…J…−†[†E…n…‘†[…hfi¯„^‚ΛžfiüŒå

Size: px
Start display at page:

Download "…J…−†[†E…n…‘†[…hfi¯„^‚ΛžfiüŒå"

Transcription

1 II

2

3 [A] D B A B A B A B

4

5 DVD

6 y = 2x + 5 x = 3 y = 11 x = 5 y = 15.

7 Google Web

8

9 (2 + 3)

10

11

12 Windows Media Player Media Player

13

14 (typed lambda calculus)

15 (computer science)

16

17

18

19 f(x) = (x + 3) 5 (x + 3) 5

20 (2 + 3) ! (2+3)!5

21 (x + 3) ! ( +3)!5

22 (x + 3) 5 x 3 5 +! (x+3)!5

23 (x + 3) 5

24 (x + 3) 5 x 3 5 +! (x+3)!5

25 x 3 5 +! (x+3)!5

26 λx.((x + 3) 5)

27 λx.((x + 3) 5) x 3 5 +! (x+3)!5

28 λx.((x + 3) 5) x (x + 3) 5

29 x 3 5 +! (x+3)!5

30 (x + 3) 5 x

31 (x + 3) 5 x x

32 (x + 3) 5 x (x + 3) 5 (x + 3) 5 x

33 x x : int x x int integer int

34 (x + 3) 5 x (x + 3) 5 : int x : int

35 λx.((x + 3) 5) x 3 5 +! (x+3)!5

36 λx.((x + 3) 5) x (x + 3) 5 λx.((x + 3) 5) : int int

37 λx.((x + 3) 5) : int int A B A B

38 x (x + 3) 5 x : int (x + 3) 5 : int λx.((x + 3) 5) : int int

39 [x : int]. (x + 3) 5 : int λx.((x + 3) 5) : int int λx.((x + 3) 5)

40 [x : A] D M : B λx.m : A B (abs) x A B M λx.m

41 [x : A] D M : B λx.m : A B (abs) λx.m A B A B x : A

42 [x : A] D M : B λx.m : A B (abs) (λ-abstraction)

43

44 λx.((x + 3) 5) : int int 2 (λx.((x + 3) 5)) 2 λx.((x + 3) 5) 2

45 2 (λx.((x + 3) 5)) 2 λx.((x + 3) 5) (λx.((x + 3) 5)) λx.((x + 3) 5)

46 2 (λx.((x + 3) 5)) M N M M M : A B 2 : int

47 (λx.((x + 3) 5)) λx.((x + 3) 5) λx.((x + 3) 5) : int int

48 M : A B N : A (app) MN : B M A B A B

49 M : A B N : A (app) MN : B

50

51 int (char) (Boolean) int int int (int int) (int int) (int int)

52 + + : int (int int) : int (int int)

53 [x : A] D M : B λx.m : A B M : A B N : A MN : B

54 λx.((x + 3) 5) + : i (i i) [x : i] x + : i i 3 : i : i (i i) (x + 3) : i (x + 3) : i i 5 : i ((x + 3) 5) : i λx.((x + 3) 5) : i i int i x 3 5 +! (x+3)!5

55

56 λx.((x + 3) 5) (typed lambda calculus)

57

58

59 (λx.((x + 3) 5)) 2

60 λx.((x + 3) 5) 3 5 +! ( +3)!5

61 (λx.((x + 3) 5)) 2 2 ( +3)!5 + 3! 5

62 (λx.((x + 3) 5)) 2 (2 + 3) ! (2+3)!5

63 x (λx.((x + 3) 5)) 2 (2 + 3) 5 = 25

64 (λx.((x + 3) 5)) 2 (2 + 3) 5 = 25 (λx.((x + 3) 5)) 3 (3 + 3) 5 = 30 (λx.((x + 3) 5)) 4 (4 + 3) 5 = 35.

