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

这是一项关于偏微分方程和断层扫描的理论和计算机辅助实践的长期研究。本基金期间，我主要进行了以下研究，得到了如下结果：(1)幂非线性热方程解爆破的Fujita型临界指数可以区分锥孔，但对于较窄的区域，如抛物面或圆柱面，不能区分锥孔，甚至不能区分有界区域。我引入了对数非线性来区分它们，并发展了一个一般理论。特别地，我引入了一个函数logu将logu和1/ logu插值，通过对非线性项取它的p次幂和u的乘积，我成功地证明了在任何定义域都可以定义爆炸的临界指数。这一新的临界指数在各自领域的具体确定有待于未来的研究。(2)从二元平面图形的两个投影重建几乎总是有许多…更多的解，并且解之间的关系或寻找好的解并没有很好地阐明。在本研究中，我通过切换操作将有向图的结构引入到解的集合中。通过研究这个图的性质，我提出了一种解决这个问题的新方法。关于由此引入的切换图，得到了许多有趣的关于各种连通性的结果，特别是关于哈米朗性质的结果。出现了许多尚未解决的问题，从而提出了一个新的研究领域。(3)开发了将代数-几何编码结合到图像中的技术，并将其应用于自恢复图像，掩模，数字足迹等。(4)反卷积是病态问题和断层扫描的典型例子。通过计算反卷积，引入一种正则化方法，对散焦照片进行了锐化处理，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褶积参数的自动选择。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