We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Nesta dissertação de mestrado o problema da estabilidade de geodésicas do tipo luz e do tipo tempo é estudado sobre o ponto de vista do formalismo de sistemas dinâmicos. Uma breve revisão bibliográfica sobre aspectos importantes de sistemas dinâmicos contínuos no tempo é realizada, bem como uma sucinta revisão de tópicos de interesse em relatividade geral. As equações de movimento para as geodésicas são deduzidas para geometrias com simetria esférica, e o caso Schwarzschild é inicialmente analisado. Em seguida, analisamos o caso das geometrias proposta por Casadio, Fabbri e Mazzacurati e um caso de buraco de minhoca assintoticamente de Sitter. A caracterização dos pontos fixos dos sistemas de interesse é feita, e a sua estabilidade é analisada sob a ótica dos métodos de Lyapunov e Jacobi, assim como bifurcações foram mapeadas. A fotosfera é caracterizada como um ciclo limite, sendo um ponto fixo estritamente instável no espaço de estados de buracos negros. A análise dos buracos de minhoca revelam a existência de uma fotosfera estável em determinadas regiões do espaço de parâmetros do sistema / In this dissertation, the problem associated with the stability of timelike and null geodesics is studied from the dynamical system point of view. A succinct bibliographical review covering important aspects of time-continuous dynamical systems is made, and a short review about some topics of interest of general relativity is also presented. The geodesic equations of motion are shown for geometries with spherical symmetry, and the Schwarzschild case is first analyzed. In the following, we analyze the geometries proposed by Casadio, Fabbri, and Mazzacurati and an asymptotically de Sitter wormhole case. The characterization of the fixed points of the system is performed, and their stability is studied from the perspective of the Lyapunov and Jacobi methods, as well as the bifurcation analysis. The photon sphere is characterized as a limit cycle, being a strictly unstable fixed point in the state space of the system. The wormhole analysis reveals the existence of a stable photon sphere in certain regions of the parameter space of the system

A colisão de buracos negros é uma das fontes mais importantes de ondas gravitacionais e, em geral, a emissão anisotrópica da radiação causa um recuo do objeto final. Este cenário já é conhecido há décadas, mas foi somente com o recente avanço na relatividade numérica que as velocidades finais dos objetos radiantes foram computadas com precisão. Os valores encontrados podem ser altos o suficiente para exercerem um importante papel no crescimento de buracos negros super massivos via coleção de galáxias e na abundância de núcleos galáticos ativos contendo buracos negros. Este é um autêntico efeito da não linearidade de Relatividade Geral e esta tese fornece uma nova metodologia estudar alguns aspectos da dinâmica da colisão de buracos negros. Consideramos o horizonte como uma tela canônica que codifica as informações da evolução temporal do espaço-tempo. Com esta hipótese, fenômenos como o anti-kick, isto é, uma súbita desaceleração do sistema antes de atingir a velocidade final, são explicado em termos da dissipação das deformações do horizonte. Estudamos primeiramente o Espaço-tempo de Robinson-Trautman. Uma das solução mais simples das equações de Einstein, esta métrica nos fornece um poderoso modelo para investigar tanto a perda de massa quanto o recuo do objeto final. Mostramos que, quando as configurações iniciais tem simetria especular, a massa do buraco negro remanescente e a energia irradiada são completamente determinadas pela condição inicial. Com isso, obtemos as expressões analíticas dos resultados numéricos obtidos anteriormente na literatura. Além disto, com o auxilio do método espectral de Galerkin, analisamos o regime não linear das equações envolvidas e verificamos que se pode estimar a velocidade de recuo final com boa precisão a partir de medidas da assimetria da condição inicial. Introduzimos na seqüência a curvatura efetiva como uma medida das deformações intrínsecas ao horizonte. Além de considerar as deformações gerais, ela também inclui as diferenças entre os hemisférios norte e sul. No espaço-tempo de Robinson-Trautman, essa quantidade se correlaciona de uma forma injetora com a velocidade final. Para superar algumas limitações dessa solução, aplicamos o mesmo procedimento nos resultados da simulação numérica de uma colisão head-on. Neste caso, a curvatura efetiva, está na realidade, correlacionada com a aceleração do sistema. Refinamentos e generalizações desta técnica são também discutidos e propostos para trabalhos futuros. / Colliding black holes are one of the most important sources of gravitational waves and the anisotropic emission of the radiation generally causes the recoil of the final hole. This scenario has been known for decades, but it is only thanks to the recent progress in numerical