教務資料アーカイブ 名古屋大学大学院多元数理科学研究科・理学部数理学科

. . . 3

I . . . 5

I . . . 7

I . . . 9

II . . . .15

II . . . .18

II . . . .23

I . . . .26

I . . . 29

I . . . 30

I . . . .31

V . . . .33

. . . 34

. . . .36

. . . .40

III . . . 43

IV . . . .44

IV . . . .46

. . . .47

. . . .50

. . . .53

. . . .57

VII . . . .65

IX . . . 68

X . . . 69

. . . .70

. . . 113

III . . . 115

III . . . 117

III . . . .118

IV . . . 120

IV . . . 122

IV . . . 124

IV . . . 125

II . . . 126

II . . . 130

II . . . 131

II . . . .133

II . . . 135

. . . .136

. . . 139

. . . 142

. . . 145

V VI . . . 156

V VI . . . 157

. . . 158

. . . 162

. . . 164

. . . 167

. . . 170

III IV . . . 177

III IV . . . .179

. . . 180

. . . 186

II . . . 188

II . . . 190

II . . . 192

II II . . . 197

II . . . 199

II . . . 200

1 . . . 203

6 19 23 SO(3) . . . 204

10 23 27 1 . . . 205

11 20 24 . . . 207

1 9 12 . . . 208

I . . . 209

5 8 12 . . . 210

. . . 212

. . . 214

. . . 215

II . . . 217

11 6 10 J- . . . 218

5 15 19

I . . . 224

5 22 26 Multiplier Hermitian metric Einstein

I . . . 226

5 29 6 2

I . . . 227

5 29 6 2 The polyhedron dual to the Coxeter arrangement

I . . . 228

6 12 16 WKB

I . . . 229

6 12 16 Mellin-Barnes

I . . . 231

10 10 13

I . . . 232

10 16 20

. . . 234 I

10 30 11 2

I . . . 235

12 4 7 Hodge

II . . . 237

1 15 19

1 I

2 I

3 VIII

I

4

1

2 I

3 IX

4

1

2

3

4

1

2

3 III X

4

1

2

4 1

2 I

3 I

4 1

2 I

3 4 1

2 I

I 3

4 1

2 I

3 I

4

5

6

5 II

6

5

6

5 I

6

5

I

I

1.5

.

Archimedes, Wallis, Leibniz, Euler

e

– Rolle

Taylor Jensen

– Riemann

–

,

2000.06.29 1.

(1) lim

n→∞

√n

n

(2) lim

p→∞





n j=1

a^p_j





1 p

= max(a1, a2, · · · , a^{n) (a1, a2}, · · · , a^{n )}

2. f (x) _{(a, ∞)} lim

x→∞(f (x+1)−f(x)) = l ^lim

x→∞

f (x) x ^{= l}

3. e =

∞ k=0

1 k!

e n n! e

4.

1. an lim

n→∞^{an= 0} n→∞^lim

a1+ · · · + an

n ^{= 0}

2. (1) lim

x→0

a^x_{− 1}

x ^{(a ) (2) lim}x→0^{x log x (3) lim}x→0

₁ x⁻

1 sin x



3. f (x) I c² - _{a ∈ I} f (a + h) = f (a) + hf^′(a + θh), (0 < θ < 1)

f”(a) = 0 ^lim

h→0^{θ =}

1 2 4.

(1)

∞ n=1

cos n

n^{2 (2)}

∞ n=2

1 n log n

5. X ≥ 0 ^{f (x)}

(1)

 _∞

0

f (x)dx xf (x) → 0 (x → ∞)

(2) (1) f (x) _{x ≥ 0} c¹ - xf^′_{(x) → 0}

I

I

1.5

,

. , .

.

2000.06.29 1. ǫ − δ

x→0lim ^{5x sin} 1 x^{3 = 0} 2.

(1) lim

n→∞

 1 −_n¹

n

(2) lim

n→∞

10ⁿ n! (3) lim

x→∞

x²_{+ x + 1 −}x²_{− x + 1}

(4) lim

x→0

x + log(1 − x) x²

(1) Tan⁻¹x (2) ^{1 + cos x}

1 + sin x ^{(3) (x}

x₎x

(4) log log x

5. f (x) =

 x²cos¹_x (x > 0)

0 _{(x ≤ 0)} ^f

I

I

1.5

1.

• ^{. .}

• ǫ-delta ^.

• ǫ-delta ^.

• ^.

2.

• ^.

• ^.

• ^.

3.

•

• ^.

• ^.

• ^.

4. 1

( )

I 2000.04.27

1. A f (x) A 1 a ǫ-δ .

2. R f (x) = −5x ^{x = a . ǫ > 0}

δ > 0 ? a ǫ .

3. R f (x) = 3x² x = a . ǫ > 0

δ > 0 ? a ǫ .

4. R f (x) = sin x x = a .

ǫ > 0 δ > 0 ? a ǫ .

5. A = R\{0} ^{f (x) =} _x^{1 ? ,}

.

6. f (x), g(x) , F (x) = f (x) + g(x) .

