µ(, y) Computing the Möbius fun tion µ(x, x) = 1 The Möbius fun tion is de ned b y and X µ(x, t) = 0 x < y if x6t6y 3
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1 ÈÖÑÙØØÓÒ ÔØØÖÒ Ò Ø ÅÙ ÙÒØÓÒ ÙÖ ØÒ ÎØ ÂÐÒ Ú ÂÐÒÓÚ Ò ÐÜ ËØÒÖÑ ÓÒ ÒÖ
2 Ì ØÛÓµ
3 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ ½ Ì ÔÓ Ø Ó ÔÖÑÙØØÓÒ ÛºÖºØº ÔØØÖÒ ÓÒØÒÑÒØ ½
4 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ Ì ÒØÖÚÐ [12,2314] ½ ¾
5 ÓÑÔÙØÒ Ø ÅÙ ÙÒØÓÒ µ(, y) ¹½ ½ ¾ ¼ ¹½ ¹½ ¹½ ½ xty µ(x, t) = 0 x < y Ì ÅÙ ÙÒØÓÒ Ò Ý µ(x, x) = 1 Ò
6 ËÓÑ ÜÑÔÐ ½ ¾ ½ ¾ ¾½ ½¾ ½ ¾ ½ ¾ ½¾ ¾½ ½¾ ¾½ ¾½ ¾ ½ ½¾ ½ ¾ ½ ¾ ½ ½¾ µ(12,2134) = 1 µ(,132) = 0 ½ ¾ µ(132, ) = 2
7 P : ½¾ ½¾ ¾½ ¾½ ½ ¾ ½ ¾ ¾½ ÀÐе Ì ÅÙ ÙÒØÓÒ Ó P ÚÒ ÌÓÖÑ ÈÐÔ i( 1) i C ÛÖ i Ý C i ÒÙÑÖ Ó Ò Ó ÐÒØ i Ø Ò P ØØ ÓÒØÒ ÓØ Ø ÑÒÑÐ Ò ÑÜÑÐ ÐÑÒØ º
8 P : ½¾ ½¾ ¾½ ¾½ Ó Ò ÐÒØ ½ ¾ ½ ¾ ¾½ ÀÐе Ì ÅÙ ÙÒØÓÒ Ó P ÚÒ ÌÓÖÑ ÈÐÔ i( 1) i C ÛÖ i Ý C i ÒÙÑÖ Ó Ò Ó ÐÒØ i Ø Ò P ØØ ÓÒØÒ ÓØ Ø ÑÒÑÐ Ò ÑÜÑÐ ÐÑÒØ º
9 P : ½¾ ½¾ ¾½ ¾½ Ó Ò ¾ ÐÒØ ½ ¾ ½ ¾ ¾½ ÀÐе Ì ÅÙ ÙÒØÓÒ Ó P ÚÒ ÌÓÖÑ ÈÐÔ i( 1) i C ÛÖ i Ý C i ÒÙÑÖ Ó Ò Ó ÐÒØ i Ø Ò P ØØ ÓÒØÒ ÓØ Ø ÑÒÑÐ Ò ÑÜÑÐ ÐÑÒØ º
10 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾
11 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾
12 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ ½¼
13 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ ½½
14 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ = 312 ½¾
15 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ = ½
16 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ = ½
17 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ = ½ ¾ ÁÒÓÑÔÓ Ð ½
18 ÔÖÑÙØØÓÒ ÓÑÔÓ Ð Ø Ø ÖØ ÙÑ Ó ÓÖ ÑÓÖ ÒÓÒÑÔØݵ ÔÖÑÙØØÓÒ ØÛÓ ½ ¾ = ½ ¾ ÁÒÓÑÔÓ Ð Ï ÛÖØ π = π 1 π 2 π n ÓÒÐÝ π i ÒÓÑÔÓ Ð ½
