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}.

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Composite materials are supplanting conventional metals in aerospace, automotive, civil and marine industries in recent times. This is mainly due to their high strength and light weight characteristics. But with all the advantages they have, they are prone to delamination or matrix cracking. These types of damage are often invisible and if undetected, could lead to appalling failures of structures. Although there are systems to detect such damage, the criticality assessment and prognosis of the damage is often more difficult to achieve. The research study conducted here primarily deals with the structural health monitoring of composite materials by analysing vibration signatures acquired from a laser vibrometer. The primary aim of the project is to develop a vibration based structural health monitoring (SHM) method for detecting flaws such as delamination within the composite beams. Secondly, the project emphasises on the method's ability to recognise the locatio n and severity of the damage within the structure. The system proposed relies on the examination of the displacement mode shapes acquired from the composite beams using the laser vibrometer and later processing them to curvature mode shapes for damage identification and characterization. Other identification techniques such as a C-scan has been applied to validate the location and size of the defects with the structures tested. The output from these plots enabled the successful identification of both the location and extent of damage within the structure with an accuracy of 96.5%. In addition to this, this project also introduces a method to experimentally compute the critical stress intensity factor, KIC for the composite beam. Based on this, a technique for extending the defect has been proposed and validated using concepts of fatigue and fracture mechanics. A composite specimen with a 40 mm wide delamination embedded within was loaded under fatigue conditions and extension of the defect by 4mm on either s ide of the specimen's loading axis was achieved satisfactorily. The experimental procedure to extend the defect using fatigue was validated using the SLV system. Displacement and Curvature mode shapes were acquired post-fatigue crack extension. Upon analysing and comparing the displacement and curvature mode shapes before and after crack extension, the extended delamination was identified satisfactorily.

Composite materials are supplanting conventional metals in aerospace, automotive, civil and marine industries in recent times. This is mainly due to their high strength and light weight characteristics. But with all the advantages they have, they are prone to delamination or matrix cracking. These types of damage are often invisible and if undetected, could lead to appalling failures of structures. Although there are systems to detect such damage, the criticality assessment and prognosis of the damage is often more difficult to achieve. The research study conducted here primarily deals with the structural health monitoring of composite materials by analysing vibration signatures acquired from a laser vibrometer. The primary aim of the project is to develop a vibration based structural health monitoring (SHM) method for detecting flaws such as delamination within the composite beams. Secondly, the project emphasises on the method's ability to recognise the locatio n and severity of the damage within the structure. The system proposed relies on the examination of the displacement mode shapes acquired from the composite beams using the laser vibrometer and later processing them to curvature mode shapes for damage identification and characterization. Other identification techniques such as a C-scan has been applied to validate the location and size of the defects with the structures tested. The output from these plots enabled the successful identification of both the location and extent of damage within the structure with an accuracy of 96.5%. In addition to this, this project also introduces a method to experimentally compute the critical stress intensity factor, KIC for the composite beam. Based on this, a technique for extending the defect has been proposed and validated using concepts of fatigue and fracture mechanics. A composite specimen with a 40 mm wide delamination embedded within was loaded under fatigue conditions and extension of the defect by 4mm on either s ide of the specimen's loading axis was achieved satisfactorily. The experimental procedure to extend the defect using fatigue was validated using the SLV system. Displacement and Curvature mode shapes were acquired post-fatigue crack extension. Upon analysing and comparing the displacement and curvature mode shapes before and after crack extension, the extended delamination was identified satisfactorily.

