. . . 3
I . . . 5
I . . . 9
I . . . .11
I . . . 13
II . . . 25
II . . . .26
II . . . .29
II . . . .34
I . . . .36
I . . . .41
I . . . .42
I . . . .44
I . . . 45
V . . . .46
. . . .48
. . . .52
. . . .57
III IV . . . 70
III IV . . . 71
III IV . . . .72
III IV . . . .74
III IV . . . .76
. . . .77
. . . .78
. . . .81
. . . .83
VII . . . .85
VIII . . . 87
IX . . . 88
X . . . .92
. . . .93
. . . .95
. . . 96
III . . . 104
III III . . . 106
III III . . . 109
III I . . . 110
I III . . . 113
III I . . . 114
I I . . . 123
I . . . 125
II . . . .128
. . . .131
III . . . 133
III . . . 135
III . . . 138
III . . . .139
IV . . . .148
IV . . . 149
IV . . . 152
IV . . . 154
II . . . 155
II . . . 164
II . . . 166
. . . 169
. . . 171
. . . 174
. . . 184
V VI . . . 188
V VI . . . 189
V VI . . . 190
V VI . . . 192
V VI . . . 193
V VI . . . 194
. . . 195
. . . .198
. . . 202
. . . 205
. . . 216
IV . . . 220
IV IV . . . 223
IV IV . . . 225
IV II . . . 233
II I . . . 241
I . . . 242
I . . . 244
III . . . 250
IV . . . .251
. . . 255
5 28 6 1 . . . 257
I . . . 259
5 7 11 . . . 260
. . . 262
. . . 264
. . . 266
II . . . 268
11 5 9 J– . . . 269
. . . 270
. . . 271
I . . . 272
4 23 27 I . . . 273
5 21 25 II . . . 275
6 25 29 F-Singularities - Splitting of Frobenius Map and Geometric Aspects of Tight Closure II . . . .276
7 2 6 I . . . 277
7 9 13 II . . . 278
11 19 23 Hodge-Arakelov I . . . 281
12 10 14 I . . . 282
1 15 18 I . . . 283
1 21 25 Directed polymers in random environment
1 I
2 I I
3 VIII III
4
1
2
3 IX III
4
1
I 2
3
4
1
2 VII
3 III III
4 IV
1
2
3 X III
4
1
2 I
3 III
4 1 2
3 III
4
1 I
2 3 4 1
2 I
3 III
4 1
2 II
3 III
4
5
6 II
5
6
5
6 I
5
6
5
I
I
1.5
1.
•
•
• 2.
•
•
•
•
•
• 3.
•
•
•
•
•
• 4.
I
•
• 5.
• Riemann
•
•
2001.06.14
1. n
(1) 1
(x + a)(x + b) ; (2) x
2ex
2. (1) lim
n→∞
an− a−n
an+ a−n (a > 0) ; (2) limn→∞n{(1 + a/n)
m− 1} (m ∈ N) ;
(3) lim
x→0
1 sin x −
1 x
3. A a1= 1, an+1=√2an+ A (n∈ N) {an}
4. (1) Rolle
(2)
f(x), g(x) [a, b] (a, b) (a, b) g′(x)6= 0
c (a < c < b)
f(b)− f(a) g(b)− g(a) =
f′(c) g′(c)
5. (1) n
11 1 1 1
I
(2)
an = 1 +1 2+
1
3 +· · · + 1
n− log n an
1 2
[ ]
2001.09.05
1. n
(1) 1
x2+ 3x + 2 ; (2) x
3ex
2. (1) lim
n→∞
an− bn
an+ bn (a, b > 0) ; (2) lim
n→∞n{(1 + a/n)
m− 1} (m ∈ N) ;
3. A a1= 1, an+1=√an+ A (n∈ N) {an}
4. a1, b1
an+1= an+ bn
2 , bn+1= panbn
{an}, {bn}
[ ]
2001.09.13 1.
(1) 1
(x + a)(x + b) ; (2) x
3ex ; (3) (cos x)3 ;
(4) (log x)3 ; (5) 1
a(cos x)2+ b(sin x)2 (ab6= 0) 2.
(1) In= Z π/2
0
sinnx dx n ;
(2) Im,n= Z π
−π
sin mx sin nx dx m, n 3.
(1) Z 1
0
log x
xα dx (0≤ α < 1) ; (2)
Z ∞
−∞
1
(x2+ a2)(x2+ b2)dx (0 < a < b) ;
4. Z 1
0
log x 1− xdx
I
5.
(1) n = 0, 1, 2, . . .
Z 1
−1
(1− x2)ndx = 2
2n+1(n!)2
(2n + 1)!
(2) Pn(x) = 1
2nn! dn dxn(x
2− 1)n
Z 1
−1
Pm(x)Pn(x) dx =
2
2n + 1 (m = n)
0 (m6= n)
[ ]
I
I
1.5
1.
