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333_pdfsam_math 54 differential equation solutions odd

# 333_pdfsam_math 54 differential equation solutions odd - 7...

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Exercises 5.7 Combining (i) and (ii) we conclude that ( x n ,v n ) ( x n ,v n )a s( x 0 ,v 0 ) ( x 0 ,v 0 ) uniformly with respect to n .T h u s ,i f( x 0 ,v 0 )isc loseto( x 0 ,v 0 ), ( x n ,v n )isc loseto( x n ,v n ) for all n . 9. (a) When x 0 =1 / 7, the doubling modulo 1 map gives x 1 = 2 7 (mod 1) = 2 7 , x 3 = 8 7 (mod 1) = 1 7 , x 2 = 4 7 (mod 1) = 4 7 , x 4 = 2 7 (mod 1) = 2 7 , x 5 = 4 7 (mod 1) = 4 7 , x 7 = 2 7 (mod 1) = 2 7 , x 6 = 8 7 (mod 1) = 1 7 , etc . This is the sequence 1 7 , 2 7 , 4 7 , 1 7 ,... .F o r x 0 = k 7 , k =2 ,..., 6, we obtain 2 7 , 4 7 , 1 7 , 2 7 ,... , 3 7 , 6 7 , 5 7 , 3 7 ,... , 4 7 , 1 7 , 2 7 , 4 7 ,... , 5 7 , 3 7 , 6 7 , 5 7 ,... ,
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Unformatted text preview: 7 , 6 7 , . . . . These sequences fall into two classes. The ±rst has the repeating sequence 1 7 , 2 7 , 4 7 and the second has the repeating sequence 3 7 , 6 7 , 5 7 . (c) To see what happens, when x = k 2 j , let’s consider the special case when x = 3 2 2 = 3 4 . Then, x 1 = 2 ± 3 4 ² (mod 1) = 3 2 (mod 1) = 1 2 , 329...
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