Σ A Σ B r Σ A (Σ A ): A r = [ A r A x r A y r z ] T Σ B : B r = [ B r B x r B y r z ] T A r = A x B B r x + A y B B r y + A z B B r z A r = A R B B r
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1 3 : Σ A = O A {X A, Y A, Z A } : Σ B = O B {X B, Y B, Z B } O B : A p B X B, Y B, Z B Σ A : A x B, A y B, A z B Σ A : A p B Σ A : { A x B, A y B, A z B } A R B = [ A x A B y A B z B ] ( A R B ) T ( A R B ) = I 3 ( A R B ) = ( A R B ) T 8
2 Σ A Σ B r Σ A (Σ A ): A r = [ A r A x r A y r z ] T Σ B : B r = [ B r B x r B y r z ] T A r = A x B B r x + A y B B r y + A z B B r z A r = A R B B r (Σ A Σ B ) B r = B R A A r r ( A R B )( B R A ) = I 3 8 2
3 Σ A Σ B 3 Σ C r Σ C C r B r = B R C C r, A r = A R C C r A R C C r = A r = A R B B r = A R B B R C C r A R C = A R B B R C, A R B = A R B = ( C R A ) T C R B ( C x A ) T C x B ( C x A ) T C y B ( C x A ) T C z B ( C y A ) T C x B ( C y A ) T C y B ( C y A ) T C z B ( C z A ) T C x B ( C z A ) T C y B ( C z A ) T C z B 8 3
4 2 Σ A, Σ B ( ) Σ A Σ B : A p B Σ A Σ B : A R B Σ B B r Σ A A r = A R B B r + A p B A r = A R B A p B B r A T B = A R B A p B [ ]: A r A r B r B r 8 4
5 A r = A T B B r Σ B Σ A Z A α A T B Σ B Σ A Y A 2 Z A A T B 8 5
6 5 4 3 B r A R B A p B 0 = I 3 A p B 0 A r = A T B B r A R B A r Σ B 0 Σ A A p B Σ A B r Σ A A R B Σ A A p B A r Σ B B r Σ A A r 8 6
7 Σ A Σ A A R B Σ A A p B Σ B Σ A Σ A A p B A R B Σ B
8 [ ] Σ A Σ A A R B Σ A A p B Σ B
9 [ 2] Σ A Σ A A p B Σ A A R B Σ B
10 3 Σ A, Σ B, Σ C Σ A Σ B A T B Σ B Σ C B T C Σ A Σ C A T C = A T B B T C A T B = A T C = I 3 A R B A p B 0 A p B 0 = A R B 0 0 I 3 A p B 0 I 3 B p C 0 A R B 0 0 B R C
11 . (a) Σ A Σ A A p B (b) A R B Σ B (c) Σ B Σ B B p C (d) B R C Σ C 2. (a) Σ A Σ A B R C (b) Σ A B p C Σ B (c) Σ B Σ A A R B (d) Σ A A p B Σ C 8
12 A T B = 3/2 /2 0 2 /2 3/ , B T C = / 2 / 2 0 / 2 / B T A = ( A T B ) = ( A R B ) T ( A R B ) T A p B 0 8 2
13 8 3
14 PUMA 8 4
15 PUMA (0 ) Σ 0 Σ Σ 0 z 0 T = C S 0 0 S C C = cos(θ ), S = sin(θ ) 8 5
16 PUMA ( 2) Σ Σ 2... Σ x π/2 z l b l d z θ 2 T 2 = = l b l d C 2 S l b l d S 2 C C 2 S S 2 C Σ y l b l d x π/2 z θ 2 8 6
17 PUMA (2 3) Σ 2 Σ 3... Σ 2 x l c z θ 3 2 T 3 = 0 0 l c = C 3 S 3 0 l c S 3 C C 3 S S 3 C
18 PUMA (3 4) Σ 3 Σ 4... Σ 3 x l e x π/2 z l f z θ 4 3 T 4 = 0 0 l e = C 4 S 4 0 l e 0 0 l f S 4 C l f C 4 S S 4 C
19 PUMA 4 T 5 = 5 T 6 = C 5 S S 5 C C 6 S S 6 C
20 ( ) θ ( ) 0 r 0 R 6 A r = A R B B r + A p B A = 5, B = 6 0 T 6 = 0 T T 2 5T
21 ( ) 0 T 6 θ 8 2
22 ( ) Z f x = f X Z, y = f Y Z Camera Coordinate System Camera Image Plane x f y cprel Z Y Object Coordinate System X s x y = f f X Y Z A = f f CCD CCD A 8 22
23 ( C T O ) Camera Coordinate System Camera Image Plane x f y World Coordinate System Z Y Object Coordinate System X O r i C r i = C T O O r i, C r i = s i x i y i = A C T O O r i = A( W T C ) W T O O r i X i Y i Z i 8 23
