(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w
|
|
- みずき うすい
- 5 years ago
- Views:
Transcription
1 S = 4π dτ dσ gg ij i X µ j X ν η µν η µν g ij g ij = g ij = ( 0 0 ) τ, σ (+, +) τ τ = iτ ds ds = dτ + dσ ds = dτ + dσ δ ij ( ) a =, a = τ b = σ g ij δ ab g g ( +, +,... ) S = 4π S = 4π ( i) = i 4π dτ dσ gg ij i X µ j X ν η µν dτ dτ dσ( τ Xµ dσ g (δ ab a X µ b X ν )η µν τ Xν + σ Xµ σ Xν )η µν τ ( ) e is e S E S E = 4π dτ dσδ ab a X µ b X ν η µν = 4π dτ dσ( τ Xµ τ Xν + σ Xµ σ Xν )η µν
2 (τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w ( g = g ww, g = g ww, g = g ww, g = g ww ) g µν = x x µ x β x ν g β g = x x w w δ + x x w w δ = τ τ w w + σ σ w w = 4 4 = 0 g = x x w w δ + x x w w δ = τ τ w w + σ σ w w = = g = g, g = g g ab = ( 0 0 ), g ab = ( 0 0 ) gdτdσ ( ) gd D x = g d D x g dτdσ g dwdw dτdσ = 4 dwdw w, w
3 S = 4π 4π dτ dσ (δ ab a X µ b X ν )η µν = 4π = π dwdw ( gab a X µ b X ν )η µν dwdw ( wx µ w X ν + w X µ w X ν )η µν ( w = w, w = w ) dwdw w X µ (w, w) w X ν (w, w)η µν w =, w = δx µ S + δs = π = π dwdw((x µ + δx µ )X µ + X µ (X µ + δx µ )) dwdw(x µ X µ + δx µ X µ + X µ δx µ ) δs δs = π = π = π dwdw(δx µ X µ + X µ δx µ ) dwdw( δx µ X µ δx µ X µ ) dwdwδx µ (X µ ) δx µ w, w δs 0 X µ (w, w) = 0 X µ (w, w) = X µ (w) + X µ (w) X µ (w) (holomorphic function) X µ (w) (anti-holomorphic function) X µ (w) = X µ (τ iσ) right moving X µ (w) = X µ (τ + iσ) left moving ( τ ± σ i(τ ± iσ)) w w right left τ, σ w, w X µ w z z = e w = e τ iσ, z = e w = e τ+iσ 3
4 τ σ z σ σ = σ + π τ ( < τ < ) σ (σ π τ ) z σ π z z = e τ e τ τ (τ = ) τ = τ z e τ τ τ + a σ σ + b τ + a e a e τ iσ = e a z σ + b e ib e τ iσ = e ib z z σ 0 l z = e τ+iσ iσ X µ (τ, σ) X µ (τ, σ) = X µ L (τ + σ) + Xµ R (τ σ) X µ (τ, σ) = X µ (τ, σ + π) X µ L (τ + σ) = xµ 0 + pµ (τ + σ) + i n µ ne in(τ+σ) X µ R (τ σ) = xµ 0 + pµ (τ σ) + i n µ ne in(τ σ) w X µ L (w = τ + iσ) = xµ 0 + pµ ( iτ + σ) + i n µ ne in( iτ+σ) = xµ 0 i pµ (τ + iσ) + i = xµ 0 i pµ w + i n µ ne nw n µ ne n(τ+iσ) z = e w X µ L (z) = xµ 0 i pµ log z + i 4 n µ nz n
