Differential Equations on Manifolds and Their Singularities
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||Akita University |
KAWAKAMI Hajime Faculty of Engineering and Resource Science associate professor, 工学資源学部, 助教授 (20240781)
KOBAYASHI Mahito Faculty of Engineering and Resource Science associate professor, 工学資源学部, 助教授 (10261645)
|Project Period (FY)
2001 – 2002
Completed (Fiscal Year 2002)
|Budget Amount *help
¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
|Keywords||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.
Report (3 results)
Research Products (6 results)