1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +
|
|
- いぶき こうい
- 6 years ago
- Views:
Transcription
1 ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University hsuzuki@icu.ac.jp
2 1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac + bc (distributive law)) R4 R 0 R 1 1x = x1 = x x R R5 (commutative ring) R5 ab = ba for all a, b R R1 R3 R4 (unital ring) R R2 R4 U(R) 0x = x0 = 0 ( 1) 0 U(R) R {0} R {0} R # (Pf.) 0 = 0x + ( 0x) = (0 + 0)x + ( 0x) = 0x + 0x + ( 0x) = 0x + 0 = 0x 1.2 U(R) = R {0} = R # (skew field) R5 (field) 1.3 R a b 0 ab = 0 [ba = 0] a (left zero divisor) [ (right zero divisor)] (zero divisor) 0 R6 ab = 0 a = 0 or b = 0 R1 R Z 1 1
3 2. Q R C 3. R R Mat n (R) 4. n Z n = Z/nZ = { 0, 1,..., n 1} n n Z R x f(x) = a 0 + a 1 x + + a n x n, a i R for i = 0, 1,..., n x R x R[x] x R f = f(x) = a 0 + a 1 x + + a n x n, g = g(x) = b 0 + b 1 x + + b m x m R[x] f + g = i fg = i (a i + b i )x i ( ) a j b i j x i j R[x] R f = f(x) = a 0 + a 1 x + + a n x n R[x], a n 0 n = deg f f f(x) = 0 deg f = deg 0 = 1.1 R f, g R[x] (1) deg(f + g) max(deg f, deg g) (2) deg(fg) = deg f + deg g R (1) (2) fg = 0 (2) = deg fg = deg f + deg g. deg f = deg g = f = 0 g = R f, g R[x] g R q, r R[x] deg r < deg g f = gq + r R q, r R[x] 1 2
4 f = a n x n + + a 1 x + a 0 a n 0 g = b m x m + + b 1 x + b 0 n < m q = 0 r = f n m n = deg f b m h = f (a n b 1 m )x n m g f deg h < n R[x] q 1, r deg r < deg g h = gq 1 +r f = g(q 1 + (a n b 1 m )x n m ) + r q = q 1 + (a n b 1 m )x n m R f = gq + r = gq + r, deg r, deg r < deg g g(q q ) = r r deg g + deg(q q ) = deg(g(q q )) = deg(r r) max(deg r, deg r) < deg g. g 0 q q = 0 r r = 0 q = q r = r n R[x 1,..., x n ] = (R[x 1,..., x n 1 ])[x n ] a i1,...,i n x i 1 1 x in n, a i1,...,i n R. i 1,...,i n R[x, y] = (R[x])[y] = (R[y])[x] 1 3
5 2 R I R/I R/I xy (x + I)(y + I) (x + I)(y + I) xy + I x = 0 y = 0 xi I Iy I 2.1 R I a, b I a + b I a I, r R ra I, [ar I]. I R [ ] A B R A + B = {a + b a A, b B}. AB = { i a i b i a i A, b B} R I, J I J I + J 2. I, J R IJ IJ I J I R I R R/I (a + I) + (b + I) = (a + b) + I, (a + I) (b + I) = (ab) + I R/I (quotient ring) a R Ra [ar] [ ] (principal) [ ] R Ra = ar (a) 2 1
