysis should incorporate the uncertain nature of climate predictions to produce statistically robust results. Extracting and tracking the positions of the maximum pressure variations that characterize the NAO therefore serves as a motivating practical application for the study of critical points in uncertain data in this work. Because uncertain data tend to be noisy, filtering is often required to separate relevant signals of variation from irrelevant fluctuations. A well-established method for extracting dominant signals from a time series of fields is to compute its empirical orthogonal functions (EOFs). In the first part of this thesis, this concept is extended to the analysis of spatiotemporal ensemble data sets to decompose their variation into modes describing the variation in the ensemble direction and modes describing the variation in the time direction. An application to different climate data sets revealed that, depending on the way an ensemble has been generated, temporal and ensemble-wise variations are not necessarily independent, making it difficult to separate these signals. Next, a computational pipeline for tracking likely locations of critical points in ensembles of scalar fields is presented. It computes leading EOFs on sliding time windows for all ensemble members, extracts regions where critical points can be expected from the resulting ensembles of EOFs for every time window, and finally tracks the barycenters of these regions over time. An application of this pipeline to sea level pressure fields over the North Atlantic revealed systematic shifts in the locations of the maximum pressure variations that characterize the NAO. These found shift were more pronounced for more extreme climate change scenarios. Existing methods for the identification of critical points in ensembles of scalar fields do not distinguish between uncertainties that are inherent in the analyzed system itself and those that are additionally introduced by using a finite sample of fields to capture these variations. In the next part of this thesis, two approaches for estimating the occurrence probabilities of critical points are presented that explicitly take into account and communicate to the viewer the additional uncertainties caused by estimating these probabilities from finite-sized ensembles. A comparison with existing works on synthetic data demonstrates the added value of the new approaches. The last part of this thesis is devoted to the question of how to characterize the spatial uncertainty of critical points. It provides a sound mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution. This ultimately leads to the notion of uncertain critical points as a generalization of critical points to uncertain scalar fields. An analysis of the theoretical properties of these structures gave conditions under which well-interpretable results can be obtained and revealed interpretational difficulties when these conditions are not met. / In Zeiten immer größerer Datensätze gewinnen Techniken zur Informationsreduktion, etwa die Extraktion und Hervorhebung charakteristischer Merkmale, zunehmend an Bedeutung für eine effiziente Datenanalyse. Besonders relevante Merkmale von Skalarfeldern sind ihre kritischen Punkte, da sie Orte in der Domäne kennzeichnen, an denen sich die Topologie der Niveaumenge eines Feldes grundlegend verändert. Es existieren etablierte Methoden, um diese Punkte in deterministischen Feldern zu lokalisieren und sie miteinander in Beziehung zu setzen. Viele Alltagsphänomene, die heute untersucht werden, sind jedoch das Ergebnis chaotischer Systeme, die sich nicht vollständig durch ein einzelnes Skalarfeld beschreiben lassen. Stattdessen wird die Variabilität solcher Systeme mit Ensemblesimulationen erfasst, die eine Vielzahl möglicher Ergebnisse des simulierten Prozesses erzeugen. Die topologische Analyse solcher Ensemble-Datensätze und unsicherer Daten im Allgemeinen ist bisher weniger gut erforscht. Insbesondere gibt es noch keine etablierte Definition für die kritischen Punkte von unsicheren Skalarfeldern. In dieser Dissertation wird daher eine Verallgemeinerung des Konzepts kritischer Punkte auf unsichere Skalarfelder angestrebt. Während ein deterministisches Feld einen einzigen Satz kritischer Punkte hat, hat jede Realisierung eines unsicheren Skalarfeldes ihre eigenen kritischen Punkte. Ein erster Schritt, um ein geeignetes Analogon für kritische Punkte in unsicheren Daten zu finden, besteht darin, die Verteilung all dieser kritischen Punkte zu untersuchen. Zu diesem Zweck werden in dieser Arbeit verschiedene Methoden vorgestellt, die es ermöglichen, die wahrscheinlichen Orte kritischer Punkte zu identifizieren und über die Zeit zu verfolgen, die lokalen Wahrscheinlichkeiten für das Auftreten kritischer Punkte zu schätzen und schließlich die räumliche Unsicherheit von kritischen Punkten zu charakterisieren. Ein bestimmender Faktor für das Winterwetter in Westeuropa ist die Nordatlantische Oszillation (NAO), die sich in Schwankungen des Druckunterschieds auf Meereshöhe zwischen dem Islandtief und dem Azorenhoch äußert. Es existieren unterschiedliche Methoden, um die Stärke dieser Oszillation zu beschreiben, von denen einige auf bestimmten Annahmen beruhen, wie etwa der fixen Position der beiden Drucksysteme. Es ist jedoch