Size: px
Start display at page:

Download ""

Transcription

1 2 2 MATHEMATICS.PDF (p n ), ( ) GL 2 (Z) SL 2 (Z) SL 2 (Z)

2

3 , a 0 + a + a b b 2 b 3 () + b n a n + b n a n., () b 2 b n a 0 + b. (2) a + a a n. a 0, a,..., a n, b, b 2,..., b n 0, (). n.,. a 0 + b a = a 0a + b a n = k, ()., b 2 b k a k + b k a k = a kb k a k a k + b k b k 2 b k b k a k a 0 + b a + a a k 2 + a k + = a 0 + b b 2 b k 2 a k b k. a + a a k 2 + a k a k + b k, n = k () n = k. n = k ()., n, (). 3

4 , a 0 + a + a a n + a n (3)., a 0 + a + a [a 0, a, a 2,..., a n ] a n ). a 0, a, a 2,..., a n (3)., [a 0, a, a 2,..., a n ] = [a 0, [a, a 2,..., a n ]] = [a 0, a, [a 2,..., a n ]] = = [a 0, a, a 2,..., a n 2, [a n, a n ]]. [a 0, a, a 2,..., a n ], 0., [a, a 2,..., a n ] 0, [a 2,..., a n ] 0,..., [a n, a n ] 0, a n 0 (4).,, [a 0, a, a 2,..., a n ]., a, a 2,..., a n, (4), [a 0, a, a 2,..., a n ]. a 0, a, a 2,..., a n, [a 0, a, a 2,..., a n ]. ) [] 2 [k 0, k,..., k n ], [a 0, a,..., a n ],. 4

5 a, a 2,..., a n 0,. n = 3, a = 0, a 0 + = a 0 + = a 0 + a 2 + a 3 a + a 2 + a 3 a 2 + a 3..2 [, 2, 3] = = = = [ 4, 3, 2] = = = = n 0. a 0, a, a 2,..., a n., [a 0, a, a 2,..., a n ], n = 0 n =, a =. n 2., s = a + a a n [a 0, a,..., a n ] = a 0 + s (5)., n 2, s >, (5).., n., n 0, n =, [a 0, a ] = a 0 + a. a >,.., n 0, n = a =..5 a, b, α, β, 0 < α <, 0 < β <., a + α = b + β, a = b α = β. 5

6 0 < α <, 0 < β <, < β < β α < α <. a + α = b + β, a b = β α, < a b <. a b, a b = 0., β α = 0..6 m, n, n m. a 0, b 0, a, a 2,..., a n, b, b 2,..., b m., [a 0, a,..., a n ] = [b 0, b,..., b m ],. (i) m = n, a 0 = b 0, a = b,..., a n = b n (ii) m = n +, a 0 = b 0, a = b,..., a n = b n +, b n+ = 0 i n i, s i = a i+ + a i t i = b i+ + a i i n 2, s i >, t i >,.5 a n b m.,.,. a i + s i = b i + t i = a i = b i a 0 = b 0, a = b, a 2 = b 2,..., a n 2 = b n 2 a n + a n = b n + t n (6) a n >, (6), t n >..5 a n = b n, a n = t n. a n,.4, m = n, a n = b n m = n +, a n = b n +, b n+ =. a n =, (6), t n =. m n + t n >, m = n, b n =. 6

7 .7 m, n, n m. a 0, b 0, a, a 2,..., a n, b, b 2,..., b m., s, t., [a 0, a,..., a n, s] = [b 0, b,..., b m, t], a 0 = b 0, a = b,..., a n = b n s = [b n+, b n+2,..., b m, t]., m = n s = t. 0 i n i,, s i >, t i >,.5 s i = a i+ + a i a n + s t i = b i+ + a i b m + t.,., a i + s i = b i + t i = a i = b i, s i = t i a 0 = b 0, a = b, a 2 = b 2,..., a n = b n a n + s = b n + b n+ + + b m + t. s >, t >,.5, a n = b n., m = n s = t. s = b n+ + b n b m + t 2 (p n ), ( ) (a n ), (p n ), ( ), p n = a n p n + p n 2, p =, p 2 = 0, = a n + 2, q = 0, q 2 = (7). n. a, a 2,..., a n, p, p 2,..., p n q, q 2,...,, p n,. 7

8 2. n a n., n (i) n (ii) < +. (i) n., q 2 =, q = 0, q 0 = a 0 q + q 2 =. a, a 2, q = a q 0 + q = a, q 2 = a 2 q + q 0 = a a n 3, k < n k k q k, a n = a n (n ) + (n 2) = 2n 3 n. (8), n n. (ii), q = a q 0 + q = a, q 2 = a 2 q + q 0 = a a 2 +. a, a 2, q < q 2. n 3, a n, (i) n 2 2, = a n (n 2) >. 2.2 = q 0 q, n 0 +., n 4 (8) 2n 3 > n, n 4 n <. 2.3 a 0, n a n., n 0 8

9 (i) n < p n (ii) p n < p n+. (i) n., p 2 = 0, p =, p 0 = a 0 p + p 2 = a 0. a, a 2, p = a p 0 + p = a a n 2, k < n k k < p k, a n p n = a n p n + p n 2 p n + p n 2 > (n ) + (n 2) = 2n 3 n., n 0 n < p n. (ii), p 0 = a 0, p = a a 0 +, a, p 0 < p. n 2, a n, (ii) n 2 < p n 2, p n = a n p n + p n 2 > p n + (n 2) > p n. 2.4 n a n., a 0, lim =. n lim p n =. n 2., n n. n. a 0, 2.3, n p n n. n p n. 2.5 n 2, (i) p n + p n+ = ( ) n 9

10 (ii) p n, gcd(p n, ) =. (i) n., p q 2 p 2 q =, n = 2 (i). n = k (i), p k+ q k p k q k+ = (a k+ p k + p k )q k p k (a k+ q k + q k ) = (p k q k p k q k ) = ( ) k = ( ) k. n = k (i)., n (i). (ii) g = gcd(p n, ) >, (i) g, g., g,. g =. 2.6 n a n > 0., p 0 q 0 < p 2 q 2 < < p 2k q 2k < p 2k+2 q 2k+2 < < p 2k+3 q 2k+3 < p 2k+ q 2k+ < < p 3 q 3 < p q., k. 2.5 p n+2 +2 p n = p n+2 p n p n+ p n + = ( ) n (9), p n+2 p n +2 = (a n+2 p n+ + p n ) p n (a n ) = (p n+ p n + )a n+2 = ( ) n a n+2. p n+2 +2 p n = ( )n a n

11 > 0, +2 > 0, a n+2 > 0, n = p n+2 +2 p n > 0 = p n+2 +2 > p n, (0) n = p n+2 +2 p n < 0 = p n+2 +2 < p n ()., n p n /, n p n /., (9) > 0, + > 0, p n+ p n = p n+ p n + = ( )n n = p n+ + p n > 0 = p n+ + > p n, (2) n = p n+ + p n < 0 = p n+ + < p n (3)., u 0 v, u < v, u < v +, (0) (3) p u q u < p v+ q v+ < p v q v, v < u, v < u +, () (2) p u q u < p u+ q u+ < p v q v.,,.. 3 (a n ), (p n ), ( ) (a n ) 2 (7). 3. n 0, 0 t, [a 0, a,..., a n, t] = tp n + p n 2 t + 2 (4)., n 2, a, a 2,..., a n. n. p =, p 2 = 0, q = 0, q 2 =, [t] = tp + p 2 tq + q 2.

12 , p 0 = a 0 p + p 2 = a 0, q 0 = a 0 q + q 2 =, [a 0, t] = a 0 + t = ta 0 + t, n = 0, (4). = tp 0 + p tq 0 + q. n 2, n (4), 3., [ [a 0, a,..., a n 2, a n, t] = a 0, a,..., a n 2, a n + ] t,, n (4). = (a n + /t)p n 2 + p n 3 (a n + /t) (a n + /t)p n 2 + p n 3 (a n + /t) = (ta n + )p n 2 + tp n 3 (ta n + ) 2 + t 3 = t(a n p n 2 + p n 3 ) + p n 2 t(a n ) + 2 = tp n + p n 2 t + 2., t a, a 2,..., a n, 0., n 0 (4). 3.2 n 0, 0 s, t, [a 0, a,..., a n, s] [a 0, a,..., a n, t] = ( ) n (s t) (s + 2 )(t + 2 )., n 2, a, a 2,..., a n. 2.5, p n 2 p n 2 = ( ) n 2 = ( ) n. 2

13 , 3., [a 0, a,..., a n, s] [a 0, a,..., a n, t] = sp n + p n 2 s + 2 tp n + p n 2 t + 2 = (sp n + p n 2 )(t + 2 ) (s + 2 )(tp n + p n 2 ) (s + 2 )(t + 2 ) = stp n + sp n 2 + tp n 2 + p n 2 2 (stp n + tp n 2 + sp n 2 + p n 2 2 ) (s + 2 )(t + 2 ) = (s t)(p n 2 p n 2 ) (s + 2 )(t + 2 ) ( ) n (s t) = (s + 2 )(t + 2 ). 3.3 n 0. n 2, a, a 2,..., a n., s, t. (a) [a 0, a,..., a n, s] = [a 0, a,..., a n, t] s = t. (b) n, [a 0, a,..., a n, s] < [a 0, a,..., a n, t] s < t. (c) n, [a 0, a,..., a n, s] < [a 0, a,..., a n, t] s > t n 0., n, a, a 2,..., a n.,. [a 0, a,..., a n ] = p n n, 3., t = a n, [a 0, a,..., a n ] = a np n + p n 2 a n + 2 = p n 3

14 ., p =, p 2 = 0, q = 0, q 2 =, p 0 = a 0 p + p 2 = a 0, q 0 = a 0 q + q 2 =,. n = 0. p 0 q 0 = a 0 = [a 0 ] 0 n m, ω = [a 0, a, a 2,..., a m ], [a 0, a, a 2,..., a n ] = p n ω n. gcd(p n, ) = > 0, p n /. 3.5 n a n. c n = [a 0, a, a 2,..., a n ], n = 0,, 2,... (c n ), ω. lim c n = ω n 3.4, n 0. c n = [a 0, a, a 2,..., a n ] = p n m, n, n m. 2.6, p m p n q m p n+ p n +., 2.5, 2., n, + n +, p n+ p n = p n+ p n + = ( )n p m p n q m ( )n. + ( ) n + n(n + ) n 2. 4