Orthokeratology (OK) is the reduction, modification or elimination of myopia through application of contact lenses. With the development of high Dk/t lens materials, overnight therapy has become the modality of choice for OK. Overnight OK lens wear has been previously investigated in terms of its efficacy to reduce myopia. However, the underlying effects of overnight OK lens wear on the human cornea have received less attention. As well as the clinical efficacy of overnight OK, this study investigated the effects of overnight OK on topographical corneal thickness and the overnight corneal edema response, and corneal tissue changes with overnight OK. Eighteen subjects participated as the OK lens-wearing group, wearing BE lenses (UltraVision, Brisbane, Queensland) in both eyes. A further ten subjects participated as control subjects, wearing conventional rigid lenses (J-Contour, UltraVision) in the right eye (RE) only. The left eye (LE) acted as a non-lens-wearing control. Both groups wore lenses overnight only, with no lens wear during the day. Measurements were conducted at baseline then on day 1, 4, 10, 30, 60, and 90 for the OK lens-wearing eyes; and up to day 30 for the control group, in the morning (after overnight lens wear) and in the evening (after 8-10 hours of lens removal). Variables measured included best vision sphere (BVS), unaided logMAR visual acuity (VA), refractive astigmatism, apical corneal power (ACP), simulated K readings (Medmont E300 corneal topographer), topographical corneal thickness (Holden-Payor optical pachometer), and keratocyte and endothelial cell densities (ConfoScan2 confocal microscope). Approximately 75% of myopia was corrected after the first night of OK lens wear and the changes in refractive error stabilised by day 10. By day 90, myopia reduction averaged 2.54 ?? 0.63 D. This was associated with significant improvement in unaided VA of about 82% after the first night of lens wear. There was no change in refractive astigmatism over the 3-month period. There was significant reduction in ACP in the OK lens-wearing eyes after the first night of lens wear, which accounted for more than 70% of the total ACP change over the 3-month period (RE: -2.16 ?? 0.53 D; LE: -2.11 ?? 0.86 D). There was significant central epithelial thinning (about 30%) and significant thickening (about 3%) in the mid-peripheral stroma in the OK lens-wearing eyes. Significant central epithelial thinning was found after the first night of lens wear while thickening in the mid-peripheral stroma reached statistical significance by day 4. Further analysis suggests that topographical corneal thickness changes account for the refractive error changes with overnight OK lens wear, rather than corneal bending. The central overnight stromal edema response was significantly reduced in the OK lens-wearing eyes (1.2 ?? 0.5%) to a level lower than in the conventional RGP (6.2 ?? 1.2%) and non-lens-wearing eyes (2.5 ?? 0.9%) in the control group. Mid-peripheral and peripheral stromal edema responses showed similar levels to those predicted based on lens Dk/t. A single overnight wear of BE and Paragon Corneal Refractive Therapy (CRT) lenses showed that the edema response to BE lens wear is significantly less than in the CRT lens-wearing eyes (BE: 2.5 ?? 0.7%; CRT 3.5 ?? 1.3%) immediately on eye opening. No significant changes were found in either central stromal keratocyte or endothelial cell densities in either OK or control groups over the study period. In conclusion, overnight OK lens wear induces significant reductions in myopia after the first night of lens wear associated with improvement in unaided VA. Overnight OK lens wear causes significant thinning in the central epithelium and significant mid-peripheral stromal thickening which results in flattening of the central cornea and steepening in the mid-periphery. Although there were no significant changes in central stromal keratocyte and endothelial cell densities, thinning of the central epithelial layer raises concerns regarding the safety of the procedure, especially with the alarming number of corneal infections reported recently in the literature.

In recent years, the use of composite structures in engineering application has increased. This is mainly due to their special advantages such as high structural performance, high corrosion resistance, tolerance of temperature; extreme fatigue resistance and high strength/weight ratio. However, some disorders like fibre breakage, matrix cracking and delaminations could be caused by operational loading, aging, chemical attack, mechanical vibration, changing of ambient conditions and shock etc. during the service. Although these disorders are hardly visible, they can severely reduce the mechanical properties and the load carrying capability of the composite structure. The aim of this research project is to develop a Vibration-based Structural Health Monitoring (SHM) method for carbon/epoxy composite beam specimens with the embedded artificial delaminations. The Laser Vibrometer Machine was used to excite the beams and gather the responses of the structure to the excitations. The physical properties such as frequency, velocity, mode shapes, and damping of the defective beams were measured. By using a C-SCAN machine, the accuracy of the positions of the delaminations was verified to be about 95% is accurate. Curvature mode shapes as a scalable damage detection parameter is calculated using an analytical model based on the Heaviside step function and the Central Difference Approximation (CDA) technique. The vibration-based damage detection method is then obtained using the difference between curvature mode shapes of the intact and damaged carbon/epoxy beams. An accurate prediction of 90% was attained. These results are proposed and discussed in detail in this study. Finally, the Fatigue Crack Propagation Test was applied on Samples with embedded delamination to extend the crack. The ASTM E399-90 standard is used for the experiment and a careful fatigue crack growth routine was designed and implemented to advance the delamination in a controlled manner. The total extension of 17 mm was observed with Microscope. The total propagation as determined by the curvature mode plots was 17.84 mm.

Orthokeratology (OK) is the reduction, modification or elimination of myopia through application of contact lenses. With the development of high Dk/t lens materials, overnight therapy has become the modality of choice for OK. Overnight OK lens wear has been previously investigated in terms of its efficacy to reduce myopia. However, the underlying effects of overnight OK lens wear on the human cornea have received less attention. As well as the clinical efficacy of overnight OK, this study investigated the effects of overnight OK on topographical corneal thickness and the overnight corneal edema response, and corneal tissue changes with overnight OK. Eighteen subjects participated as the OK lens-wearing group, wearing BE lenses (UltraVision, Brisbane, Queensland) in both eyes. A further ten subjects participated as control subjects, wearing conventional rigid lenses (J-Contour, UltraVision) in the right eye (RE) only. The left eye (LE) acted as a non-lens-wearing control. Both groups wore lenses overnight only, with no lens wear during the day. Measurements were conducted at baseline then on day 1, 4, 10, 30, 60, and 90 for the OK lens-wearing eyes; and up to day 30 for the control group, in the morning (after overnight lens wear) and in the evening (after 8-10 hours of lens removal). Variables measured included best vision sphere (BVS), unaided logMAR visual acuity (VA), refractive astigmatism, apical corneal power (ACP), simulated K readings (Medmont E300 corneal topographer), topographical corneal thickness (Holden-Payor optical pachometer), and keratocyte and endothelial cell densities (ConfoScan2 confocal microscope). Approximately 75% of myopia was corrected after the first night of OK lens wear and the changes in refractive error stabilised by day 10. By day 90, myopia reduction averaged 2.54 ?? 0.63 D. This was associated with significant improvement in unaided VA of about 82% after the first night of lens wear. There was no change in refractive astigmatism over the 3-month period. There was significant reduction in ACP in the OK lens-wearing eyes after the first night of lens wear, which accounted for more than 70% of the total ACP change over the 3-month period (RE: -2.16 ?? 0.53 D; LE: -2.11 ?? 0.86 D). There was significant central epithelial thinning (about 30%) and significant thickening (about 3%) in the mid-peripheral stroma in the OK lens-wearing eyes. Significant central epithelial thinning was found after the first night of lens wear while thickening in the mid-peripheral stroma reached statistical significance by day 4. Further analysis suggests that topographical corneal thickness changes account for the refractive error changes with overnight OK lens wear, rather than corneal bending. The central overnight stromal edema response was significantly reduced in the OK lens-wearing eyes (1.2 ?? 0.5%) to a level lower than in the conventional RGP (6.2 ?? 1.2%) and non-lens-wearing eyes (2.5 ?? 0.9%) in the control group. Mid-peripheral and peripheral stromal edema responses showed similar levels to those predicted based on lens Dk/t. A single overnight wear of BE and Paragon Corneal Refractive Therapy (CRT) lenses showed that the edema response to BE lens wear is significantly less than in the CRT lens-wearing eyes (BE: 2.5 ?? 0.7%; CRT 3.5 ?? 1.3%) immediately on eye opening. No significant changes were found in either central stromal keratocyte or endothelial cell densities in either OK or control groups over the study period. In conclusion, overnight OK lens wear induces significant reductions in myopia after the first night of lens wear associated with improvement in unaided VA. Overnight OK lens wear causes significant thinning in the central epithelium and significant mid-peripheral stromal thickening which results in flattening of the central cornea and steepening in the mid-periphery. Although there were no significant changes in central stromal keratocyte and endothelial cell densities, thinning of the central epithelial layer raises concerns regarding the safety of the procedure, especially with the alarming number of corneal infections reported recently in the literature.