7. 2 p < q .

8.

an=

 1 + ¹

n

n

, n = 1, 2, · · · ,

, (i) ; (ii) , . ( ,

, e , .)

9. 2

f (x) =



 sin ¹

x^{, x = 0,}

0, x = 0,

g(x) =





x sin¹

x^{, x = 0,}

0, x = 0,

x = 0 .

10. A f (x) A 1 a a

.

11. ?

 n

I

12. 1 n an . ( n

, 3, 6, 12, 24, 48, · · · ^.)

I 2000.05.11

1. f (x) =^√x x = 1 ǫ-δ .

2. _{an_}

a0= 1, a1=^√1 + a0, _{· · · ,} an+1=^√1 + an, · · · .

(i) {an} ^{. ( : a}n≤ 2 ^.)

(ii) {an} ^.

(iii) _{an}

.

II 2000.05.25 1. f (x) = ¹

x x = a > 0 ǫ-δ . ( ǫ > 0

δ > 0 ? a ǫ .)

2. an= 1 + ¹ 2^{2 +}

1

3^{2 + · · · +} 1

n^{2 (n ≥ 1) , {an} .}

3. .

(1) arcsin³

5^{+ arcsin} 4 5 ⁼

π 2 (2) arcsin x + arccos x = ^π

2 (−1 ≤ x ≤ 1)

4. y = arctan x ,

dⁿy

dxⁿ = (n − 1)! cos^{ny sinn y +π}₂^ .

II 2000.06.22

4. f (x) R n , f⁽ⁿ⁾_{(x) ≡ 0 (} , _{x ∈ R} f⁽ⁿ⁾(x) = 0)

f (x) _{n − 1} . (Hint: )

5. .

x 1 + x ⁼

∞ n=1

(−1)^n−1xn

x (1 − x)^{2 =}

∞ n=1

nxⁿ

x(1 + x)

(1 − x)^{3 = ( )}

log ¹ 1 − x ⁼

∞ n=1

xⁿ n

cosh x =

∞ n=0

x²ⁿ (2n)!

sinh x =

∞ n=0

x²ⁿ⁺¹ (2n + 1)!

(1 + x)e^x= ( )

arctan x =

∞ n=1

(−1)ⁿ⁻¹ 2n − 1 ^x

2n−1

6. _∞

n=0

x³ⁿ (3n)!

f (x) .

III 2000.06.29

1. ( ) , .

2. y = f (x) 2

(a) f (x) [0, 1] , f (0) = f (1);

(b) f (x) (0, 1) , x (f^′(x))²= 1;

I

3. f(x) [0, 1] C²- , 0 < x < 1 f^′′(x) = 0 , f (x)

1 .

4. x = 0 .

f (x) = ^x

1 − x^{2 g(x) = sin}

2_x

III 2000.07.13

(1)

 _dx

1 + x^{2 (2)}

 _dx

√1 − x^{2 (3)}

 _dx

√x²+ a ⁽⁴⁾



cot x dx

(5)



sinh x dx (6)

 _dx

x²_{− 4x + 1} ⁽⁷⁾

 _x

(x − a)(x − b)dx (a = b)

(8)

 _x3_{− 8x + 7}

x²_{− 4x + 5}^{dx (9)}



cos³x dx (10)

 _x

(x²_{− a}²)^3dx

(11)

 _{cos x}

√sin x^{dx (12)}

 _dx

x log x ⁽¹³⁾

 _dx

sin x ⁽¹⁴⁾

 _dx

2 sin x + 3 cos x + 4

(15)

 _dx

4 + 5 cos x ⁽¹⁶⁾

 _dx

x^√a²_{− x}^{2 (17)}

 _dx

(x²+ a)^3/2

(18)

 _dx

x(xⁿ+ 1) ⁽¹⁹⁾

 _dx

a²sin²x + b²cos²x ⁽²⁰⁾

 _dx

sin⁴x

( : )

(1) arctan x (2) arcsin x (3) _{log |x +}x²_{+ a|} (4) log | sin x| ^{(5) cosh x}

(6) ¹

2^√3^log







x − 2 −^√3 x − 2 +^√3





 ⁽⁷⁾

1

a − b(a log |x − a| − b log |x − b|)

(8) ^x

2

2 ^{+ 4x +} 3 2^{log |x}

2− 4x + 5| − 7 arctan(x − 2) ^{(9) 3}₄^{sin x +}₁₂^{1 sin 3x (10)} −_4(x2 ¹

− a²⁾²

(18) ¹ n ^log



 ^x

n

xⁿ+ 1



 ⁽¹⁹⁾ _ab^{1 arctan a}_b ^{tan x}

 (20) ₋ ^{cos x}

3 sin³x⁻ 2 3

cos x sin x

2000.09.14

1, 2, 3 , 4 7 2 .

1. f (x) R ,

 t

−t

f (x)dx t > 0 , ,

. (15 )

2. f(x) = e^xcos x x = 0 . (

) (15 )

3. an

a0=^√2, a1=^√a0+ 2, a2=^√a1+ 2, _{· · · ,} an+1=^√an+ 2, _{· · ·} .

