Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum.

Hydraulic structures with water level differences between upstream and downstream are subjected to seepage in foundation soils. Two sources of weakness are to be guarded against: (1) percolation or seepage may cause under-mining, resulting in the collapse of the whole structure, and (2) the floor of the apron may be forced upwards, owing to the upward pressure of water seeping through pervious soil under the structure. Many earlier failures of hydraulic structures have been reported due to these two reasons. The curves and charts prepared by Khosla, Bose, and Taylor still form the basis for the determination of uplift pressure and exit gradient for weir apron founded on pervious soil of infinite depth. However, in actual practice, the pervious medium may be of finite depth owing to the occurrence of a clay seam or hard strata at shallow depths in the river basin. Also, a general case of weir profile may consist of cutoffs, at the two ends of the weir apron. In addition to the cutoffs, pervious aprons are also provided at the downstream end in the form of (i) inverted filter, and (ii) launching apron. These pervious aprons may have a thickness of 2 to 5. In order to accommodate this thickness, the bed adjacent to the downstream side of downstream cutoff has to be excavated. This gives rise to the formation of step at the downstream end. Closed form theoretical solutions for the case of finite depth seepage below weir aprons with end cutoffs, with a step at the downstream side are obtained in this research. The parameters studied are : (i) finite depth of pervious medium, (ii) two cut offs at the ends, and (iii) a step at the downstream end. iv The resulting implicit equations, containing elliptic integrals of first and third kind, have been used to obtain various seepage characteristics. The results have been compared with existing solutions for some known boundary conditions. Design curves for uplift pressure at key points, exit gradient factor and seepage discharge factor have been presented in terms of nondimensional floor profile ratios. Publications resulting from the dissertation are: 1. Jain, Arun K. and Reddi, L. N. “Finite depth seepage below flat aprons with equal end cutoffs.” (Submitted to Journal of Hydraulic Engineering, ASCE, and reviewed). 2. Jain, Arun K. and Reddi, L. N. “Seepage below flat apron with end cutoffs founded on pervious medium of finite depth.” (Submitted to Journal of Irrigation & Drainage Engineering, ASCE). 3. Jain, Arun K. and Reddi, L. N. “Closed form theoretical solution for finite depth seepage below flat apron with equal end cutoffs and a downstream step.” (Submitted to Journal of Hydrologic Engineering, ASCE). 4. Jain, Arun K. and Reddi, L. N. “Closed form theoretical solution for finite depth seepage below flat apron with end cutoffs and a downstream step.” (Submitted to Journal of Engineering Mechanics, ASCE).

Differential equations are frequently used for modeling systems in the physical sciences, biology, and other important real-world disciplines. Oftentimes, however, these equations cannot be solved exactly, so suitable computer algorithms are necessary to provide an approximated solution. While these computational simulations fail to exactly represent all behaviors of the true solution, they can be constructed to exactly, or very closely, reproduce certain properties which are key to the physical or scientific applications of a problem. This paper explores a computational method specifically constructed for modeling the behavior of systems with linear damping, or a reduction of energy, introduced in them. The method was designed to be conformal symplectic, and closely reproduce dissipation of physical properties such as linear and angular momentum, mass, and energy, caused by the damping. The algorithm was constructed in such a way that it maintains low computational cost to implement. Additionally, the method demonstrates favorable accuracy and stability properties in simulation. The method can also handle more complex scenarios, such as systems with forcing terms, and nonlinear systems. In these cases, it has been shown to hold advantages over other commonly used methods in particular circumstances.

