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1 A. Ya.
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3 iii 3 A. Ya. Khinchin 3 State Press for Physics and Mathematics B.G. 0 A. N. Khovanskii Prilozhenie tsepnykh drobei i ikh obobshcheniǐ kvoprosam priblizhennogo analiza B. V. Gnedenko
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5 v 2 D. A. Glava, V. A. Venkov, I. V. Arnold A.
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7 vii a 0 + a + a 2 + a i (i ) 2
8 viii A.
9 ix
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11 a 0 + a + a 2+ regular simple continued fraction a 0, a, a 2,... a 0, a, a 2,... a, a 2,... a 0 element () a 0 + a + a a n (2) finite n n n + 2 () infinite
12 2 () [a 0 ; a, a 2,...] (3) (2) [a 0 ; a, a 2,..., a n ] (4) 0 k n k s k := [a 0 ; a, a 2,..., a k ] (4) segment k 0 s k (3) s k r k := [a k ; a k+,... a n ] (4) remainder r k := [a k ; a k+,...] (3) [a 0 ; a, a 2,..., a n ] = [a 0 ; a, a 2,..., a k, r k ] (0 k n) (5) [a 0 ; a, a 2,...] = [a 0 ; a, a 2,..., a k, r k ] (0 k n) r k
13 [a 0 ; a, a 2,..., a n ] a 0, a,..., a n 2 P (a 0, a,..., a n ) Q(a 0, a,..., a n ) p/q canonical 0 [a 0 ] = a 0 a 0 / n (5) n [a 0 ; a,..., a n ] = [a 0 ; r ] = a 0 + r r = [a ; a 2,..., a n ] (n ) r = p q [a 0 ; a,..., a n ] = a 0 + q p = a 0p + q p [a 0 ; a,..., a n ] [a 0 ; a,..., a n ] = p q r = [a ; a 2,..., a n ] = p q
14 4 p = a 0 p + q, q = p (6) α = [a 0 ; a, a 2,...] s k = [a 0 ; a, a 2,..., a k ] p k /q k α k convergent approximant n α p n q n = α n + 0,, 2,..., n. k 2 p k = a k p k + p k 2 q k = a k q k + q k 2 (7). k = 2 (7) k < n [a ; a 2,..., a n ] r p r/q r (6) p n = a 0 p n + q n q n = p n p n = a n p n 2 + p n 3 q n = a n q n 2 + q n 3
15 2. 5 a n a n [a ; a 2,... a n ] a 0 a (6) p n = a 0 (a n p n 2 + p n 3) + (a n q n 2 + q n 3 ) = a n (a 0 p n 2 + q n 2) + (a 0 p n 3 + q n 3 ) = a n p n + p n 2 q n = a n p n 2 + p n 3 = a n q n q n 2 (7) n a n 2. p = q = 0 (7) k = 2. k 0 q k p k p k q k = ( ) k (8). (7) q k 2 p k 2 q k p k p k q k = (q k p k 2 p k q k 2 ) q 0 p p 0 q =. k p k q k p k q k = ( )k q k q k (9)
16 6 3. k q k p k 2 p k q k 2 = ( ) k a k. (7) q k 2 2 p k q k p k 2 p k q k 2 = a k (q k p k 2 p k q k 2 ) = ( ) k a k 2. k 2 p k 2 q k 2 p k q k = ( )k a k q k q k 2 (0) (0) 2 a (9) 4. α α α α 2 5. k ( k n) [a 0 ; a, a 2,... a n ] = p k r k + p k 2 q k r k + q k 2 () p i, q i, r i
17 3. 7. (5) [a 0 ; a, a 2,..., a n ] = [a 0 ; a, a 2,..., a k, r k ] (k ) p k /q k k p k /q k (7) p k = p k r k + p k 2, q k = q k r k + q k 2 6. k q k q k = [a k ; a k,..., a ]. k = q /q 0 = a k > (7) q k q k 2 = [a k ; a k 2,..., a ] (2) q k = a k q k + q k 2 q k = a k + q [ k 2 = a k ; q ] k q k q k q k 2 (5) (2) q k q k = [a k ; a k,... a ] 3 [a 0 ; a, a 2,...] (3) p 0, p,..., p k,... (4) q 0 q q k
18 8 (4) α α (3) α = [a 0 ; a, a 2,...] (3) converge (4) (3) 7. (3) (3). (3) p k /q k r n p k /q k () p n+k = [a 0 ; a, a 2,..., a n+k ] = p p k n q + p k n 2 (k = 0,,...) (5) q p n+k q k n q + q k n 2 r n k p k /q k p n+k /q n+k α = p n r n + p n 2 q n r n + q n 2 (6) (5) p k /q k (6) () r n
19 (3) α k 0 α p k q k < q k q k+ (9) k = n k < n α = [a 0 ; a, a 2,..., a n ] α = p n /q n α (3) α (3) α 7 (3) i a i > 0 0. (3) a n (7) n= (9) q k q k+ (k ) (8) (8) (7) (7) 2 q k > q k 2 (k ) k 0 q k > 0 q 0 = q = a (7) 2 k > q k > 0
20 0 k q k > q k q k > q k 2 (7) 2 q k < q k q k + q k 2 k a k < (7) k k 0 q k 2 q k < a k 2 a k < q k q k < ( + a k )q k < a k k k 0 q k < a k q l l < k l k 0 q l q k < q s ( a k )( a l ) ( a r ) (9) k > l > > r k 0 s < k 0 (7) n=k 0 ( a n ) λ ( a k )( a l ) ( a r ) n=k 0 ( a n ) = λ q 0, q,..., q k0 Q (9) q k < Q λ (k k 0 )
21 3. q k+ q k < Q2 λ 2 (k k 0 ) (8) (7) q k > q k 2 k 2 q 0, q c k 0 q k c (7) 2 q k q k 2 + ca k (k 2) q 2k q 0 + c q 2k+ q + c k n= k n= a 2n a 2n+ 2k+ q 2k + q 2k+ > q 0 + q + c k q k + q k > c k n= a n n= k k q k q k 2 c k n= a n c k q k q k > c2 a n 2 n= (7) (8) 0 a n
