Degenerate Partial Differential Equations and Integral Transforms
| Project/Area Number |
17540179
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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 | Osaka Electro-Communication University |
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Principal Investigator |
MANDAI Takeshi Osaka Electro-Communication University, Faculty of Engineering, Professor (10181843)
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| Co-Investigator(Kenkyū-buntansha) |
IGARI Katsuji Ehime University, Faculty of Engineering, Professor (90025487)
TAHARA Hidetoshi Sophia University, Faculty of Science and Technology, Professor (60101028)
ASHINO Ryuichi Osaka Kyoiku University, Faculty of Education, Assistant Professor (80249490)
MORIMOTO Akira Osaka Kyoiku University, Faculty of Education, Assistant Professor (50239688)
ASAKURA Fumioki Osaka Electro-Communication University, Faculty of Engineering, Professor (20140238)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,530,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | partial differential equations / integral transforms / wavelets / singularity / Fuchsian / global solutions / blind source separation |
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| Research Abstract |
1. Existence of Global solutions: To linear partial differential equations with several Fuchsian variables, which are typical equations
Degenerating on several hypersurfaces, we succeeded to construct global solutions that are holomorphic with respect to Fuchsian variables, and of Gevrey class with respect to the other variables, with the help of Dr. Belarbi and Professor Mechab in Algeria.
2. Existence of smooth null-solutions : We proved the existence of smooth null-solutions to the above equations under some assumption of hyperbolicity in the fallowing cases.
(1) For first and second order equations under quite general conditions.
(2) For typical equations of higher order.
3. Blind source separation problem : we could confirm the efficiency of the method to use more than two wavelet functions for the problem of separating source signals from mixed observed signals. We compared the results between using two real-valued wavelet functions and using more than two real-valued wavelet functions.
As for the case when the sources are mixed with time differences, we could give a precise algorithm, a mathematical explanation, and some simulations, with the help of Professor Fumio Sasaki of Tokyo University of Science. These results will be published as a complete article in 2008.
4. During our research, we encountered an important problem of how we should define the "center" and "width" of the Fourier image of signals. We made several comparison of possible definitions from the viewpoint of Mathematics and wavelet analysis. We could get an inequality concerning a modification of Uncertainty Principle with natural new definitions.
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Report
(4 results)
Research Products
(34 results)
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[Presentation] 信号の幅と不確定性原理2008
Author(s)
萬代 武史
Organizer
ワークショップ「微分方程式とウェーブレット解析との接点」
Place of Presentation
立命館大学,びわこ草津キャンパス,エポック21・大会議室
Year and Date
2008-01-18
Description
「研究成果報告書概要(和文)」より
Related Report
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[Presentation] 時間周波数解析による信号源分離2007
Author(s)
芦野 隆一
Organizer
2007年夏の作用素論シンポジウム
Place of Presentation
青森県観光物産館アスパム
Year and Date
2007-09-20
Description
「研究成果報告書概要(和文)」より
Related Report
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