Neste trabalho estudamos uma rede hexagonal com uma cadeia de impurezas nas bordas e com supercondutividade induzida, de forma a mostrar a existência de fases com férmions de Majorana. Para tal, começamos introduzindo invariantes topológicos, número de Chern e Z2 e mostramos dois modelos para rede hexagonal. O primeiro, modelo de Haldane, fazemos como motivação histórica. O segundo, modelo de Kane-Mele, é usado como base para todo o trabalho. Seguimos introduzindo supercondutividade e como ela ocorre quando aplicada junto do Modelo de Kane-Mele, o método auto-consistente e quais as condições necessárias para termos supercondutividade apenas nas bordas. Continuamos com efeitos de impurezas magnéticas nas bordas e introduzimos férmions de Majorana que são os alvos principais dos resultados. Mostramos então, que existe fases topológicas em cadeias de impureza magnética, com momentos em espiral, contudo o diagrama de fase depende de várias condições. Por fim, mostramos que a variação da fase topológica se deve a oscilações nos níveis de energia em que o invariante topológico também varia, contrariando resultados obtidos para a rede quadrada. Concluímos esse trabalho com implicações experimentais desse resultado e possíveis caminhos que podem ser seguidos. / In this work, we study a honeycomb lattice with induced superconductivity and edge impurity in order to show the existence of a phase that host Majorana bound state. To do so, we start introducing topological invariants, Chern number and Z2, and we show two models for honeycomb lattice. The first, Haldane\'s Model, due its historical importance. The second, Kane-Mele model, because it will be used during all this work. Then we review superconductivity, showing the self-consistent method, and we apply it to Kane-Mele model, in which we find some necessary conditions to induce superconductivity only at the edges. From this point, we study the effect of magnetic impurities at the edges, and we introduce Majorana bound state, that will be the main objective of our results. In our results, we show the existence of topological non-trivial phases for spiral magnetic chain in the zigzag edge. With this we make a phase diagram. We also find oscillation in the energy spectrum and the topological phase changes with the oscillation, this is different from square lattice in which we should not have a change in the topological phase. We conclude this work with experimental implications of our result and possible developments.

by Chan Kwan Leung. / Parallel title in Chinese characters. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 168-169). / by Chan Kwan Leung. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1. --- Introduction / Chapter 1.1 --- Objective of our variational method --- p.2 / Chapter 1.2 --- Outline of the content --- p.5 / Chapter 2. --- Formulation of the new variational method / Chapter 2.1 --- Formulation --- p.14 / Chapter 2.2 --- Motivation --- p.15 / Chapter 3. --- The variational method applied to the anharmonic oscillator problem / Chapter 3.1 --- Formalism --- p.18 / Chapter 3.2 --- Relationship with usual variational method --- p.32 / Chapter 3.3 --- Relationship with W.K.B. approximation --- p.37 / Chapter 3.4 --- Perturbative corrections --- p.45 / Chapter 3.5 --- Diagonalization of non-orthogonal basis --- p.57 / Chapter 3.6 --- Perturbative corrections using the non-orthogonal basis --- p.72 / Chapter 3.7 --- Some previous works on the anharmonic oscillator problem --- p.85 / Chapter 4. --- The variational method applied to the helium-like atomic problem / Chapter 4.1 --- Previous work on the problem --- p.90 / Chapter 4.2 --- Formulation of the variational method on the problem --- p.95 / Chapter 4.3 --- Zeroth order results for atomic helium --- p.103 / Chapter 4.4 --- Diagonalization using the non-orthogonal basis --- p.109 / Chapter 4.5 --- Results for some helium-like ions --- p.136 / Chapter 4.6 --- Possibility of generalization to systems with more electrons --- p.140 / Chapter 5 --- Concluding remarks / Chapter 5.1 --- Range of applicability of our variational method --- p.164 / Chapter 5.2 --- Ground state problem --- p.165 / Chapter 5.3 --- Completeness of our 'basis' --- p.166 / References --- p.168