15 , p m p n q m n 2. ε > 0. δ = / ε, n > δ n. n 2 < δ 2 = ε m n > δ = p m p n q m < ε. (c n )., ω, lim n c n = ω. 3.6 f(x) [a, ), lim x f(x) = l. (i) f(a) < l, c f(a) < c < l, f(ξ) = c ξ > a ξ. (ii) l < f(a), c l < c < f(a), f(ξ) = c ξ > a ξ. (i) b a < b l f(b), [a, b], ξ f(ξ) = c a < ξ < b. x > a x f(x) < l, lim x f(x) = l, ε > 0, δ, x ε = (l c)/2, x > δ = f(x) l < ε. x > δ = l f(x) < l c 2 = f(x) > l + c 2 t = max{δ, a}, f(a) < c < f(t). [a, t], ξ f(ξ) = c a < ξ < t. (ii) (i). > c. 3.7 n 0. n, a, a 2,..., a n., ω. (i) [a 0, a,..., a n, a n+ ] < ω < [a 0, a,..., a n ], s, s > a n+. ω = [a 0, a,..., a n, s] 5

16 (ii) [a 0, a,..., a n ] < ω < [a 0, a,..., a n, a n+ ], s, s > a n+ ω = [a 0, a,..., a n, s]. f(x) = [a 0, a,..., a n, x]. f(x) a j + x (j = 0,,..., n) /x, (0, )., f(a n+ ) = [a 0, a,..., a n, a n+ ]., f(x) = [a 0, a,..., a n, a n + /x], lim f(x) = [a 0, a,..., a n ]. x, s s > a n+ ω = [a 0, a,..., a n, s], (i), (ii), 3.6 (i), (ii). 4 ω. ω, ω a 0, ω = a 0 + ω (5) ω. ω. ω = ω a 0 >, ω, ω a, ω = a + ω 2 (6) ω 2. ω 2. (6) (5), ω 2 = ω a > ω = a 0 + = [a 0, a, ω 2 ] a + ω 2 6

17 ., a, a 2,..., a n, ω n >, ω = a 0 + = [a 0, a, a 2,..., a n, ω n ] (7) a + a a n + ω n. n 0 (7), ω., ω n (7) n. ω, a 0 = q, q = a 0 = [a 0 ].. ω, m, n, ω = m/n, n > 0. ω,, m = a 0 n + r, 0 < r < n, n = a r + r 2, 0 < r 2 < r, r = a 2 r 2 + r 3, 0 < r 3 < r 2,, r n = a n r n + r n+, 0 < r n+ < r n, r n = a n+ r n+ a 0, a,..., a n, a n+, r, r 2,..., r n, r n+., n, r, r 2,..., r n, a, a 2,..., a n., m n = a 0 + r n = a 0 + n/r, n r = a + r 2 r = a + r /r 2, r r 2 = a 2 + r 3 r 2 = a 2 + r 2 /r 3,, r n = a n + r n+ = a n +, r n r n r n /r n+ r n = a n+ r n+.,, ω = m n = a 0 + = [a 0, a,..., a n+ ] a + a a n+.,. 7

18 4. q, a 0 a, a 2,..., a n, a n+ q = [a 0, a,..., a n, a n+ ]., ω = n/r, ω 2 = r /r 2, ω 3 = r 2 /r 3,..., ω n = r n /r n, ω n+ = r n /r n+., ω, = = = = [, 2, 3] = = + 7 = 4 + = [ 4, 3, 2] = = = = 3 + = [3, 7, 7] ω, n ω n+, ω = a 0 + a + a a n + ω n+,.., n ω n., ω, ω = + ( 2 ) = = + = 2 + ( 2 ) =

19 = + ( 3 ) = = + = + = = ( 3 ) = ω. (a n ), n a n, (c n ) c n = [a 0, a, a 2,..., a n ] (n = 0,, 2,...),., ω. lim c n = ω n ω, 4., b 0 b, b 2,..., b m ω = [b 0, b,..., b m ]. lim n c n = ω, (c 2k ) (c 2k+ ) (c n ), ω. 2.6, k 0 c 2k < ω < c 2k+. s > a n+ 3.7, n > m n, s, [b 0, b,..., b m ] = [a 0, a,..., a n, s]..7 b m = [a m, a m+,..., a n, s]., b m [a m, a m+,..., a n, s].. ω. 9

20 5, ω, ω = [a 0, a,..., a n, ω n ] (n = 0,, 2,...) ω. ω (a n )., n a n, ω n >. (a n ), (p n ), ( ),. 3.4, p n = a n p n + p n 2, p =, p 2 = 0, = a n + 2, q = 0, q 2 = [a 0, a,..., a n ] = p n. p n / ω n. 5. n 0 ω p n ( ) n = ( ω n+ + ) (8). 3.,. [a 0, a,..., a n, ω n+ ] = ω n+p n + p n ω n+ +, 2.5, ω p n = [a 0, a,..., a n, ω n+ ] p n = ω n+p n + p n ω n+ + p n = (ω n+ p n + p n ) p n (ω n+ + ). (ω n+ + ) (ω n+ p n + p n ) p n (ω n+ + ) = (p n p n ) = ( ) n = ( ) n. (8). 5.2 p 0 q 0 < p 2 q 2 < < p 2k q 2k < p 2k+2 q 2k+2 < < ω < < p 2k+3 q 2k+3 < p 2k+ q 2k+ < < p 3 q 3 < p q. 20

21 n 0. 5., > 0, > 0, ω n+ >, ω p n ( ) n = ( ω n+ + ). n = ω p n > 0 = ω > p n, n = ω p n < 0 = ω < p n., n p n / ω, n p n / ω. 2.6,. 5.3 n ω p n < ω p n (9). 3., ω n+ +, ω = ω n+p n + p n ω n+ +. (ω n+ + )ω = ω n+ p n + p n., ( ω n+ (ω p n ) = (ω p n ) = ω p ) n. ω n+,, ω p n = ω n+ ω p n. 0 < < < ω n+ (9). 0 < ω n+ <. 5.4 n 0 ω p n > (20)

22 5.3,, 2.5,, (20). ω p n+ < ω p n + p n+ p n + ω p n+ + ω p n < 2 ω p n. + p n+ p n = ( ) n p n+ p n + = p n+ p n = n 0 ω p n < +. q 2 n > 0, + > 0, ω n+ > a n+, ω n+ + > a n+ + = +., 5., ω p n = ( ) n ( ω n+ + ) = ( ω n+ + ) <. +, 2. ( 2.2), n 0 +,. 5.6 n 0, δ n,. p n = ω + δ n, δ n < δ n = (p n ω). ω., 5.5, ω p n <. q 2 n 22

23 ,,,, ω p n <. < p n ω <. < (p n ω) <., < δ n <.,. 5.7 p n lim = ω. n 5.5, n 0 ω p n <., 2.4 (n ), /qn 2 (n )., ω p n / 0 (n ). q 2 n 5.8 ω, p n / (n 0) ω. p, q, q > 0., qω p < ω p n = + q.. qω p < ω p n q < +., p n x + p n+ y = p, x + + y = q (2)., 2 p n,, 2.5 p n+ p n + = ( ) n, (p n+ p n + )y = p qp n. y = ( ) n (p qp n )., +, 2 p n+,, (p n+ p n + )x = qp n+ p+. 23

24 , x = ( ) n (qp n+ p+ ).., (2) (x, y)., x 0, y 0. x = 0, qp n+ = p gcd(p n, ) =, q k+ q. q +., x 0., x, x ω p n ω p n. y = 0, p n x = p, x = q, qω p = x( ω p n )., qω p = x ω p n ω p n.. y 0. x y., y < 0 = x = q + y > 0 = x > 0, y > 0 = + y + > q = x = q + y < 0 = x < 0. ω, 5.2 p n / < ω < p n+ /+ p n+ /+ < ω < p n /., ω p n + ω p n. x, y (2), qω p = ( x + + y)ω (p n x + p n+ y) = x( ω p n ) + y(+ ω p n+ )., x( ω p n ) y(+ ω p n ). x, qω p = x ω p n + y + ω p n+ x ω p n ω p n..,. 24

25 5.9 ω, p n / (n ) ω., p/q., p, q Z, q > 0, gcd(p, q) =., ω p q < ω p n = < q.. n, ω p/q < ω p n / q, qω p = q ω p q < ω p n = ω p n., 5.8, + q., +. n, ω, p/q., p, q Z, q > 0, gcd(p, q) =., ω p q < 2q 2, p/q ω.. ω p/q < /2q 2 p/q ω, n 0, p/q p n / qp n p, qp n p = qp n p q q q = p n p q ( ) ( = pn ω + ω p ) q ω p n + ω p q < ω p n + 2q 2. q 0 =, 2. ( ) n, k 0, q k q < q k+., 5.8, ω p k = q kω p k, qq k < q k q k qω p q k <. 2qq k = q q k ω p q ω p k + 2q 2 < + 2qq k 2q 2., q < q k, q k q. q k 25

26 5. ω. ω 2 p n /, p n+ /+, ω p n < ω p n+ < (22). 2q 2 n + 2q 2 n+ n , p n / < ω < p n+ /+ p n+ /+ < ω < p n /., p n+ p ( ) ( n pn+ = ω + ω p ) n + +, p n+ + p n > 0, p n+ + ω > 0, ω p n > 0 p n+ p n + = ω p n+ + ω p n.,., 2.5 p n+ p n + = ( ) n, p n+ p n + = p n+ p n + =. + +,, (22), + ω p n+ + ω p n = q 2 n+ + 2qn q 2 nq 2 n+, ( + ) 2 0., = +. n, 2. < +,., n = 0,, q = q 0 =., q = a q 0 + q = a a =. 5.2, p 2 q 2 < ω < p q p = a 0 + = a 0 +, q a p 2 = a 0 + q 2 a + = a 0 + a 2 + a 2 = a 0 + a 2 a 2 +, 0 < p q ω < p q p 2 q 2 = a 2 a , (22),. 26

27 5.2 ω > 0, x/y, x/y ω, y/x /ω., ω >, ω ω = a 0 + a + a a n +, /ω ω = 0 + a 0 + a + a a n +., ω > a 0. x/y ω n, x y = a 0 + a + a , y/x y x = 0 + a 0 + a + a 2 + +, /ω n +. 0 < ω <, ω > /ω., /ω ω = a 0 + a + a a n +, /(/ω) = ω ω = 0 + a 0 + a + a a n +. x/y ω n, y/x /ω n. a n a n 6 GL 2 (Z) SL 2 (Z) 2 ± GL 2 (Z) : GL 2 (Z) = P = p q p, q, r, s Z, det P = ps qr = ± r s. 6. GL 2 (Z). 2 P, Q GL 2 (Z), det P Q = det P det Q = ±, P Q GL 2 (Z). 27

