The inflationary paradigm is one which was designed to answer questions that arose from classical Hot Big Bang cosmology. The period of rapid expansion in the early Universe provides a mechanism to solve the flatness, horizon and relic problems. More importantly, since the theory was first introduced it has been realised that it also provides a mechanism to generate the initial perturbations from which structure in the Universe can grow. In the zoo of potential inflationary models there is a dominant class: slow-roll inflation. The idea that the energy density of the inflationary field is dominated by its potential highly simplifies the calculations required to predict observable quantities. This simplification relies on all the information required to know the subsequent dynamics of the field to be encoded in the space Φ-Φ̇; it must be an effective phase space. I show that Φ-Φ̇ can be considered to be such a space for the most general scalar-tensor theory which gives second-order equations of motion: Horndeski theory. There are theoretical issues associated with this reduction that are illuminated through specific examples in which they occur. A theoretical issue with inflation is that there is an overabundance of models, with some capable of predicting any value of the possible observables. The second block of work in this thesis looks at a particular set of models that make the same observational prediction. These 'attractor' models utilise a non-minimal coupling between the inflationary fields and gravity and are studied in depth, both in the case of one and several fields. Firstly, I examine the Universal Attractors, a single field subset of these models. I show, in detail, the observational prediction such a model makes in the case of a strong non-minimal coupling and then examine the constraints it would be possible to put on such a coupling if a confirmed detection of primordial gravitational waves was made. Despite the discussion existing in the literature there is a small deviation of the Universal Attractor models from the predictions of the Starobinsky model. Furthermore, the coupling, ξ is found to be constrained so that |ξ| < 1 in the case where there a level of detectable primordial tensor modes. While the attractor models have an effective one-field description in reality there are several other fields that are assumed to be fixed during the inflationary phase. This claim requires careful examination as the field-space of the models generally is not flat. This curvature can cause a destabilising effect with certain parameters and so I investigate how susceptible the α-attractors and related models are to the destabilisation. A key result of this chapter is to highlight how important it is to not rely on the slow-roll approximation when assessing the effect of the instability, as the region where the effect begins to become large corresponds with the region where slow-roll begins to break down. Assuming the slow-roll approximation is valid leads to an over-estimation of the effect that the instability mechanism has. Despite this, some of the models considered are seen to experience the instability for certain ranges of model parameters. Making the assumption that any occurrence of the instability will, at the very least, move the observational prediction of the model outside the currently constrained range allows a constraint on the model parameter in question which directly translates to a theoretical lower bound on the tensor-scalar ratio, r > 0.0005.

This thesis addresses problems in the area of mechanism design. In many settings in winch collective decisions are made, individuals' actual preferences are not publicly observable. As a result, individuals should be relied on to reveal this information. We are interested in an important application of mechanism design, which is the construction of desirable procedures for deciding upon resource allocation or task assignment. We make two main contributions. First, we propose a new mechanism for allocating a divisible commodity between a number of buyers efficiently and fairly. Buyers are assumed to behave as price-anticipators rather than as price-takers. The proposed mechanism is as parsimonious as possible, in the sense that it requires participants to report a one-dimensional message (scalar strategy) instead of an entire utility function, as required by Vickrey-Clarke-Groves (VCG) mechanisms. We show that this mechanism yields efficient allocations in Nash equilibria and moreover, that these equilibria are envy-free. Additionally, we present distinct results that this mechanism is the only simple scalar strategy mechanism that both implements efficient Nash equilibria and satisfies the no envy axiom of fairness. The mechanism's Nash equilibria are proven to satisfy the fairness properties of both Ranking and Voluntary Participation. Our second contribution is to develop optimal VCG mechanisms in order to assign identical economic "bads" (for example, costly tasks) to agents. An optimal VCG mechanism minimizes the largest ratio of budget imbalance to efficient surplus over all cost profiles. The optimal non-deficit VCG mechanism achieves asymptotic budget balance, yet the non-deficit requirement is incompatible with reasonable welfare bounds. If we omit the non-deficit requirement, individual rationality greatly changes the behavior of surplus loss and deficit loss. Allowing a slight deficit, the optimal individually rational VCG mechanism becomes asymptotically budget balanced. Such a phenomenon cannot be found in the case of assigning economic "goods."

