1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W

Size: px
Start display at page:

Download "1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W"

Transcription

1 Naoya Enomoto paper

2 1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W W G- G W V 1.4. (1) G V W V/W G ρ ρ : G GL(V/W ); g ρ g ρ g ( v) = ρ g (v) (2) G 2 (ρ 1, V 1 ), (ρ 2, V 2 ) ρ 1 ρ 2 : G GL(V 1 V 2 ); g ρ 1 (g) ρ 2 (g) ρ 1 (g) ρ 2 (g)(v 1, v 2 ) = (ρ 1 (g)(v 1 ), ρ 2 (g)(v 2 )) 1.5. G V G-V 1.6. G (ρ, V ) (τ, W ) 3 (1) V V G GL(V ); g (ρ ) g ρ g(f)(v) = ρ(f(g 1 v)) 2

3 (ρ, V ) (2) V W V W G GL(V W ); g (ρ τ) g (ρ τ) g (v w) = ρ g (v) τ g (w) (ρ τ, V W ) (3) V W Hom(V, W ) g ϕ(v) = g (ϕ(g 1 v)) G Hom(V, W ) G V g 1 ϕ W g V g ϕ W V W V W (1),(2),(3) (3) V W Hom(V, W ) V W f w ϕ f,w ϕ f,w (v) = f(v)w V v i v i v V f(v)w = 0 f = i a iv i v = v j 0 = f(v j )w = a j w w = 0 a i = 0 f = 0 f w = 0 G- ϕ (g f) (g w) (v) = (g f)(v)(g w) = f(g 1 v)(g w) (g ϕ f w )(v) = g (ϕ f w (g 1 v)) = g (f(g 1 v)w) = f(g 1 v)(g w) 1.7. G X V = e x x X G ( ) g a x e x = a x e g x x X x X G X G G G R 2 paper C C Schur 3

4 2.1. (ρ, V ) G W V G- V = W W W G- W V = W U W U p : V W p 0 = 1 σ G ρ σ p ρ 1 σ p : V W p(w ) W p 0 : V W W G-ρ 1 σ w W p ρ 1 σ (w) = ρ 1 σ p 0 (w) = 1 σ G ρ σ ρ 1 σ (w) = w p 0 V W W ρ τ p 0 ρ 1 τ = 1 σ G W = v V p 0 (v) = 0} ρ τ ρ σ p ρ 1 σ ρ 1 τ = 1 ρ τ p 0 = p 0 ρ τ w W p 0 ρ τ (w ) = ρ τ p 0 (w ) = 0 σ G ρ τσ p ρ 1 τσ = p 0 ρ τ (v) W W G G C Remark / K K G 2.1 key word 2.4. C V G-V, x, y V ρ g x, ρ g y = x, y V G- (, ) x, y := 1 (ρ g x, ρ g y) V G- 4

5 2.5. G (ρ, V ) V G-V G- W W G-G Schur 2.6. Schur V, W G ϕ : V W G- (1) ϕ 0- (2) V = W λ C ϕ = λ I I (1) ϕ : V W Ker ϕ, Im ϕ V, W G-V Ker ϕ = 0} V 0} 0-V ϕ Im ϕ = 0} W V 0} Im ϕ = W (2) C ϕ λ C v 0 (ϕ λ I)v = 0 Ker(ϕ λ I) 0}V ϕ λ I = 0 ϕ = λ I 2.7. G 1 (ρ, V ) G G g G ρ g : V V G- Schur ρ g = λ g I V ρ g 1 G 1 Remark V T i } V T i } T i } G V ρ g } V G ρ g ρ g } V 1 G 1 G G 1 ρ g } ρ g } G G G V V = U i U i V = V (V k ) V (V k ) = Ui =Vk U i V V V k 5

