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- えりか しんまつ
- 5 years ago
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3 i (1 ) (2 ) Cantor Russel
4 ii Hilbert Turing Boole Turing Lorenz Hausdorff Julia Mandelbrot Mandelbrot cypher RSA RSA
5 iii z n = α
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7 (i) ( ) (ii) (iii) (iv) (1) ( 624? 546?) ( ) (2) ( 572? 492?) mathema ( ) (3) ( 490? 429?) A B B
8 2 1 (4) ( ) mathema(ta) (5) ( 3 ) 13 (6) ( ) Eureka , 2, 3... ( ) ( ) al jabr algebra ( 2 6 ) ( ) > π > /7, 355/113 (10 14 )
9 1.1 3 (1295) 17 geometry ( ) ( 759?) ( ) ( ) (1627 ) (1642? 1708) ( ) (1637) ( ) ( ) ( ) ( ) 17 19
10 ( ) ( ) ( ) 1, 2, 3, A 5 A a, b c a = b c a b b a a c c a 1 1 (1) p 2 p (2) 2 1 (3) 2 p 1. 2 p ( p ) 2. ( ) 2 p 2( )
11 ( 3) ( p ) p 1 p 1, p 2,..., p n p n q = p 1 p 2... p n + 1 q p n. p 1, p 2,..., p n 2 n π(n) n π(n) n log n π(100) = 25, π(1000) = 168, π(10 6 ) = 78498, π(10 12 ) = , π(10 15 ) 1015 = log 1015 q q = rs (r > 1, s > 1) r s q = p α 1 1 pα pαm m (p 1, p 2,..., p m ) 3 1 p 1 < r 1, r 2,..., r n < r 1 r 2 r n p 1 < r < p p r 1 r 2 r n p i r i r i p r i < r i < p 2 p (1) a, b a b p a, b 1 p (2) m n p n m p n n : 4 : 5 4 a : b : c (a 2 + b 2 = c 2, abc 0) a = m 2 n 2, b = 2mn, c = m 2 + n 2 (n, m m, n )
12 6 1 (m 2 + n 2 ) 2 = (m 2 n 2 ) 2 + (2mn) 2 a, b, c a, b, c a 2 + b 2 = c 2 c > 1 c = m + n (m, n > 0) c 2 = (m + n ) 2 = m 2 + 2m n + n 2 = (m n ) 2 + 4m n. a = m n, c = m + n b 2 = 4m n ( ) 4 = 2 2 m n 1 p p m n m, n p c = m + n, a = m n p p m, n m, n m = m 2, n = n 2 b = 2mn m, n m 2, n 2 b 2 m 2 ± n 2 b = 10 3 = 7 7 (10 3)+3 = 7+3 = 10 = 10 7 = = 8, = 9, = = 10 $ $ c $ , 25 $ 5 30 $ 10 40, Thank you = = 5 2 n + = 0, (n = 1, 2, 3,...) 1, 2, 3, ( 4) = ( 4) 7 = + ( 4) = = = (180 12) 12 = = =
13 = a, b a = b q + r (0 r < b) (q, r) 1 q r r = 0 a b = 10 3 = = 3 4 = = ( ) 0, , 3,... n = 0 n 1 2 2,... 2 b a, d c 2 ( b a + d c )/2 1 (1) ( 0 ) (2) 2 ( ) 4 4, , (1 ) ( ) 8cm 7cm cm 2 (cm 2 ) 1cm 1cm 1cm
14 = = = = = = 0 15 / 12 = / 0 = m 5 100m/ 5 = 500m 2 100m/ ( 2) = 200m 200m 100m 2 ( 100)m/ 2 = 200m 200 m 3 ( 100)m/ ( 3) = 300m 300 m = = = p q 1 p, q ( ) ( ) 2 p = p2 q q 2 = 2 p2 = 2q 2 p 2 p p = 2p p p 2 = 4(p ) 2 = 2q 2 q 2 q p, q
15 (1) (2) x 7 = 1 π e π e (1) ( ) (2) ( ) (2) 1 3 = = 0. 1, 2 9 = 0. 2,..., 8 9 = = 1 1 =
16 = = 1 3, 0. 9 = = = = = 1 x 2 = 2 x 1, 1.4, 1.41,... x 0. 9 = (x + y) n = n k=0 ( ) n x k y n k k ( ) n n! (n k + 1)... (n 1)n = = k k!(n k)! k! 6 ( lim n n n) ( a n = 1 + n) 1 n a n < a n+1 a n = = < n ( ) ( ) k n 1 = k n n ( k! n k=0 k=0 n k=0 n+1 k=0 1 k! a n a n n k=0 ) ( 1 2 n n n(n 1)(n 2)... (n k + 1) 1 k! n k k=0 ) (... 1 k 1 ) n ( 1 1 ) ( 1 2 )... n + 1 n + 1 ( 1 k 1 n + 1 ( ) ( 1 2 ) (... 1 k 1 ) k! n + 1 n + 1 n n k! 1 + ( ) k 1 1 = /2n 2 1 1/2 3 k=1 ) = a n+1 a n e e (2 ) a n a n a n