28 GL 2 (Z) E = 0. 0 P = p q GL 2 (Z), P = r s det P P. s r q GL2 (Z) p 2 SL 2 (Z) : SL 2 (Z) = P = p q p, q, r, s Z, det P = ps qr = r s = {P GL 2 (Z) det P = }. 6.2 SL 2 (Z) GL 2 (Z) 2. P, Q GL 2 (Z) det P Q = det P det Q, GL 2 (Z) {±}, P det P. SL 2 (Z).,, GL 2 (Z)/SL 2 (Z) = {±}., [GL 2 (Z) : SL 2 (Z)] = SL 2 (Z), S =, T = det S = det T =, S, T SL 2 (Z). S, T SL 2 (Z) Γ. SL 2 (Z) Γ. p q SL 2 (Z), ps q 0 = p = s = ±., 0 s q = S q, q = S q T

29 , p q Γ., 0 s r 0 = min r p q SL 2 (Z) \ Γ r s, r 0. SL 2 (Z) \ Γ (2, )- r 0, P 0 = p 0 q 0 r 0 s 0., n, n Z, s 0 = r 0 n + n, 0 n < r 0.,, r 0, 0 s 0 r 0 n < r 0. P 0 S T = q 0 p 0 n p 0 Γ. s 0 r 0 n r 0, S T Γ, P 0 Γ.., SL 2 (Z) = Γ. 7 GL 2 (Z) C { }, P = p r, px + q rx + s, x, x s/r P x = p r, x =, x = s/r q GL 2 (Z), x C { }, r 0 s r = 0, px + q, x P x = s, x =. 7. ±q x = x ± q., ±. 0 29

30 P GL 2 (Z) P GL 2 (Z).,, x C { } P x = ( P ) x x = 0 x = /x GL 2 (Z) C { }. x C { }. E x = x. P = p q, Q = p q GL 2 (Z). r s r s x, r x + s 0, (rp + sr )x + (rq + ss ) 0, P Q x = pp + qr pq + qs x rp + sr rq + ss x =, = (pp + qr )x + (pq + qs ) (rp + sr )x + (rq + ss ), P (Q x) = P p x + q p p x + q r r x + s = x + s + q r p x + q r x + s + s = p(p x + q ) + q(r x + s ) r(p x + q ) + s(r x + s ) = (pp + qr )x + (pq + qs ) (rp + sr )x + (rq + ss ). pp + qr rp + sr, r 0 P Q = p r, r = 0 r 0, r = r = 0,,, p Q = r, r 0, r = 0 p P = r, r 0, r = 0 P p r = pp + qr rp + sr 30

31 . r x + s = 0, r = 0, s = 0, p s q r = det Q 0., x = s /r., P (Q x) = P = p/r., P Q x = p/r. (rp + sr )x + (rq + ss ) = 0, rp + sr = 0, rq + ss = 0, (pp + qr )(rq + ss ) (pq + qs )(rp + sr ) = det P Q 0., x = (rq + ss )/(rp + sr )., P Q x =., Q x = s/r, P (Q x) =., P Q x = P (Q x)., P x GL 2 (Z) C { }. x, y C { }, x y, P GL 2 (Z) x = P y. x, y C, x y, p, q, r, s. x = py + q, ps qr = ± ry + s, det P =, det P =. z, w C { }, P SL 2 (Z) w = P z , , 2 + = , 0 = 2 + = 2, 0 =. 7.5 C { }. 3

32 x, y, z C { }. ( ) x = E x, x x. ( ) x y, P GL 2 (Z), x = P y., y = E y = (P P ) y = P (P y) = P x., y x. ( ) x y y z, P, Q GL 2 (Z), x = P y y = Q z., x = P (Q z) = P Q z, x z.,. GL 2 (Z) SL 2 (Z), C { }. 7.6 (i). (ii). (iii)., P GL 2 (Z), x Q { }, P x Q { }.,.,. x, y C, P GL 2 (Z) x = P y. x, y,, x., x, y, y = P x. 7.7 Q { } 2., p/r, gcd(p, r) =, ps qr = q, s Z., P SL 2 (Z), P = p r,. q s p r = P., p /r, p/r, Q SL 2 (Z) p r = Q. 32

33 , p r = P Q p r, P Q SL 2 (Z)., p/r p /r., x 2 + bx + c = 0 2 θ = b + b 2 4c, θ = b b 2 4c 2 2., P = b, det P = θ = P θ x C { }, GL 2 (Z) x = {P GL 2 (Z) x = P x}, SL 2 (Z) x = {P SL 2 (Z) x = P x}., (i) GL 2 (Z) x GL 2 (Z). (ii) SL 2 (Z) x SL 2 (Z). (i) E, E GL 2 (Z) x., GL 2 (Z) x. P, Q GL 2 (Z) x, P Q x = P (Q x) = P x = x, P x = P (P x) = (P P ) x = E x = x., P Q, P GL 2 (Z) x. (ii) (i). 7.0 S =, T = 0., E., 0 0 (i) GL 2 (Z) S, T, E GL 2 (Z). (ii) SL 2 (Z) S, E SL 2 (Z). (i), P = S, T, E, P GL 2 (Z), P =. 33

34 P = p r, q GL 2 (Z), P =., r = 0. s ps = ps qr = det P = ±, p = ±, d = ± ( )., P = ± q 0 ± ( )., q = S q, q = T S q, 0 0 q = S q, q = T S q. 0 0, P S, T, E. (ii) det P = ± det P = (i), P SL 2 (Z) P = SL 2 (Z) S, E. 7. x C, x 2., P GL 2 (Z), x = P x P = ±E., GL 2 (Z) x = SL 2 (Z) x = {±E}., E. P = p r q GL 2 (Z), s., x = P x = px + q rx + s rx 2 + (s p)x q = 0. 34

35 x 2, r = s p = q = 0. ps qr = det P = ±, p = s = ±., P = ±E.,.,, GL 2 (Z) x, SL 2 (Z) x {±E}.,. 8 SL 2 (Z) z,, Re z, Im z : z = Re z + Im z. 8. z C, P = p q GL 2 (Z), r s. Im (P z) = det P Im z rz + s 2, z z. w = P z = pz + q rz + s.,, w = pz + q rz + s. w w = pz + q rz + s pz + q rz + s (pz + q)(rz + s) (pz + q)(rz + s) = (rz + s)(rz + s) (psz + qrz) (psz + qrz) = (rz + s)(rz + s) (ps qr)(z z) = rz + s 2. det P = ps qr, z z = 2 Im z, w w = 2 Im w, Im w = det P Im z rz + s 2. 35

36 8.2 z., S Z Z, (0, 0). R { rz + s (r, s) S \ {(0, 0)} } (23)., rz + s = 0 rz + s = 0 r = s = 0 (r, s) = (0, 0), 0 (23)., (23). x = Re z, y = Im z, rz + s = (s + rx) + ry, rz + s 2 = (s + rx) 2 + r 2 y 2. S \ {(0, 0)} =, S (r 0, s 0 ) (0, 0). R = r 0 z + s 0. R (23), (23) R., rz + s < R (r, s). rz + s < R, z, y 0., (24) 2,, (24), s + rx < R, ry < R. (24) r < R y. (25) s rx < s + rx < R., (25) ( s < R + rx < R + x ). (26) y (25), (26), rz + s < R (r, s). 8.3 z, rz + s., 8.2 (23). 36

37 , : z n, rz s < n, 0 < r n r, s Z., Dirichlet. C H = {z C Im z > 0}., C F = {z C z > /2 Re z < /2} {z C z = /2 Re z 0} SL 2 (Z). z F Im z > 0, F H. 8.4 =, Re = 0, F. ρ = ( + 3)/2 3., ρ =, Re z = /2, ρ F., ρ + = ( + 3)/2 6, ρ + =, Re z = /2, ρ + F. 8.5 z H, w F, z w. P = p r q SL 2 (Z), 8. z H, s Im (P z) = Im z rz + s 2 > 0. S = {(r, s) P SL 2 (Z)} 8.2, rz + s P. P, w 0 = P z., w 0 z H. a, w 0 + a = a w 0, w 0 + a w 0., /2 Re w 0 + a < /2 a, w = w 0 + a., Re (w 0 + a) = Re w 0 + a, Im w = Im w 0., w = w = 0 w 0 37

38 , w w. 8., Im w = Im w w 2 = Im w 0 w 2., w 2 z., Im w 0, Im w Im w 0., w 2 = Im w 0 Im w., w >, w F. w =, w ( = w =, Re ) = Re w w, w w F. 8.6 z, w F, P SL 2 (Z), w = P z., z = w., ±E, z, ρ P = ±E, ±T, z = ±E, ±T S, ±(T S) 2 z = ρ., S =, T = T S = 0, (T S) 2 = 0., ρ = ( + 3)/2 3. Im w < Im z P P, Im z Im w. P = p q, r s det P =, 8., w = P z = pz + q rz + s. (27) Im w = Im z rz + s 2. Im z Im w, rz + s = Im z. (28) Im w 38

39 Im (rz + s) = r Im z, r Im z = Im (rz + s) rz + s. F ρ, 3 = Im ρ Im z. 2, 3 r. 2, r Z, r = 0 ±. r 2 3. r = 0, ps qr =, p = s = ±. (27) z, w F, q = 0., z = w. w = z ± q. r =, P P r =., r =. r =, (28) z + s. (29) z F s Z, z > z + s > (29)., z =., s 0, (29) z = ρ s =., z =, s = 0 z = ρ, s =. z =, s = 0, ps qr =, r = q =. (27) w = p z. z =, z ( = z =, Re ) = Re z. z z, w F, p = 0, /z = p =, /z = ρ +., z = w =,, ρ + = ρ 2 = /ρ, z = w = ρ. z = ρ, s =, ps qr =, r = p q =. p = q +. (27) w =, (q + )ρ + q ρ + = q + ρ = q + (ρ + ). ρ + ρ ρ + = ρ ρ 2 = ρ = ρ2 = ρ +. w F, q =., p = 0, z = w = ρ. 39

40 8.7 z, w H., z w, z w z 0 F. ( ) 8.5, z 0 F z z 0., w 0 F w w 0. z w, z 0 w , z 0 = w 0. ( ) z z 0, z 0 w, z w. 9 2 θ 2, 2 ax 2 + bx + c = 0, gcd(a, b, c) =, a > 0 (30), θ Q., θ. 2 2, θ (30) 2 2, ax 2 + bx + c = 0, gcd(a, b, c) =, a > 0, a x 2 + b x + c = 0, gcd(a, b, c ) =, a > 0. a = au (u Q), 0 = a θ 2 + b θ + c u(aθ 2 + bθ + c) = (b bu)θ + (c cu). θ Q, b bu = c cu = 0., b = bu, c = cu. u, u = m/n, gcd(m, n) =, n > 0, a n = am, b n = bm, c n = cm., n a, b, c. gcd(a, b, c) =, n =., u., m a, b, c. gcd(a, b, c ) =, m = ±., u = ±., a > 0, a > 0 u =., a = a, b = b, c = c, θ (30) 2. 2 θ (30) 2 D θ., θ D 2. 40

