(joint work with )
|
|
|
- しょうり あさま
- 5 years ago
- Views:
Transcription
1 (joint work with ) 2018
2 1. Convenient category WHK C2 WHZ C2 WHZ C2 = WHK C2 = WH (C 2 ) Theorem 1 X X BBT- C2 : X X C2 X, x (x, x)
3 2. D C 2 WH A C 2 D WH. B f : D W D D W WH W D C λ Λ D λ D D = λ Λ D λ D D WHK C2, WHZ C2 = WH (C 2 ) WHZ(C 2 ), WH (C 2 ), WHZ(C 2 ) memo: classes
4 3. J. F. K. Proposition 2 A D WH D B C D B C D = WHK D D Corollary 3 WHK C2 A WH A WHK C2 WHK D = D WHK C2 WHZ C2 WH (C 2 ) memo: Kennison 1965, Vogt
5 4. A = {a : A X } X O X A- a A ( a 1 (O) A ) A S A a solution set a A s S b ( a = s b, b ) Proposition 4 S A O A- O S- ( O O S ) s, b ( a 1 O = b 1 s 1 O ) O a A O O O A- O O S O S a A
6 5. 2 (Prop. 2) 1 D X = { f MWH Sf D, Tf = X } X - D- D A, B, C D D X SD X SD X = A P(X ) { 1(A,O) X : (A, O) X (A, O) D } f : D X (f D X ) 1 f (D) X f f (D) : D f (D) X f (D) D (f (D), O) (f (D),O) 1 X f (f (D),O) : D (f (D), O) X B f (f (D),O) f (f (D),O) SD X [Prop. 2 ] Prop. 4 f : D X f = ε X f kd X : D k D X X 1) k D X = (X, O SDX ) D (A,O) (A, O) k 1 SD D X X X B C WHK D = D 2) ε X = 1 k DX X : k D X X (ι X ) ε X k D : WH D ι : D WH WH(ιD, X ) = D(D, k D X )( = WH(ιD, ιk D X ) ), ε X ιg g
7 C 2X X -D = C 2 X WH SC 2X = {1 L X : L X 1L X C 2X } C 2X φ = 1 φ(k) X φ φ(k) : K φ(k) X 3 P(X, Y ) { 1 dx X 1 Y : dx Y X Y } { 1 X φ : X K X Y φ C 2Y } Y WH SP(X, Y ) = { 1 dx X 1 Y : dx Y X Y } { 1 X 1 L Y 1L Y SC 2Y } BBT- X C2 Y (X Y, O SP(X,Y ) ) dx = (X, P(X )) BBT-
8 7. 4 ( ) 4 F(Y, Z) Top (, Z) C cpt(, Z) { φ : C cpt(y, Z) C cpt(k, Z) φ C 2Y } Y WH SF(Y, Z) = { (1 L Y ) : C cpt(y, Z) C cpt(l, Z) 1 L Y SC 2Y } C 2M (Y, Z) SF(Y, Z) O SF(Y,Z) C 2M (Y, Z) = (Top (Y, Z), O SF(Y,Z) ) BBT- Proposition 5 Y WH, X, Z Top e : Top (X C2 Y, Z) Top (X, C 2M (Y, Z)), f e(f ) : x f (x, ) meno: Y, Z WH C 2M (Y, Z) WH Escardo-Lawson-Simpson
9 8. e : Top (dx Y, Z) Top (dx, (Top (Y, Z), O)), g [x g(x, )] O well defined
10 9. the cetralizer of C 2 (WH, C2 ) C 2 the centralizer of C 2 ) WHZ(C 2 ) WHZ(C 2 ) = { X K C 2 ( K C2 X = X C2 K ) } WHZ(C 2 ) P(X, K) {1 X K } X C2 K = X K WHZ(C 2 ) = { X K C 2 ( K C2 X = K X ) } BBT- WHZ(C 2 ) WHZ(C 2 ) WH BBT- WH (C 2 ), WHZ C2 WH (C 2 ) = { X WH C2 : X X C2 X, x (x, x) } WHZ C2 = WHZ(C 2 ) WH (C 2 )
11 10. WHZ C2 is a convenient category WHZ C2 WH A, B, C ι : WHZ C2 = WHK WHZC2 WH k WHZC2 z C2 z C2 (X Y ) = z C2 X C2 z C2 Y = z C2 (X C2 Y ) WHZ C2 WH e : Map(X C2 Y, Z) = Map(X, Map(Y, Z)) MacLane Proposition 2 p.186 in chap. VII Haus, CGHaus, K WH, WHZ C2, z C2 Theorem 6 WHZ C2
12 11. WH (C 2 ) WHK C2!!! Lemma 7 Y WH (C 2 ) F Y C 2Y - C2 (F ) Y C2 Y (1 dy Y 1 Y ) 1 C2 (F ) = y F {y} {y} dy Y Y T 1 (1 Y φ) 1 C2 (F ) = G(φ φ 1 (F ) ) = G(φ) (Y φ 1 (F )) Theorem 1 WHK C2 = WHZ C2 = WH (C 2 ) F Y WH (C 2 ) C 2Y - F = 1 C 2 ( C2 (F )) ( WHK C2 WHZ C2 ) WH (C 2 ) WHK C2 memo: Y WH (C 2 ) Y WH C2 : Y Y C2 Y, y (y, y)
