In many everyday tasks it is necessary to compare the performance of the individuals in a population described by two or more criteria, for example comparing products in order to decide which is the best to purchase in terms of price and quality. Other examples are the comparison of universities, countries, the infrastructure in a telecommunications network, and the candidate solutions to a multi- or many-objective problem. In all of these cases, visualising the individuals better allows a decision maker to interpret their relative performance. This thesis explores methods for understanding and visualising multi- and many-criterion populations. Since people cannot generally comprehend more than three spatial dimensions the visualisation of many-criterion populations is a non-trivial task. We address this by generating visualisations based on the dominance relation which defines a structure in the population and we introduce two novel visualisation methods. The first method explicitly illustrates the dominance relationships between individuals as a graph in which individuals are sorted into Pareto shells, and is enhanced using many-criterion ranking methods to produce a finer ordering of individuals. We extend the power index, a method for ranking according to a single criterion, into the many-criterion domain by defining individual quality in terms of tournaments. The second visualisation method uses a new dominance-based distance in conjunction with multi-dimensional scaling, and we show that dominance can be used to identify an intuitive low-dimensional mapping of individuals, placing similar individuals close together. We demonstrate that this method can visualise a population comprising a large number of criteria. Heatmaps are another common method for presenting high-dimensional data, however they suffer from a drawback of being difficult to interpret if dissimilar individuals are placed close to each other. We apply spectral seriation to produce an ordering of individuals and criteria by which the heatmap is arranged, placing similar individuals and criteria close together. A basic version, computing similarity with the Euclidean distance, is demonstrated, before rank-based alternatives are investigated. The procedure is extended to seriate both the parameter and objective spaces of a multi-objective population in two stages. Since this process describes a trade-off, favouring the ordering of individuals in one space or the other, we demonstrate methods that enhance the visualisation by using an evolutionary optimiser to tune the orderings. One way of revealing the structure of a population is by highlighting which individuals are extreme. To this end, we provide three definitions of the “edge” of a multi-criterion mutually non-dominating population. All three of the definitions are in terms of dominance, and we show that one of them can be extended to cope with many-criterion populations. Because they can be difficult to visualise, it is often difficult for a decision maker to comprehend a population consisting of a large number of criteria. We therefore consider criterion selection methods to reduce the dimensionality with a view to preserving the structure of the population as quantified by its rank order. We investigate the efficacy of greedy, hill-climber and evolutionary algorithms and cast the dimension reduction as a multi-objective problem.

Problem Statement: Nowadays, multiple choice (MC) tests are very common, and replace many constructed response (CR) tests. However, literature reveals that there is no consensus whether both test formats are equally suitable for measuring students' ability or knowledge. This might be due to the fact that neither the type of MC question nor the scoring rule used when comparing test formats are mentioned. Hence, educators do not have any guidelines which test format or scoring rule is appropriate. Purpose of Study: The study focuses on the comparison of CR and MC tests. More precisely, short answer questions are contrasted to equivalent MC questions with multiple responses which are graded with three different scoring rules. Research Methods: An experiment was conducted based on three instruments: A CR and a MC test using a similar stem to assure that the questions are of an equivalent level of difficulty. This procedure enables the comparison of the scores students gained in the two forms of examination. Additionally, a questionnaire was handed out for further insights into students' learning strategy, test preference, motivation, and demographics. In contrast to previous studies the present study applies the many-facet Rasch measurement approach for analyzing data which allows improving the reliability of an assessment and applying small datasets. Findings: Results