(1) an . (20 )

(2) f(x) =^√x + 2 [^√2, 2] ǫ-δ . (10 )

(3) lim

n→∞^{an . (10 )}

4. . , a > 0 . (15 )

 a 0

√ dx x²+ a²

5. tan x y = arctan x _{. x = 0} f (x) = arctan x +

arctan¹

x , y = f (x) . (15 )

6. (f g)⁽ⁿ⁾=

n k=0

n! k!(n − k)!^f

(k)_{g(n−k) ,}

f (x) =√ ^x 1 + x

, _{n ≥ 1} x > 0 ₍₋₁₎ⁿ⁻¹f⁽ⁿ⁾(x) > 0 . (15 )

7. [0, 1] f (x) . f (1) = 2

f (0) . (15 )

I

II

II

1.5

1.

2. ,

3. .

4. 5. 6. 7. 8.

(3) [X, Y ] = En X, Y trace 2. (1)

(2) n A A²= A, A = E^{n A}

3. _



2 1 0 1 2 1 0 1 2





4. (1) n A A

(2) A, C m n

A B

0 C



2000.07.19 1. A

(1) det(A^∗) = det(A)

(2) A | det(A)| = 1

2. A, B n

det

A B

B A



= det(A + B) det(A − B)

3.

(1)





1 ω ω²

ω ω² 1

ω² 1 ω



 , ω³= 1, ω = 1, ⁽²⁾









−1 ^{1 1 1}

1 −1 ^{1 1}

1 _{1 −1} 1

1 1 _{1 −1}







^{, (3)}





1 a bc 1 b ac 1 c ab





4. (1) Vandermonde

det











1 . . . 1 a1 . . . an

a²₁ . . . a²_n ... ^... aⁿ⁻¹₁ . . . aⁿ⁻¹_n











= (−1)^n(n−1)/2

j<j

(ai− aj)

(2) n (a1, b1), . . . , (an, bn) ai_{= a}j _{(i = j)}

y = c0+ c1+ · · · + cn−1^xn−1

II

II 2000.09.13 1.

(1)









1 1 1 1

−1 ¹ 1 −1

−1 −1 ^{1 1}

−1 1 −1 ¹







^{, (2)}





1 1 1

a b c

a² b² c²





2. (1) A, B n P n A = P BP⁻¹ det(A) = det(B)

(2)A det(A) = ±1

3. (x, y) - 2

(∗)

 a1x + b1y = c1, a2x + b2y = c2

(∗) ^{A A˜ 2}

A, ˜A 4.

det















x ₋₁ 0 . . . 0 0

0 x ₋₁ . . . 0 0

0 0 x . . . _{· ·}

· · · ^{. . .} · ·

· · · . . . −1 ⁰

0 0 0 . . . x ₋₁

an a_n−1 a_n−2 . . . a1 a0















= a0xⁿ+ a1xⁿ⁻¹+ · · · + an−1^{x + an}

II

1.5

1

( )

1. A, B A + B A λ λA

. n m A m p B AB .

n m A fA: R^m_{→ R}ⁿ .

(i) n m A fA: R^m_{→ R}ⁿ .

(ii) .

.

2. R^{3 R3} f : R³_{→ R}³ 2x − y + z = 0

II

3. ^{R3 R3} f : R³_{→ R}³ x = 2t, y = −t, z = t

.

4. R³ R³ f : R³_{→ R}³ 2x − y + z = 0 x = 2t, y =

−t, z = t ^{, .}

5. V _{E, F} . _{E → F}

.

(i) V = {(xi) ∈ R^{4; x}1+x2+x3+x4= 0}, E =



















 1 1

−1

−1







^,







 2 1

−1

−2







^,







 0 1 0

−1



















 , F =



















 2 1 0

−3







^,







 1 0 0

−1







^,







 1

−2

−1 2



















 .

(ii) V = {2 }, E = {−x² − 4x + 3, x² + 2x − 2, x² + 3x − 2}, F =

{2x²+ x + 1, −x²− x + 2, 3x²+ 2x + 1}.

6. (i) R²

x y



→

2 1

−1 0

 x y

 

5 3

 ,

3 2



.

(ii) R³



 x y z



 →





1 1 ₋₁

−3 ⁹ −5

−5 13 −7







 x y z







 1 2 3



,



 1 1 1



,





−1 1 2



 ^.

(iii) R³



 x y z



 →





−1 −5 −1

2 4 0

−4 −8 ⁰







 x y z







 1

−1 2



,



 3

−2 4



,



 2

−1 3





.

7. an+2= an+1+ an a1 a2

a3, a4, · · · ^.

a1

a2



→

a2

a3



→

a3

a4



· · ·

R² .

a1

a2



0 1 1 1



R² .

(i) _∀n

 an

an+1



=

0 1 1 1

 a_n−1 an



 .

0 1^ ₂

. .

(v) (iii) .

8. A ^{R2 −→}b ^R2

Φ(−^→x ) = A−^→x +^−→b

R² .

