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1998 Fiscal Year Final Research Report Summary

Harmonic Analysis and its application to measurements

Research Project

Project/Area Number 09640219
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionKeio University

Principal Investigator

ISHIKAWA Shiro  Faculty of Science and Technology, Keio University Associate Prof., 理工学部, 助教授 (10051913)

Co-Investigator(Kenkyū-buntansha) NODERA Takashi  Faculty of Science and Technology, Keio University Associate Prof., 理工学部, 助教授 (50156212)
SHIMOMURA Shun  Faculty of Science and Technology, Keio University Associate Prof., 理工学部, 助教授 (00154328)
TANI Atsusi  Faculty of Science and Technology, Keio University Prof., 理工学部, 教授 (90118969)
KUNIMATSU Noboru  Faculty of Science and Technology, Keio University Prof., 理工学部, 教授 (70051662)
KIKUCHI Norio  Faculty of Science and Technology, Keio University Prof., 理工学部, 教授 (80090041)
Project Period (FY) 1997 – 1998
KeywordsFourier analysis / measurements / wavelet / Schrondinger eq. / fluid eq. / quantum mechanics / fuzzy information / statistical mechanics
Research Abstract

Measurement theory is a mathematical theory, which is motivated by Born's probabilistic interpretation of quantum mechanics. Particularly, it is related to the resolution of the identity, or, in terms of Fourier analysis, a theory of wavelet. Out research results are as follows.
(1) Consider an N-particle classical system (e.g., N【approximately equal】10ィイD123ィエD1) with a Hamiltonian H on the phase space RィイD16NィエD1. Moreover, we consider the class of measurements such as we can get the information concerning NィイD2oィエD2 particles in this system (e.g., NィイD2oィエD2【less than or equal】10ィイD118ィエD1<<N) . We showed that this class of measurements also induces another kind of equilibrium statistical mechanics. In this equilibrium statistical mechanics, we can assert that the "equilibrium state" exists almost everywhere as a point of the energy surface Ω({(q,p)RィイD16NィエD1:H(q,p)=E}).
(2) Quantum logic is, of course, motivated bu quantum mechanics. Thus, it is natural to consider that several logi … More c may be formulated in terms of measurements. In fact we showed that "fuzzy logic" is clarified and understood in the limit of measurement theory.
(3) The theory of "measurement errors" is an important aspect of measurement theory. In fact, we showed that the "measurement error" in Heisenberg's uncertainty relation is similar to that of classical measurement.
(4) The concept of "trajectories" is fundamental to classical mechanics. However, we showed that this concept is derived from that of simultaneous measurements.
(5) Information theory is reconstructed in terms of measurements. This is natural since "information" should be obtained by measurements.
(6) We consider that statistics is to analyze "measured data" for some purpose. Thus, it is viable to consider that statistics is described in terms of measurements. In faxt, we showed that factor analysis, regression analysis and Kalman filter are formulated in measurement theory.
As mentioned above, measurement theory covers a great region, which may be called "information science". The technological aspect of measurement theory is, of course, important. However, in our research, we devoted ourselves to the scientific and mathematical aspect of measurement theory. Less

  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] S.Ishikawa: "Fuzzy logic in measurements"Fuzzy sets and systems. 100. 291-300 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Ishikawa: "A system theoretical approach to factor analysis"The Proceedings of EUFIT'98. Vol2. 1511-1516 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Ishikawa: "Statistics in measurements"Fuzzy sets and systems. 発表予定.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Tani,T.Nishiyama: "Solvability of equations for motion at a vortex filament with or without axial flow"Publ. RIMS Kyoto. 33. 509-526 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Tani,Y.Iguchi,N.Tanaka: "On a free boundary problem for an incompressible ideal fluid in two spaces"Adv. Math. Sci. Appl.. 9. 未定 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Shimomura: "On a generalized Bessel function of two variables II"Tohoku. Math. J.. 50. 389-406 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nodera,K.Mariya: "Adaptive BiCGStab(l) Algorithm with varying l"Proceedings of IMASS'98. 34-39 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nodera,N.Tsuno: "The Parallelization of the incomplete LU Factorization on AP 1000"4th International Euro-Por Conference. 788-792 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S. Ishikawa: "Fuzzy inferences by algebraic method"Fuzzy sets and Systems. Vol 87. 181-200 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Ishikawa: "A quantum mechanical approact to a fuzzy theory"Fuzzy sets and Systems. Vol 90. 277-306 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Ishikawa: "Fuzzy logic in measurements"Fuzzy sets and Systems. Vol 100. 291-300 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Ishikawa: "A system theoretical approach to factor analysis"The proceedings of EUFIT'98. Vol 2. 1511-1516 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Ishikawa: "Statistics in measurements"Fuzzy sets and Systems. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Tani: "On the two phase free boundary problem for two-dimensional water wave"Math. Anallen. Vol 309. 199-223 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Tani, T. Nishiyama: "Solvability of equations for motion of a vertex filament without axial flow"Publ. RISM Kyoto. Vol 33. 509-526 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Tani, Y. Iguchi, N. Tanaka: "On a free boundary problem for an incompressible ideal fluid in two spaces"Adv. Math. Sci. Appl.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Shimomura: "On a generalized Bessel function of two variables"Tohoku, Math. J.. Vol 50. 389-406 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Shimomura: "Asymptotic expansions and Stokes multipliers of hypergeometric functions"J. Math. Soc. Japan. Vol 50. 1-22 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Nodera, K. Moriya: "Adaptive BiCGStab(l) Algorithm with varying l."Proceedings of IMASS'98. 34-39 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Nodera, N. Tsuno: "The Parallelization of the incomplete LU Factorization on AP 1000"4-th International Euro-Par Conference. 788-792 (1998)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23  

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