relativity that the final velocity have been accurately computed. The values found can be large enough to play an important role in the growth of supermassive black holes via mergers of galaxies and on the number of galaxies containing them. This is a genuine nonlinear effect of general relativity and this thesis provides a new methodology to study some features on the dynamics of the collision. We propose that the horizon is a canonical screen, which encodes he information of its surroundings. With this assumption, phenomena such as the anti-kick, namely the sudden deceleration before reaching the final velocity, are explained in terms of the dissipation of the horizons deformation. We first study the Robinson-Trautman spacetime. One of the simplest solutions of Einsteins equations, it provides us with a powerful toymodel to investigate both the mass loss of the system and the recoil of the final object. We show that, for the case of reflectionsymmetric initial configurations, the mass of the remnant black-hole and the total energy radiated away are completely determined by the initial data, allowing us to obtain analytical expressions for some numerical results that had appeared in the literature. Moreover, by using the Galerkin spectral method to analyze the non-linear regime of the equations involved, we found that the recoil velocity can be estimated with good accuracy from some symmetry measures of the initial data. Then we introduce the effective urvature as a measure of intrinsic deformations on the horizon. Not only does it account for overall deformation, but also for the differences on the north and south hemispheres. In the Robinson-Trautman spacetime, this quantity correlates in an injective way with the final velocity. To overcome some caveats of this solutions, we apply the same procedure to the results given by numerical simulations of a head-on collision. In the case, the effective curvature is actually correlated with the acceleration of the system. Further improvement and generalizations of this technic is also discussed and proposed for future work.

Mes travaux de recherche portent sur l'étude du rayonnement (surtouts rayons X durs) provenant des trous noirs accrétant (dans les noyaux actifs de galaxies et les binaires X). L'objectif est d'en extraire des informations sur les conditions physique régnant dans l'environnement immédiat de ces objets. Les principales question auxquelles je tente de répondre sont les suivantes: Quelle est la structure et la géométrie de la matière accrétée au voisinage du trou noir ? Comment celle -ci évolue-t-elle avec le taux d'accrétion de masse ? Quel est la relation entre les processus d'accrétion et la formation de jets souvent observés dans ces systèmes ? Mon approche est fondée sur une comparaison précise entre les observations et les prédictions des divers modèles. Je présente les efforts poursuivis depuis près de dix ans afin de développer des outils de simulation numérique pour modéliser le transfert de rayonnement dans les plasma chauds des sources compactes X. Je montre comment ces outils ont été utilisés pour modéliser le continuum haute énergie et la variabilité des trous noirs accrétants et pour contraindre la structure du flot d'accrétion. Je présente également des résultats reposant sur l'analyse et l'interprétation d'observations menées avec des télescopes spatiaux tels que XMM-Newton et INTEGRAL ainsi que sur des d'observations simultanées à plusieurs longueurs d'ondes allant de la radio aux rayons X durs.

La tesi tracta, des del punt de vista de l’anàlisi de dades, la possibilitat de detecció directa d’ones gravitatòries emeses per sistemes binaris d’objectes compactes de massa similar: forats negres, estels de neutrons, nanes blanques. En els capítols introductoris, a) es dóna una descripció detallada i exhaustiva de com passar dels patrons d’ona teòrics a la senyal detectada; b) s’introdueixen les eines més emprades en l’anàlisi de dades d’ones gravitatòries, amb especial menció a la discussió sobre les amplituds efectiva i característica. A més, els resultats originals de la tesi segueixen tres línies de recerca diferents: 1) S’ha predit la precisió amb la que el futur detector interferomètric espacial LISA, estimarà els paràmetres (posició, masses, velocitat de rotació, paràmetres cosmològics…) de les observacions de xocs entre dos forats negres supermassius en la fase “inspiral”. 2) S’ha desenvolupat un algorisme propi de cerca de senyals gravitatòries procedents de sistemes binaris estel•lars, basat en teories de probabilitat Bayesiana i MCMC. Aquest algorisme distingeix alhora milers de senyals superposades en una única sèrie temporal de dades, extraient paràmetres individuals de cadascuna d’elles. 