22 2 4 a, a 2,... a 0 0 (a n ) r n = a n + n [a 0 ; a, a 2,..., a n, ] (n ) [a 0 ; a, a 2,..., a n + ] 0 [] 2 5 p, q, p 0, q 0 (7). irreducible (8) p n q n q n p n p n q n (7) 2 k 2 q k > q k q, q 2,... q k,... q k 2. 2 k 2. k 2 q k 2 k 2 q k a k q k + q k 2 q k + q k 2 2q k 2 2 q k k
23 4. 3 q 2k 2 k q 0 = 2 k, q 2k+ 2 k q 2 k intermediate fractions k 2 i p k (i + ) + p k 2 q k (i + ) + q k 2 p k i + p k 2 q k i + q k 2 ( ) k [q k (i + ) + q k 2 ][q k i + q k 2 ] k i 0 p k 2 q k 2, p k 2 + p k q k 2 + q k, p k 2 + 2p k q k 2 + 2q k,..., p k 2 + a k p k q k 2 + a k q k = p k q k (20) k k 4 a k > intermediate fractions 2 mediant a 2 b c d a + c b + d. 2. a/b c/d bc ad 0 a + c b + d a bc ad = b b(b + d) 0, a + c b + d c ad bc = d b(b + d) 0
24 4 (20) (20) p k 2 /q k 2 p k /q k p k /q k 4 p k q k p k /q k p k 2 /q k 2 α p k /q k p k /q k p k 2 /q k 2 p k /q k α (20) α p k /q k (p k + p k 2 )/(q k + q k 2 ) p k /q k α p k 2 /q k 2 p k /q k a k p k /q k α 2 p k /q k p k /q k α α p k 2 /q k 2 p k /q k p k /q k α p k 2 /q k 2 α (p k + p k )/(q k + q k ) α α (p k /p k+ )/(q k + q k+ ) p k /q k α α p k /q k α p k > p k + p k+ p k q k + q k+ = q k ( k + q k+ ) q k q k α = p k + p k+ q k + q k+ p k q k = p k+2 q k+2, a k+2 = α
25 k 0 α p k > q k (q k+ + q k ) q k (2) α (p k /q k ) 9
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27 α α α. α a 0 α α = a 0 + r (22) r r = α a 0 < r > r n a n r n r n+ r n = a n + r n+ (23) r n r n > n (22) α = [a 0 ; r ] α = [a 0 ; a, a 2,..., a n, r n ] (24) (5) (23) α = [a 0 ; a, a 2,..., a n, a n, r n+ ] i a i > 0
28 8 2 (24) n r, r 2,..., r n α r n r n = a/b r n a n = a ba n b r n a n < c < b (23) = c b r n+ = b c c 0 r n r n r n+ r n r, r 2,... r n = a n (24) α a n = r n > α r n [a 0 ; a, a 2,..., a n ] = p n q n p n /q n q n > 0 (24) (6) α = p n r n + p n 2 q n r n + q n 2 (n 2) p n q n = p n a n + p n 2 q n a n + q n 2 α p n = (p n q n 2 q n p n 2 )(r n a n ) q n (q n r n + q n 2 )(q n a n + q n 2 ) α p n < (q n r n + q n 2 )(q n a n + q n 2 ) < q n q 2 n pn q n α n
29 5. 9 [a 0 ; a, a 2,...] α α α 4 α = [a 0 ; a, a 2,...] = [a 0; a, a 2,...] 2 [x] x a 0 = [α] a 0 = [α] a 0 = a 0 a i = a i (i = 0,, 2,..., n) } p i = p i (i = 0,, 2,..., n) q i = q i (6) α = p nr n+ + p n = p nr n+ + p n q n r n+ + q n q nr n+ + = p nr n+ + p n q n q n r n+ + q n r n+ = r n+ a n+ = [r n+ ] a n+ = [r n+] a n+ = a n+ 2 a n+ r n = a n + a n [r n ] 0
30 20 2 0
31 α α 9 3 q n (q n + q n+ ) < α p n q n q n q n+ α α 2 α M. V. Ostrogradskiǐ E. Ya. Remez O znakoperemennykh ryadakh, kotorye mogut byt svyazany s dvumya algorifmami M. V. Ostrogradskogo dlya priblizheniya irratsional nykh chisel M. V. Ostrigradskiǐ 2
32 22 2 a/b b > 0 α 0 < d b a/b c/d α c > α a d b α best approximation 5. α 2 p = q = , 0,, 2, 3, 4 3. a/b α a/b a 0 a/b < a 0 a 0 / a/b b a/b α a/b a b a 0 + a 0 < (a/b) < a 0 + a/b = a 0 a/b = a 0 + a 0 / = p 0 /q 0 (a 0 + )/ = (p 0 + p )/(q 0 + q ) α a/b α 2 k r k > 0 0 r < a k+ k = 0 r < a p k r + p k q k r + q k, Uspekhi matematicheskikh nauk, 6, No. 5(45), (95) Remez Ostrogradskiǐ B.G.
33 6. 23 p k (r + ) + p k q k (r + ) + q k a b p kr + p k q k r + q k < p k (r + ) + p k p kr + p k q k (r + ) + q k q k r + q k = {q k (r + ) + q k }{q k r + q k } a b p kr + p k q k r + q k = m b(q k r + q k ) m b(q k r + q k ) < {q k (r + ) + q k }{q k r + q k } q k (r + ) + q k < b b p k (r + ) + p k q k (r + ) + q k (25) p k r + p k q k r + q k (26) α 3 α a/b α a/b (25) (26) 5 a/b α α (a/b) bα a b a/b α
34 24 2 b 5 best approximation of the first kind a/b b > 0 α 2 best approximation of the second kind c/d a/b 0 < d b dα c > bα a 2 α a/b bα a 2 α c α a ( c d b d a ) b, d b 3 dα c bα a a/b < 3 5 ( < 3) a/b α = [a 0 ; a, a 2,...]