41 (30), b + D, 2a b D 2a 2. 2., 2 θ, 2 θ, θ. θ 2, θ 2 (30), 0. D = b 2 4ac 2 2, 2 2., θ 2, θ., θ θ, θ θ., θ, 3 : (i) θ 2. (ii) θ 2 ax 2 + bx + c = 0, a 0, D = b 2 4ac 0. (iii) θ 2 x 2 + b x + c = 0, D = b 2 4c 0., 2 θ, θ, 3 : (i) θ, θ 2. (ii) θ, θ 2 ax 2 + bx + c = 0, a 0, D = b 2 4ac 0. (iii) θ, θ 2 x 2 + b x + c = 0, D = b 2 4c 0. (3),. (i) (ii). (ii) (iii) ax 2 + bx + c = 0 θ, aθ 2 + bθ + c = 0. a, θ 2 + b θ + c = 0, b, c Q., b = b/a, c = c/a., D = b 4a c = b2 4ac a 2 = D a 2 4

42 , D 0, D 0. (iii) (i) x 2 + b x + c = 0 θ, θ 2 + b θ + c = 0. b = b /b 2, c = c /c 2 (b, b 2, c, c 2 Z, b 2 > 0, c 2 > 0), b 2 c 2, 2 aθ 2 + bθ + c = 0, a = b 2 c 2 > 0, b = b c 2, c = b 2 c. g = gcd(a, b, c), g >, aθ 2 + bθ + c = 0 g., b 2 4ac g 2 = b2 c 2 2 4b 2 2c c 2 g ( 2 b 2 = b2 2c 2 2 g 2 b 2 2 = b2 2c 2 2 g 2 D ) 4 c c 2, D 0, (b 2 4ac)/g 2 0., (iii) (i),, θ, θ (30) 2 ( ) D 2 θ, D 0, D 0 (mod 4). D 2 θ, a, b, c, aθ 2 + bθ + c = 0, gcd(a, b, c) =, a > 0, θ = b ± D, D = b 2 4ac. 2a θ 2, D 0., 0 (mod 4), b D b 2 (mod 4), b., D 0 (mod 4), b, D b 2 4. a =, c = b2 D, θ = b + D

43 , b 2 4ac = D, aθ 2 + bθ + c = 0, gcd(a, b, c) =, a > 0. D 0, θ., θ D 2. D (mod 4), b. (30) 2 θ, θ ax 2 bx + c = 0., /θ cx 2 + bx + a = 0., θ /θ 2.,. 9.3 θ 2, P GL 2 (Z) ω = P θ., ω 2, ω = P θ. P = p q pθ + q, ω = P θ = r s rθ + s, ω = P θ = pθ + q rθ + s, ω.. θ ω + ω = pθ + q rθ + s + pθ + q rθ + s (pθ + q)(rθ + s) + (pθ + q)(rθ + s) = (rθ + s)(rθ + s) = 2prθθ + qr(θ + θ) + 2qs r 2 θθ + rs(θ + θ) + s 2., ωω (pθ + q)(pθ + q) = (rθ + s)(rθ + s) = p2 θθ + pq(θ + θ) + q 2 r 2 θθ + rs(θ + θ) + s 2. θ + θ, θθ Q, ω + ω, ωω Q., ω, ω 2 x 2 + bx + c = 0, b = (ω + ω ), c = ωω. ω, 2 0., 9., ω, ω 2. ω = ω, θ = P ω = P ω = θ., θ 2, θ θ., ω ω., ω, ω. 43

44 9.4 θ, θ 2, θ + θ θθ, 2., + 2, 3 x 2 2x = 0, x 2 3 = 0, D 0, D 0 (mod 4). θ, D 2 ax 2 + bx + c = 0, D = b 2 4ac, θ = b + e D, e = ± 2a., ω θ, P = p r θ = P ω = pω + q rω + s q GL 2 (Z) s (32)., a = ap 2 + bpr + cr 2, b = 2apq + b(ps + qr) + 2crs, (33) c = aq 2 + bqs + cs 2, ω D 2 a x 2 + b x + c = 0, D = b 2 4a c, ω = b + e D 2a, e = e det P., gcd(a, b, c) = gcd(a, b, c ) =. D, D ( 9.2). A = a b/2, b/2 c [ ] θ A θ = aθ 2 + bθ + c = 0., (32), P ω = pω + q = (rω + s) θ. rω + s 44

45 , A = a b /2, (33), b /2 c t P AP = A. (34), [ ] a ω 2 + b ω + c = ω A ω [ = t ω ] P AP ω = t P ω A P ω [ ] = (rω + s) 2 θ A θ = 0., ω 2 a x 2 + b x + c = 0.. 2, b 2 4a c = 4 det A = 4 det t P AP = 4(det P ) 2 det A = 4 det A = b 2 4ac P = s det P r q = ± s p r = D. q (32), p ω = P θ = sθ q rθ + p. θ, ω θ, ω 2, 9.3, ω ω = sθ q rθ + p sθ q rθ + p (ps qr)(θ θ) = (p rθ)(p rθ) = det P a(θ θ) a. 45

46 , (p rθ)(p rθ) = p 2 pr(θ + θ) + r 2 θθ ( = p 2 pr b ) + r 2 c a a = ap2 + bpr + cr 2 a = a a., e = e det P, a (ω ω) = det P a(θ θ) = e D.,, (34), ω = b + e D 2a, ω = b e D 2a. A = t (P )A P, P = s det P r q. p, a = a s 2 b rs + c r 2, b = 2a qs + b (ps + qr) 2c pr, c = a q 2 b pq + c p 2. g = gcd(a, b, c ), g a, b, c., g > gcd(a, b, c) >.,, gcd(a, b, c) = g = , 2 2, 2 2., θ 2, θ θ, < θ < 0, < θ. 46

47 0. D, D 0 (mod 4). (i) D 0 (mod 4), r D/4, θ = r + D/4 D 2. (ii) D (mod 4), r D/4, θ = (r + D)/2 D 2. (i) θ 2 x 2 2rx + r 2 D/4 = 0, ( 2r) 2 4(r 2 D/4) = D. 2 θ = r D/4, r < θ < 0 < θ. (ii) θ 2 x 2 rx + (r 2 D)/4 = 0, ( r) 2 4(r 2 D)/4 = D. 2 θ = (r D)/2, r ω θ > 0., 0 < D r < 2, < θ < m. m 2 x 2 m = 0, 4m. m, m m 2., m + m 2 (x m ) 2 m = 0, 4m. m m, 2 2. m m + m 2., m + m < m m < 0, < m + m, D > 0, 2 ax 2 + bx + c = 0, D = b 2 4ac (35) 2. θ 2, 2 (35). a < 0, θ, ( b) 2 4( a)( c) = b 2 4ac = D., a > 0. θ θ 2, < θ < 0, < θ. (36) 47

48 , a > 0, ( b D)/2a < ( b + D)/2a., (37), θ = b + D 2a b a = θ + θ > 0,, θ = b D. (37) 2a c = θθ < 0. a a > 0, b < 0, c < 0., c = c, D = b 2 + 4a c > b 2., b < D. (38), b = b, (37), θ = b + D 2a, θ = b D. 2a, (36),, a > 0,, (38), θ < < θ aθ < a < aθ. D b D + b < a < < a < D. (39) D, (38), (39) b a., θ = ( b + D)/2a.. A = a c b GL 2 (Z), 0 < d < c., a/c d a c = [a 0, a,..., a n ], n det A = ad bc = ( ) n 48

49 ., a/c p k /q k (0 k n).,. A = p n p n,, n., a n =, [a 0, a,..., a n, a n ] = [a 0, a,..., a n + ], a n 2,., [a 0, a,..., a n ] = [a 0, a,..., a n, ] ad bc = ( ) n n. p n /, > 0., ad bc = ± gcd(a, c) =, c > 0, a = p n, c =., p n d b = ad bc = ( ) n = p n p n., p n (d ) = (b p n ). (40) gcd(p n, ) =, (d )., d d = 0., 0 < d < c, > 0, d < d < c =., d = 0., d =., > 0, (40) b = p n..2 θ >, a, b, c, d, 0 < d < c, ad bc = ± ω = aθ + b cθ + d., θ ω. 49

50 . 3., ω = a c b θ = p n d p n θ = p nθ + p n θ + = [a 0, a,..., a n, θ]. θ >, θ ω..3 2 θ, ω, θ ω,,, ζ > n 0, m 0. θ = [a 0, a,..., a n, ζ], ω = [b 0, b,..., b m, ζ] (4) ( ) θ (4). p n / θ n, θ = [a 0, a,..., a n, ζ] = p nζ + p n ζ + p n p n = ±, θ ζ., ω ζ., θ ω. ( ) θ ω, a, b, c, d, ω = aθ + b, ad bc = ±. (42) cθ + d a, b, c, d, cθ + d > 0. θ, θ = [a 0, a,..., a n, θ n ] = p n ζ + p n 2 ζ + 2., θ n >, p n / θ n. (42),, ω = a θ n + b c θ n + d. a = ap n + b, b = ap n 2 + b 2, c = cp n + d, d = cp n 2 + d 2. 50

51 a, b, c, d, a d b c = (ad bc)(p n 2 p n 2 ) = ±. 5.6, δ, δ,, p n = θ + δ, δ <, p n 2 = 2 θ + δ 2, δ <. c = (cθ + d) + d = (cθ + d) 2 + cδ, cδ 2. cθ + d > 0 0 < 2 <, n 0 < c < d. n ζ = θ n, ζ, θ,.2 ω. 2 2.,.,.. ω ω = [a 0, a, a 2,..., a n, ω n ] (n = 0,, 2,...) (43), n 0 m, ω n0 = ω n0 +m = = ω n0 +jm = (j = 0,, 2,...) (44), ω., ω. (44) m ω., n 0 = 0, ω., ω. 5

52 ω (43), (44) ( m ), ω n0., ω = [a 0, a, a 2,..., a n0, ω n0 ] = [a 0, a, a 2,..., a n0, a n0,..., a n0 +m, ω n0 ] = [a 0, a, a 2,..., a n0, a n0,..., a n0 +m, a n0,..., a n0 +m, ω n0 ] =, a n0,..., a n0 +m.., ω, ω = [a 0, a,..., a n0, ȧ n0,..., ȧ n0 +m ] ω = [ȧ 0,..., ȧ m ] = = [2,,,, 4] = [2,,,, 4,,, ]. 7, = = [ 4,,, ] ω, ω = [ȧ 0, a,..., ȧ n ], n. a 0, a 0., ω >., ω = [a 0, a,..., a n, ω] = p n ω + p n 2 ω