2. ( - )
3. ( Taylor , Taylor
),
4. ( )
2001.09.13
1. f(x) =
xacos1x (x > 0)
0 (x≤ 0)
(i) f(x) x = 0 a
(ii) f(x) x = 0 a
2. Sin−1
1
√5
+ Sin−1
2
√5
I
3. (i) Tan−1
ra− b a + btan
x 2
!
(a > b > 0)
(ii) (sin x)Sin−1x (Hint: )
4. (i)
Z e 1
(log x)2
x dx (ii)
Z π2
0
dx 1 + sin x
5. r = max{0, a sin 3θ} (0 ≤ θ ≤ 2π, a > 0)
1.5
(
: ε− N δ?)
an a a
:
p.21
:
: 1− 1/2 + 1/3 − 1/4 + ...
: ε – δ
limx→0sin x/x = 1
: Bolzano-Weierstrass
:
:
:
:
:
:
:
1. n 0
(i)
Z 1
ex(ex+ 1)dx (ii) Z ∞
0
xne−xdx 2.
(i) lim
x→0
√x + 1− 1 −x2
x2 (ii) limx→0
1 x2 −
1 sin2x
3. (1) [0,1] f(x)
Z π 0
xf(sin x)dx = π Z π2
0
f(sin x)dx (2)
Z π 0
x 2 + sin xdx
4. n
f(x) = 1 x+
1
x− 1 +· · · + 1 x− n
f(x) = 0 n ( : y = f(x) )
I
I
1.5
. .
(1) 1 (6 )
(2) 1 (6 )
( )
1
an=
1 + 1
n
n
(n = 1, 2, . . . ) .
(1) an, an+1 , k an< an+1(n = 1, 2, . . . ) .
(2) n! > 2n−1 an < 3 (n = 1, 2, . . .) .
{an} , , .
2 x = sin 10◦ . sin 30◦= 3 sin 10◦− 4 sin310◦
I
, 3x− 4x3= 1/2 ,
f(x) = 4x3− 3x + 1 2 .
(1) y = f(x) , , .
(2) (a, f(a)) y = f(x) x .
(3) x0= 0 x1, x2, . . .
(xn, f(xn)) y = f(x) x (xn+1, 0)
(n = 0, 1, 2, . . .) . x1, x2 .
sin 10◦ = 0.173648177· · · . x3 .
.
3
f(x) =
( x2sin1x (x6= 0)
0 (x = 0)
R , f′(x) .
4
(1) cos x
cos x = α0+ α1x + α2x2+ δ(x), lim
x→0
δ(x) x2 = 0 , α0, α1, α2 .
(2) .
x→0lim
1− cos x x2
(3) .
x→0lim
e2x− ex− x x2
e2x, ex (1) .
5 (1) f(x) = sin x f(n)(x), f(n)(0) , sin x 0 .
(2) limn→∞Rn(x) = 0 , sin x 0 .
(3) , cos x 0 .
I
f(x) [a, b] , (a, b) , m .
(⋆) f(b)− f(a) (x− a)m =
f′(c) m(c− a)m−1
c a b .
(1) A
F (x) = f(x)− f(a) − A(x − a)m
F (b) = 0 .
(2) , F (x) ,
(⋆) .
7
(1) sin−1x .
(2) cos−1x .
(3) √ 1
a2− x2 (a > 0) .
x = au , .
(4)∗ 1
x3+ 1 .
, 1
x3+ 1= a x + 1+
bx + c
x2− x + 1 , a, b, c .
8 (1)
x5+ 1 x3+ 1
( ) +( )
x3+ 1
, .
(2) (1) , .
Z x5+ 1
x3+ 1dx
(3)∗ . Z
x6 x4+ x2+ 1dx
9
Z b a
f(x) dx = lim
n→∞
b− a n
Xn k=1
f(ak−1)
ak = a + kb− a n
.
I
(1) Z 1
0
x2dx
(2) Z 1
0
exdx
10 . 0 < ε < 1, R > 1 . , ε→ 0 R→ ∞
. (1)
Z 1 ε
dx
xs (s > 0) (2)
Z R 1
dx
xs (s > 0) (3)
Z 1−ε
0
√ dx 1− x2 (4)
Z R 0
dx x2+ 1
11 (1) p < 1
Z 1/2 0
xp−1(1− x)q−1dx 0 . p > 0 ,
. (2) s < 1
Z 1 0
e−xxs−1dx 0 . s > 0 ,
. (3)
Z ∞
1
e−xxs−1dx s .
, 7 19 .
12 f(x) = 1
x [a, 1] (0 < a < 1) ,
a≤ cn< dn≤ 1 (n = 1, 2, . . .), dn− cn→ 0 (n → ∞)
. cn, dn
cnmax≤x≤dn
f(x)− min
cn≤x≤dn
f(x)→ 0 (n → ∞) .
13
(1) Γ(s) = (s− 1)Γ(s − 1) (s > 1) . , , n
Γ(n) = (n− 1)! .
(2) B(p, q) = q− 1
p B(p + 1, q− 1) (p > 0, q > 1) . , , (p− 1)!(q − 1)!
I
, p, q > 0 ,
B(p, q) =Γ(p)Γ(q) Γ(p + q)
. p, q . ,
.
