On the Unramified Fontaine-Mazur Conjecture and its generalizations

Diese Dissertation untersucht Galois-Erweiterungen von Zahlkörpern und die Unverzweigte Fontaine-Mazur-Vermutung für p-adische Galois-Darstellungen und deren Verallgemeinerungen. Wir beweisen viele grundlegende Fälle der Vermutung und liefern einige nützliche Kriterien zur Überprüfung. Darüber hinaus schlagen wir mehrere verschiedene Strategien vor, um die Vermutung anzugreifen und auf einige spezielle Fälle zu reduzieren. Wir beweisen auch viele neue Ergebnisse der Vermutung im zweidimensionalen Fall. Als Anwendung beweisen wir die Endlichkeit der unverzweigten Galois-Deformationsringe unter der Annahme eines speziellen Falles der Vermutung und geben einige Gegenbeispiele zur sogenannten Dimension-Vermutung für Galois-Deformationsringe unter der Annahme der Vermutung.This thesis studies Galois extensions of number fields, and the Unramified Fontaine-Mazur Conjecture for p-adic Galois representations and its generalizations. We prove many basic cases of the conjecture, and provide some useful criterions for verifying it. In addition, we propose several different strategies to attack the conjecture and reduce it to some special cases. We also prove many new results of the conjecture in the two-dimensional case. As an application, we prove the finiteness of unramified Galois deformation rings assuming a special case of the conjecture, and we give some counterexamples to the so-called dimension conjecture for Galois deformation rings assuming the conjecture

Image processing techniques for high-speed atomic force microscopy

Atomic force microscopy (AFM) is a powerful tool for imaging topography or other characteristics of sample surfaces at nanometer-scale spatial resolution by recording the interaction of a sharp probe with the surface. Dispute its excellent spatial resolution, one of the enduring challenges in AFM imaging is its poor temporal resolution relative to the rate of dynamics in many systems of interest. This has led to a large research effort on the development of high-speed AFM (HS-AFM). Most of these efforts focus on mechanical improvement and control algorithm design. This dissertation investigates a complementary HS-AFM approach based on the idea of undersampling which aims at increasing the imaging rate of the instrument by reducing the number of pixels in the sample surface that need to be acquired to create a high-quality image. The first part of this work focuses on the reconstruction of images sub-sampled according to a scheme known as μ path patterns. These patterns consist of randomly placed short and disjoint scans and are designed specifically for fast, efficient, and consistent data acquisition in AFM. We compare compressive sensing (CS) reconstruction methods with inpainting methods on recovering μ-path undersampled images. The results illustrate that the reconstruction quality depends on the choice of reconstruction methods and the sample under study, with CS generally producing a superior result for samples with sparse frequency content and inpainting performing better for samples with information limited to low frequencies. Motivated by the comparison, a basis pursuit vertical variation (BPVV) method, combing CS and inpainting, is proposed. Based on single image reconstruction results, we also extend our analysis to the problem of multiple AFM frames, in which higher overall video reconstruction quality is achieved by pixel sharing among different frames. The second part of the thesis considers patterns for sub-sampling in AFM. The allocation of measurements plays an important role in producing accurate reconstructions of the sample surface. We analyze the expected image reconstruction error using a greedy CS algorithm of our design, termed simplified matching pursuit (SMP), and propose a Monte Carlo-based strategy to create μ-path patterns that minimize the expected error. Because these μ path patterns involve a collection of disjoint scan paths, they require the tip of the instrument to be repeatedly lifted from and re-engaged to the surface. In many cases, the re-engagements make up a significant portion of the total data acquisition time. We therefore extend our Monte Carlo design strategy to find continuous scan patterns that minimize the reconstruction error without requiring the tip to be lifted from the surface. For the final part of the work, we provide a hardware demonstration on a commercial AFM. We describe hardware implementation details and image a calibration grating using the proposed μ-path and continuous scan patterns. The sample surface is reconstructed from acquired data using CS and inpainting methods. The recovered image quality and achievable imaging rate are compared to full raster-scans of the sample. The experimental results show that the proposed scanning combining with reconstruction methods can produce higher image quality with less imaging time

Experimental study on shear performance of saw-tooth rock joint with weak interlayer under different moisture contents and filling degrees

Rock joints are important factors affecting natural disasters such as landslides, and weak interlayer is the decisive factor for rock joint stability. Therefore, it is necessary to carry out shear test research on rock joints with weak interlayer. In this paper, the red sandstone was used as rock joint material, and consistent saw-tooth rock joints were produced by the 3D carving technology. According to the on-site composition analysis, the weak interlayer used in the test was obtained through remodeling. Weak interlayer with different moisture contents and filling degrees were produced by the Constant Dry Mass Method (CDMM). Shear strength test of rock joint with weak interlayer of different filling degree and moisture content under multi-level normal stress was carried out by the self-developed direct shear apparatus. The test results show that peak shear strength of rock joint with weak interlayer increases firstly and then decreases with the raise of moisture content. Peak shear strength of rock joint with weak interlayer decreases firstly and then tends to a stable value with the raise of filling degree. As the filling degree of weak interlayer changes from small (less than 1.0) to large (more than 1.0), dilatancy mode of saw-tooth rock joint changes from dilatation to contraction during shearing. Peak shear strength of rock joint with weak interlayer increases with the raise of normal stress