6 0 = {0} R R R I R [ ] I = R U(R) I. (Pf.) I = R 1 I U(R) U(R) I u U(R) I r R R I I = R r = r(u 1 u) = (ru 1 )u RI I. 2.1 R R R [ ] 0 R ( ) I 0 R a I {0} a U(R) I = R ( ) a 0 a Ra Ra 0 1 R = Ra R b ba = 1 b 0 R = Rb R c cb = 1 c = c1 = c(ba) = (cb)a = 1a = a ab = ba = 1 R 0 R X (well-ordered set) (PID : principal ideal domain) 2. R (well-ordered set) X ρ : R X R (Euclidean domain) (a) 0 a R ρ(0) < ρ(a). (b) a, b R (a 0) b = aq + r, ρ(r) < ρ(a) q, r R 2.2 R I R I = 0 I 0 {ρ(x) 0 x I} X ρ(a) a I b I b = aq + r ρ(r) < ρ(a) q, r R r = b aq I a r = 0 b Ra b I = Ra 2 2
7 ρ : Z {0} N ρ(a) = a Z 2.2 Z Z Z 2. K ρ : K[x] {, 0} N ρ(f) = deg f 1.2 K[x] 2.2 K[x] 2 3
8 3 3.1 R R f : R R f(a + b) = f(a) + f(b), f(ab) = f(a)f(b), f(1 R ) = f R f R R ((ring) homomorophism) f R R 3.1 I R (R I) 3.2 R S f : R R/I, (a a + I) a, b S a b S, ab S, 1 R S S R (subring) R S (extension ring) f : R R (1) Kerf = {a R f(a) = 0} (2) Imf = {f(a) a R} R R R f : R R R/Kerf Imf. 3.1 R/Kerf Imf f : R/Kerf Imf, (a + Kerf f(a)) well-defined f((a + Kerf)(b + Kerf)) = f(ab + Kerf) = f(ab) = f(a)f(b) = f(a + Kerf) f(b + Kerf) f(1 R/Kerf ) = f(1 + Kerf) = f(1 R ) = 1 R. f f(a) = f(b) f(a b) = 0 a b Kerf a + Kerf = b + Kerf f well-defined 3 1
9 3.2 K L α L φ : K[x] L (f(x) f(α)) Imφ K[α] K[α] = {f(α) f(x) K[x]} K[x]/Kerφ K[α] Kerφ K[x] p(x) K[x] Kerφ = K[x]p(x) = (p(x)) p(x) = 0 Kerφ = 0 α K p(x) 0 p(x) 1 p(x) Kerφ α K p(x) α K K = Q L = C e π Q Q (transcendental number) α = 1 Kerφ = (x 2 + 1) Q[ 1] = {a + b 1 a, b Q} α = 3 2 Kerφ = (x 3 2) Q[ 3 2] = {a + b c( 3 2) 2 a, b Q} 3.3 A Mat n (C) ψ : C[x] Mat n (C) (f(x) f(a)) Hamilton-Cayley Kerψ (det(xi A)) 0 monic Kerψ = p A (x) p A (x) p A det(xi A) A = ( ), B = ( ), C = ( p A (x) = (x 1)(x + 2), p B (x) = (x 2) 2, p C (x) = x 2 ) 3 2
10 4 R I R/I I 4.1 (1) R I( R) ab I a I b I I (prime ideal) (2) R I( R) I J : R I = J J = R I (maximal ideal) 4.1 R I (1) R/I I (2) R/I I (3) (1) R/I ā, b R/I, ā b = 0 ā = 0 b = 0 a, b R, (a + I)(b + I) = ab + I = I a + I = I b + I = I a, b R, ab I a I b I (x + I = y + I x y I ) (2) 2.1 R/I R/I 0 R/I R J I R I R (3) I R/I R/I I R (0) R (0) R 4.2 R (PID) I R I (0) I I 4 1
11 4.1 I = (a) (0) I J I R R J = (b) a (a) = I J = (b) a = bc c R I = (a) b I c I b I J = (b) (a) = I J I c I = (a) c = ad d R a = bc = bad = abd a(bd 1) = 0 a 0 bd = 1 R = (1) (b) = J J = R I 4.3 n Z (n) (n) n (n) (m) m n (m) = (n) m = ±n (m) (m) (n) (n) = (m) (n) = (1) n ±n ±1 (1) = Z (n) n Z 4.2 Z n = Z/(n) n 4.2 R 1 f(x) R[x] f(x) = g(x)h(x) deg g > 0 deg h > 0 R R 4.4 K f(x) K[x] (f(x)) (f(x)) f(x) f(x) deg f(x) 1 K[x] (f(x)) f(x) 4 2