(i) .

(ii) 2 3 . 2 3

(3 ).

9. R² 2 A ^−→b

Φ(−^→x ) = A−^→x +^−→b

R² . 2 3

. 2 ( ) 3 .

10. .

(i) R⁴ 3 _





 1 1

−1 2







^,







 2 1 0

−2







^,







 0 1

−2 6









.

(ii) R⁴ (xi) _









x1+ 2x2+ 3x3− x4= 0 2x1+ x2+ x3= 0

− 3x^{1+ x3}− x^{4= 0} .

(iii) 3 ^R- V3 (V3 4 .)

1 + x + x²+ x³, 1 + x²+ 2x³_{, x − x}³ .

(iv) V3 (ii) . V3 f

 1

−1

f (x)dx = 0 .

(v) V3 (ii) . V3 f f (−1) = f(1) = 0 ^.

II

(ii) _{D : V → V} D(f ) = f^′(= _dx^df) _{D : V → V} .

(iii) _{{1, x, x}², x³, x⁴_} D .

(iv) 4 5 0 . (iii) A

A⁵= 0 .

1. .

(i) f : R³_{→ R}³ .

(ii) f : R³_{→ R}³ Im f R³ .

(iii) f : R³_{→ R}³ Ker f ^R3 .

(iv) f : R³_{→ R}³ .

2. V , W _{E = {e}1, · · · , en}, E^′= {e^′1, · · · , e^′m} ^{V , W .}

T : V → W ^{m n A = (a}ij)_1≤i≤m

1≤j≤n

T (ej) =

m i=1

e^′_iaij (j = 1, · · · , n)



⇔ ^{T (e}1, · · · , en) = (e^′₁, · · · , e^′m^)A



. A T _{E, E}^′ _{. V ∋ v =}ⁿ_j=1vjej

E ^v T (v) ∈ W E^{′ T (v) =m}i=1^vi^′e′i

. , _



 v^′₁

... v^′_m





 = A





 v1

... vn







.

3. 0 2 V .

(i)

0 1 0 0

 ,

0 0 1 0

 ,

1 0 0 −1



V .

(ii) A =

a b c d



X → AXA⁻¹ (X ∈ V ) ^V .

(iii) _{X → AXA}⁻¹ (i) .

4. x + y + z = 0 R³ V .

(i) _{  } V _{ } .



 c f

a d

b e



 =



 a d b e c f





x y

z u



2 X =

x y

z u



. V _{f |}V

(i) . _{f |}V f V

.

II

II

1.5

1.

•

•

• 2.

•

•

•

•

• 3.

•

•

•

2000.06.28

1. ( A¹, A², A³, · · · , A^ν, · · · )

(1)





1 1 0

1 1

0

1



 ⁽²⁾

x y

z _−x



2. A A²− a + E = 0 ^A

3. 2 X =

x1 x2

x3 x4



4 2 A =

a1 a2

a3 a4



LA: X → AX, RA:→ XA 4

4.

(1)









1 _{2 1 −2}

−1 ^{1 2 1}

0 _{1 3 −2}

1 −1 0 ³







 ⁽²⁾









4 −7 6 1

1 0 5 2

−1 ^{5 5 3}

0 1 2 1









5.

(1)





−3 ^{2 2}

−2 ^{2 1}

2 −1 −1



 ⁽²⁾





3 −4 −3

−2 ^{2 1}

2 −1 ⁰





6.

(1)







3x1_{− 2x}2+ x3 + x5= 2 x1− x2+ x3− 2x4+ x5= 1 2x1+ x2− 3x3+ x4+ 3x5= 2

(2)







ax + y + z = 1

x + ay + z = a (a ) x + y + az = a²

2000.09.13

1. 5

(1)

1 2 3 2 3 1



(2)

1 2 3 4

3 4 2 1



i j (ij)

II

(1)









1 0 2

−1 −1 1

2 1 2









(2)









a b c

c a b

b c a









(3)









sin θ cos ϕ sin θ sin ϕ cos θ r cos θ cos ϕ r cos θ sin ϕ −r sin θ

−r sin θ sin ϕ r sin θ cos ϕ ⁰









3. 7

(1)











2 _{1 −1 −1}

−2 −3 ^{1 2}

−2 −1 ^{3 1}

−1 −4 ^{1 2}











(2)











1 a b c + d 1 b c d + a 1 c d a + b 1 d a b + c











(3)











0 a b c

−a ^{0 d e}

−b −d ^{0 f}

−c −e −f ⁰











(4)















2 −1 ^{0 . . . 0}

−1 2 −1 ^{. .. ...}

0 −1 ^{2 . .. 0}

... . .. ... ... −1

0 . . . _{0 −1} 2















(n ) (5)















x a1 a2 . . . an

a1 x a2 . . . an

a1 a2 x . . . an

... ^... ^... . .. ... a1 a2 a3 . . . x















4. 10

(1)

x + y + z = 1 ax + by + cz = d a²x + b²y + c²z = d²





 ⁽²⁾













−x^{1+ x2+ x3+ x4= 1} x1− x2+ x3+ x4= 0 x1+ x2− x3+ x4= 0 x1+ x2+ x3_{− x}4= 0 a, b, c

5. A A⁻¹ A

±1 ¹⁵

I

http://www.math.nagoya-u.ac.jp/~kanai/

1.1 1.2 1.3 1.4 1.5

Brouwer

Brouwer 2, 3

Brouwer Sperner

Brouwer

I

2.1 2.2 2.3 2.4

2.5 Sperner 2.6

2000.05.08

.

