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Functional analytic and numerical study of partial differential equation and tomography

偏微分方程和断层扫描的泛函分析和数值研究

基本信息

批准号：
12640158
负责人：
KANEKO Akira
金额：
$ 2.24万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

项目摘要

(1)Liouville's theorem, famous in the theory of functions, was generalized by L.Schwartz in the form "there exists a solution of polynomial growth if and only if the total symbol of the equation has no real zeros", employing the tempered distributions. In this research I studied existence of solutions with quasi-exponential and infra-exponential growth, employing Fourier hyperfuncitons and its ultra-distribution analogues. Fourier hyperfunctions, contrary to Schwartz distributions, may have support only at infinity. Therefore the validity of Liouville type assertion depends on the approaching speed of the complex zeros to the real axis at infinity. (2)Generalizing the explicit construction by Morimoto-Yoshino of Fourier hyperfunctions supported by one point at infinity, I succeeded in constructing such objects in any quasi-exponential growth, employing the Ahlfors distortion theorem. (3)In collaboration with A.L.Balandin in Irkutsk, we gave a theory to reconstruct the solenoidal part of a 3D vector field from a small number of projection data of the inner product with the obserbed directions, by using expansion by vector spherical harmonics. (4)As a provisionary work to apply the theory of D-modules to error-correcting codes, we implemented a new example of algebraic-geometric codes, and applied it to color raster images. (5) As a provisionary work of application of microlocal analysis to image analysis, we developped socket communication protocol and GUI on unix Xlib, and using them made a graphical implementation of fair coin flipping, its Japanese version (janken), and mental poker protocol for three players. We also gave a definition of wave-front-set of images and analyzed its fundamental behaviour.

（1）函数论中著名的刘维尔定理由L.Schwartz推广为“存在多项式增长的解当且仅当方程的全符号没有真实的零点”，并利用调和分布。在这项研究中，我研究了解的存在性与拟指数和亚指数增长，采用傅立叶超泛函及其超分布类似。傅立叶超函数，与施瓦茨分布相反，可能只有在无穷远处才有支撑。因此Liouville型断言的有效性取决于复零点在无穷远处向真实的轴的逼近速度。（2）推广了Morimoto-Yoshino关于无穷远处一点支撑的Fourier超函数的显式构造，我利用Ahlfors畸变定理成功地构造了任何准指数增长的此类对象。(3)In在伊尔库茨克与A. L. Balandin的合作中，我们给出了一个理论，通过使用矢量球谐函数展开，从少量的内积投影数据中重建三维矢量场的螺线管部分。(4)As作为将D-模理论应用于纠错码的一个预备性工作，我们实现了一个新的代数几何码的例子，并将其应用于彩色光栅图像。(5)作为微局部分析应用于图像分析的预备性工作，我们在unix Xlib上开发了socket通信协议和图形用户界面，并利用它们实现了一个公平抛硬币游戏、日本版（janken）游戏和三人心理扑克游戏协议的图形化实现。给出了像的波前集的定义，分析了它的基本性质。