17 (10 10 ) = = A B x y y/2 x x : y = y/2 : x x 2 = y 2 /2 x : y = 1 : 2 A (ISO ) A0 1m 2 A0 l 2l 1 2 l 2 = 1 l = 1/ m 2l = m 2 1/ 2 A /( 2) 4 = /4 = , /4 = m B (B4 ) A1 B1 A0 B0 ( ) = ( 1000) 0.1 (2 5 ) 0.1 = 2 = ( ) 3 1 = / / / = ( ) ( 12 2)% ( 5/4, 4/3, 3/2) E
18 12 1 a n a m = a n+m, (a n ) m = a m n a 0 = 1, a 1 = 1/a, a 1/n = n a 2 = 10 x x log 10 2 a + b ab 2 log ab = log ab 1/2 log a + log b = x 2 = x 3 = 3px + 2q x = 3 q + q 2 p q q 2 p 3 ( ) x 3 7x + 6 = (x 1)(x 2)(x + 3) = 0 1, 2, 3 p = 7 ( 1/2 3, q = ) ( 400/ /2 6 ) 400/27 1, 2, i 1 a + ib (a, b ) i 2 = 1 x, y i i j a + ib (a + ib)(c id) (a + ib)(c id) = = c + id (c + id)(c id) c 2 + d 2 i ( ) α = a + ib (a, b) ( ) α = r(cos θ + i sin θ) α = a + ib a = r cos θ, b = r sin θ (a + ib) + (c + id) = (a + c) + i(b + d)
19 (a, b) (c, d) (a + c, b + d) 4 α β = r(cos ϕ + i sin ϕ) s(cos θ + i sin θ) = rs(cos(ϕ + θ) + i sin(ϕ + θ)) 1 α β αβ O E A B C OEA OBC α 90 iα 3 α = a + ib α α = a 2 + b α n = r n (cos nθ + i sin nθ) 7 n f(z) = a 0 + a 1 z + + a n 1 z n 1 + a n z n = 0 (a n 0) a 0 = 0 z = 0 f(z) = 0 a 0 0 ( ) f(z) = z n 1 a n + a n 1 z + + a 1 0 z n max{a n 1,..., a 0 } = A, z = M > 1 a 1 n 1 z + + a 1 0 z n a n 1 1 z + + a 1 0 z n n A M z 0 z n a 0 0 z = 0 z 0 f(z) f(z) = DVD
20 14 1 4
21 ( ) : 4 : Fermat Galois 5 4 f F = f F f 8 f F b a f dx = F (b) F (a) ( ) 2 1
22 a, b ( a > b) a b r 1 r 1 b r 1 < b r 1 b 1 b r 1 r 2 r 2 < r 1 r 2 b 2 r 1 r 2 r q 1 a = b q 1 + r 1 q 2 b = r 1 q 2 + r 2 r n r n 1 q n r n 1 = r n q n 1 q n 1 r n 2 = r n 1 q n 1 + r n r n 1 r n r n 2 r n r n 3 r n r 1 r n b r n a r n r n a, b a, b d r 1 = a b q 1 a, b d r 1 d r 2 = b r 1 q 2 r 2 d r n 1 d r n d r n = d q r n d r n a, b (meta language) a, b a b d 1:{ r = a mod b a b 2: if r <> 0 then {a = b; b = r 3: goto 1 ;} 4: else d = b; return d ; 5:} 10 2 a, b d m, n am + bn d d = ah + bk h, k a b q 1 r 1 r 1 = a b q 1 r 1 = a h 1 + b k 1 b r 1 q 2 r 2 r 2 = b r 1 q 2 = b (a h 1 +b k 1 ) = a ( h 1 )+b (1 k 1 ) r 2 = a h 2 +b k 2 r m = a h m +b k m d = r n = a h n + b k n h n h k n k
23 a b b (a b) a b b 25a (26b 25a) a 79b a 79b b 101a b 101a a 184b a 184b b 278a b 278a a 473b = = 91 1 x 3 2x 2 5x + 6 a x 3 + 0x 2 7x + 6 b x + 1 x 3 + 0x 2 7x + 6 b x 3 + 0x 2 x + 0 x + 1 (a b) 2 2x 2 + 2x + 0 a b 6x + 6 f(x) x x 2 x (a b) x f(x) x 2 x f(x) = 1 6 (b (a b)(x + 1)) = 1 ((x + 1)a (x 1)b) 12 = 1 ( (x 4 x 3 7x 2 + x + 6) (x 4 x 3 7x x 6) ) = x A B (x Ω ) P(x) Q(x) P(x) Q(x) (deduction) (induction)
24 18 2 A A = 3, = 7, = 31 p 2 p 1 ( ) UFO ( ) ( )
25 ( ) 2 (1) (2) a a 11 3 n n n 1 2 n 1 3 A,B,C n A C 1. n 1 B 2. n C 3. B n 1 C n = n 1 n = = 2 1 = 1 n = 1 n > 1 n 1 2 n 1 1 n n 1 B n C B n 1 C (2 n 1 1) (2 n 1 1) = 2 2 n 1 1 = 2 n 1 n n
26 ( ) (1) (2) (1) (2) (1) (1) (2) (2) (1) (2) (Stoicheia) Hilbert (1) 2 (2) (3) (4) (5) ( ( )
27 ( ) (5) ( ) (paradox) (i) (ii) 2 (antinomy)...
