<p>In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.</p> <p>The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.</p> <p>One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.</p> / Dissertation

A study of double-variable-curvature microlenses (DVCM) for promoting coupling efficiency between the high-power 980-nm laser diodes and the single-mode fibers has been proposed. The purpose of the fiber microlens fabrication was to make the mode field match between the laser beam and the fiber as the beam propagating through the fiber microlens. To make the mode match, the shapes of the fiber microlens demanded nothing else but the offset and the curvature radii in minor and major axes. The double-variable-curvature fiber endface (DVCFE) was manufactured through a single-step fully automation grinding process and had less average offset of 0.3£gm, consequently. The radii of curvature in minor and major axis were controlled as an average of 1.2£gm and 33.6£gm, respectively. In the fusing procedure, the slight arc fusion was mainly applied for fine polishing merely instead of reshaping for the reason that the fabricated DVCFE was very close to the ideal shape. Hence, the fabrication time was reduced and the yield was promoted due to the withdrawn step of tip elimination. Furthermore, while the fusion parameters were set to be: fusing distance: 10£gm, arc intensity: 3bits, and fusing time: 200ms in the slight fusion process, the offset was reduced to 0.2£gm due to the shape constraint and surface tension of the DVCFE. And the radii of curvature increased 1.7£gm to 2.9£gm in the minor axis and increased 4.5£gm to 38.1£gm in the major axis, respectively. Owing to the controls of the fully automated grinding procedure and the omission of the tip elimination, the coupling efficiency and yield were improved. As a result, in the experiment, the average and maximum coupling efficiency of 83% and 88% were demonstrated, respectively. And the coupling efficiencies of the 20 samples were higher than 80%. In other words, the proposed DVCM structure of this study was a high coupling efficiency, a high yield output, and reproducible and fully automated single-step grinding process.

This study is to improve the coupling efficiency between 980nm high-power pump laser diode and single-mode fiber. In this study, we use the third generation of fiber grinding machine which is designed by Cheng Shiu University, professor Ying-Chien Tsai. This machine is fully automatic. we use it to fabricate the hyperbola microlenses. The advantages about hyperbola microlenses structure are a single-step fabrication, grinding steps to simplify, reduce the grinding time and will greatly reduce the offset of fiber. In the fusing procedure, the slight arc fusion was mainly applied for fine polishing merely instead of reshaping for the reason that the fabricated hyperbola fiber endface was very close to the ideal shape. The fabrication reproducibility and yield increase, and can reduce the cost of grinding. The fiber end shape is similar to the math on the hyperboloid, and the length of the axis of the hyperboloid profile shows a hyperbola. By mathematical properties of hyperbola, we derivation the parameter of radius of curvature for hyperbola microlenses. The definition of the radius of curvature of the hyperbolic vertex and the mode field diameter (the MFD) = 4.2£gm point of intersection with the hyperbola, the characteristics of the formation of this three o'clock round the curvature is the radius of curvature we have said. The radius of curvature (R) is a semi-consistent axial length (a) and two progressive line angle (£c) function, it means we can through the control of ¡§a¡¨ and £c to control the R, but £c is fixed after grinding process. So we choose control parameter ¡§a¡¨ by fusing process, via control ¡§a¡¨ to achieve the purpose of the control R. By various fusing parameters to adjust the gain of ¡§a¡¨, we can control the R in an ideal 2.6-2.8£gm. This process indeed improves the coupling efficiency. This method gives a low offset of the fiber it easier for more than 80%. And larger offset of the fiber by this method can achieve to 70% even 80%.