4

2000.06.12

. (A) (B) D = {(x, y) ∈ R^{2: x2+ y2}≤ 1}

(A) ϕ(P ) = P , (∀P ∈ ∂D) ϕ : D → D

(B) _{ψ : D → D}

2000.07.03

. (1) 3 n

3 3

3 3

(2) Sperner (1) (2 ) Sperner 1 Sperner

I

I

I

20 2 10 10

.

I

I

culture shock

absorber ,

I

I, II

I

1.

V

V

1.5

1

1 Cauchy

1.

•

• ^Euler

2.

• Cauchy-Riemann

•

• 3. Cauchy

•

• Cauchy 4. Cauchy

•

•

3. Caylay-Hamilton

4. Jordan

5. 6. 7. 8.

2000.06.21

1. A

A =





4 _{−3 −2}

8 _{−6 −4}

−4 ^{3 2}



 ^{A =}





0 1 1

2 0 ₋₁

0 −1 ⁰





(1) A²

(4)

6v = P16v + P26v (6v )

P16v, P26v A P1, P2

2. A =

 6 10

−1 −1



Aⁿ

3. (1) _

∞ 0

xⁿe^−xdx = n! _{n ≥ 0} (2)

f(x), g(x) =

 _∞

0

f (x)g(x)e^−xdx 1, x, x², x³ Schmidt

2000.09.20

1. A Jordan P⁻¹AP Jordan P

A =





0 0 4

1 0 2

0 2 −2





2. A P⁻¹AP P

A =





−1 −^{√2 √2}

−^{√2 0 1}

√2 1 0





3. (1) A

(2) 4. (1) (2)

(3) Cayley-Hamilton

( )

1.

•

• ǫ-δ

•

• ǫ-δ

• ^{( )}

2.

•

•

•

• 3.

• ^{( )}

•

•

• 4.

•

• ^{( )}

•

•

• ^{( )}

• Γ- ^B-

I 2000.05.25

1. _{an_} :

ǫ > 0 , N ,

m, n > N =⇒ |a^m− aⁿ| < ǫ. (1)an= 1 + ¹

2^{2 +} 1

3^{2 + · · · +} 1

n^{2 (n ≥ 1) , {an} .}

(2) |aⁿ⁺¹− aⁿ| ≤ ¹_n ^{? , .}

2. .

(1) arcsin³

5^{+ arcsin} 4 5 ⁼

π 2 (2) arcsin x + arccos x = ^π

2 (−1 ≤ x ≤ 1) 3. x ∈ R

n→∞lim

 lim

m→∞(cos(n!πx))^2m .

4. .

x²y

f (x, y) =







 xy

x²+ y^2, (x, y) = (0, 0)

0, (x, y) = (0, 0)

, .

(1) f (x, y) (0, 0) .

(2) f (x, y) (0, 0) .

(3) f (x, y) (0, 0) .

2. (1) 1 ( ) .

(2) .

1 −^x

2

2 < cos x < 1 −^x

2

2 ⁺ x⁴

24^{, x = 0.}

(3) _∞

n=0

x³ⁿ (3n)!

f (x) .

3. S , .

2000.09.14

5 4 . ,

.

1. [0, 1] f (x) . f (1) = 2

f (0) .

2. R² C²- f (x, y)

(x,y)→(0,0)lim f (x, y) x²+ y^{2 ≡ α} ,

f (0, 0) = ^∂f

∂x^{(0, 0) =}

∂f

∂y(0, 0) = 0, ∆f (0, 0) = 4α,

. , f (x) C²- , 2 , ∆

∆f (x, y) = ^∂

2_f

(x, y) +^∂

2_f

(x, y)

3. n ≥ 1 ^{. 2 (r, θ)}

D = {(r, θ) ; 0 ≤ r ≤ | sin nθ|, 0 ≤ θ ≤ 2π} .

(1) D .

(2) D .

4. f (x, y) = x³+ 3xy²_{− 3x} .

(1)f (x, y) .

(2)c , f (x, y) = c .

5. D = {1 ≤ x^{2+ y2}≤ 2}, a > 0, b > 0 ^{, .}

 

D

dxdy a²x²+ b²y²

,

Hausdorff Rⁿ

I. 1. 2.

(4/17) 3.

4.

5.

( ) (4/24)

Map(A,B)

6. ( )

Source ,

7. ,

(5/8)

9. Bernstein

10. (5/22)

Cantor

11. Hasse ,

, a.c.c. (5/29)

II. (6/5)

III. 1. 2. 3.

4. (6/12)

5. Rⁿ

6. ,

(6/19) 7.