Buoyant flows are characterized with unsteady large-scale structures and thus time-dependent large eddy simulation (LES) is generally favored. In this dissertation, to further explore LES for buoyant flow, an LES code based on a collocated grid system is first developed. A multigrid solver using a control strategy is developed for the pressure Poisson equations. The control strategy significantly accelerated the convergence rate. A temperature solver using a fourth-order Runge-Kutta approach is also developed. The LES code is extensively tested before it is applied. Although the collocated grid system will introduce conservation errors, in tests of a steady lid-driven cavity flow and transient start-up flow, the effect of the non-conservation of the collocated grid system was not significant. <p>In LES, the effect of SGS scales is represented by SGS models. A novel dynamic nonlinear model (DNM) for SGS stress is tested using isothermal channel flow at Reynolds number 395. The kinetic energy dissipation and geometrical characteristics of the resolved scale and SGS scale with respect to the DNM are investigated. In general, the DNM is reliable and has relatively realistic geometrical properties in comparison with the conventional dynamic model in the present study. In contrast to a pure advecting velocity field, a scalar (temperature) field displays very different characteristics. The modelling of SGS heat flux has not been as extensively studied as that of SGS stress partly due to the complexity of the scalar transport. In this dissertation, LES for a turbulent combined forced and natural convection is studied. The DNM model and a nonlinear dynamic tensor diffusivity model (DTDM-HF) are applied for the SGS stress and heat flux, respectively. The combined effect of the nonlinear models is compared to that of linear models. Notable differences between the nonlinear and linear SGS models are observed at the subgrid-scale level. At the resolved scale, the difference is smaller but relatively more distinguishable in terms of quantities related to the temperature field. <p>Finally, the geometrical properties of the resolved velocity and temperature fields of the thermal flow are investigated based on the LES prediction. Some universal geometrical patterns have been reproduced, e.g. the positively skewed resolved enstrophy generation and the alignment between the vorticity and vortex stretching vectors. The present research demonstrates that LES is an effective tool for the study of the geometrical properties of a turbulent flow at the resolved-scales. The wall imposed anisotropy on the flow structures and orientation of the SGS heat flux vector are also specifically examined. In contrast to the dynamic eddy diffusivity model, the DTDM-HF successfully predicts the near-wall physics and demonstrates a non-alignment pattern between the SGS heat flux and temperature gradient vector.

In any research program we begin with a hypothesis and when our expected results do not concur with the observed results we must try and understand the dynamics behind the changed process. In this study we were trying to understand the flux between regional groundwater systems, surface waters and sedimentation processes in order to predict the fate of contaminants entering one of the larger bodies of water in the world- Lake Ontario. This lake has increased levels of tritium due to anthropogenic inputs. Our first approach to the problem was to look at tritium fluxes within the system . Hydrological balances were constructed and a series of sediment cores were taken longitudinally and laterally across the lake. The second approach was to quantify the sediment accumulation rate (SAR) within the depositional basins and zones of erosion in order to improve the linkage between erosion control (sedimentation) and the water quality program. In the last chapter the movement of tritium, by molecular diffusion, through the clayey-silts of Lake Ontario is quantified in terms of an effective diffusion coefficient. In these sediments effective diffusion equals molecular diffusion. In a laboratory experiment four cores of lake sediment were spiked with tritium . The resulting concentration gradient changes in the sediment porewaters after six weeks could be modeled by an analytical one- dimensional diffusive transport equation. Results calculated the average lab diffusion coefficient to be 2. 7 x 10 - 5cm 2. sec -1 which is twice that determined by Wang et al, 1952 but still reasonable. Short cores (50 cm) from lake Ontario had observed tritium concentrations with depth that reflected a variable diffusive profile. The increases and decreases in tritium with depth could be correlated between cores. Monthly tritium emission data was obtained and correlations between peaks in the tritium profile and emissions were observed. Monthly variations in release emissions corresponded to approximately a one centimeter slice of core. An average calculated diffusion coefficient of theses cores was 1. 0 x 10 -5 cm 2. sec -1 which compares to Wang's coefficient of 1. 39 x 10 -5 cm 2. sec -1. This implies that tritium is moving through the sediment column at a rate equal to diffusion. The results were obtained for smoothed values. It was not possible to model the perturbations of the data with a one dimensional model. The dynamics of the system imply that tritium could be used as a biomonitor for reactor emissions, mixing time and current direction scenarios and that a better understanding of this process could be gained by future coring studies and a new hypothesis.

This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.

