• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Mathematical Research on Mathematical Models of Quantum Computing

Research Project

Project/Area Number 11440028
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTohoku University (2001)
Nagoya University (1999-2000)

Principal Investigator

OZAWA Masanao  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (40126313)

Co-Investigator(Kenkyū-buntansha) IHARA Shunsuke  Nagoya University, School of Informatics, Professor, 情報文化学部, 教授 (00023200)
TSUKAJI Tasuie  Nagoya University, Graduate School of Human Informatics, Research Associate, 大学院・人間情報学研究科, 助手 (70291961)
YASUMOTO Masahiro  Nagoya University, Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (10144114)
MATSUBARA Yo  Nagoya University, Graduate School of Human Informatics, Associate Professor, 大学院・人間情報学研究科, 助教授 (30242788)
MATSUMOTO Hiroyuki  Nagoya University, Graduate School of Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (00190538)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥10,500,000 (Direct Cost: ¥10,500,000)
Fiscal Year 2001: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2000: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥3,900,000 (Direct Cost: ¥3,900,000)
Keywordsquantum computing / quantum Turing machines / quantum circuits / quantum complexity theory / quantum gates / quantum operation / fault-tolerant quantum computing / controlled NOT gate / 加法的保存量 / 誤り訂正理論 / 万能量子論理素子 / 制御否定素子 / 計算基底 / 閾値定理 / 量子オラクルTuring機械 / 量子アルゴリズム / 量子コンピュータ / 完全正写像 / 量子測定 / 量子回路族 / 量子チューリング機械 / 量子計算量 / 量子力学 / 量子情報
Research Abstract

The following results have been obtained on mathematical foundations on quantum Turing machine and quantumcircuits: (1) Local transition functions of quantum Turing machines (QTM) are generally characterized including multitape cases. (2) The notion of uniform quantum circuit families (UQCF) was introduced for the first time and developed their complexity theory nd proved the computational equivalence between QTMs and UQCF in Monte Caro type computations. (3) In order to solve the halting problem for QTMs, it has been proved that under a refined halting protocol measurements of halting flag do not disturb the probability distribution of the output of computations.
The following results have been obtained on physical implementations of quantum logicgates: (1) Conservation laws limit theaccuracy of physical implementations of elementary quantum logic gates. (2) Although the SWAP gate has no conflict with the conservation law, the controlled-NOT gate, which is one of the universal quantum … More logic gates, cannot be implemented by any 2-qubit rotationally invariant unitary operation within error probability 1/16.. (3) If the computational basis is represented by a component of spin and physical implementations obey the angular momentum conservation law, any physically realizable quantum logicgates with n qubit ancilla cannot implement the controlled-NOT gate within the error probability 1/(4n^2). (4) An analogous relation holds for bosonic ancillae with the size defined through the average number of photons. Any set of universal gates inevitably obeys a related limitation with error probability O(n^<-2>). (5) The current theory demands the threshold error probability 10^5 10^6 for each quantum gate. Thus, a single controlled-NOT gate would not be in reality a unitary operation on a 2-qubitsystem but would be a unitary operation on a system with at least 100 qubits. (6) The present investigation suggests that the current choice of the computational basis should be modified so that the computational basis commutes with the conserved quantity. Less

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (81 results)

All Other

All Publications (81 results)