28 22 2 W.Shakespeare Hamlet To be or not to be, that is a question.
29 Cantor Cantor(1895 ) Cantor 5 Cantor (1) (2) ( ) (3) 1 (1) A a a A a A (2) 2 a = b a b (3) A B A = B A B A B A B Russel (B.Russell) {n n } {x x } {f f } {x x } { } N N N = {x x x} A = {x x x} N A N N N N N A N N N A A N N N N A
30 ( ) ( ) ( ) Hilbert 17 ( ) 6 ( ) (1) (2) (3) 3 (1931)
31 Turing 1 M = (Q, Σ, δ) Q: Σ: δ: Q Σ Q Σ {left, right} δ(p, X) = (q, Y, D) (D = left or right) p X q Y δ δ (1936) (universal Turing machine) ( ) ( ) ( ) (abacus) abacus calculator calcul (i) ( ) (ii) TV
32 ( ) ( ) ( ) (1674) Boole = 1 10 = = = , = 10, = 11, = 10 + (1 + 1) = = 100, = 101, = (1 + 1) = = 110, = 111, = (1 + 1) = = ( ) = = 1000,... 1, 10, 11, 100, 101, 110, 111, 1000, , 1, 2,..., 8, 9, a, b, c, d, e, f (16 ) N d n d n 1... d 1 d 0 (d i = 0, 1, ) t m t m 1... t 1 t 0 (t i = 0, 1, 2) b l b l 1... b 1 b 0 (b i = 0, 1) h r h r 1... h 1 h 0 (h i = 0, 1, 2..., f) N = d n 10 n + d n 1 10 n d d 0 = t m 3 m + t m 1 3 m t t 0 = b l 2 l + b l 1 2 l b b 0 = h r 16 r + h r 1 16 r h h 0 ( ) 10 2
33 b 0 2 b , 4, 8, 16, M d 1 d 2 d 3... (d j = 0, 1, 2,..., 9), 0.t 1 t 2 t 3... (t j = 0, 1, 2), 0.b 1 b 2 b 3... (b j = 0, 1), 0.h 1 h 2 h 3... (h j = 0, 1, 2..., f) M = d d d = t t t = b b b = h h h b 1 2 b = 0.2 b 1 = = 0.4 b 2 = = 0.8 b 3 = = 1.6 b 4 = = 1.2 b 5 = = 0.4 d 6 = (10) = (2) = (2) (2) = 1 2, 0.01 (2) = 1 4, (2) = 1, (1854), {0, 1} ( ) 1 not ( ) and, or, eor(exclusive or) not and 0 1 or 0 1 eor
34 x + y = c(x, y) 2 + s(x, y) c(x, y) = x and y s(x, y) = x eor y on 1 off 0 and or not eor on/off Stibitz(1936) Zuse Turing (EDVAC 1947 )
35 p 1, p 2, p 3,... p n p n 1 7 x 1.x 2,..., x n,... φ(x) x n+1 = φ(x n ) φ(x) x 0 x 1, x 2,..., x n,... ( ) [0, 1] φ(x) φ(x) = { 2x (0 x 1/2) 2 2x (1/2 x 1) 0 x 1 0 φ(x) 1 1 [0, 1] 2
36 30 3 φ(x) φ(x) x 0, x 1, x 2,... x n 0 x n 1/2 A 1/2 < x n 1 B A B x 0 12 φ(x) (0, 1) A B (1) φ(x) (2) (3) (1) ABABABAB... x 0 A 0 < x 0 < 1/2 x 1 = 2x 0 B 1/2 < x 1 < 1 x 2 = 2 2x 1 = 2 2(2x 0 ) = 2 4x 0 x 0 x 0 x 1 ABABAB x 0 = x 0 x 0 = 2/5 (2) ABAABAABA... x 1 = 2x 0, x 2 = 2 2x 1 = 2 4x 0, x 3 = 2x 2 = 4 8x 0 = x 0 x 0 = 4/9 (3) BBABBABBA... x 1 = 2 2x 0, x 2 = 2 2x 1 = 2 2(2 2x 0 ) = 2 + 4x 0, x 3 = 2x 2 = 4 + 8x 0 = x 0 x 0 = 4/7 φ(x) Malthus 17 18
37 N 0 n N n r N 1 = N 0 + N 0 r = N 0 (1 + r), N 2 = N 0 (1 + r) 2,..., N n = N 0 (1 + r) n t N(t) N(t + t) N(t) t t 0 dn dt = rn N = N 0 e rt = rn(t) N 0 N n N(t) N(t) 1 N n Berharst 1840 ( dn dt = r 1 N ) N = rn r K K N 2 2 (dn/dt) (N) rn 2 r K 0 3 N 2 N 0 KN 0 e rt N(t) = N 0 e rt + K N 0 N 0 0 K N 0 K 1941