65 β (λx.m)n M[x := N] (β-reduction)

66 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]

67 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]

68 .. λy.n 1 : A B. N 2 : A λx.m : B C (λy.n 1 )N 2 : B (λx.m)((λy.n 1 ) N 2 ) : C λy.n 1 N 2 λx.m

69 (λx.m)((λy.n 1 ) N 2 ) λy.n 1 N 2 λx.m 1 (λx.m)((λy.n 1 )N 2 ) (λx.m)(n 1 [y := N 2 ]) λx.m N 1 [y := N 2 ]

70 .. λx.m : B C N 1 [y := N 2 ] : B? (λx.m)(n 1 [y := N 2 ]) : C? N 1 [y := N 2 ] B λx.m

71 (λx.m)n : B M[x := N] : B

72 (λx.m)n M[x := N] (λx.m)n. D 1 λx.m : A B N : A (λx.m)n : B

73 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A

74 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A

75 (λx.m)n M[x := N] D 1 N : A D 0 [x := N] M[x := N] : B

76 [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B x N x N

77

78 x 3 5 +! (x+3)!5

79 [x : A] D M : B λx.m : A B M : A B N : A MN : B

80 + : i (i i) [x : i] x + : i i 3 : i : i (i i) (x + 3) : i (x + 3) : i i 5 : i ((x + 3) 5) : i λx.((x + 3) 5) : i i

81 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B

82 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]

83

84

85 Recall

86 [x : A] D M : B λx.m : A B [A] D B A B M : A B N : A MN : B A B A B

87

88 [x : A] D M : B λx.m : A B [A] D B A B λx.m

89 M : A B N : A MN : B A B A B M N

90 A x : A

91 [A (B C)] [A] [A B] [A] B C B C A C (A B) (A C) (A (B C)) ((A B) (A C))

92 [x : A (B C)] [y : A] [z : A B] [y : A] xy : B C zy : B (xy)(zy) : C λy.((xy)(zy)) : A C λzλy.((xy)(zy)) : (A B) (A C) λxλzλy.((xy)(zy)) : (A (B C)) ((A B) (A C))

93 = = =

94 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B

95 [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B

96 [A] D 0 B A B B D 1 A D 1 A D 0 B

97 [A] D 0 B A B B D 1 A

98 [A] D 0 B A B B D 1 A D 1 A A A

99 [A] D 0 B A B B D 1 A [A] D 0 B A B A B A B

100 [A] D 0 B A B B D 1 A A B A B

101 1. A 2. A B A B 3. A B A B B 1. A 2. A B B

102 [A] D 0 B A B B D 1 A D 1 A D 0 B A B

103 [A] D 0 B A B B D 1 A D 1 A D 0 B (reduction)

104

105 [A] D 0 B A B B D 1 A B

106 [A] D 0 B A B B D 1 A B A B A A B A B

107 [A] D 0 B A B B D 1 A A D 0 B B B A A B

108 [A] D 0 B A B B D 1 A D 1 A D 0 B B

109

110 [x : A] D M : B λx.m : A B [A] D B A B M : A B N : A MN : B A B A B

111 [x : A] D 0 M : B λx.m : A B (λx.m)n : B [A] D 0 B A B B D 1 N : A D 1 A D 1 N : A D 0 [x := N] M[x := N] : B D 1 A D 0 B

112 = = =

113

114 [2005]

115

116

117 [2005]

118

add1 2 β β - conversion (λx.x + 1(2 β x + 1 x λ f(x, y = 2 x + y 2 λ(x, y.2 x + y 1 λy.2 x + y λx.(λy.2 x + y x λy.2 x + y EXAMPLE (λ(x, y.2

add1 2 β β - conversion (λx.x + 1(2 β x + 1 x λ f(x, y = 2 x + y 2 λ(x, y.2 x + y 1 λy.2 x + y λx.(λy.2 x + y x λy.2 x + y EXAMPLE (λ(x, y.2 output: 2011,11,10 2.1 λ λ β λ λ - abstraction λ λ - binding 1 add1 + add1(x = x + 1 add1 λx.x + 1 x + 1 add1 function application 2 add1 add1(2 g.yamadatakahiro@gmail.com 1 add1 2 β β - conversion (λx.x

More information

I: 2 : 3 +

I: 2 : 3 + I: 1 I: 2008 I: 2 : 3 + I: 3, 3700. (ISBN4-00-010352-0) H.P.Barendregt, The lambda calculus: its syntax and semantics, Studies in logic and the foundations of mathematics, v.103, North-Holland, 1984. (ISBN

More information

Int Int 29 print Int fmt tostring 2 2 [19] ML ML [19] ML Emacs Standard ML M M ::= x c λx.m M M let x = M in M end (M) x c λx.

