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Degenerate Partial Differential Equations and Integral Transforms

Research Project

Project/Area Number 17540179
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOsaka Electro-Communication University

Principal Investigator

MANDAI Takeshi  Osaka Electro-Communication University, Faculty of Engineering, Professor (10181843)

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)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
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)
Keywordspartial differential equations / integral transforms / wavelets / singularity / Fuchsian / global solutions / blind source separation
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.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (34 results)

All 2008 2007 2006 2005

All Journal Article (22 results) (of which Peer Reviewed: 10 results) Presentation (12 results)

  • [Journal Article] ウェーブレット解析のシステム同定ヘの応用2007

    • Author(s)
      芦野 隆一-萬代 武史-守本 晃
    • Journal Title

      応用数理 17:1

      Pages: 2-13

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] 特異値分解とウェーブレットを使った画像処理2007

    • Author(s)
      芦野 隆一-萬代 武史-守本 晃
    • Journal Title

      数理解析研究所講究録 1529

      Pages: 26-41

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global Fuchsian Goursat problem in the class of holomorphic-Gevrey functions2007

    • Author(s)
      Belarbi, Malika - Mandai, Takeshi - Mechab, Mustapha
    • Journal Title

      Osaka J.Math. 44

      Pages: 255-283

    • NAID

      120004841971

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application2007

    • Author(s)
      H.Tahara
    • Journal Title

      Publ.Res.Inst.Math.Sci. 43

      Pages: 535-583

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Blind source separation using time-frequency information matrix given by several wavelet transforms2007

    • Author(s)
      Ryuichi Ashino - Keiko Fujita -Takeshi Mandai - Akira Morimoto - Kiyoaki Nishihara
    • Journal Title

      Information 10

      Pages: 555-568

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Blind source separation using time-frequency analysis2007

    • Author(s)
      R.Ashino, T.Mandai, A.Morimoto and F.Sasaki
    • Journal Title

      Fields Institute Communications 52

      Pages: 401-414

    • NAID

      110007003253

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] An application of wavelet analysis And the sistem identification(in Japanese).2007

    • Author(s)
      R. Ashino, T. Mandai, A. Morimoto
    • Journal Title

      Bulletin of Japan Soc. For ind. Appl. 17:1

      Pages: 2-13

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Image processing using singular value decompositions and wavelets(in Japanese).2007

    • Author(s)
      R. Ashino, T. Mandai, A. Morimoto
    • Journal Title

      RIMS Kokyuroku 1529

      Pages: 26-41

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global Fuchisan Goursat problem In the class of holomorphic-Geverey functions.2007

    • Author(s)
      M.Belaribi, t. Mandai, M. Mechab
    • Journal Title

      Osaka J. Math. 44

      Pages: 255-283

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Couping of two partial differential equations and its application.2007

    • Author(s)
      H. tahara
    • Journal Title

      Publ. res. Inst. Math. Sci. 43

      Pages: 535-583

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Blind source separation using time-frequency information matrixz given by several wavelet transforms.2007

    • Author(s)
      R.Ashino. K. Fujita, T. Mandai, A.Morimoto, K. Nishihara
    • Journal Title

      Information 15:5

      Pages: 555-568

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Blind source separation using time-frequency analysis.2007

    • Author(s)
      R. Ashino, T. Mandai, A. Morimoto, F. Sasaki
    • Journal Title

      Fields Institute Communications 52

      Pages: 401-414

    • NAID

      110007003253

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global Fuchsian Goursat problem in the class of holomorphic-Gevrey functions2007

    • Author(s)
      Belarbi, Malika-Mandai, Takeshi-Mechab, Mustapha
    • Journal Title

      Osaka J.Math 44

      Pages: 255-283

    • NAID

      120004841971

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application2007

    • Author(s)
      H.Tahara
    • Journal Title

      Publ.Res.Inst.Math.Sci 43

      Pages: 535-583

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blind source separation using time-frequency information matrix given by Several wavelet transforms2007

    • Author(s)
      Ryuichi Ashino-Keiko Fujita Takeshi Mandai-Akira Morimoto-Kiyoaki Nishihara
    • Journal Title

      Information 10

      Pages: 555-568

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] ウェ-ブレット解析のシステム同定への応用2007

    • Author(s)
      芦野 隆一, 萬代 武史, 守本 晃
    • Journal Title

      応用数理 17.1

      Pages: 2-13

    • NAID

      110006242208

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 特異値分析とウェ-ブレットを使った画像処理2007

    • Author(s)
      芦野 隆一, 萬代 武史, 守本 晃
    • Journal Title

      数理解析研究所講究録 1529

      Pages: 26-41

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Blind source separation using several types of time-frequency information given by wavelet transforms2006