möglich, dass der Klimawandel die Lage der Hauptdruckschwankungen und somit die Gültigkeit dieser Beschreibungsmethoden beeinträchtigt. Ein alternativer Ansatz basiert auf der führenden empirischen Orthogonalfunktion (EOF), welche aus den Druckfeldern auf Meereshöhe über dem Nordatlantik berechnet wird. Die kritischen Punkte dieses Feldes entsprechen den tatsächlichen Orten maximaler Druckschwankungen. Sie können daher verwendet werden, um die Auswirkungen des Klimawandels auf diese Orte zu bewerten und dadurch die Gültigkeit von Methoden, die feste Positionen zur Charakterisierung der Stärke der NAO verwenden, zu beurteilen. Da das Klima durch ein chaotisches System beschrieben wird, sollte eine solche Analyse die Unsicherheit von Klimavorhersagen berücksichtigen, um statistisch zuverlässige Ergebnisse zu erhalten. Die Extraktion und Verfolgung der für die NAO charakteristischen Positionen maximaler Druckschwankungen dient daher als motivierende praktische Anwendung für die Untersuchung kritischer Punkte in unsicheren Daten in dieser Arbeit. Da unsichere Daten oft verrauscht sind, ist meist zunächst eine Filterung erforderlich, um relevante Signale von irrelevanten Fluktuationen zu trennen. Ein etabliertes Konzept zur Extraktion dominanter Signale aus Zeitreihen von Skalarfeldern ist die empirische Orthogonalfunktionsanalyse (EOF-Analyse). Im ersten Teil dieser Arbeit wird dieses Konzept auf die Analyse von zeitabhängigen Ensemble-Datensätzen erweitert, um deren Variation in Moden zu zerlegen, die die jeweiligen Schwankungen in Ensemble- und Zeitrichtung beschreiben. Eine Anwendung auf verschiedene Klimadatensätze hat gezeigt, dass je nachdem, wie ein Ensemble generiert wurde, zeitliche und ensemblebezogene Variationen nicht zwangsläufig unabhängig sind, was eine Trennung dieser Signale erschwert. Im weiteren wird eine Berechnungspipeline zur Verfolgung der wahrscheinlichen Positionen kritischer Punkte in Ensemblen von Skalarfeldern vorgestellt. Sie berechnet zunächst die führenden EOFs auf gleitenden Zeitfenstern für jedes Ensemblemitglied, extrahiert dann aus den resultierenden Ensemblen von EOFs an jedem Zeitfenster Regionen, in denen kritische Punkte zu erwarten sind, und verfolgt schließlich die Baryzentren dieser Regionen über die Zeit. Die Anwendung dieser Pipeline auf die nordatlantischen Meeresspiegeldruckfelder hat eine systematische Verschiebungen der für die NAO charakteristischen Orte der maximalen Druckvariationen offenbart. Dabei führten extremere Klimawandelszenarien zu stärkeren Verschiebungen. Vorhandene Methoden zur Identifikation von kritischen Punkten in Ensemblen von Skalarfeldern unterscheiden nicht zwischen Unsicherheiten, die dem analysierten System selbst innewohnen, und solchen, die durch die Verwendung einer endlichen Stichprobe von Feldern zur Erfassung dieser Variationen zusätzlich verursacht werden. Im nächsten Teil dieser Arbeit werden daher zwei Ansätze zur Schätzung der Auftrittswahrscheinlichkeiten kritischer Punkte vorgestellt, die explizit auch die zusätzlichen Unsicherheiten berücksichtigen, die durch die Schätzung dieser Wahrscheinlichkeiten aus endlichen Ensemblen entstehen, und diese an den Betrachter kommunizieren. Der Mehrwert der neuen Verfahren wurde in einem Vergleich mit bestehenden Arbeiten auf synthetischen Daten demonstriert. Der letzte Teil dieser Arbeit ist der Frage gewidmet, wie sich die räumliche Unsicherheit kritischer Punkte charakterisieren lässt. Es wird eine fundierte mathematische Formulierung des Problems der Suche nach kritischen Punkten mit räumlicher Unsicherheit und der Berechnung ihrer räumlichen Verteilung erbracht. Das führt schließlich zum Begriff unsicherer kritischer Punkte als Verallgemeinerung von kritischen Punkten auf unsichere Skalarfelder. Eine Analyse der theoretischen Eigenschaften dieser Strukturen hat Bedingungen ergeben, unter denen einfach zu interpretierende Ergebnisse erzielt werden können, und offenbarte Interpretationsschwierigkeiten, die entstehen, wenn diese Bedingungen nicht erfüllt sind.

With regard to the space field, the number of the sensors has grown for a middle-sized spacecraft from more than 500 at the beginning of the twenty-first century [1] to several thousands for nowadays applications. Meanwhile, Reusable Launch Vehicles (RLVs) moved their steps from demonstrators to commercial working systems. As a result, Health Monitoring (HM) is conquered its own space in the field and sensors are the primary elements required for implementing a monitoring unit. The innovative concept of reusable rockets requires, from the point of view of HM implementation, not only the evaluation of the vehicle health status but also the prediction of the reusability of the individual subsystems w.r.t. the next launch cycle. Therefore, the goal of this work is divided in two parts. The former is to identify the most critical points for the development of reusable rockets, focusing on theoretical working conditions and analysis or failures. The latter is to discuss the sensing units useful for addressing the defined points, describing the possible innovative approaches for sensing the system conditions. Among them, piezoelectric units, fiber optics, imaging units, and conductive layers can be identified for enhancing the comprehension of the system working conditions.