N = Q R = R N < R
|
|
|
- きみとし おうじ
- 7 years ago
- Views:
Transcription
1
2 N = Q R = R N < R
3 1 2 P, Q P P P Q P Q P Q P Q P = Q P Q P Q P Q Q P x 2 + y 2 = z 2 x, y, z = 5 2 x 3 + y 3 = z 3 x, y, z 4 40 x, y, z x, y, z x 2 + y 2 = z 2 x, y, z x, y, z x 3 + y 3 = z 3 x y z(x, y, z (x 2 + y 2 = z 2 )) x y z(x, y, z = (x 3 + y 3 = z 3 )) x, y, z x, y, z Z + 3
4 x y z(((x Z + ) (y Z + ) (z Z + )) (x 2 + y 2 = z 2 )) x y z(((x Z + ) (y Z + ) (z Z + )) = (x 3 + y 3 = z 3 )) P Q Q P (1) P Q Q P (2) P (Q R) (P Q) R (3) P (Q R) (P Q) R (4) P (Q R) (P Q) (P R) (5) P (Q R) (P Q) (P R) (6) P ( P ) (7) (P Q) ( P ) ( Q) (8) (P Q) ( P ) ( Q) (9) P = Q ( P ) Q (10) ( xp ) x( P ) (11) ( xp ) x( P ) (12) x yp y xp (13) x yp y xp (14) x(p Q) ( xp ) ( xq) (15) x(p Q) ( xp ) ( xq) (16) P x x(p Q) P ( xq) (17) P x x(p Q) P ( xq) (18) P P = Q Q (19) P Q Q P (P Q) (Q P ) 2.1 4
5 2.1.1 P Q P, Q P Q (6)(8)(9) (8) x x x = x= x = x x x x x (8) x x x = x= x = x x = (8),,,,,,,, = = = = = = = = = = = (6) = P, Q R 5
6 P (Q R) ( ) ( ) = = (P Q) (P R) ( ) ( ) ( ) ( ) = = (8)(9) = (10) P = Q ( P ) Q P Q P Q (P Q) (8) (7) ( P ) Q P P Q P = Q P P Q ( P ) Q (11)(12) ( xp ) P x x( P ) x P ( xp ) x P x( P ) P x 6
7 ( xp ) x P x( P ) cm 5 A a A a A A a a A a A a / A A a 6 7 x(x A x B) A B A = B A B A B A B x(x A = x B) A B A B A B B A A B A B A B A B B A A A A x(x = x A) x x 4 (10) x = x A x A 7
8 A B B A A = B A B B C A C 8 A = {,, } B = {x x 3 2 } N = {0, 1, 2, 3, 4,... } 9 {x x A x B} A B A B {x x A x B} A B A B {x x A x / B} A B A B A B A B = A A B (20) A B = A A B (21) A B = B A (22) A B = B A (23) A (B C) = (A B) C (24) A (B C) = (A B) C (25) A (B C) = (A B) (A C) (26) A (B C) = (A B) (A C) (27) A (B C) = (A B) (A C) (28) A (B C) = (A B) (A C) (29) (26) (20) (26) (28) (20) (= ) x A A = A B B x B A B ( =) A B C A B A A A B x A A B x B x A B A A B 8
9 (26) A (B C) = (A B) (A C) x A (B C) x (A B) (A C) x A (B C) (x A x (B C)) (x A (x B x C)) 4 (5) (x A (x B x C)) ((x A x B) (x A x C)) ((x A x B) (x A x C)) (x (A B) x (A C)) x (A B) (A C) (28) A (B C) = (A B) (A C) x A (B C) x (A B) (A C) (= ) x A (B C) x A x (B C) x (B C) x (B C) x (B C) x B x C x (B C) (x B x C) 4 (8) x (B C) x B x C x A (x B x C) 4 (5) x (A x B) (A x C) x (A B) (A C) ( =) U U U X U X U X def = {x x U X(X U x X)} U (30) X def = {x X(X U x X)} U (31) A, B U U = {A, B} U U X U X U X = A B (32) X = A B (33) 9