19 ÄØ σ = σ 1 σ 2 σ m Ò π = π 1 π 2 π n ½
20 ÄØ σ = σ 1 σ 2 σ m Ò π = π 1 π 2 π n ÄØ π >i = π i+1 π i+2 π n غ ÓÖ π i, σ >i, σ i ½
21 ÄØ σ = σ 1 σ 2 σ m Ò π = π 1 π 2 π n ÄØ π >i = π i+1 π i+2 π n غ ÓÖ π i, σ >i, σ i ÊÙÖÖÒ ÄØ l 0 Ò k 1 ÑÜÑÐ Ó ØØ Ö Ø σ 1 = σ 2 = = σ l ½ Ò = π 1 = π 2 = = π k ½º ÌÒ = µ(σ, π) = 0 l k 2 µ(σ k, π >k ) l = k 1 µ(σ >k, π >k ) µ(σ k, π >k ) l k ½
22 ÄØ σ = σ 1 σ 2 σ m Ò π = π 1 π 2 π n ÄØ π >i = π i+1 π i+2 π n غ ÓÖ π i, σ >i, σ i ÊÙÖÖÒ ÄØ l 0 Ò k 1 ÑÜÑÐ Ó ØØ Ö Ø σ 1 = σ 2 = = σ l ½ Ò = π 1 = π 2 = = π k ½º ÌÒ = µ(σ, π) = 0 l k 2 µ(σ k, π >k ) l = k 1 µ(σ >k, π >k ) µ(σ k, π >k ) l k ÜÑÔÐ µ(132, ) = 0 µ(132,126453) = µ(21,4231) = 2 µ(132,13524) = µ(21,2413) µ(132,2413) = 3 ( 1) = 4 ¾¼
23 ÌÓÖÑ ËÙÔÔÓ π ÅÒ 1 ½º ÄØ k ÑÜÑÐ 1 ØØ π Ó 1 = π 2 = = π k ÌÒ º µ(σ, π) = m k i=1 j=1 µ(σ i, π 1 )µ(σ >i, π >j ) ¾½
24 ÌÓÖÑ ËÙÔÔÓ π ÅÒ 1 ½º ÄØ k ÑÜÑÐ 1 ØØ π Ó 1 = π 2 = = π k ÌÒ º µ(σ, π) = m k i=1 j=1 µ(σ i, π 1 )µ(σ >i, π >j ) Á σ = a b Ò π = c d ÛÖ c, d ½, c d ÓÖÓÐÐÖÝ ØÒ µ(σ, π) = µ(a, c) µ(b, d) ¾¾
25 ÌÓÖÑ ËÙÔÔÓ π ÅÒ 1 ½º ÄØ k ÑÜÑÐ 1 ØØ π Ó 1 = π 2 = = π k ÌÒ º µ(σ, π) = m k i=1 j=1 µ(σ i, π 1 )µ(σ >i, π >j ) Á σ = a b Ò π = c d ÛÖ c, d ½, c d ÓÖÓÐÐÖÝ ØÒ µ(σ, π) = µ(a, c) µ(b, d) Á σ ÒÓÑÔÓ Ð Ó m = 1µ ØÒ ÓÖÓÐÐÖÝ µ(σ, π) = µ(σ, π 1 ) π = π 1 π 1 π 1 µ(σ, π) = µ(σ, π 1 ) π = π 1 π 1 π 1 ½ π 1 ½µ µ(σ, π) = 0 ÓØÖÛ ¾
26 ¾ ½ ¾
27 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÙÑ Ò Û ÙÑ º ÖØ = = (3124 1) 231 = ¾ ½ ¾
28 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÙÑ Ò Û ÙÑ º ÖØ = = (3124 1) 231 = ¾ ½ ¾
29 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ¾
30 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ¾