24 Camera Coordinate System Camera Image Plane x f y World Coordinate System Z Y Object Coordinate System X ( W T O ) ( O r i ) (x i, y i ) (A) ( W T C ) s i x i y i = A( W T C ) W T O O r i 8 24
25 ( ) Z f x = f X Z, y = f Y Z Camera Coordinate System Camera Image Plane x f y cprel Z Y Object Coordinate System X s x y = f f X Y Z A = f f CCD CCD A 8 25
26 Camera Coordinate System Image Plane y Object Coordinate System s x y = A ( C T O ) X Y Z Camera x f Z Y X O r i C r i = C T O O r i, C r i = X i Y i Z i World Coordinate System s i x i y i = AC T O O r i = A( W T C ) W T O O r i 8 26
27 Camera Coordinate System Image Plane y Object Coordinate System s i x i y i = A( W T C ) W T O O r i Camera x f Z Y X : W T O : O r i : x i, y i (i =,..., N) World Coordinate System : A : W T C 8 27
28 Camera Coordinate System Camera Image Plane x f y World Coordinate System Z Y Object Coordinate System X s i x i y i = A( W T C ) W T O = P W T O O r i O r i P P = αp 2 6 P 8 28
29 : : Find P R 3 4 s.t. : i s i s i i x i y i x i y i = P W T O P X i Y i Z i 2 O r i min P = [P jk ] s i x i (P X i + P 2 Y i + P 3 Z i + P 4 ) s i y i (P 2 X i + P 22 Y i + P 23 Z i + P 24 ) s i (P 3 X i + P 32 Y i + P 33 Z i + P 34 ) 8 29
30 P P = [P, P 2,..., P 34 ] T s i [ ] Xi Y i Z i x i X i x i Y i x i Z i x i P [ ] Xi Y i Z i y i X i y i Y i y i Z i y i P P P 34 = ˆP = [P, P 2,..., P 33 ] T i =,, N x y x 2 y 2. y N X Y Z x X x Y x Z X Y Z y X y Y y Z X 2 Y 2 Z x 2 X 2 x 2 Y 2 x 2 Z X 2 Y 2 Z 2 y 2 X 2 y 2 Y 2 y 2 Z X N Y N Z N y N X N y N Y N y N Z N 2 min x y Mx 2 ˆP 8 30
31 : x i s i y i = P W T O O r i f x α c x 0 A = 0 f y c y , P = A( W T C ) = [B 0] ( W T C ) = C T W = C R W C t W P = A( W T C ) = [ B C R W B C t W ] 8 3
32 : 80cm 64 :.5 pixel B = tc =.0e+03 * Rc = P = Phat =
33 : 40cm 64 :.5 pixel B = tc = Rc = P = Phat =
34 P = A( W T C ) 8 34
35 Image Plane Object Z Camera f x D B X x 2 B x 2 2 ξ = x x 2, x = f B D, x 2 = f B D p = X Z 8 35
36 Image Plane Object Z Camera f x D B X x 2 B x ξ p = f D f D fb D 2 fb D 2 J = ξ p 8 36
37 D = 000, B = 00, f = 00 (J = UΣV T ) U = J =, Σ = , V T = JV = UD, Jv = σ u, Jv 2 = σ 2 u
38 J 0 = , J 0 = Image Plane Object Z Camera f x D B X x 2 B Z x Image Jacobian x 2 u X v 2 x u 2 v Camera Motion Feature Motion 8 38
39 Image Plane Object Z Camera f x D B X x 2 B x u (= X) (v ) Image Plane x x
40 u 2 (= Z) (v 2 ) Image Plane x x 2 X Z 8 40
41 n : q R n : r R m : r = f(q) : J = r q : ṙ = J q : q q ṙ : : q 2 = J + v + (I J + J)k 2 v T (J + ) T J + v, v ImJ : w = det JJ T (m = n w = det J ) 8 4
42 Y l 2 θ 2 l θ X : r : J r = l C + l 2 C 2 l S + l 2 S 2, J = r θ = l S l 2 S 2 l 2 S 2 l C + l 2 C 2 l 2 C 2 : w w = det JJ T = l l 2 S 2 l, l 2, θ θ 2 = ±π/2 8 42
43 l = l 2 =, θ 2 = 2θ
44 R C T 2 2 A A 2 R C T 2 R t