35 p k /q k a/b < a 0 α a 0 < α a bα a ( b) b a/b 2 a/b a 0 a/b 2 p k /q k p k+ /q k+ p /q a b p k q k bq k a b p k q k < p k p k q k q k = q k q k b > q k (27) α a p k+ b a q k+ b bα a q k+ q k α p k q k+ bq k+ q k α p k bα a (28) (27) (28) a/b 2 2 a/b > p /q α a > p b a q b bq bα a > q = a α a 0 a
36 26 2 bα a > α a 0 ( b) a = < ( < 2) α = a 0 + 2, p 0 = a 0 q 0 α = a p 0/q 0 = a 0 / α (a 0 + ) = α a 0 2. yα x (29) y, 2,..., q k x y 0 x (29) y y y 0 x 0 y 0 a x x α x 0 = α x 0 (x 0 x 0) y 0 y 0
37 6. 27 α = x 0 + x 0 2y 0 x 0 + x 0 = lp 2y 0 = lq l > l < 2 q < y 0, α = p, qα p = 0 q y 0 l = 2 q = y 0 qα p = y 0 α p = 0 < y 0 α x 0 x 0 α α = p n q n, p n = x 0 = x 0, q n = 2y 0 = a n q n + q n 2, a n 2 a n > 2 a n = 2 n > q n < y 0 q n α p n = q n = 2y 0 2 y 0α x 0 y 0 n = a n = 2 α = a y 0 = y 0 x 0 x 0 /y 0 α 2 bα a y 0 α x 0, a b x 0 y 0, b y 0 x 0 y 0 6 x 0 = p s, y 0 = q s (s k) s = k s < k q s α p s >, q k α p k q s + q s+ q k + q k q k+
38 28 2 p s = x 0 q s = y 0 q s α p s q k α p k q k + q k < q k+ q k+ < q k + q k q k 7 Huygens 2 α α α 7 α (p k /q k ) α (p k /q k ) q k 9 3 α p k q k < qk 2 (30) 3 α = p k /q k q k+ 9 (30)
39 7. 29 q k f(q k ) n f(n) < n 2 (30) k α ε > 0 α p k > ε q k q 2 k α = [0; n,, n] = n + n(n + 2) p =, q = n, p 3 = n +, q 3 = n(n + 2) α p = p 3 p q 3 = n(n + 2) = q 2( + 2 n ) q n q + 2 n > ε α p > ε q q 2 k α 8. α k > 0 2 α p k <, α p k < q k 2q 2 k q k 2q 2 k
40 30 2. α p k /q k p k /q k α p k q k + α p k q k = p k p k q k q k = < q k q k+ 2qk 2 + 2qk 2 /qk 2 /q2 k q k = q k 9. α a < b 2b 2 α. 6 a/b α 2 dα c bα a < 2b ( d > 0, α c < d 2bd c d a ) b c d a α c + α a < b d b 2bd + 2b 2 = b + d 2b 2 d c/d a/b c d a b bd (3) ab < b + d 2b 2 d d > b a/b α (3)
41 α k > 3 α p k q k <, 5q 2 α p k k q k <, 5q 2 k α p k 2 q k 2 < 5q 2 k 2. k qk 2 = ϕ k, q k ϕ k + r k = ψ k 2. k 2 ψ k 5 ψ k 5 ϕ k > 5 2. = q n = a n + ϕ n (32) ϕ n+ q n r n = a n + r n+ + = ϕ n + r n = ψ n ϕ n+ r n+ ϕ k + r k 5, ϕ k + r k 5 ( ( ) 5 5 ϕ k ) ϕ k 5 ) 5 (ϕ k + ϕk > 0 4 I. I. Zhogin Variant dokazatel stva odnoi teoremy iz teoriǐtsepnykh drobei, Uspekhi matematicheskikh nauk, 2, No. 3, (957) ϕ k
42 32 2 ϕ k > 0 ( ) ϕ k < ϕ k < 5 2, ϕ k > 2 α p n (n = k, k, k 2) 5q 2 n q n (6) α p n = p n r n+ + p n p n q n r n+ + q n q n = q n q n (q n r n+ + q n ) = q 2 n(r n+ + ϕ n+ ) = q 2 nϕ n+ ϕ n+ 5 (n = k, k, k 2) 5 5 ϕ k >, ϕ k+ > 2 2 (32) a n = 2 5 ϕ k < = ϕ k α = [;,,...] α = + (/r ) r = α α = + α, α2 α = 0
43 8. 33 α = n r n = α α = p kα + p k q k α + q k α p k q k = q k (q k α + q k ) = ( α + q k 6 q 2 k q k q k = [;,,..., ] α (k ) q k q k = α + ε k = α p k q k = ( 5+ qk 2 q k ) 5 + ε k (ε k 0 as k ) ) = 2 + ε k qk 2( 5 + ε k ) c < (/ 5) k α p k > c q k q 2 k 20 / 5 α k α α = 2 ( 5 + ) k
44 34 2 c α α p q < c q 2 (33) p, q q > 0 2. α c (/ 5) (33) p, q q > 0 c < (/ 5) α (33). 20 α a/b 2 n =, 2, 3,... q = nb p = na c < (/ 5) 7 α = = [;,,...] 2 p q q > 0 (33) 9 p/q α 7 c < (/ 5) (33) α q/( 5q 2 ) 5 5
45 ϕ(q) α p q < ϕ(q) p, q q > 0 α. a k+ > qk 2ϕ(q k) α a 0 k 0 α p k q k < = q k q k+ q k (a k+ q k + q k ) a k+ qk 2 < ϕ(q k ) q k (q k + q k+ ) < α p k q k q k q k+ ( ) < qk 2 a k+ + + q α p k k q k qk 2a k+ + q k q q k k q k (a k+ + 2) < α p k q k qk 2a k+ (34) a 0, a,..., a k a k+ p k /q k α (34)
46 = [;,,...] 2 a α c α p q < c q 2 p, q q > 0 α c > 0 (33) /q 2. α c k a k+ > c (34) 2 α p k < c q k q 2 k k 2 M > 0 a k < M (k =, 2,...) (34) k 0 α p k > (M + 2) q k q 2 k
47 8. 37 p q q > 0 k q k < q q k α p q α p ( ) k 2 q k > qk 2(M + 2) = q q 2 (M + 2) q k ( ) 2 ( ) 2 qk q k > q 2 = (M + 2) q k q 2 (M + 2) a k q k + q k 2 > q 2 (M + 2) (a k + ) 2 > (M + 2)(M + ) 2 q 2 c < (M + 2)(M + ) 2 (33) p q q > 0 α (p/q) 6 qα p 2 x, y αx y = 0 (35) α x = y = 0 x y αx y (35) 2 / 5 C αx y < C (36) x x y y > 0 (35) αx y = β (37)
48 38 2 β x y x y αx y β P. L. Chebyshev β x y αx y β α αx y α b > 0 a α = a/b β = /2b x, y αx y β = 2(ax by) 2b 2b 2(ax by) α αx y β 24. α β αx y β < 3/x x y x > / p/q α α = p q + δ q 2 (0 < δ < ) (38) β t qβ t 2 6 Printsip Dirikhle v teorii diofantovykh, Uspekhi matematicheskikh nauk, priblizheniǐ 3, No. 3, 7 8 (948) O zadache Chebysheva, Izvestiya akad. nauk SSSR, ser. matem., 0, (946) B.G.