12 f(x) f(x) = g(x)h(x) deg g(x) > 1 deg h(x) > 1 (f(x)) (g(x)) K[x] U(K[x]) = U(K) = K (f 1 (x)) = (f 2 (x)) f 1 (x) = cf 2 (x) (c K ). (f(x)) (g(x)) (f(x)) K[x] f(x) = g(x)h(x) f(x) 4.1 x 2 +1 x 2 2 Q (x 2 +1) (x 2 2) Q[x] Q[ 1] Q[x]/(x 2 +1) Q[ 2] Q[x]/(x 2 2) 4.5 [Eisenstein ] p f(x) = a n x n + + a 1 x + a 0 Z[x] a n 0 (mod p), a n 1 a 1 a 0 0 (mod p), a 0 0 (mod p 2 ) f(x) Z f(x) = g(x)h(x), r = deg g > 0, s = deg h > 0, g(x) = b r x r + + b 0, h(x) = c s x s + + c 0 a 0 = b 0 c 0 p p 2 p b 0 c 0 a n = b r c s p c s p c 0 p i c i p c 0 c 1 c i 1 0 c i (mod p). a i = b 0 c i + b 1 c i b i c 0 b 0 c i 0 (mod p). n = i < s = deg h f(x) Z 7.7 Q x n 2 x 3 3x 2 9x 6 Z Q 4 3
13 5 5.1 R 1, R 2,..., R n R = R 1 R 2 R n = {(a 1, a 2,..., a n ) a i R i, i = 1,..., n} (a 1,..., a n ) + (b 1,..., b n ) = (a 1 + b 1,..., a n + b n ) (a 1,..., a n ) (b 1,..., b n ) = (a 1 b 1,..., a n b n ) R R R 1,..., R n R = R 1 R 2 R n. 1 R = (1 R1, 1 R2,..., 1 Rn ) 0 R = (0 R1, 0 R2,..., 0 Rn ) Ri = {(0,..., 0, a, 0,..., 0) a R i } i 0 Ri R R I, J I + J = R I J I + J = R x + y = 1 x I, y J Z (m) (n) (m, n) = 1 m n 1 x + y = 1 x (m), y (n) am + bn = 1 a, b Z (m, n) = (Chinise Remainer s Theorem) 5.1 [ ] R I 1, I 2,..., I n i j I i + I j = R a 1, a 2,..., a n R x a i (mod I i ) i = 1, 2,..., n x R n = 2 1 = c 1 + c 2 c 1 I 1, c 2 I 2 x = a 1 c 2 + a 2 c 1 (mod I 1 ) x a 1 c 2 + a 2 c 1 a 1 c 2 a 1 (1 c 1 ) a 1 a 1 c 1 a 1 5 1
14 x a 2 (mod I 2 ) n > 2 i x i R x i 1 (mod I i ), j i x i 0 (mod I j ). i = 1 j 2 I 1 + I j = R c (j) 1 + c j = 1 c (j) 1 I 1 c j I j n 1 = (c (j) 1 + c j ) c 2 c n (mod I 1 ) j=2 1 c 2 c n = c 1 c 1 I 1 R = I 1 + I 2 I n I 1 I 2 I n n = 2 x 1 R x 1 1 (mod I 1 ), x 1 0 (mod I 2 I n ) j 2 I 2 I n I j x 1 0 (mod I j ) i x i x = a 1 x a n x n x a 1 x a n x n (mod I i ) a i x i (mod I i ) a i (mod I i ) 5 2
15 6 6.1 R S (i) a, b S ab S (ii) 1 S, 0 S S R (multiplicative subset) R 2. P R R P R S R S (a, s) (a, s ) (as a s)t = 0 t S (a, s) a/s S 1 R S 1 R (a 1 /s 1 ) + (a 2 /s 2 ) = (a 1 s 2 + a 2 s 1 )/s 1 s 2 (a 1 /s 1 )(a 2 /s 2 ) = (a 1 a 2 /s 1 s 2 ) S 1 R R S (quotient ring) 1. 0 S 1 R = 0/1 1 S 1 R = 1/1 (a/s) = ( a)/s s S s/1 U(S 1 R) 2. S a/s = a /s as a s = 0 3. φ S : R S 1 R (a a/1) φ S S S R S 1 R R (ring of total quotients) 2. R R (quotient field) Q(R) (a) Q(Z) = Q 6 1
