1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5.
A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6) f : Z Z f(n) = n π = h f h : Z Z/ Z/ 1
A1 C, V = M 3 (C) 3, V C A M 3 (C), V f A : V V f A (X) = AX + XA (1) α C f A αi = f A 2α I f A 2α 2α V 2α (2) A, B M 3 (C), P B = P 1 AP, φ : V V f B = φ f A φ 1 (3) T J 1 1 1 T = 1 1 1 2 2 4, P 1 T P = J P (4) (3) T, f T : V V A2 a > 0 f a (x) = { ax + x 2 sin 1 x x 0 0 x = 0. (1) f a (x). (2) f a (x) C 1 -. (3) a > 1., f a 0. (4) 0 < a 1., 0 f a. 2
A3 f : R R 2 f(t) = ( ) 4t 3 + t, 4t 2 4 3 + t 4 f R R 2 (1) (1, ) R f f(1, ) R 2 (2) f R f(r) R 2 (3) f (4) f f : R f(r) f 1 : f(r) R f(r) R 2 A4 m. X k, k = 1, 2,..., m 1. (, x > 0 P (X k x) = e x.) (1) 0 < a < b. X 1 > a X 1 > b. (2) S m m X k. k=1 (3) λ. S m λ. 3
A5 Pascal ( ) function f(n: integer; a, b, c: boolean) : boolean; var ta, tb, tc : boolean; var r : integer; begin ta := false; tb := false; tc := false; r := n mod 4; n := n div 4; if c and (r = 1) then tc := true; if b then begin case r of 1 : tc := true; 2 : begin tb := true; tc := true; end end end; if a then begin case r of 1 : tc := true; 2 : begin tc := true; tb := true; end; 3 : begin ta := true; tb := true; tc := true; end end end; if (n > 0) then f := f(n, ta, tb, tc) else if (r = 0) then f := true else f := tc; end; (1) f(22, false, true, true) f(22, false, false, true) (2) x integer x 0 f(x, true, true, true) true x 4
B1 F 3 = Z/3Z 3 F 3 2 GL 2 (F 3 ) SL 2 (F 3 ) { ( ) } a b GL 2 (F 3 ) = A = c d a, b, c, d F 3, det(a) 0 SL 2 (F 3 ) = { A GL 2 (F 3 ) det(a) = 1 } (1) GL 2 (F 3 ), SL 2 (F 3 ) G = SL 2 (F 3 ) G S, T, U ( ) ( ) ( ) 1 1 0 1 1 1 S =, T =, U = 1 1 1 0 0 1 (2) S, T (3) G S, T, U (4) G (Sylow) 3 (5) G D(G) 5
B2 1 R R 2 I, J R- M = (R/I) (R/J) = {(a + I, b + J) a, b R} f : R/I M, g : R/J M φ : R M f(1 R + I) = (1 R + I, J), g(1 R + J) = (I, 1 R + J), φ(1 R ) = (1 R + I, 1 R + J) R- σ : R/I R/(I + J) τ : R/J R/(I + J) σ(1 R + I) = 1 R + (I + J), τ(1 R + J) = 1 R + (I + J) R- (1) ψ : M R/(I + J) ψ f = σ ψ g = τ R- a, b R ψ (a + I, b + J) M (2) ψ Im φ = Ker ψ (3) a R a + (I J) a + (I + J) R/(I J) R/(I + J) a + I a + J R/I R/J S x y S xy = 0 y = 0 6
B3. F : R 3 R F (x, y, z) = x 2 + y 2 z 2, q M q = F 1 (q) (1) M q q 0. q 0. (2) M q p 0 = (x 0, y 0, z 0 ) M q. {(x, y, z) x 0 x + y 0 y z 0 z = q} (3) a, b, c a 2 + b 2 + c 2 = 1, h : M q R h(x, y, z) = ax + by + cz. h a, b, c., h, (ξ, η) h/ ξ = h/ η = 0. (4) φ : R ( π, π) M 1. M 1 2 φ φ ω. φ(t, θ) = (cosh t cos θ, cosh t sin θ, sinh t) ω = xdy dz + ydz dx zdx dy B4 R 3 X, Y X = {(x, y, z) R 3 x 2 + y 2 + z 2 = 1}, Y = {(x, y, z) R 3 x 2 + y 2 + z 2 = 1/4} {(0, 0, z) R 3 1/2 z 1} Z = X Y (1) X H 0 (X; Z), H 1 (X; Z), H 2 (X; Z) (2) Y H 0 (Y ; Z), H 1 (Y ; Z), H 2 (Y ; Z) (3) Z H 0 (Z; Z) (4) Z H 1 (Z; Z), H 2 (Z; Z) H 1 (Z; R), H 2 (Z; R) 7