8. 9.

(6/26)

Rⁿ Euclid R

10. (7/3)

R 11.

12.

, Tychonoff (7/10)

13. Hausdorff Hausdorff Hausdorff T1 ,

Hausdorff Rⁿ

, (7/17)

(1)f (f⁻¹(f (P ))) = f (P ) (2) f⁻¹(f (f⁻¹(Q))) = f⁻¹(Q)

2. A _{f : A → A} n fⁿ= f ◦ f ◦ · · · ◦ f (n ⁾

(1) _{n ∈ N} fⁿ f

(2) _{n ∈ N} fⁿ f

(3)A

(i) f (B) = B

(ii) n fⁿ_{(A) ⊂ B}

A B

3. A, B _{f : A → B} A f

4. R _{∼ a ∼ b a − b}

(1)∼ (2)R/ ∼

(3) R_{/ ∼ ℵ}

III

III

3

IX

,

( )

, ( ) , ,

, ( ).

(I) : , , ,

. , .

(II) ( ) , .

( , ) , 30 ,

. ,

. ( ).

• ^{( , , , )}

• ^{, (} , Bernstein , , , )

• ^{, Zorn}

( ) , , .

. , (30 )

. , ( , , )

. , , :

3 , , , , , ,

, , , , , ,

ǫ-δ , , , , , , , ,

, , , ,

, ℓ² , , R , , ,

( ) ,

• ^{f−1 .}

IX

• ^.

• ^.

2000.09.22

1. R ( ) ,

• B1= {(a, b) | a, b ∈ R}

• B2= {[a, b] | a, b ∈ R}

• B3= {(a, b] | a, b ∈ R}

• B4= {R \ A | A ^R }

i = 1, 2, 3, 4 _{, O}i Bi ( ) _{, (R, O}i)

(i = 1, 2, 3, 4) ,

(1) ,

(2) , 1 ,

(3) 1 , , 0 C0

2. (a),(b)

(a) S¹:= {(x, y) ∈ R²| x^{2+ y2}= 1} ^{R2 ( ) compact}

(b) , compact ,

IX

1.

• 2. 3. 4.

•

• 5.

•

• ^exp

2 2

Hilbert 90

90

1.

2.

5.

6.

7.

8.

9.

R-

R-

T.A.Springer Invariant Theory

I

1. G H _{G × {1} G × H} G × H/G × {1} ^H

2. G S

NG(S) := {g ∈ G | gSg⁻¹= S}

ZG(S) := {g ∈ G | s ∈ S ^gsg−1 = s}

NG(S) G ZG(S) NG(S)

3. ^{Q Z}

Q_{= { },} Z_{= { }}

4. A =

 2 −1

−1 ²



E = {

x y



| x, y ∈ Z}, F = {A

x y



| x, y ∈ Z}

E/F E, F

II

1. R, S _{f : R → S} f (r + r^′ = f (r) + f(r^′), f (rr^′) =

f (r)f (r^′) r, r^′_{∈ R} ker f := {r ∈ R| f(r) = 0} ^R

2. I := {(x²− 1)f(x)| f(x) ∈ C[x]} ^{C[x] I}

3. K 1k _{{n ∈ Z| n1}k _{= 0} {0}} ^Zp p

4. x²+ x + 1 = 0 ω Q(ω) := {a + bω| a, b ∈ Q}

Poincar´e-Bendixon

1.

•

•

• (Cauchy-Peano )

•

•

•

•

•

• 2.

•

•

•

• 3.

•

• ^{( )}

•

•

•

•

• ω-limit set limit cycle

• Poincar´e-Bendixon

2000.09.19

1. Rⁿ Rⁿ f L

|f(x) − f(y)| ≤ L|x − y|, x, y ∈ Rⁿ

x1(t), x2(t) (t ∈ R^{n) x′}(t) = f (x(t))

|x¹(t) − x²(t)| ≤ |x¹(0) − x²(0)|e^L|t|, t ∈ Rⁿ

2.

x” + 4x = 0 x = x(t)

(ii)

A =





2 1 0 0 2 0 0 0 1





exp(tA) 5.



 x^′= y

y^′= −6x − y − 3x²

,

1. 1,5 4/19 4/26

• motivation

• σ-

• Lebesgue-Stieltjes

• · · · ^σ

2. 0,5 4/26

•

•

• Borel

•

3. 1 5/10

•

•

•

• E.Hopf

•

6. Fubini 1 5/24 5/31

•

•

• Fubini

7. Borel Borel 1,5 5/31 6/7

•

• Borel

• Borel

• R^d convolution

8. L^p 1 6/14

•

• L^{p Lp}

• H¨older ^Minkowski

• Banach ^Lp

9. 6/28 7/5

• ^(6/28)