14 . . s
. (1)
Z ∞
e
dx x(log x)s (2)
Z 1/e 0
dx x(− log x)s
( u = log x .)
15
Z ∞ 0
dx
(x2+ 1)n (n ) , .
( , 8 (2) .)
16 .
Z 1
−1
dx x2 =
−x1
=−2
17 .
(1) f(x) = 1
|x| + 1 (2) g(x) =
( x
1+e1/x (x6= 0)
0 (x = 0)
18 x3− 6x + 11 = 0 , 3 3.1 .
, .
19 0 .
(1) f(x) = x 1− x2 (2) g(x) = sin2x
( .)
20 .
(1) sin−13 5+ sin
−14
5 = π 2 (2) sin−1x + cos−1x = π 2 (3) tan−11
2 + tan
−11
3 = π 4
I
(4) tan−1x + tan−1 1 x=
( π
2 (x > 0)
−π2 (x < 0)
1 (1)
(1 + x)2/3= 1 + ax + bx2+ cx3+· · ·
3 , x, x2, x3 a, b, c .
(2) α
αCk =α(α− 1) · · · (α − k + 1) k!
. 2/3C1,2/3C2,2/3C3 , (1) a, b, c .
2
f(x) = x2− 2 ,√2 . x0= 2
x1 x5 , .
3 α > 0 .
f(x) =
( xα (x≥ 0)
−(−x)α (x < 0)
R , α , f′(x) .
4
(1) sin x, cos x 4 (1) , .
x→0lim
sin x− x cos x x(1− cos x)
(2) log(1 + x) 4 (1) , .
x→0lim
x− log(1 + x) x2
5 f(x) = log(1 + x) 0
log(1 + x) = x−1 2x
2+1
3x
3· · · + (−1)n n−2
− 1 x
n−1+ R n(x) Rn(x) = (−1)n−1
n(1 + θx)nx
n (0 < θ < 1)
n = 6 log 2 .
I
(2) x = 1/3 R6(x) , log43 . , R6(x)
, .
(3) x =−1/3 R6(x) , log23 . , R6(x)
, .
(4) (2), (3) log 2 .
.
(5) f(x) = log(1 + x) 0 .
, 5 .
sin x 0
sin x = x− 1 3!x
3+ R
5(x), R5(x) = cos(θx) 5! x
5 (0 < θ < 1)
sin 10◦ , . , π
.
+α x = 10◦= π/18 R5(x) = R5(π/18) , sin 10◦ π
18− 1 3!
π 18
3
= 0.173646829
. , 0 < θ× π/18 < π/6 √3/2 < cos(θ× π/18) < 1, 1
5!×
√3 2 ×
π 18
5
< R5(π/18) < 1 5!×
π 18
5
,√3 > 1.7 R5(π/18) (= − )
1.14716× 10−6< R5(π/18) < 1.34960× 10−6
. sin 10◦ 0.173647976 0.173648178 .
. sin 10◦= 0.173648177· · ·
, . 0 < cos(θ× π/18) < 1 ,
0 < R5(π/18) < 1.34960× 10−6 .
6
(log(1 + x))′ = 1 1 + x , f(x) = log(1 + x) 0
(⋆) log(1 + x) = x−1 2x
2+1
3x
3+
· · ·(−1)nn−1xn (−1 < x ≤ 1)
I
. (1) x6= −1
1
1 + x = 1− x + x
2− · · · + (−1)n−1xn−1+(−1)
nxn
1 + x .
(2) −1 < x ≤ 1
n→∞lim Z x
0
tn
1 + tdt = 0 .
0≤ x ≤ 1 −1 < x < 0 . 0≤ x ≤ 1 , 0≤ t ≤ x
1/(1 + t)≤ 1 . .
(3) (1) (2) , (⋆) .
7 (1) t = tanx
2
sin x = 2t
1 + t2, cos x = 1− t2
1 + t2, tan x = 2t
1− t2, dx = 2 1 + t2dt .
, . p.34
(v) ,
Z dx sin x =
Z 1 + t2
2t 2 1 + t2dt
= Z dt
t = log|t|
= log tanx 2 .
(2) 1
sin x cos x . p.34 (iv)
(3) cot x
= 1
tan x
. p.34 (vi)
(4) sin x
1 + sin x . p.34 (vii)
, 1
x√1− x . t =
√1− x x = 1− t2, dx =−2t dt , Z
dx x√1− x =
Z −2t
(1− t2)tdt .
(5) . p.37 (i)
I
1
x√x2+ x + 1 . t =
√x2+ x + 1 + x
√x2+ x + 1 = t− x ,
x = t
2− 1
2t + 1, dx =
2(t2+ t + 1) (2t + 1)2 dt
. .
(7) . p.37 (ii)
(8) 1
x√x2− x + 2 . p.37 (v)
(9) √ 1
3− 2x − x2 . p.37 (vii)
x2 , (7), (8) .
√ 1
3− 2x − x2 = p 1
(1− x)(3 + x) = 1 (1− x)
r1− x 3 + x
(5), (6) . , 7 (3) .