indicate that CR tests are equal to MC tests with multiple responses if Number Correct (NC) scoring is used. An explanation seems straight forward since the grader of the CR tests did not penalize wrong answers and rewarded partially correct answers. This means that s/he uses the same logic as NC scoring. All other scoring methods such as the All or-Nothing or University-Specific rule neither reward partial knowledge nor penalize guessing. Therefore, these methods are found to be stricter than NC scoring or CR tests and cannot be used interchangeably. Conclusions: CR tests can be replaced by MC tests with multiple responses if NC scoring is used, due to the fact that the multiple response format measures more complex thinking skills than conventional MC questions. Hence, educators can take advantage of low grading costs, consistent grading, no scoring biases, and greater coverage of the syllabus while students benefit from timely feedback. (authors' abstract)

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

A interação intermolecular nos clusters N2 C5H5, He CsHs e Ne C5H5 é estudada através de métodos ab initio de teoria de perturbação de muitos corpos. Estes sistemas formam uma interessante família onde as diferentes contribuições intermoleculares desempenham papéis complementares para a compreensão das forças intermoleculares envolvidas. De um modo geral, nestes sistemas neutros e apelares as interações de carga e dipolo são inexistentes. Assim, as contribuições eletrostáticas advém de multipolos superiores. Tais clusters devem sua ligação às chamadas forças de van der Waals que tem sua origem nos momentos induzidos. Para o sistema N2 C5H5, único destes que foi observado experimentalmente, sua existência se deve à força de dispersão de London. Entretanto, a interação eletrostática do tipo quadrupolo-quadrupolo pode ser atrativa ou repulsiva dependendo da orientação dos monômeros. Cálculos realizados em teoria de perturbação de segunda ordem, com funções base extensas, indicam que o sistema é ligado com energia de ligação de cerca de 400 cm-1 e que a estrutura mais estável consiste do N2 paralelo ao plano do C5H5 a 3,4 Á, em concordância com os resultados inferidos por estudos experimentais. No caso dos complexos He C5H5 e Ne C5H5 tem-se a ausência da interação quadrupolar. As forças de indução envolvendo momentos multipolares permanentes do CsHs com momentos induzidos do He ou Ne são muito pequenas. As interações são portanto dominadas pelas forças de dispersão de London. No caso He C5H5 nossos resultados obtidos por teoria de perturbação de segunda ordem indicam que o complexo não é ligado ou é ligado por uma energia inferior a 50 cm-1. Já no caso Ne CsHs a interação é suficiente para criar um complexo estável. Nossa melhor estimativa indica uma energia de ligação de cerca de 100 cm-1 Estes resultados podem ser entendidos qualitativamente através da análise das polarizabilidades de dipolo calculadas para estes sistemas. As análises das possíveis geometrias do C5H5 (C2v ou D5h) e das funções base utilizadas são feitas. No caso do C5H5 isolado, cálculos realizados em alto nível de teoria de perturbação mostram que o estado fundamental tem simetria 2 B2 (C2v) decorrente da distorção Jahn¬Teller do estado 2 E(D5h) A diferença de energia é encontrada ser da ordem de 0,2 eV. / The intermolecular interaction betwen N2 C5H5 , He C5H5 and Ne C5H5 is analyzed using second-order many-body perturbation theory. For these systems the interaction is dominated by the van der Waals contribution of the London dispersion force. For N2 C5H5 it is found that the system is stable by rv400 cm-1 at an intermolecular distance of 3,4 A. Analysis of the electrostatic quadrupole-quadrupole interaction shows that the system is stabilized when N2 lies parallel to the C5H5 plane and is destabilized when N2 lies perpendicular to the C5H5 plane. Therefore, although the dispersion force is the major contributor to the stabilization it is the quadrupole-quadrupole interaction that is held responsible for the favourable structure. In the case of both He C5H5 and Ne C5H5 there is no direct electrostatic interaction as the permanent moments of He and Ne are zero. For the He C5 H5 it is found a very small binding energy, if any. As for the Ne C5H5 we find a binding energy of around 100 cm-1. This different behavior for He and Ne is analyzed using the calculated values for the dipole polarizabilities. The influence of the different geometries for C5H5 (C 2v or D5h) as well as the so-called basis set superposition error are analyzed. For isolated C5H5 high-order calculations show that the 2B2 ground state for symmetry C2v lies around 0,2 eV below the 2E ground state for symmetry D5h. This energy lowering comes from the expected Jahn-Teller distortion.