5 X µ R X µ R (z) = xµ 0 i pµ log z + i n µ nz n µ [x µ 0, pν ] = [xµ 0, ν 0] = iη µν [ µ m, ν n] = [ µ m, ν n] = mη µν δ m+n,0 η µν µ n, µ n n µ n = na µ n, µ n = na µ n µ n = n(a µ n), µ n = n(a µ n) n, µ µ n 0 µ n 0 = µ n 0 = 0, 0 µ n = 0 µ n = 0 (n ) p µ p µ 0 = 0 X µ ( )G X(τ, σ ), X(τ, σ ) τ, τ ( ) T (X µ (τ, σ )X ν (τ, σ )) = : X µ (τ, σ )X ν (τ, σ ) : +G(τ, σ ; τ, σ ) T : : G(τ, σ ; τ, σ ) = T (X µ (τ, σ )X ν (τ, σ )) : X µ (τ, σ )X ν (τ, σ ) : τ, σ z, z G(z, z ; z, z ) = T (X µ (z, z )X ν (z, z )) : X µ (z, z )X ν (z, z ) : () τ > τ z, z τ z > z c c τ > τ ( z > z ) 0 X µ (z, z )X ν (z, z ) 0 X µ X µ L + Xµ R 5
6 X µ (z, z )X ν (z, z ) = (X µ L (z ) + X µ R (z ))(X ν L(z ) + X ν R(z )) = (X µ L (z )X ν L(z ) + X µ R (z )X ν R(z ) + X µ L (z )X ν R(z ) + X µ R (z )X ν L(z )) () X µ L (z )X ν L (z ) 0 X µ L (z )X ν L(z ) 0 = 4 0 xµ 0 xν log z log z 0 p µ p ν 0 i 4 log z 0 x µ 0 pν 0 + i i 4 log z 0 p µ x ν m 0 m 0 m, m z m 0 x µ 0 ν m 0 m z m 0 p µ ν m 0 + i n z n 0 µ nx ν i n z n 0 µ np ν 0 = 4 0 xµ 0 xν 0 0 m, mn z n z m 0 µ n ν m 0 mn z n z m 0 µ n ν m 0 i 4 log z 0 p µ x ν 0 0 (3) x 0, p µ n ( n n > 0 n < 0 ) 0 µ n ν m 0 m, mn z n z m 0 µ n ν m 0 = = = m= m= m= = η µν n= mn z n zm 0 µ n ν m 0 mn z n zm 0 ([ µ n, ν m] + ν m µ n) 0 mn z n zm nη µν δ n m,0 ( 0 0 = ) n z n zn n m = 0 n n 0 p µ x ν p µ x ν 0 0 = 0 ([p µ, x ν 0] + x ν 0p µ ) 0 = 0 ( iη µν ) 0 = iη µν 0 X µ L (z )X ν L(z ) 0 = 4 0 xµ 0 xν ηµν log z + ηµν 6 n= n (z z ) n (4)
7 X µ R (z )X ν R (z ) z z 0 X µ R (z )X ν R(z ) 0 = 4 0 xµ 0 xν ηµν log z + ηµν n= n (z z ) n (5) X µ L (z )X ν R (z ) 0 µ n ν m 0 µ n ν m µ n ν m x µ 0 xν 0 xµ 0 pν 0 X µ L (z )X ν R(z ) 0 = 4 0 xµ 0 xν 0 0 i 4 log z 0 x µ 0 pν 0 = 4 0 xµ 0 xν ηµν log z (6) X µ R (z )X ν L (z ) 0 X µ R (z )X ν L(z ) 0 = 4 0 xµ 0 xν ηµν log z (7) () (4) (7) 0 X µ (z, z )X ν (z, z ) 0 = 0 x µ 0 xν ηµν log z + ηµν 4 ηµν log z + ηµν n= 4 ηµν log z 4 ηµν log z n (z z ) n n= n (z z ) n = 0 x µ 0 xν 0 0 ηµν log z ηµν log z + ηµν n= n (z z ) n + ηµν n= n (z z ) n n= x n n = log[ x] n= n (z z ) n = log[ z z ] = log z [z z ] = log z log[z z ] 7
8 ηµν log z ηµν log z + ηµν n= n (z z ) n + ηµν n= n (z z ) n = ηµν log z ηµν log z + ηµν (log z log[z z ]) + ηµν (log z log[z z ]) = ηµν log[z z ] ηµν log[z z ] 0 X µ X ν 0 0 X µ (z, z )X ν (z, z ) 0 = 0 x µ 0 xν 0 0 ηµν log[(z z )(z z )] (8) z, z (τ, τ ) (µ, ν z, z ) 0 X µ (z, z )X ν (z, z ) 0 = 0 X ν (z, z )X µ (z, z ) 0 (8) () () : p µ x ν 0 := x ν 0p µ, : m n := n m, : m n := n m (m, n ) (3) 0 x µ 0 xν 0 0 (8) 0 : X µ (z, z )X ν (z, z ) : 0 = 0 x µ 0 xν X µ (z, z )X ν (z, z ) 0 0 : X µ (z, z )X ν (z, z ) : 0 = ηµν log[(z z )(z z )] = ηµν log z z G(z, z ; z, z ) = ηµν log[(z z )(z z )] = ηµν log z z (9) z z (9) z, z z z G(z, z ; z, z ) = z X µ (z, z ) z X ν (z, z ) = ηµν z z z = ηµν (z z ) 8