• ^{Fubini Lp (7/5)}

10. Lebesgue-Stieltjes 1.5 7/12 7/19

•

• E.Hopf

• Lebesgue-Stieltjes

•

11. 0.5 7/19

• R^d

I 2000.06.28

1. an, bn, n ∈ N [−∞, ∞] ∞ − ∞

lim sup

n→∞ ^{(an+ bn}) ≤ lim sup

n→∞ ^{an+ lim sup}n→∞ ^{bn (1)}

lim sup

n→∞ ^{(an+ bn}) < lim sup

n→∞ ^{an+ lim sup}n→∞ ^{bn (2)}

2. (X, B, µ) ^µ

{ ∈ ; µ({x}) > 0} ⁽³⁾

3. (X, B, µ) ^{µ(X) = 1 A}n∈ B ^∞n=1^µ(An) < ∞

µ(lim sup

n→∞ ^{An) = 0 (4)}

4. δa a delta

δa(A) =

 1 a ∈ A 0 a /_{∈ A}

f fdδa

II 2000.07.05

1. _{f : R → R R}|f(x)|dx < ∞ ^{ϕ(ξ) =}R^e

iξx_{f (x)dx}

(2) _{B = {z inR}^d; z = x − y for some x, y ∈ A} ^.

4. M

|fn(x)| ≤ M for all n, x (1)

 lim inf

n→∞ ^fndµ ≤ lim inf

n→∞



fndµ (2)

2000.09.13

1. f : R → R R|f(x)|(1 + |x|)dx < ∞ ^{ϕ(ξ) =}R^e

iξx_{f (x)dx}

2. (X, B, µ) ^fn

n→∞lim ^fn(x) = ∞ for all x ∈ X ⁽¹⁾

n→∞lim



X

fndµ =



X

n→∞lim ^{fndµ (2)}

3. (X, B, µ) ^{f , fn}

n→∞lim ^fn(x) = f (x) for all x ∈ X ⁽³⁾

:= sup

n



X^|f

n|²dµ < ∞ ⁽⁴⁾

n→∞lim



X

fndµ =



X

f dµ (5)

(a) µ(X) = ∞ ⁽⁵⁾

(b) µ(X) < ∞ ⁽⁵⁾

4. _{f : R → R} −∞ < a < b < ∞ _a^bf (x)dx ≥ 0

f (x) ≥ 0, a.e.-x

A. Mathematica

1 2

1. Introduction

•

2.

• ^Bezier

3.

•

sn, cn, dn, 4.

• Gerono

cissoido of Diocles Cassinian oval

Lituus

8.

• ^R3

9.

• Coons (hyperbolic paraboloid) (Monkey saddle) (el-

lipsoid) (torus) (paraboloid) (figure eight surface)

(Whitney umbrella) (catenoid) (helicoid)

(Enneper’s surface) (Scherk’s minimal surface)

(Henneberg’s minimal surface) (Catalan’s minimal surface)

(tractoid) (Kuen’s surface)

10.

• ^,

1 1 2

2 1

1 1

11.

• 12.

• 13.

•

2. 3. 4.

5. Bezier deCasteljau

6. Bezier Bezier Bezier

7.

8. Gerono 9.

10. 4

11. 2 3

1 Principia 1

12.

13. 3 cissoid of Diocles

14. clothoid 15.

16. Lissajous 17.

18. base rollomg curve pole

19.

26.

27. sn Euler-Lagrange

28. 8 29. 30. 31. 4 32.

33. a, b

34. Coons h0(u), h1(u), h2(u), h3(u) 35.

36.

37. 4 reflection 1 4

38. 39. 40. 41.

42. 2

43.

44. 1

45.

I 2000.04.28

α(t) = (sin⁻¹(t) e^t), t ∈ (−0.5, 0.5)

(2) t

(3) arc length parameter

β(t) = (Rcost, Rsint), t ∈ (0, 2π) (0, R)

II 2000.05.25 1.

2 2.

III 2000.06.15 .

(1) (2)

IV 2000.07.07 .

X(u, v) = (a cos v cos u, b cos v sin u, c sin v)

A _{(1 0)} (0 R)

(1 R)

(1)A = (0, 0) B = (1, 2R)

(2) C^∞

2. t ∈ [0, 2π] α(t) = (sin t, sin t cos t)

(1) C^∞

(2) (3)

3. (1) (1, 0) 1 _{r(θ) =? −}^π

2 ^{≤ θ ≤} π 2

(2) (1) (x(t), y(t))

x 4.

(1)

(2) r

r

5. A = (0, a), B = (1, b) f (0) = a, f (1) = b [0, 1]

(1)f

(2)(1) f

6. y = x³ (1)

(2)

1. D = {(r, θ) : 0 ≤ r ≤ R, 0 ≤ r ≤ Θ} ^{R Θ 2}

D X(r, θ)

2. R

φ, −^π₂ < φ < ^π

2^, λ, 0 < λ < 2π (Rλ, f (φ)) f

f^′ > 0

(1) f⁻¹ φ λ

(R cos φ cos λ, R cos φ sin λ, R sin φ)

(2) E, F, G

(3)E = G, F = 0 f (4)EG − F^{2= 1 f} (5)(3),(4)

3.

