Bulletin of the Japan Society for Industrial and Applied Mathematics

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New approaches are presented to the problem of topological inconsistency caused by geometric algorithms implemented in finite-precision arithmetic. In geometric computation numerical errors often create inconsistency in topological structures and thus cause theoretically correct algorithms to fail. To overcome this problem two approaches are considered for the case of constructing the Voronoi diagram as an example. In the first approach, higher-precision arithmetic is used to construct a closed world in which topological structures are judged always precisely, and the symbolic perturbation technique is employed to avoid complicated branches of processing for degenerate cases. In the second approach, the highest priority is placed on the maintenance of topological consistency and numerical results are used as lower-priority information; the resultant algorithm is robust in the sense that inconsistency never arises and is correct in the sense that the output converges to the true solution as the precision becomes higher.

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Crystal growth is a typical pattern formation in nature. It is very interesting that the complicated shapes are formed in almost uniform circumstances like the formation snow flakes. In order to understand such phenomena, we must consider the inter facial motion accompanying the phase transition. Mathematical models and their simulations are introduced. The interface equation can express the motion caused by the thermodynamical driving force and surface tension. Also the effect of anisotropy is given to it. It will be shown that the model equations can describe the dendritic crystal growth realistically.

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The Weibull distribution in which all the three parameters, scale η, shape β and location γ, are unknown is widely used, but its property still remains unrevealed. For instance, some data case has infinity parameters. The Weibull distribution does not seem to express the data correctly in the case. However, the other distribution does it well if we reconsider that the Weibull distribution is derived from the extreme-value distributions. The Gumbel (doubly exponential) distribution has finite parameters and represents the case. In the paper, I will describe the three, (1) the derivation of the Weibull distribution from the extreme-value distribution, (2) maximum likelihood parameter estimation, and (3) the way of dealing with the Weibull distribution in the generalized extreme-value distribution.

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