1 3 2 ( )
2
3 1 5 1.1.......................... 5 1.2.................... 8 2 4 13 2.1.......................... 14 2.2.......................... 17 2.3 I......................... 20 3 5 23 3.1 I............................ 24 3.2 I............................ 27 3.3 IV........................... 30 3.4 II......................... 33 3.5 I.......................... 36 3.6 I.......................... 39 3.7.............. 42 3.8........................... 45 4 49 4.1............................ 50
4 4.2.............................. 51 4.3............................ 52 4.4 Web............................ 53 4.5......................... 53 5 55 5.1......................... 55 5.2......................... 56
5 1 1.1 119 (1 ) 104 (1,2 )
6 1 117 (1 ) 3 204 (2 ) I I ( 1 ) 222 (2 ) 266 13 ( )
1.1 7
8 1 1.2 4,5 56 4 2 18 14 10 4 4 5 2 16 4 I I 5 I I I I IV I I
1.2 9 7 XA XB 8 4 4 1,2 4 1,2 4 I 1,2 3 1 3 2 *1 U-Task 4 *1
10 1 5 5 8:30 10:00 IV I II I 10:40 12:15 13:00 I 14:30 14:40 16:10 16:20 17:50 I 16:55 5 I I 12:10 I 15:15 15:25
1.2 11 I 10 14 30 4 1 5 I PC 6 6 5 6 20 1 7,8 XA 7 1 XB
12 1 7,8 4 7 XA 4 3 5 4 3 ( )
13 2 4 1 3 1 3 3 ( )
14 2 4 2.1 ( ) ( ) (2009 ) II ( ) 1,2 rank 1
2.1 15 3 *1 ( ) II ( ) ( Jordan ) 3 IV Jordan 4,5 4 5 6 5 I *1
16 2 4 7,8 well-defined 8 2 ( )
2.2 17 2.2 ( ) ( ) (2009 ) ( ) ( 1 ) Zorn *2 *2 ( )
18 2 4 *3 R R ( x, y R d (x, y) = x y ) S T f f T S ε δ 4 Kuratowski Zorn 10 Zorn Kuratowski-Zorn Kuratowski *3 Hahn-Banach
2.2 19 6 ( ) ( ) ( ) ( ) Topology for Analysis (Albert Wilansky Dover Publications) ( ) ( )
20 2 4 2.3 I ( ) ( )
2.3 I 21 (Taylor ) ( ) Cauchy f (z) = 1 f (ζ) 2πi C ζ z dζ. f (z) C ( ) Taylor Cauchy
22 2 4 Cauchy ( )
23 3 5 2 2 ( 2 ) 2 ( n 2 n+1 1 ) ( )
24 3 5 3.1 I I 5 (Cauchy Ultrafilter in nonstandard method ) R I ( ) G 2 : G G G 3 G (group) 1. a, b, c G (a b) c = a (b c) 2. e G a G e a = a e = a ( e G ) 3. a G a G a a = a a = e Fermat Lagrange (Fermat *1 ) *1 p a {1, 2,..., p 1} modp {a, 2a,..., (p 1) a} = {1, 2,..., p 1}
3.1 I 25 p x, y Z x 2 + y 2 = p p 1 (mod 4) Z [ 1 ] ( ) 0 Web Lemma TA TA
26 3 5 5 21 9 7 30 I ( ) ( ) ( ) nonstandard analysis Applied Nonstandard Analysis (Martin Davis Dover Publications) ( )
3.2 I 27 3.2 I 2 R 2
28 3 5 1. 20 ( I ( ) ) ( ) 2. 3. 4. Riemann II
3.2 I 29 ( )