The dissertation focuses on the analysis, through combined analytical and numerical techniques, of the partial differential equations arising from a new approach to modeling brittle fracture, based on extension of continuum mechanics to the nanoscale. The main part of this work deals with the analysis of several fracture models. Integral transform methods are used to reduce the problem to a Cauchy singular, linear integro-differential equation. It is shown that ascribing constant surface tension to the fracture surfaces and using the appropriate crack surface boundary condition, given by the jump momentum balance, leads to a sharp crack opening profile at the crack tip, in contrast to the classical theory of brittle fracture. However, such a model still predicts singular crack tip stress. For this reason a modified model is studied, where the surface excess property is responsive to the curvature of the fracture surfaces. It is shown that curvature-dependent surface tension, together with boundary conditions in the form of the jump momentum balance, leads to bounded stresses and a cusp-like opening profile at the crack tip. Further, an alternative approach, based on asymptotic analysis, which is suitable to apply in cases when the model includes a mutual body force correction term, is considered. The nonlinear nonlocal problem, resulting from the proposed model, is simplified which allows us to approximate the crack opening profile and derive asymptotic forms for the cleavage stress in a neighborhood of the crack tip. Finally, two possible fracture criteria, in the context of the new theory, are discussed. The first one is an energy based fracture criterion. Classically the energy release rate arises due to singular fields, whereas in the case of the modeling approach adopted here, a notion analogous to the energy release rate arises through a different mechanism, associated to the rate of working of the surface excess properties at the crack tip. Due to the fact that the proposed modeling approach allows us to fully resolve the stress in a neighborhood of the crack tip, without the customary singularity, a second fracture criterion, based on crack tip stress, is possible.

A simulation algorithm is proposed in this thesis to investigate the effects of lensed fiber parameters on the variation of radius of curvature of the melted lens and the coupling efficiency of butterfly type laser diode transiver module. Two different endface shapes, i.e. the taper and the conical-wedge type lensed fibers, will be studied. The effect of endface shapes, sizes, and the melting zone volume on the coupling efficiency of lensed fibers are simulated and discussed. In the study on the conical type lensed fiber, the MARC¡¦s elastic-plastic-thermal finite element model is employed to simulate the melting and the solidification processes at the fiber tip endface with different conical angles. The temperature dependent material properties are used to calculate the melting zone and the post-melten deformation during the heating process. The Surface Evolver Software has also been employed to simulate the solidified lens shapes. The variation of radius of curvature of the tip lens is analyzed. The ZEMAX optical analysis software is applied to explore the relation between the coupling efficiency and the distribution of the radius of curvature. The variation of laser signal coupling efficiency introduced from different conical lensed fibers is simulated numerically. A good agreement between the published measured data and the simulated results indicate the proposed simulation model is feasible. The effect of endface shape and molten zone size on the conical wedge type lensed fiber has been studied in a similar way. The coherence between the shape of solidified elliptical lens at fiber tip and the coupling efficiency for the 980nm LD will be explored. Different endface shapes will also be investigated by using the simulation model proposed previously. Different aspect ratio of the conical-wedge type tip will be introduced to compensate the elliptical LD ray model and to recover the coupling efficiency loss. The agreement between the results simulated using the proposed model and the measured data is examined. The simulated results indicate that the coupling efficiency of a butterfly type laser diode transever can be improved significantly by controlling the shape of the lens introduced in this type lensed fiber. The optimal grinding parameters and the melting parameters used to fabricate the lensed fibers will also be studied. The effects of the shape parameters, i.e. the conical taper angle, the wedge angle and the size of molten zones on the curvature variation of the lens will also be studied. A better understanding about the design and fabrication of the lensed fiber of a laser diode based transever module is expected from the results presented in this thesis.

Robustness of a system has been defined in various ways and a lot of work has been done to model the system robustness , but quantifying or measuring robustness has always been very difficult. In this research we consider a simple system of a linear estimator and then attempt to model the system performance and robustness in a geometrical manner which admits an analysis using the differential geometric concepts of slope and curvature. We try to compare two different types of curvatures, namely the curvature along the maximum slope of a surface and the square-root of the absolute value of sectional curvature of a surface, and observe the values to see if both of them can alternately be used in the process of understanding or measuring system robustness. In this process we have worked on two different examples and taken readings for many points to find if there is any consistency in the two curvatures.

The response surface methodology is a useful method to find the optimum response in an experiment. In this work, a new test statistic with only few replicates at the center point for curvature detection is discussed. The performance of the new statistic is investigated through simulation. In the second part of this work, when the response variable is of the proportional type taking only values between 0 and 1, some analysis methods are compared based on the predicted variances on the design region.

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-128). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 129-131). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 129-131). Also available on the Internet.