On Applications of Global Variation Theory

论整体变分理论的应用

• 批准号: 63460003
• 负责人: SHIKATA Yoshihiro
• 金额: $ 4.42万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1989
• 资助国家: 日本
• 起止时间: 1989 至 1990
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-63460003/
• 关键词: (invariant) superpotential; topological classification of phenomena; inverse problem of differential equation; models with necessary condition; multiplicity of phenomena for potential; 雪結晶の理論; 液滴の振動現象; 変分法的な手法; 負の二項分布; 発生における周期性; 1@fノイズと非線形性; 変

This research was started from the purely mathematical problem to reassemble the potential theory, which played the main role in the theory closed geodesics, into a fundamental technique for the mathematical inverse problems.On the way, we noticed the technique is also effective for the problems in the field of biophysics, material science, biology and physiology, as a new method to construct a model with necessity.We joined into active groups in these field to develop further the mathematical theory and tried to establish a theoretical breakthrough in the fields, because the conventional models are only those with sufficient conditions, in the other words, the existing models are taken for granted only when they yield the solution coincidie with the observed phenomena.The logical approach of this type requiring only the sufficient condition can not deny the existence of the other models which also gives the same solution, resulting too many models in some cases. Therefore it is necessary to construct a theory of models with necessity, reversing the logics above, which may distinguish the real models.We are getting some results along this line as follows ;1) We get a partial success in the theory of alpha rhythm of EEG and a relation to ECG.2) We solved the dendritic growth problem of snow crystal as an inverse problem of the partial differential equation.3) We get a model with necessity for growth of the yeast and sea-urchin.4) We considered the 1/f- noise problem in our line to get the necessary characteristic for the filter.5) We treated the motion of vorteces and get a evaluation of some singularities.

本研究从纯数学问题出发，将封闭测地线理论中起主要作用的势理论重新组合为数学反问题的基本方法，并注意到该方法对生物物理、材料科学、生物学和生理学等领域的问题同样有效。作为一种新的构造具有必然性的模型的方法，我们加入了这些领域的活跃团体，进一步发展数学理论，试图在这些领域建立理论突破，因为传统的模型只是具有充分条件的模型，换句话说，现有的模型只有当它们给出的解与观测现象相一致时才被认为是理所当然的，这种只要求充分条件的逻辑方法并不能否认其他模型也给出了同样的解，从而导致在某些情况下模型过多。因此，有必要建立一个具有必然性的模型理论，颠倒上述逻辑，这可能会区分真实的的模型。1）在脑电α节律理论及其与心电图的关系方面取得了部分成功。2）将雪晶的枝晶生长问题作为偏微分方程的反问题求解。4）考虑了1/f-噪声问题，得到了滤波器的必要特征。5）对涡旋运动进行了处理，得到了一些奇异性的估计。