In this thesis we study the central limit theorem (CLT) for nonuniformly hyperbolic dynamical systems. We examine cases in which polynomial decay of correlations leads to a CLT with a non-standard scaling factor of √ n ln n. We also formulate an explicit expression for the the diffusion constant σ in situations where a return time function on the system is a certain class of supermartingale. We then demonstrate applications by exhibiting the CLT for the return time function in four classes of dynamical billiards, including one previously unproven case, the skewed stadium, as well as for the linked twist map. Finally, we introduce a new class of billiards which we conjecture are ergodic, and we provide numerical evidence to support that claim.

This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.

We study a Ginzburg--Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg--Landau parameter tends to infinity. The inhomogeneity is represented by a potential term which vanishes when the order parameter equals a given smooth function, the pinning term, which is assumed to become negative in finitely many smooth subdomains, the ''normally included'' regions. For large exterior magnetic field, we study the Gamma-limit of this inhomogeneous Ginzburg-Landau functional. The vanishing of the given smooth function near the inner boundaries imply that the associated operators are strictly but not uniformly elliptic, leading to many questions to be resolved near the boundaries of the normal regions. The method we use is an extension of many techniques including the product estimate from Sandier-Serfaty, Jacobian estimates from Jerrard-Soner and an appropriate Hodge decomposition adapted to our problem. To resolve these problems, we first study the Gamma-limit in the simpler case when the pinning term is varying but bounded below by a positive constant. Second, we consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain is multiply connected and has a small characteristic thickness, and we consider the simultaneous limit as the thickness tends to zero and the Ginzburg-Landau parameter to infinity. We do this when the applied field is strong in its components tangential to the film domain. Finally, we study the Gamma-limit of the inhomogeneous superconducting Ginzburg-Landau model with the pinning term vanishing on the boundary of the normal regions. / Thesis / Doctor of Science (PhD)

<p> Triples and the categories of triple algebras are relativized by a full faithful functors. The Tripleability Theorem in [1] is correspondingly relativized. The concept of the rank of a triple becomes intrinsic in this setting. Preservation of non-existing limits is interpreted in terms of limit-colimit commutation property. This is used to account for the usual description of the category of algebras as the cateeory of all product preserving setvalued functors on the opposite category of free algebras. </p> / Thesis / Doctor of Philosophy (PhD)

Unlike other thermo-mechanical soil responses, the effects of temperature on residual shear strength of soils are not well understood. Previous studies on temperature effects on residual shear strength show some contradictory findings that might be attributed to the sample's mineralogical composition and the testing procedure. This thesis aims to contribute to the understanding of (1) the temperature effects on the liquid limit of cohesive soils, (2) the impact of testing procedure on temperature-dependent residual friction angle, and (3) temperature effects on residual friction angle of soils. The fall cone tests are used to determine temperature effects on the liquid limit, while a temperature-modified ring shear apparatus is used to evaluate the residual friction angle in this study. To assess the impact of the testing procedure, the temperature is changed to 50°C at three different instants: before consolidation, before preshearing, and after preshearing; the resulting residual friction angles are assessed and compared. The effects of temperature on residual friction angle of soils are also investigated by changing the temperature in the ring shear apparatus to 10°C, 20°C, 40°C, and 50°C before consolidation. The study found that the impacts of temperature on liquid limit is mineralogy dependent. Also, the instant at which temperature change occurs in ring shear tests was found to be insignificant in terms of the residual friction angle. Moreover, the findings of the ring shear experiments suggest that clay mineralogy is important in the study of temperature-dependent residual friction angle of cohesive soils. Antigorite-rich soils may experience up to 50% changes in their residual friction angle, while soils with other clay minerals may experience less than 20% variations over a temperature range from 10 to 50 °C. / Master of Science / The increase in the frequency of landslides was found to be attributed to seasonal variation in temperature and an increase in global temperature due to climate change. To anticipate, mitigate and adapt to this costly natural disaster, understanding soil response to temperature change is an essential step. The residual shear strength of a soil is a parameter used to analyze stability of landslides. The relationship between this residual shear strength and temperature is not well understood. Previous studies on temperature effects on residual shear strength show some contradictory findings that need to be better understood for a more robust assessment of the climate change impacts on the stability of natural and man-made slopes. This thesis represents a first step to fill the knowledge gap in identifying the temperature effects on the residual shear strength of soils so that the impact of climate change and seasonal variation in temperature on slopes can be assessed more rigorously. This study consists of three tasks. The first task is to assess the effects of temperature on liquid limit, a parameter widely used to estimate the residual shear strength. The second task is to investigate the impacts of the testing procedures on residual shear strength, representing three field conditions where temperature change takes place at three different instants: when the soils is consolidating under applied load, after the soil consolidated and before development of a failure plan, and after failure initiated. The last task is to assess the effects of temperature on residual shear strength of soils. From the study, it was found that the effects of temperature on liquid limits and residual shear strength are dependent on the soil's mineralogical composition. It was also found that the instant in which the temperature changes in the testing procedure does not substantially impact the residual friction angle of the soil.

No description available.