38 32 3 τ N(τ), N(2τ), N(3τ),... N n = N(nτ) N n+1 = N((n + 1)τ) ( ) 1 N n+1 = N n b + cn n τ N 0 N 0, N 1, N 2,... τ ( ) 1 N n+1 = σ N n b + cn n σ b, c, σ N(t + t) N(t) t = rn(t) r K N(t)2 = r N(n t) = N n N n+1 = x n = ( (1 + r t) r t ) K N n N n r tn n, a = (1 + r t) K(1 + r t) (K N(t)) N(t) K ( ) x n+1 = a(1 x n )x n ( ) f a (x) = a(1 x)x 0 < a < 4 a/4 0 < x < 1 x 0 < f a (x) < 1 ( ) x n+1 = f a (x n ) (0 < x 0 < 1) 0 < x < 1 a 0 < a < 1 0 < x 0 < 1 lim n x n = 0
39 a 2 0 < x 0 < 1 lim n x n = 1 1/a 1 1/a y = x y = a(1 x)x 2 < a < 3 0 < x 0 < 1 1 1/a < x N 1 1/a < x N x N+1 < 1 1/a < x N+2 1 1/a lim n x n = 1 1/a 3 a < < x 0 < 1 1 1/a < x N 1 1/a a < a c a c a c < x 0 < 1 a 4 a 8... ( 2 n ) a c < a 4 0 < x 0 < 1 φ(x) f a (X) Lorenz 1963 X Y Z 0 dx dt = σx + σy, dy dt = XZ + rx Y, dz dt = XY bz σ r, b 2 r Z P 1, P 2, P 3,... (P 1, P 2 ), (P 2, P 3 ), (P 3, P 4 ),..., (P n, P n+1 ), n 8 x n+1 = f(x n ) x 0, x 1, x 2,... (1) n x n = x n+1 = x n+2 =... x n (2) n x n x n+1 x n+2 = x n = x n+4 =..., x n+3 = x n+1 =... x n x n, x n+1 x n 2
40 34 3 (3) x n, x n+1,..., x n+p 1 x n+p = x n x n x n, x n+1,..., x n+p 1 p 13 f(x) [0, 1] [0, 1] 4 p, q, r, s s p < q < r f(p) = q, f(q) = r, f(r) = s x n+1 = f(x n ) (1) n n x 0 [0, 1] (2) ( ) [0, 1] [0, 1] 1 1 (space filling curve)
41 Hausdorff r N D 1 = Nr D D = log N log r (1) n n N = n, r = 1/n D = log n/ log n = 1 (2) 1 n n 2 N = n 2, r = 1/n D = log n 2 / log n = 2 (3) 1/2 4 D = log 4/ log 2 = 2 (4) r = 1/3 N = 4 D = log 4/ log 3 = (5) [0, 1] 1/3 [0, 1/3], [2/3, 1] 1/3 1 2/3 ( ) lim n (2/3) n = 0
42 36 3 [0, 1] 3 0, 2 a = 0.a 1 a 2... a n... b = 0.b 1 b 2... b n... a i = 0 b i = 0, a i = 2 b i = 1 b [0, 1] 2 D = log 2/ log 3 = (6) Julia Mandelbrot f(x) = a n x n + a n 1 x n a 0 = 0 14 f(x) [a, b] 2 (i) f(a) < 0, f(b) > 0 (ii) f (x) > 0, f (x) > 0 (a x b) ( ) f(x) = 0 [a, b] 1 ξ {c n } c 1 = b, c n+1 = c n f(c n) f (c n ) (n > 1) lim n c n = ξ c n+2 c n1 < c n+1 c n 4 11 f(z) = a n z n + a n 1 z n a 0 {ζ n } J f = C {ζ C ζ 0 = ζ, f( lim n ζ n) = 0} z 0 f(z) = 0 (1) (i) (ii) ε (iii) (iv) ( ) (2) f(z 0 ) < ε 0 ( ) (3) z 1 f(z 1 ) < ε 1 (4) z 2 f(z 2 ) < ε 2
43 (5) ( ) ( ) Mandelbrot g(z) z n+1 = g(z n ) z ( ) 12 µ f µ (z) = z 2 + µ z n+1 = f µ (z n ) = zn 2 + µ {µ C z 0 = 0, z n+1 = f µ (z n ) n z n } z 0 = 0 µ (1970 ) µ ( )
44
45 (1) ( ) (2) ( ) (3) ( 2 ) (4) ( ) (5) (6) ( ) A, C, G, T 4 13 ( ) (encode) (decode) (code breaking) steganography
46 40 4 code BBC, ( ) (skytale) cypher cypher cypher
47 ( ) 3 e n g u n w o, h a k e n s i t a h q j x q z r c k d n h q v l w d 1 a b, a c,..., a z 26 1 angounosuuri a b c d e f g h i j k l m n o p q r s t u v w x y z a n g o s u r i b c d e f h j k l m p q t v w x y z e n g u n w o h a k e n s i t a s h r t h w j i a d s h p b q a ( ) 9 (1) (2) (3) (4) 1 ( ) e q the a e 3 the... al (angonosuuri) a= 0, b= 1,..., z= u+o = = 8 i 26