Orthokeratology (OK) is the reduction, modification or elimination of myopia through application of contact lenses. With the development of high Dk/t lens materials, overnight therapy has become the modality of choice for OK. Overnight OK lens wear has been previously investigated in terms of its efficacy to reduce myopia. However, the underlying effects of overnight OK lens wear on the human cornea have received less attention. As well as the clinical efficacy of overnight OK, this study investigated the effects of overnight OK on topographical corneal thickness and the overnight corneal edema response, and corneal tissue changes with overnight OK. Eighteen subjects participated as the OK lens-wearing group, wearing BE lenses (UltraVision, Brisbane, Queensland) in both eyes. A further ten subjects participated as control subjects, wearing conventional rigid lenses (J-Contour, UltraVision) in the right eye (RE) only. The left eye (LE) acted as a non-lens-wearing control. Both groups wore lenses overnight only, with no lens wear during the day. Measurements were conducted at baseline then on day 1, 4, 10, 30, 60, and 90 for the OK lens-wearing eyes; and up to day 30 for the control group, in the morning (after overnight lens wear) and in the evening (after 8-10 hours of lens removal). Variables measured included best vision sphere (BVS), unaided logMAR visual acuity (VA), refractive astigmatism, apical corneal power (ACP), simulated K readings (Medmont E300 corneal topographer), topographical corneal thickness (Holden-Payor optical pachometer), and keratocyte and endothelial cell densities (ConfoScan2 confocal microscope). Approximately 75% of myopia was corrected after the first night of OK lens wear and the changes in refractive error stabilised by day 10. By day 90, myopia reduction averaged 2.54 ?? 0.63 D. This was associated with significant improvement in unaided VA of about 82% after the first night of lens wear. There was no change in refractive astigmatism over the 3-month period. There was significant reduction in ACP in the OK lens-wearing eyes after the first night of lens wear, which accounted for more than 70% of the total ACP change over the 3-month period (RE: -2.16 ?? 0.53 D; LE: -2.11 ?? 0.86 D). There was significant central epithelial thinning (about 30%) and significant thickening (about 3%) in the mid-peripheral stroma in the OK lens-wearing eyes. Significant central epithelial thinning was found after the first night of lens wear while thickening in the mid-peripheral stroma reached statistical significance by day 4. Further analysis suggests that topographical corneal thickness changes account for the refractive error changes with overnight OK lens wear, rather than corneal bending. The central overnight stromal edema response was significantly reduced in the OK lens-wearing eyes (1.2 ?? 0.5%) to a level lower than in the conventional RGP (6.2 ?? 1.2%) and non-lens-wearing eyes (2.5 ?? 0.9%) in the control group. Mid-peripheral and peripheral stromal edema responses showed similar levels to those predicted based on lens Dk/t. A single overnight wear of BE and Paragon Corneal Refractive Therapy (CRT) lenses showed that the edema response to BE lens wear is significantly less than in the CRT lens-wearing eyes (BE: 2.5 ?? 0.7%; CRT 3.5 ?? 1.3%) immediately on eye opening. No significant changes were found in either central stromal keratocyte or endothelial cell densities in either OK or control groups over the study period. In conclusion, overnight OK lens wear induces significant reductions in myopia after the first night of lens wear associated with improvement in unaided VA. Overnight OK lens wear causes significant thinning in the central epithelium and significant mid-peripheral stromal thickening which results in flattening of the central cornea and steepening in the mid-periphery. Although there were no significant changes in central stromal keratocyte and endothelial cell densities, thinning of the central epithelial layer raises concerns regarding the safety of the procedure, especially with the alarming number of corneal infections reported recently in the literature.

Adviser: Peter B. Gilkey. ix, 128 leaves / A print copy of this title is available through the UO Libraries under the call number: MATH QA613 .Z43 2000 / Relative to a non-degenerate metric of signature (p, q), an algebraic curvature tensor is said to be IP if the associated skew-symmetric curvature operator R(π) has constant eigenvalues and if the kernel of R(π) has constant dimension on the Grassmanian of non-degenerate oriented 2-planes. A pseudo-Riemannian manifold with a non-degenerate indefinite metric of signature (p, q) is said to be IP if the curvature tensor of the Levi-Civita connection is IP at every point; the eigenvalues are permitted to vary with the point. In the Riemannian setting (p, q) = (0, m), the work of Gilkey, Leahy, and Sadofsky and the work of Ivanov and Petrova have classified the IP metrics and IP algebraic curvature tensors if the dimension is at least 4 and if the dimension is not 7. We use techniques from algebraic topology and from differential geometry to extend some of their results to the Lorentzian setting (p, q) = (1, m – 1) and to the setting of metrics of signature (p, q) = (2, m – 2).

x, 164 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the topology of the space of metrics of positive scalar curvature on a compact manifold. The main tool we use for constructing such metrics is the surgery technique of Gromov and Lawson. We extend this technique to construct families of positive scalar curvature cobordisms and concordances which are parametrised by Morse functions and later, by generalised Morse functions. We then use these results to study concordances of positive scalar curvature metrics on simply connected manifolds of dimension at least five. In particular, we describe a subspace of the space of positive scalar curvature concordances, parametrised by generalised Morse functions. We call such concordances Gromov-Lawson concordances. One of the main results is that positive scalar curvature metrics which are Gromov-Lawson concordant are in fact isotopic. This work relies heavily on contemporary Riemannian geometry as well as on differential topology, in particular pseudo-isotopy theory. We make substantial use of the work of Eliashberg and Mishachev on wrinkled maps and of results by Hatcher and Igusa on the space of generalised Morse functions. / Committee in charge: Boris Botvinnik, Chairperson, Mathematics; James Isenberg, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Phillips, Member, Mathematics; Michael Kellman, Outside Member, Chemistry