Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the propo...Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.展开更多
Based on the negative exponential pair-potential （I/R）n, the interaction potential between the micro/nano planar curve and the particle located outside the curve is studied. We verified that, whatever the value of e...Based on the negative exponential pair-potential （I/R）n, the interaction potential between the micro/nano planar curve and the particle located outside the curve is studied. We verified that, whatever the value of exponent n may be the potential of particle/plane-curve is always of unified curvature form. Furthermore, we proved that the driving forces acted on the particle may be induced by the highly curved micro/nano curve, and the curvature and gradient of curvature are confirmed to be the essential factors forming the driving force. Through the idealized numerical experiments, the accuracy and reliability of the curvature-based potential are examined.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.10872114,11072125,and 11272175)the National Natural Science Foundation of Jiangsu Province(No.SBK201140044)the Fundation of Tutor for Doctor Degree of Higher Education of China(No.20130002110044)
文摘Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.
基金supported by the National Natural Science Foundation of China(Nos.11072125 and 11272175)the NSF of Jiangsu Province(Nos.BK2011075 and BK20130910)the research found for doctor student education
文摘Based on the negative exponential pair-potential （I/R）n, the interaction potential between the micro/nano planar curve and the particle located outside the curve is studied. We verified that, whatever the value of exponent n may be the potential of particle/plane-curve is always of unified curvature form. Furthermore, we proved that the driving forces acted on the particle may be induced by the highly curved micro/nano curve, and the curvature and gradient of curvature are confirmed to be the essential factors forming the driving force. Through the idealized numerical experiments, the accuracy and reliability of the curvature-based potential are examined.