A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete measures such as the degree of polynomials, and complexity of Pfaffian functions. A generalisation of the Thom-Milnor Bound [17, 19] for sets defined by the conjunction of equations and non-strict inequalities is presented, in the new context of sets definable in o-minimal structures using the axiomatic complexity measure. Next bounds are produced for sets defined by Boolean combinations of equations and inequalities, through firstly considering sets defined by sign conditions, then using this to produce results for closed sets, and then making use of a construction to approximate any set defined by a Boolean combination of equations and inequalities by a closed set. Lastly, existing results [12] for sets defined using quantifiers on an open or closed set are generalised, using a construction from Gabrielov and Vorobjov [11] to approximate any set by a compact set. This results in a method to find a general bound for any set definable in an o-minimal structure in terms of the axiomatic complexity measure. As a consequence for the first time an upper bound for sub-Pfaffian sets defined by arbitrary formulae with quantifiers is given. This bound is singly exponential if the number of quantifier alternations is fixed.

Wireless video capsule endoscopy has been in use for over a decade and it uses radio frequency (RF) signals to transmit approximately fifty five thousands clear pictures of inside the GI tract to the body-mounted sensor array. However, physician has no clue on the exact location of the capsule inside the GI tract to associate it with the pictures showing abnormalities such as bleeding or tumors. It is desirable to use the same RF signal for localization of the VCE as it passes through the human GI tract. In this thesis, we address the accuracy limits of RF localization techniques for VCE localization applications. We present an assessment of the accuracy of cooperative localization of VCE using radio frequency (RF) signals with particular emphasis on localization inside the small intestine. We derive the Cramer-Rao Lower Bound (CRLB) for cooperative location estimators using the received signal strength(RSS) or the time of arrival (TOA) of the RF signal. Our derivations are based on a three-dimension human body model, an existing model for RSS propagation from implant organs to body surface and a TOA ranging error model for the effects of non-homogenity of the human body on TOA of the RF signals. Using models for RSS and TOA errors, we first calculate the 3D CRLB bounds for cooperative localization of the VCE in three major digestive organs in the path of GI tract: the stomach, the small intestine and the large intestine. Then we analyze the performance of localization techniques on a typical path inside the small intestine. Our analysis includes the effects of number of external sensors, the external sensor array topology, number of VCE in cooperation and the random variations in transmit power from the capsule.

Recently, indoor localization becomes a hot topic no matter in industry or academic field. Smart phones are good candidates for localization since they are carrying various sensors such as GPS, Wi-Fi, accelerometer, barometer and etc, which can be used to estimate the current location. But there are still many challenges for 3D indoor geolocation using smart phones, among which the map selection and 3D performance evaluation problems are the most common and crucial. In the indoor environment, the popular outdoor Google maps cannot be utilized since we need maps showing the layout of every individual floor. Also, layout of different floors differ from one another. Therefore, algorithms are required to detect whether we are inside or outside a building and determine on which floor we are located so that an appropriate map can be selected accordingly. For Wi-Fi based indoor localization, the performance of location estimation is closely related to the algorithms and deployment that we are using. It is difficult to find out a general approach that can be used to evaluate any localization system. On one hand, since the RF signal will suffer extra loss when traveling through the ceilings between floors, its propagation property will be different from the empirical ones and consequently we should design a new propagation model for 3D scenarios. On the other hand, properties of sensors are unique so that corresponding models are required before we analyze the localization scheme. In-depth investigation on the possible hybrid are also needed in case more than one sensor is operated in the localization system. In this thesis, we firstly designed two algorithms to use GPS signal for detecting whether the smart device is operating inside or outside a building, which is called outdoor-indoor transition detection. We also design another algorithm to use barometer data for determining on which floor are we located, which is considered as a multi-floor transition detection. With three scenarios designed inside the Akwater Kent Laboratory building (AK building) at Worcester Polytechnic Institute (WPI), we collected raw data from an Android phone with a version of 4.3 and conducted experimental analysis based on that. An efficient way to quantitatively evaluate the 3D localization systems is using Cramer-Rao Lower Bound (CRLB), which is considered as the lower bound of the estimated error for any localization system. The characteristics of Wi-Fi and barometer signals are explored and proper models are introduced as a foundation. Then we extended the 2D CRLB into a 3D format so that it can fit the our 3D scenarios. A barometer-assisted CRLB is introduced as an improvement for the existing Wi-Fi Receive Signal Strength (RSS)-only scheme and both of the two schemes are compared with the contours in every scenario and the statistical analysis.

