1 ( 12 11 44 7 20 10 10 1
1 ( ( 2
10 46 11 10 10 5 8 3 2 6 9 47 2 3
48 4 2 2 ( 97 12 ) 97 12 -Spencer modulus moduli (modulus of elasticity) modulus (le) module modulus module 4
b θ a q φ p 1: 3 (le) module R 3 3 1 2 3 2 (a, b, θ) =(p, q, φ) (a, b, θ) =(q, p, φ) 3 3 3 Δ(a, b, θ) Δ(p, q, φ) Δ(a, b, θ) Δ(p, q, φ) (a, b, θ) =(p, q, φ). 3 (a, b, θ) (a, b > 0, 0 <θ<180 ) 3 3 (a, b, θ) 3 (= 3 {3 } = {(a, b, θ) R 3 ; a, b, > 0, 0 <θ<180 }. 3 3 { {3 } = (a, b, c) R 3 a, b, c > 0,a+ b>c } ; b + c > a, c + a>b 5
c a, b, θ c = a 2 + b 2 2ab cos θ. 3 (3 ) 1637 5 (1642-1727) 22 2 ( ) 1664 300 ( (1639 or 1642-1708) ) ( 262?-?) 3 ( 2) (x, y, z) a >0 C C = {(x, y, z) R 3 ; x 2 + y 2 = a 2 z 2 } (0, 0, 0) z = h (0, 0,h) ah z = h 6
z C y =0 x y 2: y =0 C x 2 = a 2 z 2 C y =0 x = az, x = az z C C (0, 0, 0) z = sx + ty + u u 0 z = sx + ty + u, x 2 + y 2 = a 2 (sx + ty + u) 2 z = sx + ty + u x y x 2 + y 2 = a 2 (sx + ty + u) 2 x y 2 px 2 +2qxy + ry 2 sx 2 + ty 2 (s, t 0) sx 2 (s 0) ( 7
s, t s, t s, t 4 sx 2 + ty 2 =1 (s, t > 0) (1) sx 2 ty 2 =1 (s, t > 0) (2) y = rx 2 (r 0) (3) sx 2 ty 2 =0 (s, t > 0) (4) (4) u 0 s =1/a 2, t =1/b 2 3 : (1) (x/a) 2 +(y/b) 2 =1 (a, b > 0) ( ) (2) (x/a) 2 (y/b) 2 =1 (a, b > 0) ( ) (3) y = rx 2 (r>0) ( ) { } = {(1) } {(2) } {(3) }. {(1) } = {(a, b) R 2 ; a, b > 0}, {(2) } = {(a, b) R 2 ; a, b > 0}, {(3) } = {r R; r>0}. (1) (a, b) (b, a) 3 1900 ) 3 2 : 3 2 ax 2 0 + bx 2 1 + cx 2 2 +2px 0 x 1 +2qx 1 x 2 +2rx 2 x 0 8
x i y j ( ) y j x 0 a 00 a 01 a 02 y 0 x 1 = a 10 a 11 a 12 y 1 (5) x 2 a 20 a 21 a 22 y 2 det(a ij ) 0. x 0 =1,x 1 = x, x 2 = y a ij det(a ij ) 0 3 (1),(2),(3) (1) u = x, v = 1y (1) (2) (2) x 2 y 2 =1 (2) u = x y, v = x + y uv =1 (1) (2) xy =1 2 x 2 0 x 1x 2 (3) y = x 2 2 x 0 x 2 x 2 1 x2 0 x 1x 2 x 0 x 2 x 2 1 (4) x 2 0 x 0x 1 : x 0 x 2 x 2 1, x2 0, x 0x 1 2 ( ) 1 x 0 x 2 x 2 1 = 0 (6) x 0 =1,x 1 = x, x 2 = y x 0 x 2 x 2 1 =0 y = x2 (1) (2) (3) ( 1 ) x 0 x 2 x 2 1 =0 x 0,x 1,x 2 9
a + b 1(a, b : ) u = x, v = 1y x, y u v x, y u, v (1) (2) (1) (2) x, y y = x 2 x 2 (1) (2) (3) 1 (6) 2 4 3 ( )3 58 3 3 C Z a + b 1(a, b : ) 2 3 3 x 0 =1,x 1 = x, x 2 = y ( ) E : y 2 + xy = x 3 36 j 1728 x 1 j 1728 (7) j j 0, 1728. 3 (8) E : y 2 = x 3 (c 4 /48)x c 6 /864 (8) j(e) =c 3 4 /Δ, Δ=(c3 4 c2 6 )/1728 (9) (7) j(e) =j 3 E E j(e) =j(e ). 10