31 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ¾
32 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ¼
33 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ½
34 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ËÔÖÐ ¾
35 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ËÔÖÐ ÔÖÑÙØØÓÒ ÔÖÐ ÓÒÐÝ Ø ÚÓ Ø Ôع Ò ØÖÒ ¾½ Ò ½¾º
36 ÔÖÑÙØØÓÒ ÔÖÐ Ò ÒÖØ ÖÓÑ ½ Ý Ø ÖØ ÙÑ Ò Û ÙÑ º = = (3124 1) 231 = ¾ ½ ËÔÖÐ ÔÖÑÙØØÓÒ ÔÖÐ ÓÒÐÝ Ø ÚÓ Ø Ôع Ò ØÖÒ ¾½ Ò ½¾º ¾ ½ ÆÓØ ÔÖÐ
37 + + q : p : σ = π = 3,1,2,6,4,5,9,7,8,10,13,11,12 σ Ò π ÔÖе Ì ÔÖØÒ ØÖ Ó σ Ò π
38 ÍÒÔÖ ÓÙÖÖÒ Ó σ = 123 Ò π
39 ÌÓÖÑ Á σ Ò π Ö ÔÖÐ ÔÖÑÙØØÓÒ ØÒ µ(σ, π) = X (1) parity(x) ÛÖ Ø ÙÑ ÓÚÖ ÙÒÔÖ ÓÙÖÖÒ Ó σ Ò πº
40 ÌÓÖÑ Á σ Ò π Ö ÔÖÐ ÔÖÑÙØØÓÒ ØÒ µ(σ, π) = X (1) parity(x) ÛÖ Ø ÙÑ ÓÚÖ ÙÒÔÖ ÓÙÖÖÒ Ó σ Ò πº Ì ÓÑÔÙØ µ(σ, π) Ò ÔÓÐÝÒÓÑÐ ØÑ ÐØÓÙ ÆÓØ Þ Ó Ø ÒØÖÚÐ [σ, π] ÑÝ ÖÓÛ ÜÔÓÒÒØÐÐݺ Ø
41 ÌÓÖÑ Á σ Ò π Ö ÔÖÐ ÔÖÑÙØØÓÒ ØÒ µ(σ, π) = X (1) parity(x) ÛÖ Ø ÙÑ ÓÚÖ ÙÒÔÖ ÓÙÖÖÒ Ó σ Ò πº Ì ÓÑÔÙØ µ(σ, π) Ò ÔÓÐÝÒÓÑÐ ØÑ ÐØÓÙ ÆÓØ Þ Ó Ø ÒØÖÚÐ [σ, π] ÑÝ ÖÓÛ ÜÔÓÒÒØÐÐݺ Ø Á π ÔÖÐ ØÒ ÓÖÓÐÐÖÝ µ(σ, π) σ(π) ÛÖ σ(π) Ø ÒÙÑÖ Ó ÓÙÖÖÒ Ó σ Ò πº
42 ÌÓÖÑ Á σ Ò π Ö ÔÖÐ ÔÖÑÙØØÓÒ ØÒ µ(σ, π) = X (1) parity(x) ÛÖ Ø ÙÑ ÓÚÖ ÙÒÔÖ ÓÙÖÖÒ Ó σ Ò πº Ì ÓÑÔÙØ µ(σ, π) Ò ÔÓÐÝÒÓÑÐ ØÑ ÐØÓÙ ÆÓØ Þ Ó Ø ÒØÖÚÐ [σ, π] ÑÝ ÖÓÛ ÜÔÓÒÒØÐÐݺ Ø Á π ÔÖÐ ØÒ ÓÖÓÐÐÖÝ µ(σ, π) σ(π) ÛÖ σ(π) Ø ÒÙÑÖ Ó ÓÙÖÖÒ Ó σ Ò πº ÁÒ ÔÖØÙÐÖ Á π ÚÓ ½ ¾ ØÒ µ(σ, π) σ(π) ¼
43 ÁÒØÓÒ π[π ÄØ 1,..., π n Ø ÔÖÑÙØØÓÒ ÓØÒ Ý ÖÔÐÒ ] Ò i Ý π π i ÒÖÑÒØÒ Ø ÐØØÖ Ó π ØÖ i ØØ ØÝ Ó ÐÖÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π Ö i 1 Ø ÒØ Ò Ò ÑÐÐÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π ÔÖÑÙØØÓÒ i+1 º ½
