The orientation of fractures with transpressional and transtensional wrenches in pre-existing faults has not been quantitatively determined. Based on Coulomb failure criterion and Byerlee’s frictional sliding criteri...The orientation of fractures with transpressional and transtensional wrenches in pre-existing faults has not been quantitatively determined. Based on Coulomb failure criterion and Byerlee’s frictional sliding criterion, this paper has indicated quantitative geometric relationships between the pre-existing fault and the local induced principal stress axes caused by the rejuvenation of the pre-existing fault. For a hidden pre-existing fault with some cohesion, the angles between the local induced principal stress axes and the pre-existing fault quantitatively vary with the applied stress and the cohesion coefficient, the ratio of the thickness of the cover layer to the thickness of the whole wrench body, whether transpressional or transtensional wrenches occur. For a surface pre-existing fault with zero cohesion, the angles between the pre-existing fault and the local induced principal stress axes are related to the rock inner frictional angle regardless of both the applied stress and the cohesion coefficient where transpressional wrenches occur, and the local induced maximum principal stress axis is identical with the applied maximum principal stress axis where transtensional wrenches occur. Therefore, the geometric relationships between the pre-existing faults and their related fractures are defined, because the local induced principal stress axes determine the directions of the related fractures. The results can be applied to pre-existing weak fabrics. They can help to understand and analyze wrench structures in outcrops or subsurface areas. They are of significance in petroleum exploration.展开更多
Analyzing the geometric relationships among genomic sequences from a mathematical perspective and revealing the laws hidden within these relationships is a crucial challenge in bioinformatics.The natural vector method...Analyzing the geometric relationships among genomic sequences from a mathematical perspective and revealing the laws hidden within these relationships is a crucial challenge in bioinformatics.The natural vector method constructs a genome space by extracting statistical moments of k-mers to illuminate the relationships among genomes.This approach highlights a fundamental law in biology known as the convex hull principle,which states that natural vectors corresponding to different types of biological sequences form distinct,non-overlapping convex hulls.Previous studies have validated this important principle across various datasets.However,they often focused on specific kingdoms and did not thoroughly analyze the significance of the dimensions required for the convex hull separation.In this study,we integrate all reliable sequences from different kingdoms to construct the grand biological universe,within which we comprehensively validate the multi-level convex hull principle.We demonstrate that the separation of convex hulls arises from biological properties rather than mathematical characteristics of high-dimensional spaces.Furthermore,we develop suitable metrics within the grand biological universe to facilitate efficient sequence classification.This research advances the convex hull principle through both theoretical development and experimental validation,making significant contributions to the understanding of the geometric structure of genome space.展开更多
文摘The orientation of fractures with transpressional and transtensional wrenches in pre-existing faults has not been quantitatively determined. Based on Coulomb failure criterion and Byerlee’s frictional sliding criterion, this paper has indicated quantitative geometric relationships between the pre-existing fault and the local induced principal stress axes caused by the rejuvenation of the pre-existing fault. For a hidden pre-existing fault with some cohesion, the angles between the local induced principal stress axes and the pre-existing fault quantitatively vary with the applied stress and the cohesion coefficient, the ratio of the thickness of the cover layer to the thickness of the whole wrench body, whether transpressional or transtensional wrenches occur. For a surface pre-existing fault with zero cohesion, the angles between the pre-existing fault and the local induced principal stress axes are related to the rock inner frictional angle regardless of both the applied stress and the cohesion coefficient where transpressional wrenches occur, and the local induced maximum principal stress axis is identical with the applied maximum principal stress axis where transtensional wrenches occur. Therefore, the geometric relationships between the pre-existing faults and their related fractures are defined, because the local induced principal stress axes determine the directions of the related fractures. The results can be applied to pre-existing weak fabrics. They can help to understand and analyze wrench structures in outcrops or subsurface areas. They are of significance in petroleum exploration.
基金supported by grants from the National Natural Science Foundation of China(12171275)the Tsinghua University Education Foundation fund.
文摘Analyzing the geometric relationships among genomic sequences from a mathematical perspective and revealing the laws hidden within these relationships is a crucial challenge in bioinformatics.The natural vector method constructs a genome space by extracting statistical moments of k-mers to illuminate the relationships among genomes.This approach highlights a fundamental law in biology known as the convex hull principle,which states that natural vectors corresponding to different types of biological sequences form distinct,non-overlapping convex hulls.Previous studies have validated this important principle across various datasets.However,they often focused on specific kingdoms and did not thoroughly analyze the significance of the dimensions required for the convex hull separation.In this study,we integrate all reliable sequences from different kingdoms to construct the grand biological universe,within which we comprehensively validate the multi-level convex hull principle.We demonstrate that the separation of convex hulls arises from biological properties rather than mathematical characteristics of high-dimensional spaces.Furthermore,we develop suitable metrics within the grand biological universe to facilitate efficient sequence classification.This research advances the convex hull principle through both theoretical development and experimental validation,making significant contributions to the understanding of the geometric structure of genome space.
