臺灣博碩士論文加值系統

The generalized spectral radius ( ) of a set complex matrices is ( ) =
, where = sup{ ( ): each }. The main object of this paper is to study the following problems. Finiteness Conjecture: For each
finite set of n n complex matrices, there is some finite k such that
( ) = . Effective Finiteness Conjecture: For any finite set of n n matrices with rational entries, there is some finite k such that ( ) = .
Question: Are the Finiteness Conjecture and the Effective Finiteness Conjecture true? Via the undecidability of the Effective Finiteness Conjecture, we show that the answer to the Finiteness Conjecture is negative.

The generalized spectral radius ( ) of a set complex matrices is ( ) =
, where = sup{ ( ): each }. The main object of this paper is to study the following problems. Finiteness Conjecture: For each
finite set of n n complex matrices, there is some finite k such that
( ) = . Effective Finiteness Conjecture: For any finite set of n n matrices with rational entries, there is some finite k such that ( ) = .
Question: Are the Finiteness Conjecture and the Effective Finiteness Conjecture true? Via the undecidability of the Effective Finiteness Conjecture, we show that the answer to the Finiteness Conjecture is negative.

Content
1. Introduction………………………………………………2
2. Undecidability Problem………………………………………3
3. Undecidability of Effective Finiteness Conjecture…8
4. Disproof of Finiteness Conjecture……………………8
5. Normed Finiteness Conjecture……………………………9
6. Normed Finiteness Conjecture for Euclidean Norm……11
7. Application of Theorem 6.1…………………………16
8. A Research Problem…………17

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