Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.
This paper presents an assessment of weathering effect to the rock mass structure by studying the joint characteristics of tropically weathered granite. Joint survey was performed by scanline method and the results were analyzed statistically by stereographic projection plots. The overall trend of mean joint spacing followed a sharp decrement from fresh to moderately weathered zone and then a slight increment to highly and completely weathered zones, whereas the overall trend of mean joint trace length showed a gradual decrement over progress of weathering. In addition, the degree of joints inclination and weathering zones revealed an increasing trend in the percentage of horizontal joints from fresh to completely weathered rocks, while no specific relation was found between the numbers of major joint set and different weathering zones. The results of this study may contribute to understanding the behavior and better classification of weathered granitic rock mass as a heterogeneous mass in engineering works.
Text: Basic principles of anisotropy of magnetic susceptibility (AMS); Figure S1: Photographs of supplementary shearing indicators of the structural kinematics; Figure S2: Complete magnetic hysteresis data set for specimen selected from 34 sites; Figure S3: Measurements of isothermal remanent magnetization of specimen selected from 34 representative sites of the Bikou Terrane; Figure S4: Thermo-magnetic experiments of specimen selected from 12 representative sites; Figure S5: Mrs/Ms versus Hcr/Hc diagram from the selected specimen of the representative sites to define the size of magnetite; Figure S6: Stereographic projection of the AMS directional fabrics and structural fabrics.
Random Walks with Invariant Loop Probabilities: Stereographic Random Walks