8
(1) Z
dx x2+ 1=
x x2+ 1+
Z 2x2
(x2+ 1)2dx
, Z
dx (x2+ 1)2 .
(2)
In(x) =
Z dx
(x2+ 1)n
. (1) , In+1(x) In(x)
2nIn+1(x) = (2n− 1)In(x) + x (x2+ 1)n .
(3) I3(x), I4(x) .
Z b 9
a
f(x) dx
b− a n
Xn k=1
f(ak−1)
ak= a + kb− a n
. , f(x) , f′(x)
,|f′(x)| [a, b] M .
(1) f(x) F (x) ,
F (ak)− F (ak−1) = f(ak−1)(ak− ak−1) +1 2f
′(c
k)(ak− ak−1)2 (ak−1< ck< ak)
I
. ,
(⋆)
Z b a
f(x) dx−b− a n
Xn k=1
f(ak−1) ≤
(b− a)2
2n M
. (2)
Z 2 1
1
xdx n = 10 . . , (⋆)
.
1 v1, v2
sin α1
v1
= sin α2 v2
. α1, α2 , .
. A, B . , A, B
(0, a), (l,−b) (a, b, l > 0) . A B , x
x (0 < x < l) .
(1) sin α1, sin α2 a, b, x, l .
(2) A B T (x) .
(3) T′(x) , T′(x) = 0 .
(4) T′(x) = 0 x T (x) . ,
, A B .
2 ex 0
ex= 1 + x + 1 2!x
2+
· · · +n!1xn+ e
θx
(n + 1)!x
n+1 (0 < θ < 1)
, e .
(1) , 2 < e < 3 .
e < 3 e−1> 1/3 .
(2) e .
e = m/n (m, n ) . x = 1 , n!
.
3 z ,
1 + z +z
2
2! +· · · + zn n! +· · ·
. z ,
ez .
I
. V . ,
, ,
, .
(1) x
eix = cos x + i sin x ( , )
. i .
(2) , x, y
ei(x+y)= eixeiy
.
x = π
eiπ=−1
, π, e, i .
4 f(x) n , n f(n)(x) . ,
f(x) = f(a) + f′(a)(x− a) + f′′(a) 2! (x− a)
2+
· · · +f(n(n−1)(a)
− 1)! (x− a)n−1+ Rn(x)
Rn(x) = Z x
a
f(n)(t) (n− 1)!(x− t)
(n−1)dt
.
f(x) = f(a) + Z x
a
f′(t) dt
= f(a)− Z x
a
f′(t)(x− t)′dt
= f(a)−hf′(t)(x− t)ix
a+
Z x a
f′′(t)(x− t) dt
= · · ·
5 In=
Z π2
0
sinnx dx .
(1) sinnx = sinn−1x· sin x , In= n− 1
n In−2 .
(2) I0, I1 , (1) In .
(n .)
.
I
(3) .
π
2 = limn→∞
22
1· 3· 42 3· 5·
62 5· 7·
82 7· 9·
102 9· 11· · ·
(2n)2 (2n− 1) · (2n + 1)
( , )
.
√2 = lim
n→∞
22
1· 3· 62 5· 7·
102 9· 11· · ·
(4n− 2)2 (4n− 3) · (4n − 1)
( , )
2001.09.13
1 ( )
(1) lim
n→∞
1
√n2+ n+
√ 1
n2+ 2n+· · · +
√ 1
n2+ n· n
(2) n 2
log(n + 1) < 1 + 1
2+· · · + 1
n < 1 + log n
2 ( )
(1) lim
ε→0
Z 1 ε
dx
xs s
(2)
Z ∞
1
dx
10√
x11+ x9
3 ( )
Z ∞
0
dx (x2+ 1)n
4 ( )
(1)
f(x) =
x (x≥ 0)
x
1 + e1/x (x < 0)
x = 0 f′(x) x6= 0
(2) f′(x) x = 0 lim
t→∞
t et = 0
II
II
1.5
— —
2 (m, n) 1
II
II
1.5
1.
•
• 2.
•
•
• 3 × 3
•
• 3.
•
•
•
•
•
•
•
• Ax = b
II
•
• 5.
•
•
•
1 A
(1) rank A (A )
(2) det A (A ) A
(3) Ax = 0 rank A det A
2 (1) v1= 1
2
!
v2= 3 4
! R2
(2) R4 x + 2y− 3z − w = 0
3 R4
v1=
1
−1 0 0
v2=
1 0
−1 0
v3=
1 0 0
−1
v4=
0 1
−1 0
v1, v2, v3, v4
4
x + 2y + z = 0
−2x + y + Cz = 0 x− 2y − 7z = 0
II
(1) C
(2) (1)
x + 2y + z = 4
−2x + y + Cz = B x− 2y − 7z = −8 B
(3) (2)
II
II
1.5
(I)( (I))
[1] (9),(10) ζn= 1
(1)
sin θ cos φ sin θ sin φ cos φ r cos θ cos φ r cos θ sin φ −r sin φ
−r sin θ sin φ r sin θ cos φ 0
(2)
1 a2− bc a3 1 b2− ca b3 1 c2− ab c3
(3)
1 a b c + d 1 b c d + a 1 c d a + b 1 d a b + c
(4)
a + b + c −c −b
−c a + b + c −a
−b −a a + b + c
(5)
1 a a2 0
0 1 a a2
a2 0 1 a
a a2 0 1
(6)
a + b + 2c a b
c b + c + 2a b
c a c + a + 2b
II
(7)
a + b a a · · · a
a a + b a · · · a
. . . .