53 , ω 2 + ( 2 p n )ω p n 2 = 0., ω 2., f(x) = x 2 + ( 2 p n )x p n 2, n >, 2., < p n 2 < p n, 0 < 2, n =, p =, p 0 = a 0, q = 0, q 0 =, f(0) = p n 2 < 0, f( ) = 2 + p n p n 2 > 0., f(x) = 0 < x < 0., f(x) = 0 ω ω 2. ω >, < ω < 0., ω ω, ω = [a 0, a,..., a n, ȧ n, a n+,..., ȧ n+k ]. ω, 4.7 ω., ω = [ȧ n, a n+,..., ȧ n+k ], ω, , 3. ω = [a 0, a,..., a n, ω ] = p n ω + p n 2 ω + 2., ω ω., 9.6, ω 2. ω ω = [a 0, a, a 2,..., a n, ω n ] (n = 0,, 2,...), n, n 2 ω n = ω n2, n < n 2 53

54 , ω., ω n = [a n, a n+, a n+2,..., a n2, ω n2 ] = [a n, a n+, a n+2,..., a n2, ω n ] = [ȧ n, a n+, a n+2,..., ȧ n2 ], ω = [a 0, a, a 2,..., a n, ω n ] = [a 0, a, a 2,..., a n, ȧ n, a n +, a n +2,..., ȧ n2 ] θ 2, 2 ax 2 + bx + c = 0, gcd(a, b, c) =, D = b 2 4ac > 0., θ θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...), p n / θ, 9.5, θ = p n θ n + p n 2 θ n + 2. A n = ap 2 n + bp n + cqn, 2 B n = 2ap n p n 2 + b(p n 2 + p n 2 ) + 2c 2, C n = ap 2 n 2 + bp n cqn 2 2, θ n 2 A n x 2 + B n x + C n = 0, B 2 n 4A n C n = D. n, 5.6, δ n,, p n = θ + δ n, δ n <. ( A n = a θ + δ ) 2 ( n + b θ + δ ) n + cqn 2 = (aθ 2 + bθ + c)qn 2 + 2aθδ n + a δ2 n qn 2 + bδ n = 2aθδ n + a δ2 n qn 2 + bδ n. 54

55 ,, C n = A n, A n < 2 aθ + a + b. C n < 2 aθ + a + b., B 2 n 4A n C n = D > 0, B 2 n 4 A n C n + D < 4(2 aθ + a + b ) 2 + D., B n < 4(2 aθ + a + b ) 2 + D. A n, B n, C n n, (A n, B n, C n )., n, n 2, n 3, n < n 2 < n 3 (A n, B n, C n ) = (A n2, B n2, C n2 ) = (A n3, B n3, C n3 ). (A, B, C)., θ n, θ n2, θ n3 2 Ax 2 + Bx + C = 0, 2., θ. 2.5 θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...), θ n 2. θ 2 f(x) = ax 2 + bx + c = 0, θ. θ 2, < θ < 0, < θ. (f n (x)) ( f 0 (x) = f(x), f n (x) = x 2 f n a n + ) x (n =, 2,...), n 0, f n (x) 2, f n (θ n ) = 0 55

56 ., (θ n ) θ 0 = θ, θ n =, n 0. θ n a n (n =, 2,...) f n (θ n ) = 0, n. n 0, θ n > θ n, < θ n < 0. n = 0 θ 0 = θ. θ n, < θ n < 0. a n, θ n a n <.,, < θ n a n < 0. < θ n < 0., θ n 2., n θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...). 2.4, 2, n, m θ n = θ m, n < m. n, θ n = a n + θ n = a n θ n + θ n, θ m = a m + θ m = a m θ m + θ m. θ n θ n 2,, 9.3 θ n = a n + θ n, θ m = a m + θ m. 56

57 θ n = θ m θ n = θ m, θ n θ m = a n a m. 2.5, θ n, θ m 2, θ n, θ m 0., < θ n < θ n θ m < θ m <., < a n a m <. a n a m, 0., a n = a m., θ n = θ m.,., θ. θ 0 = θ m n 2.7 2, 2. θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...). 2.4, 2, n, m θ n = θ m, n < m., θ = [a 0, a, a 2,..., a n, θ n ] = [a 0, a, a 2,..., a n, a n,..., a m, θ m ] = [a 0, a, a 2,..., a n, a n,..., a m, θ n ],.7, θ n = [a n,..., a m, θ n ]., θ n. 2.2, θ n 2. n, 3., θ = p n θ n + p n 2 θ n

58 , 2.5, p n 2 p n 2 = ( ) n 2 =., θ θ n. n, 2.5, 2.2, θ n+ 2., θ n+ n, θ θ n a, b, c, x, y f(x, y) = ax 2 + bxy + cy 2 (45) 2 2.,, 2. gcd(a, b, c) =, f(x, y). 2,, 2. (45), f(x, y) = x (ax + b2 ) y = = [ x [ x ( b + y ] ax + b y 2 y b 2 x + cy ) 2 x + cy ] y a b/2 x b/2 c y, 2 a b/2 (46) b/2 c 2 f. D = b 2 4ac 2 f. f (46) A, D = (4ac b 2 2a b ) = = 4 det A (47) b 2c. 3. f(x, y) = ax 2 + bxy + cy 2 2, D., 4af(x, y) = (2ax + by) 2 Dy 2. 58

59 D = b 2 4ac, 4af(x, y) = 4a 2 x 2 + 4abxy + 4acy 2 = (4a 2 x 2 + 4abxy + b 2 y 2 ) (b 2 y 2 4acy 2 ) = (2ax + by 2 ) 2 (b 2 4ac)y 2 = (2ax + by) 2 Dy 2. 2 f(x, y), (i) f f(x, y). (ii) f x, y Z f(x, y) 0. (iii) f x, y Z f(x, y) 0. (iv) f f., x, y Z, f(x, y) = 0 x = y = 0. (v) f f., x, y Z, f(x, y) = 0 x = y = 0.., f,. 3.2 f(x, y) = ax 2 + bxy + cy 2 2, D = b 2 4ac f. a 0, (i) D > 0 f. (ii) D = 0 a > 0 f. (iii) D = 0 a < 0 f. (iv) D < 0 a > 0 f. (v) D < 0 a < 0 f. a = 0, (i) D > 0 f. (ii) D = 0 c > 0 f, 0. (iii) D = 0 c < 0 f, 0. (iv) D = 0 c = 0 f 0. a 0 : (i) ( ) 3., f(x, y). (ii) ( ) a > 0, D = 0 3., 4af(x, y) = (2ax + by) 2. a > 0, f(x, y)., 2ax + by = 0 f(x, y) = 0, f. 59

60 (iii) ( ) (iv) ( ), (ii). 3., D < 0 a > 0, f(x, y)., x, y Z f(x, y) = 0 = 2ax + by = y = 0 = x = y = 0., f. (v) ( ) (iv). (i) (v),, ( ). a = 0 : (i) ( ) D = b 2, b 0., f(x, y) = bxy + cy 2. (ii) ( ) D = b 2, b = 0., f(x, y) = cy 2., y = 0 x f(x, y) = 0, f., y 0 f(x, y) 0, f 0. (iii) ( ) (ii). (iv) ( ) D = b 2, b = 0., a = b = c = 0., f 0. (i) (iv)., a = 0 D = b 2 0.,., ( ). 3.3 f(x, y) = ax 2 + bxy + cy 2 2, D = b 2 4ac f.,. (i) f D > 0. (ii) f, 0 D = 0 a > 0 c > 0. (iii) f, 0 D = 0 a < 0 c < 0. (iv) f 0 D = 0 a = c = 0. (v) f D < 0 a > 0. (vi) f D < 0 a < 0. (ii) ( ), D = 0 c > 0, 2 f(x, y) x, y, D = 0 a > 0. (iii) ( )., f(x, y) = ax 2 + bxy + cy 2, y =, x 2 g(x) = f(x, ) = ax 2 + bx + c (48) 60

61 ., g(x) = ax 2 + bx + c, ( ) x f(x, y) = y 2 g = ax 2 + bxy + cy 2 y., g(x) f(x, y) 2, f(x, y) g(x) 2., f(x, y) g(x) [ ] f(x, y) = ax 2 + bxy + cy 2 = x y A x, A = a b/2 y b/2 c, P = p q GL 2 (Z) r s x = P y x y, (49),, f(x, y) = x = px + qy, y = rx + sy [ x y ] t P AP x y = a x 2 + b x y + c y 2., a = ap 2 + bpr + cr 2 (= f(p, r)), b = 2apq + b(ps + qr) + 2crs, c = aq 2 + bqs + cs 2 (= f(q, s)). (50), a, b, c., (49), a x 2 + b x y + cy 2, f(x, y) P [ ] f(x, y) = ax 2 + bxy + cy 2 = x y A x, A = a b/2, (5) y b/2 c ] f(x, y ) = a x 2 + b x y + c y 2 = [x y A x, A = a b /2 (52) b /2 c y 6

62 f = f, f = f a = a, b = b, c = c., f = f A = A., f f, P GL 2 (Z) A = t P AP (53)., t P P. f(x, y) f (x, y ) f f., (49) f(x, y) f (x, y ). P = p q, 2 (53) (50) r s., det P =, det P = f(x, y) = ax 2 + bxy + cy 2, f (x, y ) = a x 2 + b x y + c y 2, f = f, n, n 2 Z f(n, n 2 ) = f (n, n 2 ). ( ) f = f, a = a, b = b, c = c, n, n 2 Z f(n, n 2 ) = an 2 + bn n 2 + cn 2 2 = a n 2 + b n n 2 + c n 2 2 = f (n, n 2 ). ( ), a = f(, 0) = f (, 0) = a, c = f(0, ) = f (0, ) = c., a + b + c = f(, ) = f (, ) = a + b + c., b = b. 4.2 f, f 2. f f f, f. 62

63 f, f A, A (5), (52), f f, P = p q GL 2 (Z), (50). r s g = gcd(a, b, c). (50) g a, b, c. g >, f., g = f, f 2, D, D f, f., A, A f, f. f f, P GL 2 (Z) A = t P AP., (47), D = 4 det A = 4 det t P AP = 4 det t P det A det P = 4(det P ) 2 det A = 4 det A = D f(x, y), f (x, y ), f (x, y ) 2, A, A, A. ( ) E = 0, A = t EAE. 0 ( ) f f, P GL 2 (Z), A = t P AP., A = t P A P, f f. ( ) f f f f., P GL 2 (Z), A = t P AP.,, P GL 2 (Z), A = t P A P., A = t (P P )A(P P ), f f. D 0, D 0 (mod 4). D 2 ax 2 + bx + c, D = b 2 4ac, 2 ax 2 + bx + c = 0 b + D, 2a b D 2a,, 2. 9., 2. 63