3) S’ha definit de manera matemàtica rigorosa com determinar el rang de validesa (per a extracció de paràmetres i detecció) de models aproximats de patrons d’ones gravitatòries, aplicant-ho a un cas concret de models semi-analítics / La tesis trata, desde el punto de vista del análisis de datos, la posibilidad de detección directa de ondas gravitacionales emitidas por sistemas binarios de objetos compactos de masa similar: agujeros negros, estrellas de neutrones, enanas blancas. En los capítulos introductorios, a) se desarrolla una descripción detallada y exhaustiva de como pasar de los patrones de onda teóricos a la señal detectada; b) se introducen las herramientas más utilizadas en el análisis de datos de ondas gravitacionales, con especial mención a la discusión sobre las amplitudes efectiva y característica. Además, los resultados originales de la tesis siguen tres líneas de investigación diferentes: 1) Se ha predicho la precisión con la que el futuro detector interferométrico espacial LISA, estimará los parámetros (posición, masas, velocidad de rotación, parámetros cosmológicos…) de las observaciones de choques entre dos agujeros negros supermasivos en la fase “inspiral”. 2) Se ha desarrollado un algoritmo propio de búsqueda de señales gravitacionales procedentes de sistemas binarios estelares, basado en teorías de probabilidad Bayesiana y MCMC. Este algoritmo distingue a la vez miles de señales superpuestas en una única serie temporal de datos, extrayendo parámetros individuales de cada una de ellas. 3) Se ha definido de manera matemática rigurosa como determinar el rango de validez (para extracción de parámetros y detección) de modelos aproximados de patrones de ondas gravitacionales, aplicándolo a un caso concreto de modelos semi-analíticos. / In this PhD thesis one studies, from the data analysis perspective, the possibility of direct detection of gravitational waves emitted by similar mass compact binary objects: black holes, neutron stars, white dwarfs. In the introductory chapters, a) a detailed and exhaustive description about how to derive the detected strain from the theoretical emitted waveform predictions is given; b) the most used gravitational wave data analysis results are derived, being worth pointing out the discussion about effective and characteristic amplitudes. Moreover, three different research lines have been followed in the thesis: 1) It has been predicted the parameter estimation (position, masses, spin, cosmological parameters…) of supermassive black hole binary inspiral signals, observed with the future interferometric space detector, LISA. 2) A new algorithm, based on Bayesian probability and MCMC techniques, has been developed in order to search for gravitational wave signals from stellar-mass binary systems. The algorithm is able to distinguish thousands of overlapping signals from a single observed time series, allowing for individual parameter extraction. 3) It has been, mathematically and rigorously, defined how to compute the validity range (for parameter estimation and detection purposes) of approximated gravitational waveform models, applying it to the particular case of closed-form models

In this thesis we address several analytical and numerical problems related with the general relativistic study of black hole space-times and boson stars. We have developed a new centered finite volume method based on the flux splitting approach. The techniques for dealing with the singularity, steep gradients and apparent horizon location, are studied in the context of a single Schwarzschild black hole, in both spherically symmetric and full 3D simulations. We present an extended study of gauge instabilities related with a class of singularity avoiding slicing conditions and show that, contrary to previous claims, these instabilities are not generic for evolved gauge conditions. We developed an alternative to the current space coordinate conditions, based on a generalized Almost Killing Equation. We performed a general relativistic study regarding the long term stability of Mixed-State Boson Stars configurations and showed that they are suitable candidates for dark matter models. / En esta tesis abordamos varios problemas analíticos y numéricos relacionados con el estudio de agujeros negros relativistas y modelos de materia oscura. Hemos desarrollado un nuevo método de volúmenes finitos centrados basado en el enfoque de la división de flujo. Discutimos las técnicas para tratar con la singularidad, los gradientes abruptos y la localización del horizonte aparente en el contexto de un solo agujero negro de Schwarzschild, en simulaciones tanto con simetría esférica como completamente tridimensionales. Hemos extendido el estudio de una familia de condiciones de foliaciones evitadoras de singularidad y mostrado que ciertas inestabilidades no son genéricas para condiciones de gauge dinámicas. Desarrollamos una alternativa a las prescripciones actuales basada en una Almost Killing Equation generalizada. Hemos realizado también un estudio con respecto a la estabilidad a largo plazo de configuraciones de Mixed-State Boson Stars, el cual sugiere que estas podrían ser candidatas apropiadas para modelos de materia oscura.