9 X(z, z )X(z, z ) z X(z, z ) z X(z, z ) ((8) ) ( ) z z λz ϕ(z, z) ϕ(z, z) ϕ (z, z) = λ h λ h ϕ(λz, λz) h, h (conformal dimension) (h h ) A = ab a λa, b λb A λλa z f(z) ϕ(z, z) ϕ (z, z) = ( f z )h ( f z )h ϕ(f(z), f(z)) (0) primary field ϕ primary field quari-primary field z = z + n ϵ n ( z n+ ), z = z + n ϵ n ( z n+ ) l n = z n+ z, l n = z n+ z z n+ z = 0 z = n ( ) n z = 0 z = z = /v z = v z v = z v = v v l n = z n+ z = ( v )n+ ( v ) v = ( v )n v z = v 0 n + l n l, l 0, l + l, l 0, l +, l, l 0, l + (globally) primary field quasi-primary field primary fieldϕ (u), ϕ (v) (u, v ) ϕ (u)ϕ (v) n = u = u + ϵ = u + a, v = v + a ϕ (u)ϕ (v) ϕ (u + a)ϕ (v + a) ϕ (u)ϕ (v) 9
10 D(u, v) = ϕ (u)ϕ (v) D(u, v) = D(u v) n = 0 u = u + ϵ 0 ( u) = λu, v = λv primary field ϕ (u)ϕ (v) λ h ϕ (λu)λ h ϕ (λv) = λ h +h ϕ (λu)ϕ (λv) λ h +h ϕ (λu)ϕ (λv) = λ h +h D(λ(u v)) λ h+h D(λ(u v)) = D(u v) D(u v) d D(u v) = () (u v) h+h ( (λu λv) h +h = λ (h +h ) ) (u v) h +h d n = + u = u + ϵ + ( u ) = ( cu)u, v = ( cv)v (x /x) u = u, v = v D(u v) = ϕ (u)ϕ (v) D( u v ) = ϕ ( u )ϕ ( v ) = ϕ ( u u)ϕ ( v v) = u h v h ϕ ( u )ϕ ( v ) () 0
11 u h v h ϕ ( u )ϕ ( v ) = u h v h d ( u + v )h+h D(u v) = D( u v ) d (u v) h +h = u h v h d ( u + v )h +h h = h = h d u h v h ( u + = d v )h u h v h (u v/uv) h = d (u v) h n =, 0, + u = u + a, v = v + a u = λu, v = λv u = ( cu)u, v = ( cv)v ϕ (u)ϕ (v) ϕ (u)ϕ (v) = d (u v) h l, l 0, l + quasi-primary field 3 (Möbius transformation) z z = az + b cz + d (ad cb =, ( cz)z z cz + ) G(z, z ; z, z ) = X µ (z, z )X ν (z, z = ηµν log[(z z )(z z )] X µ (z, z) (quasi)-primary field (0) z X µ z X µ (z, z ) z X ν (z, z ) = ηµν (z z ) z X µ (z, z) primary field (0)
『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
More informationZ: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
More informationTOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................
More information(yx4) 1887-1945 741936 50 1995 1 31 http://kenboushoten.web.fc.com/ OCR TeX 50 yx4 e-mail: yx4.aydx5@gmail.com i Jacobi 1751 1 3 Euler Fagnano 187 9 0 Abel iii 1 1...................................