Int Int 29 print Int fmt tostring 2 2 [19] ML ML [19] ML Emacs Standard ML M M ::= x c λx.m M M let x = M in M end (M) x c λx. 1, 2 1 m110057@shibaura-it.ac.jp 2 sasano@sic.shibaura-it.ac.jp Eclipse Visual Studio ML Standard ML Emacs 1 ( IDE ) IDE C C++ Java IDE IDE IDE IDE Eclipse Java IDE Java Standard ML 1 print (Int. 1 Int

More information

24 201170068 1 4 2 6 2.1....................... 6 2.1.1................... 6 2.1.2................... 7 2.1.3................... 8 2.2..................... 8 2.3................. 9 2.3.1........... 12

More information

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc

More information

untitled

untitled 20 7 1 22 7 1 1 2 3 7 8 9 10 11 13 14 15 17 18 19 21 22 - 1 - - 2 - - 3 - - 4 - 50 200 50 200-5 - 50 200 50 200 50 200 - 6 - - 7 - () - 8 - (XY) - 9 - 112-10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 -

More information

untitled

untitled 19 1 19 19 3 8 1 19 1 61 2 479 1965 64 1237 148 1272 58 183 X 1 X 2 12 2 15 A B 5 18 B 29 X 1 12 10 31 A 1 58 Y B 14 1 25 3 31 1 5 5 15 Y B 1 232 Y B 1 4235 14 11 8 5350 2409 X 1 15 10 10 B Y Y 2 X 1 X

More information

FX ) 2

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

More information

C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 新装版 1 刷発行時のものです.

C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.  このサンプルページの内容は, 新装版 1 刷発行時のものです. C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009383 このサンプルページの内容は, 新装版 1 刷発行時のものです. i 2 22 2 13 ( ) 2 (1) ANSI (2) 2 (3) Web http://www.morikita.co.jp/books/mid/009383

More information

koba/class/soft-kiso/ 1 λ if λx.λy.y false 0 λ typed λ-calculus λ λ 1.1 λ λ, syntax τ (types) ::= b τ 1 τ 2 τ 1

koba/class/soft-kiso/ 1 λ if λx.λy.y false 0 λ typed λ-calculus λ λ 1.1 λ λ, syntax τ (types) ::= b τ 1 τ 2 τ 1 http://www.kb.ecei.tohoku.ac.jp/ koba/class/soft-kiso/ 1 λ if λx.λy.y false 0 λ typed λ-calculus λ λ 1.1 λ 1.1.1 λ, syntax τ (types) ::= b τ 1 τ 2 τ 1 τ 2 M (terms) ::= c τ x M 1 M 2 λx : τ.m (M 1,M 2

More information

疎な転置推移確率行列

疎な転置推移確率行列 B E05(2019-05-15 Tue) : Time-stamp: 2019-05-17 Fri 16:18 JST hig http://hig3.net ( ) E05 B(2019) 1 / 11 x = 0,..., m 1 m. p(t), p(x, t) 1 double p [m] = { 1. 0, 0. 0,...., 0. 0 } ; /. m. / 2 / {p ( 0,

More information

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ( ) 24 25 26 27 28 29 30 ( ) ( ) ( ) 31 32 ( ) ( ) 33 34 35 36 37 38 39 40 41 42 43 44 ) i ii i ii 45 46 47 2 48 49 50 51 52 53 54 55 56 57 58

More information

untitled

untitled i ii (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (2) (3) (1) (2) (3) (1) (1) (1) (1) (2) (1) (3) (1) (2) (1) (3) (1) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3)

More information

23 15961615 1659 1657 14 1701 1711 1715 11 15 22 15 35 18 22 35 23 17 17 106 1.25 21 27 12 17 420,845 23 32 58.7 32 17 11.4 71.3 17.3 32 13.3 66.4 20.3 17 10,657 k 23 20 12 17 23 17 490,708 420,845 23

More information

平成18年度「商品先物取引に関する実態調査」報告書

平成18年度「商品先物取引に関する実態調査」報告書 ... 1.... 5-1.... 6-2.... 9-3.... 10-4.... 12-5.... 13-6.... 15-7.... 16-8.... 17-9.... 20-10.... 22-11.... 24-12.... 27-13... 29-14.... 32-15... 37-16.... 39-17.... 41-18... 43-19... 45.... 49-1... 50-2...

