O objetivo deste trabalho é apresentar alguns testes de cointegração fracionária para séries integradas de ordem d (dR), i.e., séries I(d), comparando-os com os testes de cointegração, cujo parâmetro d assume valores inteiros. O procedimento para os testes de cointegração fracionária utiliza reamostragens de bootstrap com reposição para gerar séries sob a hipótese nula de não cointegração. Estas reamostragens são então utilizadas para calcular os p-valores de algumas estatísticas de testes de regressão, tais como a estatística de Durbin-Watson e a estimativa do parâmetro de memória longa (d) residual. O poder destes testes é apresentado e comparado com os testes de cointegração, mostrando sua consistência. A aplicação destes testes a dados reais compara o modelo de correção de erros de cointegração com o modelo de correção de erros de cointegração fracionária utilizando a medida de erros quadráticos médios dos modelos ajustados. / The purpose of this project is to present some fractional cointegration tests for integrated time series of order d (dR), i.e., I(d) time series, comparing them to cointegration tests, where the parameter d assumes integer values. The tests procedure is done by using bootstrap samples to obtain series under the null hypothesis of non-cointegration. These samples are then used to estimate the p-value of some regression-based test statistics, such as the Durbin-Watson statistic and estimates of residual d parameter. The application of these tests to real series compares the error correction model of cointegration to the error correction model of fractional cointegration by evaluating the mean squared errors over the residuals from the fitted models.

O objetivo deste trabalho é apresentar alguns testes de cointegração fracionária para séries integradas de ordem d (dR), i.e., séries I(d), comparando-os com os testes de cointegração, cujo parâmetro d assume valores inteiros. O procedimento para os testes de cointegração fracionária utiliza reamostragens de bootstrap com reposição para gerar séries sob a hipótese nula de não cointegração. Estas reamostragens são então utilizadas para calcular os p-valores de algumas estatísticas de testes de regressão, tais como a estatística de Durbin-Watson e a estimativa do parâmetro de memória longa (d) residual. O poder destes testes é apresentado e comparado com os testes de cointegração, mostrando sua consistência. A aplicação destes testes a dados reais compara o modelo de correção de erros de cointegração com o modelo de correção de erros de cointegração fracionária utilizando a medida de erros quadráticos médios dos modelos ajustados. / The purpose of this project is to present some fractional cointegration tests for integrated time series of order d (dR), i.e., I(d) time series, comparing them to cointegration tests, where the parameter d assumes integer values. The tests procedure is done by using bootstrap samples to obtain series under the null hypothesis of non-cointegration. These samples are then used to estimate the p-value of some regression-based test statistics, such as the Durbin-Watson statistic and estimates of residual d parameter. The application of these tests to real series compares the error correction model of cointegration to the error correction model of fractional cointegration by evaluating the mean squared errors over the residuals from the fitted models.

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For the fractional-N frequency synthesizers using delta-sigma modulation (DSM) techniques, higher PLL bandwidth is highly desirable in order to achieve faster settling time. As the PLL bandwidth is increased, more quantization noises pass through the PLL so that the output phase noise performance is degraded. There is a tradeoff between phase-noise performance and PLL bandwidth. To improve the problem, the thesis studies the quantization noise cancellation technique. With this technique, the PLL bandwidth can be increased without the cost of degrading phase-noise performance. With the help of Agilent EEsof¡¦s ADS, the phase-noise performance of the studied fractional-N frequency synthesizers can be predicted. For demonstration, this research implements a 2.6 GHz fractional-N frequency synthesizer hybrid module, and compares the measured phase noises with and without the technique under considering various combinations of MASH DSM orders and PLL bandwidth. Another demonstration of this thesis is to design a PLL IC using TSMC 0.18 £gm CMOS process, and make a discussion on the testing performance of the PLL IC.