13 12. WHK C2 = WHZ C2 = WH (C 2 ) WHK C2 WHZ C2 WH (C 2 ) WH WH
14 13. some colimits WH Top WHZ C2 (= WHK C2 ) McCord (1968 Transaction) WHK C2 Proposition 8 f : (X, A) (Y, B) Top X, B WHZ C2 A, Y WHZ C2 Proposition 9 X X 0 X 1 X n X n X n WHZ C2 X WHZ C2
15 14. a Lemma Lemma 10 f : (X, A) (Y, B) 8 f L X f L : L X Y 1) f 1 (f (L)) = f 1 (f (L)) L = (f 1 (f (L A)) f 1 (f (L A))) L = f 1 (f (L A)) L L A L C 2 L A C 2 fb L A C 2B B WHZ C2 f (L A) = fb L A (L A) B Y f (L) Y 4) L = 1-point f Y T 1 5) F L F C 2 f (F ) f L Y T 1 memo: f (A) f (X A) B (Y B) = def. : A f A = f 1 f (A) X A f
16 15. Prop. 8 Proposition 8 f : (X, A) (Y, B) Top X, B WHZ C2 A, Y WHZ C2 1) 8 1 B Y, f : (B X, B A) (Y, B) f 2) A WHZ C2 B Y WH Y WHZ C2 3) Y WH φ : K Y G(φ) = { (k, φ(k)) k K } K Y [ ] (φ, 1 K ) : K Y K = G(φ), [ ] p Y : Y K Y p Y (G(φ)) = φ(k) 4) K 1 K f : K X K Y (1 K f ) 1 G(φ) K X
17 16. 5) K X = K C2 X (1 dk K 1 X ) 1 (1 K f ) 1 G(φ) = k K {k} f 1 (φ(k)) dk X Lemma 8 Y T 1 6) (1 K 1 L Y ) 1 (1 K f ) 1 G(φ) = (1 K f L ) 1 G(φ) K L 7) f L : L X Y (1 K f L ) G(φ) (1 K f L ) 1 G(φ) K L K L
18 17. Proposition 11 (1) X, Y WH k C2 (X Y ) X C2 Y X Y (2) X, Y WHK C2 k C2 (X Y ) = X C2 Y (1) O X Y P(X, Y )- C 2X Y - (2) SP(X, Y ) B, C X C2 Y WHK C2 (1) WHK C2 WHZ C2 WH
19 18. K C 2, Y WH SP(K, Y ) (dk Y ) (K k C2 Y ) K C2 Y
20 19. 1 X A = {a : A X } A a A p(a) = p(a, x 1,, x N ) A = { a p(a) } O X A- q(o) { O q(o) } O A = { O P(X ) q(o) } O A X, Y X C2 Y = (X Y, O P(X,Y ) ) X C2 Y C2
21 20. 2 A = C 2 φ, K, O 1 X, O φ, K, O 1 C 2 X,O φ, K, O 1, X, O MTop K, O 1 C 2 X, O O P(X ) C 2 X,O - q(o) φ K O 1 ( φ, K, O 1 C 2 X,O φ 1 O O 1 )
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1.2 R A 1.3 X : (1)X (2)X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1. R A 1.3 X : (1)X ()X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f 1 (A) f X X f 1 (A) = X f 1 (A) = A a A f f(x) = a x
Akito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete sp
1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete space T1 T2 =, X = X, X X = X T3 =, X =, X X = X 2 X
Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2
Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 C II,,,,,,,,,,, 0.2. 1 (Connectivity) 3 2 (Compactness)
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
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
7310_J_Print.qxd
J 1 2 3 4 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 1 2 3 1 2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 1 2 1 2 1 2 1 2 3 1 2 3 1 1 2 2 1 2 3 4 1 H P C 1 2 3 4 1 2 1 2 1 2 1 2 3 4 1
OM_J_MFS2B_3.5B_body_CS_151023N.indd
1 2 13 14 25 26 3 15 27 4 5 16 17 28 6 18 7 8 19 20 9 21 10 11 22 23 12 24 1. 2. 5. 3. 4. !!! 1. 2. 1. 2. 1. !!!! 1. 2. 1. 1. 2. 3.! 1. 2. 3. !! !!!! 1. 2. 1. 2. 1. !!!!! 1. 1. 1.! 1. 2. 3. 1. 1. 2.!