More information医系の統計入門第 2 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, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More information.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
More informationmeiji_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
More informationI 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
More information4 Mindlin -Reissner 4 δ T T T εσdω= δ ubdω+ δ utd Γ Ω Ω Γ T εσ (1.1) ε σ u b t 3 σ ε. u T T T = = = { σx σ y σ z τxy τ yz τzx} { εx εy εz γ xy γ yz γ
Mindlin -Rissnr δ εσd δ ubd+ δ utd Γ Γ εσ (.) ε σ u b t σ ε. u { σ σ σ z τ τ z τz} { ε ε εz γ γ z γ z} { u u uz} { b b bz} b t { t t tz}. ε u u u u z u u u z u u z ε + + + (.) z z z (.) u u NU (.) N U
More informationtomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
More information1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
More informationO x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
More informationIII 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
More informationI z n+1 = zn 2 + c (c ) c pd L.V. K. 2
I 2012 00-1 I : October 1, 2012 Version : 1.1 3. 10 1 10 15 10 22 1: 10 29 11 5 11 12 11 19 2: 11 26 12 3 12 10 12 17 3: 12 25 1 9 1 21 3 1 I 2012 00-2 z n+1 = zn 2 + c (c ) c http://www.math.nagoya-u.ac.jp/~kawahira/courses/12w-tenbou.html
More information( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
More information(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z
B 4 24 7 9 ( ) :,..,,.,. 4 4. f(z): D C: D a C, 2πi C f(z) dz = f(a). z a a C, ( ). (ii), a D, a U a,r D f. f(z) = A n (z a) n, z U a,r, n= A n := 2πi C f(ζ) dζ, n =,,..., (ζ a) n+, C a D. (iii) U a,r
More informationTOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................
More informationW u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
More informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More informationZ: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
More information50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
More informationgr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
More informationCIII CIII : October 4, 2013 Version : 1.1 A A441 Kawahira, Tomoki TA (Takahiro, Wakasa 3 )
III203 00- III : October 4, 203 Version :. Kawahira, Tomoki TA (Takahiro, Wakasa 3 ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/3w-kansuron.html pdf 0 4 0 0 8 2 0 25 8 5 22 2 6 2 3 2 20 0 7 24 3
More information( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
More informationDirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp
More informationl µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r
2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
More information(1) (2) (3) (4) 1
8 3 4 3.................................... 3........................ 6.3 B [, ].......................... 8.4........................... 9........................................... 9.................................
More informationN cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
More information= = = = = = 1, 000, 000, 000, = = 2 2 = = = = a,
2011 10 26 1. 1627 27, 682, 574, 402 = 2 7 7 7 7 7 7 7 7 7 7 7 7 2 2 7 2 7 7..., 100 100 99 2 100 100 two to the hundred n n, n 2, n 3,... n 10... 10 10 1 = 10 10 2 = 10 10 = 100 10 3 = 10 10 10 = 1000
More informationx (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
More informationprime number theorem
For Tutor MeBio ζ Eite by kamei MeBio 7.8.3 : Bernoulli Bernoulli 4 Bernoulli....................................................................................... 4 Bernoulli............................................................................
More information,,..,. 1
016 9 3 6 0 016 1 0 1 10 1 1 17 1..,,..,. 1 1 c = h = G = ε 0 = 1. 1.1 L L T V 1.1. T, V. d dt L q i L q i = 0 1.. q i t L q i, q i, t L ϕ, ϕ, x µ x µ 1.3. ϕ x µ, L. S, L, L S = Ld 4 x 1.4 = Ld 3 xdt 1.5
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More information転位の応力場について
y.koyama f ( F ( F( f( xp( i d f( F( xp( i ( π xp( i d δ ( F( δ ( f( δ ( xp( i f ( δ ( F( f ( xp( i d xp( i d F( f( xp( i d ' F( xp( i xp( i ' ' d ' F( xp{ i( } d ' ' ' d ' ' F( ' xp{ i( } d ' ' ' F( '
More informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
More informationnewmain.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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information中央大学セミナー.ppt
String Gas Cosmology References Brandenberger & Vafa, Superstrings in the early universe, Nucl.Phys.B316(1988) 391. Tseytlin & Vafa, Elements of string cosmology, Nucl.Phys.B372 (1992) 443. Brandenberger,
More information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)
More informationD = [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..