10 [3] def = def def X def = {x X(X U x X)} U = X U X U X x U U = X U X U X = U X U U U U U U U U = X = X U 11 A {X X A} A A P(A) A = {,, } P(A) = {, { }, { }, { }, {, }, {, }, {, }, {,, }} P( ) P( ) = { } { } = { } 12 a, b {{a}, {a, b}} (a, b) a b (a, b) (c, d) ((a, b) = (c, d)) (a = c b = d) [2] ((a, b) = (c, d)) (a = c b = d) a b (a, b) a b 13 (a, b, c) = ((a, b), c) (a, b, c) a, b, c (a 1, a 2,, a n 1, a n ) = ((a 1, a 2,, a n 1 ), a n ) n ((a 1, a 2,, a n ) = (b 1, b 2,, b n )) (a i = b i (i = 1, 2,, n)) 14 10
![d) ((a, b) = (c, d)) (a = c b = d) [2] ((a, b) = (c, d)) (a = c b = d) a b (a, b) a b 13 (a, b, c) = ((a, b), c) (a, b, c) a, b,](/docs-images/41/22619179/images/page_10.jpg "c (a 1, a 2,, a n 1, a n ) = ((a 1, a 2,, a n 1 ), a n ) n ((a 1, a 2,, a n ) = (b 1, b 2,, b n )) (a i = b i (i = 1, 2,, n)) 14")
11 A, B {(x, y) x A y B} A B A B A A A 2 A 2 A A 3 B A n 1 A A n A = A = (A B = B A) ((A = B) (A = ) (B = )) A 15 A A A A R A a, b A (a, b) R arb = 16 A, B G A B (G, A, B) A B G A B (G, A, B) {a a A b B (a, b) G} (G, A, B) {b b B a A (a, b) G} (G, A, B) 17 (G, A, B) G {(b, a) b B a A (a, b) G} G G 1 (G 1, B, A) B A (G, A, B) (G 1 ) 1 = G G G 1 18 (G, A, B) f A B f : A B A f B f A B f = (G, A, B) Domf Imf (G 1, B, A) f 1 (f 1 ) 1 = f f f 1 f : A B G f U A {b b B u U (u, b) G} f(u) f U Imf = f(domf) = f(a) V B f 1 (V ) f 1 V f V a A f({a}) f(a) 11
12 19 A A R R A a A ara (34) a, b A ((arb bra) = a = b) (35) a, b, c A ((arb brc) = arc) (36) a, b A arb bra (37) R < > (34) 20 A A 19 R A (A, ) (A, ) (, ) (34) a A ara a(a A = ara) R = a a R 3 2 P P = Q Q a a R = ara a(a R = ara) (A, ) U A U = (U U) U U (U, U ) (A, ) U (U, U ) U U (A, ) A U A U (U, U ) (U, ) U 12
13 21 A A R R a A ara (38) a, b A (arb = bra) (39) a, b, c A ((arb brc) = arc) (40) 22 A R A a A {x x A xra} R a C R (a) C(a) C(a) A a A a C(a) C(a) (41) a, b A (C(a) = C(b) arb) (42) a, b A (C(a) C(b) C(a) C(b) = ) (43) C(x) = A C(x) A 28 (44) x A (43) C(a) = C(b) C(a) C(b) = (41) = C(a) C(b) x x C(a) C(b) x x C(a) x C(b) xra xrb (39) (40) arb (42) C(a) = C(b) (43) (44) A C(x)(x A) A {C(x) x A} R A A/R C(b) C(a) C(c) C(d) A 13
14 23 (G, X, Y ) (G, X, Y ) x X y Y (x, y) G (45) x X (((x, y 1 ) G (x, y 2 ) G) = y 1 = y 2 ) (46) (45) x X y Y (G, X, Y ) X (46) y Y x X y Y y Y x X X f : X Y (45) (46) x X y Y f(x) = {y} f(x) = y X X f f(x) {y} y y = f(x) x X y Y x y Y Y 24 f X Y U X u U u f(u) U Y (45) (46) f U f U G = {(u, f(u)) u U} (G, U, Y ) U Y f U f U f U 14