49 8. 39 β = t q + δ 2q ( δ ) (39) p q ± x y q 2 x < 3q, px qy = t 2 r/s p/q qr ps = ε = ±, k q(εrt) p(εst) = tε 2 = t p kq εst) q(kp εrt) = t k q 3q x = kq εst < 2 2 (38) (39) αx y β = xp q + xδ q 2 y t q δ 2q = xδ q 2 δ 2q < x q 2 + 2q q > 2 3 x αx y β < 9 4x + 3 4x = 3 x q x /2 x (37) x y αx y β 24 n α β x > 0 y αx y β < n (40)
50 /n N n α β (40) x N 24 x x β = n α x y 0 < x n, αx y < n. α q/b 0 < b < n x = b y = a α n k q k n < q k+ p k /q k α k α p k < q k q k+ q k n q k αq k p k < n, 0 < q k n (37) α C 0 < x Cn, αx y β < n
51 8. 4 n β x y C = C α α αx y β β α αx y β 24 β x y α αx y αx y β α 26. n β 2 x, y x > 0 x Cn, αx y β < n C α. α = [a 0 ; a, a 2,...] a i < M i =, 2,... m β α p k /q k k q k m < q k+ α p k q k < < q k q k+ mq k α = p k + δ q k mq k ( δ ) (4) t βq k t 2 β = t q k + δ 2q k ( δ ) (42)
52 x y xp k yq K = t, 0 < x q K (43) (4) (42) (43) αx y β = xp k y t + xδ δ q k q k mq k 2q k = xδ δ mq k 2q k < x + mq k 2q k m + ( ) qk+ 2q k+ q k < m + 2m (a k+ + ) < m + M + 2m = M + 3 2m m n m = 2 (M + 3) m > x y 0 < x q k m = M + 3 n 2 αx y β < n 2 α a k 23 ε q > 0 p α p q < ε2 q 2 α = p q + δε2 q 2 ( δ < ) n = q/ε β = /2q x y 0 < x Cn αx y β = xp q y 2q + xδε2 q 2 = 2(xp yq) 2q > 2(xp yq) 2q xε2 q 2 2q Cε q + xδε2 q 2 = 2Cε 2q = 2Cε 2ε C ε [( 2Cε)/2ε] > x y 0 < x Cn n αx y β > n
53 (37) normal C α /n x < Cn n β = supernormal ε > 0 n /n x < εn x > 0 y 26 C β C β 2 9 f(x) = a 0 + a x + + a n x n (44) a 0, a,..., a n n α α = a/b bx a = 0 n f(x) = 0 n x transcendental e π Liouville 27. n α C
54 44 2 p q q > 0 α p q > C q n. α (44) f(x) = (x α)f (x) (45) f (x) n f (α) 0 f (α) = 0 f (x) x α f(x) (x α) 2 f (x) x α f (α) = 0 f (x) α n f (α) 0 δ f (x) 0 (α δ x α + δ) p q q > 0 α (p/q) δ f (p/q) 0 x = p/q (45) p q α = f( p q ) f ( p q ) = a p 0 + a ( q ) + + a n( p q )n f ( p q ) = a 0q n + a pq n + + a n p n q n f ( p q ) α = p/q α (α δ, α + δ) f (x) M α p q Mq n α p q > δ α p q > δ q n
55 C δ /M q > 0 p α p q > C q n C > 0 n p q q > 0 α p q C q n (46) α a 0, a,..., a k p k /q k a k+ > q k k α p k q k < < q k q k+ qk 2a k+ < q k+ k C > 0 n k (46) α α C α p q < C q 2 p q q >
56 46 2 Lagrange 2 2 α = [a 0 ; a, a 2,...] periodic k 0 h k k 0 a k+h = a k 0 α = [a 0 ; a, a 2,..., a k0, a k0, a k0 +,..., a k0 +h ] (47) (47) r k+h = r k (k k 0 ) (6) k k 0 α = p k r k + p k 2 q k r k + q k 2 = p k+h r k+h + p k+h 2 q k+h r k+h + q k+h 2 = p k+h r k + p k+h 2 q k+h r k + q k+h 2 (48) pk r k + p k 2 q k r k + q k 2 = p k+h r k + p k+h 2 q k+h r k + q k+h 2 r k 2 2 (48) α 2 α 2 aα 2 + bα + c = 0 (49) α n α = p n r n + p n 2 q n r n + q n 2
57 (6) r n A n r 2 n + B n r n + C n = 0 (50) A n, B n, C n A n = ap 2 n + bp n q n + cq 2 n, B n = 2ap n p n 2 + b(p n q n 2 + p n 2 q n ) + 2cq n q n 2, C n = ap 2 n 2 + bp n 2 q n 2 + cq 2 n 2 (5) C n = A n (52) B 2 n 4A n C n = (b 2 4ac)(p n q n 2 q n p n 2 ) 2 = b 2 4ac (53) (50) n (49) α p n < q n (5) q 2 n p n = αq n + δ n q n ( δ n < ) ( A n = a αq n + δ ) ( n + b αq n + δ ) n q n + cqn 2 q n q n = (aα 2 + bα + c)qn 2 + 2aαδ n + a δ2 n qn bδ n (49) A n = 2aαδ n + a δ2 n + bδ n < 2 aα + a + b q 2 n (52) C n = A n < 2 aα + a + b
58 48 2 (50) A n C n n (53) B n n (50) r n k h r k = r k+h α 2 2 2
59 49 3 p/q /q () (2) 2 measure arithmetic of the continuum
60 50 3 a 4 = 2 q measure theory of continued fractions 0 a 0 = 0 2 α = [a 0 ; a, a 2,...] a n α α a n = a n (α) a 0 = 0 α = [a 0 ; a, a 2,...] α = [a, a 2,...]