44 ÁÒØÓÒ π[π ÄØ 1,..., π n Ø ÔÖÑÙØØÓÒ ÓØÒ Ý ÖÔÐÒ ] Ò i Ý π π i ÒÖÑÒØÒ Ø ÐØØÖ Ó π ØÖ i ØØ ØÝ Ó ÐÖÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π Ö i 1 Ø ÒØ Ò Ò ÑÐÐÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π ÔÖÑÙØØÓÒ i+1 º ÜÑÔÐ 2413[312,1,21,12] = ¾
45 ÁÒØÓÒ π[π ÄØ 1,..., π n Ø ÔÖÑÙØØÓÒ ÓØÒ Ý ÖÔÐÒ ] Ò i Ý π π i ÒÖÑÒØÒ Ø ÐØØÖ Ó π ØÖ i ØØ ØÝ Ó ÐÖÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π Ö i 1 Ø ÒØ Ò Ò ÑÐÐÖ ØÒ ØÓ ÓÖÖ ÔÓÒÒ ØÓ π ÔÖÑÙØØÓÒ i+1 º ÜÑÔÐ 2413[312,1,21,12] = Á π ÑÔÐ Ó ÐÒØ n Ò ÒÓÒ Ó π ÌÓÖÑ 1,..., π n π ØÒ ÓÒØÒ µ(π, π[π 1,..., π n ]) = n i=1 µ(1, π i )
46 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C
47 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C Û¹ÐÓ σ π C
48 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C Û¹ÐÓ σ π C ÐÓ ÙÖ Ð(C) Ó C Ø ÑÐÐ Ø ÙѹÐÓ Ò Û¹ Ì Ð ÓÒØÒÒ Cº ÐÓ
49 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C Û¹ÐÓ σ π C ÐÓ ÙÖ Ð(C) Ó C Ø ÑÐÐ Ø ÙѹÐÓ Ò Û¹ Ì Ð ÓÒØÒÒ Cº ÐÓ Ð({1}) Ø Ø Ó ÔÖÐ ÔÖÑÙØØÓÒ º
50 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C Û¹ÐÓ σ π C ÐÓ ÙÖ Ð(C) Ó C Ø ÑÐÐ Ø ÙѹÐÓ Ò Û¹ Ì Ð ÓÒØÒÒ Cº ÐÓ Ð({1}) Ø Ø Ó ÔÖÐ ÔÖÑÙØØÓÒ º ËÙÔÔÓ σ ÒØÖ ÓÑÔÓ Ð ÒÓÖ Û¹ ÌÓÖÑ ÄØ C ÒÝ Ø Ó ÔÖÑÙØØÓÒ º ÌÒ ÓÑÔÓ Ðº max{ µ(σ, π) : π Ð(C)} = max{ µ(σ, π) : π C}.
51 Ð C Ó ÔÖÑÙØØÓÒ ÙѹÐÓ σ, π C σ π C Û¹ÐÓ σ π C ÐÓ ÙÖ Ð(C) Ó C Ø ÑÐÐ Ø ÙѹÐÓ Ò Û¹ Ì Ð ÓÒØÒÒ Cº ÐÓ Ð({1}) Ø Ø Ó ÔÖÐ ÔÖÑÙØØÓÒ º ËÙÔÔÓ σ ÒØÖ ÓÑÔÓ Ð ÒÓÖ Û¹ ÌÓÖÑ ÄØ C ÒÝ Ø Ó ÔÖÑÙØØÓÒ º ÌÒ ÓÑÔÓ Ðº max{ µ(σ, π) : π Ð(C)} = max{ µ(σ, π) : π C}. ÓÑÔÙØØÓÒ Ó µ(σ, π) ÓÖ π Ð(C) Ò ÒØÐÝ Ì ØÓ Ø ÓÑÔÙØØÓÒ Ó Ø ÚÐÙ µ(σ, τ) ÓÖ τ Cº ÖÙ
52 ÅÓÖ Ö ÙÐØ ¼
53 µ( k¹1 2k...42, n 1 2n...42) = ( ) n+k 1 n k ÅÓÖ Ö ÙÐØ ½
54 µ( k¹1 2k...42, n 1 2n...42) = ( ) n+k 1 n k ÅÓÖ Ö ÙÐØ ÁÒ ÔÖØÙÐÖ µ(σ, π) ÙÒÓÙÒ ¾