0120-37-2269 0120-78-2269 1 2 3 4 Memo 
0120-37-2269 0120-78-2269 1 2 3 4 Memo Memo 
1
1 3 2 5 4 7 6 8 9 11 10 12 13 14 15 16 17 19 18 20 21 23 22 25 24 27 26 29 28 31 30 33 32 35 34 37 36 39 38 40 41 43 42 45 44 46 47 49 48 51 50 52 53 55 54 57 56 59 58 60 61 63 62 64 65 67 66 68 69 70
0120-37-2269 0120-78-2269 1 2 3 4 Memo 
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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
homes01_P027-084_chousa_sai.indd
1 027 1 028 029 030 031 032 033 2 034 035 3 3-1 1 036 2 3 037 038 4 039 3-2 1 040 041 2 category 1 042 043 category 2 044 045 category 3 046 047 category 4 048 049 category 5 050 051 category 6 052 053
1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)
![1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)](/thumbs/94/118328149.jpg "1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)")
1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p)
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
< E D834F E696E6464>
EXTENSION EXTENSION PROGRAM 1 2 3 4 5 6 4 6 6 6 3 8 5 6 6 5 6 5 4 6 8 6 4 2 5 4 5 5 5 1 2 2 3 4 4 2 4 4 3 3 6 5 1 1 1 1 1 1 1 3 6 5 1 1 1 1 1 1 1 5 5 1 1 1 1 1 1 1 4 5
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
![c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l](/thumbs/85/91372642.jpg "c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l")
c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC
1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC
( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x
![( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x](/thumbs/84/89665763.jpg "( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x")
Z Z Ẑ 1 1.1 (X, d) X x 1, x 2,, x n, x x n x(n) ( ) X x x ε N N i, j i, j d(x(i), x(j)) < ε ( ) X x x n N N i i d(x(n), x(i)) < 1 n ( ) X x lim n x(n) X x X () X x, y lim n d(x(n), y(n)) = 0 x y x y 1
Tricorn
Triorn 016 3 1 Mandelbrot Triorn Mandelbrot Robert L DevaneyAn introdution to haoti dynamial Systems Addison-Wesley, 1989 Triorn 1 W.D.Crowe, R.Hasson, P.J.Rippon, P.E.D.Strain- Clark, On the struture
微分積分 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, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
Lecture 12. Properties of Expanders
Lecture 12. Properties of Expanders M2 Mitsuru Kusumoto Kyoto University 2013/10/29 Preliminalies G = (V, E) L G : A G : 0 = λ 1 λ 2 λ n : L G ψ 1,..., ψ n : L G µ 1 µ 2 µ n : A G ϕ 1,..., ϕ n : A G (Lecture
PowerPoint プレゼンテーション
003.10.3 003.10.8 Y 1 0031016 B4(4 3 B4,1 M 0 C,Q 0. M,Q 1.- MQ 003/10/16 10/8 Girder BeamColumn Foundation SlabWall Girder BeamColumn Foundation SlabWall 1.-1 5mm 0 kn/m 3 0.05m=0.5 kn/m 60mm 18 kn/m
ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
株主通信：第18期 中間
19 01 02 03 04 290,826 342,459 1,250,678 276,387 601,695 2,128,760 31,096 114,946 193,064 45,455 18,478 10,590 199,810 22,785 2,494 3,400,763 284,979 319,372 1,197,774 422,502 513,081 2,133,357 25,023