61 2. 5 [a, a 2,...] = a + a α a α = a + a 2 + a = [/α] a /α a =, a = 2, a = 3, a = k, α < 2 2 < α 2 α < 3 3 < α 2 3 α < 4 4 < α 3 k α < k + k + < α k a = a (α) /α α α 0 3. a k + < α k intervals of the first rank 0 a (α) dα = + k= ( k k ) = k + k= k +
62 : a 2 (α) k + < α k a = k α = k + r 2 r 2 < a 2 = [r 2 ] r 2 α /(k + ) /k a 2 =, a 2 = 2, a 2 = 3, r 2 < 2 k + < α 2 r 2 < 3 k r 2 < 4 k + 3 k + 2 < α k + 3 < α k + 4
63 2. 53 a 2 = l, l r 2 < l + k + l < α k + l+ a 2 (α) : a 2 (α) ( k +, l ) k + l+ 2 2 a = k a 2 = l 2 a = k 2 a = k, a 2 = l n a (α), a 2 (α),..., a n (α) a = k, a 2 = k 2,..., a n = k n (54)
64 54 3 n J n J n a n+ (α) α α = [k, k 2,..., r n, r n+ ] (55) r n+ r n+ < r n+ < (55) (54) α J n a n+ = [r n+ ] n a n+ (α) α p k /q k α = p nr n+ + p n q n r n+ + q n α J n r n+ p n, q n, p n, q n a, a 2,..., a n J n r n+ = r n+ J n p n + p n q n + q n p n q n α p n = p nr n+ + p n p n ( ) n = q n q n r n+ + q n q n q n (q n r n+ + q n ) α (, ) r n+ r n+ a n+ J n = ( pn, p ) n + p n q n q n + q n α α J n a n+ (α), 2, 3,... J n n + n n a n (α) (0, ) 0 n n + n n a n+ (α) n = 0,, 2,... n +
65 2. 55 n a = k, a 2 = k 2,..., a n = k n n a m = k, a m2 = k 2,..., a ms = k s (0, ) a n = k (0, ) ( ) n, n 2,..., n s E k, k 2,..., k s a n = k, a n2 = k 2,..., a ns = k s n i k i n i ( ), 2,..., n E k, k 2,..., k n n a i = k i (i =, 2,..., n) ( ) n,..., n l, n l, n l+,..., n s E k,..., k l, k l k l+,..., k s ( ) n,..., n l, n l+,..., n s = E k,..., k l, k l+,..., k s k l = (56)
66 56 3 ME E n ( ), 2,..., n J n = E k, k 2,..., k n n + ( ) J (s) n+ = E, 2,..., n, n + k, k 2,..., k n, s J n p n q n p n + p n q n + q n p k /q k [k, k 2,..., k n ] k J (s) n+ a n+ = [r n+ ] = s s r n+ < s + J n α = p nr n+ + p n q n r n+ + q n s r n+ < s + n+ n+ p n s + p n p n(s + ) + p n q n s + q n q n (s + ) + q n J (s) MJ n = p n p n + p n q n q n + q n = q n (q n + q n ) = qn( 2 + q n q n ) J (s)
67 2. 57 MJ (s) n+ = p n s + p n p n(s + ) + p n p n s + q n q n (s + ) + q n = [q n s + q n ][q n (s + ) + q n ] = ) ( qns ( q n sq n + s + q n MJ (s) n+ = + ( MJ n s 2 + qn q n q n ) ( sq n + s + qn sq n ) + q n q n + qn sq n + s + q n sq n < 3 sq n ) (s) MJ n+ < < 2 3s2 MJ n s 2 (57) n a n+ = s n + /s 2 (57) k, k 2,..., k n n s MJ n 3s 2 < MJ (s) n+ < 2MJ n s 2 n J n k, k 2,..., k n ( ) (s) MJn =, MJ n+ = ME n + s ( ) 3s 2 < ME n + < 2 s s 2 s /3s 2 2/s 2 /s 2
68 (0, ) 0. (0, ) M E M J n n a i < M (i =, 2,..., M) (58) J n a n+ = k n + n+ (57) J (k) MJ (k) n+ > 3k 2 MJ n M k M J (k) n+ > 3 MJ n k 2 k M M > 3 MJ n k<m i= (M + i) 2 > 3 MJ du n M+ u 2 = 3(M + ) MJ n k= J (k) n+ > { τ = J (k) n+ = J n } MJ n = τmj n (59) 3(M + ) 3(M + ) M > 0 τ < E (n) M (58) (0, ) (59) n J n E (n+) M τmj n n E (n) M
69 3. 59 (58) E (n+) M (59) E (n) n ME (n+) M < τme (n) M (60) ME (n+) M < τ n ME () M (n ) τ < ME (n) M 0 (n ) E M ME M = 0 E M = E M= ME ME M = 0 M= E (n) M M E M E 2 23 α p q < C q 2 (6) α (6)
70 ϕ(n) n n= /ϕ(n) a n = a n (α) ϕ(n) (62) α n n= /ϕ(n) α n ϕ(n) M J n+m m + n E M a m+i < ϕ(m + i) (i =, 2,..., n) (63) a, a 2,..., a m 29 (59) M k<ϕ(m+n+) J (k) m+n+ < { 3( + ϕ(m + n + )) } MJ m+n (63) m + n (0, ) E m,n { } ME m,n+ < ME m,n 3( + ϕ(m + n + )) ME m,n < ME m, i=2 { } 3( + ϕ(m + i)) n= /ϕ(n) i=2 3( + ϕ(m + i))
71 3. 6 m i=2 3( + ϕ(m + i)) n 0 m ME m,n 0 (n ) a m+i < ϕ(m + i) (i =, 2,...) α E m,n (n =, 2,...) E m 0 E + E E m + = E ME = 0 (62) α m E m E n= /ϕ(n) J n n a n+ = k n + (57) 2 M k>ϕ(n+) J (k) n+ < 2MJ n 2MJ n MJ (k) n+ < 2 k 2 MJ n k>ϕ(n+) i=0 k 2 {ϕ(n + ) + i} 2 { < 2MJ n ϕ(n + ) + ϕ(n+) } du u 2 = 4MJ n ϕ(n + ) J (k) n+
72 62 3 (0, ) a n ϕ(n) F n n MF n+ < 4 ϕ(n + ) MJn = F, F 2,..., F n,... (0, ) F n F 2 MF = 0 F n (62) B n q n = q n (α) < e Bn 4 2 α q n n 3 α 2 a A < a < A (0, ) α n a < n q n < A 2 F m n=m Fn n=m MFn m
73 4. 63 γ n qn γ (n ) q n. (0, ) a a 2... a n g E n (g) n > 0, g 0 n 2 p n p n + p n q n q n + q n = q n (q n + q n ) < < (a n a n... a 2 a ) 2 q n > q n q n q 2 n q n > a n a n... a 2 a ME n (g) < a 2 na 2 n... a2 2 a2 a a 2...a n g (64) a a 2... a n g a, a 2,... a n n a 2 i= i = n ) ( + ai n a i (a i + ) 2n a i (a i + ) i= = 2 n n i= ai+ a i dx i x 2 i i= = 2 n a+ a a2+ a 2... an+ a n dx dx 2... dx n x 2 x x2 n Zur metrischen Kettenbruchtheorie, Composite Mathematica, 3, No. 2, (936) P. Lévy γ ln γ = π 2 /(2 ln 2) P. Lévy, Théorie de l addition des variables aléatoires, Paris, 937, p. 320 B.G.
