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This research, initiated by a problem excerpted from Baidu, one of the most famous internet search engines in China, explores two types of movements, “converging curve” and “tracing curve” named by this research group, which really act in a regular pattern. In the research, by applying scores of mathematical methods, we found that the key to converging curves is to spot the invariant elements from the changeful motions. For tracing curves, we adopt estimating equations to get the solution. Due to the limit of time and the relevant knowledge of the research group, many unexpected problems constantly appeared which are quite far beyond our ability. Yet we still tried to apply some promising mathematical thoughts, and have achieved some exciting results. In the process of discovering and rediscovering, we did not succeed in finding a perfect solution to the problem. After all, as a group of middle school students fascinated by mathematics, creating and discovering more are the greatest delight of us all.

In this paper, we firstly re-present the results of Bondar et al. (2013). Then, following the approach taken in it, we use the method of Operational Dynamic Modeling (ODM) to deal with the situation where thebackground mechanics is varying between quantum and classical mechanics.

The optimal mass transportation problem, proposed by Monge, is dominated by the Monge-Ampere equation. In general, as the Monge-Ampere equation is highly non-linear, this type of partial dierential equations is beyond the solving ability of a conventional nite element method. This diculty led Gu et al. alternatively to the discrete optimal mass transportation problem. They developed variational principles for this problem and reported the calculation for the optimal mapping, based on theorem that among all possible cell decompositions, with constrained measures, the trans- portation cost of the discrete mapping from cells to the corresponding discrete points is minimized by the decomposition induced by a power Voronoi diagram. Their research inspired us to consider a similar discrete optimal mass transportation problem. Here we replace the L2 Euclidean distance by the length of the shortest path connecting two points on a line network. To solve the optimal trans- portation problem on a line network, we study a type of Voronoi diagram on undirected and connected networks and propose an elegant construction algorithm. We further consider weighted distances in a network and develop a method to compute the centroidal Voronoi tessellation (CVT) for a network. By using real geographic data, the method proposed is proved ecient and eective in several practical applications, including charging station distribution in a trac network, and trash can distribution in a park.

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