15 25 X Y f Y Z g x X x g(f(x)) X Z (45) (46) g f gf f g f : X Y g : Y Z g f : X Z g f X Z f g 7 Y 26 f X Y f(x) = Y f x 1, x 2 X (x 1 x 2 = f(x 1 ) f(x 2 )) f f f 1 (f 1 ) 1 = f f f 1 U X u U u u u U X U X U X U = X U X X X X id X 27 Λ a A a Λ A Λ a(λ) Λ λ Λ a(λ) a λ a (a λ ) λ Λ a(λ) {a λ } λ Λ {a λ } λ Λ {a λ } λ Λ 28 {X λ } λ Λ λ Λ X λ (X λ ) λ Λ Λ 15
16 λ Λ λ Λ λ Λ X λ def = X λ def = X λ def = X (X λ ) λ Λ (47) X {X λ } λ Λ X (X λ ) λ Λ (48) X {X λ } λ Λ { f f Λ f } X λ λ Λ f(λ) X λ (X λ ) λ Λ (49) λ Λ Λ = {1, 2,, n} X λ λ Λ X λ λ Λ n X λ X λ λ=1 Λ = {1, 2} λ=1 λ Λ n X λ λ=1 n X λ λ=1 2 2 X λ = X 1 X 2 X λ = X 1 X 2 λ=1 2 X λ X 1 X 2 12 ((x 1, x 2 ) = (y 1, y 2 )) λ=1 (x 1 = y 1 x 2 = y 2 ) {1, 2} X 1 X 2 ; (1 x 1, 2 x 2 ) 2 X λ X 1 X 2 n X λ n n λ=1 λ= *1 1, 2, 3, A A {,, } = 3 0 1, 2, 3, A, B A B A B A = B A B A = B A B B A A B B A *1 0 16
17 A B A B (50) B A A B (51) A B (50) ( A B B A ) ( A = B ) (52) ( A B B C ) = ( A C ) (53) A, B A B B A (54) ( A B ) (B A ) (55) A f B B f 1 A (52) = A f B B g C A g f C (53) (52) = (54) (55) [3] (52) (54) (55) (50) (51) A B B A A B A = B A < B B > A 30 N = Q N Q N = Q N Q N Q N Q Q N 29 (52) N = Q Q ± n m n, m n m 2n 3 m n (56) m 3n 5 m Q N Q N 17
![= (54) (55) [3] (52) (54) (55) (50) (51) A B B A A B A = B A < B B > A 30 N = Q N Q N = Q](/docs-images/41/22619179/images/page_17.jpg "N Q N Q N Q Q N 29 (52) N = Q Q ± n m n, m 0 +0 1 + n m 2n 3 m n (56) m 3n 5 m Q N Q N 17")
18 N ℵ 0 a N Q N = N 2 Q = Q 2 31 R = R 2 R R = R 2 a, b R a < b {x x R a < x < b} (a, b) a, b R (0, 1) = R (0, 1) R x (0, 1) x tan(π(x 1/2)) (0, 1) = R (0, 1) 2 = R 2 R = R 2 (0, 1) = (0, 1) 2 (0, 1) x (x, x) (0, 1) 2 (0, 1) (0, 1) 2 x (0, 1) x = 0.x 1 x 2 x 3 (0, 1) 2 (0.x 1 x 2 x 3, 0.y 1 y 2 y 3 ) 0.x 1 y 1 x 2 y 2 x 3 y 3 (0, 1) (57) (0, 1) (0, 1) 2 (0, 1) = (0, 1) 2 R = R 2 c = = / (0, 1) 32 N < R N < R A A < P(A) N < R N R N R 31 (0, 1) = R N (0, 1) 31 (0, 1) 31 (0, 1)
19 0.1 9 = = 1 / (0, 1) 0. 0 = 0 / (0, 1) (0, 1) {x x = } (0, 1) N (0, 1) 1 0.a 11 a 12 a 13 a 14 a 15 a 16 a a 21 a 22 a 23 a 24 a 25 a 26 a a 31 a 32 a 33 a 34 a 35 a 36 a a 41 a 42 a 43 a 44 a 45 a 46 a a 51 a 52 a 53 a 54 a 55 a 56 a a nn 1 a n = a = 0.a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 2 a nn = 1 2 a (0, 1) a n a nn a N (0, 1) A < P(A) A x {x} P(A) A P(A) A P(A) A P(A) N < R A P(A) A f P(A) B = {x x A x / f(x)} B A B P(A) f a A f(a) = B B a a B B a / f(a) = B a / B = f(a) a a A a / f(a) a B 33 (A, ) a A x A x a a A a A x A a x = a = x a A a A x A a x a A a A x A x a = a = x a A A maxa mina 19