"Localization inside the human body using Radio Frequency (RF) transmission is gaining importance in a number of applications such as Wireless Capsule Endoscopy. The accuracy of RF localization depends on the technology adopted for this purpose. The two most common RF localization technologies use Received Signal Strength (RSS) and Time-Of-Arrival (TOA). This research first provides bounds for accuracy of localization of a Endoscopy capsule inside the human body as it moves through the gastro-Intestinal track with and without randomness in transmit power using RSS based localization with a triangulation algorithm. It is observed that in spite of presence of a large number of anchor nodes; the localization error is still in range of few cm, which is quite high; hence we resort to TOA based localization. Due to lack of a widely accepted model for TOA based localization inside human body we use a computational technique for simulation inside and around the human body, named Finite Difference Time Domain (FDTD). We first show that our proprietary FDTD simulation software shows acceptable results when compared with real empirical measurements using a vector network analyzer. We then show that, the FDTD method, which has been used extensively in all kinds of electromagnetic modeling due to its versatility and simplicity, suffers seriously because of its demanding requirement on memory storage and computation time, which is due to its inherently recursive nature and the need for absorbing boundary conditions. In this research we suggest a novel computationally efficient technique for simulation using FDTD by considering FDTD as a Linear Time Invariant (LTI) system. Then we use the software to simulate the TOA of the narrowband and wideband signals propagated inside the human body for RF localization to compare the accuracies of the two using this method. "

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes bibliographical references. / by Jens Hoppe. / Ph.D.

Orientador: Antonio Soares de Castro / Banca: Denis Dalmazi / Banca: Manuel Máximo B. Malheiro de Oliveira / Resumo: O estudo dos estados ligados das equações relativísticas têm muitas aplicações em física nuclear e em outras áreas da física, portanto as soluções que apresentam estados ligados usando as equações de Dirac e Klei-Gordon(KG) desempenham um papel importante no interesse dos pesquisadores. A redução a uma dimensão espacial das equações relativisticas permite melhorar o entendimento dos problemas equivalentes em três dimensões de uma maneira fisicamente mais transparente. Esta redução espacial traz como consequência alterações nas estruturas relativísticas dos potenciais que devem ser estudadas. A tarefa de achar estados ligados dessas emoções não é fácil para formas funcionais gerais para os potenciais externos. Neste trabalho as equações de Dirac e KG em um dimensão espacial são investigadas para diferentes tipos de acoplamento e formas funcionais para o potenciais externos que apresentam estados ligados.Um resultado relevantes que para certas misturas convenientes dos potenciais externos as equações de Dirac e KG apresentam as mesmas auto-energias mas diferentes autofunções. O problema em geral pode ser mapeado num problema de Sturm-Liouville encontrando-se soluções de estados ligados exatamente. Discutimos detalhadamente o comportamento das autofunções e auto-energias para partículas e antipartículas obtidas do problema de Sturm-Liouvillee as possíveis soluções isoladas no caso da equação de Dirac. Uma aparente violação do princípio da incerteza em algns casos é remediada com a introdução do conceito de comprimento de onda Compton efetivo, mostrando que a partícula pode ser localizada numa região do espaço arbitrariamente pequena sem a produção de pares partícula-antipartícula. / Abstract: The study of bound states of the relativistic equations as many applications in nuclear physics and other areas of the physics, therefor the solutions that present bound states for the Dirac and the Klein-Gordon (KG) equations play an important role in the interest of the researches. The reduction to one space dimension of the relativistic equations allows to improve our understanding of the equivalent problems in three dimensionin a physically more transparent way. This space reduction brings as a consequence alterations in the relativistic structures of the potentials that must be studied. The task of findingof these equations is not easy for general functional forms for the external potentials. In this work the Dirac and the KG equationsin one space dimensions for different tpes of coupling and functional forms for the external potentials that present bound states are investigated. A relevant result is that for certain convenient mixtures of the external potentials the Dirac and the KG equations present the same eigenvalues but different eigenfunctins. The problem in general can be mapped into an exactly solvable Sturm-Liouville problem. The behavior of the eigenfunctions and eigenvalues for particles and antiparticles of the Sturn-Liouville problem and the possible isolated solutions in the case of the Dirac equation is discussed in detail. An apparent breaking of the uncertainty priniple in some cases is remedied by the introduction of the concept of effective Compton wavelength, showing that the particle can be located into a region of space arbitrarily small without producing particle-antiparticle pairs. / Mestre