Distillation technology has been evolving for many decades for a variety of reasons, with the most important ones being energy efficiency and cost. As a part of the evolution, semicontinuous distillation was conceived, which has the advantages of both batch and continuous distillation. The economic benefits of this intensified process compared to batch and continuous distillation were expounded in many of the previous studies. Semicontinuous distillation of ternary mixtures, which is the main focus of this thesis, is carried out in a single distillation column with a tightly integrated external middle vessel and the operation is driven by a control system. The system operation does not include any start-up or shut-down phases of the column and has three periodically repeating operating modes. In the status quo design procedure, called the ‘sequential design methodology,’ an imaginary continuous distillation system design was used to design the semicontinuous distillation system. In this methodology, dynamic simulations of the process were used to find the values of the controller tuning parameters based on the design of the continuous system. Afterwards, black-box optimization was used to find better controller tuning parameter values that minimized cost. However, after analyzing the dynamics of the system for different cases, it was found that the heuristics used in this design methodology yielded suboptimal designs. Therefore, the primary goal of the thesis is to improve these heuristics by incorporating more knowledge of the system and thereby develop a better design methodology. Firstly, the setpoint trajectories generated by the ideal side draw recovery arrangement for side stream flowrate control, which was standard in most semicontinuous distillation studies, was modified. In this thesis, the performance of the status quo as compared to the modified version, based on the criteria, cycle time and cost for different case studies, was presented. Results showed that the modified-ideal side draw recovery arrangement for side stream flowrate control performed better with a 10-20% lower separating cost while maintaining product purities. Furthermore, to reap more cost benefits, dynamic optimization was used to seek the flow rate trajectory that minimized cost. However, it was found that the additional cost savings, which is in addition to the benefits gained by using the modified version, were at the most 2% from different case studies. Subsequently, the impact of changing the imaginary continuous distillation system design on the nature of the semicontinuous distillation limit cycle, specifically, its period was studied. Results revealed the necessity for a new design procedure, and thus the back-stepping design methodology was proposed. This design methodology was used to find better limit cycles of zeotropic ternary semicontinuous distillation using the aspenONE Engineering suite. The proposed methodology was applied to three different case studies using feed mixtures with different chemical components. A comparison with the sequential design methodology for the two case studies indicates that the new method outperforms the state-of-the-art by finding limit cycles that were 4% to 57% lower in terms of cost. Furthermore, the designs obtained from this procedure were guaranteed to have feasible column operation with stable periodic steady-state behaviour. Semicontinuous distillation design using the design methodology with heuristic components involves guessing, checking and then using black-box optimization to find the values of the design variables to meet some performance criteria. Furthermore, mathematical guarantees of either local or global optimality of the designs obtained from the design procedure do not exist. Therefore, to address these issues, in this thesis, the application of using the shooting method for designing the semicontinuous distillation process was demonstrated using two case studies, which involve the separation of hexane, heptane and octane. This method has the potential to be combined with gradient-based optimization algorithms for optimization of the process design in the future. / Thesis / Doctor of Philosophy (PhD)

No description available.

Complex nonlinear aspects of aeroelastic phenomena include unsteady nonlinear aerodynamic loads, structural nonlinearities, as well as nonlinear couplings between the flow and the structural response. Nonlinearities in aerodynamic loads originate from unsteady shocks and/or flow separation. Structural nonlinearities are geometric, or a result of free play. Nonlinear fluid structure couplings result from nonlinear resonance between the aerodynamic load and structural modes. Under different conditions, one or a combination of these aspects could yield flutter or Limit Cycle Oscillations (LCO). The overall goal of this work is to develop the capabilities to quantify the role that these different nonlinear mechanisms could play in observed flutter and LCO. The realization of such a goal would help in providing a benchmark for the detection of nonlinear aeroelastic instabilities and possibly effective means for obtaining improved performance and reduced uncertainties through operation beyond conventional boundaries that are based on linear analysis. Additionally, this effort will provide a benchmark for the validation of computational methodologies. In this thesis, wavelet-based higher order spectra are applied to identify different nonlinear aeroelastic phenomena as encountered in two experiments. First, the analysis is applied to a set of experiments involving a flexible semispan model (FSM) of a High Speed Civil Transport (HSCT) wing configuration conducted by Silva et al. (Experimental Steady and Unsteady Aerodynamic and Flutter Results for HSCT Semispan Models; AIAA/ASME/ASCE/AHS/ASC 41st Structures, Structural Dynamics, and Materials Conference, 2000). The interest is in the identification of nonlinear aeroelastic phenomena associated with a high dynamic response region which was measured over a large range of dynamic pressures around Mach number 0.98. At the top of this region is a ``hard'' flutter point that resulted in the loss of the model. The results show that ``hard'' flutter is related to intermittent nonlinear coupling between the shock motion and large amplitude structural motions. Second, the analysis is applied to identify nonlinear aspects of LCO encountered during test flights of an F-16 aircraft. The results show quadratic and cubic couplings in the acceleration signals of the under-wing launchers and high quadratic coupling levels between flaperon motions and wing oscillations. The implications of applying these techniques in the capacity of a ``flutterometer'' are also discussed. / Ph. D.

Lean-premixed combustion has the advantage of low emissions for modern gas turbines, but it is susceptible to thermoacoustic instabilities, which can result in large amplitude pressure oscillations in the combustion chamber. The thermoacoustic limit cycle is generated by the unsteady heat release dynamics coupled to the combustor acoustics. In this dissertation, we focused on reduced-order modeling of the dynamics of a laminar premixed flame. From first principles of combustion dynamics, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method. In addition, the describing function for the flame was measured experimentally and used to identify an empirical nonlinear flame model. Furthermore, a linear acoustic model was developed and identified for the Rijke tube experiment. Closed-loop thermoacoustic modeling using the first principles flame model coupled to the linear acoustics successfully reproduced the linear instability and predicted the thermoacoustic limit cycle amplitude. With the measured experimental flame data and the modeled linear acoustics, the describing function technique was applied for limit cycle analysis. The thermoacoustic limit cycle amplitude was predicted with reasonable accuracy, and the closed-loop model also predicted the performance for a phase shift controller. Some problems found in the predictions for high heat release cases were documented. / Ph. D.