6 V = U i = W j 2 G V k } V (V k ) = Ui =Vk U i, V (V k ) = Wj =Vk W j V = k V (V k ) = k V (V k ) U i, W j V (V 1 ) = U 1 U p, V (V 1 ) = W 1 W q π j : V W j U i π j Ui = θ ij 1 i p j > q θ j 0 G- Schur 0- U i W j 1 i p, j > q 0- V u i U i u W j u U i u = π 1 (u) + π 2 (u) + π q (u) + π q+1 (u) + u = θ i1 (u) + θ i2 u + + θ iq u + θ i,q+1 (u) + θ ij = 0 (1 i p, j > q) u = θ i1 (u) + θ i2 u + + θ iq u + θ i,q+1 (u) V (V 1 ) 1 i p U i V (V 1 )V (V 1 ) = p i=1 U i V (V 1 ) V (V 1 ) = V (V 1 )V (V k ) = V (V k ) V 3 G G (ρ, V ) χ V : G C χ V (g) = tr(ρ g ) (ρ, V ) 3.2. G (ρ, V ) χ V (1) χ V G (2) χ V (g 1 ) = χ V (g) χ V (hgh 1 ) = tr(ρ hgh 1) = tr(ρ h ρ g ρ 1 h ) = tr(ρ g) = χ V (g) ρ g λ i ρ g 1 λ i = λ 1 i χ V (g) = i λ i = i λ i 1 = tr(ρ 1 g ) = χ V (g 1 ) 6

7 3.3. G G V χ V G 3.4. V, W G (1) χ V W = χ V + χ W (2) χ V W = χ V χ W (3) χ V = χ V G (ρ, U) U G = v V ρ g (v) = v} ϕ := 1 ρ g End (U) 3.5. ϕ : U U G U Im(ϕ) u = ϕ(v) ( ) 1 ρ h (u) = ρ h ρ g (v) = 1 ρ hg (v) = 1 ρ g (v) = ϕ(v) = u Im(ϕ) U G u U G ϕ(u) = 1 ρ g (u) = 1 u = u U G Im ϕ 3.6. dim(u G ) = 1 χ V (g) U G U dim(u G ) = tr(ϕ) 3.7. dim(u G ) = tr(ϕ) = 1 tr ρ g = 1 χ V (g) Hom(V, W ) G = f : V W : G- } 7

8 Hom(V, W ) G g ϕ(v) = g (ϕ(g 1 v)) g ϕ = ϕ G ρ g ϕ = ϕ ρ g G- g ϕ = ϕ V, W Schur dim(hom(v, W ) G 1 V ) = = G W 0 V = W 1 χ Hom(V,W ) (g) = dim(hom(v, W ) G ) = Hom(V, W ) = G V W 1 V = G W 0 V = W χ Hom(V,W ) (g) = χ V (g) χ W (g) = χ V (g) χ W (g) 3.8. V, W G χ V, χ W = 1 χ V (g)χ W (g) = G a, b : G C a, b = 1 b(g)a(g) G 1 V = G W 0 V = W 3.9. (1) (2) G V χ V, χ V = 1 (3) V V i V i a i = χ V, χ Vi (4) G (5) = V i (dim(v i)) 2 (1) (2) G V V V = V a i i χ V = V a i iχ Vi χ V, χ V = i a2 i 1 (3) (4) G R χ R (g) g e 0g = e χ R (g) = R R = i V a i i a i = χ Vi, χ R = 1 (χ V i (e) ) = dim(v i ) (5) = dim(r) = i (dim(v i)) 2 8

9 G a : G C G (ρ, V ) ϕ a,v = a(g) ρ g : V V G G- a G a G ϕ a,v (ρ h (v)) = a(g)ρ g (ρ h (v)) = a(hgh 1 )ρ hgh 1(ρ h (v)) ( g hgh 1 ) = ρ h ( a(hgh 1 )ρ g (v) = ρ h ( a(g)ρ g (v) = ρ h ϕ a,v (v) ) ) ( a ) G- a h 1, h 2, h G h 1 h 1 h = h 2 a(h 1 ) a(h 2 ) G R ρ h ϕ a,v (e 1 h ) = ρ h( a(g)ρ g (e 1 h )) a G = a(g)e hgh 1 = a(h 1 gh)e g ( hgh 1 g ) ϕ a,v ρ h (e 1 h ) = a(g)ρ g (e 1 ) = a(g)e g e g g G} α(h 1 ) α(h 1 h 1 h) = α(h 2 ) ϕ a,v G G G G a 0 a 9