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Piezoceramic fiber composite (PFC) actuators and sensors offer many advantages over conventional monolithic piezoceramic devices. Conformable, durable and, when equipped with interdigitated electrodes (IDEs), more responsive than regular monolithic devices, PFCs promise to revolutionize the application of piezoelectric materials. Developed by the NASA-Langley Research Center, the Macro-Fiber Composite (MFC) actuator and sensor is the most sophisticated PFC device yet invented. With superior qualities among PFCs in performance, behavior repeatability and manufacturability, the MFC has spawned great interest in the commercial and academic community as a tool in multitudinous engineering applications. While the MFC's characteristics render it a singularly useful device, limited characterization and modeling research on the MFC exists. Empirically designed and assembled, the MFC is poorly understood, especially in terms of its underlying operating principles, its dependence on design parameters and its electrical properties. The majority of published MFC studies focus on experimental quantification of MFC mechanical and actuation properties, and the research that attempts to model the MFC relies totally on finite element analysis. Published works widely assume that analytical models of the MFC are totally impossible. Rectifying gaps in the current body of MFC research, this study presents the first accurate analytical model of the static electrical field properties of the MFC. Implementing the techniques of conformal mapping, a branch of complex analysis, the following chapters derive a closed-form, exact analytical solution describing the electrical potential field and electrical field of the MFC's dual-IDE structure. Based on the conformal mapping solution for the MFC's electrical field, the electrical field of the commercially available MFC is examined and analyzed, introducing an intuitive knowledge of the MFC's operation. Demonstrating the utility of this solution in modeling the MFC, this work also predicts the capacitance and induced strain properties of a continuum of potential MFC designs and offers final suggestions on improving the current commercial MFC design. After establishing the theoretical underpinnings of the analytical MFC model, this report derives the conformal mapping solutions for the MFC, discusses the computational application of the resulting equations and then presents the results of numerical analyses executed using the new analytical model. / Master of Science

This thesis constitutes a study in the field of approximation theory and is restricted to sets defined on the complex plane. The main objective is to present means that have been used to determine whether a sequence of polynomials can be considered as being uniformly convergent to a given analytic function and if convergence can be considered stronger than uniform. Background material is given in Chapter I. This includes definitions of point sets and of measures of approximation. Also basic theorems concerning both approximation and conformal mapping are given. In Chapter II properties of conformal mappings are established. The theorems discussed lead to a statement of necessary and sufficient conditions for a mapping of a simply connected region onto the interior of the unit circle to be homeomorphic on the closure of the region. The bulk of the work presented in Chapter II is based on definitions and theorems given by Caratheodory and Markushevich. The last chapter puts to use the theorems given in Chapters I and II to prove Walsh's Theorem and Farrell's Theorem. Other theorems originally presented by Walsh and Mergelyan are also discussed in Chapter III. The thesis concludes with examples of sequences uniformly convergent but not convergent in a given measure of approximations in order to show the reader that the latter property is stronger. / M.S.

This thesis begins with the fundamentals of conformal field theory in three dimensions. The general properties of the conformal bootstrap are then reviewed. The three-dimensional Ising model is presented from the perspective of the renormalization group, after which the conformal field theory aspect at the critical point is discussed. Finally, the bootstrap programme is applied to the three-dimensional Ising model using numerical techniques, and the results analysed.

When data are stored in different locations and pooling of such data is not allowed, there is an informational loss when doing predictive modeling. In this thesis, a new method called Non-Disclosed Conformal Prediction (NDCP) is adapted into a regression setting, such that predictions and prediction intervals can be aggregated from different data sources without interchanging any data. The method is built upon the Conformal Prediction framework, which produces predictions with confidence measures on top of any machine learning method. The method is evaluated on regression benchmark data sets using Support Vector Regression, with different sizes and settings for the data sources, to simulate real life scenarios. The results show that the method produces conservatively valid prediction intervals even though in some settings, the individual data sources do not manage to create valid intervals. NDCP also creates more stable intervals than the individual data sources. Thanks to its straightforward implementation, data owners which cannot share data but would like to contribute to predictive modeling, would benefit from using this method.

Definição, classificação, propriedades e exemplos básicos da teoria de superálgebras conformes e módulos conformes. / Definition, classification, properties and basic examples about conformal superalgebras and conformal modules.

Definição, classificação, propriedades e exemplos básicos da teoria de superálgebras conformes e módulos conformes. / Definition, classification, properties and basic examples about conformal superalgebras and conformal modules.