48 42 4 e n g u n w o h a k e n s i t a a n g o n o s u u r i a n g o n e a m j a k g b u b m n b o h n 1 26 b r 1 17 = = 10 k 0, 1, 2,..., 9 a, b,..., z 26 ( ) ( ) ASCII(American Standard for Communication and Information Industory) JIS-X2022(Japan Industory Standard) ISO-8808 (International Standardization Organization) K= K K (one time pad) one time pad
49 U (colossus ) ( ) ( eavesdrop ) (1) secured route ( ) K (2) K (3) (i) (ii) K (iii) AES 3DES MD5 SSL SSL Secured Socket Layer secured route
50 44 4 (public key cryptography) 1974 (Martin E. Hellman) (Whitefield Diffie) (Rolph Merkle) (Eo, Do) 1 Eo ( PKI(public key initiative) ) 1 Do (1) Eb (2) Eb (3) Db Eb Do (1) Da (2) Ea Ea (1) (Ea, Da) (Eb, Db) (2) Da (3) Eb (4) Db (5) Ea
51 Da Eb Db Ea Eb Da Ea Db 2 Ea, Eb (1) (2) (3) 3 5 ( ) (1) (i) (ii) (iii) (algorithm) (2) ( ) n ( ) n O(n)
52 46 4 n m m n 3 n (1) 1 1 (2) (3) 2 n 1 (TSP) ( ) (SAT) n TSP NP N P a, b d = gcd(a, b) begin p := max(a, b); q := min(a, b); while q 0 do begin r := p mod q; p := q; q := r { } end; d := p { } end. p = q s 1 + r 1 0 r 1 < q, q = r 1 s 2 + r 2 0 r 2 < r 1, r 1 = r 2 s 3 + r 3 0 r 3 < r 2,.
53 q > r 1 > r 2 > r 3 n r n+1 = 0 r n 2 = r n 1 s n + r n, r n 1 = r n s n+1 r n 1 r n r n 2 r n p, q r n r n p, q p q s 1 = r 1, q r 1 s 2 = r 2, r 1 r 2 s 3 = r 3,. r n 2 r n 1 s n = r n p, q e r 1 e r 2 e r n e r n e. r n p, q 5 (a, b) d = gcd(a, b) e, f a e + b f d d = a h + b k h, k 1 d = gcd(a, b) a h + b k = a b b (a b) a b b 25a (26b 25a) a 79b a 79b b 101a b 101a a 184b a 184b b 278a b 278a a 473b 0 gcd(112385, ) = 91, = = a b (mod p) p a b (a b p ) a = b + p q q a b (mod p) a b p
54 48 4 mod 2 n n 0 (mod 2) n 1 (mod 2) mod (mod 2), (mod 2), (mod 2) mod (mod 6), 2 4 (mod 4), 3 3 (mod 6), 4 2 (mod 6), 5 1 (mod 6) mod (mod 7), 2 5 (mod 7), (mod 7), 1 5 (mod 7) 3 6 (1) a b (mod q) c a ± c b ± c (mod q), a c b c (mod q). (2) a b (mod q), c d (mod q) a ± c b ± d (mod q), a c b d (mod q). (3) a, b, k b a + k (mod q) a b k (mod q). mod (mod 6), mod p, a (0 < a < p) ab 1 (mod p) 0 < b < p 0 p 2. p ab 0 (mod p) a 0 (mod p) b 0 (mod p)
55 p a b, q ab + pq = 1 0 < b < p ab 1 (mod p) b < 0 b > p a(b ± p) + p(q a) = 1 ( ) 0 < b < p b p 2. a, b p p a, b p ah+pq = 1 h, q ah = 1 pq ah 1 (mod p) b bk 1 (mod p) k 2 (ab)(hk) 1 (mod p) (1) a b (mod p) a, b p (2) Z p = {0, 1, 2,..., p 1} p (3) p Z p p char(z p ) = p Q R Z 0 char(q) = 0 (4) Z p 0 Z p p, g Z p {g, g 2,..., g p 1 } = Z p g p (5) Z p g 1 a Z p gt a (mod p) t (6) Z p a a + b 0 (mod p) b p a p = 29, g = 18 g, g 2, g 3,..., g , 5, 3, 25, 15, 9, 17, 16, 27, 22, 19, 23, 8, 28, 11, 24, 26, 4, 14, 20, 12, 13, 2, 7, 10, 6, 21, p g Z p g, g 2,..., g p 1 32 p = g = 16807, , , g, g 2, g 3,... g s, g s+1, g s+2,... s seed 64 p = ( ) p h, h 2, h 3,... mod p 1 h p 1 1 (mod p) 15 p 0 < a < p a a p 1 1 (mod p)