64 4.5 f(x, y) = ax 2 + bxy + cy 2, f (x, y ) = a x 2 + b x y + c y 2 D 2., θ f 2 ax 2 + bx + c, θ f 2 a x 2 + b x + c. (i) f f θ = θ. (ii) f f θ θ. (iii) f f θ θ., θ = ( b + D)/2a, θ = ( b + D)/2a. (i) ( ) f f, 2, θ = θ. ( ) θ = θ,, a, b, a, b Z D Q, b + D 2a = b + D 2a. (a b ab ) + (a a ) D = 0. a b a b = a a = 0., a = a, b = b., f f, b 2 4ac = b 2 4a c., c = c., f = f. (ii) A, A f, f, A = a b/2, A = a b /2. b/2 c b /2 c ( ) f f., P = p q SL 2 (Z), r s A = t P AP. (33). ω = P θ, 9.5 ω = θ., θ = P θ. ( ) θ θ., P = p r, a = ap 2 + bpr + cr 2,, 9.5, θ = ( b + D)/2a., b + D 2a = b + D 2a. q SL 2 (Z), θ = P θ. s b = 2apq + b(ps + qr) + 2crs, (54) c = aq 2 + bqs + cs 2 64

65 , (a b a b ) + (a a ) D = 0. a, b, a, b Z D Q, a b a b = a a = 0., a = a, b = b., (54) A = t P AP., f f. (iii) (ii) , f ( ), a > 0 ( a < 0)., 2., 2 2, ( ) ( )., 4 (50), P GL 2 (Z), ( ) 2 P 2 ( )., f(x, y) 2, f(x, y) 2. f(x, y) P f (x, y ), P f(x, y) f (x, y )., f, f A, A, A = t P AP A = t P ( A)P., 2. 2 f(x, y) = ax 2 + bxy + cy 2, a > 0, c 0 b 2 4ac 0, c > 0. 2 f(x, y) = ax 2 + bxy + cy 2 2, a < b a < c 0 b a = c (55) f(x, y) 2 g(x) θ., f 2, θ >, 2 Re θ < 2 θ =, 2 Re θ 0., F SL 2 (Z),. f 2 θ F 65

66 θ θ, g(x) 2., 2 Re θ = θ + θ = b a, θ = θθ = c a. (56) f, (55) 0 b a = c a < b a < c a. (57), (56) θ F., θ F, (56) (57)., (55), f. 5.2 D < 0 2. f(x, y) = ax 2 + bxy + cy 2 D 2, (55) b a c., D = 4ac b 2 4b 2 b 2 = 3b 2., b D 3.., b., b, 4ac = b 2 D (a, c)., a, c., D, D f, f 2. f(x, y) = ax 2 + bxy + cy 2., D = b 2 4ac f, g(x) = ax 2 + bx + c f 2, θ = ( b + D)/2a g(x). D < 0 a > 0, θ H. F SL 2 (Z), θ θ 0 F ( 8.5). 9.5, θ 0 2 g 0 (x ). g 0 (x ) 2 f 0 (x, y ), 5., f 0., θ θ 0, 4.5, f f ,. 66

67 f(x, y) = ax 2 + bxy + cy 2 f (x, y ) = a x 2 + b x y + c y 2 2., g(x), g (x ) f, f 2, θ g(x), θ g (x ). f f, 4.5, θ θ., F SL 2 (Z), 5., θ, θ F., 8.6, θ = θ., 4.5, f = f. 5.5 f, f 2., f f, f f 2 f 0. ( ) 5.3, 2 f 0 f f 0., 2 f 0 f f , f 0 = f 0. ( ) f f 0, f 0 f, f f. 6 2 D, 0. D 2 f(x, y) = ax 2 + bxy + cy 2 2, f a > 0, a b + c > 0, a + b + c < 0, c < 0 (58). f(x, y) 2 g(x) = ax 2 + bx + c, g(0) = c, g() = a + b + c, g( ) = a b + c, (58) a > 0, g( ) > 0, g(0) < 0, g() < 0 (59)., 2b = g() g( ), f b < 0. f, 0, g. 6. f(x, y) 2, g(x) f 2, θ g(x)., f(x, y) 2, θ 2,, < θ < 0, < θ (60)., θ θ, g(x) 2. 67

68 f(x, y) = ax 2 +bxy+cy 2, D = b 2 4ac, g(x) = ax 2 +bx+c, θ = ( b+ D)/2a, θ = ( b D)/2a., g(x), g(x) = a(x θ)(x θ). (6) f, (59). θ < θ a > 0, (6), x, < 0, θ < x < θ, g(x) = = 0, x = θ x = θ, > 0, x < θ θ < x. g( ) > 0, g(0) < 0, g() < 0, < θ < 0 < θ., (60)., (60), D a = θ θ > 0 a > 0., (6), g( ) > 0, g(0) < 0, g() < 0., (59). 6.2 D 0., D 2. f D 2, f(x, y) = ax 2 + bxy + cy 2, 6. 0 < b + D 2a < < b + D 2a,, 0 < b + D < 2a < b + D. b < D., b., b, 4ac = b 2 D (a, c)., a, c., D, D D > 0 2 f, f 2. 68

69 f(x, y) = ax 2 + bxy + cy 2., D = b 2 4ac f, g(x) = ax 2 + bx + c f 2, θ = ( b + D)/2a g(x). D 0, θ , θ 2 θ , θ 0 2 g 0 (x ). g 0 (x ) 2 f 0 (x, y ), 6., f 0., θ θ 0, 4.5, f f n, f(x, y) = ax 2 + bxy + cy 2 2. x, y ax 2 + bxy + cy 2 = n (62) (x, y), n 2 f., gcd(x, y) =,, n f. (62) g = gcd(x, y) > (x, y), g 2 n, (62) g 2, a ( ) 2 x + b x g g y ( ) 2 y g + c = n g g 2, n/g 2 f(x, y). P = p q GL 2 (Z), r s x = P y, (50) f(x, y) = ax 2 + bxy + cy x y f (x, y ) = a x 2 + b x y + c y 2., x = P x, P = s y y det P r q, det P = ± p, f(x, y) = n (x, y) f (x, y ) = n (x, y )., n 2 f, n f P 2 f., n, f(x, y), f (x, y ) 2, n f, f. 69

70 7. n D 2, z z 2 D (mod 4n) (63). (x 0, y 0 ) (62). gcd(x 0, y 0 ) =, z 0, w 0 Z x 0 z 0 + y 0 w 0 =., x = P y x y, P = x 0 w 0 y 0 z 0, P GL 2 (Z), 4 (50) f(x, y) = ax 2 + bxy + cy f (x, y ) = nx 2 + b x y + c y 2. f f, 4.3 f, f., b 2 4nc = D., z = b (63)., (63) z = b, c Z b 2 4nc = D., f(x, y) = nx 2 + bxy + cy 2, f D, (62) (x, y) = (, 0). 70

71 [] : 2,, 97. [2] :,, 972. [3] :,, 98. [4] :,, 992. [5], : UBASIC,, 994. [6] G. M., E. M. : I,,

72 , , , , ,

13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1 I, A 25 8 24 1 1.1 ( 3 ) 3 9 10 3 9 : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 6 3 9 10 3 9 : 6 3 + 3 2 + 1 = 25 25 10 : 6 3 + 3 3

More information

O E ( ) A a A A(a) O ( ) (1) O O () 467

O E ( ) A a A A(a) O ( ) (1) O O () 467 1 1.0 16 1 ( 1 1 ) 1 466 1.1 1.1.1 4 O E ( ) A a A A(a) O ( ) (1) O O () 467 ( ) A(a) O A 0 a x ( ) A(3), B( ), C 1, D( 5) DB C A x 5 4 3 1 0 1 3 4 5 16 A(1), B( 3) A(a) B(b) d ( ) A(a) B(b) d AB d = d(a,

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law

(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................

More information

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >

More information

untitled

untitled .. 3. 3 3. 3 4 3. 5 6 3 7 3.3 9 4. 9 0 6 3 7 0705 φ c d φ d., φ cd, φd. ) O x s + b l cos s s c l / q taφ / q taφ / c l / X + X E + C l w q B s E q q ul q q ul w w q q E E + E E + ul X X + (a) (b) (c)

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

y a y y b e

y a y y b e DIGITAL CAMERA FINEPIX F1000EXR BL01893-100 JA http://fujifilm.jp/personal/digitalcamera/index.html y a y y b e 1 2 P 3 y P y P y P y P y P Q R P R E E E E Adv. SP P S A M d F N h Fn R I P O X Y b I A

More information

量子力学A

量子力学A c 1 1 1.1....................................... 1 1............................................ 4 1.3.............................. 6 10.1.................................. 10......................................

More information

FinePix Z5fd 使用説明書

FinePix Z5fd 使用説明書 http://fujifilm.jp/ BL00582-100(1) q 2 A B 3 ab a 4 P. 5 6 6 P. AC AC P.P. P. P. xd xd 7 P.79 8 30 a ON 54 b 36 36 74 75 81 89 9 - e d * j p p p S S T H 10 p V r w U 11 12 e s ep e H DISP/BACK 13 .30.52

More information

122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin

122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin 121 6,.,,,,,,. 2, 1. 6.1,.., M, A(2 R).,. 49.. Oxy ( ' ' ), f Oxy, O 1 x 1 y 1 ( ' ' ). A (p q), A 0 (p q). y q A q q 0 y 1 q A O 1 p x 1 O p p 0 p x 6.1: ( ), 6.1, 122 6 A 0 (p 0 q 0 ). ( p 0 = p cos

More information

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000 1 ( S/E) 006 7 30 0 (1 ) 01 Excel 0 7 3 1 (-4 ) 5 11 5 1 6 13 7 (5-7 ) 9 1 1 9 11 3 Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) 17 31 17 3 18 33 19 34 35 36 Excel 3 4 (11-13 ) 5 41 5 4

More information

i

i 14 i ii iii iv v vi 14 13 86 13 12 28 14 16 14 15 31 (1) 13 12 28 20 (2) (3) 2 (4) (5) 14 14 50 48 3 11 11 22 14 15 10 14 20 21 20 (1) 14 (2) 14 4 (3) (4) (5) 12 12 (6) 14 15 5 6 7 8 9 10 7

More information

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

More information

FinePix Z3 使用説明書

FinePix Z3 使用説明書 http://fujifilm.jp/ BL00546-100(1) q 2 A B 3 ab a 4 OPEN OPEN P. 5 6 AC P. P.P. P. P. xd xd 7 DISP/BACK P.74 W T MENU /OK 8 - e d * j p p p S S T H 9 p V r w U 10 11 DISP/BACK MENU K W T e sauto 43464747

More information

BL01622-100 JA DIGITAL CAMERA FINEPIX F770EXR http://fujifilm.jp/personal/digitalcamera/index.html y a y y b e y DISP/BACK 1 2 P 3 y P y P y P y P y P Q R P R E d F N h Fn b R I P O X Y n E E E I Adv.