16 (b) Q(K[x 1,..., x n ]) = K(x 1,..., x n ) = {f/g f, g K[x 1,... x n ], g 0} (c) P R (R P ) 1 R R P R P (localization) 6.2 R M R (local ring) (0) R M I R Zorn I R I M M R 6.1 R [(1) (2) ] (1) R (2) R U(R) R (1) (2) M R M R M U(R) = M R U(R) a R U(R) Ra R a Ra M R U(R) M M = R U(R) (2) (1) J R J R I = R U(R) J U(R) = J I I R 6.2 P R R P P = {a/s a P, s P } P R P (Pf.) (a/s) + (b/t) = (at + bs)/st a, b P, s, t P at + bs P st P (at + bs)/st P r R (r/t)(a/s) = ar/ts P P R P a/s P a P. (Pf.) ( ) a P a/s P ( ) a/s P a/s = a /s a P s P (as a s)t = 0 t P as t = a st P a P 6 2
17 R P P = U(R P ) (Pf.) ( ) a/s P a P s/a R P a/s U(R P ) ( ) 1 P 1/1 P a/s U(R P ) P a P (a/s)(b/t) = 1/1 b R t P t P abt = stt P P U(R P ) P = R P P = U(R P ) 6 3
18 7 7.1 R a, b R (a) (b) a = bc c R b a (a) = (b) a = bu u U(R) a b R p 0 p = uv u U(R) v U(R) p 7.1 R R (UFD = Unique Factorization Domain) (i) a R a = p 1 p 2 p r (p i ) (ii) a = p 1 p 2 p r = q 1 q 2 q s (p i, q j r = s p i q i 7.1 R 0 p R (1) (p) p (2) R UFD (p) p (1) p = ab a (p) b (p) a (p) (a) (p) = (ab) (a) a p p = au u U(R) a(b u) = p p = 0 R b = u U(R) (2) p ab (p) a b U(R) ab = pc a = p 1 p r b = q 1 q s c = v 1 v t p 1 p r q 1 q s = pv 1 v t p p i p q j p p i a = p 1 p r (p i ) = (p) p q j b = q 1 q s (q j ) = (p) R 7.2 R p 0 (1) p 7 1
19 (2) (p) (3) (p) 4.2 (2) (3) 7.1 (2) (1) (1) (3) (p) I = (q) R p = qa q a (q) = R (p) = (q) (p) 7.3 R 0 a R U(R) (a) R (a) (p 1 ) 7.2 p 1 (a) (p 1 ) a = p 1 a 1 p 1 U(R) (a 1 ) (a) a 1 U(R) p 2 a 1 = p 2 a 2 (a = p 1 p 2 a 2 ) a i (a) (a 1 ) (a 2 ) (a i ) i=1 (a i ) R i=1 (a i ) = (d) i d (a i ) (a i ) = (a i+1 ) r a r p r a r a = p 1 p 2 (p r a r ) a = p 1 p 2 p r = q 1 q 2 q s r s r q 1 q 2 q s = a (p 1 ) (p 1 ) q i (p 1 ) i (q i ) (p 1 ) q i p 1 q 1 = p 1 u u U(R) p 1 p 2 p r = p 1 uq 2 q s p 2 p r = uq 2 q s r = s p i q i Z[x] Q[x 1,, x n ] 7.1 Z[ 5] = {a + b 5 a, b Z} 2 (2) α = a + b 5 N(α) = αᾱ = a 2 + 5b 2 α U(Z[ 5]) N(α) = 1 α = ±1. (Pf.) ±1 U(Z[ 5]) αβ = 1 1 = N(αβ) = N(α)N(β) a 2 + 5b 2 = N(α) = 1 a, b Z b = 0 a = ±1 7 2
20 2 (Pf.) 2 = αβ N(α) 1 N(β) 1 α = a + b 5 4 = N(2) = N(αβ) = N(α)N(β) a 2 + 5b 2 = N(α) = 2 N(α) = 1 N(β) = 1 α, β (2) (Pf.) (1 + 5)(1 5) = 6 (2) 1 ± 5 (2) 1 ± 5 = 2γ 6 = N(1 ± 5) = 4N(γ) 1 ± 5 (2) (2) 7 3