株主通信 第16 期 報告書
10 15 01 02 1 2 3 03 04 4 05 06 5 153,476 232,822 6,962 19,799 133,362 276,221 344,360 440,112 412,477 846,445 164,935 422,265 1,433,645 26,694 336,206 935,497 352,675 451,321 1,739,493 30,593 48,894 153,612
p01.qxd
2 s 1 1 2 6 2 POINT 23 23 32 15 3 4 s 1 3 2 4 6 2 7003800 1600 1200 45 5 3 11 POINT 2 7003800 7 11 7003800 8 12 9 10 POINT 2003 5 s 45700 3800 5 6 s3 1 POINT POINT 45 2700 3800 7 s 5 8 s3 1 POINT POINT
ワタベウェディング株式会社
1 2 3 4 140,000 100,000 60,000 20,000 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 5 6 71 2 13 14 7 8 9 10 11 12 1 2 2 point 1 point 2 1 1 3 point 3 4 4 5 6 point 4 point 5 point 6 13 14 15 16 point 17
untitled
1 2 3 4 5 6 7 Point 60,000 50,000 40,000 30,000 20,000 10,000 0 29,979 41,972 31,726 45,468 35,837 37,251 24,000 20,000 16,000 12,000 8,000 4,000 0 16,795 22,071 20,378 14 13 12 11 10 0 12.19 12.43 12.40
![[商品カタログ]ゼンリン電子地図帳Ｚｉ16](/thumbs/48/24409875.jpg "[商品カタログ]ゼンリン電子地図帳Ｚｉ16")
ヤフー株式会社 株主通信VOL.16
01 260,602264,402 122,795125,595 64,84366,493 107110 120,260123,060 0 500 300 400 200 100 700 600 800 39.8% 23.7% 36.6% 26.6% 21.1% 52.4% 545 700 0 50 200 150 100 250 300 350 312 276 151 171 02 03 04 POINT
-- 0 500 1000 1500 2000 2500 3000 () 0% 20% 40% 60%23 47.5% 16.0% 26.8% 27.6% 10,000 -- 350 322 300 286 250 200 150 100 50 0 20 21 22 23 24 25 26 27 28 29 -- ) 300 280 260 240 163,558 165,000 160,000
1003shinseihin.pdf
1 1 1 2 2 3 4 4 P.14 2 P.5 3 P.620 6 7 8 9 10 11 13 14 18 20 00 P.21 1 1 2 3 4 5 2 6 P7 P14 P13 P11 P14 P13 P11 3 P13 7 8 9 10 Point! Point! 11 12 13 14 Point! Point! 15 16 17 18 19 Point! Point! 20 21
重力方向に基づくコントローラの向き決定方法
( ) 2/Sep 09 1 ( ) ( ) 3 2 X w, Y w, Z w +X w = +Y w = +Z w = 1 X c, Y c, Z c X c, Y c, Z c X w, Y w, Z w Y c Z c X c 1: X c, Y c, Z c Kentaro Yamaguchi@bandainamcogames.co.jp 1 M M v 0, v 1, v 2 v 0 v
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
¿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
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
B2 ( 19 ) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercia
B2 ( 19) Lebesgue ( ) ( 19 7 12 ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purposes. i Riemann f n : [0, 1] R 1, x = k (1 m
(note-02) Rademacher 1/57
(note-02) Rademacher 1/57 (x 1, y 1 ),..., (x n, y n ) X Y f : X Y Y = R f Y = {+1, 1}, {1, 2,..., G} f x y 1. (x 1, y 1 ),..., (x n, y n ) f(x i ) ( ) 2. x f(x) Y 2/57 (x, y) f(x) f(x) y (, loss) l(f(x),
入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
![Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.](/thumbs/93/113887245.jpg "Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.")
Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)
線形空間の入門編 Part3
Part3 j1701 March 15, 2013 (j1701) Part3 March 15, 2013 1 / 46 table of contents 1 2 3 (j1701) Part3 March 15, 2013 2 / 46 f : R 2 R 2 ( ) x f = y ( 1 1 1 1 ) ( x y ) = ( ) x y y x, y = x ( x y) 0!! (