30 3 5 3.3 IV 1 ( ) 7 1 0 ( ) 1 0 3 ( 0 0 1 ) ( )
3.3 IV 31 ( ) ( ) 2 ( ) ( ) ( ) (50%) (30%) (20%)
32 3 5 IV( ) Lp 2 5 1 1 TA 4 5 ( )
3.4 II 33 3.4 II II I I I I II C( Ĉ) ( ) I
34 3 5 ( ) ( ) D Ĉ D 2 a D f : D (1) f (a) = 0, f (a) > 0 (a ), df z d z (0) > 0 (a = ) 1 ( ( ) ) C
3.4 II 35 ( ) ϕ (z) = z i z + i ϕ : H (1) ( ) H C n D f : H D < a 1 < < a n + b j = f (a j ) D α j π (0 < a j < 2) f (z) f (z) = C 1 z n j=1 z n 1 f (z) = C 1 j=1 C 1 ( 0) C 2 (z a j ) α j 1 dz + C 2 (a n + ) (z a j ) α j 1 dz + C 2 (a n = + ) ( ( ) ) ( )
36 3 5 3.5 I
3.5 I 37 7 1 TEX 6 Jacobi SOR Newton Newton-Cotes Gauss Euler Heum Runge-Kutta 1 30 90 gnuplot TEX 4
38 3 5 4 UNIX GCC gnuplot TEX 2 ( )
3.6 I 39 3.6 I TA TA PC Lebesgue PC PC 1. PC PC
40 3 5 PC 2. Windows Vista PC Windows Vista 10 (Adobe Reader ) 3. Knoppix/Math Knoppix/Math CD/DVD Linux Octave Maxima Reduce R 4. Ubuntu Linux Ubuntu perl 5. TEX 6. SSH 7. 1 PDF
3.6 I 41 TEX PC II ( )
42 3 5 3.7 ( ) ( ) 8 13 22 *2 5 *2
3.7 43 ( ) ( ) Maxima( ) UMV
44 3 5 1 2 ( 3,4 ) ( )
3.8 45 3.8 ( ) 2 1 2 10 1 10 1 ( 1 8 ) 2,3 1 2
46 3 5 B 1 4,5 5,6
3.8 47 ( ) 4 3 4 9 9 3 2 3 5
48 3 5 I or II 2 3 5 11 2 5 10 4 ( )
49 4 3 1 10 e i π 9 ( )
50 4 4.1 4 5 ( ) ( 2 ) ( ) 1 1
4.2 51 ( ) 4.2 ( ) ( )
52 4 ( ) ( ) 4.3 ( )
4.4 Web 53 4.4 Web Web Web Web Web Web URL http://www4.atpages.jp/math2008/index.html Web math2009.web@gmail.com yahoo ( ) 4.5
54 4 TEX TEX 1 *1 1 1 2 ( ) *1 4,5
55 5 5.1 ( ) 2003 17 21 2 6 1 3 7 38 2004 19 25 0 2 2 0 6 35 2005 24 30 5 5 0 0 10 45 2006 23 25 4 5 0 0 3 33 2007 20 23 3 4 0 1 7 35 2008 22 27 2 5 3 0 7 42
56 5 5.2 7 9 1 9 3 3 3 4 9 1 10:00 12:00 2 1 1 A 1 13:00 16:00 7 2 5 2 4 3 2 4 B 2 11:00 15:00 4 18 3 4 5 Web 3 http://www.ms.u-tokyo.ac.jp/kyoumu/examination1.html
5.2 57 ( ) 1 130 63 40 40 20 ( 1 ) 9 1 2 Web ( ) A B
58 5 A 2 B 3 3 3 4 2 4 4 30 2 6 (4 ) 1 ( ) 2 ( 2 ) 8 ( )
59
60 1 3 l, m, n l A A m 60 (m 2 A 60 60 ) m m n C A C 60 B A, B, C l, m, n AB = AC, BAC = 60 ABC 2 N = p i q j (p < q) p i+1 1 p i+1 p i qj+1 1 q j+1 q j = 2 p 2 p 3, q 5 ( ) < p p 1 q q 1 3 2 5 4 = 15 8 < 2
61 p = 2 2 i+1 1, qj+1 1 q 1 ( 2 i+1 1 ) q j+1 1 q 1 = 2 i+1 q j 2, q 2 i+1 1 = q j, q j+1 1 = 2 i+1 (q 1) i j = 1, q = 2 i+1 1 N = 2 i ( 2 i+1 1 ) 3 9 e πi 9 + π (e) i 9 i π(e!) 2 π ( x x ) 3 (e! = Γ (e + 1) = 4.26082 )
62 3 8 PDF 5 7 8 (266 ) ( ) ( )