応用数理

Automatic drawings of graphs have attracted much attention due to its numerous applications in information visualization, visual larguages, graphical user interfaces, visual network managements, etc. In this paper, we survey recent advances and results on automatic drawings of graphs.

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We survey the recent research in the field of cake divisions and their procedures. The question is how to divide a cake among n players, so that a certain fairness is achieved, where players have individual measures on the cake, and each player only knows his own measure. The model has very wide applications, such as dividing up the property in an estate, and even in determining the border in an international dispute. We first review mathematical definitions of various concepts of fairness. Although the existence of fair divisions is proved under some mathematical conditions, their dividing procedures are not known for all cases. We summarize several existing division procedures and classify them according to their methods and purposes. Finally, we mention some related topics and describe possible future research directions.

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Recently the role of nonlinear oscillation in nature has been recognized not only in physical systems but also in biological systems. Among them, the synchronization and its breakdown are one of origins of the diversity of structures and complex behaviors in nature. The synchronization of two periodic oscillators are nowadays known as the Huygens phenomenon. When the synchronization of chaotic oscillators breaks down, an intermittency called the on-off intermittency is observed. In the present report, a simple interpretation on the onset of mechanism of on-off intermittency is briefly discussed by giving several concrete models exhibiting on-off intermittency. The intermittency is characterized by several characteristic statistics. It is shown that they can be derived by introducing a multiplicative stochastic process by rigorously solving the Fokker-Planck equation. Furthermore, results based on a solvable model of on-off intermittency recently proposed are compared with those of the stochastic model and other dynamical models.

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Travel demand analysis and modelling are necessary for evaluating transport infrastructure projects and transport system management strategies. Since transport systems are usually described using network configuration, transport network flow analysis is one of the most important process in the travel demand analysis and modelling. Network flows and the level of service in the network are the resuluts of the interaction between travel demand and system performance. This paper shows the User Equilibrium formulation of network flow applying nonlinear optimization framework. The UE flow model can be incorporated into the system evaluation model of networks under degraded conditions due to natural disasters. The expected value of the consumer's surplus represents the performability measure of the degraded network. An approximation algorithm and numerical examples are presented.

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