  • [Publications] M.Ozawa: "Entanglement measures and the Hilbert-Schmidt distance"Phys.Lett.A. 268. 158-160 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Operational characterization of simultaneous measurements in quantum mechanics"Phys.Lett.A. 275. 5-11 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.yamashita, M.Ozawa: "Nonstandard representations of the canonical commutation relations"Rev.Math.Phys.. 12. 1407-1427 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Measurements of nondegenerate discrete observables"Phys.Rev.A. 62. 062101(1)-062101(13) (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa, H.Nishimura: "Local transition functions of quantum Turing machines"Theoret.Informatics and Appl.. 34. 379-402 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Operations, disturbance, and simultaneous measurability"Phys.Rev.A. 63. 032109(1)-032109(15) (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Controlling quantum state reduction"Phys.Lett.A. 282. 336-342 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Quantum computing and Church s thesis"J.Japan Assoc.Phil.Sci.. 28. 59-65 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Conservation laws, uncertainty relations, and quantum limits of measurements"Phys.Rev.Lett.. 88. 050402(1)-050402(4) (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Nishimura, M.Ozawa: "Computational complexity of uniform quantum circuit families and quantum Turing machines"Theor.Comput.Sci.. 276. 147-181 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Position measuring interactions and the Heisenberg uncertainty principle"Phys.Lett.A. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Conservative quantum computing"Phys.Rev.Lett.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Ihara: "Information transmission over continuous-time Gaussian channels with feedback"Problems of Information Transmission. 35. 10-24 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Kubo, S.Ihara: "Error exponent for coding of memoryless Gaussian sources with a fidelity criterion"IEICE Trans.Fundamentals. E83-A. 1891-1897 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Ihara: "Large deviation theorems for Gaussian processes and their applications in information theory"Acta Applicandae Math.. 63. 165-174 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Iriyama, S.Ihara: "The error exponent and minimum achievable rates for the fixed-length coding of general sources"IEICE Trans.Fundamentals. E84-A. 2466-2473 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Kubo, S.Ihara: "The asymptotics of string matching probabilities for Gaussian random sequences"Nagoya Math.J.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Ikeda, H.Matsumoto: "Brownian motion on the hyperbolic plane and Selberg trace formula"J.Func.Anal.. 163. 63-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "A version of Pitman s 2M-X theorem for geometric Brownian motion"C.R.Acad.Sc.Paris, S-erie I. 328. 1067-1074 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "Some changes of probabilities related to a geometric Brownian version of Pitman s 2M-X theorem"Elect.Comm.Prob.. 4. 15-23 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "An analogue of Pitman s 2M X theorem for exponential Brownian functionals, Part I : A time inversion approach"Nagoya Math.J.. 159. 125-166 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Dontati-Martin, H.Matsumoto, M.Yor: "On positive and negative moments of the integrals of geometric Brownian motions"Stat.Prob.Lett.. 49. 45-52 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, N.Ueki: "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space"J.Math.Soc.Japan. 52. 269-292 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Donati-Martin, H.Matsumoto, M.Yor: "On striking identities about the exponetial functionals of the Brownian bridge and Brownian Motion"Periodica Math.Hung.. 41. 103-119 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration"Osaka J.Math.. 38. 383-398 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "An analogue of Pitman s 2M X theorem for exponential Brownian functionals, Part II : the role of generalized inverse Gaussian distributions"Nagoya Math.J.. 162. 65-86 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Donati-Martin, H.Matsumoto, M.Yor: "Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motion"Publ.RIMS, Kyoto Univ.. 37. 295-326 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one"Bull.Sci.math.. 125. 553-581 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Matsubara, S.Shelah: "Nowhere precipitousness of the non-stationary ideal over P_kλ"Journal of Mathematical Logic. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Matsubara: "Stronger ideals over P_kλ"Fundamanta Mathematicae. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Tsukiji, H.Mahmoud: "A limit law for outputs in random recursive circuits"Algorithmica. 31. 403-412 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Tsukiji: "P-comp versus P-samp questions on average polynomial domination."ICICE Trans.Information an Systems. E84-D. 1402-1410 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Entanglement measures and the Hubert-Schmidt distance"Phys.Lett.. 268. 158-160 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Operational characterization of simultaneous measurements in quantum mechanics"Phys.Lett.. 275. 5-11 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Yamashita, M.Ozawa: "Nonstandard representations of the canonical commutation relations"Rev.Math.Phys.. 12. 1407-1427 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Measurements of nondegenerate discrete observables"Phys.Rev.A. 62. 062101-1-062101-13 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa, H.Nishimura: "Local transition functions of quantum Turing machines"Theoret.Informatics and Appl.. 34. 379-402 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Operations, disturbance, ands imultaneous measurability"Phys.Rev.. 63. 032109-1-032109-15 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Controlling quantum state reduction"Phys.Lett.. 282. 336-342 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Ozawa: "Quantum computing and Church s thesis"J.Japan Assoc.Phil.Sci.. 28. 59-65 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Conservation laws, uncertainty relations, and quantum limits of measurements"Phys.Rev.Lett.. 88. 050402-1-050402-4 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Nishimura, M.Ozawa: "Computational complexity of uniform quantum circuit families and quantum Turing machines"Theor.Comput.Sci.. 276. 