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

3/4/8:9 { } { } β β β α β α β β

3/4/8:9 { } { } β β β α β α β β α β : α β β α β α, [ ] [ ] V, [ ] α α β [ ] β 3/4/8:9 3/4/8:9 { } { } β β β α β α β β [] β [] β β β β α ( ( ( ( ( ( [ ] [ ] [ β ] [ α β β ] [ α ( β β ] [ α] [ ( β β ] [] α [ β β ] ( / α α [ β β ] [ ] 3

More information

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi II (Basics of Probability Theory ad Radom Walks) (Preface),.,,,.,,,...,,.,.,,.,,. (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

( ) FAS87 FAS FAS87 v = 1 i 1 + i

( ) FAS87 FAS FAS87 v = 1 i 1 + i ( ) ( 7 6 ) ( ) 1 6 1 18 FAS87 FAS87 7 1 FAS87 v = 1 i 1 + i 10 14 6 6-1 - 7 73 2 N (m) N L m a N (m) L m a N m a (m) N 73 9 99 18 4-2 - 4 143 2 145 3 37 4 37 4 40 6 40 6 41 10 41 10 13 10 14 4 24 3 145

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

04年度LS民法Ⅰ教材改訂版.PDF

04年度LS民法Ⅰ教材改訂版.PDF ?? A AB A B C AB A B A B A B A A B A 98 A B A B A B A B B A A B AB AB A B A BB A B A B A B A B A B A AB A B B A B AB A A C AB A C A A B A B B A B A B B A B A B B A B A B A B A B A B A B A B

More information

BL01479-100 JA DIGITAL CAMERA FINEPIX F600EXR http://fujifilm.jp/personal/digitalcamera/index.html y a y y b 6 y DISP/BACK 1 2 3 P y P y P y P y P y P Q R P R E O E E Adv. SP M A S P d F N h b R I P O

More information

FnePix S8000fd 使用説明書

FnePix S8000fd 使用説明書 http://fujifilm.jp/ BL00677-100(1) 2 27 aon b 39 39 88 88 100 3 4 B N < M > S e> d * j p p p S T H G p p p V r w U 5 6 e> B N ep ep 10 80 1. 2m 3. 2 m 1 cm 10 cm 60 mm35mm 30 cm3.0 m > e r DISP/BACK

More information

ÿþ

ÿþ I O 01 II O III IV 02 II O 03 II O III IV III IV 04 II O III IV III IV 05 II O III IV 06 III O 07 III O 08 III 09 O III O 10 IV O 11 IV O 12 V O 13 V O 14 V O 15 O ( - ) ( - ) 16 本 校 志 望 の 理 由 入 学 後 の

More information

学習の手順

学習の手順 NAVI 2 MAP 3 ABCD EFGH D F ABCD EFGH CD EH A ABC A BC AD ABC DBA BC//DE x 4 a //b // c x BC//DE EC AD//EF//BC x y AD DB AE EC DE//BC 5 D E AB AC BC 12cm DE 10 AP=PB=BR AQ=CQ BS CS 11 ABCD 1 C AB M BD P

More information

「産業上利用することができる発明」の審査の運用指針(案)

「産業上利用することができる発明」の審査の運用指針(案) 1 1.... 2 1.1... 2 2.... 4 2.1... 4 3.... 6 4.... 6 1 1 29 1 29 1 1 1. 2 1 1.1 (1) (2) (3) 1 (4) 2 4 1 2 2 3 4 31 12 5 7 2.2 (5) ( a ) ( b ) 1 3 2 ( c ) (6) 2. 2.1 2.1 (1) 4 ( i ) ( ii ) ( iii ) ( iv)

More information

44 4 I (1) ( ) (10 15 ) ( 17 ) ( 3 1 ) (2)

44 4 I (1) ( ) (10 15 ) ( 17 ) ( 3 1 ) (2) (1) I 44 II 45 III 47 IV 52 44 4 I (1) ( ) 1945 8 9 (10 15 ) ( 17 ) ( 3 1 ) (2) 45 II 1 (3) 511 ( 451 1 ) ( ) 365 1 2 512 1 2 365 1 2 363 2 ( ) 3 ( ) ( 451 2 ( 314 1 ) ( 339 1 4 ) 337 2 3 ) 363 (4) 46

More information

i ii i iii iv 1 3 3 10 14 17 17 18 22 23 28 29 31 36 37 39 40 43 48 59 70 75 75 77 90 95 102 107 109 110 118 125 128 130 132 134 48 43 43 51 52 61 61 64 62 124 70 58 3 10 17 29 78 82 85 102 95 109 iii

More information

i ii iii iv v vi vii ( ー ー ) ( ) ( ) ( ) ( ) ー ( ) ( ) ー ー ( ) ( ) ( ) ( ) ( ) 13 202 24122783 3622316 (1) (2) (3) (4) 2483 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 11 11 2483 13

More information

o 2o 3o 3 1. I o 3. 1o 2o 31. I 3o PDF Adobe Reader 4o 2 1o I 2o 3o 4o 5o 6o 7o 2197/ o 1o 1 1o

o 2o 3o 3 1. I o 3. 1o 2o 31. I 3o PDF Adobe Reader 4o 2 1o I 2o 3o 4o 5o 6o 7o 2197/ o 1o 1 1o 78 2 78... 2 22201011... 4... 9... 7... 29 1 1214 2 7 1 8 2 2 3 1 2 1o 2o 3o 3 1. I 1124 4o 3. 1o 2o 31. I 3o PDF Adobe Reader 4o 2 1o 72 1. I 2o 3o 4o 5o 6o 7o 2197/6 9. 9 8o 1o 1 1o 2o / 3o 4o 5o 6o

More information

example2_time.eps

example2_time.eps Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank

More information

10:30 12:00 P.G. vs vs vs 2

10:30 12:00 P.G. vs vs vs 2 1 10:30 12:00 P.G. vs vs vs 2 LOGIT PROBIT TOBIT mean median mode CV 3 4 5 0.5 1000 6 45 7 P(A B) = P(A) + P(B) - P(A B) P(B A)=P(A B)/P(A) P(A B)=P(B A) P(A) P(A B) P(A) P(B A) P(B) P(A B) P(A) P(B) P(B

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi I (Basics of Probability Theory ad Radom Walks) 25 4 5 ( 4 ) (Preface),.,,,.,,,...,,.,.,,.,,. (,.) (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios,

More information

1 2 3

1 2 3 BL01604-103 JA DIGITAL CAMERA X-S1 http://fujifilm.jp/personal/digitalcamera/index.html 1 2 3 y y y y y c a b P S A M C1/C2/C3 E E E B Adv. SP F N h I P O W X Y d ISO Fn1 Fn2 b S I A b X F a K A E A Adv.

More information

178 5 I 1 ( ) ( ) 10 3 13 3 1 8891 8 3023 6317 ( 10 1914 7152 ) 16 5 1 ( ) 6 13 3 13 3 8575 3896 8 1715 779 6 (1) 2 7 4 ( 2 ) 13 11 26 12 21 14 11 21

178 5 I 1 ( ) ( ) 10 3 13 3 1 8891 8 3023 6317 ( 10 1914 7152 ) 16 5 1 ( ) 6 13 3 13 3 8575 3896 8 1715 779 6 (1) 2 7 4 ( 2 ) 13 11 26 12 21 14 11 21 I 178 II 180 III ( ) 181 IV 183 V 185 VI 186 178 5 I 1 ( ) ( ) 10 3 13 3 1 8891 8 3023 6317 ( 10 1914 7152 ) 16 5 1 ( ) 6 13 3 13 3 8575 3896 8 1715 779 6 (1) 2 7 4 ( 2 ) 13 11 26 12 21 14 11 21 4 10 (

More information

+ + + + n S (n) = + + + + n S (n) S (n) S 0 (n) S (n) 6 4 S (n) S (n) 7 S (n) S 4 (n) 8 6 S k (n) 0 7 (k + )S k (n) 8 S 6 (n), S 7 (n), S 8 (n), S 9 (

+ + + + n S (n) = + + + + n S (n) S (n) S 0 (n) S (n) 6 4 S (n) S (n) 7 S (n) S 4 (n) 8 6 S k (n) 0 7 (k + )S k (n) 8 S 6 (n), S 7 (n), S 8 (n), S 9 ( k k + k + k + + n k 006.7. + + + + n S (n) = + + + + n S (n) S (n) S 0 (n) S (n) 6 4 S (n) S (n) 7 S (n) S 4 (n) 8 6 S k (n) 0 7 (k + )S k (n) 8 S 6 (n), S 7 (n), S 8 (n), S 9 (n), S 0 (n) 9 S (n) S 4

More information

α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2

α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2 1. 2. 3. 4. 5. 6. 7. 8. N Z 9. Z Q 10. Q R 2 1. 2. 3. 4. Zorn 5. 6. 7. 8. 9. x x x y x, y α = 2 2 α x = y = 2 1 α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn

More information

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p a a a a y y ax q y ax q q y ax y ax a a a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p y a xp q y a x p q p p x p p q p q y a x xy xy a a a y a x

More information

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980 % 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2006.11.20 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory

More information

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-7 O1-8 O1-9 O1-10 O1-11 O1-12 O1-13 O1-14 O1-15 O1-16 O1-17 O1-18 O1-19 O1-20 O1-21 O1-22 O1-23 O1-24 O1-25 O1-26 O1-27 O1-28 O1-29 O1-30 O1-31 O1-32 O1-33 O1-34 O1-35

More information

エラーコード 一 覧 コード 名 称 : 審 査 チェックエラーコード => 返 戻 事 由 と 共 有 する コード 概 要 : 審 査 において 一 次 チェック 資 格 チェックを 行 ったときにエラーとなった 項 目 に 設 定 するコード 及 び 返 戻 一 覧 に 出 力 する 返 戻