More information

1. A0 A B A0 A : A1,...,A5 B : B1,...,B

1. A0 A B A0 A : A1,...,A5 B : B1,...,B 1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A B f : A B 4 (i) f (ii) f (iii) C 2 g, h: C A f g = f h g = h (iv) C 2 g, h: B C g f = h f g = h 4 (1) (i) (iii) (2) (iii) (i) (3) (ii) (iv) (4)

More information

( )

( ) Web Web 1 3 1 21 11 22 23 24 3 2 3 4 5 1 1 11 22 9 2 3 15 11 22 2 11 21 4 5 ( ) 102 ( ) 1 ( 1 2001 Web 1 5 4 1 1 - 7 - [] - 7 10 11 12 12 1 10 1 12 - [] 1 1 2 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 3 1 47

More information

夏目小兵衛直克

夏目小兵衛直克 39(1906)1222 14(1817) 3(1832)1514(1843) 2628 6 (1853) (1854)3727 3(1856) 1 / 13 5(1858)6(1859) 5(1853) () () () () () () 3(1867)29 504111( 2 / 13 )98 23 18 2(1869)310283 100 50() 58 226 3313200982 5033

More information

nenkin.PDF

nenkin.PDF 1 31 1 WEB 10 3,544 429 13 10 22 11 7 WEB 1 2 41.0 15 80.0 20 46.7% 1000 55.8 1000 34.4 21 18.2 1000 23 25 41.0 49.2 29 90.6 42.7 33 56.4% 79.2% 67.4 51.7 37 39 83.7 1 91.0 93.6 9 2 3 1000 96.3 300 1000

More information

-1-

-1- -1- -2- -3-1 8 6% 2 4 6 8 1 48 63 43 6 55 38 78 58 2 88 67 11 22 78 1 56 22 89 47 34 36 32 38 4 34 26 7 -4- 18-5- 3 25 28 (6.%) (6.%) (.9%) 2 15 18 158 1 (3.8%) (56.4%) 5 2 137 27 8 1 68 119 26 71 28 65

More information

( ) ( ) ( ) D VD

( ) ( ) ( ) D VD DVD ( ) ( ) (FAX ) DVD DVD ( http://www.chibakenshakyo.com) 9:001700 (12:0013:00 ) 260-8508 4 3 1F TEL043-204-6010 FAX043-204-6015 ( ) ( ) ( ) D VD 001XX 3 18 VHS 3 XX 2 : CS 1 6 10 13 15 18 21 ( ) 27

More information

働く女性の母性健康管理、母性保護に関する法律のあらまし

働く女性の母性健康管理、母性保護に関する法律のあらまし 17 1 3 3 12 3 13 10 19 21 22 22 23 26 28 33 33 35 36 38 39 1 I 23 2435 36 4/2 4/3 4/30 12 13 14 15 16 (1) 1 2 3 (2) 1 (1) (2)(1) 13 3060 32 3060 38 10 17 20 12 22 22 500 20 2430m 12 100 11 300m2n 2n

More information

応用数学III-4.ppt

応用数学III-4.ppt III f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( ) ! xf ( x) = = n n!! ( n

More information

....02.p65

....02.p65 LAN () (ABS) 10 8 6 4 2 0-2 -4 86 88 90 92 94 96 98 2000 30 20 10 0-10 -20-30 86 88 90 92 94 96 98 2000 14000 12291 12000 10000 8000 5737 5968 6000 4226 4000 2000 0 4.5 6.8 5.8 8.5 1.7 13.7 4 0

More information

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4 20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d

More information

koji07-02.dvi

koji07-02.dvi 007 I II III 1,, 3, 4, 5, 6, 7 5 4 1 ε-n 1 ε-n ε-n ε-n. {a } =1 a ε N N a a N= a a