    • Author(s)
      A.Morimoto, K.Fujita, R.Ashino, T.Mandai
    • Journal Title

      Proc.of The Fourth International Conference on Information and The Fourth Irish Conf.on MFCSIT

      Pages: 164-167

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Blind source separation using several types of time-frequency information given by wavelet transorms.2006

    • Author(s)
      A. Morimoto, K. Fujita, R. Ashino, T. Mandai
    • Journal Title

      Proc. Fourth International Conf. on Information and Fourth Irish Conf. on MFCSIF

      Pages: 164-167

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Blind source separation using several types of time-frequency information given by wavelet transforms2006

    • Author(s)
      A.Morimoto, et al.
    • Journal Title

      Proceedings of The Fourth International Conference on Information and The Fourth Irish Conference on MFCSIT

      Pages: 164-167

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Note on discrete phenomena in uniqueness in doubly characteristic Cauchy problems2005

    • Author(s)
      Igari, K.
    • Journal Title

      J.Math.Kyoto Univ. 45:4

      Pages: 831-839

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Note on discrete phenomena in uniqueness in doubly characteristic Cauchy problems.2005

    • Author(s)
      K. Igari
    • Journal Title

      J. Math. Kyoto Univ. 45:2

      Pages: 831-839

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 時間周波数解析に基づくブラインド信号源分離のための位相情報を利用した高速アルゴリズム2008

    • Author(s)
      守本 晃
    • Organizer
      日本応用数理学会 研究部会連合発表会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2008-03-09
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Fast algorithm for blind source separation using time-frequency information based on time-frequency analysis.2008

    • Author(s)
      A. Morimoto
    • Organizer
      Joint Conference of Research Activity Groups, Japan Society for Industrial and Applied Mathematics
    • Place of Presentation
      Tokyo Metropolitan Universioty
    • Year and Date
      2008-03-09
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [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
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] The width of a signal and uncertainty principle(in Japanese).2008

    • Author(s)
      T. Mandai
    • Organizer
      Workshop : Interaction of Differential Equations and Wavelet Analysis
    • Place of Presentation
      Ritsumeikan Univ
    • Year and Date
      2008-01-18
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [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
      2007 Final Research Report Summary
  • [Presentation] Blind source separation by time-frequency analysis.(in Japanese).2007

    • Author(s)
      R. Ashino
    • Organizer
      2007 Summer Symposium on Operator Theory
    • Place of Presentation
      ASPAM, Aomori
    • Year and Date
      2007-09-20
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 時間周波数情報を利用したブラインド信号源分離について2007

    • Author(s)
      守本 晃
    • Organizer
      日本応用数理学会2007年度年会
    • Place of Presentation
      北海道大学
    • Year and Date
      2007-09-17
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Blind source separation using time-frequency information(in Japanese).2007

    • Author(s)
      A. Morimoto
    • Organizer
      2007 Annual Conference of Japan Society for Industrial and Applied Mathematics
    • Place of Presentation
      Hokkaido Univ
    • Year and Date
      2007-09-17
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 複数のフックス型変数を持つ偏微分方程式の滑らかな零解2006

    • Author(s)
      萬代 武史
    • Organizer
      微分方程式の解の解析性とその周辺
    • Place of Presentation
      金沢大学サテライトプラザ3階大講義室
    • Year and Date
      2006-10-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Smooth null-solutions to PDEs with several Fuchsian variables(in Japanese).2006

    • Author(s)
      T. Mandai
    • Organizer
      Analyticity of solutions to differential equations and related topics
    • Place of Presentation
      Satellite Plaza of Kanazawa Univ
    • Year and Date
      2006-10-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Smooth null-solutions to PDEs with several Fuchsian variables2006

    • Author(s)
      MANDAI, Takeshi
    • Organizer
      International Congress of Mathematicians 2006
    • Place of Presentation
      Palacio Municipal de Congresos,Madrid,Spain
    • Year and Date
      2006-08-26
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Smooth null-solutions to PDEs with several Fuchsian Variables.2006

    • Author(s)
      T. Mandai
    • Organizer
      International Congress of Mathematicians, 2006
    • Place of Presentation
      Palacio Municipal de Congresos, Madrid, Spain
    • Year and Date
      2006-08-26
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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