20 (A, ) (U, ) a A x U x a a U U A U A a A x U a x a U U A U A A supa infa U U U U U U U A U A U A U A U A U A a A a a x(x = x a) a {1, 2} (A, ) a A {x x A x a x a} A a A a a A B = A a B A A a B A A a B A A a = a A 35 (A, ) A A A a A a b A a a b x A a x b x = a x = b U = {x x A a x a x} a A A U A U minu a N N n N n + 1 Z Z Z 20
21 (, ) 36 0 ( 1) (+1) ( 2) (+2) ( 3) (+3) (Z, ) R A (A, ) A(A ) (A, ) A(A ) (A, ) 38 [3] 37 N N N N N ω Ω = N {ω} n N n ω ω ω Ω (Ω, ) (Ω, ) ω Ω A A A A 38 21
22 [3] 39 X X X P(X) P(X) X P(X) X P(X) X X < P(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 40 [1] I [2] [3] [4] I.N.Bronstein K.A.Semenjaef G.Groche V.Ziegler D.Ziegler 22
Solutions 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)
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
x : = : x x
x : = : x x x :1 = 1: x 1 x : = : x x : = : x x : = : x x ( x ) = x = x x = + x x = + + x x = + + + + x = + + + + +L x x :1 = 1: x 1 x ( x 1) = 1 x 2 x =1 x 2 x 1= 0 1± 1+ 4 x = 2 = 1 ± 5 2 x > 1
¿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
~ ~.86 ~.02 ~.08 ~.01 ~.01 ~.1 6 ~.1 3 ~.01 ~.ω ~.09 ~.1 7 ~.05 ~.03 ~.01 ~.23 ~.1 6 ~.01 ~.1 2 ~.03 ~.04 ~.01 ~.1 0 ~.1 5 ~.ω ~.02 ~.29 ~.01 ~.01 ~.11 ~.03 ~.02 ~.ω 本 ~.02 ~.1 7 ~.1 4 ~.02 ~.21 ~.I
1 1 2 65
3 3 2000 6 14 2 30 4 2 1 1 2 65 1!?? < > 3 2 2 100 19 19 100 100 100 < > 19 2 2 2 2 < > 2000 2000 50 1945 5 50 1945 5 45 20 20 4 1945 4 5 5 5 100 50 20 5 2 20 5 20 5 5 6 20 6 19 5 5 6 5 6 2 20 6 21
EL-G37
EL-G37 A A A A A A A A A A A A A A GT F543210A CE CA 4 00. 00 23.80 2 015.6250 77.14286 96. 91. 102. 25. 9. 1 700. 2 720. 2.5 7. 20. 25. 220. 400. 4 096. 4.685 0.084 M 125. M 28. M 85.
Taro13-第６章（まとめ）.PDF
% % % % % % % % 31 NO 1 52,422 10,431 19.9 10,431 19.9 1,380 2.6 1,039 2.0 33,859 64.6 5,713 10.9 2 8,292 1,591 19.2 1,591 19.2 1,827 22.0 1,782 21.5 1,431 17.3 1,661 20.0 3 1,948 1,541 79.1 1,541 79.1
15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = N N 0 x, y x y N x y (mod N) x y N mod N mod N N, x, y N > 0 (1) x x (mod N) (2) x y (mod N) y x
A( ) 1 1.1 12 3 15 3 9 3 12 x (x ) x 12 0 12 1.1.1 x x = 12q + r, 0 r < 12 q r 1 N > 0 x = Nq + r, 0 r < N q r 1 q x/n r r x mod N 1 15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = 3 1.1.2 N N 0 x, y x y N x y
109
108 1 2 3 109 110 111 112 1 c c < c c< B N a d 1 2 3 5 8 9 0 ; 2c< rc a h 3c 113 1p 2c< 3c< a r 4cc 1p 2 c 3c< c p a c 4p 5 c rr c c Cr m e 114 1 c c < c< B N d a 1 3 5 9 0 ; 2c< B ; p c< BN50; ar h 3c
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
?