エラーコード 一 覧 コード 名 称 : 審 査 チェックエラーコード => 返 戻 事 由 と 共 有 する コード 概 要 : 審 査 において 一 次 チェック 資 格 チェックを 行 ったときにエラーとなった 項 目 に 設 定 するコード 及 び 返 戻 一 覧 に 出 力 する 返 戻 12.エラーコード 一 覧 表 エラーコードは 随 時 更 新 されます 最 新 のエラーコード 表 については 本 会 ホームページより ダウンロードできます エラーコード 一 覧 コード 名 称 : 審 査 チェックエラーコード => 返 戻 事 由 と 共 有 する コード 概 要 : 審 査 において 一 次 チェック 資 格 チェックを 行 ったときにエラーとなった 項 目 に 設 定 するコード

More information

(1) (François Viète : ) 1593 (Eectionum Geometricarum Canonica Recensio) 2 ( 1 p.372 pp ) 3 A D BAC CD CE DE BC F B A F C BF F D F C (

(1) (François Viète : ) 1593 (Eectionum Geometricarum Canonica Recensio) 2 ( 1 p.372 pp ) 3 A D BAC CD CE DE BC F B A F C BF F D F C ( 12 (Euclid (Eukleides : EÎkleÐdhc) : 300 ) (StoiqeÐwsic) ( ) 2 ( ) 2 16 3 17 18 (Introductio in Analysin Innitorum : 1748 ) 120 1 (1) (François Viète : 15401603) 1593 (Eectionum Geometricarum Canonica

More information

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

( ) >

( ) > (Ryūta Hashimoto) α α p q < p/q α q Lagrange 0 0. 3.4.4.96.5.5.5.4 <

More information

2008 (2008/09/30) 1 ISBN 7 1.1 ISBN................................ 7 1.2.......................... 8 1.3................................ 9 1.4 ISBN.............................. 12 2 13 2.1.....................

More information

道路施設基本データ作成入力書式マニュアル(中国地方整備局版)平成20年10月

道路施設基本データ作成入力書式マニュアル(中国地方整備局版)平成20年10月 道 路 BOX 等 に 関 する 調 査 表 記 入 マニュアル D080 D080 道 路 B O X 基 本 この 調 査 表 は 道 路 BOX 等 に 関 する 基 本 的 データを 登 録 するためのものであ る なお ここで 取 扱 う 道 路 BOX 等 とは 管 理 する 道 路 に 対 し 平 行 ( 縦 断 方 向 ) しているアンダーパス 等 の 箇 所 などに 設 けられたボックスカルバート

More information

http://know-star.com/ 3 1 7 1.1................................. 7 1.2................................ 8 1.3 x n.................................. 8 1.4 e x.................................. 10 1.5 sin

More information

分科会(OHP_プログラム.PDF

分科会(OHP_プログラム.PDF 2B-11p 2B-12p 2B-13p 2B-14p 2B-15p 2C-3p 2C-4p 2C-5p 2C-6p 2C-7p 2D-8a 2D-9a 2D-10a 2D-11a 2D-12a 2D-13a 2E-1a 2E-2a 2E-3a 2E-4a 2E-5a 2E-6a 2F-3p 2F-4p 2F-5p 2F-6p 2F-7p 2F-8p 2F-9p 2F-10p 2F-11p 2F-12p

More information

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e No. 1 1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e I X e Cs Ba F Ra Hf Ta W Re Os I Rf Db Sg Bh

More information

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

untitled

untitled i ii iii iv v 43 43 vi 43 vii T+1 T+2 1 viii 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 a) ( ) b) ( ) 51

More information

AccessflÌfl—−ÇŠš1

AccessflÌfl—−ÇŠš1 ACCESS ACCESS i ii ACCESS iii iv ACCESS v vi ACCESS CONTENTS ACCESS CONTENTS ACCESS 1 ACCESS 1 2 ACCESS 3 1 4 ACCESS 5 1 6 ACCESS 7 1 8 9 ACCESS 10 1 ACCESS 11 1 12 ACCESS 13 1 14 ACCESS 15 1 v 16 ACCESS

More information

繖 7 縺6ァ80キ3 ッ0キ3 ェ ュ ョ07 縺00 06 ュ0503 ュ ッ 70キ ァ805 ョ0705 ョ ッ0キ3 x 罍陦ァ ァ 0 04 縺 ァ タ0903 タ05 ァ. 7

繖 7 縺6ァ80キ3 ッ0キ3 ェ ュ ョ07 縺00 06 ュ0503 ュ ッ 70キ ァ805 ョ0705 ョ ッ0キ3 x 罍陦ァ ァ 0 04 縺 ァ タ0903 タ05 ァ. 7 30キ36ヲ0 7 7 ュ6 70キ3 ョ6ァ8056 50キ300 縺6 5 ッ05 7 07 ッ 7 ュ ッ04 ュ03 ー 0キ36ヲ06 7 繖 70キ306 6 5 0 タ0503070060 08 ョ0303 縺0 ァ090609 0403 閨0303 003 ァ 0060503 陦ァ 06 タ09 ァ タ04 縺06 閨06-0006003 ァ ァ 04 罍ァ006 縺03 0403

More information

2

2 1 2 3 4 5 6 7 8 9 10 I II III 11 IV 12 V 13 VI VII 14 VIII. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 _ 33 _ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 VII 51 52 53 54 55 56 57 58 59

More information

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 () - 1 - - 2 - - 3 - - 4 - - 5 - 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

More information

10月表紙BB.indd

10月表紙BB.indd Q F Q Q Q Q Q Q Q F QF F QF QF F Q F 三 Q QF Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q P F F F F F F F F QF Q F PNAYQ RX Y F A P P P P Q F P Q F R P Y F Q R QF P QF P A Q F P A F Q P A F A Y F Q P A F P Q F PNAYQ RX

More information

憲法h1out

憲法h1out m n mnm mnn m m m m m m. x x x ax bxc a x x bb ac a fxax bxc fxx x ax bxca b ac x x ax bxca x x x.x x x x x x xxx x x xxx x x xxx x x xx x x x axbcxdacx adbcxbd x xxx m n mnm mnn m m m m m m m m

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

B

B B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T

More information

橡6 中表紙(教育施設).PDF

橡6 中表紙(教育施設).PDF 10 i ii 14 1 2 52 53 55 56 57 63 2 3 5 6 13 13 14 14 15 1 3 10 3 4 50 11 7 6 1 45,813.68 19,120.95 3 17,979.75 63,793.43 2,501.74 1,458.27 23 4,226.95 76 2,277.28 17 1,264.41 5 1,768.60 26 2,378.96 9 2,126.92

More information

『共形場理論』

『共形場理論』 T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3

More information

Γ Ec Γ V BIAS THBV3_0401JA THBV3_0402JAa THBV3_0402JAb 1000 800 600 400 50 % 25 % 200 100 80 60 40 20 10 8 6 4 10 % 2.5 % 0.5 % 0.25 % 2 1.0 0.8 0.6 0.4 0.2 0.1 200 300 400 500 600 700 800 1000 1200 14001600

More information

2 1 17 1.1 1.1.1 1650

2 1 17 1.1 1.1.1 1650 1 3 5 1 1 2 0 0 1 2 I II III J. 2 1 17 1.1 1.1.1 1650 1.1 3 3 6 10 3 5 1 3/5 1 2 + 1 10 ( = 6 ) 10 1/10 2000 19 17 60 2 1 1 3 10 25 33221 73 13111 0. 31 11 11 60 11/60 2 111111 3 60 + 3 332221 27 x y xy

More information

nsg04-28/ky208684356100043077

nsg04-28/ky208684356100043077 δ!!! μ μ μ γ UBE3A Ube3a Ube3a δ !!!! α α α α α α α α α α μ μ α β α β β !!!!!!!! μ! Suncus murinus μ Ω! π μ Ω in vivo! μ μ μ!!! ! in situ! in vivo δ δ !!!!!!!!!! ! in vivo Orexin-Arch Orexin-Arch !!

More information

- - - - - - - - - - - - - - - - - - - - - - - - - -1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2...2...3...4...4...4...5...6...7...8...

- - - - - - - - - - - - - - - - - - - - - - - - - -1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2...2...3...4...4...4...5...6...7...8... 取 扱 説 明 書 - - - - - - - - - - - - - - - - - - - - - - - - - -1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2...2...3...4...4...4...5...6...7...8...9...11 - - - - - - - - - - - - - - - - -

More information

1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b

More information

c a a ca c c% c11 c12 % s & %

c a a ca c c% c11 c12 % s & % c a a ca c c% c11 c12 % s & % c13 c14 cc c15 %s & % c16 c211 c21% c212 c21% c213 c21% c214 c21% c215 c21% c216 c21% c23 & & % c24 c25 c311 c312 % c31 c315 c32 c33 c34 % c35 c36 c37 c411 c N N c413 c c414c

More information

i

i i ii iii iv v vi vii viii ix x xi ( ) 854.3 700.9 10 200 3,126.9 162.3 100.6 18.3 26.5 5.6/s ( ) ( ) 1949 8 12 () () ア イ ウ ) ) () () () () BC () () (

More information

86 7 I ( 13 ) II ( )

86 7 I ( 13 ) II ( ) 10 I 86 II 86 III 89 IV 92 V 2001 93 VI 95 86 7 I 2001 6 12 10 2001 ( 13 ) 10 66 2000 2001 4 100 1 3000 II 1988 1990 1991 ( ) 500 1994 2 87 1 1994 2 1000 1000 1000 2 1994 12 21 1000 700 5 800 ( 97 ) 1000

More information

橡魅力ある数学教材を考えよう.PDF

橡魅力ある数学教材を考えよう.PDF Web 0 2 2_1 x y f x y f f 2_2 2 1 2_3 m n AB A'B' x m n 2 1 ( ) 2_4 1883 5 6 2 2_5 2 9 10 2 1 1 1 3 3_1 2 2 2 16 2 1 0 1 2 2 4 =16 0 31 32 1 2 0 31 3_2 2 3_3 3_4 1 1 GO 3 3_5 2 5 9 A 2 6 10 B 3 7 11 C

More information

48 * *2

48 * *2 374-1- 17 2 1 1 B A C A C 48 *2 49-2- 2 176 176 *2 -3- B A A B B C A B A C 1 B C B C 2 B C 94 2 B C 3 1 6 2 8 1 177 C B C C C A D A A B A 7 B C C A 3 C A 187 187 C B 10 AC 187-4- 10 C C B B B B A B 2 BC

More information

入門ガイド

入門ガイド ii iii iv NEC Corporation 1998 v P A R 1 P A R 2 P A R 3 T T T vi P A R T 4 P A R T 5 P A R T 6 P A R T 7 vii 1P A R T 1 2 2 1 3 1 4 1 1 5 2 3 6 4 1 7 1 2 3 8 1 1 2 3 9 1 2 10 1 1 2 11 3 12 1 2 1 3 4 13

More information