2000 Fiscal Year Final Research Report Summary
Boundary behavior of analytic functions and harmonic functions
| Project/Area Number |
11640187
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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 |
Basic analysis
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| Research Institution | Daido Institute of Technology |
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Principal Investigator |
SEGAWA Shigeo Daido Institute of Technology, Engineering, Professor, 工学部, 教授 (80105634)
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| Co-Investigator(Kenkyū-buntansha) |
UEDA Hideharu Daido Institute of Technology, Engineering, Professor, 工学部, 教授 (20139968)
TADA Toshimasa Daido Institute of Technology, Engineering, Professor, 工学部, 教授 (90105635)
IMAI Hideo Daido Institute of Technology, Engineering, Professor, 工学部, 教授 (00075855)
NAKAI Mitsuru Nagoya Institute of Technology, Professor emeritus, 名誉教授 (10022550)
NARITA Junichiro Daido Institute of Technology, Engineering, Assistant Professor, 工学部, 助教授 (30189211)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Martin boundary / positive harmonic function / unlimited covering surface / Picard principle / polyharmonic function / meromorphic function / Myrberg phenomenon / interpolating sequence |
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| Research Abstract |
1. Segawa showed that every positive harmonic function on a finitely sheeted unlimited covering surface of an open Riemann surface of positive boundary is a pullback of a positive harmonic function on the base surface by the projection map if and only if the Martin compactification of the covering surface is isomorphic to that of the base surface via the projection map. Segawa proved an analogous result of the above for bounded harmonic functions in terms of Martin boundary. Segawa determined the Martin boundaries of m-sheeted cyclic unlimited covering surfaces of the complex plane. Nakai showed that Royden p-compactifications for 1<p<d of two d-dimensional Riemannian manifolds (d【greater than or equal】2) are homeomorphic if and only if there exists a almost quasiisometric homeomorphism between these Riemannian manifolds. 2. Nakai and Tada determined the maximal growth of a rotation free density which is an exceptional perturbation for Picard principle. Tada and Nakai showed that if the Picard principle is valid for a rotation free density P, then there exists an essential set of P which is arbitrarily small and rare in a sense. 3. Nakai and Tada proved an extension of the Liouville theorem for a class of functions which properly contains polyharmonic functions. 4. Ueda showed that for a family of entire functions, the zeros of each function in the family are of odd order. Ueda generalized Nevanlinna's three-function theorem. 5. Narita gave a sufficient condition for bounded domains in order that a harmonic interpolating sequence is also an interpolating sequence. Narita gave a sufficient condition for bounded domains without irregular boundary points in order that there exists a harmonic interpolating sequence which is not interpolating. Nakai showed that the uniqueness theorem is sufficient but not necessary for the occurrence of the Myrberg phenomenon.
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