10 a ϕ a,v G- Schur ϕ a,v = λ I V n = dim(v ) nλ = tr(ϕ a,v ) = a(g) tr(ρ g ) = a(g)χ V (g) = a, χ V ( χ V (g) = χ V (g)) = 0 n 0 λ = 0 V V G R 0 = ϕ a,v (e 1 ) = a(g)ρ g (e 1 ) = a(g)e g e g g G} g G a(g) = G G (1) χ 1, χ 2 G χ 1 (g)χ 2 (g 1 ) = (2) X(G) G χ X(G) χ(g)χ(h 1 ) = (χ 1 = χ2 ) 0 (χ 1 = χ2 ) C G (g) (g h) 0 (g h) (1) (2) f g (h) = 1 (g h) 0 (g h) f g = i λ iχ i λ i = f g, χ i = 1 f g (h)χ i (h) ( ) h G = 1 χ i (h) ( f g (h) g h 1 0) h C(g) = C(g) χ i (g) ( χ i ) 10

11 f g (x) = i λ i χ i (x) = C(g) χ i (g)χ i (x) i χ X(G) χ(g)χ(x) = C(g) f g(x) = C G (g) g x 0 g x [Serre] J.P.Serre Linear Representation of Finite Groups Springer-Verlag [FH] Representation Theory A First Course(Fulton-Harris Springer-Verlag GTM129) 11

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}

More information

20 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

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.   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

II Time-stamp: <05/09/30 17:14:06 waki> ii

II Time-stamp: <05/09/30 17:14:06 waki> ii II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................

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.   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 information

数学Ⅱ演習(足助・09夏)

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

More information

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n

More information

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

More information

meiji_resume_1.PDF

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

More information

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h 2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 + 2016 12 16 1 1 2 2 2.1 C s................. 2 2.2 C 3v................ 9 3 11 3.1.............. 11 3.2 32............... 12 3.3.............. 13 4 14 4.1........... 14 4.2................ 15 4.3................

More information

20 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 information

S 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.

S 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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

LLG-R8.Nisus.pdf

LLG-R8.Nisus.pdf d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =

More information

『共形場理論』

『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 information

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..........

More information

I , : ~/math/functional-analysis/functional-analysis-1.tex

I , : ~/math/functional-analysis/functional-analysis-1.tex I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................

More information

振動工学に基礎

振動工学に基礎 Ky Words. ω. ω.3 osω snω.4 ω snω ω osω.5 .6 ω osω snω.7 ω ω ( sn( ω φ.7 ( ω os( ω φ.8 ω ( ω sn( ω φ.9 ω anφ / ω ω φ ω T ω T s π T π. ω Hz ω. T π π rad/s π ω π T. T ω φ 6. 6. 4. 4... -... -. -4. -4. -6.

More information

5.................................... 5.2............................. 6.3.......................... 8.4..............................5................................... 4 2 5 3 2 3....................................

More information

Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)

Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x) 3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,

More information

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18 2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1

More information

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,

More information

I

I I 2008 I i 1 1 1.1.............................. 1 1.2................................. 7 1.3......................... 13 2 23 2.1......................... 23 2.2............................... 31 3 37

More information

12 2 e S,T S s S T t T (map) α α : S T s t = α(s) (2.1) S (domain) T (codomain) (target set), {α(s)} T (range) (image) s, s S t T s S

12 2 e S,T S s S T t T (map) α α : S T s t = α(s) (2.1) S (domain) T (codomain) (target set), {α(s)} T (range) (image) s, s S t T s S 12 2 e 2.1 2.1.1 S,T S s S T t T (map α α : S T s t = α(s (2.1 S (domain T (codomain (target set, {α(s} T (range (image 2.1.2 s, s S t T s S t T, α s, s S s s, α(s α(s (2.2 α (injection 4 T t T (coimage

More information

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+ R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x

More information

all.dvi

all.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 information

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

More information

*.....J.....S.q..2013B_....