In multi-hop wireless network (MWN), the mobile nodes relay others’ packets for enabling new applications and enhancing the network deployment and performance. However, the selfish nodes drop the packets because packet relay consumes their resources without benefits, and the malicious nodes drop the packets to launch Denial-of-Service attacks. Packet drop attacks adversely degrade the network fairness and performance in terms of throughput, delay, and packet delivery ratio. Moreover, due to the nature of wireless transmission and multi-hop packet relay, the attackers can analyze the network traffic in undetectable way to learn the users’ locations in number of hops and their communication activities causing a serious threat to the users’ privacy. In this thesis, we propose efficient security protocols for thwarting packet drop attacks and preserving users’ privacy in multi-hop wireless networks. First, we design a fair and efficient cooperation incentive protocol to stimulate the selfish nodes to relay others’ packets. The source and the destination nodes pay credits (or micropayment) to the intermediate nodes for relaying their packets. In addition to cooperation stimulation, the incentive protocol enforces fairness by rewarding credits to compensate the nodes for the consumed resources in relaying others’ packets. The protocol also discourages launching Resource-Exhaustion attacks by sending bogus packets to exhaust the intermediate nodes’ resources because the nodes pay for relaying their packets. For fair charging policy, both the source and the destination nodes are charged when the two nodes benefit from the communication. Since micropayment protocols have been originally proposed for web-based applications, we propose a practical payment model specifically designed for MWNs to consider the significant differences between web-based applications and cooperation stimulation. Although the non-repudiation property of the public-key cryptography is essential for securing the incentive protocol, the public-key cryptography requires too complicated computations and has a long signature tag. For efficient implementation, we use the public-key cryptography only for the first packet in a series and use the efficient hashing operations for the next packets, so that the overhead of the packet series converges to that of the hashing operations. Since a trusted party is not involved in the communication sessions, the nodes usually submit undeniable digital receipts (proofs of packet relay) to a centralized trusted party for updating their credit accounts. Instead of submitting large-size payment receipts, the nodes submit brief reports containing the alleged charges and rewards and store undeniable security evidences. The payment of the fair reports can be cleared with almost no processing overhead. For the cheating reports, the evidences are requested to identify and evict the cheating nodes. Since the cheating actions are exceptional, the proposed protocol can significantly reduce the required bandwidth and energy for submitting the payment data and clear the payment with almost no processing overhead while achieving the same security strength as the receipt-based protocols. Second, the payment reports are processed to extract financial information to reward the cooperative nodes, and contextual information such as the broken links to build up a trust system to measure the nodes’ packet-relay success ratios in terms of trust values. A node’s trust value is degraded whenever it does not relay a packet and improved whenever it does. A node is identified as malicious and excluded from the network once its trust value reaches to a threshold. Using trust system is necessary to keep track of the nodes’ long-term behaviors because the network packets may be dropped normally, e.g., due to mobility, or temporarily, e.g., due to network congestion, but the high frequency of packet drop is an obvious misbehavior. Then, we propose a trust-based and energy-aware routing protocol to route traffics through the highly trusted nodes having sufficient residual energy in order to establish stable routes and thus minimize the probability of route breakage. A node’s trust value is a real and live measurement to the node’s failure probability and mobility level, i.e., the low-mobility nodes having large hardware resources can perform packet relay more efficiently. In this way, the proposed protocol stimulates the nodes not only to cooperate but also to improve their packet-relay success ratio and tell the truth about their residual energy to improve their trust values and thus raise their chances to participate in future routes. Finally, we propose a privacy-preserving routing and incentive protocol for hybrid ad hoc wireless network. Micropayment is used to stimulate the nodes’ cooperation without submitting payment receipts. We only use the lightweight hashing and symmetric-key-cryptography operations to preserve the users’ privacy. The nodes’ pseudonyms are efficiently computed using hashing operations. Only trusted parties can link these pseudonyms to the real identities for charging and rewarding operations. Moreover, our protocol protects the location privacy of the anonymous source and destination nodes. Extensive analysis and simulations demonstrate that our protocols can secure the payment and trust calculation, preserve the users’ privacy with acceptable overhead, and precisely identify the malicious and the cheating nodes. Moreover, the simulation and measurement results demonstrate that our routing protocols can significantly improve route stability and thus the packet delivery ratio due to stimulating the selfish nodes’ cooperation, evicting the malicious nodes, and making informed decisions regarding route selection. In addition, the processing and submitting overheads of the payment-reports are incomparable with those of the receipts in the receipt-based incentive protocols. Our protocol also requires incomparable overhead to the signature-based protocols because the lightweight hashing operations dominate the nodes’ operations.

In this bachelor's thesis the concept of microscopical black hole production at colliders is investigated. By using extra dimensional models where the value of the Planck mass can be reduced to the TeV scale, gravity can be made stronger at small distances and the Hierarchy problem can be solved. Since gravity is much stronger already at the TeV-scale, there is a possibility that microscopical black holes are produced at the LHC. In this thesis the possibility to produce Quantum Black Holes, black holes with masses around the Planck mass, is implemented in the event generator PYTHIA. Events where the Quantum Black Holes decay into two particles are simulated and studied. A main contribution is successful colour connections between the final states. Something to solve in future simulations is how to give the black holes a spectrum of masses.