56 50 4 a 0 < a < p 7 a, 2a, 3a,..., (p 1)a ia ja (mod p)(0 < i < j < p) (i j)a 0 (mod p) a p i j p 0 i j < p i = j a, 2a, 3a,..., (p 1)a p 1, 2, 3,..., (p 1) a p 1 (p 1)! (p 1)! (mod p) (p 1)! p a p 1 1 (mod p) 1 r 1 (mod p 1) a r a (mod p) r 1 (mod p 1) m r = 1 + m(p 1) a r = a 1+m(p 1) = a (a p 1 ) m a (mod p) onetime pad Purdy p = f(x) = x a 1 x a 2 x 3 + a 3 x 2 + a 4 x + a 5 a i 19 50!/( ) ( ! y = f(x) x y y x ( ) 2. (1)Deffie Hellman (2)RSA
57 NP Diffie Hellman p Z p g 1. a h g a (mod p) (p, g, h) 2. b k g b (mod p) (p, g, k) 3. k R = k a = (g b ) a g ab (mod p) 4. h R = h b = (g a ) b g ab (mod p) ElGamal 1. p Z p g b k g b (mod p) (p, g, k) 2. a M (M < p) h g a (mod p), L Mk a (mod p) (h, L) 3. h b (mod p) L/h b M (mod p) h b = (g a ) b = (g b ) a k a (mod p) L/h b = Mk a /k a M (mod p) Z p 7 3 h b = k a = g ab = R R MR g, h = g a, k = g b a, b RSA Diffie Hellman MIT( ) (Rivest) (Shamir) (Adleman) 1977 RSA RSA 2 p, q ( ) n n = p q r (p 1), (q 1) ( ) (n, r) (n, r) n a b a r (mod n) 8 e a, b e ab
58 52 4 e a, b ah + ek = 1, bs + et = 1 h, k, s, t bs 1 + et = 1 1 bs(ah + ek) + et = 1. ab(sh) + e(bsk + t) = 1. q ab ab = qr q(rsh) + e(bsk + t) = 1 e ab e d (d1) l = lcm(p 1, q 1) a l 1 (mod pq) 1 (mod n) m = gcd(p 1, q 1) l = (p 1)((q 1)/m) = (q 1)((p 1)/m) 2 a p 1 1 (mod p) a l = (a p 1 ) (q 1)/m 1 (mod p) a q 1 1 (mod q) a l = (a (q 1) ) (p 1)/m 1 (mod q). a l p, q 8 pq a l 1 (mod pq) 1 (mod n) lcm(p 1, q 1) (d2) ed + hl = 1 d, h d e (p 1), (q 1) 8 l d, h (d3) b b d (d1,2) ed = 1 hl b d = (a e ) d = a ed = a 1 hl = a a lh a (mod n) (d2) d < 0 l + d > 0 b l+d = (a e ) l+d = a ed a el = a 1 hl a el a a ( h+e)l a (mod n) RSA n RSA RSA 1024 ( ) 4096 secured route
59 RSA RSA x, n x n (1) n 2 (2 ) n = d k 2 k + d k 1 2 k d d 0. d k = 1, d i = 1 0 (0 i < k). (2) x 0 = x, x 1 = x 2 0 = x 2, x 2 = x 2 1 = x 22,..., x i+1 = x 2 i = x2i+1,..., x k = (x k 1 ) 2 = x 2k (3) x n = x d k2 k +d k 1 2 k 1 + +d 1 2+d 0 = x d k k xd k 1 k 1 x d 1 1 xd 0 0. d i = 0 1 d i = 1 x n = x k x i x j... RSA p = 11, q = 13 n = = 143 p 1 = 10, q 1 = 12 10, 12 r = 7 (143, 7) a = 5 b = a 7 = (mod 143) 47 (d1) p 1 = 10, q 1 = 12 l = lcm(10, 12) = 60. (d2) e = 7 ed + hl = 1 7d + 60h = 1 d, h d = 17, h = 2 (d3) d = 17 < 0 l + d = = 43 b l+d = = b 2 = (mod 143), b 8 = (b 2 ) (mod 143) 92 2 (mod 143) 27 (mod 143), b 32 = ((b 8 ) 2 ) 2 (27 2 ) 2 (mod 143) 14 2 (mod 143) 53 (mod 143) a = (mod 143) (mod 143) (mod 143) 5 (mod 143) Y 2 = X 3 + ax + b 18 E