*.....J.....S.q..2013B_.... 1 1 2 2 3 3 4 4 5 6 5 7 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

More information

2 Part A B C A > B > C (0) 90, 69, 61, 68, 6, 77, 75, 20, 41, 34 (1) 8, 56, 16, 50, 43, 66, 44, 77, 55, 48 (2) 92, 74, 56, 81, 84, 86, 1, 27,

2 Part A B C A > B > C (0) 90, 69, 61, 68, 6, 77, 75, 20, 41, 34 (1) 8, 56, 16, 50, 43, 66, 44, 77, 55, 48 (2) 92, 74, 56, 81, 84, 86, 1, 27, / (1) (2) (3) ysawano@tmu.ac.jp (4) (0) (10) 11 (10) (a) (b) (c) (5) - - 11160939-11160939- - 1 2 Part 1. 1. 1. A B C A > B > C (0) 90, 69, 61, 68, 6, 77, 75, 20, 41, 34 (1) 8, 56, 16, 50, 43, 66, 44,

More information

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2 7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)

More information

* n x 11,, x 1n N(µ 1, σ 2 ) x 21,, x 2n N(µ 2, σ 2 ) H 0 µ 1 = µ 2 (= µ ) H 1 µ 1 µ 2 H 0, H 1 *2 σ 2 σ 2 0, σ 2 1 *1 *2 H 0 H

* n x 11,, x 1n N(µ 1, σ 2 ) x 21,, x 2n N(µ 2, σ 2 ) H 0 µ 1 = µ 2 (= µ ) H 1 µ 1 µ 2 H 0, H 1 *2 σ 2 σ 2 0, σ 2 1 *1 *2 H 0 H 1 1 1.1 *1 1. 1.3.1 n x 11,, x 1n Nµ 1, σ x 1,, x n Nµ, σ H 0 µ 1 = µ = µ H 1 µ 1 µ H 0, H 1 * σ σ 0, σ 1 *1 * H 0 H 0, H 1 H 1 1 H 0 µ, σ 0 H 1 µ 1, µ, σ 1 L 0 µ, σ x L 1 µ 1, µ, σ x x H 0 L 0 µ, σ 0

More information

Armstrong culture Web

Armstrong 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

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)

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

.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 information

Îã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹

Îã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹ (kaji@math.sci.fukuoka-u.ac.jp) 2009 8 10 R 3 R 3 ( wikipedia ) (Schubert, 19 ) (= )(Ehresmann, 20 ) (Chevalley, 20 ) G/P: ( : ) W : ( : ) X w : W X w W G: B G: Borel P B: G/P: 1 C n ( ) Fl n := {0 V

More information

u Θ u u u ( λ + ) v Θ v v v ( λ + ) (.) Θ ( λ + ) (.) u + + v (.),, S ( λ + ) uv,, S uv, SH (.8) (.8) S S (.9),

u Θ u u u ( λ + ) v Θ v v v ( λ + ) (.) Θ ( λ + ) (.) u + + v (.),, S ( λ + ) uv,, S uv, SH (.8) (.8) S S (.9), ML rgr ML ML ML (,, ) σ τ τ u + + τ σ τ v + + τ τ σ + + (.) uv,,,, σ, σ, σ, τ, τ, τ t (Hook) σ λθ + ε, τ γ σ λθ + ε, τ γ σ λθ + ε, τ γ λ, E ν ν λ E, E ( + ν)( ν) ( + ν) Θ Θ ε + ε + ε (.) ε, ε, ε, γ, γ,

More information

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, σ,..., σ N ) i σ i i n S n n = 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, 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

II 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 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 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

. 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 information

修士論文

修士論文 SAW 14 2 M3622 i 1 1 1-1 1 1-2 2 1-3 2 2 3 2-1 3 2-2 5 2-3 7 2-3-1 7 2-3-2 2-3-3 SAW 12 3 13 3-1 13 3-2 14 4 SAW 19 4-1 19 4-2 21 4-2-1 21 4-2-2 22 4-3 24 4-4 35 5 SAW 36 5-1 Wedge 36 5-1-1 SAW 36 5-1-2

More information

重力方向に基づくコントローラの向き決定方法

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

More information

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N 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

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 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 information

n (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

n (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 information

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

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

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

i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................