45 : (X, Y, Z) R T 2 2 R T : (u, v ), (u 2, v 2 ) : P, P 2 C A A 2 R t C 2 2 s i u i v i = P i X Y Z 8 45
46 , 2 s i u i v i = P i X Y Z i =, 2 P, P 2, s, s 2 (u, v ), (u 2, v 2 ) X, Y, Z X Y Z = P + y y = s u s v s s 2 u 2 s 2 v 2 s 2, P = s i P P
47 2 M : A, A 2 : (R, T ), (R 2, T 2 ) m m 2 2 : R = R T R 2, t = R T (T 2 T ) : M = (X, Y, Z, ) T C e R t e 2 C 2 : m = (u, v, ) T, m 2 = (u 2, v 2, ) T R T R 2 T 2 s m = A [ R T R T T ] M s 2 m 2 = A 2 [ R T 2 R T 2 T 2 ] M 8 47
48 C m L e R t M Σ l l 2 R T R 2 T 2 L e 2 2 m 2 C 2 C, C 2, M Σ m, m 2 l, l 2 C, C 2 e, e 2 ( ) 8 48
49 : M s m = A [ R T R T T ] M s 2 m 2 = A 2 [ R T 2 R T 2 T 2 ] M M = [X, Y, Z] T, M = [ M T ] T m m 2 s (A R T ) m = M T s 2 (A 2 R T 2 ) m 2 = M T 2 C e R t e 2 C 2 M s R A m s 2 R 2 A 2 m 2 = T 2 T R T R 2 T 2 R = R T R 2, t = R T (T 2 T ) s A m s 2 RA 2 m 2 = t 3 A m, RA 2 m 2, t ( ) 8 49
50 M (m, m 2 ) (A, A 2 ) (R, t) ( ) s A m s 2 RA 2 m 2 = t A m t (RA 2 m 2 ) ( ) C m e R t e 2 m 2 C 2 t = (t x, t y, t z ) t x = T x T = 0 t z t y t z 0 t x t y t x 0 R T R 2 T 2 (fundamental equation): m T (A ) T T RA 2 m 2 = 0 F (fundamental matrix) m T F m 2 = 0, F = (A ) T T RA
51 x 2 Rx 2 (essential matrix) E = T R m 2 x (Rm, m 2, t ) m m T 2 (t Rm ) = 0 m T 2 Em = 0 8 5
52 x 2 Rx 3 x ki = x 2i = Rx i + t for i =,... N k X i k Y i k Z i for k =, 2, i =,... N m kj = u kj v kj w kj = fx kj /Z kj fy kj /Z kj f m 2 m x m T 2iEm i = 0 for i =,... N E = e e 2 e 3 e 2 e 22 e 23 e 3 e 32 e
53 E = T R, T = t 3 3 Q (= ) : t = t z Q = UΣV T, Q E Σ = diag{λ, λ, 0} QQ T = T RR T T T = T T T = T 2 T 2 [x y z] = t t [x y z] = t 2 [ x y 0] = [x y z]diag{ t 2, t 2, 0} QQ T = T 2 = [x y z]diag{ t 2, t 2, 0}[x y z] T Q = [x y z]diag{ t, t, 0}V T ( =) : Q = UΣ 0 V T R z = Rot(z, π/2) Q = UΣ 0 R T z U T UR z V T = T 0 R 0, T 0 = UΣ 0 R T z U T, R 0 = UR z V T R T 0 R 0 = V R T z U T UR z V T = I T T 0 = UR z Σ 0 U T = UΣ 0 R z U T = T 0 (R T z = R z ) 8 53
54 (8 ) m T 2iEm i = 0 for i =,... N E = e e 2 e 3 e 2 e 22 e 23 e 3 e 32 e 33 x 2 m 2 m Rx x (3 3 ) 8 Be = 0, B = b. b N, e = e e 2. e 33, b i = [ u 2j u j u 2j v j u 2j s j v 2j u j v 2j v j v 2j s j s 2j u j s 2j v j s 2j s j ] 8 54
55 (8 ) (3 3 ) e = N 8 Be = 0, B = b. b N, e = b i = [ u 2j u j u 2j v j u 2j s j v 2j u j v 2j v j v 2j s j s 2j u j s 2j v j s 2j s j ] e e 2. e 33 Find e R 9 s.t. Be min with e = B e E E λ, λ, 0, 8 55
56 E = UΣV T t = σ u 3 (u 3 U 3 ) R = UR z V T 8 56
57 8 I. {(m i, m 2i ), i =,..., N} B B=zeros(N,9); for i=:n u=m(,i); v=m(2,i); w=m(3,i); u2=m2(,i); v2=m2(2,i); w2=m2(3,i); B(i,:)=[u2*u, u2*v, u2*w, v2*u, v2*v, v2*w, w2*u, w2*v, w2*w]; end 2. Be min e ( e = ) [U D V]=svd(B); e=v(:,9); 3. e E tracee T E = 2 E=sqrt(2)*[q(:3) ; q(4:6) ; q(7:9) ]; # residual=trace(m2 *E*m); 8 57