X(u, v) = (u −^u

3

3 ^{+ uv}

2_{, v −} ^v3

3 ^{+ vu}

2_{, u}2_{− v}2₎

(1) (2)

(3) E, F, G

∆

VII

VII

2 2

2 , , ,

. 2 , 10 . , 70 1

. 8 , :

1. 2 (2 )

• ^{, , , .}

• ^{, , , , .}

2. , (1 )

• ^{, , , , , .}

3. (2 )

• ^{, ( 1 , ).}

• ^{( , ).}

4. 3 )

• ^{, , , , ,}

• ^{( , ,} , Eisenstein ).

• ^{( , , , , ).}

2000.06.16

1. n . ^C n ^{Cn C}-

EndC(Cⁿ) . EndC(Cⁿ) f , g

(f + g)(v) := f (v) + g(v) _{(v ∈ C}ⁿ), (f ◦ g)(v) := f(g(v)) (v ∈ Cⁿ⁾

, EndC(Cⁿ) + _{, ◦} . ( ,

.) EndC(Cⁿ) _{n × n} Mn(C)

.

2. ( ) .

(1)

1 −1

1 3



(2)

−2 −1

3 1



3. n , 2 Dn:= a, b| a^{n= e, b2}= e, ab = ba⁻¹_{ 2 × 2}

O2(R) := _{g ∈ GL}2_(R)|^t_{g · g = I}2^!

(= ) . , g ^tg g .

2000.07.14

.

(A) 6 16 , 18 .

=⇒ ^.

(B) 3 .

=⇒ {4 − ( )}

.

(C) .

=⇒ ^{. (A), (B)}

. . 49

(1)–(5), 65 (1)–(4), 66, 6 30 , 7 7 .

: 9 1 ( )

VII

(A-1) _{2 × 2} A =

a b c d



{g, (ad − bc)/g} ( ^{g a, b, c, d )} .

(A-2) :

GL2(Z) :=

a b c d

 _



a, b, c, d ∈ Z, ad − bc = ±1

"

(= n Aⁿ = I2 A) 1, 2, 3, 4, 6

. n = 1, 2, 3, 4, 6 n _{2 × 2} .

:

(1) A 2 Aⁿ= I2 A

ΦA(T ) Tⁿ_{− 1} .

(2) Tⁿ_{− 1} 2 _{n (≥ 2)} . (

: 78 (3))

(3) 1 n (= n 1 ) 2

n (≥ 2) ^.

(4) (3) n Aⁿ = I2 2 × 2 ^{A .}

(A-3) X = {P, Q, R} ^{3 .}

(1) X C ( 67 ) C_{× C × C}

.

(2) X .

(3) X , C . , X C

C_{× C × C} .

(4) X {φ, X, {P }, {P, Q}} ^{. (C .) X C}

.

(5) X {φ, X, {P }, {Q}, {P, Q}} ^{. (C .) X}

C .

( : X (1) ^C_{× C × C} .)

IX

L^p

X

X

. 1.

2. 3. 4. 5. 6. 7. 8. 9.

.

Galois

2 3

Galois

1. 2.5

§1

§2

§3

§4 ^Q

§5

2. 3.5

§6

§7

§8

§9

§10

§11

3. Galois 2.5

§12 Galois ^Aritin

§13 Galois

§14

4. Galois (3.5

§16 1

§17

§18 ²

§19

§20 ^{Galois Galois Abel}

§21

§22 3 ⁴

1

2000.09.18 1.

(1)K/F [K : F ] K/F

(2)Q K [K : Q] = ∞ [K : Q] = ∞

(3)Q(^√42)/Q

(4)Fp(T )/Fp(T^p) Fp p p

F_{p(T )} F_{p 1}

2. X³_{− 2} Q K

(1)K Q 1

(2)Galois Gal(K/Q) (3)K/Q

3. F2 1 ^F2[X] (X⁴+X +1) K = F2[X]/(X⁴+X +1)

(3)β = sin(2π/5) ∈ K

(4)Q(β)/Q Galois Galois Gal(Q(β)/Q)

( )

L²

1. Weierstrap Mittag-Leffler Runge

2. Cauchy Hartogs

3. Cousin

4. ¯∂

5. Koszul ∂^¯ Dolbeault

6. ∂^¯

7. L² ∂^¯

8. H¨ormander Hilbert

9. ¯∂ L²

10. L²

I

B.Kernighan, D.Richie, ,

.

, .

, ,

. .

, , UNIX

, , . ,

, , ,

, , .

• ^.

• ^.

, .

•

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• ^{UNIX .}

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◦ ^{, .}

◦ ^.

◦ ^.

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•

•

◦ N × N ^.

◦ N × N ^.

◦ ^{atoi .}

◦ strcat, strrchr .

•

•

•

◦ ^{, 100000} .

• ^,

•

• ^,

•

◦ unsigned int ^{, b}

. , b . ,

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1. , unsigned int .

2. unsigned int , b .

3. , .

4. , .

◦ ^,

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qsort .

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, WEB http://www.math.nagoya-u.ac.jp/~naito/lecture/2000 SS/ .

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