147-181 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Position measuring interactions and the Heisenberg uncertainty principle"Phys.Lett.A. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ozawa: "Conservative quantum computing"Phys.Rev.Lett.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Ihara: "Information transmission over continuous-time Gaussian channels with feedback"Problems of Information Transmission. 35. 10-24 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Kubo, S.Ihara: "Error exponent for coding of memoryless Gaussiansources with a fidelity criterion"IEICE Trans.Fundamentals. E83-A. 1891-1897 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Ihara: "Large deviation theorems for Gaussian processes and their applications in information theory"Acta Applicandae Math.. 63. 165-174 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Iriyama, S.Ihara: "The error exponent and minimum achievable rates for the fixed-length coding of general sources"IEICE Trans.Fundamentals. E84-A. 2466-2473 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kubo, S.Ihara: "The asymptotics of string matching probabilities for Gaussian random sequences"Nagoya Math.J.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Ikeda, H., Matsumoto: "Brownian motion on the hyperbolic plane and Selberg trace formula"J.Func.Anal.. 163. 63-110 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "A version of Pitman s 2M-X theorem for geometric Brownian motions"C.R.Acad.Sc.Paris, Serie I. 328. 1067-1074 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, H.Yor: "Some changes of probabilities related to a geometric Brownian version of Pitman s 2M-X theorem Elect."Comm.Prob.. 4. 15-23 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Hatsumoto, H.Yor: "An analogue of Pitman s 2M X theorem for exponential" Brownian functionals, Part I: A time inversion approach"Nagoya Math.J.. 159. 125-166 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Donati-Martin, H.Hatsumoto, H.Yor: "On positive and negative moments of the integrals of geometric Brownian motions"Stat.Prob.Lett.. 49. 45-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, N.Ueki: "Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space"J.Math.Soc.Japan. 52. 269-292 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Donati-Hartin, H.Matsumoto, H.Yor: "On striking identities about the exponential functionals of the Brownian bridge and Brownian motion, Periodica"Math.Hung.. 41. 103-119 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration"Osaka J. Math.. 38. 383-398 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto, M.Yor: "An analogue of Pitman s 2M-X theorem for exponential Brownian functionals, Part II: the role of generalized inverse Gaussian distributions"Nagoya Math. J.. 162. 65-86 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.Donati-Martin, H.Matsumoto, M.Yor: "Some absolute continuity relationship's for certain anticipative transformations of geometric Brownian motions"Publ.RIMS, Kyoto Univ.. 37. 295-326 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one"Bull.Sci.math.. 125. 553-581 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Matsubara, S.Shelah: "Nowhere precipitousness of the non-stationary ideal over PκA"Journal of Mathematical Logic. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Matsubara: "Stronger ideals over P_κλ"Fundamenta Mathematicae. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Tsukiji, H.Mahmoud: "A limit law for outputs in random recursive circuits"Algorithmica. 31. 403-412 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Tsukiji: "P-compversus P-samp questions on average polynomialdomination"ICICE Trans. Information and Systems. E84-D. 1402-1410 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Ozawa: "Controlling quantum state reduction"Phys.Lett.A. 282. 336-342 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小澤正直: "量子計算とチャーチの提唱"科学基礎論研究. 28・2. 5-11 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Ozawa: "Conservation laws, uncertainty relations and quantum limits of measurements"Phys.Rev.Lett. 88. 050402(1)-050402(4) (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Ozawa: "Quantum measurement, information, and completely positive maps"Quantum Communication, Computing, and Measurement(Plenum, New York). 3. 97-106 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Ozawa: "Operations, disturbance, and simultaneous measurability"Phys.Rev.A. 63・6. 032109(1)-032109(15) (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Ozawa and H.Nishimura: "Local transition functions of quantum Turing machines"RAIRO Theor.Inform.Appl.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Nishimura and M.Ozawa: "Computational complexity of uniform quantum circuit families and quantum turing machines"Theoret.Comput.Sci.. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Ozawa: "Operations, disturbance, and simultaneous measurability"Phys.Rev.A. 63・6. 032109(1-15) (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Ozawa: "Measurements of nondegenerate discrete observables"Phys.Rev.A. 62・6. 062101(1-13) (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Yamashita and M.Ozawa: "Nonstandard representations of the canonical commutation relations"Rev.Math.Phys.. 12・11. 1407-1427 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Ozawa: "Operational characterization of simultaneous measurements in quantum mechanics"Phys.Lett.A. 275. 5-11 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M. Ozawa: "Entanglement measures and the Hilbert-Schmidt distance"Phys. Lett. A. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Ozawa: "Measuring Processes and Uncertainty Principle"J. Japan Ass. Phil. Sci.. 27. 79-85 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] D. Burke: "The extent of the strength in the club filters"Israel J. Math.. 114. 253-263 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S Ihara: "Information transmission over continuous time Gaussian channel with feedback."Prob. Inform. Trans.. 35. 13-29 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H. MATSUMOTO: "A version of Pitman's 2M-X theorem for geometric Brownian motions"C. R. Acad. Sci. Paris. 328,I. 1067-1074 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H. Yamashita: "Nonstandard Representations of the Canonical Commutation Relations"Rev. Math. Phys.. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report

URL: 

Published: 1999-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi