| Project/Area Number |
16540117
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University (2005-2006)
Tokai University (2004) |
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Principal Investigator |
KIKYO Hirotaka Kobe University, Faculty of Engineering, Professor, 工学部, 教授 (80204824)
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| Co-Investigator(Kenkyū-buntansha) |
ITAI Masanori Tokai University, School of Science, Professor, 理学部, 教授 (80266361)
WATANABE Junzo Tokai University, School of Science, Professor, 理学部, 教授 (40022727)
TSUBOI Akito University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院数理物質科学研究科, 教授 (30180045)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | generic automorphism / generic predicate / generic structure / amalgamation property / Peano Arithmetic / definable model / Lefschetz Property / complete intersection / Peano Arithmetic / definable model / Lefschetz Property / Artinian Gorenstein Algebra / central gap number / symmetric group / 局所次元 / 自由融合性 / 不安定 / 加群の自己同型 / quasi-minimal structure |
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| Research Abstract |
1. The amalgamation property for automorphisms is preserved under the addition of generic predicates to a theory. Therefore, if we get an unstable theory by the addition of generic predicates to a stable theory, then the class of the generic automorphisms of the resulting theory is not elementary.
2. The omega-power of the additive group of the rational numbers equipped with a shift function is a quasi-minimal structure. Its theory can be axiomatized by sentences expressing a kind of genericity. Let K be a field. The class of generic automorphisms of an infinite K-vector space is elementary. Its theory can be axiomatized in the same manner as above. Z-power of a countable infinite K-vector space equipped with a shift function is a quasi-minimal model of this theory. It is omega-stable with Morley rank omega.
3. Consider a typical predimension function on the finite structures for a finite relational language and the class of finite structures in which the empty set is closed with respect to this predimension. Then the generic structure of this class has a universal-existential theory. This is a rather general solution to a problem of Baldwin-Shelah.
4. If a model N of PA is definable without parameters in an elementary extension M of the structure of natural numbers and N and M are elementarily equivalent, then N is definably isomorphic to M. But if we allow N to be definable with parameters then there are examples such that M and N are elementarily equivalent but non-isomorphic, or M and N are isomorphic but not definably isomorphic.
5. For graded Artinian K-algebras, we gave several characterizations of the strong and weak Lefschetz Properties. With these results, we found new classes of complete intersections with the strong Lefschetz Property.
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