Microlocal,FunctionalAnalytic and Numerical Study of Partial Differential Equations and Tomography

这是一个长期的研究，无论是理论和计算机辅助实践，偏微分方程和层析成像。（1）幂非线性热方程解爆破的Fujita型临界指数可以区分锥的孔径，但对于抛物面或柱面等较窄的区域，不能区分锥的孔径，甚至不能区分有界区域。我引入了对数非线性来区分它们，并发展了一个通用的理论。特别地，我引入了一个函数logg u内插log u和1/log u，并采取的非线性项的产品，其p次幂和u，我成功地表明，一个临界指数的blowup是定义任何区域。具体确定这一新的临界指数为各自的领域是留给未来的研究（2）重建从两个投影的二元平面图形几乎总是有许多 关于我们 解决方案之间的关系或寻求良好的解决方案没有得到很好的澄清。在本研究中，我借由切换运算，将一个有向图的结构引入到解的集合中。通过对这类图的性质的研究，我发展了一种新的方法来解决这一问题。关于这样引入的开关图，在各种连通性特别是Haminlonian性质上得到了许多有趣的结果。许多开放的问题出现，从而提出了一个新的刺激性的研究领域。(3)我开发的技术，将代数几何代码的图像，并给出了它的应用，自恢复图像，掩蔽，数字足迹等。（4）反卷积是不适定问题的典型例子，以及层析成像。通过引入一种正则化的反卷积计算，尝试对离焦照片进行锐化处理，并验证了其实用性。这项研究将发表后，我推进它纳入适当的反卷积参数的自动选择。少

This is a long lasting research, both theoretical and computer aided practical, on partial differential equations and tomography. For the period of this fund, I executed mainly the following study and got results: (1) Fujita type critical exponent for the blowup of solutions of heat equations with power non-linearity could distinguish the aperture of cones, but for narrower domains such as paraboloidal or cylindrical ones it could not separate them and even bounded ones. I introduced a logarithmic non-linearity to distinguish them and developed a general theory. Especially, I introduced a function logg u interpolating log u and 1/log u, and by taking for the non-linear term the product of its p-th power and u, I succeeded in showing that a critical exponent of blowup is defined for any domain. Concrete determination of this new critical exponent for respective domains is left for future research (2) Reconstruction from the two projections of binary plane figures has almost always many … More olutions, and relations between the solutions or the search for good solutions is not well clarified. In this research, I introduced a structure of digraph to the set of the solutions by means of switching operation. Through the study of the properties of this graph, I developed a new means of approach to this problem. About thus introduced switching graph, many interesting results were obtained on various connectivity especially the Haminlonian property. Many open problems arose, thus presenting a new stimulating research field. (3) I developed techniques to incorporate algebraic-geometric codes to images, and gave its applications to self restoring images, masking, digital footprints etc. (4) Deconvolutions are typical examples of ill-posed problems as well as the tomography. By calculating deconvolution introducing a kind of regularization, I tried sharpening of defocused photos, and verified its practical use. This study will be published after I advance it up to incorporate an automatic choice of appropriate deconvolution parameters. Less