a a a · · · a + b
(n× n) (8)
1 + x2 x 0 · · · 0
x 1 + x2 x · · · 0
0 x 1 + x2 . .. 0
... ... . .. . .. x
0 0 · · · x 1 + x2
(n× n)
(9)
1 ζ · · · ζn−1 ζn ζ ζ2 · · · ζn 1 . . . . ζn−1 1 · · · ζn−3 ζn−2
ζn 1 · · · ζn−2 ζn−1
(10)
1 ζ · · · ζn−2 ζn−1 ζ ζ2 · · · ζn−1 1 . . . . ζn−1 1 · · · ζn−3 ζn−2
(11)
a0 −1 0 · · · 0
a1 x −1 . .. ...
a2 0 x . .. ... ... ... ... . .. ... ... 0
an−1 0 · · · 0 x −1
an 0 · · · 0 x
(12)
0 1 2 · · · n
1 0 1 · · · n − 1
2 1 0 . .. n− 2
· · · . .. . .. ... · · ·
n− 1 n − 2 · · · 1 0 1
n n− 1 n − 2 · · · 1 0
(13)
a b c d
−b a −d c
−c d a −b
−d −c b a
(14)
x1 a2 a3 · · · an
a1 x2 a3 · · · an
a1 a2 x3 · · · an
... ... ... . .. ... a1 a2 a3 · · · xn
(15)
1 0 · · · 0 a1
0 1 · · · 0 a2
... ... . .. ... ... 0 0 · · · 1 an
a1 a2 · · · an b
(16)
1 1 · · · 1 a 1 1 · · · a 1 ... ... ... ... 1 a · · · 1 1 a 1 · · · 1 1
( (1) r2sin θ, (2) (a + b + c)2(a− b)(b − c)(c − a)), (3) 0, (4) (b + c)(c + a)(a + b),
(5) 1 + a4+ a8, (6) 2(a + b + c)3, (7) (na + b)bn−1, (8) 1 + x2+ x4· · · + x2n, (9) (−1)n(n−1)/2(1− ζ)n), (10) 0, (11) a0xn+ a1xn−1+· · · an−1x + an, (12) (−1)n2n−1n, (13) (a2+ b2+ c2+ d2)2
(14) (x1− a1)(x2− a2)· · · (xn− an)(1 +Pi=1n ai/(xi− ai)), (15) b−Pni=1a2i, (16) (−1)n(n−1)/2(a− 1)n−1(a + n− 1)
(II)( (II), )
[1] n ζ = cos2πn +√−1 sin2πn
det
x0 x1 x2 · · · xn−1
xn−1 x0 x1 · · · xn−2
x x x · · · x
=n−1Y(x + ζkx + +ζ2kx +· · · + ζ(n−1)kx )
II
[2] 4 (xi, yi, zi) 1≤ i ≤ 4 det
1 1 1 1
x1 x2 x3 x4
y1 y2 y3 y4
z1 z2 z3 z4
= 0
[3] A, B, C, D n× n (1) det A B
B A
!
= det(A + B)· det(A − B)
(2) A det A B
C D
!
= det A· det(D − CA−1B)
(3) A, B det A −B
B A
!
=| det(A +√−1B)|2
[4] A n× n v∈ Rn Av = v A
[6] 1≤ i, j ≤ n Ei,j (i, j) 1 0 n× n
Ei,j· Ek,l=
Ei,l if j = k, 0 otherwise. [7] 2× 2 A A2= O
[8] n
(1) x y
z −x
! (2)
a b c
0 a d
0 0 a
( (1) n (x2+ yz)n/2 1 0
0 1
!
, n (x2+ yz)(n−1)/2 x y z −x
!