More information

土木工事の

土木工事の .1 ... 2 -...2 -...3 -...4 -...5 -...7... 8 -...8 -...9... 10 -... 10 -... 13 -... 14... 17 -... 17... 19... 19... 24... 24... 26 -... 26... 27... 27... 29... 30... 31 -... 31 -... 33 -... 35 1 1 2 3 4

More information

Chap9.dvi

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

More information

ランダムウォークの境界条件・偏微分方程式の数値計算

ランダムウォークの境界条件・偏微分方程式の数値計算 B L06(2018-05-22 Tue) : Time-stamp: 2018-05-22 Tue 21:53 JST hig,, 2, multiply transf http://hig3.net L06 B(2018) 1 / 38 L05-Q1 Quiz : 1 M λ 1 = 1 u 1 ( ). M u 1 = u 1, u 1 = ( 3 4 ) s (s 0)., u 1 = 1

More information

II (No.2) 2 4,.. (1) (cm) (2) (cm) , (

II (No.2) 2 4,.. (1) (cm) (2) (cm) , ( II (No.1) 1 x 1, x 2,..., x µ = 1 V = 1 k=1 x k (x k µ) 2 k=1 σ = V. V = σ 2 = 1 x 2 k µ 2 k=1 1 µ, V σ. (1) 4, 7, 3, 1, 9, 6 (2) 14, 17, 13, 11, 19, 16 (3) 12, 21, 9, 3, 27, 18 (4) 27.2, 29.3, 29.1, 26.0,

More information

02

02 54 163116831 02 1 168 54 158 53 162 53 148 52 152 52 10,000 0 40,000 30,000 20,000 50,000 70,000 60,000 1,000 500 1,500 2,000 0 2,500 3,000 4,000 3,500 4,500 168 54 158 53 162 53 148 52 152 52 03 52148

More information

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552 3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n

More information

x V x x V x, x V 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 =(+( ))x =x +( )x = x +( )x ( )x = x x x R

x V x x V x, x V 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 =(+( ))x =x +( )x = x +( )x ( )x = x x x R V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x

More information

1.1 1 A

1.1 1 A . A..2 2 2. () (xyz) ( xyz) ( xy z) = (x x)yz ( xy z) = yz ( xy z) = y(z ( x z)) = y((z x)(z z)) = y( x z) (2) (3) M aj (x, y, M aj ( x, ȳ, z)) = xy ȳm aj ( x, ȳ, z) M aj ( x, ȳ, z)x M aj (x, y, z) x =

More information

BA-100_Ver3.05

BA-100_Ver3.05 JA Windows 1 2 3 4 5 6 7 8 10 1 1 2 11 1 2 3 12 1 1 1 13 1 2 3 1 2 3 14 4 1 5 6 7 2 3 15 16 17 18 19 20 21 22 1 2 3 23 24 25 26 1 2 3 1 2 27 1 2 3 1 28 2 3 4 1 2 1 2 3 4 29 5 6 1 2 3 1 2 3 4 1 2 3 30

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa

A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa 1 2 21 2 2 [ ] a 11 a 12 A = a 21 a 22 (1) A = a 11 a 22 a 12 a 21 (2) 3 3 n n A A = n ( 1) i+j a ij M ij i =1 n (3) j=1 M ij A i j (n 1) (n 1) 2-1 3 3 A A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

More information

1... 5 2... 6 3... 7 4... 8 4.1... 8 4.2... 9 4.3... 9 4.4 SMTP... 10 5 2... 11 5.1 FP Mail... 11 5.2 MX... 11 6 3 /... 12 6.1... 12 6.2... 12 6.3...

1... 5 2... 6 3... 7 4... 8 4.1... 8 4.2... 9 4.3... 9 4.4 SMTP... 10 5 2... 11 5.1 FP Mail... 11 5.2 MX... 11 6 3 /... 12 6.1... 12 6.2... 12 6.3... 1... 5 2... 6 3... 7 4... 8 4.1... 8 4.2... 9 4.3... 9 4.4 SMTP... 10 5 2... 11 5.1 FP Mail... 11 5.2 MX... 11 6 3 /... 12 6.1... 12 6.2... 12 6.3... 12 6.4... 12 6.5... 12 7 4... 13 7.1 URL... 13 8 5...

