2007 Fiscal Year Final Research Report Summary
Microlocal,FunctionalAnalytic and Numerical Study of Partial Differential Equations and Tomography
| Project/Area Number |
16540140
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Ochanomizu University |
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Principal Investigator |
KANEKO Akira Ochanomizu University, Graduate School of Humanities and Sciences, Professor (30011654)
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| Project Period (FY) |
2004 – 2007
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| Keywords | partial differential equations / tomography / critical exponent of blowup / switching graph / Hamiltonian graph / algenraic-geometric codes / inverse convolution / electronic fingerprint |
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| Research Abstract |
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
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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
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