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.

Plasma turbulence on a uniform density gradient with unfavorable magnetic curvature is investigated extensively in the Helimak device. The turbulence is strong with density and electrostatic potential fluctuation levels in excess of 40%. Measurements of the dispersion relation, k[subscript z], and k[subscript parallel lines] identify the the fluctuations with drift-waves, which propagate in the poloidal direction at the diamagnetic drift velocity and have a small, but nite parallel wavenumber. A non-zero phase shift between the density and potential fluctuations gives rise to turbulent cross-eld particle transport, which is measured using spectral techniques. In addition, the electrostatic drift-wave fluctuations have a small magnetic component that is driven by the turbulent parallel current [scientific symbols]. An examination of nonlinear processes associated with the plasma turbulence uncovers high levels of intermittency near the plasma edge and long-time persistence of the density fluctuations on the order of the parallel confinement time. An analysis of the bispectrum conrms the existence local and nonlocal three-wave interactions between unstable drift-waves although the turbulent saturation of the density fluctuation spectrum is likely due nonlinear processes acting in the time domain. / text

Adaptive optics (AO) systems have become an essential part of ground-based telescopes and enable diffraction-limited imaging at near-IR and mid-IR wavelengths. For several key science applications the required wavefront quality is higher than what current systems can deliver. For instance obtaining high quality diffraction-limited images at visible wavelengths requires residual wavefront errors to be well below 100 nm RMS. High contrast imaging of exoplanets and disks around nearby stars requires high accuracy control of low-order modes that dominate atmospheric turbulence and scatter light at small angles where exoplanets are likely to be found. Imaging planets using a high contrast corona graphic camera, as is the case for the Spectro-Polarimetric High-contrast Exoplanet Research (SPHERE) on the Very Large Telescope (VLT), and the Gemini Planet Imager (GPI), requires even greater wavefront control accuracy. My dissertation develops a highly sensitive non-linear Curvature Wavefront Sensor (nlCWFS) that can deliver diffraction-limited (λ/D) images, in the visible, by approaching the theoretical sensitivity limit imposed by fundamental physics. The nlCWFS is derived from the successful curvature wavefront sensing concept but uses a non-linear reconstructor in order to maintain sensitivity to low spatial frequencies. The nlCWFS sensitivity makes it optimal for extreme AO and visible AO systems because it utilizes the full spatial coherence of the pupil plane as opposed to conventional sensors such as the Shack-Hartmann Wavefront Sensor (SHWFS) which operate at the atmospheric seeing limit (λ/r₀). The difference is equivalent to a gain of (D/r₀)² in sensitivity, for the lowest order mode, which translates to the nlCWFS requiring that many fewer photons. When background limited the nlCWFS sensitivity scales as D⁴, a combination of D² gain due to the diffraction limit and D² gain due to telescope's collecting power. Whereas conventional wavefront sensors only benefit from the D² gain due to the telescope's collecting power. For a 6.5 m telescope, at 0.5 μm, and seeing of 0.5", the nlCWFS can deliver for low order modes the same wavefront measurement accuracy as the SHWFS with 1000 times fewer photons. This is especially significant for upcoming extremely large telescopes such as the Giant Magellan Telescope (GMT) which has a 25.4 m aperture, the Thirty Meter Telescope (TMT) and the European Extremely Large Telescope (E-ELT) which has a 39 m aperture.

In this thesis we study three different problems: convex ancient solutions to the power-of-mean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from -\infty) to the power-of-mean curvature flow, when the power is strictly greater than \frac{1}{2}. As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x), locally uniformly converges to \frac{1}{1+\alpha}|x|^{1+\alpha} as h\rightarrow\infty. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space \mathbb{R}^{2}, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the $n$-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality. In the first section of the final chapter, under some suitable condition, we prove that the solution to the principal-agent problem must be C^{1}. The proof is based on a perturbation argument. The second section represents joint work with Emanuel Indrei; the main result is that, under (A3S) condition on the cost and c-convexity condition on the domains, the free boundary in the optimal partial transport problem is C^{1,\alpha}.