This dissertation comprises two essays that attempt to determine, empirically, the fiscal response of governments’ to international assistance. The first essay examines whether an increasingly popular recommendation in international aid policy to switch from tied foreign assistance to untied foreign assistance affects investment in critical development expenditure sectors by developing countries. In the past, most international aid has been in the form of tied assistance as donors believed that tying aid will improve its effectiveness. It has been argued, that if tied aid is well designed and effectively managed then its overall effectiveness can be improved. On the contrary, it is also believed that tied aid acts as an impediment to donor cooperation and the building of partnership with developing countries. In addition, it is also argued that it removes the ‘feeling’ of ownership and responsibility of projects from partner countries in aid supported development. Two other more popular arguments used to challenge the effectiveness of foreign aid is that it is compromised when tied to the goods and services of the donor countries because almost 30 percent of its value is eliminated and also because it does not allow recipient countries to act on their priorities for public spending. These problems bring into question whether tied aid is truly the most effective way to help poor countries. A recommendation by the international community is that a switch to untied aid would be necessary. With untied aid, the recipient country is not obligated to buy the goods of the donor country neither is it compelled to pursue the public expenditure priorities of donors. Instead with untied aid they will have greater flexibility over spending decisions and can more easily pursue the priorities of their countries as they see fit. Hence, one could expect that a one dollar increase in untied aid will increase spending in the critical priority sectors by more than a one dollar increase in tied assistance. The question therefore is whether national domestic priorities coincide or not with what the international community has traditionally deemed should be priority. Empirically, we test this prediction using country-by-country data for 57 countries for the period 1973 to 2006. The results suggest that on average untied aid has a greater impact on pro-poor spending than do tied aid. In addition, the results also suggest that fungibility is still an issue even after accounting for the effects of untied aid. However, one could argue that fungibility may not be as bad as it appears since the switch to untied aid improves spending in the sectors that are essential for growth and development. The second essay explores the hypothesis that the expectations of debt forgiveness can discourage developing countries from attaining fiscal independence through an improvement of their tax effort. On the one hand, the international financial community typically advises poor countries to improve revenue mobilization but, on the other hand, the same international community routinely continues to bail-out poor countries that fail to meet their loan repayment obligations. The act of bailing-out these countries creates an expectation on the part of developing country governments that they will receive debt forgiveness time and again in the future. Therefore, the expectation of future bail outs creates a moral hazard that leads to endemic lower tax efforts. The key prediction of our simple theoretical model is that in the presence of debt forgiveness, tax ratios will decline and this decline will be stronger the higher the frequency and intensity of the bailouts. Empirically, we test this prediction using country-level data for 66 countries for the period 1989 to 2006. The results strongly suggest that debt forgiveness plays a significant role in the low tax effort observed in developing countries. Our empirical model allows for the endogeneity of tax effort and debt forgiveness. Interestingly we find that more debt forgiveness is actually provided to countries with lower tax effort. The results are robust to various specifications.

This thesie discusses uniquely circular colourable and uniquely fractional colourable graphs. Suppose G = (V;E) is a graph and r ¸ 2 is a real number. A circular r-colouring of G is a mapping f : V (G) ! [0; r) such that for any edge xy of G, 1 ¡P jf(x) ¡ f(y)j ¡P r ¡ 1. We say G is uniquely circular r-colourable if there is a circular r-colouring f of G and any other circular r-colouring of G can obtained from f by a rotation or a ¢Xip of the colours. Let I(G) denote the family of independent sets of G. A fractional r-colouring of G is a mapping f : I(G) ! [0; 1] such that for any vertex x, Px2I f(I) = 1 and PI2I(G) f(I) ¡P r. A graph G is called uniquely fractional r-colourable if there is exactly one fractional r-colouring of G. Uniquely circular r-colourable graphs have been studied extensively in the literature. In particular, it is known that for any r ¸ 2, for any integer g, there is a uniquely circular r- colourable graph of girth at least g. Uniquely fractional r-colouring of graphs is a new concept. In this thesis, we prove that for any r ¸ 2 for any integer g, there is a uniquely fractional r-colourable graph of girth at least g. It is well-known that for any graph G, Âf (G) ¡P Âc(G). We prove that for any rational numbers r ¸ r0 > 2 and any integer g, there is a graph G of girth at least g, which is uniquely circular r-colourable and at the same time uniquely fractional r0-colourable.

In the first, a time-variant modulus phase lock loop(PLL) model is established. Applying the model, Theorems of fractional-N synthesizers are introduced. We also explain theorems and simulations of Closed-Loop Modulation and Two-Point Delta Sigma Modulation with the model. In the end, a 2.4GHz FHSS transmitter using Two-Point Delta Sigma Modulation which meets Bluetooth specifications is demonstrated.

Modern cellular systems feature increasingly dense base station deployments, augmented by multiple tiers of access points, in an effort to provide higher network capacity as user traffic, especially data traffic, increases. The primary limitation of these dense networks is co-channel interference. The primary source of interference is inter-cell and cross-tier interference, which is especially limiting for users near the boundary of the cells. Inter-cell interference coordination (ICIC) is a broad umbrella term for strategies to improve the performance of the network by having each cell allocate its resources such that the interference experienced in the network is minimized, while maximizing spatial reuse. Fractional frequency reuse (FFR) has been proposed as an ICIC technique in modern wireless networks. The basic idea of FFR is to partition the cell’s bandwidth so that (i) cell-edge users of adjacent cells do not interfere with each other and (ii) interference received by (and created by) cell-interior users is reduced, while (iii) improving spectral reuse compared to conventional frequency reuse. It is attractive for its intuitive implementation and relatively low network coordination requirements compared to other ICIC strategies including interference cancellation, network MIMO, and opportunistic scheduling. There are two common FFR deployment modes: Strict FFR and Soft Frequency Reuse (SFR). This dissertation identifies and addresses key technical challenges associated with fractional frequency reuse in modern cellular networks by utilizing an accurate yet tractable model of both the downlink (base station to mobile) and uplink (mobile to base station) based on the Poisson point process for modeling base station locations. The resulting expressions allow for the development of system design guidelines as a function of FFR parameters and show their impact on important metrics of coverage, rate, power control, and spectral efficiency. This new complete analytical framework addresses system design and performance differences in the uplink and downlink. Also, this model can be applied to cellular networks with multiple tiers of access points, often called heterogeneous cellular networks. The model allows for analysis as a function of system design parameters for users under Strict FFR and SFR with closed and open access between tiers. / text