74 64 3 { n } 2 n J n (g) a a 2...a n g a 2 i= i J n (g) n dx dx 2... dx n... x 2 x x2 n x i (i =, 2,..., n) x x 2... x n g g x i < i =, 2,..., n { } n dx J n (g) = x 2 = (65) g > J n (g) = g n i=0 (ln g) i i! (66) n = 0 dx x 2 = g n = k ( ) dx k+ g J k+ (g) = x 2 J k = g J k (u) du k+ x k+ g 0 = { g } J k (u) du + J k (u) du g 0 J k (u) (65) 2 (66) n = k, g { } J k+ (g) = k (ln g) i+ + = k (ln g) i g (i + ) g i! i=0 i=0
75 4. 65 ME n (g) < 2n g n i=0 (ln g) i i! A > g = e An n ME n (e An ) < e n(ln 2 A) i=0 (An) i (An) n n! Stirling ME n (e An ) < e n(ln 2 A) n (An)n n! < C e n(ln 2 A) n(an) n n n e n n < C 2 ne n(a ln A ln 2 ) C C 2 A A ln A ln 2 > 0 ME n (e An ) n ME n (e An ) n= 0 (0, ) E n (e An ) (0, ) n a a 2... a n < e An i! q n = a n q n + q n 2 < 2a n q n q n < 2 n a n q n... a 2 a
76 66 3 n q n < 2 n e An = e Bn B = A + ln f(x) x xf(x) α p q < f(q) q α p q q > 0 c c (67) f(x) dx (68) (68) (67) α p q q > 0 32 α p q < q 2 ln q α p q < q 2 ln +ε q ε > 0 0. (68) ϕ(x) = e Bx f(e Bx )
77 4. 67 B 3 A > a > 0 A a ϕ(x) dx = b BA Ba f(u) du A ϕ(x) ϕ(n) n= 30 a i+ ϕ (i) i α p i a i+qi 2 q i q i+ q i 3 i q i < e bi i > ln q i B (69) ( ) α p i ϕ ln qi q i B = f(q i) q i q 2 i i i ϕ(i) q 2 i 2 (68) f(n) n= (0, ) k α k n < f(n) n (69)
78 68 3 α E n E n /n, 2/n,..., (n )/n 2f(n)/n (0, f(n)/n) ( f(n)/n, ) ME n 2f(n) < f(n) > 2 ME n n= (0, ) α E n (0, ) α α p q f(q) q q p 2 5 Gauss α = [0; a, a 2,..., a n,...] r n = r n (α) = [a n ; a n+, a n+2,...] [0; a n+, a n+2,...] 4 R. O. Kuz min Ob odnoǐzadache Gaussa, Doklady akad. nauk, ser. A, (928) P. Lévym Sur les lois de probabilité dont dependent les quotients complets et incomplets d une fraction continue, Bull. Soc. Math., 57, (929) B.G.
79 5. 69 z n = z n (α) z n = r n = a n (0, ) α 0 z n < z n (α) < x m n (x) Laplace lim m ln( + x) n(x) = n ln 2 n ln( + x) m n (x) (70) ln 2 Kuz min (70) 5 m 0 (x), m (x), m 2 (x),..., m n (x),... { ( ) ( )} m n+ (x) = m n m n k k + x k= (0 x, n 0) (7) a n = a n+ + z n+ a n+ < x 5
80 70 3 k k + x < z n k ( ) ( ) m n m n k k + x (7) C ϕ(x) = ϕ(x) = C ln( + x) k= { ( ) ( )} ϕ ϕ k k + x m n (x) n (7) m n+(x) = k= ( ) (k + x) 2 m n k + x (72) z 0 (α) = α m 0 (x) = x m 0(x) = m n(x) n (72) (0, ) m n+(x) (72) (72) (7) 33. f (x), f 2 (x),..., f n (x),... (0, ) f n+ (x) = k= ( ) (k + x) 2 f n k + x 0 x 0 < f 0 (x) < M (72) (n 0) (73)
81 5. 7 f 0(x) < µ f n (x) = a + x + n θae λ (0 x ) a = ln 2 0 f 0 (z) dz θ < λ A M µ 3. n 0 (n) ( ) pn + xp n f n (x) = f0 q n + xq n (q n + xq n ) 2 (74) (p n /q, (p n + p n )/(q n + q n ) n n a, a 2,..., a n. n = 0 (74) (0, ) p 0 = 0, q 0 =, p =, q = 0 (74) n (73) f n+ (x) = ( ) (k + x) 2 f n k + x k= ( (n) pn + k+x = p ) n f0 (k + x) 2 q n + k+x q n k= (n) = k= ( ) (pn k + p n ) + xp n f 0 (q n k + q n ) + xq n = ( q n + k+x q n ) 2 {(q n k + q n ) + xq n } 2 (n+) ( ) pn+ + xp n f0 q n+ + xq n (q n+ + xq n ) 2
82 n 0 f n(x) < µ + 4M 2n 3. (74) f n(x) = (n) f 0 (u) ( )n (n) (q n xq n ) 4 2 q n f0 (u) (q n + xq n ) 3 u = p n + xp n q n + xq n 0 x (q n + xq n ) 2 < 2 q n (q n + q n ) 2 q n (q n + q n ) > q 2 n > 2 n (n) (n) q n (q n + q n ) = p n p n + p n q n q n + q n = 33 f n(x) < µ + 4M 2n 3 5. t + x < f n(x) < T + x t + x < f n+(x) < T + x (0 x ) (0 x )
83 (73) t k= k= t + (k + x) k+x 2 < f n+(x) < (k + x)(k + x + ) < f n+(x) < T t k= ( ) k + x < f n+ (x) < T k + x + k= T + (k + x) k+x 2 k= k= t + x < f n+(x) < T + x (k + x)(k + x + ) ( ) k + x k + x f n (z) dz = 0 f 0 (z) dz (n = 0,, 2,...). (73) n > 0 0 f n (z) dz = = ( ) dz f n k + z (k + z) 2 k= k= k k+ f n (u) du = 0 f n (u) du 33. f 0 (x) 0 x m m f 0 (x) < M 0 x m 2( + x) < f 0(x) < 2M + x g + x < f 0(x) < G + x (0 x ) (0 x )