55 µ( k¹1 2k...42, n 1 2n...42) = ( ) n+k 1 n k ÅÓÖ Ö ÙÐØ ÁÒ ÔÖØÙÐÖ µ(σ, π) ÙÒÓÙÒ Á π ÔÖÐ ØÒ µ(½, π) {0,1, 1}
56 µ( k¹1 2k...42, n 1 2n...42) = ( ) n+k 1 n k ÅÓÖ Ö ÙÐØ ÁÒ ÔÖØÙÐÖ µ(σ, π) ÙÒÓÙÒ Á π ÔÖÐ ØÒ µ(½, π) {0,1, 1} Á σ ÒÓÑÔÓ Ð Ò π = π 1 1 π 2 ØÒ µ(σ, π) = 0
57 ÇÔÒ ÔÖÓÐÑ
58 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) = 0
59 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) = 0 ÏÒ µ(σ, π) = σ(π)
60 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) = 0 ÏÒ µ(σ, π) = σ(π) ÏØ max{µ(½, π) : π = n}
61 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) 0 = ÏÒ µ(σ, π) σ(π) = ÏØ max{µ(½, π) : π n} = Á Ò σ Ö ÑÔÐ τ µ(σ, τ) 0
62 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) 0 = ÏÒ µ(σ, π) σ(π) = ÏØ max{µ(½, π) : π n} = Á Ò σ Ö ÑÔÐ τ µ(σ, τ) 0 Ò Û ÖÐØ µ(σ, ØÓ Ø ÓÑÓÐÓÝ Ó π) π] [σ, ¼
63 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) 0 = ÏÒ µ(σ, π) σ(π) = ÏØ max{µ(½, π) : π n} = Á Ò σ Ö ÑÔÐ τ µ(σ, τ) 0 Ò Û ÖÐØ µ(σ, ØÓ Ø ÓÑÓÐÓÝ Ó π) π] [σ, Ï ÒØÖÚÐ [σ, Ö ÐÐÐ π] ½
64 ÇÔÒ ÔÖÓÐÑ ÏÒ µ(σ, π) 0 = ÏÒ µ(σ, π) σ(π) = ÏØ max{µ(½, π) : π n} = Á Ò σ Ö ÑÔÐ τ µ(σ, τ) 0 Ò Û ÖÐØ µ(σ, ØÓ Ø ÓÑÓÐÓÝ Ó π) π] [σ, Ï ÒØÖÚÐ [σ, π] Ö ÐÐÐ ÓÒØÙÖ max π Sn µ(½, π) ÙÒÓÙÒ ÙÒØÓÒ Ó n ¾
65 ÌÓ Ó ØØÖ ÒÖÐ ØÓÖÑ ØØ ÓÐÚ ÓÑ Ó Ø ÓÔÒ Ò Ò ÙÒÝ ÓÑ Ó ØÓ Ò Ø ÓÚ Ö ÙÐØ ÔÖÓÐÑ º
F(jω) = a(jω p 1 )(jω p 2 ) Û Ö p i = b± b 2 4ac. ω c = Y X (jω) = 1. 6R 2 C 2 (jω) 2 +7RCjω+1. 1 (6jωRC+1)(jωRC+1) RC, 1. RC = p 1, p
ÓÖ Ò ÊÄ Ò Ò Û Ò Ò Ö Ý ½¾ Ù Ö ÓÖ ÖÓÑ Ö ÓÒ Ò ÄÈ ÐØ Ö ½¾ ½¾ ½» ½½ ÓÖ Ò ÊÄ Ò Ò Û Ò Ò Ö Ý ¾ Á b 2 < 4ac Û ÒÒÓØ ÓÖ Þ Û Ö Ð Ó ÒØ Ó Û Ð Ú ÕÙ Ö º ËÓÑ Ñ ÐÐ ÕÙ Ö Ö ÓÒ Ò º Ù Ö ÓÖ ½¾ ÓÖ Ù Ö ÕÙ Ö ÓÖ Ò ØÖ Ò Ö ÙÒØ ÓÒ
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