(2)
an nan−1b nan−1c +n(n−1)2 an−2bd
0 an nan−1d
0 0 an
)
(III)( , )
[1]
(1)
2x1+ 2x3+ 3x4= 0 3x2+ 5x3+ 2x4= 1 2x1+ 3x2+ 7x3+ 5x4= 1 4x1+ 3x2+ 9x3+ 8x4= 1
(
x1
x2
x3
x4
=
0 1/3
0 0
+ α
−3
−5 3 0
+ β
−9
−4 0 6
)
II
(2)
2x2+ 4x3+ 2x4= 2c
−x1+ x2+ 3x3+ 2x4= 2c x1+ 2x2+ 3x3+ x4= bc
−2x1− x2+ ax4= c
(a, b, c )
( Case b6= 1: , Case b = 1, a = 1:
x1
x2
x3
x4
=
−1 1 0 0
+ α
1
−2 1 0
+ β
1
−1 0 1
,
Case b = 1, a6= 1:
x1
x2
x3
x4
=
−1 1 0 0
+ α
1
−2 1 0
)
[2]
(1)
1 2 1 1
2 −1 1 −1
2 3 2 1
3 5 4 2
(2)
1 a c
0 1 b
0 0 1
( (1) 12
3 1 0 −1
−3 −1 4 −1
−2 0 −2 2
7 1 −6 1
(2)
1 −a ab − c
0 1 −b
0 0 1
)
[3] A =
1 2 1
0 1 1
1 0 −2
B =
3 4 3
−1 1 −2
4 8 6
A−1B (
6 −4 8
−2 7 −3
1 −6 1
)
[4] λ
λx1+ x2+ x3+ x4= 0 x1+ λx2+ x3+ x4= 0 x1+ x2+ λx3+ x4= 0 x1+ x2+ x3+ λx4= 0
( λ = 1:
x1
x2
x3
x4
= α
1
−1 0 0
+ β
1 0
−1 0
+ γ
1 0 0
−1
, λ =−3:
x1
x2
x3
x4
= α
1 1 1 1
)
[5] α, β, γ ABC
cos 2α· x1+ cot α· x2+ x3= 0 cos 2β· x1+ cot β· x2+ x3= 0 cos 2γ· x1+ cot γ· x2+ x3= 0
(IV)( , , )
[1]
II
(1){α
1 1
−1 2
+ β
2 1 0 2
+ γ
0 1
−2 6
| α, β, γ ∈ R} ⊂ R4 ( 2 <
1 1
−1 2
,
2 1 0 2
>)
(2){
x1
x2
x3
x4
∈ R
4|
x1+ 2x2+ 3x3− x4= 0 2x1+ x2+ x3= 0
−3x1+ x3− x4= 0
}} ( 2 <
1
−5 3 0
,
−1 2 0 3
>)
(3){A ∈ M(n)| trA = 0} ( : n2− 1 )
(4){A ∈ M(n)|A =tA} ( : n(n + 1)/2 )
(5){A ∈ M(n)|A +tA = 0} ( : n(n− 1)/2 )
(6){f(x) ∈ R[x]| R−11 f(x) = 0, deg f(x)≤ 3} ( : 3 < x, x2−13, x3>)
(7){(an)n≥0| an = an−1− an−2+ an−3} ( : 3 )
(8){y = f(x) : R → R| f(x) , ddx3y3 = d2y
dx2 −dydx+ y} ( : 3 )
[2] n V n v0= f(x) n
vk= f(k)(x)(k ) , < v0, v1, . . . , vn> V eb
[3] m× n A, B rank(A + B) ≤ rank(A) + rank(B)
[4] l× m A m× n B rank(AB)≤ min{rank(A), rank(B)}
[5] n× n A, B AB = O rank(A) + rank(B)≤ n
[6] n× n A AB = O rank(A) + rank(B) = n n× n B
[7] n× n A, B A + B =En AB = O rank(A) + rank(B) = n
[8] m× n A rank(A) = 1 m× 1 B 1× n C A = BC
II
II
1.5
“ ”
“ ”
2 2
1.
•
• 2.
• 3. 4.
•
5. ( )
1.
1 3 2
2 6 3
II
2. a, b .
2x2+ 4x3+ 2x4 = 2
−x1+ x2+ 3x3+ 2x4 = 2 x1+ 2x2+ 3x3+ x4 = b
−2x1− x2 + ax4 = 1
3. n× n A i j B B B−1 A−1
4.1. , x θ
4.2. , , ,
.
I
I
3 G. Polya
19
1 4 16
I
3 2
4 23
5 7
sin(π/6)
5 14
5 21
5 28
heuristic
6 4
I
6 11
5 cos(2π/5) 5
cos(2π/17) 17
6 18
y2= x4− 1
6 25
x = Z y
0
1p1− x2dx y = sin x
x = Z y
0
1p1− x4dx
y = sn x
I
1
π/2
7 9
modulo mod 5
mod 6 mod mod 11
7 16
5 17
7 23
t = Z x
0
1p1− x2dx
x = sin t x = sin t, y = (sin t)′
cos sin
cos sin
3 5
n2
I
3
I
I
I II
• —
• —
• —
• —
I II
I
I
1.
•
•
• 2.
•
• 3.
• 4.
•
• 5.
•
•
I
I
I
( )
( )
( )
1.
2. ( )
3. ( )
4. (I)
5. (II)
6. 7. 8.