In any research program we begin with a hypothesis and when our expected results do not concur with the observed results we must try and understand the dynamics behind the changed process. In this study we were trying to understand the flux between regional groundwater systems, surface waters and sedimentation processes in order to predict the fate of contaminants entering one of the larger bodies of water in the world- Lake Ontario. This lake has increased levels of tritium due to anthropogenic inputs. Our first approach to the problem was to look at tritium fluxes within the system . Hydrological balances were constructed and a series of sediment cores were taken longitudinally and laterally across the lake. The second approach was to quantify the sediment accumulation rate (SAR) within the depositional basins and zones of erosion in order to improve the linkage between erosion control (sedimentation) and the water quality program. In the last chapter the movement of tritium, by molecular diffusion, through the clayey-silts of Lake Ontario is quantified in terms of an effective diffusion coefficient. In these sediments effective diffusion equals molecular diffusion. In a laboratory experiment four cores of lake sediment were spiked with tritium . The resulting concentration gradient changes in the sediment porewaters after six weeks could be modeled by an analytical one- dimensional diffusive transport equation. Results calculated the average lab diffusion coefficient to be 2. 7 x 10 - 5cm 2. sec -1 which is twice that determined by Wang et al, 1952 but still reasonable. Short cores (50 cm) from lake Ontario had observed tritium concentrations with depth that reflected a variable diffusive profile. The increases and decreases in tritium with depth could be correlated between cores. Monthly tritium emission data was obtained and correlations between peaks in the tritium profile and emissions were observed. Monthly variations in release emissions corresponded to approximately a one centimeter slice of core. An average calculated diffusion coefficient of theses cores was 1. 0 x 10 -5 cm 2. sec -1 which compares to Wang's coefficient of 1. 39 x 10 -5 cm 2. sec -1. This implies that tritium is moving through the sediment column at a rate equal to diffusion. The results were obtained for smoothed values. It was not possible to model the perturbations of the data with a one dimensional model. The dynamics of the system imply that tritium could be used as a biomonitor for reactor emissions, mixing time and current direction scenarios and that a better understanding of this process could be gained by future coring studies and a new hypothesis.

Buoyant flows are characterized with unsteady large-scale structures and thus time-dependent large eddy simulation (LES) is generally favored. In this dissertation, to further explore LES for buoyant flow, an LES code based on a collocated grid system is first developed. A multigrid solver using a control strategy is developed for the pressure Poisson equations. The control strategy significantly accelerated the convergence rate. A temperature solver using a fourth-order Runge-Kutta approach is also developed. The LES code is extensively tested before it is applied. Although the collocated grid system will introduce conservation errors, in tests of a steady lid-driven cavity flow and transient start-up flow, the effect of the non-conservation of the collocated grid system was not significant. <p>In LES, the effect of SGS scales is represented by SGS models. A novel dynamic nonlinear model (DNM) for SGS stress is tested using isothermal channel flow at Reynolds number 395. The kinetic energy dissipation and geometrical characteristics of the resolved scale and SGS scale with respect to the DNM are investigated. In general, the DNM is reliable and has relatively realistic geometrical properties in comparison with the conventional dynamic model in the present study. In contrast to a pure advecting velocity field, a scalar (temperature) field displays very different characteristics. The modelling of SGS heat flux has not been as extensively studied as that of SGS stress partly due to the complexity of the scalar transport. In this dissertation, LES for a turbulent combined forced and natural convection is studied. The DNM model and a nonlinear dynamic tensor diffusivity model (DTDM-HF) are applied for the SGS stress and heat flux, respectively. The combined effect of the nonlinear models is compared to that of linear models. Notable differences between the nonlinear and linear SGS models are observed at the subgrid-scale level. At the resolved scale, the difference is smaller but relatively more distinguishable in terms of quantities related to the temperature field. <p>Finally, the geometrical properties of the resolved velocity and temperature fields of the thermal flow are investigated based on the LES prediction. Some universal geometrical patterns have been reproduced, e.g. the positively skewed resolved enstrophy generation and the alignment between the vorticity and vortex stretching vectors. The present research demonstrates that LES is an effective tool for the study of the geometrical properties of a turbulent flow at the resolved-scales. The wall imposed anisotropy on the flow structures and orientation of the SGS heat flux vector are also specifically examined. In contrast to the dynamic eddy diffusivity model, the DTDM-HF successfully predicts the near-wall physics and demonstrates a non-alignment pattern between the SGS heat flux and temperature gradient vector.