More information

量子力学 問題

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

Dirac 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 + 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 information

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1 t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.

More information

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 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 information

λ 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

λ 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 information

ver Web

ver Web ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3

More information

φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m

φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m 2009 10 6 23 7.5 7.5.1 7.2.5 φ s i m j1 x j ξ j s i (1)? φ i φ s i f j x j x ji ξ j s i (1) φ 1 φ 2. φ n m j1 f jx j1 m j1 f jx j2. m j1 f jx jn x 11 x 21 x m1 x 12 x 22 x m2...... m j1 x j1f j m j1 x

More information

1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +

1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac + ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 7.1....................... 7 1 7.2........................... 7 4 8

More information

all.dvi

all.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

O 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

O 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 information

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i [ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk

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

8 300 mm 2.50 m/s L/s ( ) 1.13 kg/m MPa 240 C 5.00mm 120 kpa ( ) kg/s c p = 1.02kJ/kgK, R = 287J/kgK kPa, 17.0 C 118 C 870m 3 R = 287J

8 300 mm 2.50 m/s L/s ( ) 1.13 kg/m MPa 240 C 5.00mm 120 kpa ( ) kg/s c p = 1.02kJ/kgK, R = 287J/kgK kPa, 17.0 C 118 C 870m 3 R = 287J 26 1 22 10 1 2 3 4 5 6 30.0 cm 1.59 kg 110kPa, 42.1 C, 18.0m/s 107kPa c p =1.02kJ/kgK 278J/kgK 30.0 C, 250kPa (c p = 1.02kJ/kgK, R = 287J/kgK) 18.0 C m/s 16.9 C 320kPa 270 m/s C c p = 1.02kJ/kgK, R = 292J/kgK

More information

262014 3 1 1 6 3 2 198810 2/ 198810 2 1 3 4 http://www.pref.hiroshima.lg.jp/site/monjokan/ 1... 1... 1... 2... 2... 4... 5... 9... 9... 10... 10... 10... 10... 13 2... 13 3... 15... 15... 15... 16 4...

More information

X 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 information

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 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

More information

phs.dvi

phs.dvi 483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....

More information

第5章 偏微分方程式の境界値問題

第5章 偏微分方程式の境界値問題 October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ

More information

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys

More information

線形空間の入門編 Part3

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!! (

More information

等質空間の幾何学入門

等質空間の幾何学入門 2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................

More information

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) = 2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I

More information

untitled

untitled 13,000 1 35,400 8,000 1 35,400 2 ~6 46,000 2 ~6 46,000 7 55,000 7 55,200 2,750 2,750 4,910 4,910 149,000 149,000 45,000 75,000 45,000 75,000 189,000 165,400 151,200 132,300 33 29 4 33 29 4 150,410 128,100

More information

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 ±

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 ± 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 information

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í

‚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 information

arxiv: v1(astro-ph.co)

arxiv: 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 information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

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

More information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e ( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )

More information

SO(n) [8] SU(2)

SO(n) [8] SU(2) SO(n) [8] 1 2 1.1.............................. 3 1.2.............................. 6 1.3 SU(2)............................. 7 1.4 -...................... 10 1.5 SO(3).............................. 11

More information

TOP URL 1

TOP 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 information

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

More information

2016 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 information

K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................

More information

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij

More information

Microsoft Word - 表紙.docx

Microsoft Word - 表紙.docx 黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3 .69, 3.8 4 (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i

More information

main.dvi

main.dvi SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1

More information

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................

More information

gr09.dvi

gr09.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 information

1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1

1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg)

More information

2011 (2011/02/08) 1 7 1.1.................................... 7 1.2..................................... 8 1.3.................................. 9 1.4.................................. 10 1.5..................................

More information

( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( )

( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( ) n n (n) (n) (n) (n) n n ( n) n n n n n en1, en ( n) nen1 + nen nen1, nen ( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( ) ( n) Τ n n n ( n) n + n ( n) (n) n + n n n n n n n n

More information

I. (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, ( ) 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 information