(a) (b)
(c) ( )
(d) ( )
I
I
55 55
•
•
•
• 2× 2
•
•
•
•
•
•
•
•
•
V
V
1.5
I III
•
•
•
• exp(z)
•
•
T.A.
log z,√z
2001.06.12
1 (35) r > 0 Cr a∈ C r > 0 C
n
Z
Cr
(z− a)ndz
Z
Cr
1 (z− a)ndz
V
2 (35) M , δ, R0 |z| > R0 f |f(z)| ≤ M|z|−1−δ
R→∞lim Z
|z|=R
f(z) dz = 0
3 (30)
Z ∞
0
1 1 + x4dx
Z ∞
0
1 1 + x2dx
2001.09.11 1 (30)
Z 2π 0
1 5 + 4 sin xdx
2 (30)
Z ∞
−∞
1 x2+ 1dx
3 (30)
Z ∞
−∞
e−√−1x x2+ 1 dx
4 (30)
Z ∞
0
log x x3+ 1dx
2
1
1. ( )
(mathematical structure)
Part 1.
Kernel Image
Part 2.
(the set theory) the category theory)
Hom
Part 3.
2.
4/18 Part 1-1 . 0. . 1. ( , ) 2. ( ,
Q, R, C ) 3. ( , )
4/25 Part 1-2 / . 1. ( , , , ,
) 2. ( , , )
5/2 .
5/9 Part 1-3 . 1. ( ( ), , Kn , , ) 2.
( , ( ) ) 3. ( , , )
5/16 Part 1-4 . 1. (0 0, ) 2. ϕ : Kn→ Km
( , ϕ : Kn→ Km ) 3. Im ϕ Ker ϕ ( , , rank,
rank )
5/23 Part 1-5 / . 1. , , ( , , ( ),
, ϕ ) 2. ( , , ,
) 3. ( , )
5/30 Part 1 Part 2-1 1. ( , , , ) 2.
( , , well-defined, , )
6/6 Part 2-2 / . 1. ( , , , , ) 2. (
, , , , )
6/13 Part 2-3 / . 1. ( , , , , , , ,
) 2. ( ) 3.
6/20 Part 2 Part 3-1 . 1. ( ,
) 2. ( , ) 3. ( ,
) . ( . . 40 )
6/27 Part 3-2 . 1. ( ) 2. ( , ,
) 3. ( , )
7/4 Part 3-3 Jordan (1). 1. ( , , ,
) 2. Jordan (Jordan block, Jordan , Jordan block Jordan chain)
7/11 Part 3-4 Jordan (2). 1. Jordan chain ( , , Jordan block
) 2. ( , Jordan chain , (
)) 3. Jordan chain ( ) 4. Jordan ( )
7/12 Part 3 Part 3-5 Summary. 1. Part 3 (Jordan
) 2. Lecture ( ) : A.
B. Jordan chain
9/12 .
, .
.
( )
1. ( ) ( , )
C
A = 2 0
1 0
!
, B = 2 −1
1 0
!
.
2. (Jordan ) ( , )
(1) C V ϕ : V → V .
(i) dim V = 4.
(ii) ϕ λ .
, A V ϕ , Jordan block .
A . , Jordan block .
(2) (i), (ii) ,
(iii) N(k)={~v ∈ V | (ϕ − λ)k(~v) = ~0} , dim N(1)= 2, dim N(2)= 4.
. A . (Hint: Part 3
)
3. ( , ) ( , )
1, 2, 3 .
. V n , W V k (1≤ k ≤ n − 1) . , V
n− k W′
V = W ⊕ W′ (⊕ )
.
. k = n− 1 . ~a1, . . . , ~an−1 W . , ~b (6= ~0) W
2. V =h~a1, . . . ,~an−1,~bi
. ~a1, . . . , ~an−1, ~b V . , W′:=h~bi 3. V = W⊕ W′
. k = n− 1 . ( )
4. ( , ) ( , )
V K , ~a1, . . . , ~an V .
(1) V∗ V , ~a∗1, . . . , ~a∗n ∈ V∗
~a∗i(~aj) = δij (δij: Kronecker )
, ~a∗1, . . . , ~a∗n V∗ (~a1, . . . , ~an ). ,
~a∗1, . . . , ~a∗n .
(2) ϕ : V → V , ~a1, . . . , ~an ϕ A = (aij) . ,
φ : V∗→ V∗ , ϕ
φ : V∗ → V∗ f 7→ f ◦ ϕ
. , ~a∗1, . . . , ~a∗n φ A tA . (Hint:
(1) V∗ 0 ~0V∗ ~0V∗ : ~v 7→ 0 (~v ∈ V ) . (2) φ C = (cij) ,
φ(~a∗j) =Pnk=1ckj~a∗k . ~ai ...)
J.
–
I IV
{(1 + n−1)n} e
Cauchy
ǫ-δ
1
1. 4 16 23
• ǫ-N .
• .
• Cauchy .
• .
2. 5 7 14
• .
• .
3. 5 21 28
• ǫ-δ .
• .
• .
4. 5 28 6 4
• .
• .
• ( ).
5. Taylor 6 4
• Rolle .
• .
• Taylor .
• .
6. 6 11 18 25
• .
• .
• ( ) .
• .
• .
7. 6 25 7 2
• .
• .
• .
8. 7 9 16
• .
• Darboux .
• .
• .