21 7.2 R K = Q(R) d a 1,..., a n R 1. d a i, i = 1, 2,..., n 2. c a i, i = 1, 2,..., n c d l a 1,..., a n R 1. a i l, i = 1, 2,..., n 2. a i m, i = 1, 2,..., n l m a 1, a 2,..., a n 1 a 1, a 2,..., a n (coprime) a 0, a 1,..., a n f(x) = a 0 + a 1 x + + a n x n R[x] (primitive polynomial) R R R = Z 4 6 ±2 7.4 R R[x 1, x 2,..., x n ] 7.5 f(x) K[x] c K f 0 (x) R[x] f(x) = cf 0 (x) c R I(f) f(x) = (b 0 /a 0 ) + (b 1 /a 1 )x + + (b n /a n )x n 0 a i, b j R m a 0, a 1,..., a n m = a i c i d b 0 c 0, b 1 c 1,..., b n c n de i = b i c i e 0, e 1,..., e n f(x) = (b 0 /a 0 ) + (b 1 /a 1 )x + + (b n /a n )x n = 1 m (b 0c 0 + b 1 c 1 x + + b n c n x n ) = d m (e 0 + e 1 x + + e n x n ) c = d/m f 0 (x) = e 0 + e 1 x + + e n x n f(x) = cf 0 (x) = c f 0(x) f 0 (x) f 0(x) R c = b/a c = b /a a b a b R a bf 0 (x) = ab f 0(x) 7 4
22 f 0 (x) f 0(x) a b = ab u u U(R) c = b/a = (b /a )u = c u K c, c c = cu u U(R) c c f(x) K[x] f(x) R[x] I(f) R f(x) I(f) (1) (2) f(x), g(x) K[x] I(fg) I(f)I(g) (1) f(x) = a 0 + a 1 x + + a l x l g(x) = b 0 + b 1 x + + b m x m h(x) = f(x)g(x) = c 0 + c 1 x + + c n x n, p c i, ; i = 0, 1,..., n p a i p i i 0 b j p j j 0 c i0 +j 0 = a 0 b i0 +j a i0 1b j a i0 b j0 + a i0 +1b j a i0 +j 0 b 0 a i0 b j0 (mod (p)) 0 (mod (p)) (2) f(x) = I(f)f 0 (x) g(x) = I(g)g 0 (x) f 0 (x) g 0 (x) f(x)g(x) = I(f)I(g)f 0 (x)g 0 (x) f 0 (x)g 0 (x) (1) I(f)I(g) I(fg) 7.7 f(x) R[x] f(x) R[x] K[x] f(x) K[x] R[x] K[x] f(x) = g(x)h(x) g(x), h(x) K[x] g(x) = I(g)g 0 (x) h(x) = I(h)h 0 (x) f 0 (x), g 0 (x) f(x) = I(g)I(h)g 0 (x)h 0 (x) f(x) R[x] I(g)I(h) I(gh) R deg g 0 = deg g = 0 deg h 0 = deg h = 0 K[x] 7.8 f(x) R[x] (i) deg f = 0 f R (ii) deg f > 0 f 7 5
23 U(R[x]) = U(R) (i), (ii) f(x) f = gh g, h U(R[x]) = U(R) f R deg f = 0 f R deg f > 0 f f = I(f)f 0 I(f) U(R) f (ii) 7.4 R[x 1,..., x n 1, x n ] = (R[x 1,..., x n 1 ])[x n ] n = 1 R R[x] 0 f(x) R[x] deg f deg f = 0 R 7.8 (i) R[x] deg f > 0 f = gh deg g > 0, deg h > 0 deg g < deg f deg h < deg f g h f f = I(f)f 0 f 0 f (ii) I(f) R R[x] f = p 1 p k f 1 f l = q 1 q m g 1 g n f p 1,..., p k, q 1,..., q m R f 1,..., f l, g 1,..., g n 1 f 1 f l g 1 g n 7.6 I(f) p 1 p k q 1 q m u U(R) up 1 p k = q 1 q m R K[x] uf 1 f l = g 1 g n c i f i = g i c i K I(g i ) = 1 c i R g i c i U(R) 7 6
24 8 8.1 R M R M M, (r, m) rm M R- R- r(x + y) = rx + ry, (r + s)x = rx + ry, (rs)x = r(sx), 1x = x (x, y M, r, s R) R- R R- N M R- N rx N r R x N RN N N R f : M M R- f(a + b) = f(a) + f(b), f(ra) = rf(a), (r R, a, b M) f(ra) = rf(a) f R Z- 2. R R- I R- R I R 3. K K- K M R- S M { } < U >= r i u i r i R, u i U i U R- 2. U < U M =< U > M R- u 1, u 2,..., u n M = Ru 1 + Ru Ru n 3. r 1 u 1 + r 2 u r n u n = 0 (r i R) r 1 = r 2 = = r n = 0 u 1, u 2,..., u n R- M U R- U R- M U R- 8 1