60 P E (x, y) E (x, y) (x ) P 2. P, Q E, (Q P, P ) 2 P, Q E y ( x ) R P + Q = R 3. P + P = 2P P E E y 4. P, P y E O P P = O E Q Q y E Q Q + O = Q O E Q aq = h h a 4.5.2
61 Cardano 3 z 3 = 3pz + 2q α = 3 q + q 2 p 3, β = 3 q q 2 p 3 α + β, ωα + ω 2 β, ω 2 α + ωβ ω z 3 = 1 1 ω = 1 + i 3 2 z = x + y (x + y) 3 = 3xy(x + y) + (x 3 + y 3 ) = 3p(x + y) + 2q 2q = x 3 + y 3, p = xy X = x 3, Y = y 3 X + Y = 2q, XY = p 3 X = q ± q 2 p 3 x 3 = X = q + q 2 p 3 α = q 3 + q 2 p 3 x = α, ωα, ω 2 α y = p x 3 ax 3 + bx 2 + cx + d = 0 (a 0) z = x + b/(3a) Ferrari 4 z 4 + pz 2 + qz + r = 0 2
62 56 5 ξ 3 pξ 2 4rξ + (4pr q 2 ) = 0 1 ξ z 2 ± ξ 0 p ( ) q z + ξ 0 2(ξ 0 p) 2 = 0 2 z 4 + pz 2 + qz + r = (z 2 + αz + β)(z 2 αz + γ) β + γ α 2 = p α(γ β) = q βγ = r α 2 p + α 2 = ξ 2 (β γ) 2 = (β + γ) 2 4βγ = ξ 2 4r = q2 α 2 = q2 ξ p ξ 3 pξ 2 4rξ + (4pr q 2 ) = 0 1 ξ 0 α = ± ξ 0 p, β + γ = ξ 0, γ β = q α β, γ α 4 ax 4 + bx 3 + cx 2 + dx + e = 0 (a 0) z = x + b/(4a) z n = α 18 z n = 1 ω = cos 2π n + i sin 2π n ω, ω 2,..., ω n = 1 19 z n = α α = r(cos θ + i sin θ) ( χ = n r cos θ n + i sin θ ) n χ, χω, χω 2,..., χω n Abel, Galois n 5 n group
63 ( ) H 4 (1) 90, 180, 270, 360 (2) 2 2 (3) f, g g f f = 90 g = g f = (1) (1, 2, 3, 4) , (1, 3)(2, 4) 270 (1, 4, 3, 2) E (2) (1, 2)(3, 4), (1, 4)(2, 3) 2 (3) (1)(3)(2, 4), (1, 3)(2)(4) 2 1, 3 2 (2, 4), (1, 3) 90 (1, 2)(3, 4) (1, 2, 3, 4) = (1)(2, 4)(3) = (2, 4) E (1, 2)(3, 4) (1, 2)(3, 4) = E 2 n n (1) (2)
64 58 5 (3) (1) GL(n, R) ( ) GL(n, C) ( ) (2) AB = A B 1 SL(n, R) SL(n, C). (3) 1 SL(n, Z) (4) O(n) U(n). Ω = {1, 2,..., n} Ω (1) (2) (3) n ( ) S n G 2 a, b a b G (1) 3 a, b, c (a b) c = a (b c) ( ) (2) G a a e = e a = a ( ) (3) G a a g = g a = e G g g a a a G, H ϕ : G H G 2 a, b ϕ(a b) = ϕ(a) ϕ(b) ϕ G H H 4 H 4 {1, 2, 3, 4} S 4 (1, 1), ( 1, 1), ( 1, 1), (1, 1) 90 (1, 1) ( 1, 1) ( 1, 1) (1, 1) (1, 1) ( )
65 ( ) ( ) (1, 3)(2, 4), (1, 4, 3, 2), (1, 2)(3, 4) ( ) ( ) ( (1, 4)(2, 3), (2, 4), (1, 3) (1, 2)(3, 4) (1, 2, 3, 4) = (2, 4) ( ) ( ) = ( ) ( ) ), E (1) (2) (3) (4) (1) Q, R, C Q Q (2) Q( 2) = {a + b 2 a, b Q} (3) Q(i) = {a + bi a, b Q} 22 K K K ϕ K K A(K)
66 60 5 (1) C A(C) (2) Q( 2) ϕ(a + b 2) = a b 2 23 (1) 2 E F F E (F E ) E F (2) E F G(E/F ) = {σ σ A(E) σ F = id F } E/F (3) F f(x) F F f(x) = 0 (C ) F E f(x) (1) R Q (2) C R G(C/R) (3) f(x) = x 3 1 Q(ω) = {a + bω a, b, c Q} ( ω 3 = 1) A(Q(ω)) ϕ(a + bω) = a + bω 2 = a + b ω (4) g(x) = x 3 2 α, αω, αω 2 (ω 3 = 1, ω 1, α = 3 2) g(x) E = Q(α, αω, αω 2 ) E = Q(α, ω) g(x) Q Q(α) = {a + bα + cα 2 a, b, c Q} 21 f(x) F C f(x) E G(E/F ) ( ) E F G(E/F ) = E : F G(C/R), G(Q(i)/Q), G(Q(ω)/Q) 2 22 f(x) F f(x) = 0 E n G(E/F ) n S n n! E = Q(α, ω) G(E/Q) = E : Q = E : Q(α) Q(α) : Q = 3 E : Q(α) > 3 G(E/Q) 3 S F f(x) F E f(x) F = B 0 B 1 B r = E B i = B i 1 (χ i ) (χ n i = a i B i 1 ) f(x) = 0 F