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information73
73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ
More information: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
More informationuntitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
More information1 0/1, a/b/c/ {0, 1} S = {s 1, s 2,..., s q } S x = X 1 X 2 X 3 X n S (n = 1, 2, 3,...) n n s i P (X n = s i ) X m (m < n) P (X n = s i X n 1 = s j )
(Communication and Network) 1 1 0/1, a/b/c/ {0, 1} S = {s 1, s 2,..., s q } S x = X 1 X 2 X 3 X n S (n = 1, 2, 3,...) n n s i P (X n = s i ) X m (m < n) P (X n = s i X n 1 = s j ) p i = P (X n = s i )
More informationJuly 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
More information7 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 ±
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
More informationBayesfiI‡É“ÅfiK‡È−w‘K‡Ì‡½‡ß‡ÌChow-Liu…A…‰…S…−…Y…•
1 / 21 Kruscal V : w i,j R: w i,j = w j,i i j Kruscal (w i,j 0 ) 1 E {{i, j} i, j V, i i} 2 E {} 3 while(e = ϕ) for w i,j {i, j} E 1 E E\{i, j} 2 G = (V, E {i, j}) = E E {i, j} G {i,j} E w i,j 2 / 21 w
More information基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
More information変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
More information2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
More informationn (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
More informationall.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
More information‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í
Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I
More informationn ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
More information( ) ,
II 2007 4 0. 0 1 0 2 ( ) 0 3 1 2 3 4, - 5 6 7 1 1 1 1 1) 2) 3) 4) ( ) () H 2.79 10 10 He 2.72 10 9 C 1.01 10 7 N 3.13 10 6 O 2.38 10 7 Ne 3.44 10 6 Mg 1.076 10 6 Si 1 10 6 S 5.15 10 5 Ar 1.01 10 5 Fe 9.00
More informationUntitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
More information( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
More information. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More information1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More informationn=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
More information6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4
35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m
More information1 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 informationx () 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 informationuntitled
1 ( 12 11 44 7 20 10 10 1 1 ( ( 2 10 46 11 10 10 5 8 3 2 6 9 47 2 3 48 4 2 2 ( 97 12 ) 97 12 -Spencer modulus moduli (modulus of elasticity) modulus (le) module modulus module 4 b θ a q φ p 1: 3 (le) module
More informationZ: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
More informationMicrosoft Word - 11問題表紙(選択).docx
A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx
More informatione a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
More information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,
More information( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
More informationarxiv: v1(astro-ph.co)
arxiv:1311.0281v1(astro-ph.co) R µν 1 2 Rg µν + Λg µν = 8πG c 4 T µν Λ f(r) R f(r) Galileon φ(t) Massive Gravity etc... Action S = d 4 x g (L GG + L m ) L GG = K(φ,X) G 3 (φ,x)φ + G 4 (φ,x)r + G 4X (φ)
More informationI 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第1章 微分方程式と近似解法
April 12, 2018 1 / 52 1.1 ( ) 2 / 52 1.2 1.1 1.1: 3 / 52 1.3 Poisson Poisson Poisson 1 d {2, 3} 4 / 52 1 1.3.1 1 u,b b(t,x) u(t,x) x=0 1.1: 1 a x=l 1.1 1 (0, t T ) (0, l) 1 a b : (0, t T ) (0, l) R, u
More informationK E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
More informationφ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
More information2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
More information30 I .............................................2........................................3................................................4.......................................... 2.5..........................................
More information構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
More information20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
More informationII (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 informationall.dvi
72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
More informationI II III IV V
I II III IV V N/m 2 640 980 50 200 290 440 2m 50 4m 100 100 150 200 290 390 590 150 340 4m 6m 8m 100 170 250 µ = E FRVβ β N/mm 2 N/mm 2 1.1 F c t.1 3 1 1.1 1.1 2 2 2 2 F F b F s F c F t F b F s 3 3 3
More informationz f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z
More informationλ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More informationn ( (
1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128
More information液晶の物理1:連続体理論(弾性,粘性)
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
More informationEinstein ( ) YITP
Einstein ( ) 2013 8 21 YITP 0. massivegravity Massive spin 2 field theory Fierz-Pauli (FP ) Kinetic term L (2) EH = 1 2 [ λh µν λ h µν λ h λ h 2 µ h µλ ν h νλ + 2 µ h µλ λ h], (1) Mass term FP L mass =
More informationI. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
More informationuntitled
2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0
More informationArmstrong culture Web
2004 5 10 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1 Armstrong culture Web 1 3 1.1................................. 3 1.2.................................
More information/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat
/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiation and the Continuing Failure of the Bilinear Formalism,
More information211 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 informationm(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
More informationS K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.
() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
More information