25 V K K- V K- 8.3 R R- A A R R- a, b A, r R (ra)b = a(rb) = r(ab) R R 2. G = {1 = u 1, u 2,..., u n } G R- R[G] = Ru 1 Ru n ( n ) n n α i u i β j u j = α i β j u i u j. i=1 j=1 i,j=1 G A = C[G] V A- g G φ(g) : V V, (v gv) φ(g) GL(V ) φ : G GL(V ), (g φ(g)) φ : G GL(V ) V A M R- M 0 M M 8.1 (Schur s Lemma) M N R- (1) f : M N R- 0 f (2) End R (M) M M End R (M) f R- Kerf Imf R- (1) f 0 Kerf M Imf 0 Kerf = 0 Imf = N f (2) (1) 8 2
26 R-( ) M R- [ ] M [ ] 2. R R-( ) [ ] R ( )- [ ] 3. M R- M 1 M 2 M i (M 1 M 2 M i ) n M n = M n+1 = M [ ] 9.1 R- M [ ] M [ ] R- M M 1 M 2 {M i i N} M n M n = M n+1 = S M 1 M 2 M i M i S M i M i+1, M i M i R- M (i) M (ii) M R- R- (i) (ii) N M R- S N R- R- S N 0 N N 0 x N N 0 Rx + N 0 N 0 N 0 N = N 0 N (ii) (i) M 1 M 2 M N = i M i R- N =< u 1, u 2,..., u n > N M m u 1, u 2,..., u n N M m M m+1 N. M 9.1 M
27 1 9.4 R R[x 1, x 2,..., x n ] n = 1 I R[x] I i = {r R f(x) = a i x i + + a 1 x + a 0 I a i = r } R f(x) = a i x i + + a 1 x + a 0 I xf(x) = a i x i a 1 x 2 + a 0 x I I 0 I 1 I 2 R 9.1 I r = I r+1 = r 9.2 I 0, I 1,..., I r a i1,..., a isi I i (i = 0, 1,..., r) R f ij a ij I i I r s i I = R[x]f ij (x). i=0 j=1 f = a m x m + + a 1 x + a 0 I m = deg f m = 0 f = a 0 I 0 = s 0 j=1 Ra 0j = s 0 j=1 Rf 0j m > 0 r < m e = m r r m e = 0 a m I m = I m e = a m = s m e j=1 c j a (m e)j s m e deg(f(x) x e j=1 s m e j=1 Ra (m e)j c j f (m e)j (x)) < deg f(x) f r i=0 si j=1 R[x]f ij (x) R[x] 9.2 R[x] S R s 1,..., s n S R {s 1,..., s n } S S {s 1,..., s n } R- S Z[x] x Z- Z[x] Z- Z + Zx R R- R R 9 2
2014 (2014/04/01)
2014 (2014/04/01) 1 5 1.1...................................... 5 1.2...................................... 7 1.3...................................... 8 1.4............................... 10 1.5 Zorn...........................
More informationAI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
More information16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
More informationALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,
More informationJacobson Prime Avoidance
2016 2017 2 22 1 1 3 2 4 2.1 Jacobson................. 4 2.2.................... 5 3 6 3.1 Prime Avoidance....................... 7 3.2............................. 8 3.3..............................