ǫ-δ
ǫ-δ (ǫ-N )
ǫ-δ
{a(1)n }, {a(2)n } a(1), a(2) ǫ N
N < n
|a(1)n − a(1)| < ǫ, |a(2)n − a(2)| < ǫ
1
90 2
2001.07.23
1( ) I = [0, 1] (0 1 ) f : I → I
0 < r < 1 r x, y∈ I |f(x) − f(y)| < r|x − y|.
(1) a∈ I a1 = f(a) an an+1 = f(an) {an}
Cauchy (20 )
2 ( ) f(x), g(x) (a, b)
(1) f(x) + g(x) ǫ-δ (10 )
(2) h(x) = max{f(x), g(x)} x∈ (a, b) f(x) g(x)
h(x) ǫ-δ (10 )
(3)|f(x)| (5 )
3 ( ) {(x, y) : x2+ y2 ≤ 1} X(x, y) P (1/2, 1), Q(1/2,−1),
R(−3/2, 0)
XP2+ XQ2+ XR2
4 ( )
(1) xy D ={(x, y) : 0 < x, 0 < y, x + y < 1} x = u− uv, y = uv
D uv (10 )
(2) x = u− uv, y = uv Jacobi
xu xv
yu yv
!
(5 ) (3)
Z Z
D
ex−yx+ydxdy (10 )
2001.09.14
1( ) {an}
0 < r < 1 |an+1− an| < r|an− an−1|.
(1) n > 1 |an+1− an| < rn−1|a2− a1|
(2) m > n > 1
|am− an| < 1rn−1
− r|a2− a1|
((1),(2) 5 )
(3){an} Cauchy ǫ N
N≤ n < m |am− an| < ǫ N (10 )
(4) {an} a1= 1
an+1= 1 2 + an
an (10 )
2 ( ) f(x), g(x) (a, b)
(1) α, β (α6= 0, β 6= 0 ) αf(x) + βg(x) ǫ-δ
(10 )
(2) a1, a2, b1, b2 a1< b1, a2< b2 min{a1, a2} < min{b1, b2}
ǫ ( ) min{a1+ ǫ, a2+ ǫ} = min{a1, a2} + ǫ
(5 )
(3) h(x) = min{f(x), g(x)} x∈ (a, b) f(x) g(x)
h(x) ǫ-δ (10 ) (2)
3 ( ) f(x, y) = x2+ xy + y2− 4x − 2y (1) f(x, y)
(2) {(x, y) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 2} f(x, y)
4 ( )
(1) xy D = {(x, y) : 0 < y, x2 + y2 < x} x = r cos θ,
y = r sin θ D rθ (10 )
(2)
Z Z
D
x2+ y2dxdy (15 )
· 30
30
( )
1. (2.5 )
•
• ( )
• N, Z, Q, R
•
• (1 de Morgan
2. (2 )
•
•
•
•
• :
3. (1.5 )
•
•
• (2) :
• well-defined
• Z/nZ, R/Z, R2/Y (Y : )
• R R/Z
• R/Z
4. R 1.5
• : R
• R R (1)
• : Cauchy-Schwartz
•
• :
5. 1.5
•
•
• R R (2)
•
•
• (1): ǫ-δ
6. 2
• ǫ-N
•
•
• (2) :
•
• :
7. compact
• compact : compact compact
• [0, 1]
• compact
• Cauchy : compact ( )
2001.05.31
N ={1, 2, 3, . . .} , R .
(1) a) A, B A B f
, .
b) N N , .
c) f : A→ B, g : B → C , g◦ f : A → C
.
(2) a) X , A, B, C X X .
A∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
b) A, B . A× B Γ⊂ A × B f : A→ B
: Γ = {(a, f(a)) ∈ A × B, a ∈ A}. g : B → A
Γ′={(b, a) ∈ B × A, (a, b) ∈ Γ} .
(3) a) 3
f(X) = X3+ aX2+ cX + d , Ff: R→ R
Ff(x) = f(x) (x∈ R)
. Ff .
b) a2< 3c Ff .
(4) a) A ∼ , .
b) N× N ∼Q
(m, n)∼Q (m′, n′)⇔ nm′ = mn′
. ∼Q .
c) A N× N/ ∼Q , (m, n)∈ N × N (m, n) . F
F : A× A → A
F ((m, n), (m′, n′)) = (mm′, mn′+ nm′)
well-defined .
(5) , .
2001.09.20
(1)-(5) . (4) (4-1) (4-2) .
: N ={1, 2, 3, . . .} , Z ={0, ±1, ±2, ±3, . . .} , Q , R
. Rn
.
(1) a) .
b) (X, dX), (Y, dY) X Y f
, . , R R ,
.
c) b) , f : X→ Y , g : Y → Z g◦ f .
X, Y , Z dX, dY, dZ .
(2) (a) (X, dX) , .
(b) R2 .
(b-1) R2
(b-2) {(x, y) ∈ R2, x, y∈ Z}
(b-3) {(x, y) ∈ R2, x, y∈ Q}
(3) a) f, g : R2→ R . .
(a-1) f + g (a-2) fg
b) x, y f(x, y) =P0≤i,j≤Naijxiyj (aij ∈ R) R2 .
c) ( ) R2 .