58 8 II 4. T R Ê = T R Ê Ê E Ê [U D V]=svd(E); D=diag((D()+D(2))/2, (D()+D(2))/2, 0); hate=u*d*v ; 5. Ê R T [U, D, V]=svd(hatE); t=u(:,3); Rz=[0 0; - 0 0;0 0 ]; R=U*Rz*V ; R2=U*Rz *V ; 8 58
59 f = 3; noizelevel = 0.3; rpy = [0 30 5]; t = [50 0 3]; D = 000; objsize =
60 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
61 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
62 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
63 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
64 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
65 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
66 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
67 Essential Matrix: R: Re: rpy: rpye: t: normlized t: te:
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
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,
数学Ⅱ演習（足助・09夏）
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0
![C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0](/thumbs/92/108844778.jpg "C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0")
1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
![24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x](/thumbs/94/118744578.jpg "24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x")
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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Workbook E-mail: kawahira@math.nagoya-u.ac.jp 2004 8 9, 10, 11 1 2 1 2 a n+1 = pa n + q x = px + q a n better 2 a n+1 = aan+b ca n+d 1 (a, b, c, d) =(p, q, 0, 1) 1 = 0 3 2 2 2 f(z) =z 2 + c a n+1 = a 2
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0
(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
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Onsager SOLUTION OF THE EIGENWERT PROBLEM (O-29) V = e H A e H B λ max Z 2 Onsager (O-77) (O-82) (O-83) Kramers-Wannier 1 1 Ons
Onsager 2 9 207.2.7 3 SOLUTION OF THE EIGENWERT PROBLEM O-29 V = e H A e H B λ max Z 2 OnsagerO-77O-82 O-83 2 Kramers-Wannier Onsager * * * * * V self-adjoint V = V /2 V V /2 = V /2 V 2 V /2 = 2 sinh 2H
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数学演習：微分方程式
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7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
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January 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
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OHP.dvi
7 2010 11 22 1 7 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2010 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/18 3. 10/25 2, 3 4. 11/ 1 5. 11/ 8 6. 11/15 7. 11/22 8. 11/29 9. 12/ 6 skyline 10. 12/13
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
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OHP.dvi
t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F
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