This study analyzes a turbulent passive scalar field using two-point and three-point correlations of the fluctuating scalar field. Multipoint correlation functions are investigated because they retain scaling property information and simultaneously probe the concentration field for the spatial structure of the scalar filaments. Thus, multipoint correlation functions provide unique information about the spatial properties of the concentration filaments. The concentration field is created by the iso-kinetic release of a high Schmidt number dye into a fully developed turbulent boundary layer of an open channel flow. The concentration fields were previously measured using the planar laser-induced fluorescence technique. The two-point correlations of the fluctuating scalar field indicate that as the scalar field evolves downstream, the anisotropic influence of the tracer injection method diminishes, and the scalar field becomes dominated by the mean velocity shear. As the scalar filaments align with the mean velocity gradient, the elliptical shape associated with the contours of the correlation function tilts in the direction of the mean velocity gradient. As a result, the two-point correlation contours of the concentration fluctuations indicate that anisotropic conditions (i.e. the tilted, asymmetric, elliptical shape) develop as a consequence of the mean velocity shear. Three-point correlations of the fluctuating scalar field are calculated based on configuration geometries defined by previous researchers. The first configuration follows Mydlarski and Warhaft (1998), which employs two cold-wire measurements and Taylor's frozen turbulence hypothesis. The three-point correlation contours of the concentration fluctuations associated with the cold-wire measurements exhibit a symmetric characteristic V-shape. Similar symmetric properties are observed in the current study. The second set of configurations follows on recent theoretical predictions, which indicate that the three-point correlation of the fluctuating scalar field is dependent on the size, shape, and orientation of the triangle created by the three points. The current study analyzes two geometric configurations (isosceles and collinear). The geometric configurations are defined to ensure that the influence of the shape remains constant as the configuration is rotated, translated, and dilated. Additionally, the scaling exponent in the inertial-convective regime is calculated to determine the dependence of the correlation function on the size of the triangle pattern.

An important advantage of Elliptic Curve Cryptosystem (ECC) is the shorter key length in public key cryptographic systems. It can provide adequate security when the bit length over than 160 bits. Therefore, it has become a popular system in recent years. Scalar multiplication also called point multiplication is the core operation in ECC. In this thesis, we propose the ECC architectures of two different irreducible polynomial versions that are trinomial in GF(2167) and pentanomial in GF(2163). These architectures are based on Montgomery point multiplication with projective coordinate. We use polynomial basis representation for finite field arithmetic. All adopted multiplication, square and add operations over binary field can be completed within one clock cycle, and the critical path lies on multiplication. In addition, we use Itoh-Tsujii algorithm combined with addition chain, to execute binary inversion through using iterative binary square and multiplication. Because the double and add operations in point multiplication need to run many iterations, the execution time in overall design will be decreased if we can improve this partition. We propose two ways to improve the performance of point multiplication. The first way is Minus Cycle Version. In this version, we reschedule the double and add operations according to point multiplication algorithm. When the clock cycle time (i.e., critical path) of multiplication is longer than that of add and square, this method will be useful in improving performance. The second way is Pipeline Version. It speeds up the multiplication operations by executing them in pipeline, leading to shorter clock cycle time. For the hardware implementation, TSMC 0.13um library is employed and all modules are organized in a hierarchy structure. The implementation result shows that the proposed 167-bit Minus Cycle Version requires 156.4K gates, and the execution time of point multiplication is 2.34us and the maximum speed is 591.7Mhz. Moreover, we compare the Area x Time (AT) value of proposed architectures with other relative work. The results exhibit that proposed 167-bit Minus Cycle Version is the best one and it can save up to 38% A T value than traditional one.

In asymmetric cryptosystems, an important advantage of Elliptic Curve Cryptosystem (ECC) is the shorter key lengths than other cryptosystems. It can provide a level of security when the bit length over than 160 bits. So it has become a popular public key cryptographic system in recent year. Multiplier needs to run many times in scalar multiplication and it plays an essential role in ECC. Since the registers in multiplier are shifted every iteration, it will consume a lot of power in the computing process. So in this thesis, we propose five methods to save multiplication¡¦s energy consumption based on a scalable serial-parallel algorithm[1]. The first method is to design a low-power shift-register by modifying shift-register B to reduce the frequency of registers shifted. The second method is to use a frequency divider circuit. It can make registers to access a value every two clock cycles by modifying RA units. The third method is to introduce the gated clock circuit, and the clock signal of register will be disabled if its value is the same. The fourth method is to skip redundant operations and it can decrease the number of clock cycles for completing a multiplication operation. The last method raises multiplier¡¦s throughput by modifying RA units. The former three methods focus on low-power design, and the latter two methods emphasize on improving performance. Reducing power consumption and improving performance will save multiplication¡¦s energy consumption. Finally, we propose a Half Cycles schedule to raise scalar multiplication¡¦s performance. It is based on Montgomery scalar multiplication algorithm with projective coordinate[22][26]. For the hardware implementation, TSMC 0.13um library is employed and all modules are organized in a hierarchy structure. The implementation results show that the proposed multipliers have less energy consumption than traditional multiplier. It can get 5% ~ 24% energy saving. For Montgomery scalar multiplication, it can also reduce 12% ~ 47% energy consumption and is suitable for portable electronic products because its low area complexity and low energy.