B5 C r D r = {z C z < r}, D 1 f(z) f(z) = 1 2 log 1 + z 1 z., f(0) = 0., w Im w. (1) f(z) z = 0,. (2) f : D 1 C, Ω = f(d 1 ). (3) 0 < r < 1, sup z D r Im f(z). (4) g(z) D 1 g(0) = 0, g(d 1 ) Ω,. g (0) 1 B6 y(x) x 0. (1) (y (x)) 2 = 1 2 y(x)4 y(x) 2 + 1 2, y(0) = 2,. x 0 y(x) > 1 y (x) 0. (2) y (x) = y(x) 3 y(x), y(0) = 0, y (0) = 1 2. 8
B7 [0, 1] µ H 2 H f, g = 1 0 f(x)g(x) dµ(x) T : H H (T f)(x) = (1) T (2) n = 1, 2,... f H (T n+1 f)(x) = 1 n! x 0 x 0 f(y) dµ(y) (3) n = 1, 2,... T n+1 1/n! (4) T 0 (5) T T (T f + T f)(x) = (x y) n f(y) dµ(y) 1 0 f(y) dµ(y) 9
B8 X j, j {1, 2,...} p. ( j P (X j = 1) = p, P (X j = 0) = 1 p (p (0, 1)).) m {1, 2,...} Y m +, Ym, Z m Y + m = m X j, Ym = j=1 m j=1 ( 1) j 1 X j, Z m = min{n Y + n m}.. (1) Y + m Y m,. (2) m, n {1, 2,...} Cov(Y + m, Y n ). (3) k {1, 2,...} P (Z 1 = k) = (1 p) k 1 p. (4) k, m {1, 2,...} P (Z m = k) B9 (1) (X, Y ) E[X] X E[X Y ] Y X.. E [E[X Y ]] = E[X] (2) S, T U = E[T S]. θ W 2 MSE(W ) = E[(W θ) 2 ]. (i) MSE(U) MSE(T ). (ii) L( ) θ E[L(U)] E[L(T )]. 10
B10. = φ φ (logically valid). (1) R Q 2 1. φ ( x y R(x, y) z Q(z)) (prenex normal form) θ. θ = φ θ, (quantifier) Q i =, θ 0 θ (Q 1 x 1 Q n x n θ 0 ) (n = 0, 1,...). (2) φ(x, y) = x yφ(x, y) = xθ(x) θ(x). (3) L 2. L φ F (φ) F, φ = φ = F (φ). F., L-. B11 F 2 F 2 [X] 15 C C = {f(x) F 2 [X] f(α 6 ) = f(α 7 ) = 0, deg f < 15} deg f f α X 4 + X + 1 F 2 [X] (1) C F 2 (2) C 5 C f(x) = g(x) = 0 i 14 g i X i d(f(x), g(x)) = #{0 i 14 f i g i } (3) C [15, 7] 7 0 i 14 f i X i, 11
B12 Scheme (define (concat ll) (if (null? ll) () (append (car ll) (concat (cdr ll))))) (define (substr s l h) (list s l h)) (define (str s) (car s)) (define (low s) (cadr s)) (define (high s) (caddr s)) (define (len s) (- (high s) (low s))) (define (inclow n s) (substr (str s) (+ (low s) n) (high s))) (define (inchigh n s) (substr (str s) (low s) (+ (high s) n))) (define (dechigh n s) (substr (str s) (low s) (- (high s) n))) (define (output r s) (cons r s)) (define (result o) (car o)) (define (next o) (cdr o)) (define (rfun1 r) 1) (define (rfun2 r1 r2) (+ r1 r2)) (define (a lit) (lambda (s) (let ((n (string-length lit))) (if (and (<= n (len s)) (string=? (substring (str s) (low s) (+ (low s) n)) lit)) (list (output (rfun1 lit) (inclow n s))) ())))) (define (seq p1 p2) (lambda (s) (let ((f (lambda (rs1) (map (lambda (rs2) (output (rfun2 (result rs1) (result rs2)) (next rs2))) (p2 (next rs1)))))) (concat (map f (p1 s)))))) (define (alt p1 p2) (lambda (s) (append (p1 s) (p2 s)))) 12
(define (pe s) ((alt (seq (a "+") (seq pe pe)) (a "")) s)) (define (run p s) (filter (lambda (rs) (= (low (next rs)) (high (next rs)))) (p (list s 0 (string-length s))))) : Scheme " 0 ( ) string-length 2 string=? s l h 1 (s l h) ( 0 ) (substring s l h) Scheme (1) ((a "1") ("12" 0 2)) (2) ((seq (a "1") (a "2")) ("12" 0 2)) (3) (run pe "++") (4) n s n + n (length (run pe s n )) a n a n a i (0 i n 1) 13