84 74 3 g = m 2, G = 2M f n (x) g q + x = ϕ n(x) (0 x, n = 0,, 2,...) 5 F (x) = g/( + x) F (x) = F k= ( ) k + x (k + x) 2 ϕ 0 (x), ϕ (x),..., ϕ n (x),... (73) (74) ϕ n (x) = u = p n + xp n q n + xq n (n) ϕ0 (u) (q n + xq n ) 2 q n + xq n q n + q n < 2q n ϕ 0 (u) > 0 ϕ n (x) > 2 (n) ϕ0 (u) q n (q n + q n ) (75) 2 0 ϕ 0 (z) dz = 2 ϕ 0(u ) q n (q n + q n ) u (p n /q n, (p n +p n )/(q n +q n ) /[q n (q n + q n )] (75) (76) (76) ϕ n (x) 2 0 ϕ 0 (z) dz > 2 (n) {ϕ0 (u) ϕ 0 (u )} q n (q n + q n ) (77)
85 5. 75 ϕ 0 (x) f 0(x) + g < µ + g (0 x ) ϕ 0 (u) ϕ 0 (u ) < (µ + g) u u < < µ + g q 2 n < µ + g 2 n (77) ϕ n (x) > 2 f n (x) > 0 µ + g q n (q n + q n ) ϕ 0 (z) dz µ + g 2 n = l µ + g 2 n l = 2 0 ϕ 0 (z) dz g + x + l µ + g 2 n > g + l 2 n+ (µ + g) = g q + x + x ψ n (x) = G + x f n(x) (n = 0,, 2,...) f n (x) < G l + 2 n+ (µ + G) + x l = 2 0 ψ 0 (z) dz = G + x l > 0 l > 0 n g < g < G < G G g < G g (l + l ) + 2 n+2 (µ + G) l + l = 2 0 G g + z dz = (G g)ln 2 2
86 76 3 G g < (G g)δ + 2 n+2 (µ + G) δ = ln 2 2 < n g + x < f 0(x) < G + x, f 0(x) < µ g + x < f n(x) < G + x g < g < G < G, G g < (G g)δ + 2 n+2 (µ + G) f 0 (x) f n (x) µ g 2 + x < f 2n(x) < G 2 + x g < g 2 < G 2 < G, G 2 g 2 < δ(g g ) + 2 n+2 (µ + G ) f n(x) < µ (0 x ) g r + x < f rn(x) < G r + x (0 x ; r = 0,, 2,... r > 0 g r < g r < G r < G r G r g r < δ(g r g r ) + 2 n+2 (µ r + G r ) (78) µ r f (r )n (x) < µ r (0 x )
87 µ r = µ + 4M (r = 0,, 2,...) 2rn 3 n µ r < 5M (r =, 2,...) (78) r =, 2,..., n G n g n < (G g)δ n + 2 n+2 {(µ + 2M)δ n + 7Mδ n 2 + 7Mδ n Mδ + 7} δ < G n g n < Be λn λ > 0 B > 0 M µ lim G n = lim = a n n f n 2(x) a + x (0 x ) (79) 0 f n 2(z) dz a ln 2 (n ) 6 a = ln 2 0 f 0 (z) dz n 2 N < (n + ) 2 (79) a 2Be λn + x < f n 2(x) < a + 2Be λn + x
88 a 2Be λn + x < f N (x) < a + 2Be λn + x f N(x) a + x < 2Be λn < Ae λ(n+) < Ae λ N A = 2Be λ N A N 0 33 f n (x) = m n(x) (0 x ) f 0 (x) 33 m n(x) ( + x) ln 2 < Ae λ n (0 x ) (80) m ln( + x) n(x) ln 2 < Ae λ n (0 x ) A λ 6 n a n = k a n = k ( ) ( n ME = m n k k k + < z n k ) ( ) m n = k + k k+ m n (x) dx 6 B.G. m n (x) ln( + x) ln 2 < Ae λn (0 x )
89 5. 79 (80) ( ) { } ME n ln + k(k+2) k ln 2 < A n k(k + ) e λ (8) ( ) n 3 ME k ( ) { } n ln + k(k+2) ME (n ) k ln 2 a n = n ln 4 ln 3 ln 2 33 k z k+n < x M n (x) M n (x) (0, ) a = r, a 2 = r 2,..., a k = r k ; z k+n < x (82) r, r 2,..., r k n 0 x 0 x (82) a = r, a 2 = r 2,..., a k = r k ; r + x < z k+n r r { ( ) ( )} M n (x) = M n M n (n, 0 x ) r r + x r= M 0(x), M (x),..., M n(x),... (73) (p k /q k, (p k + p k )/(q k + q k )) α α = p kr k+ + p k q k r k+ + q k
90 80 3 z k = /r k+ α = p k + z k p k q k + z k q k z k < x α p k /q k (p k + xp k )/(q k + xq k ) M 0 (x) = p k p k + xp k q k q k + xq k = x q k (q k + q k x) ( ), 2,..., k M n (x) = ME χ n (x) (n 0, 0 x ) r, r 2,..., r k χ 0 (x), χ (x),..., χ n (x),... χ n(x) M n(x) (73) ( ), 2,..., k ME = p k r, r 2,..., r p k + p k k q k q k + q k = q k (q k + q k ) (83) χ 0 (x) = (q k + q k )x q k + q k x χ 0(x) = q k(q k + q k ) (q k + q k x) 2 χ 0(x) = 2q kq k (q k + q k ) (q k + q k x) 3 2 < χ 0(x) < 2, χ 0(x) < 4 (0 x ) (83)
91 χ n(x) A λ r, r 2,..., r k χ M n(x) = n(x) ( ) =, 2,..., k ( + x) ln 2 + n θae λ, θ < ME r, r 2,..., r k r /(r + ) /r θ < ( M ) ( ) n r Mn r+ ( ) =, 2,..., k ME r, r 2,..., r k { } ln + r(r+2) + θ A n ln 2 r(r + ) e λ ( ) ( ) ( ), 2,..., k, k + n + M n M n = ME r r + r, r 2,..., r k, r ( ), 2,..., k, k + n + ME r, r 2,..., r k, r { } = ln + r(r+2) + θ Ae λ ( ) n, 2,..., k ME ln 2 r(r + ) r, r 2,..., r k r, r 2,..., r k, 2,..., k n, n 2,..., n t A λ n < n 2 < < n t < n t+ r, r 2,..., r t, r ( ) n, n 2,..., n t, n t+ { } ME r, r 2,..., r t, r ln + r(r+2) ( ) A n, n 2,..., n t ln 2 < r(r + ) e λ n t+ n t ME r, r 2,..., r t
92 82 3 (0, ) a n = r n f(r) r C δ f(r) < Cr 2 δ (r =, 2,...) (0, ) 0 { } n ln + r(r+2) lim f(a k ) = f(r) (84) n n ln 2 k= r= (84) f(r). 0 f(a k ) dα = u k, 0 0 {f(a k ) u k } 2 dα = b k, {f(a i ) u i }{f(a k ) u k } dα g ik n {f(a k ) u k } = s n = s n (α) k= f(r) {f(a k )} 2 dα = {f(r)} 2 < C 2 r 2δ ( ) {f(r)} 2 k ME < 2C 2 r r 2δ r 2 = C 0 r= r= 7 Metrische Kettenbruchprobleme, Composito Mathematica,, (935) B.G.