240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................
FX ) 2
(FX) 1 1 2009 12 12 13 2009 1 FX ) 2 1 (FX) 2 1 2 1 2 3 2010 8 FX 1998 1 FX FX 4 1 1 (FX) () () 1998 4 1 100 120 1 100 120 120 100 20 FX 100 100 100 1 100 100 100 1 100 1 100 100 1 100 101 101 100 100
... 3... 3... 3... 3... 4... 7... 10... 10... 11... 12... 12... 13... 14... 15... 18... 19... 20... 22... 22... 23 2
1 ... 3... 3... 3... 3... 4... 7... 10... 10... 11... 12... 12... 13... 14... 15... 18... 19... 20... 22... 22... 23 2 3 4 5 6 7 8 9 Excel2007 10 Excel2007 11 12 13 - 14 15 16 17 18 19 20 21 22 Excel2007
程蘊(ていうん).indd
1963 1964 3 1963 1 2 3 1 2 3 1963 1964 1962 LT 1963 4 5 9 30 6 7 10 8 9 10 26 10 10 27 11 12 13 14 15 1 2 34 16 1963 10 7 17 18 19 10 8 20 8 9 10 16 21 22 17 22 10 24 23 10 27 24 28 25 30 26 27 11 20 UNHCR
A B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3
π 9 3 7 4. π 3................................................. 3.3........................ 3.4 π.................... 4.5..................... 4 7...................... 7..................... 9 3 3. p
N N 1,, N 2 N N N N N 1,, N 2 N N N N N 1,, N 2 N N N 8 1 6 3 5 7 4 9 2 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11 7 11 23 5 19 3 20 9 12 21 14 22 1 18 10 16 8 15 24 2 25 4 17 6 13 8 1 6 3 5 7 4 9 2 1 12 13
1W 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
http://banso.cocolog-nifty.com/ 100 100 250 5 1 1 http://www.banso.com/ 2009 5 2 10 http://www.banso.com/ 2009 5 2 http://www.banso.com/ 2009 5 2 http://www.banso.com/ < /> < /> / http://www.banso.com/
2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h
009 IA I, 3, 4, 5, 6, 7 7 7 4 5 h fx) x x h 4 5 4 5 1 3 1.1........................... 3 1........................... 4 1.3..................................... 6 1.4.............................. 8 1.4.1..............................
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... (
MultiWriter 5600C 活用マニュアル
1 *1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 9 1 2 3 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 a b c 26 27 28 C *1 *2 *2 29 2 2 2 2 2 2 2 2 2 30 *1 *2 ± *1 C C 31 32 33 34 35 36 M C Y K 1 2 3 4 5 6
2
1 2 10 14 945 3000 2012 3 10 4 5 6 7 8 9 10 11 12 2011 11 21 12301430 (1215 ) 13 6 27 17 () ( ) ( ) (112360) 2 (1157) (119099) ((11861231) )( ) (11641205) 3 (1277) 3 4 (1558) (1639)() 12 (1699)( ) 7 (1722)
n=360 28.6% 34.4% 36.9% n=360 2.5% 17.8% 19.2% n=64 0.8% 0.3% n=69 1.7% 3.6% 0.6% 1.4% 1.9% < > n=218 1.4% 5.6% 3.1% 60.6% 0.6% 6.9% 10.8% 6.4% 10.3% 33.1% 1.4% 3.6% 1.1% 0.0% 3.1% n=360 0% 50%
.1 1,... ( )
1 δ( ε )δ 2 f(b) f(a) slope f (c) = f(b) f(a) b a a c b 1 213 3 21. 2 [e-mail] nobuo@math.kyoto-u.ac.jp, [URL] http://www.math.kyoto-u.ac.jp/ nobuo 1 .1 1,... ( ) 2.1....................................