A very wide-ranging subgroup of vehicle routing problems from the graph theory is a common and frequent problem handled daily by transport companies, airline businesses, hi-tech companies with planning drilling of printed circuits boards or other companies from different industries. During numerous previous researches of these problems a lot of analyses were made and many solutions proposed -- of which an outline is in this paper. Some of them giving better or worse results in longer or shorter computing time. In spite of the fact that the processors and new technologies performance is increasing, with some algorithms we cannon compute the result in a reasonable time. That is why this paper is asking a question, if there can be found a fitting algorithm which could be applied on different and faster processing unit structures so it could be ensured a multiple computing speed increase so far. The analysis was carried out using computer experiments on a new build and implemented branch and bound algorithm with a matrix rate reduction.

For many years scientists have been working on more advanced methods to determine the structure of DNA under various conditions. However, research on the study of the shape of DNA segments bound by protein (protein-bound DNA) has been limited by the accuracy and dependability of laboratory techniques. These techniques vary from gel electrophoresis experiments, which can determine the topological shape of unbound DNA, to experiments that crystallize protein-bound DNA in a fixed state in order to determine the spatial location of non-hydrogen atoms. Each method comes with its own benefits and drawbacks. One drawback is experiments used to determine the exact geometry for protein-bound DNA often fail for large complexes. So when laboratory techniques fail to yield a geometric description of the DNA bound by protein, we turn to modeling software. We introduce software that takes known protein-bound DNA topology and determines a potential geometry. An n-string tangle consists of n strings embedded in a three-dimensional ball with endpoints fixed on the boundary. The strings represent the DNA segments bound by protein. Starting with a topological tangle file from KnotPlot (a knot and tangle visualization program), our software minimizes over geometric parameters while keeping the topology fixed. Our modeling software assumes the base pairs to be rigid bodies (or rectangles). Our routine minimizes over an energy function established by Dr. Wilma Olson et al. for the software package, 3DNA. Our program differs from 3DNA in that it can determine potential geometric structures while remaining consistent with the known topology. Thus, for protein-bound DNA whose topology has been identified, we can associate a likely geometry. The energy function over which we minimize is given in terms of dimer geometric parameters derived from crystal structures of protein-bound DNA. A dimer refers to two consecutive DNA base pairs. Thus, not only is our solution topologically relevant, but the geometric solution is DNA sequence specific. Currently only two-dimensional models for protein-bound DNA tangles are easily available. Thus, one of the main benefits of our software is that it offers three-dimensional visualization of the protein-bound DNA segments. This modeling software is a great starting point for determining potential geometries for protein-bound DNA, analyzing the geometrical shape of the bound DNA, and learning more about how the topology and geometry of protein-bound DNA structures are associated.