67 F Q χ 1 m E = F (χ) G(E/F ) S 2, S 3, S 4 ( ) 24 f(x) Q Q G(E/F ) 25 5 Q f(x) geometry geometry (1637 ), (1) L = {(x, y) ax + by + c = 0} (2), 90,, 2 (x 1, y ), (x 2, y 2 ) (x 1 x 2 ) 2 + (y 1 y 2 ) 2 (3),, 1 (4) y = Ax A A A t A = t AA = E (4) O(n)
68 ax 2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 (x, y) 2 x 2 + y 2 = a 2 a 2 + y2 = 1 (a b) b2 x 2 a 2 y2 b 2 = 1 y = px 2 x 2,, 26,, x 2 + y 2 = z 2 (1), z = a (a 0),, x 2 + y 2 = a 2 (2), x = a (a 0),, a 2 + y 2 = z 2, z 2 y 2 = a 2 (3), x + z = 1,, x 2 + y 2 = (1 x) 2 y 2 = 1 2x (4) x = 2z+1,, (2z+1) 2 +y 2 = z 2 3(z + 2/3) 2 + y 2 = 1/3, 1 ( ) 1 ( ),,, 2, 2, 27 2, x 2 + y 2 = a 2 a 2 + y2 = 1 (a b) b2 x 2 a 2 y2 b 2 = 1 y = px 2 x 2,
69 ax 2 + 2bxy + cy 2 + 2dx + 2ey + f = (x, y, 1) a b d b c e x y d e f 1 A 2 1 x = X + u, y = Y + v X, Y { au + bv + d = 0 bu + cv + e = 0 I. ac b ax 2 + 2bxy + cy 2 + f = 0 28 ac b α, β P ( ) ( ) t a b α 0 P P = = D (αβ 0) b c 0 β (ξ, η) = (x, y)p ( ( ax 2 + 2bxy + cy 2 x = (x, y)a = (x, y)p D y) t x P y) ( ( ) α 0 ξ = (ξ, η) = αξ 0 β) 2 + βη 2 η 0 α, β (α = β ) (α β αβ > 0) (αβ < 0 ) II. ac b 2 = 0 A 1 0 x, y ( ) (1) (2) (3) (1) (2) 3 2 1
70 64 5 (3) 2 x 2 +y 2 = 1( ) x 2 y 2 = 1( ) y = x 2 ( ) (1) (2) 2 2 l P 2 (1 ) ( ) 3 28 P = (x, y, z) P = (x, y, z ) k 0 x = kx, y = ky, z = kz P, P P P (1) (x, y) {(x, y, 1) } (2) l = {(x, y, 0) } (3) P GL(2, R) = GL(3, R)/{RE} 30 2 ax 2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 ax 2 + cy 2 + fz 2 + 2bxy + 2eyz + 2dzx = 0 2 x 2 + y 2 z 2 = 0
71 x 2 +y 2 = 1, x 2 y 2 = 1, y = x 2 x 2 + y 2 = z 2, x 2 y 2 = z 2, yz = x 2 yz 45 y 2 z 2 = x M 2 ρ M (1) ρ(x, y) 0 ρ(x, y) = 0 x = y ( ) (2) ρ(x, y) = ρ(y, x) ( ) (3) ρ(x, z) ρ(x, y) + ρ(y, z) ( ) 30 M ρ M l l 3 x, y, z ρ(x, z) = ρ(x, y) + ρ(y, z) l ( ) ρ 2 ( ) ( )
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3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
1 1 n 0, 1, 2,, n n 2 a, b a n b n a, b n a b (mod n) 1 1. n = (mod 10) 2. n = (mod 9) n II Z n := {0, 1, 2,, n 1} 1.
1 1 n 0, 1, 2,, n 1 1.1 n 2 a, b a n b n a, b n a b (mod n) 1 1. n = 10 1567 237 (mod 10) 2. n = 9 1567 1826578 (mod 9) n II Z n := {0, 1, 2,, n 1} 1.2 a b a = bq + r (0 r < b) q, r q a b r 2 1. a = 456,
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n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................