93 6. 83 Bunyakovskiǐ Schwarz b k = {f(a k )} 2 dα u 2 k < C, 0 u k = f(a k ) dα < {f(a k )} 2 dα < (85) C 0 k > i ( ) i, k g ik = f(a i )f(a k ) dα u i u k = f(r)f(s)me u i u k (86) r, s 0 0 r,s= 34 2 ( ) { } ME i, k ln + ( ) s(s+2) i ME r, s ln 2 r ( ) k i < Ae λ i ME s(s + ) r ( ) ( ) < 3Ae λ k i i k ME ME r s (87) ( ) { } ME k ln + ( ) s(s+2) k s ln 2 < Ae λ < 3Ae λ k k ME (88) s(s + ) s ( ) i (88) ME (87) r ( ) ( ) ( ) ( ) ( ) ME i, k i k ME ME r, s r s < k i i k 6Ae λ ME ME r s (86) ( ) ( ) g i k ik f(s)f(s)me ME + u i u k r s r,s= ( ) ( ) < 6Ae λ k i i k f(s)f(s)me ME r s r,s=
94 84 3 ( ) ( ) i k f(s)f(s)me ME = u i u k r s r,s= (85) 2 g ik < 6Ae λ k i u i u k < 6AC e λ k i (89) (85) (89) n > m > 0 (s n s m ) 2 dα = 0 0 = = { n n k=m+ + 2 k=m+ n 0 i=m+ k=i+ n k=m+ b k AC n (f(a k ) u k )} 2 dα {f(a k ) u k } 2 dα n int 0{f(a i ) u i }{f(a k ) u k } dα n i=m+ k=i+ i=m+ k=i+ n g ik < C (n m) e λ k i < C 2 (n m) (90) C 2 e n (0, ) s n εn ε 0 s 2 n dα s 2 n dα ε 2 n 2 Me n e n (90) m = 0 Me n 0 s2 n dα ε 2 n 2 < C 2 ε 2 n
95 6. 85 Me n 2 n= (0, ) n =, 2, 3,... e n 2 (0, ) n s n 2 n 2 < ε ε s lim n 2 n n 2 = 0 (9) n 2 N < (n + ) 2 (90) 0 (s N s n 2) dα < C 2 (N n 2 ) < C 2 (2n + ) 3C 2 n (0, ) s N s n 2 εn 2 e n,n (n+) 2 N=n 2 e n,n E n n 2 N < (n + ) 2 (s N s n 2) 2 dα (s N s n 2) 2 dα > ε 2 n 4 Me n,n 0 e n,n ME n (n+) 2 Me n,n < ec 2 ε 2 n 3 Me n,n < 3C 2(2n + ) ε 2 n 3 9C 2 ε 2 n 2 N=n 2 n= ME n (0, ) E n e n,n (0, ) s N s n 2 > εn 2 n n 2 N < (n + ) 2 s N s n 2 n 2 < ε
96 86 3 n n 2 N < (n + ) 2 ε s N n 2 s n 2 n 2 0 (n, n2 N < (n + ) 2 ) (9) s N n 2 0 (n, n2 N < (n + ) 2 ) S N N 0 (N ) N N f(a k ) N k= N u k 0 (N ) (92) (8) { } ln + u r(r+2) k f(r) ln 2 = r= ( ) f(r) ME k ln r r= { + r(r+2) ln 2 k= } < Ae λ k r= f(r) r(r + ) < A e λ k A { } ln + r(r+2) u k f(r) (k ) ln 2 r= N N u k k= r= ln f(r) (92) N N f(a k ) k= r= { } + r(r+2) ln 2 (N ) { } ln + r(r+2) f(r) ln 2 (0, ) 35
97 6. 87 f(r) = r = k f(r) = 0 r k k ψ n (k) = n f(a i ) i= n k ψ n (k) n = n n f(a i ) i= n k ψ n (k) lim = d(k) n n k f(n) 35 k d() = ln 4 ln 3, d(2) = ln 2 ln 9 ln 8 ln 6 ln 5, d(3) = ln 2 ln 2 f(r) = ln r (r =, 2, 3,...) 35 n n ln a i i= r= { } ln + r(r+2) ln(r) ln 2 (n )
98 88 3 n a a 2... a n r= { } ln r ln 2 + r(r + 2) n n r= { } ln r ln 2 + = r(r + 2) 35 n n a i (93) i= f(r) = r 35 (93) 30 3 n a n > n ln n n a i > n ln n i= n a i < ln n n i= (93)
さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
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II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
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