Primitive prime divisors, rings of integers and class numbers in arithmetic dynamics

Let K be a number field, f(x) in K(x) be a rational function of degree at least 2, and a in K. By applying f to a repeatedly, we have a sequence a, f(a) , f^2(a), ..., where f^n denotes the n-th iterate of f.This thesis concerns arithmetic properties of sequences that are similar to the above sequence. We look into primitive prime divisors problems in sequences (φ_{n} o ... o φ_{1}(a) - b)_{n >= 1}, where (φ_{i})_{i >= 1} is a sequence of rational functions, sequences (a_{n}-b)_{n >= 1}, where (a_{n})_{n >= 1} lives in the backward orbit of dynamical systems, and sequences involving roots of unity. We also study the rings of integers and class numbers of number fields generated by sequences in backward orbits of dynamical systems

FGO-ILNS: Tightly Coupled Multi-Sensor Integrated Navigation System Based on Factor Graph Optimization for Autonomous Underwater Vehicle

Multi-sensor fusion is an effective way to enhance the positioning performance of autonomous underwater vehicles (AUVs). However, underwater multi-sensor fusion faces challenges such as heterogeneous frequency and dynamic availability of sensors. Traditional filter-based algorithms suffer from low accuracy and robustness when sensors become unavailable. The factor graph optimization (FGO) can enable multi-sensor plug-and-play despite data frequency. Therefore, we present an FGO-based strapdown inertial navigation system (SINS) and long baseline location (LBL) system tightly coupled navigation system (FGO-ILNS). Sensors such as Doppler velocity log (DVL), magnetic compass pilot (MCP), pressure sensor (PS), and global navigation satellite system (GNSS) can be tightly coupled with FGO-ILNS to satisfy different navigation scenarios. In this system, we propose a floating LBL slant range difference factor model tightly coupled with IMU preintegration factor to achieve unification of global position above and below water. Furthermore, to address the issue of sensor measurements not being synchronized with the LBL during fusion, we employ forward-backward IMU preintegration to construct sensor factors such as GNSS and DVL. Moreover, we utilize the marginalization method to reduce the computational load of factor graph optimization. Simulation and public KAIST dataset experiments have verified that, compared to filter-based algorithms like the extended Kalman filter and federal Kalman filter, as well as the state-of-the-art optimization-based algorithm ORB-SLAM3, our proposed FGO-ILNS leads in accuracy and robustness

A model for describing and predicting the creep strain of rocks from the primary to the tertiary stage

Rocks under applied stresses can exhibit more or less degree of creep. Over the years, a large number of creep models have been proposed for rocks. However, few models account for friction angle and time to failure. In most cases, curve fitting technique is applied to all of the available experimental results to obtain the required model parameters. The ability of the calibrated model (i.e. the model with the obtained model parameters) to predict the rheological behavior under untested stress conditions remains unknown. In this paper, a new model, called ubiquitous-corrosion-Coulomb (UCC) creep model, is proposed. Distinction is made between reversible and irreversible creep strains. Subcritical crack growth is related to the irreversible creep strain and delayed failure of rocks. The effect of friction angle and confining stresses on the rate of irreversible creep strain and time to failure has been considered. With the UCC model, the failure plane in creep tests making an angle of 45°−ϕ/2 with the major principal stress is explained by the fact that among the numerous micro cracks, the cracks along this orientation are the first ones becoming unstable. To test the capability of the UCC creep model against experimental results available in the literature, the required model parameters are first obtained by applying the curve-fitting technique on a part of the available experimental results. The predictability of the calibrated model is then tested against another part of the available experimental results, which are not used in the previous curve-fitting process. The results showed that the proposed UCC creep model can be used to describe and predict the creep strain and time to failure of rocks

Scaling Up, Scaling Deep: Blockwise Graph Contrastive Learning

Oversmoothing is a common phenomenon in graph neural networks (GNNs), in which an increase in the network depth leads to a deterioration in their performance. Graph contrastive learning (GCL) is emerging as a promising way of leveraging vast unlabeled graph data. As a marriage between GNNs and contrastive learning, it remains unclear whether GCL inherits the same oversmoothing defect from GNNs. This work undertakes a fundamental analysis of GCL from the perspective of oversmoothing on the first hand. We demonstrate empirically that increasing network depth in GCL also leads to oversmoothing in their deep representations, and surprisingly, the shallow ones. We refer to this phenomenon in GCL as long-range starvation', wherein lower layers in deep networks suffer from degradation due to the lack of sufficient guidance from supervision (e.g., loss computing). Based on our findings, we present BlockGCL, a remarkably simple yet effective blockwise training framework to prevent GCL from notorious oversmoothing. Without bells and whistles, BlockGCL consistently improves robustness and stability for well-established GCL methods with increasing numbers of layers on real-world graph benchmarks. We believe our work will provide insights for future improvements of scalable and deep GCL frameworks.Comment: Preprint; Code is available at https://github.com/EdisonLeeeee/BlockGC