Dans cette thèse, nous proposons une méthodologie de conception d’architectures de communication dédiées aux réseaux de capteurs basées sur la technique de radio impulsionnelle pour les transmissions ultralarge bande (ULB). La technique impulsionnelle proposée ici repose sur la modulation de forme d’impulsion. L’approche de conception architecturale présentée dans cette thèse se focalise plus particulièrement sur la forme des impulsions et leur génération, qui revêt un intérêt majeur puisqu’elle constitue le support de l’information échangée. L’étude sur le choix de la forme d’impulsion nous a conduit à proposer deux architectures différentes. Une première architecture repose sur les polynômes orthogonaux, et plus particulièrement sur les polynômes d’Hermite, pour la génération des impulsions, et sur une architecture de corrélation pour la détection et la reconnaissance des trains d’impulsions transmis. La deuxième architecture est basée sur la transformée en paquets d’ondelettes discrète et peut être exploitée selon deux modes d’utilisation différents, mono et multiutilisateurs. L’utilisation d’une architecture de synthèse à l’émission et d’analyse à la réception ouvre une nouvelle orientation pour les communications numériques, permettant à la transformée en ondelettes d’assurer à la fois la génération des impulsions à l’émission et leur reconnaissance à la réception. Un intérêt immédiat de la technique proposée permet notamment de faciliter l’accès multiutilisateurs au canal ultralarge bande, et d’autoriser des communications simultanées (Many-to-one, des nœuds vers le puits) ou du broadcast (One-to-many, du puits vers les nœuds) sans surcharger la couche MAC. L’architecture proposée s’inscrit donc à l’interface des couches PHY et MAC et permet de relâcher les contraintes de conception spécifiques à ces couches / In this thesis, we propose a design methodology for communication architectures dedicated to wireless sensor network based on impulse radio techniques for UWB communications. The impulse technique proposed in this work relies on pulse shape modulation. The architecture design approach proposed in this thesis focuses on pulses shape and their generation, which is of major interest as it constitutes the carrier of the information exchanged. The study on the choice of pulse shape led us to propose two different architectures. The first one is based on orthogonal polynomials, more especially on the Hermite polynomials, for impulse generation, and on a correlation architecture for detection and recognition of transmitted impulses. The second architecture is based on discrete wavelet packet transform and can be used according two different modes, mono and multi-users. The use of both synthesis and analysis architectures for emitter and receiver, respectively, offers a new way for digital communications and allows the wavelet transform to ensure the impulses generation on the transmitter and their recognition on the receiver. A major interest point of the proposed technique is to facilitate the multi-users access to the ultra-wideband channel and to allow simultaneous communications (many-to-one, from the sensors to the sink) or broadcast (one-to-many, from the coordinator to the nodes) but without overloading the MAC layer. Hence, the proposed architecture is part of the interface between both PHY and MAC layers, and allows to release their specific design constraints

Neste trabalho estudamos a extensão da aproximação de campo médio, dada uma condição inicial, para a evolução temporal de um sistema composto de férmions e bósons que interagem. Para isto usamos uma técnica de projeção dependente do tempo através do qual obtemos equações de movimento do tipo cinético para o conjunto de variáveis dinâmicas de um corpo. Na primeira parte do trabalho aplicamos a técnica para um sistema descrito pelo modelo de Jaynes Cummings, o qual descreve a interação da matéria, representada por um sistema de dois níveis, com a radiação, representada por um modo normal do campo quantizado. Obtemos a dinâmica de campo médio e a seguir usando a técnica de projeçào, calculamos correções à esta descrição de campo médio. Além de ser um modelo exatamente solúvel, o que nos permite comparar nossos resultados com a solução exata, o modelo de Jaynes-Cummings corresponde ao plasma escalar relativístico em zero dimensões espaciais. Na segunda parte deste trabalho estudamos o modelo do plasma escalar relativístico. Esta teoria quântica de campos descreve a interação de campos bosônicos escalares e fermiônicos de spin-1/2 através de uma interação do tipo Yukawa. Para o sistema do plasma escalar relativístico obtemos as equações que descrevem a dinâmica de campo médio e a partir das soluções estacionárias, renormalizamos a teoria. Finalmente, estudamos o regime de pequenas oscilações em torno do equilíbrio, obtendo soluções analíticas para a evolução de nossas variáveis. Analisamos também as condições para existência de estados ligados neste regime. / In this work we study the extension of the mean-field approximation, given an initial condition, to the time evolution of a fermion-boson system. We use a time-dependent projection where we obtain kinetic-type equations for the set of one-body variables. First, we study the Jaynes-Cummings model which describes the interaction of the matter represented by the two-level system with the radiation represented by the normal mode of the quantized radiation field. We obtain the mean-field dynamics of the system and using the projection technique, we evaluate corrections to this mean-field description. Relevance of the Jaynes-Cummings model stems from the fact that, besides being soluble which possibilities compare our results with the exact solution, it can be seen as corresponding to the relativistic scalar plasma in zero spatial dimensions. Next, we study the relativistic scalar plasma. This quantum field theory describes a system of spin-1/2 fermions interacting through the exchange of scalar particles via a Yukawa-type interaction. In the study of the relativistic scalar plasma, we obtain the mean-field dynamics and from the static solution, we renormalize the theory. Finally, we study the small oscillations regime obtaining analytical solution for one-body variables. We have also examined the condition for the existence of bound-state in this case.

Yu, Wing Chi = 多體系統中的量子相變動力學 / 余詠芝. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 94-99). / Abstracts in English and Chinese. / Yu, Wing Chi = Duo ti xi tong zhong de liang zi xiang bian dong li xue / Yu Yongzhi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Quantum phase transitions --- p.1 / Chapter 1.2 --- Schemes detecting QPTs --- p.3 / Chapter 1.2.1 --- Traditional schemes --- p.3 / Chapter 1.2.2 --- Quantum Entanglement --- p.4 / Chapter 1.2.3 --- Quantum fidelity --- p.4 / Chapter 1.2.4 --- Loschmidt echoes --- p.5 / Chapter 1.2.5 --- Quench dynamics --- p.6 / Chapter 1.3 --- Motivation --- p.7 / Chapter 2 --- Theoretical framework --- p.9 / Chapter 2.1 --- Quantum Zeno effect --- p.9 / Chapter 2.2 --- Mathematical formulation --- p.11 / Chapter 2.3 --- Remarks --- p.14 / Chapter 3 --- Analysis on the One-dimensional Transverse-field Ising model --- p.17 / Chapter 3.1 --- The model --- p.17 / Chapter 3.2 --- Diagonalization of the Hamiltonian --- p.20 / Chapter 3.2.1 --- Jordan-Wigner transformation --- p.20 / Chapter 3.2.2 --- Fourier Transformation --- p.24 / Chapter 3.2.3 --- Bogoliubov transformation --- p.26 / Chapter 3.3 --- Quantum Zeno dynamics in the model --- p.28 / Chapter 3.3.1 --- Analytical calculation of the Zeno susceptibility --- p.28 / Chapter 3.3.2 --- Validity of the analytical result --- p.31 / Chapter 3.3.3 --- Scaling behavior of the Zeno susceptibility --- p.33 / Chapter 3.3.4 --- Zeno susceptibility around the critical point --- p.35 / Chapter 3.4 --- Conclusion and experimental outlook --- p.38 / Chapter 4 --- Analysis on the Lipkin-Meshkov-Glick Model --- p.40 / Chapter 4.1 --- The model --- p.41 / Chapter 4.2 --- Diagonalization of the Hamiltonian --- p.46 / Chapter 4.2.1 --- Holstein-Primakoff transformation --- p.46 / Chapter 4.2.2 --- Bogoliubov transformation --- p.49 / Chapter 4.3 --- Quantum Zeno dynamics in the model --- p.51 / Chapter 4.3.1 --- Analytical form of the Zeno susceptibility and its scaling behavior --- p.51 / Chapter 4.3.2 --- Validity of the analytical result --- p.54 / Chapter 4.3.3 --- Numerical analysis of the Zeno susceptibility --- p.55 / Chapter 4.4 --- Conclusion --- p.60 / Chapter 5 --- Analysis on the Kitaev model on a honeycomb lattice --- p.61 / Chapter 5.1 --- The model --- p.61 / Chapter 5.2 --- Diagonalization of the Hamiltonian --- p.63 / Chapter 5.2.1 --- Jordan-Wigner transformation for two-dimensional systems --- p.64 / Chapter 5.2.2 --- Majorana fermion representation --- p.68 / Chapter 5.2.3 --- Fermions on the 之-bonds --- p.71 / Chapter 5.2.4 --- Bogoliubov transformation --- p.73 / Chapter 5.3 --- Energy spectrum --- p.75 / Chapter 5.4 --- Quantum Zeno dynamics in the model --- p.77 / Chapter 5.4.1 --- Coupling along the Jx = Jy line --- p.77 / Chapter 5.4.2 --- Coupling along the line with constant Jz --- p.83 / Chapter 5.5 --- Conclusion --- p.90 / Chapter 6 --- Conclusion and outlook --- p.91 / Bibliography --- p.94 / Chapter A --- Perturbative form of the Loschimdt Echo --- p.100 / Chapter B --- Hellmann-Feynman theorem --- p.107 / Chapter C --- Commutation relations in the Jordan-Wigner transformation --- p.108

In chapter 1, we give an introduction to QPT, and take one-dimensional XXZ model as an example to illustrate the QPT therein. Through this simple example, we would show that when the tunable parameter is varied, the system evolves into different phases, across two quantum QPT points. The distinct phases exhibit very different behaviors. Also a schematic phase diagram is appended. / In chapter 2, we are engaged in research on ordered phases. Originating in the work of Landau and Ginzburg on second-order phase transition, the spontaneous symmetry breaking induces nonzero expectation of field operator, e.g., magnetization M in the Ising model, and then we say long range order (LRO) exists in the system. LRO plays a key role in determining the ordered-disorder transition. Thereby, we investigate two-dimensional 120° orbital-only model to present how to extract the information of LRO in a pedagogical manner, by applying the reflection positivity method introduced by Dyson, Lieb, and Simon. We rigorously establish the existence of an anti-ferromagnetic like transverse orbital long-range order in the so called two-dimensional 120° model at zero temperature. Next we consider possible pairings in the family of FeAs-based ReO1--xFxFeAs (Re=La, Nd, Ce, Pr, etc.) high-temperature superconductors. We build some identities based on a two-orbital model, and obtained some constraints on a few possible pairings. We also establish the sufficient conditions for the coexistence of two superconducting orders, and we propose the most favorable pairings around half filling according to physical consideration. / In chapter 3, we present a quantum solvation process with solvent of fermion character based on the one-dimensional asymmetric t-J-Jz model. The model is experimental realizable in optical lattices and exhibits rich physics. In this work, we show that there exist two types of phase separations, one is driven by potential energy while the other by kinetic energy. In between, solvation process occurs. Analytically, we are able to obtain some rigorous results to understand the underlying physics. Numerically, we perform exact diagonalization and density matrix renormalization group calculations, accompanied by detailed finite size analysis. / In chapter 4, we explore several characterizations of QPT points. As distinguished from the methods in condensed-matter physics, we give much attention to understand QPT from the quantum information (QI) point of view. The perspective makes a new bridge between these two fields. It no only can facilitate the understanding of condensed-matter physics, but also provide the prominent playground for the quantum information theory. They are fidelity susceptibility and reduced fidelity susceptibility. We establish a general relation between fidelity and structure factor of the driving term in a Hamiltonian through fidelity susceptibility and show that the evaluation of fidelity in terms of susceptibility is facilitated by using well developed techniques such as density matrix renormalization group for the ground state, or Monte Carlo simulations for the states in thermal equilibrium. Furthermore, we show that the reduced fidelity susceptibility in the family of one-dimensional XY model obeys scaling law in the vicinity of quantum critical points both analytically and numerically. The logarithmic divergence behavior suggests that the reduced fidelity susceptibility can act as an indicator of quantum phase transition. / Quantum Phase Transition (QPT) describes the non-analytic behaviors of the ground-state properties in a many-body system by varying a physical parameter at absolute zero temperature - such as magnetic field or pressure, driven by quantum fluctuations. Such quantum phase transitions can be first-order phase transition or continuous. The phase transition is usually accompanied by a qualitative change in the nature of the correlations in the ground state, and describing this change shall clearly be one of our major interests. We address this issue from three prospects in a few strong correlated many-body systems in this thesis, i.e., identifying the ordered phases, studying the properties of different phases, characterizing the QPT points. / The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions (QPTs), driven by experimental advance on the cuprate superconductors, the heavy fermion materials, organic conductors, Quantum Hall effect, Fe-As based superconductors and other related compounds. It is clear that strong electronic interactions play a crucial role in the systems of current interest, and simple paradigms for the behavior of such systems near quantum critical points remain unclear. Furthermore, the rapid progress in Feshbach resonance and optical lattice provides a flexible platform to study QPT. / You, Wenlong = 強關聯多體體系中的量子相變 / 尤文龍. / Adviser: Hai Qing Lin. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 104-115). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307. / You, Wenlong = Qiang guan lian duo ti ti xi zhong de liang zi xiang bian / You Wenlong.

Orientador: Hércules de Araújo Feitosa / Banca: Edélcio Gonçalves de Souza / Banca: Mauri Cunha do Nascimento / Resumo: Dentre as diversas lógicas não-clássicas, que complementam o cálculo de predicados de primeira ordem, destacamos as lógicas moduladas. As lógicas moduladas são caracterizadas pela inclusão de um novo quantificador, chamado modulado, que tem a incumbência de interpretar aspectos indutivos de quantificadores das linguagens naturais. Como um caso particular de lógica modulada, a lógica do muito formaliza a noção intuitiva de "muitos". O quantificador do muito é representado por G. Assim, uma sentença do tipo Gxα(x) deve ser entendida como "muitos indivíduos satisfazem a propriedade α". Semanticamente, a noção de muitos está associada a uma estrutura matemática denominada família fechada superiormente e própria. Seja E um conjunto não vazio. Uma família própria fechada superiormente F em E é tal que: (i) F ⊆ P(E); (ii) E ∈ F; (iii) ∅ ∉ F; (iv) A ∈ F e A ⊆ B ⇒ B ∈ F. Intuitivamente, F caracteriza os conjuntos que possuem 'muitos' elementos. E, assim, o universo E possui muitos elementos; o ∅ não possui muitos elementos; e se A possui muitos elementos, então todo conjunto que contém A também possui muitos elementos. Com elementos sintáticos que caracterizam linguisticamente estas propriedades de F, pode-se verificar que a lógica do muito é correta e completa para uma estrutura de primeira ordem estendida por uma família própria fechada superiormente. A lógica do muito foi originalmente introduzida em um sistema dedutivo hilbertiano, baseado apenas em axiomas e regras de dedução. Neste trabalho, desenvolvemos um outro sistema dedutivo para a lógica do muito, porém num sistema de tablôs. Demonstramos, naturalmente, que esse novo sistema é equivalente ao sistema axiomático original. / Abstract: Among the several non classical logics that complement the classical first-order logic, we detach the Modulated Logics. This class of logics is characterized by extending the classical logic by the introduction of a new generalized