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

(1) PQ (2) () 2 PR = PR P : P = R : R (2) () = P = P R M = XM : = M : M (1) (2) = N = N X M 161 (1) (2) F F = F F F EF = F E

(1) PQ (2) () 2 PR = PR P : P = R : R (2) () = P = P R M = XM : = M : M (1) (2) = N = N X M 161 (1) (2) F F = F F F EF = F E 5 1 1 1.1 2 159 O O PQ RS OR P = PQ P O M MQ O (1) M P (2) P : P R : R () PR P 160 > M : = M : M X (1) N = N M // N X M (2) M 161 (1) E = 8 = 4 = = E = (2) : = 2 : = E = E F 5 F EF F E 5 1 159 (1) PQ (2)

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

II

II II 16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 +

More information

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2 1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2

More information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915

More information

1. A0 A B A0 A : A1,...,A5 B : B1,...,B

1. A0 A B A0 A : A1,...,A5 B : B1,...,B 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)

More information

web web web web -1- 1 2 1 2-2- 1 1 2 3-3- 4 5 6 1 2 3 4 5 6 CECIPA http://www2.edu.ipa.go.jp/gz/ NICER http://www.nicer.go.jp/ http://www2.jyose.pref.okayama.jp/cec/ -4- http://gakuen.gifu-net.ed.jp/ Wonder

More information

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, .1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K 1 3 3 lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x,

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

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

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

More information

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-7 O1-8 O1-9 O1-10 O1-11 O1-12 O1-13 O1-14 O1-15 O1-16 O1-17 O1-18 O1-19 O1-20 O1-21 O1-22 O1-23 O1-24 O1-25 O1-26 O1-27 O1-28 O1-29 O1-30 O1-31 O1-32 O1-33 O1-34 O1-35

More information

Flash Player ローカル設定マネージャー

Flash Player ローカル設定マネージャー ADOBE FLASH PLAYER http://help.adobe.com/ja_jp/legalnotices/index.html iii................................................................................................................. 1...........................................................................................................

More information

79!! 21

79!! 21 79!! 21 94 94 138 94 8 15 1 100 10 25 65 138 30 94 1 138 94 !! 1570 km 1.5 79 7 12 138 11 72 11 72 15 11 72 11 72 11 15 11 72 15 12 138 11 72 12 21 12 94 12 21 12 12 94 12 138 12 138 11 79 12 12 138

More information

PowerPoint プレゼンテーション

PowerPoint プレゼンテーション Version:090910 1 5 3 4 5 11 13 14 16 22 26 29 2 1 3 4 5 6 7 1 11 8 2 225 2009M 9 225225miniTOPIX 2009/6225mini225 225mini225 10 3 11 12 13 14 15 16 17 18 19 20 21 22 . 115 15 23 . 24 .2 225 1,000 10 225mini

More information

インターネットを活用した公開天文台の可能性とその展望.PDF

インターネットを活用した公開天文台の可能性とその展望.PDF 17 - 2 - ...- 4-1....- 5-2....- 7-2.1....- 7-2.2....- 7-2.3....- 8-3....- 9-3.1....- 9-3.2....- 10-3.3....- 13-3.4....- 14-4....- 15-4.1....- 15-4.2....- 16-4.3....- 18-4.4....- 19-4.4.1....- 19-4.4.2....-

More information

Microsoft Word - hozon-fujimura-HP-伊勢工業高校における造船教育の歴史から学ぶ20160713

Microsoft Word - hozon-fujimura-HP-伊勢工業高校における造船教育の歴史から学ぶ20160713 1233 1 600 300 1578 1636 1800 Google 2 3 1853 1854 2000 1842 1848 300 1800 4 300 800 5 6 7 8 9 10 14 6 2 4 3 1 2 1 13 3 3 3 1 3 3 1 1 1 10 1 14 14 2 23 3 136 1 72 1 6 1 22 2 236 12 236 11 10% 7% 16% 17%

More information

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f ,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c

More information