Difference of sequence topologies

We argue that topology can be interpreted as an area of mathematics studying preserved properties under an equivalence relation, and representation, classification and comparison of the corresponding equivalence classes. With this understanding, we can generalize ideas in topology to non-geometric objects. In this paper, which presents an example of such generalization, we define a sequence topology to be an equivalence class of finite integer sequences of the same length under relabeling or permutations. The difference for a set of finite integer sequences of the same length is defined to be the number of mismatches in the sequences. While the difference for a set of sequence topologies is defined to be the minimum difference over all sets of sequences constructed by choosing one sequence from each sequence topology. We count the number of different sequence topologies of a given length and a set of possible labels and determine the minimum upper bound of the difference for sequence topologies. Finally, we compute the exact difference for a set of sequence topologies of the same length.Comment: 11 page

The Optimal Design Method and Standardized Mathematical Model of Tooth Profile Modification of Spur Gear

The paper reports a tooth profile modification method of spur gear. After establishing a standardized mathematical model for optimized tooth profile and simulating meshing process with ANSYS finite element analysis, we obtained 625 groups of gear models with different modification parameters. The group with minimum transmission errors owns the optimal parameters. Genetic algorithm was adopted in the entire process for the purpose of reducing the variation of transmission errors in meshing process. The arc and parabolic modification were doing the same processing. After comparing the transmission errors fluctuation produced by the meshing process of gear of nonmodification with arc modification and parabolic modification, we found that the best modification effects of arc modification and parabolic modification were both reduced by 90%. The modification method makes the gear drive process more stable and efficient, and it is also promising in general application for gear drive