Differential Equations on Manifolds and Their Singularities

流形及其奇点的微分方程

• 批准号: 13640059
• 负责人: KAWAKAMI Hajime
• 金额: $ 0.7万
• 依托单位: Akita University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640059/
• 关键词: Holder continuity; C^∞ smoothing; diffusion equation; inverse problem; Gaussian curvature; stable map; discriminant; computer network; 熱方程式; ガウス曲率; 平面への投影; smoothing

Head investigator Kawakami has studied the following. In the first year:He and Prof. Tsuchiya (Kanazawa Univ.) proved that any Cr,α manifold (manifold pair) has a C∞ smoothing by using a method of J.R. Munkres.He and Dr. Murayama (Shobi Univ.) and others studied about a means of giving teaching-materials of mathematics through a computer network.In the second year:He and Prof. Tsuchiya (Kanazawa Univ.) have studied a generalization of "Kurt Bryant and Lester F. caudill Jr., Inverse Problem 14 1429-1453 (1998)". They proved that the data in a finite time-interval uniquely determine the shape of the back surfice.He conjectured that the Gauss-Bonnet formula gives a necessary and sufficient condition for the existence of a metric deformation to obtain a positive/negative Gaussian curvature on a disk. He gave a partial answer of the conjecture.Investigator Kobayashi worked on studying the curious relation of generic maps to their discriminants. The main results in the first year are;a characterization of the 'folding into four' action in general dimensions by the discriminant of the folding map,finding of an infinite to one correspondence of maps of a fixed closed 4-manifolds to their discriminants,providing a family of discrimiants of stable maps of closed manifolds.Those in the second year are;a characterization of plane curves which are the critical value set of a generic projection of a closed surface into the plane;study of planar projections of sphere bundles over spheres.

首席调查员Kawakami研究了以下内容。第一年：他和土屋教授（金泽大学）利用J. R. Munkres.他和Murayama博士（Shobi大学）第二年：他和土屋教授（金泽大学）在日本东京大学（东京大学）进行了数学教材的研究。研究了“库尔特·布莱恩特和莱斯特·F.小考迪尔，反问题14 1429-1453（1998）"。他们证明了在有限时间间隔内的数据唯一地决定了后表面的形状。他证明了高斯-博内公式给出了一个度量变形存在的充分必要条件，以获得一个正/负高斯曲率的磁盘。他给出了部分答案的猜想。调查员小林工作研究的好奇关系的一般地图，他们的判别式。第一年的主要结果是：用折叠映射的判别式刻画了一般维的“折叠成四”作用，找到了固定闭4-流形的映射与其判别式的无穷对一对应，给出了闭流形稳定映射的一族判别式。一种平面曲线的特征，它是一个闭曲面在平面上的一般投影的临界值集;研究球面丛在球面上的平面投影。

项目成果

Hajime Kawakami: "C∞ Smoothing of Manifolds of Fractional Order and Basic Properties of the Whitney Topology on the Spaces of Holder Maps"International Journal of Applied Mathematics. 6-3. 319-340 (2001)

Hajime Kawakami：“分数阶流形的 C∞ 平滑和持有人映射空间上惠特尼拓扑的基本属性”国际应用数学杂志 6-3（2001）。

Hajime Kawakami: "C^<∞> Smoothing of Manifolds of Fractional Order and Basic Properties of the Whitney Topology on the Spaces of Holder Maps"International Journal of Applied Mathematics. 6. 319-340 (2001)

Hajime Kawakami：“C^<∞> 分数阶流形的平滑和持有人映射空间上惠特尼拓扑的基本性质”国际应用数学杂志 6. 319-340 (2001)。

Hajime Kawakami: "C^∞ Smoothing of Manifolds of Fractional Order and Basic Properties of the Whitney Topology on the Spaces of Holder Maps"International Journal of Applied Mathematics. 6・3. 319-340 (2001)

Hajime Kawakami：“Holder 映射空间上分数阶流形的 C^∞ 平滑和惠特尼拓扑的基本属性”国际应用数学杂志 6・3（2001 年）。