( [1]) (1) ( ) 1: ( ) 2 2.1,,, X Y f X Y (a mapping, a map) X ( ) x Y f(x) X Y, f X Y f : X Y, X f Y f : X Y X Y f f 1 : X 1 Y 1 f 2 : X 2 Y 2 2 (X 1
![( [1]) (1) ( ) 1: ( ) 2 2.1,,, X Y f X Y (a mapping, a map) X ( ) x Y f(x) X Y, f X Y f : X Y, X f Y f : X Y X Y f f 1 : X 1 Y 1 f 2 : X 2 Y 2 2 (X 1](/thumbs/91/105288755.jpg "( [1]) (1) ( ) 1: ( ) 2 2.1,,, X Y f X Y (a mapping, a map) X ( ) x Y f(x) X Y, f X Y f : X Y, X f Y f : X Y X Y f f 1 : X 1 Y 1 f 2 : X 2 Y 2 2 (X 1")
2013 5 11, 2014 11 29 WWW ( ) ( ) (2014/7/6) 1 (a mapping, a map) (function) ( ) ( ) 1.1 ( ) X = {,, }, Y = {, } f( ) =, f( ) =, f( ) = f : X Y 1.1 ( ) (1) ( ) ( 1 ) (2) 1 function 1 ( [1]) (1) ( ) 1:
OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P
4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e
![かんたん操作ガイド[arrows M02]](/thumbs/39/20071242.jpg "かんたん操作ガイド[arrows M02]")
![かんたん操作ガイド[arrows RM02]](/thumbs/39/20475147.jpg "かんたん操作ガイド[arrows RM02]")
![かんたん操作ガイド[arrows M03]](/thumbs/42/22941141.jpg "かんたん操作ガイド[arrows M03]")
ii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
( )
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...............................
2 (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
項 目
1 1 2 3 11 4 6 5 7,000 2 120 1.3 4,000 04 450 < > 5 3 6 7 8 9 4 10 11 5 12 45 6 13 E. 7 B. C. 14 15 16 17 18 19 20 21 22 23 8 24 25 9 27 2 26 6 27 3 1 3 3 28 29 30 9 31 32 33 500 1 4000 0 2~3 10 10 34
記号と準備
tbasic.org * 1 [2017 6 ] 1 2 1.1................................................ 2 1.2................................................ 2 1.3.............................................. 3 2 5 2.1............................................
[A](2): [B](4): DNA [B] [B](4):
: [B](4): DNA [B] [B](4):](/thumbs/91/104797401.jpg "[A](2): [B](4): DNA [B] [B](4):")
3 [A](2): 7 6 19.4 16.7 [A] 0 0.0 1 2.8 1 2.8 13 [A](1): 36.1 1974 1 3.0 2 5.6 1975 1 3.0 2 5.6 1980 1 3.0 4 11.1 1990 1 3.0 0 0.0 1993 1 3.0 36 100.0 1994 1 3.0 1999 1 3.0 [A](2): 2000 1 3.0 2001 5 15.2
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
ユニセフ表紙_CS6_三.indd
16 179 97 101 94 121 70 36 30,552 1,042 100 700 61 32 110 41 15 16 13 35 13 7 3,173 41 1 4,700 77 97 81 47 25 26 24 40 22 14 39,208 952 25 5,290 71 73 x 99 185 9 3 3 3 8 2 1 79 0 d 1 226 167 175 159 133
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,
HITACHI 液晶プロジェクター CP-AX3505J/CP-AW3005J 取扱説明書 -詳細版- 【技術情報編】
B A C E D 1 3 5 7 9 11 13 15 17 19 2 4 6 8 10 12 14 16 18 H G I F J M N L K Y CB/PB CR/PR COMPONENT VIDEO OUT RS-232C LAN RS-232C LAN LAN BE EF 03 06 00 2A D3 01 00 00 60 00 00 BE EF 03 06 00 BA D2 01
Lightブログ_Mikata.qxd
46 Ω 1 3 2 15 Ω 47 6 4 5 7 5 CONTENTS 4 6 8 1 2 3 4 1 12 14 16 18 20 5 6 7 8 9 10 11 2 22 24 26 28 30 34 36 12 38 40 2 3 4 13 42 20 58 14 44 21 60 15 16 17 18 19 46 48 50 52 54 22 23 24 25 64 66 68 72
