1993 Fiscal Year Final Research Report Summary
Computation Algorithms Development and its Theory
| Project/Area Number |
03640233
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Hosei University |
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Principal Investigator |
NAGASAKA Kenji Hosei University, College of Eng., Professor, 工学部, 教授 (40000187)
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| Co-Investigator(Kenkyū-buntansha) |
FUSE Mitso Hosei University, College of Eng., Lecturer, 工学部, 講師 (00120832)
HIRAMATSU Toyokazu Hosei University, College of Eng., Professor, 工学部, 教授 (40029674)
ANDO Shiro Hosei University, College of Eng., Professor, 工学部, 教授 (60061016)
TANAKA Hisao Hosei University, College of Eng., Professor, 工学部, 教授 (70061025)
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| Project Period (FY) |
1991 – 1993
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| Keywords | Chinese remainder theorem / Finite field / Multiplicative function / Error-free computation / Computational complexity / Oracle / Pascal's triangle / Goppa code |
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| Research Abstract |
We propose newly developed fast algorithm of the Chinese remainder theorem, which is one of most important and basic computing algorithms and examined numerically the efficiency of our algorithm in comparison with the usual method and confirm the effectiveness of our method. This fast algorithm is proved to be valid on polynomial rings over finite fields.
Under the regularity conditions, or the difference conditions, we show that multiplicative arithmetical functions are constant multiple of power functions with some exponent.
We also obtain that integral row operations of matrices admit the one-to-one mapping between Farey-N-fractions and subsets of finite precision in quadratic rationals. This result allows us to solve integral linear equations (congruences) and hence leads to an error-free computation method.(Nagasaka and Fuse)
As for computational complexity theory, we consider it from the scope of relitivisation. Then we give the solution of Bennet-Gill's problem in the classification of Kleene's hierarchy under the existence of oracles and determine the levels of certain classes. Some results on BPP are also obtained.(Tanaka)
We investigate the configuration of generalized binomial and multinomial coefficients and get remarks on the proof of GCD and LCM equalities and also another generalization of David's theorem. These results are generalized in more general set up and give necessary and sufficient conditions on a certain configuration.(Ando)
Goppa code is a fundamental tool in coding theory and we mention the link between Goppa code and number theory then summarize unsolved problems, some of which are partially solved.(Hiramatsu)
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