quantifier, called modulated quantifier, that has the attribution of interpreting some inductive aspects of quantifiers in any natural language. As a particular case of Modulated Logic, the Logic of Many formalize the intuitive notion of "many". The quantifier of many is represented by G. Thus, a sentence of the type Gxα(x) must be understood like "many individuals satisfy the property α". Semantically, the notion of many is associated with a mathematical structure named proper superiorly closed family. Let E be a non empty set. A proper superiorly closed family F in E is such that: (i) F ⊆ P(E); (ii) E ∈ F; (iii) ∅ ∉ F; (iv) A ∈ F e A ⊆ B ⇒ B ∈ F. Intuitively, F characterizes the sets which have "many" elements. The empty set ∅ does not have many elements. And if A has many elements, then any set which contains A, also has many elements. The logic of many has syntactical elements that caracterize linguisticaly these properties of F. We can verify that the Logic of Many is correct and complete for a first order structure extended by a proper superiorly closed family. The Logic of Many was originally introduced in a Hilbertian deductive system, based only on axioms and rules. In this work, we developed another deductive system for the Logic of Many, but in a tableaux system. We proof that this new system is equivalent to the original one. / Mestre

Wei, Bobo = 基於Bethe Ansatz解的兩個一維多體模型的研究 / 魏勃勃. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 62-68). / Abstracts in English and Chinese. / Wei, Bobo = Ji yu Bethe Ansatz jie de liang ge yi wei duo ti mo xing de yan jiu / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Cold atoms systems --- p.1 / Chapter 1.1.1 --- Optical lattice --- p.2 / Chapter 1.1.2 --- Feshbach resonance --- p.4 / Chapter 1.2 --- Outline of this work --- p.6 / Chapter 2 --- Review of Bethe ansatz method --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Coordinate Bethe ansatz: One-dimensional Bose gas --- p.10 / Chapter 2.2.1 --- N = 2 bosons case --- p.11 / Chapter 2.2.2 --- N = 3 bosons case --- p.13 / Chapter 2.2.3 --- Arbitrary N bosons case --- p.15 / Chapter 3 --- Persistent currents in the one-dimensional mesoscopic Hubbard ring --- p.18 / Chapter 3.1 --- Introduction --- p.18 / Chapter 3.2 --- The model and its Bethe ansatz soluiton --- p.20 / Chapter 3.3 --- The charge persistent current --- p.23 / Chapter 3.3.1 --- The charge persistent current and the on-site interaction U --- p.24 / Chapter 3.3.2 --- The charge persistent current and the system size L --- p.28 / Chapter 3.4 --- The spin persistent current --- p.30 / Chapter 3.4.1 --- The spin persistent current and the on-site interaction U --- p.30 / Chapter 3.4.2 --- The spin persistent current and the system size L --- p.32 / Chapter 3.5 --- Conclusions --- p.33 / Chapter 4 --- Exact results of two-component ultra-cold Fermi gas in a hard wall trap --- p.36 / Chapter 4.1 --- Introduction --- p.36 / Chapter 4.2 --- The model and its exact solution --- p.37 / Chapter 4.3 --- The Theoretical Background --- p.41 / Chapter 4.4 --- N = 2 --- p.44 / Chapter 4.4.1 --- Single-particle reduced density matrix and Position density distributions --- p.44 / Chapter 4.4.2 --- Momentum density distributions --- p.45 / Chapter 4.5 --- N = 3 --- p.46 / Chapter 4.5.1 --- Single-particle reduced density matrix --- p.46 / Chapter 4.5.2 --- Natural orbitals and their populations --- p.48 / Chapter 4.5.3 --- Momentum density distribution --- p.51 / Chapter 4.5.4 --- Two-particle density distributions --- p.53 / Chapter 4.6 --- Conclusions --- p.53 / Chapter 5 --- Summary and prospects --- p.54 / Chapter 5.1 --- Summary --- p.54 / Chapter 5.2 --- Prospects for further study --- p.55 / Chapter 5.2.1 --- Recent experimental advancements on realization of quantum gas --- p.55 / Chapter 5.2.2 --- Some recent work on FTG gas --- p.57 / Bibliography --- p.62 / Chapter A --- Explicit form of Bethe ansatz wave function for N = 2 fermions --- p.69 / Chapter B --- "Simplified form of Bethe ansatz wave function for N = 3, M=1 fermions" --- p.73 / Chapter C --- Explicit form of Single-particle reduced density matrix for free fermions --- p.79