2 3 j = c 3 4 /Δ (7) (8) j =0 j = 1728 c 4 =0,c 6 0 c 4 0,c 6 =0 3 (5) 3 E E { 3 } = { j C } M ( 3 ) { j C } = C ( ) 3 2 3 M 3 z τ x(z) = 1 z 2 + 1 { (z mτ n) 2 1 (mτ + n) 2 } (m,n) (0,0) (0, 0) (m, n) z pτ +q (p, q Z) ( ) y(z) =dx(z)/dz g 2,g 3 y(z) 2 =4x(z) 3 g 2 x(z) g 3 g 2, g 3 τ z z (x(z),y(z)) C(τ) : y 2 =4x 3 g 2 x g 3 τ Im(τ) ( Im(τ) > 0 ) 3 C(τ) 11
0 y τ 1+τ E(τ) 1 x 3: E(τ) z z z + mτ + n (m, n Z) ( ) E(τ)( 3) z 3 E(τ) 3 E(τ) C(τ) z (x, y) =(x(z),y(z)) : τ = aτ + b cτ + d C(τ) =C(τ ) ( ) a b SL(2, Z). a, b, c, d Z, ad bc =1. c d M τ : M = { j C } { } = τ C ;Im(τ) > 0 / (10) τ τ aτ + b cτ + d, ( a c ) b SL(2, Z). d g 2, g 3 τ g 2, g 3 τ q = e 2πτ q q =0 g 2, g 3 4π 4 /3, 8π 6 /27 12
(11) 1 n 4 = (2π)4 1440, 1 n 6 = (2π)6 60480. (11) n=1 a 2 =4π 4 /3, a 4 =8π 6 /27 E 4 = g 2 /a 2, E 6 = g 3 /a 3 q =0 1 n=1 E 4, E 6 q E4 3 E2 6 1728 E4 3 E2 6 = 1728 q (1 q n ) 24 (12) j (9) Δ 1728 (= 12 3 ) 12 2 3 2 3 j(e) =c 3 4 /Δ 4 2 3 : n=1 E4 3 E6 2 1728? 1728 τ ( 2, 3 ) M M 2 (10) (13) 3 E 3 4 j = 1728 E4 3 E2 6 = 1 q + 744 + 196884 q + 21493760 q2 + (13) (13) 196884 21493760 (13) 13
3 5 (7) : E : y 2 + xy = x 3 36 j 1728 x 1 j 1728 j = s =1/j E : y 2 + xy = x 3 36 s 1 1728 s x s 1 1728 s (14) s =0 E : y 2 + xy = x 3 (15) (x, y) =(0, 0) x, y y 2 + xy y 2 + xy =0 2 y 2 + xy = x 3 4 E j E E ( ) ( ) x, y s u, v u = y + 1 2 (x + x 2 +4x 6 ), v =(1 1728s)(y + 1 2 (x x 2 +4x 6 ))/(1 + 36x) (14) uv + s = 0 (16) s =0 2 u =0 v = 0 s 0 uv = uv = s 14
y y =0 C x y + x =0 4: E F (u, v) =0( F (0, 0) = 0 ) C[u, v]/(f, F/ u, F/ v) (17) F (u, v) =0 u, v <ε ε 1 C[u, v] u, v F/ u, F/ v u, v (17) C[u, v] F, F/ u, F/ v C[u, v] F, F/ u, F/ v F (u, v) =uv s u, v G(u, v, s) s s <ε s =0 G(u, v, 0) = F (u, v) G(u, v, s) =0 u, v <ε G(u, v, s) uv + suv =(1+s)uv F (u, v) suv uv =0 15
( ) uv + su = u(v + s) v =0 v + s =0 (17) F, F/ u, F/ v s, t uv + su + tv =(u + t)(v + s) st uv st (16) s st (16) (17) F/ u = v, F/ v = u (17) F = uv (17) = {s C} (18) (16) (17) (17) F (u, v) =u 2 + v 3 (17) 1 v s + tv (s, t C) F (u, v) =0 u 2 + v 3 + s + tv = 0 (19) -Spencer 6 3 3 3 10 : x 3 0, x3 1,x3 2, x 0x 1 x 2,x 2 0 x 1,x 2 1 x 2,x 2 2 x 0,x 0 x 2 1,x 1x 2 2, x 2x 2 0. 16
x 0 =1,x 1 = x, x 2 = y 1, x 3,y 3, xy, x, x 2 y, y 2, x 2,xy 2,y. 9 10 9=1 3 1 4 j 4 -Spencer 3 3 : E : H(x 0,x 1,x 2 )=0 F (x 0,x 1,x 2 )=0 E (17) 3 /(x i H/ x j ) i,j=0,1,2 (20) 1 (20) h H + sh = 0 (21) E -Spencer 3 4 4 ( ) (19) (20) (21) 7 K3 17
-Spencer - Spencer -Spencer -Spencer 97 12 18