More information13 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 informationII Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
More information2012 A, N, Z, Q, R, C
2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
More informationx, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
More informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More informationA11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
More informationx = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................
More informationORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information1 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,
More information1990 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 information1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
More informationS K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.
() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
More informationSolutions to Quiz 1 (April 20, 2007) 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T T T T T F T F F F T T F T F T T T T T F F F T T F
Quiz 1 Due at 10:00 a.m. on April 20, 2007 Division: ID#: Name: 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F T 2. 1.1 (1) (7) p.44 (1)-(4)
More information( )
18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
More information14 (x a x x a f(x x 3 + 2x 2 + 3x + 4 (x 1 1 y x 1 x y + 1 x 3 + 2x 2 + 3x + 4 (y (y (y y 3 + 3y 2 + 3y y 2 + 4y + 2 +
III 2005 1 6 1 1 ( 11 0 0, 0 deg (f(xg(x deg f(x + deg g(x 12 f(x, g(x ( g(x 0 f(x q(xg(x + r(x, r(x 0 deg r(x < deg g(x q(x, r(x q(x, r(x f(x g(x r(x 0 f(x g(x g(x f(x g(x f(x g(x f(x 13 f(x x a q(x,
More informationone way two way (talk back) (... ) C.E.Shannon 1948 A Mathematical theory of communication. 1 ( ) 0 ( ) 1
1 1.1 1.2 one way two way (talk back) (... ) 1.3 0 C.E.Shannon 1948 A Mathematical theory of communication. 1 ( ) 0 ( ) 1 ( (coding theory)) 2 2.1 (convolution code) (block code), 3 3.1 Q q Q n Q n 1 Q
More information2, Steven Roman GTM [8]., [3].,.
( ) : 28 7 22 2, Steven Roman GTM [8]., [3].,. 1 5 1.1........................................................... 5 1.2......................................................... 6 1.3....................................................
More informationkoji07-01.dvi
2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?
More information2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
More information行列代数2010A
a ij i j 1) i +j i, j) ij ij 1 j a i1 a ij a i a 1 a j a ij 1) i +j 1,j 1,j +1 a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a, ij i j 1,j 1,j +1 ij 1) i +j a
More information2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =
2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I
More informationI A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
More information1 1.1 n 3 X n + Y n = Z n Fermat Fermat Diophantus 2 Bachet x 2 + y 2 = z 2 Fermat Wiles 4 Kummer 5 Dedekind 6 ζ n 1 n ζ n =
2013 2 26 2 26 1 2 2 5 3 8 4 11 5 13 6 19 6.1...................................... 19 6.2...................................... 39 6.3...................................... 47 6.4 Noether Dedekind............................
More information熊本県数学問題正解
00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (
More informationuntitled
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
More information2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
More information(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n
. 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d = 99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n
More informationArmstrong culture Web
2004 5 10 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1 Armstrong culture Web 1 3 1.1................................. 3 1.2.................................
More information18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
More information1 Groebner hara/
1 Groebner 2005 1 sinara@blade.nagaokaut.ac.jp http://blade.nagaokaut.ac.jp/ hara/ 2005 7 19 3 1 1 1.................................................. 1 2 1..............................................
More informationp-sylow :
p-sylow :15114075 30 2 20 1 2 1.1................................... 2 1.2.................................. 2 1.3.................................. 3 2 3 2.1................................... 3 2.2................................
More information2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a
More information.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +
.1 n.1 1 A T ra A A a b c d A 2 a b a b c d c d a 2 + bc ab + bd ac + cd bc + d 2 a 2 + bc ba + d ca + d bc + d 2 A a + d b c T ra A T ra A 2 A 2 A A 2 A 2 A n A A n cos 2π sin 2π n n A k sin 2π cos 2π
More informationn ( (
1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128
More informationA S- hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A %
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A S- 3.4.5. 9 phone: 9-8-444, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office
More informationr 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
More information数学Ⅱ演習(足助・09夏)
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
More informationB ( ) :
B ( ) : 29 2 6 2 B. 5............................................................ 5......................................................... 5..2..................................................... 5..3........................................................