This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional. / text

When electrons are confined to a two dimensional layer with a perpendicular applied magnetic field, such that the ratio of electrons to flux quanta \((\nu)\) is a small integer or simple rational value, these electrons condense into remarkable new phases of matter that are strikingly different from the metallic electron gas that exists in the absence of a magnetic field. These phases, called integer or fractional quantum Hall (IQH or FQH) states, appear to be conventional insulators in their bulk, but behave as a dissipationless metal along their edge. Furthermore, electrical measurements of such a system are largely insensitive to the detailed geometry of how the system is contacted or even how large the system is... only the order in which contacts are made appears to matter. This insensitivity to local geometry has since appeared in a number of other two and three dimensional systems, earning them the classification of "topological insulators" and prompting an enormous experimental and theoretical effort to understand their properties and perhaps manipulate these properties to create robust quantum information processors. The focus of this thesis will be two experiments designed to elucidate remarkable properties of the metallic edge and insulating bulk of certain FQH systems. To study such systems, we can use mesoscopic devices known as single electron transistors (SETs). These devices operate by watching single electrons hop into and out of a confining box and into a nearby wire (for measurement). If it is initially unfavorable for an electron to leave the box, it can be made favorable by bringing another charge nearby, modifying the energy of the confined electron and pushing it out of the box and into the nearby wire. In this way, the SET can measure nearby charges. Alternatively, we can heat up the nearby wire to make it easier for electrons to enter and leave the box. In this way, the SET is a sensitive thermometer. First, by operating the SET as an electrometer, we measure the local charge of the \(\nu = 5/2\) FQH state. An immediate consequence of measuring fractionally quantized conductance plateaus is that the charge of local excitations should be a fraction of \(e\), the charge of an electron. The simplest charge that would be expected at \(\nu = 5/2\) would \(e/2\). However, if the charged particles that condense into the \(\nu = 5/2\) FQH state are paired, the expected local charge becomes \(e/4\). By watching these local charges being added to compressible puddles at \(\nu = 5/2\) and \(\nu = 5/7\), we find that the local charge at \(\nu = 5/2\) is indeed \(e/4\), indicating that objects of charge \(e\) are pairing to form the ground state of the system. This has implications for the future possibility of detecting non-Abelian braiding statistics in this state, and is described in detail in Chapter 2. By further monitoring how eagerly these \(e/4\) particles enter puddles as we increase the temperature, we can attempt to identify the presence of some excess entropy related to an unconventional degeneracy of their ground state. Such an entropy would be expected if the \(\nu = 5/2\) state exhibited non-Abelian braiding statistics. Progress on these experiments and prospects for building a quantum computer are presented in Chapter 3. Next, by operating the SET as a thermometer, we monitor heat flow along the compressible edge and through the bulk of IQH and FQH states. As an edge is heated and charge on that edge is swept downstream by the external magnetic field, we expect that charge to carry the injected energy in the same downstream direction. However, for certain FQH states, this is not the case. By heating an edge with a quantum point contact (QPC) and monitoring the heat transported upstream and downstream, we find that heat can be transported upstream when the edge contains structure related to \(\nu = 2/3\) FQH physics. Surprisingly, this can be present even when the bulk is in a conventional insulating (IQH) state. Additionally, we unexpectedly find that the \(\nu = 1\) bulk is capable of transporting heat, while the \(\nu = 2\) and \(\nu = 3\) bulk are not. These experiments are presented in Chapter 4. Finally, in Chapter 5, we describe preliminary work on a very different type of topological material, the quantum spin Hall (QSH) insulator. Here, the spin of electrons takes the place of the external magnetic field, creating edge states that propagate in both directions. Each of these edges behaves as an ideal one-dimensional mode, with predicted resistance \(h/e^2\). By creating well-defined regions where these modes can exist, we identify and characterize the conductance associated with topological edges. / Physics