Operation scales of the travel agencies can be giant or tiny in term of sizes. No matter in Taiwan or abroad, most trvel agents are SMEs (small and medium enterprises), while some large-scale foreign or mainland travel agencies are even bigger than ordinary banks or airlines. Travel agents play a role as an important channel for travelers. They collect a considerable huge amount of money from travelers and pay to the suppliers just like a temporary God of Wealth. Airline tickets and tour fares end up to some hundred millions or even some billions of turn over (in total amount counting method) commonly. This is an industry which charges in advance before providing services. In addition to the shorter term of airline credit, six elements of travel - meals, accommodations, transportations, sightseeing, shopping, and entertainments, are allowed routinely credited periodically, especially tours fares often accumulated to an astronomical figures. This sevice industry is quite different from other manufacture suffered from the "inventories" problem, but with something in common, it has a huge amount of outstanding accounts remain unsettled, or even worse to cause the phoenomenon of financial black hole with "assets virtually inflated and liabilities virtually reduced". Several large tours wholesalers in Taiwan went bankruptcy might be owing to these reasons. Even the subsidiary of TUI, the world's largest tours group, outbroke similar news incident in early 2011. It was not singly but in pairs, Reuters reported that Europe's second largest travel company Thomas Cook facing financial crisis in late 2011, were forced to acquire ¢G 2 billion loan from banks to survive in the coming low winter season. A great amount of accumulated money often causes a crisis of credit expansion, if any ambitious entrepreneur seeks to gain more profits from other reinvestments, intends to get real estates, accquitions, mergers, and other risky investments, possibly resulted in large scale damage with both investments and foreign currency exchange losses, when facing an economy downturn or political turmoils. "Financial management" is always the primary course of business running. Emphasis on the "financial problems management" is not just a slogon, and not just to watch bank deposits closely, but also to realize how and what sholuld be seriously paid attention to. Revenue increasing and cost cutting are always the true value of financial management on business development. The differences of corporate finance from time to time are as same as different figures of human pulse, blood presure, or blood suguar that varies at diffent times or in different conditions. Financial management of a company is just like personal health management, such as people in different stages faces different situations and needs. So companies must grasp the pulse of business development in different stages to plan for the future direction. Many companies seek to maximize profits at the same time, yet ignoring the strong managing and decision-guiding functions of financial management. Newspapers and magazines often report how many employees and consumers are impacted by the closing of plants or travel agencies, many of them are wellknown enterprises. Although the global economic downturns, the failure of business diversification or reinvestment hold-up losses and other factors might take the blames. But as the old saying: "There is no sunset industry but only sunset enterprises." How to survive in the harsh environment? We must strengthen to master financial management, through analysis and control of "financial problems management" to generate necessary resources and to cut the costs to enhance the business performance. "The management of financial problems among travel industry" is really the weakest point of this industry. Practical financial management books or informations for travel agencies are not easy to be found in stores or other resources. In past decades we have seen the rotation of rises and falls in the industry, we strongly believe "Accounting and Financial Management" is the basis for all kinds of management. This thesis contains the examples of Phoenix Tours International and other travel agents among Taiwan travel industry. The real experiences of daily practices are the most important contents. I attempt to integrate academic theories into daily practices, a little theoretical but more practical argument, to provide better healthy informations for the industry. Despite the company accounting system, I had also tried to establish a basic financial training material for my management team. Wish all our colleagues will not repeat the same financial errors of their predecessors. Under the guideline of Taiwan Tourism Bureau which is committed to developing tourism with ¡§Vanguard for excellences and piloting for all projects¡¨, I tried to do my best to write down personal experiences and researches knowledge as basic works. Hope to bring up some echoes and brainstormings, to lead a brighter future and better progress for the industry. All the appendixes to this thesis are important references to the main text. Many of them are also my writings in the past. At last, IFRSs regulations will soon be implemented, we have to find out its impacts on the enterprises that should not be underestimated. Not only affect financial and accounting areas, information system, performance reward systems, investor relations, tax and legal aspects will also have significant impacts. IFRSs conversion or convergence may not only be an easy task, but provides an excellent opportunity for companies to re-examine the existing financial strategies, accounting policies, management systems, process efficiencies, risk managements, internal controls, system functions, corporate governances and investor relations practices, such as whether to meet current and future needs, and take this opportunity to enhance their strengths by improving inadequacies.