More information1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
More informationy π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
More informationBasic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)
More information21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
More information, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
More information1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th
1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2
More information20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
More information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
More information1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1.2 R A 1.3 X : (1)X (2)X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1. R A 1.3 X : (1)X ()X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f 1 (A) f X X f 1 (A) = X f 1 (A) = A a A f f(x) = a x
More informationx () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
More information,2,4
2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................
More informationi I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
More informationA µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x A (1)
7 2 2.1 A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x 1 2.1.1 A (1) A = R x y = xy + x + y (2) A = N x y = x y (3) A =
More informationLINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
More information1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
More informationx V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
More informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
More information¿ô³Ø³Ø½øÏÀ¥Î¡¼¥È
2011 i N Z Q R C A def B, A B. ii..,.,.. (, ), ( ),.?????????,. iii 04-13 04-20 04-27 05-04 [ ] 05-11 05-18 05-25 06-01 06-08 06-15 06-22 06-29 07-06 07-13 07-20 07-27 08-03 10-05 10-12 10-19 [ ] 10-26
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More information2016
2016 1 G x x G d G (x) 1 ( ) G d G (x) = 2 E(G). x V (G) 2 ( ) 1.1 1: n m on-off ( 1 ) off on 1: on-off ( on ) G v v N(v) on-off G S V (G) N(v) S { 3 G v S v S G G = 1 OK ( ) G 2 3.1 u S u u u 1 G u S
More informationI, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
More informationさくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n
1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
More informationIII 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
More informationMathematical Logic I 12 Contents I Zorn
Mathematical Logic I 12 Contents I 2 1 3 1.1............................. 3 1.2.......................... 5 1.3 Zorn.................. 5 2 6 2.1.............................. 6 2.2..............................
More information2014 x n 1 : : :
2014 x n 1 : : 2015 1 30 : 5510113 1 x n 1 n x 2 1 = (x 1)(x+1) x 3 1 = (x 1)(x 2 +x+1) x 4 1 = (x 1)(x + 1)(x 2 + 1) x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) 1, 1,0 n = 105 2 1 n x n 1 Maple 1, 1,0 n 2
More informationII 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
More informationAkito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
More informationII (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
More information高校生の就職への数学II
II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................
More informationax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
More informationii-03.dvi
2005 II 3 I 18, 19 1. A, B AB BA 0 1 0 0 0 0 (1) A = 0 0 1,B= 1 0 0 0 0 0 0 1 0 (2) A = 3 1 1 2 6 4 1 2 5,B= 12 11 12 22 46 46 12 23 34 5 25 2. 3 A AB = BA 3 B 2 0 1 A = 0 3 0 1 0 2 3. 2 A (1) A 2 = O,
More information9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
More informationver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
More information微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
More informationIMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
More information18 5 10 1 1 1.1 1.1.1 P Q P Q, P, Q P Q P Q P Q, P, Q 2 1 1.1.2 P.Q T F Z R 0 1 x, y x + y x y x y = y x x (y z) = (x y) z x + y = y + x x + (y + z) = (x + y) + z P.Q V = {T, F } V P.Q P.Q T F T F 1.1.3
More information6.1 (P (P (P (P (P (P (, P (, P.101
(008 0 3 7 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........
More informatione a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
More information2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m
2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2
More information1 I
1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More information6.1 (P (P (P (P (P (P (, P (, P.
(011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.
More informationII
II 16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 +
More informationi 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35
More informationI. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
More information( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
More information12 2 e S,T S s S T t T (map) α α : S T s t = α(s) (2.1) S (domain) T (codomain) (target set), {α(s)} T (range) (image) s, s S t T s S
12 2 e 2.1 2.1.1 S,T S s S T t T (map α α : S T s t = α(s (2.1 S (domain T (codomain (target set, {α(s} T (range (image 2.1.2 s, s S t T s S t T, α s, s S s s, α(s α(s (2.2 α (injection 4 T t T (coimage
More information1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
More informationSAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
More information2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (
(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
More informationさくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a
... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c
More information2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
More informationMazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ
Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R
More informationuntitled
yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
More information直交座標系の回転
b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx
More information17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
More information