4 Examples for safe symbol existing Case
4.3 Checking all cases in section4.2 can let A n be prim- prim-itive for all n 2
We separate these cases into four subsections.
4.3.1 s2 = b11 or b44 and one column and one row of A are all zero We consider the case1 and case2 in section4.2.
Example 14 Consider A2 = 0
By (2:1) ; it is easily checked that
(1) A2 and A3 =
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
Example 15 Consider A2 =
By (2:1) ; it is easily checked that
(1) A2 and A3 =
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
4.3.2 A is full matrix
We consider the case3 in section4.2.
Example 16 Consider A2 = 0
By (2:1) ; it is easily checked that
(1) A2 and A3 =
(2) there exists
0 = 1; 1 = 1; 2 = 1 and
s2 = b11; s3 = b11b11= s2b11
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
4.3.3 s2 = b11 or b44 and two column and row of A are all zero We consider the case4~case9 in section4.2.
Example 17 Consider
By (2:1) ; it is easily checked that
(1) A2 and A3 =
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
Example 18 Consider
By (2:1) ; it is easily checked that
(1) A2 and A3 =
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
By (2:1) ; it is easily checked that
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
4.3.4 s2 = b14; s3 = b14b44 and two column and row of A are all zero We consider the case10 and case11 in section4.2.
Example 20 Consider
By (2:1) ; it is easily checked that
(1) A2 and A3 = 0 BB BB BB BB BB
@
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 CC CC CC CC CC A
have safe symbols.
(2) there exists
0 = 1; 1 = 4; 2 = 1 and
s2 = b14; s3 = b14b41= s2b41
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
4.4 Conclusion
In the section we describe some Remarks and examples related to the prim-itivity of An:
Remark 21 For the examples in the section4.3; we can …nd all cases for H2 and V2 which satisfying the Theorem5.8 related to the extensively weak mixing property in [33] can all let An be primitive for all n 2: Therefore, we can cover the Theorem5.8 in [33], i.e., the Theorem12 in our paper can be used to show more examples that An is primitive therein for all n 2 than the Theorem5.8 in [33].
Remark 22 Observing the examples in the section4.3; we …nd one of H and V 2 1 1
1 1 ; 1 1
1 0 ; 0 1
1 1 ;
In fact, if H and V satisfy one of the follow situations, and using the intro-duced method to get H2 and V2; then for A2 = one of H2 and V2; Theorem12 is applied to show that An is primitive for all n 2:
(1) one of H and V = E; and the other =2 fOg ;
(2) one of H and V = G; and the other 2 fU; L; I; T1; T2; K1g ; (3) one of H and V = G0; and the other 2 fU; L; I; T3; T4; K4g ;
(4) one of H and V 2 fU; L; Ig ; and the other 2 fK1; K2; K3; K4g ; Next, we give one example to show the statement given above.
Example 23 (from(1)) If
H = E = 1 1
By (2:1) ; it is easily checked that
(1) A2 and A3 =
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
Remark 24 If A2 is not constructed from H and V; then Theorem12 is also applied to show that An is primitive for all n 2:
we give one example follow.
Example 25 (Simpli…ed Golden Mean) Consider
A2 =
By (2:1) ; it is easily checked that
(1) A2 and A3 =
and
s2 = b11; s3 = b11b11= s2b11
such that (2)-(a)~(2)-(c) of Theorem12 hold, then Theorem12 is applied to show that An is primitive for all n 2:
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