応用数学特論.dvi
1 1 1.1.1 ( ). P,Q,R,.... 2+3=5 2 1.1.2 ( ). P T (true) F (false) T F P P T P. T 2 F 1.1.3 ( ). 2 P Q P Q P Q P Q P or Q P Q P Q P Q T T T T F T F T T F F F. P = 5 4 Q = 3 2 P Q = 5 4 3 2 P F Q T P Q T
●70974_100_AC009160_KAPヘ<3099>ーシス自動車約款（11.10）.indb
" # $ % & ' ( ) * +, -. / 0 1 2 3 4 5 6 7 8 9 : ; < = >? @ 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 " # $ % & ' ( ) * + , -. / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
取扱説明書 -詳細版- 液晶プロジェクター CP-AW3019WNJ
B A C D E F K I M L J H G N O Q P Y CB/PB CR/PR COMPONENT VIDEO OUT RS-232C LAN RS-232C LAN LAN BE EF 03 06 00 2A D3 01 00 00 60 00 00 BE EF 03 06 00 BA D2 01 00 00 60 01 00 BE EF 03 06 00 19 D3 02 00
4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx
4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan
X 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
Chap9.dvi
.,. f(),, f(),,.,. () lim 2 +3 2 9 (2) lim 3 3 2 9 (4) lim ( ) 2 3 +3 (5) lim 2 9 (6) lim + (7) lim (8) lim (9) lim (0) lim 2 3 + 3 9 2 2 +3 () lim sin 2 sin 2 (2) lim +3 () lim 2 2 9 = 5 5 = 3 (2) lim
untitled
280 200 5 7,800 6 8,600 28 1 1 18 7 8 2 ( 31 ) 7 42 2 / / / / / / / / / / 1 3 (1) 4 5 3 1 1 1 A B C D 6 (1) -----) (2) -- ()) (3) ----(). ()() () ( )( )( )( ) ( ) ( )( )( )( ) () (). () ()() 7 () ( ) 1
橡matufw
3 10 25 3 18 42 1 2 6 2001 8 22 3 03 36 3 4 A 2002 2001 1 1 2014 28 26 5 9 1990 2000 2000 12 2000 12 12 12 1999 88 5 2014 60 57 1996 30 25 205 0 4 120 1,5 A 1995 3 1990 30 6 2000 2004 2000 6 7 2001 5 2002
O
11 2 1 2 1 1 2 1 80 2 160 3 4 17 257 1 2 1 2 3 3 1 2 138 1 1 170 O 3 5 1 5 6 139 1 A 5 2.5 A 1 A 1 1 3 20 5 A 81 87 67 A 140 11 12 2 1 1 1 12 22 1 10 1 13 A 2 3 2 6 1 B 2 B B B 1 2 B 100 B 10 B 3 3 B 1
取扱説明書 [F-01D]
![取扱説明書 [F-01D]](/thumbs/39/20094991.jpg "取扱説明書 [F-01D]")
F-01D 12.7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 a b 18 c d e 19 a b c d e f a b j k l g h i f m n o p a b c d e f g h i j k l m n o p a t y b X 20 a b a b a b c a b c 21 a b a b c a b c a b 22 23
.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
A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
Unknown
1 1 4 2014 6 2014 10 12 16 60 18 20 22 24 26 28 34 38 38 46 50 51 1 52 53 55 2 55 3 56 4 57 5 58 6 58 7 58 8 58 9 59 10 59 11 60 12 61 14 61 15 62 16 62 17 63 18 64 19 64 20 64 10 35 10,000 10 4,090 1
III
III http://www.manabino-academ.com . = k...................................... = k p + q................................. = a + b c + d.................................. 4.4..........................................
1 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 =
-1 - -2 - -3 - -4-2000 -5 - -6 - -7 - -8 - -9 - - 10 - -11-60 200 2,000 1980 24-12 - - 13 - - 14 - - 15 - - 16 - - 17 - 1998 '98 593'98.4. 604'99.3. 1998 '98.10.10 11 80 '98.11. 81'99.3. 49 '98.11. 50